ESE 330 5-12 Methods: Mathematics

ESE 331 Middle School Methods: Mathematics

Augsburg College, spring 06

These courses, taught by Kay Shager, were combined. The middle-school preservice teachers met with the high-school preservice teachers for the first half of a 14-week semester.

All players encountered five simulations. the high-school players then encountered the last two simulations.

The sequence of simulations was designed to help players move from a rather traditional conception of mathematics and teaching to a more standards-based conception. We tried to counter these misconceptions, among others:

 

The simulations were:

1. Combining negatives. To begin challenging players to think about their assumptions, in this simulation the player is surreptitiously observing a group of students helping out one friend with homework concerning the product of two negative numbers. They offer various explanations and mnemonics, with varying degrees of success. The player is presented with questions—both within the simulation and in later discussion questions—to begin to counter some of the misconceptions listed above. One underlying question is, ÒHow would you teach this student?Ó

2. Coins. The problem was a typical one in which the number of pennies and nickels and their total value are given and the challenge is to find how many there are of each type. Players are given two major choices: Whether to use a lecture or a very structured worksheet for groups, and whether or not to allow calculators. Many choices involve subtle differences in wording of explanations. Effective play keeps students from getting too distracted with calculations and leads to enough understanding that one student will solve the problem with clever thinking.

3. Catchup. This simulation is like the versions described above except more focused and somewhat briefer. Many choices are between making statements or asking questions. Players see how questions can lead to better learning, as reflected in one studentÕs thinking about closing speeds.

4. Feet and yards. Like earlier versions, this incarnation has the player working with groups. It is less complex than before, with no classroom management issues, so that it can focus on the difference between closed and open-ended questions. The rewards for good play are as described above. With this simulation, the software engine began to change, allowing the script to take paths based on states of the class as well as on choices made.

5. Monk. This simulation was essentially unchanged. The discussion questions were very much affected by comments from players in previous courses.

6. Bouncing. This version was not altered much, except that it was adapted for a much more complex engine. Scoring was broken down to show various states of the class: Understanding of the mathematics, the Nature of math (including problem solving), Independence of student thinking, Comfort level at taking intellectual risks, and Engagement in the problem.

7. Hole digger. This simulation changed considerably. Although it was still a single group working on the problem, the player was given three visits to the group. After leaving, the player could view the results on the group of his or her choices while visiting but could not intervene again until the next visit. Poor choices led to a group that got nowhere. Excellent choices led to the no-symbols approach by one student in the group.

In addition, the Catchup simulation was revisited late in the course, to help us look for growth.

To Kay and me, the latest versions of simulations 4, 6, and 8 felt much more like a game than  had previous versions. Players were most successful if they carefully created a class atmosphere (as measured by the I, C, and E scores) in which learning could take place. Then, with appropriate choices, the U and N scores shot up.

Kay has compiled responses of her 15 players to the discussion questions asked for each simulation they played. (About half the class were middle-school pre-service teachers, who stopped after simulation 4 or 5.)  The quotations here are from the files turned in electronically. As noted, many reports are still on paper.

 

Student # 2 

For this student Coins and CatchUp revisited is in the paper packet.  This student did not do the first simulation.  Nor did they do Bouncing.

 

 

Simulation #4: Feet/Yards

Mathematical Content:

A lesson plan would say that the mathematical objectives of the lesson were to use variables to solve equations in order to find the amount of feet in yards or yards in feet. However in this case and many other cases in teaching, the students may not fully understand the concept by the time class is over. In this case, the objective was not reached and the teacher may have felt like s\he did not accomplish the overall goal.

In the eyes of a student, 3f = y makes sense since it seems to show that there are 3 feet in a yard, which is true. However in the eyes of a teacher 3y = f is correct since when it is multiplied, f will represent how many feet there are in one yard because there are 3 feet in one yard. I think answers generally hold true for math problems but how the answer is derived can vary in many different ways.

Problem Solving

I believe the students were involved with problem solving because they were on their own in groups to try and decide how to find the value of feet and yards. They had to come up with an equation that expressed that and made sense. I think in order to problem solve, the teacher needs to give a minimal amount of knowledge but give the proper amount of materials and background information to be able to construct their own knowledge base on a given subject.

Depending on the concept I think math problems can have more than one solution, especially since students will use their own approaches and each approach will be different.

Some students began to think outside the box when they began incorporating other ideas, such as jeans and legs. Using that, the students thought of new variables while still using a similar approach. I think problem solving often times requires the student to think outside the box since the students have to make sense of the problem and come up with an answer.

Communication:

By just looking at the student, the teacher communicates that the student has a choice in answering the question or not. In that way however, the student might feel an obligation to speak—more so than if his\her name was said aloud. I think just looking at a student is fine if the teacher knows that the student has some sort of answer or question. Body language is everything in communication. It can be accidental or on purpose and conveys many feelings or attitudes in which the teacher must be aware.

Curriculum:

This lesson would fit into curriculum on measurement by converting among units and also into an algebra standard in which students are using representation in a mathematical relationship. A curriculum is a plan of what subjects will be covered and how each subject or unit fits with particular standards.

Learning and Teaching:

Some groups are motivated enough to start a project on their own. These groups might get more attention because they might ask questions or they may need more help to keep them going. Others do not have motivation or simply like to be off the subject. In this case, the first group needs to be pushed along primarily and then the other groups can be helped so that they can begin the task at hand.

I think this is a good idea if for one; the students are not interested in the subject of the lesson (in this case feet and yards). Secondly, in this case the teacher needs to know that the concept is understood (3 feet in a yard) by the group or individual. If not, the teacher needs to stay strict with the concept of the subject as well.

When students hear the problem read out loud and can see it on their papers, they are using more than sensory skill to understand the problem. It also breaks them apart from staring at the same problem.

 

 

 

Simulation #5 Monk

Problem Solving

The students solved the problem by brainstorming their own ideas and then arguing them.  The guest helped to organize the discussion and push for peopleÕs ideas, but otherwise the students used their own thoughts to solve the problem. I liked that each student added something new, for example the issues of changing sunrise and sunset and changing pace all added for more ideas into the discussion. I think many of the students had decent justification but did not always prove that there was a meeting point. In that regard, Sara and Maurice had a very good understanding of the problem.

Reasoning and Proof

In any case, a model might seem or look like a good representation to a problem but it does not prove a problem. The model and graph only serve to justify their answers.

Communication

If the teacher had intended to try and boost the students listening and memory skills, I donÕt think that the problem should have been written. However for more visual learners, a written problem would have been easier to understand. One example of poor communication is when the teacher led the student by how the teacher phrased the question. She asked better questions when she left out leading phrases and asked ÒthinkingÓ questions.

Representation

Sara and Maurice chose to graph a model of two monks that were walking each path on the mountain. Another student made a time chart to show that there must be some number (in between 12 and 1) that the monks must meet. Some other students talked about the changing times of day and how that might affect the point, if there was one. Personally, I liked the graph demonstrating the 2 paths of the monks to demonstrate the problem.

Learning and teaching

The teacher needs to be a ÒchoreographerÓ in the case of a discussion to make sure that the students stay on task and that each student gets a fair chance to speak. The teacher plans movements in such a way to act as more of a director or as another student. By standing in front of the room she acts as director, but by sitting among the students she acts as another student. In a manner teaching is acting because it must be directed and carefully drawn out. In this case, if the teacher were not wary of the wording then the problem would be given away right away (and so in acting if the script is no good, the audience would figure out the end of the play).

Part of the point of the lesson was to show students that intuition is not always proof for a mathematical problem. Intuition does sometimes serve the purpose to separate ideas and to make a person think about reasoning to certain ideas and proofs.

The students worked in pairs so that they could feed off of each otherÕs ideas and give feedback. The teacher probably walked around from group to group to make sure the students were staying on task and to see if they needed help to get going.

I donÕt think that every student in a particular classroom is always engaged. For example, I may be in a classroom and look engaged but my thoughts are totally off of the subject. Being engaged is important because the student misses out on what others are learning.

In this situation I think it was okay that the teacher said that she was flattered because she used sarcasm throughout the lesson. I think otherwise, using reinforcement like that is not always a wise choice since it may make others feel bad that they did not have as good of answers. 

9. I think that confusion leaves room for more questioning or more inquiry by the students. I think that in a state of confusion the teacher should step in the discussion to bring in more ideas to see if the students could agree upon some answer that they could justify.

 

Simulation: Hole Digger #2

Mathematical Content

2. Sean said that they didnÕt need any variables to solve the equation, and for some people, they may I think that variables may not be necessary. I think some people can envision a problem in their head, without having to use a system of equations. I personally need equations as visual help, but I donÕt think that is necessary for everyone. I would say that SeanÕs manner of thinking was more problem solving since he didnÕt use a system of equations.

3. I think that SeanÕs idea is valid because he thought logically about the situation and found numbers that fit the problem. He may not have come up with the answer the teacher had wanted, but he still found a ÒsystemÓ that worked.

Nature of Mathematics

If you tell students to try a method for finding a solution that is ÒeasierÓ, I think that you are communicating that math is hard and there are better ways to find a solution than the one they had tried. I think there are manners that are faster to find solutions in mathematics, but I donÕt think a teacher should advertise that some ways of doing math is easier. I think if you do, the student will always look for the easiest method, rather than perhaps the correct method.

By finding many solutions to a problem in mathematics communicates that there is not just one answer to any problem. Mathematics is about coming up with different ideas and theories. Also, by finding more than one solution, it shows students that there are not always wrong answers in math; that a different variation of the solution is not incorrect.

Many times when I am trying to write proofs (especially in geometry), I have to try and change my intuition to fit the logic. I will write statements that seem obviously true, but I cannot always back them up. It is hard for me and will probably be hard for students to change their intuition because sometimes math seems so obvious, but does not always provide proof.

 

 

Student # 3

 

Student # 3

 

Paper copies of Combining Negatives, Coins, The Monk, Bouncing 2, Catchup Revisited, and Hole Digger 1 are in the packet.  I do not have electronic copies of these for Student # 3.

 

Simulation #3 Catch Up

2-13-2006

 

 

1.  At the end you had a chance to ask Kelly, ÒWould the time be the same if the car were going 65 and the truck 60?Ó  What do you think?  No, the time would not be the same if the car were going 65 and the truck 60.  The higher the rate of the truck, the longer amount of time it will take the car to catch up.  This is explained by looking at the distance the truck traveled in that first 15 minutes.  For example if the truck were driving 30mph, he would have gone 7.5 miles, if he were traveling 55mph, he would have gone 13.75 mile, and if he were driving 100mph, he would have gone 25 miles.  However if the car is only  going 5mph faster than the truck in each situation, it will obviously take longer to close the distance gap, the larger it gets.

 

2.  One choice included ÒAnd now for the magic.Ó Do you believe that mathematics is magic?  Do you think others believe that mathematics is magic?  I do not believe that mathematics is magic, but I view it as a science.  Unlike other subjects such as English and Literature, mathematics is not subjective.  The numbers do not lie and they are right or wrong.  I do not believe that others think of mathematics as magic either.  Many people feel that they are not proficient in mathematics and refer to math as being magic.  But I believe that most know it is an exact science.  Even someone who only understood basic math would be able to see that there is a process and a right and wrong.

 

3.  Are the Catch-up problem and the other problems of this simulation real math problems?  Are problems different from exercises?  Were all the problems solved during the class session?  Were the students engaged in problem solving?  Yes, this problem and the others were real math problems.  They help the students to understand mathematical concepts and tools.  The catch-up problem in particular is a good problem, because it is a Òreal-lifeÓ situation.  Using a real-life example shows the students a practical use for what they are learning.  Problems are different from exercises in that problems are when a concept is first presented to the students, and exercises are practice on something that has already been explained. The goal of the class was to get through the initial problem.  This goal was completed.  However not all additional problems were finished.  For example just as the class ended, the teacher asked would the answer be the same if the truck was driving 60 and the car 65, and no discussion was given to that question.  Due to the questions and openness of this lecture, the students were engaged in the problem solving.  Since they were allowed to present ideas and suggestions, their interest was sparked in what was going on in the class.

 

4.  Was this lesson taught the way you were taught mathematics?  This lesson was not taught exactly the way that I was taught mathematics.  My teacher would not have asked for brainstorming ideas, although she would have gone over the different methods.  Also, to the best of my memory, my algebra class was taught in the lecture style with no questions to the students.  The only questions we were asked were about our answers. 

 

5.  Is a curriculum a textbook or standards? Does it matter what textbook you use?  I believe that a curriculum has been made to represent standards.  Textbooks should be used as a guide toward those standards.  I believe that the curriculum should involve the textbook, but also incorporate other methods of teaching to enable the teacher to help all the students understand the concept being taught.  However, I have seen in too many classes where the teacher strictly abides by the ways of the textbook and does not use any other supplements to aid the lesson plan.  For teachers to be successful, they must be creative and use various resources to help the students understand not only math, but any subject. 

 

6.  Was this lesson more effective than the lecture-based lesson of the Coins class?  Are questions more effective than statements for teaching?  This lesson did seem to be more effective than the lecture-based lesson from the Coins class.  The questions seemed to make the teaching more effective.  In the coins lecture, many of the students were bored or didnÕt understand, but did not interrupt the lecture to ask questions.  However in the Catch-up lecture, by asking questions, it made the lecture seem much more Òopen forumÓ, which engaged the students and let them feel free to ask questions and give opinions.

 

7.  What kinds of questions get more points than others?  In general, what was the difference between calling on a student before and after asking a question?  Can you think of exceptions to the general principle?  The questions that got more points were the ones where the question was asked first, and then a personÕs name was called.  This is a good method of calling on students, because by stating the question first, it appears to be a question to the whole class.  Then calling on the student is just picking a particular person out of the class to answer the question.  By calling on a student first, it seems that you are picking them out of the group for some reason.  This might make many students nervous and feel that you are picking on them for a reason.  One exception to this rule might be to get a studentÕs attention if they are disturbing the classroom.    

 

8. Would you ever ask one student what another student might be thinking?  Honestly, I am not sure.  I have never considered this as a teaching technique for teaching a lesson.  I think using this method could make the class very interesting in how the students would react to such questions, but I worry that it could backfire as well.  You might have someone hurting another studentÕs feelings and then parents might get involved over what the other student said.  However, I think it could be a strategy worth risking every now and then throwing the students off to get their attention.  Teachers may be able to explain a concept very well to a class, but if the teachers donÕt have the attention of the class, then that great explanation was a lost cause.  

 

9.  Why did you get fewer points from choosing ÒGoodÓ than from ÒThank youÓ?  This situation happened while the students were making their initial guesses.  If a teacher says good to one of the guesses, the students can pick up on that and will assume that it was the correct answer or very close.  This might keep the students who think that it was a very different answer from sharing their thoughts. 

 

10.  Should students have been allowed to use graphing calculators to make tables in their class?  For this particular problem, I think it is important for the students to work the problem and tables out by hand.  I think that teachers need to make sure that the students actually understand how to solve the problem and know how the problem works before letting them have a calculator do the work for them.  I believe after the students get the hang of solving such problems, the teacher could then assign more complex problems where making a table could be used as a time saver, not just a way of solving the problem.

Simulation #4

Feet and Yards

2-20-2006

 

1.  Which of the equations 3f = y and of 3y = f would you say is right?

Do mathematical problems always have right answers?

I would say that 3y = f is the right equation.  It might seem more natural the other way because there are 3 feet in a yard but that does not automatically make the equation 3f = y.  This would only be correct if you were just making a statement and f and y were just units (i.e. 3 feet = 1 yard).  Whereas if you knew that you had 4 yards of something and you wanted to figure how many feet were in it, you would use the equation 3y = f or 3x4=f and f = 12.  However, mathematical problems do not always have right answers.  Answers can be based on how the student perceives the question, and often there is more than one solution to a problem.

 

2.  Were the students in this class engaged in problem solving?

What makes a problem a problem rather than an exercise?

Most of the students in the class were engaged in the problem solving.  Some of them were harder to get motivated, but overall the students were actively working to get an answer.  A problem is when a student is first introduced to an idea.  An exercise is when they have seen it before and are working on examples. 

 

3.  At one point you have a choice between asking, ÒYiscah?Ó or just looking at Yiscah. Under what conditions, if any, is it good teaching just to look at someone?

What are various ways a teacher might communicate with body language?

By just looking at someone, as opposed to calling on them, it allows the student to decide on their own what if anything they want to say.  If the student is completely lost in what they are trying to do, they can ask for help.  However if a student is working through something and doesnÕt want help, just looking at them allows them not to say anything and continue to solve the problem on their own.  Not calling on a student can help to keep the learning environment open.  However, teachers will have to be careful when using this method.  Some quieter students will not ask for help even if they do need it, so it would be necessary for the teacher to probe the student into asking their questions.  Looking at students is a good way to communicate a variety of things verbally.  It can show interest in what a student is saying or doing, and a stern look can bring a disruptive student back into the lesson.  Teachers might use various hand motions or gestures to communicate with a student without the rest of the class knowing it. 

 

4.  Where might this lesson fit into a curriculum?  What is a curriculum? 

This problem would fit into the curriculum under units of measure.  A curriculum is a set of concepts in a given area that a teacher feels should be covered in a school year or semester.  However, the concepts chosen by the teacher are greatly influenced by standards that are set up by both the nation and the state. 

 

5.  You got the most points by encouraging some students to change the problem into something in which they were interested.  Under what circumstances, is this a good idea? 

It is a good idea to encourage students to change the problem into something they are interested in to motivate them to learn to do the work.  It is okay to do this, when the problem itself will not be changed.  For example by substitution football players and teams for feet and yards, the problem is still essentially the same, but it is about a more interesting subject.

 

 

6.  Group 3 was stuck until they reread the problem aloud.  Why might reading the problem aloud for a second time have helped a group get unstuck?

Sometimes rereading the problem can show you more details that you missed the first time.  This is especially helpful for long problems or problems with a lot of information.  It is often hard to ascertain what is important from one read through of the problem.

 

7.  In Group 4, after talking for a while with Sean (who was working alone), you said, ÒMaybe your group has ideas you can use.Ó Was that a good thing to say to a loner?  Should loners be required to contribute to groups?

Suggesting that a group might have ideas to help is a good thing to tell a loner.  While some students do work better by themselves, by working in groups, the students can work together to determine a solution.  Sometimes someone else can explain something to the students that they hadnÕt thought of before.  I think that loaners should be required to contribute to groups.  While it might not be necessary every time, besides learning the Òmath at handÓ, school is a way to prepare students for the future.  It is very likely that everyone at some point in their future will have to work in some sort of a group.  It is better to introduce this type of learning now to acclimate ÒlonerÓ students to working with a group.

 

8.  A number of times you were given a choice to wait.  Under what conditions, if any, is this a good teaching strategy?

If students are in the middle of figuring something out on their own, it is a good teaching strategy to wait and see what happens.  The more a student can figure out as opposed to just being told, the more they will remember.  So if a student or group seems to be coming up with valid solutions or possibilities, it is better to wait and see what they come up with than to suggest things.  However, if you notice that the group starts to go far off from the right path, you would need to speak up and point them back in the right direction.  Even if students are actively working and seeking the right answer, it would be fruitless to let them go in the wrong direction for an entire class period, when they might have only needed a nudge in the right direction. 

 

9.  It was possible in this class to avoid saying anything during a visit to a group. Under what conditions, if any, would that be good teaching?  What conception of mathematics, teaching, and learning is behind your answer?

 Listening to what a group is doing, or waiting to see what they are going to say, is a very good teaching strategy when a group is making productive forward progress.  If a group is working on solving a problem, and they are getting closer to that goal, it is often better to let them continue on than to interrupt.  If a group is working and you do interrupt, it could possibly eliminate some of their desire to find out the answer on their own.  However on the other hand, if a group is not moving forward in solving the problem, it may be necessary for the teacher to step in and point them in the right direction.

 

10.  Toward the end of the simulation Jesse might have said ÒWow!Ó How does that response from a student make you feel?  What assumptions about teaching and learning does that feeling represent?

ÒWowsÓ are what teaching is all about.  In this simulation particularly, the students have figured something out on their own.  This is a major accomplishment.  It is one thing when a student works a problem and gets the correct answer, but being able to work and understand something on their own is a major accomplishment.  It also goes against the assumption that learning and teaching is boring work and not fun.  This wow just proves that learning can be fun.  It can give the students a feeling of accomplishment.  I would think that this would make any teacher feel proud of themselves and their students.

 

Simulation #6 Bouncing

3-13-2006

 

1. At some point you asked Terry, ÒDo you think there might be more than one way to look at this situation?Ó What messages about the nature of mathematics are conveyed by that question? This portrays the message that there is often more than one way to solve a problem.  While there might not always be more than one way, and often one way is the best, many problems have multiple ways of getting an answer.

 

2. A couple of times you had the opportunity to ask, ÒDoes your answer make sense to you?Ó What about the nature of mathematics does that question communicate?  This question entices the student to think about whether or not they actually understand what they are doing.  It is important to convey to the students the logic behind what they are doing, and to help them realize that they can understand mathematics, not just memorize some steps to how to solve a problem.

 

3. What, if any, problem solving were you able to elicit in this class session? One student proposed using patty paper to draw the line and flip it over to see if it would hit or miss.  Another student came up with the idea to use a protractor to solve the problem.

 

4. At one point two of your choices were

Please don't come up unless I ask you to.

Please don't come up unless I call on you.

How does the slight difference in wording affect what is communicated?  This affects how you want your classroom to run.  Saying the word ask keeps things open in the sense that it is a more ÒopenÓ style classroom.  This gives off more of the impression that class is somewhat of an open forum.  Using the word ÒcallÓ indicates that you are in control of the class at all times, and things shouldnÕt happen unless you specifically say they should.

 

 

5. Why do you say ÒI like controversyÓ? What are strengths and weaknesses of having controversy in a classroom discussion?  Having controversy in a classroom discussion allows the students to voice their own opinions or thoughts as to how something should be done.  This will allow them to grow mathematically.  However a negative aspect to controversy would be if the controversy inhibits the learning process.

 

 

6. What was the best way to deal with JustinÕs eagerness in this class session?  If a student knows a lot about something and is eager to answer questions, they should be given some opportunity to answer, but should not be allowed to dominate the class.  If a student is eager to participate, it should be encouraged by the teacher; however, since the main goal is to engage all students, the teacher should make sure that other students get a chance to field some questions or propose ideas.

 

 

7. There were times when you clicked for a long time without getting any choices.

Was it good to have yourself as teacher not the center of attention?  Yes, it is often good for the teacher not to be the center of attention.  As the teacher from the monk simulation thought, telling is not often teaching.  As long as the students were working constructively on understanding the situation and making forward progress, it was beneficial to the class.

 

8. Were any of the What if? questions worth the effort it took to elicit them?  Some of the what if questions were worth the effort it took to elicit them.  For example, one student asked if the ball would ever go in a pocket, which turned out to be the homework problem, and another student asked what the difference would be if the table was bent.  These showed understanding of the reasoning behind the problem.  So of the questions were pointless, such as what if the ball were a cube and didnÕt roll at all.

 

9. Was it a better idea to ask the class about the meaning of obtuse or simply to tell them?  Asking a class the meaning of something is a good way to see what they know.  If you are sure the class should know the meaning of something, asking them is a way to see if they actually do know it, or at least make them think about it.  If you are fairly certain that something will be new to the class, it would be more appropriate to just tell them.  However in most situations, asking the class if beneficial.

 

10. Why was writing ÒReflectionsÓ not such a good choice as asking, ÒWhat about reflections?Ó  Asking the students about reflections allowed you to see how much the students understood about it.  By simply writing it on the board and moving on, students who didnÕt understand might not ask.  If you arenÕt sure how well your students understand something, it is usually a good idea to ask.

 

Simulation #7 

HoleDigger

Week 11

 

  1. Would it have been more appropriate to teach the elimination method than the substitution method for solving this pair of equations?  Under what circumstances is each method preferred?

 

I donÕt believe that the elimination method would have benefited the students more in solving this problem.  I think the substitution would serve best because students can probably make better since out of the problem by using the substitution method.  I think that the elimination method would serve best in situations where two variables are presented in two different scenarios and one of the variables happens to be the same value in both scenarios.  Even this type of problem might be confusing to some students.  The substitution method is best used when two scenarios are given and one variable is present in both. 

 

  1. At one point, you had the option of saying, ÒIn doing math, you often have to change your intuition to fit the logic.Ó  In your own experiences, have you done that?  Under what circumstances might students resist changing their intuition?

 

I believe that one main situation where you need to change your intuition to fit the logic would be when you get a right answer but the equation you used would not work for every situation.  This can be hard for students to understand sometimes.  If they work a problem and get a right answer, they probably will not want to find another method.  This is a situation where I have had to change my intuition and one that most students will experience in their math careers.

 

  1. What problem solving strategies did this group use?

 

This group used several problem solving strategies to solve the problem.  They used pictures, variables and equations in that order.   Their first intuition was to draw a picture.  Then they realized that they didnÕt know everything that they needed for drawing the picture and used a variable to complete the picture.  Then they were able to use the picture with the variable in it to set up and solve an equation.

 

  1. What non-pictorial representations of the situation did the group generate?  Which of these representations do you prefer?

 

This group used equations as a non-pictorial representation of the situation.  They used the equation and the variables that they discovered to solve the problem.  I prefer using pictorial representations.  I think it helps people, including myself, to grasp what they are trying to determine.  It shows you what you are looking for and what you know.  I feel that most problems should start with a picture of the known and unknown information.

 

  1. From Lao-Tsu: ÒAs for the best leaders, the people do not notice their existenceÉ When the best leaderÕs work is done the people say, ÒWe did it ourselves!Ó  To what extent, if any, does this aphorism apply to teachers and students?

 

This aphorism applies greatly to teachers and students.  I feel that a teacher is there to guide the students to think for themselves.  It cannot always be the case.  Not all ÒtraditionalÓ lectures can be removed from the classroom, but whenever possible teachers should try to be a guide for the students.  The students will retain much more of the information that they discover on their own than information that is ÒtaughtÓ to them. 

 

 

 

 

 

Student # 4  Coins for this student in is the paper packet.  I am missing several of their assignments.  This person was very late and I did not get all of their work copied.  They did not do The Monk.

 

 

Simulation Questions:  Catchup

 

And now for the magicÉ

I wish that I could say that right now in my analysis class however we must be able to show students and get across to students that math should not feel like magic it should start to feel like a logical sequence of procedures that gets us to a logical answer.  I think that we joke too much about math being magic that we send the wrong message tostudents.

Would you ever ask a student what another might be thinking?

It depends on the circumstance.  If it looks like they might be able to contribute some valuable information that might lead the class or group in the right direction or be able to bring a few concepts together.  I donÕt think I would ask what is ___thinking.  Something more along the lines of if the understand or can explain in more detail what someone else has shown. 

Is it ever okay to make a mistake and just keep moving on?

I do it on a regular basis.  The biggest part to this is to make sure it eventually gets corrected.  I always find that students start to neglect checking their answers if the know it will always be right.  I especially like to see if students can spot their own mistakes when the answer does not make sense.  It also gives the student more practice with evaluating whether the answer makes sense or not.  I donÕt think that it is okay to just leave a mistake hanging out there because it could leave the wrong impression and even cause problems in the future. 

Making connections

I canÕt say I would make a connection like we did by lying to the students, rather make up a problem that is actually real or something that pertains to them.  This usually works very well to keep students interested in what is happening in the classroom.  The given problem could be a car trip, but with different units and/or numbers it could be a track practice where someone needed a drink of water. 

Ask if the truck were going 60 and car 65.

It would be a great problem for students to start to build connections and realize the actual relationships between the two answers.  It would also give the student more practice on that type of problem.  Another thing it does is reinforce the connection to the real world.  Cars can go whatever speed even if it is over the limit.  They can begin to toy with that. 

Unit

This would fit perfectly with a system of linear equations unit.  My placement is working on systems of linear equations right now.  They work with the three ways to solve linear equations, graphing, substitution, and elimination.  We have an equation for the car and equation for the truck I think it would fit perfectly there.

Curriculum

A curriculum is neither a textbook or standards.  It is the long term plan of action for the teacher.  A curriculum may have a text book it may have more than one text book.  It could even have no textbook at all.  The standards are then implemented through the curriculum.  The curriculum includes not only what is going to be taught, it includes objectives and how those objectives will be assessed.  It includes policies that the teacher will have and many other things.  It shouldnÕt matter what textbook a teacher uses however it is easier for the teacher if the textbook covers all of the content needed for the standards.  Beyond that it is nice if the text is set up in the order that the teacher wants to present the information. 

Questions versus Statements

I think that questions are often a better way to help a student.  It helps the teacher assess where the student is at and what they need help with.  It also allows the student to express in more detail where their question is or what they donÕt understand.  Trying to have patience and figure out where the student is at lets the student show what they know and can also avoid a lot of unwanted confusion.  Today I had a student who understood the concept of how to solve the equation using substitution.  It didnÕt look like he did.  He just needed a short review in how to get things to the other side.  Now I give him more credit than if I would have started from the beginning and taught him what he already knew about systems of equations.  By asking questions we can also encourage students to be confident in their answers and find problems themselves.  I always love the student who asks ÒIs this right?Ó for every answer of 5 problems that are solved the same way with a method of checking.  By making them show me why they think it is right they must explain what they did and then they can possibly spot their errors. 

The way it was taught

The way it was taught seems quite familiar.  The teacher states a question and the class collectively tries to figure it out.  It keeps more of the students more engaged than lecture however it may take longer. 

Graphing calculators

Graphing calculators could have been useful for the students to actually plot points and see where they intersected.  However, the goal of this was probably to figure out the equations and solve them or use a table and I think that the students were fully capable of figuring out the table.  If calculators were brought in I would bring them in as a way of checking our answers by plugging in the two equations and seeing where they intersected.  My thought is that the students are not all that efficient with graphing calculators so it would have been more of a waste of time than a helpful tool.  I guess the teacher could have graphed it on an overhead or mathematica. 

 

 

SIMULATION #4: Feet and Yards

After reading Chapter 1, 2, 3, & 5 in the text and viewing the simulation several times, address at least ten of the following questions:

Mathematical content

What would a lesson plan say were the mathematical objective(s) of this lesson?

Are there any disadvantages to having objectives governing your teaching?

Something makes me want to say an objective might have been exploring variables.  If it was they did a good job of incorporating that.  The student will be able to work with equations to convert two different units of measure.  When you have objectives you must teach to those objectives.  Sometimes students come up with some insightful information and you often donÕt have enough time to explore their interests and meet your objectives however if you have objectives you usually have thought about what is going to come and there is a logical structure for all of the students.  I think that it is far better to have objectives than to not have objectives. 

 

Which of the equations 3f = y and of 3y = f would you say is right? both

Do mathematical problems always have right answers? They usually have right answers but often have more than one right answer. 

 

About these students one player wrote, ÒTheir words are right but theyÕre not paying attention to units so theyÕre getting the wrong equation.Ó To whom might she have been referring? What might she have meant about words and units? Do you agree?

Nature of mathematics

Ultimately the emphasis of the problem was in making sense of the two equations.

Is this the same sense making as, say, that it makes sense to you that the graph of a linear equation is a straight line?

Problem solving

 Were the students in this class engaged in problem solving? Yes, after a little help to change the question a bit.

What makes a problem a problem rather than an exercise?

 

The students found several solutions to the problem of making sense of both equations.

Can a legitimate math problem have more than one solution? It can indeed!! Especially when we make our own equation.   There are also many times when a problem does have two correct solutions. 

 

About this class one teacher wrote, ÒThese equations kind of make them have to think outside of the box, and they usually fight that.Ó   Did you see signs of Òthinking outside the box?Ó   To what extent does problem solving require thinking outside the box? It requires thinking outside the box because they have to work a lot with interpretation and how they are going to look at the variables and how they will set up the problem. 

Communication

At one point you have a choice between asking, ÒYiscah?Ó or just looking at Yiscah. Under what conditions, if any, is it good teaching just to look at someone? To nudge them to speak without making them feel nervous.  Also to see if they look like they are going to speak.  Calling on someone often draws extra attention to the student and some students donÕt want the extra attention. 

What are various ways a teacher might communicate with body language? Obvious ones might include pointing and gestures.  Most importantly is the height of the teacher (and not being short).  If the teacher is standing, seated or at the studentÕs level.  If the teacher is standing that is body language for authority and sitting is more relaxed and the students are equals with the teacher.  Its like when the teacher joins a group if he or she continues to stand they are expecting all attention and not much response.  Sitting at the table with the students means that the students should respond. 

Representation

What mathematical representations arose in the class? Were F and Y variables or were they units of measurement?  They were both. That is why we got two different equations.  When we had 3F=1Y those were units.  If we look at the equation 3Y=F they would be variables.  When we replace Y( the number of yards) with  1 we get three feet.  That works out.  That is the problem we were having with labeling our variables and understanding the difference.  From my experience this is not an uncommon error. 

Curriculum

 Where might this lesson fit into a curriculum?  What is a curriculum? This would fit well with something in understanding what a variable is.  I know in the MAT 105 class they are constantly going over variables and labeling the dependent and independent variables. 

Learning and teaching

Why were you required to visit the working groups in the order that you did?  Does it matter what order you visit groups?

 

You got the most points by encouraging some students to change the problem into something in which they were interested.  Under what circumstances is this a good idea?  When the problem is fairly easy to change and get the same concepts and where it is easy to then apply their new problem back to the old one. 

 

Group 3 was stuck until they reread the problem aloud.  Why might reading the problem aloud for a second time have helped a group get unstuck?

 

In Group 4, after talking for a while with Sean (who was working alone), you said, ÒMaybe your group has ideas you can use.Ó Was that a good thing to say to a loner?  Should loners be required to contribute to groups?

 

A number of times you were given a choice to wait.  Under what conditions, if any, is this a good teaching strategy?

 

It was possible in this class to avoid saying anything during a visit to a group. Under what conditions, if any, would that be good teaching?  What conception of mathematics, teaching, and learning is behind your answer?

 

Toward the end of the simulation Jesse might have said ÒWow!Ó How does that response from a student make you feel?  What assumptions about teaching and learning does that feeling represent?

Assessment

What can you say, if anything, about how well any of the students understand the use of measurement units in equations like these?  How much assessment can you do by watching students at work?  Do you think this assessment will be more or less accurate than that during later quizzes and exams?  You can do some valuable assessment however the students do need practice and will later show you that they can apply what they have learned.  Once they can apply what they have learned you know that they have learned what you have taught them.  I think that at the point of this observation it is too early to assess students for final understanding.  They have not had the opportunity to practice with this type of problem.  Assessment at this stage should be used to improve the teacherÕs teaching and the student understanding.  I think that the teacher must continue to ask for explanations of answers and procedure so that he or she knows that the student knows.  This is one problem that I run into tutoring MAT 105.  I find that the students choose different variables than I would.  There is no rule that T has to be time and it canÕt be H.  Sometimes it helps to have the students there to explain what they know.  If a teacher can get enough explanation out of the students then it is fine. 

 

 

SIMULATION #6: Bouncing 

After reading Chapter 1, 2, 3, 5, 6, 8 and 9 in the text and viewing the simulation several times, address at least ten of the following questions:

 

 

Nature of mathematics

At some point you asked Terry, ÒDo you think there might be more than one way to look at this situation?Ó What messages about the nature of mathematics are conveyed by that question?  I think that the nature of mathematic is that there are many ways to come up with the same solution.  They are somewhat similar, but often a different twist.  I think that it encourages students to look at the many different ways.  Every time new math is discovered it is discovered because either someone was looking for a new way to solve an old problem or someone asked another Òwhat ifÓ question.

 

A couple of times you had the opportunity to ask, ÒDoes your answer make sense to you?Ó What about the nature of mathematics does that question communicate?

I think that it communicates that it is important to understand your own work and understand what your solution means.  Like the car problem did you start timing when the car or the truck started out?  The solutions meant the same things however were different solutions.  Also, one must think about checking ones answer and seeing if it makes sense in the context.  There are some impossibilities in mathematics such as a percentage over 100.  Encouraging students to become confident with their answers and justifying them will get them a long way.  

 

 

Communication

Under what, if any, conditions would it be appropriate to ask, ÒWhat the heck is that, Dula?Ó  I donÕt think this is ever appropriate.  It seems so difficult to get students to share their ideas in a mathematics classroom.  Saying things like this will prevent Dula from ever speaking in your classroom and probably other math classrooms.  Often when students speak or do something wrong they are not the only students thinking that way.  It is an important tool for the teacher when he or she sees something wrong before the exam.  The teacher will have the time to clear things up and give more students a better chance at doing well on the test.  Anyway Communication is so key I wouldnÕt risk it. 

 

Connections

What connections did students make?

Students made connections between incidence and reflection.  They also made connections to other problems such as how long the ball could keep going and things like that.  I think that one of the biggest connections for a class like this was the connection between physics and mathematics.  I think students often forget that life is interdisciplinary. 

 

Representation

What different representations of the problem arose?

We had our regulars like drawing pictures and things, but this time we had an often forgotten representation; the demonstration.  I think that it engaged a lot of the students and will probably stick better in their minds than it would had the teacher just flat out told them what the answer was. 

 

 

Curriculum

Does this problem belong in a mathematics curriculum or only in a physics curriculum?

I think it belongs in any classroom we can fit it into.  Like I said life is interdisciplinary.  That means that we canÕt just shut off sections of our brains when we get to certain times of day.  It is great when we can intermingle the many different subjects. 

 

Learning and teaching

How did beginning with a problem affect the various levels?

Is beginning with a problem what is known as inquiry-based teaching? Starting off with a problem is not necessarily inquiry based teaching.  The students can just solve the problem and be done.  With inquiry based teaching the teacher tries to get the students more and more involved and ask questions and pose problems and opinions of their own.  Often when we pre-set the problem we have narrowed down the responses if we will get any.  

 

Why do you say ÒI like controversyÓ? What are strengths and weaknesses of having controversy in a classroom discussion? I think that the strengths of having controversy in your classroom is that you get to explore and show students why things donÕt happen and why other things do happen.  If everyone agrees who gets something out of bouncing a ball against a wall?  However as a teacher you must make sure that all students hear and remember what you discovered when the controversy is settled.  Another issue you may deal with is that a student may or may not take it too far.  You must make sure to set up an environment where students question one another and question you when they donÕt understand, but taking it personally is not a good thing. 

 

What was the best way to deal with JustinÕs eagerness in this class session?

Use him to get involved!!!  When there is one eager student you must involve the rest of the class, but continue to return to the excited member of you class.  He or she may not be excited about anything else all year.  This may be how you foster the love for mathematics.  If Justin is over eager all of the time and disrupting everyone a conversation reminding him of the number of students in the class and that they have the right to learn too is not always a bad thing.  He may have forgotten that. 

 

There were times when the class was in ÒhubbubÓ or gathered around a demonstration on the floor. Was the class out of control?  No!!! I hate that.  Some of the best experiences that I have had in a classroom is where the class may look out of control.  We love to see students eager to learn and explore.  Hands on may look out of control but we are physically doing something that they will remember.  It reminds me of the unconventional methods of Robin WilliamsÕ character in Dead PoetÕs Society.  He really reached those students.  Very few students are reached in a straight laced situation. 

 

There were times when you clicked for a long time without getting any choices.

Was it good to have yourself as teacher not the center of attention?

For sure!!  We must let students explore for themselves.  When the students explore they may find and understand things that we would not teach them.  They also learn what it is like to explore mathematics and that is how new mathematics has come about.  When they then explain things to their peers it becomes an opportunity for them to learn it better and have a better grasp on the situation.  I always learned that you really know it when you can teach it to someone else. 

 

 

Why was writing ÒReflectionsÓ not such a good choice as asking, ÒWhat about reflections?Ó

Writing it only shows them how to spell it.  Asking about it gives meaning to the word.  They need meaning at this point not spelling. 

 

Was there a summary before starting homework? When is it good to do a summary toward the end of a class session?  I think it is good to do a summary when you have covered a great deal of information.  When there have been many people not understanding and when there have been many ideas found wrong.  Often students will need a reminder that the class may have disproved something or they may remember another student talking about something you disproved or be a little confused because you covered so much information.  I donÕt think a little summary hurts anything.

 

SIMULATION #6: Bouncing (Revisited)

 

Spend time again on the simulation Bouncing.  Report your scores for both Week # 7 and Week # 8.  During Week #8, try to improve both your total score and your component scores. Answer each of the following questions and then address three additional questions of your choice from the original list in Week #7 (The list is also given below).  Do not address the same questions that you did during week 7.

 

Week #8 Questions: 

 

1.  In the simulation Bouncing, identify a question, if possible, at each of the various levels of BloomÕs Taxonomy. Check the BloomÕs Taxonomy sheet for further examples of terms used at the various levels.

The angle of incidence is equal to the angle of ________

Is this possible?

Where does the ball go next? 

 

Would it keep going forever, or would it eventually go in a pocket? (use above diagram)

Given a specific set up of balls, not direction is it possible to get the ball in the pocket with only one hit on the side. 

 

2.  Write two objectives for the Bouncing lesson.

The student will be able to draw a line of reflection given the line of incidence.

The student will be able to incorporate hands on trials to discover the correlation between the angle of incidence to the line of reflection. 

 

3. Identify the standards from both NCTM and MN Standards that this lesson would address. (Access the web sites from weeks 1 and 2 to find the 9 – 12 Standards of each.  We will address these more in class after break.)

 

4.  What were the choices that led to DulaÕs having a Big Idea?  In general, what kinds of choices lead to creative problem solving?

The kind of choices that lead to creative problem solving are being open to students ideas and encouraging them to try even if they end up figuring out that they are wrong. 

 

5.  Was the amount of closure for this lesson appropriate?  Why or why not?  If not, what would you include?  I think that it could have used a little more closure because I think that the many different ideas that came out of the discussion may have confused some students.  Students were bringing in their own ideas and thoughts before they discovered and it is important that when we help students discover that we also remind them of what they discovered and what they found wrong because surprisingly they somehow seem to forget. 

 

 

After reading Chapter 1, 2, 3, 5, 6, 8, 9 and 4 in the text and viewing the simulation several times, address three of the following questions:

 

Mathematical content

Under some conditions, Pat asked, ÒWhat if you have only one ball, and you want to bounce it off the cushion to go into the pocket. Where would you aim?Ó How would you solve that problem?  First of all I would praise him for bringing up such a great question.  Then I would proceed to ask him how he would solve it or challenge him to solve it.  I would begin by drawing a diagram as I usually do.  Then, I know that we can make a triangle with the top vertex as the point that it hits the side and bounces off.  We will know the base or the distance straight from the point where the ball is and the pocket we want it to go into.  Then we can draw up a system of equations to solve the triangle knowing that one angle will be theta, another one 180-2theta and the third will be unknown.  We can also set up a couple of trig equations and solve for the variables. 

 

 

 

Equity

Was there any pattern to the kinds of students who were more involved and those who were less involved?  I think that it is pretty obvious that the students who played pool were much more involved.  I think it is great to involve students in this way however I also think that it is important to remember that you need to change it up because if you stick to one special area where a certain group of students is not familiar or not engaged they can lose a lot.  I know this from experience in my geography class that ended up being based on football teams.  I like the game, but do not keep track of where professional teams are from.  It is a great one-time activity however make sure to have a great variety and make sure students who do not understand the game still have an equal opportunity to do well. These pool questions did not seem to have a great deal dependence on knowing more than the basic concepts of the game. 

 

 

Learning and teaching

 

 

Were any of the What if? questions worth the effort it took to elicit them?

We seemed to struggle with getting what if questions however I think that it is very important to allow students to ask these types of questions.  It allows them to further their understanding and helps them to explore mathematics.  I think we do too much talking at the students and not allowing them to figure things out in their own way.  Also students are more intrigued by the questions that they ask themselves. 

 

 

SIMULATION #7: Revisit the Holedigger one more time.  Address 5 of the questions below.  Did your attitude change on any of the questions that you addressed last week?  If so, please explain.

Mathematical content

Would it have been more appropriate to teach the elimination method than the substitution method for solving this pair of equations?  Under what circumstances is each method preferable?

 

Sean said, ÒWe didnÕt need any variables.Ó Do you agree?

Is this method algebraic?

 

Is SeanÕs ÒweirdÓ idea valid?

Nature of mathematics

If you choose the option ÒWould you try that and let me know if you think it's easier?Ó what are you communicating about the nature of teaching and learning and mathematics?  You are communicating that there are many ways to do a problem and they all come to the same solution.  It also encourages students to look around at different ways to do a problem.  This is how mathematics has advanced. 

Problem solving

What problem-solving strategies did this group use?  My trusty re-read the problem.  I also saw them in a way picking out the important information.  Finally checking your answer. 

Communication

What difference to communication does it make if you stand or sit? When you are at eye level with the students it gains their attention but, more than that you are working with the students.  That is you are equals and they should not only be listening, but contributing. 

 

One of the questions was, "Should loners like Sean be forced to work with a group?" When you answered that, did you automatically defend your opinion, even though the question did not include instructions to?  Yes I did automatically defend that Sean should be forced to work in a group to improve social skills and share a new idea with other students. 

 

How might you as a teacher create a climate in which students will automatically "show their work" without being instructed to?  I would have a small number of points assigned to the actual answer and a larger percentage for the process.  If you think about mathematics in the real world if you can identify the process, there are many tools and proof readings to make sure you have come up with the correct answer if the student does not know what process to use there is little or no chance that they will ever get the right answer.  If you simply donÕt allow the students to get the points without work they will begin to show work as second nature. 

 

From Lao-Tsu: "As for the best leaders, the people do not notice their existence.... When the best leader's work is done the people say, 'We did it ourselves!'" To what extent, if any, does this aphorism apply to teachers and students?  This applies a lot.  Students take joy in what they learn by themselves.  If a teacher does not let students explore for themselves they become bored.  Students learn to love to learn. 

Assessment

How would you authentically asses what you hope students learned from working on this problem?  I think that if you spent as much time with each group as you did with this group you would be able to assess what the students were thinking of and how they solved the problem.  If not, the quality of their explanations would be another good route to assess included in that the ability to answer questions you have about their solution and other ways they had thought about it. 

 

Student #5

 A paper copy of COINS for Student # 5 is in the packet.  I did not have an electronic copy.  Student #5 did not do Feet/Yards or The Monk.

 

 

Simulation #1 (Combining Negatives)

 

I disagree with the father of student #2. It is possible to make sense of math, but it depends on how it is taught and how the student approaches it. The idea that Òmath doesnÕt make sense Éyou just get used to it,Ó is common among many people. Going through school I know many times I just memorized how to do a problem, but I know that in order to understand better or understand at all, students have to know the answers to questions like why and how it works.

 

I agree and also disagree with Blair when he says ÒBut math is math. There is only one answer.Ó Most problems only have one right answer. The very last answer a whole class writes is usually the exact same thing. The part that I disagree with is that math is not just an answer. It is a method of solving problems that can be applied to real life. In life there are many different ways to find solutions, there may only be one answer but the exciting part is the many different ways to get to a solution.

 

I definitely think the students were ÒdoingÓ math. Thinking about a problem and different ways to find a solution and why there are those solutions are the main thing that students need to get out of math. Just writing down answers is part of doing math, but another important part is figuring out the process.

 

The question about students being Ògood at mathÓ is a hard one. The student (#2) who Òjust gets used to [math]Ó may get good grades in math class because at his level he is able to memorize and mimic a problem. But student #1 is trying to understand why and how negative and positive numbers work and when he finally grasps the concept he will be better off when doing lots of other problems. Also the fact that they are all talking about the problem and remembering what they were taught makes them good at math.

 

An explanation that is good is one that helps learners get a solution and leads to understanding, but it also has to answer questions like how and why the solution works. It needs to allow people to take the explanation and think in different ways about the problem.

 

In some instances the students were communicating clearly. For example when the student wrote down the table as a visual aid in what he was talking about. But, at other times, the bad communication didnÕt allow them to demonstrate their skills. For example student #1 was often confused about the difference between adding and multiplying negatives and positives.

 

A connection that was made in the simulation was the realization of the difference between adding and multiplying negative and positive numbers. They also used patterns (which probably were connected from earlier math) in the table to understand. The students tried to find other ideas so they could interconnect and build on one another.

 

Many different mathematical representations can be found in the students reasoning. The example of running the movie forward and backward is one, and the number line is another. A last representation is the table with repeating patterns to understand the negatives and positives.

I donÕt know if a student would normally think of using a table to help understand the problem, but he or she may have understood that method of approaching the problem really well and it stuck with them. Another idea that I remember from my math classes was using manipulatives  and grouping them to deal with positives and negatives.

 

 

Simulations: Catchup

 

            I do believe mathematics is magic in the sense of the wonderment and interesting things that can be done using it. It often happens that students may believe itÕs magic because they donÕt understand how an answer to some problem is obtained. The confusion that people experience leads them to think that the teacher is just a magician and only they know the tricks.

One connection in the simulation was made when a student asked if the situation really happened. The most points were given when the teacher told a story about driving to Wisconsin when the situation actually happened. Making connections serves to help students relate what they are doing to the real world and to let them care more about the math.

When answering the question if a student is ÒgoodÓ at math, it depends on how you look at the situation. Some students seem to be able to understand more quickly or get better grades in math class, so they could be considered ÒgoodÓ at math. But, other students may want to know what is behind the math, so they could be Ògood.Ó I think all students can learn mathematics they, along with the teacher, simply need to find the best way they can learn it and also practice learning in other styles.

This lesson was taught in a similar way to the way I was taught mathematics. It depended on the teacher and class, but some of my teachers would present problems and help the class along in discovering how to solve them. Other times we would be taught definitions and methods of solving problems and then have to do homework on our own.

This problem would fit into an algebra section of a math curriculum. It would help to fulfill a standard where students use symbolic algebra to represent situations and solve problems.

A curriculum is a set of objectives that students should master and standards are those objectives expanded to help define the objectives. A textbook works into all of this by being there to support and fulfill the curriculum. It does matter what textbook we use because it can help students who are visual learners and also it can support our different types of teaching.

I believe this lesson was more effective than the lecture based coins lesson because the students are active and involved with the problem. They also will remember the process and the information better if they have to answer direct questions and think about the problem.

I would ask a student what another might be thinking because it involves more of the class and helps the students work together to find a solution. Also it can show that there are different ways of thinking about a problem and that everyone goes about doing things differently.

I think fewer points were given for ÒGoodÓ than ÒThank youÓ because good can be repeated over and over and become meaningless. The word good, or other small praises, can also make it seem that other answers are not good, and that certain students are good at math, but others are not.

            Students should be allowed to use graphing calculators in this class to make tables along with learning how to solve it with out the technology. It will give them a different view of the problem and help them to understand even better.

 

 

 

Week 7 Assignment

Bouncing Simulation Questions

 

Score: 18700

Choices made:

Let's consider a problem.

Thank you, Jesse. Now, how about some gut reactions? No need to justify. What's your intuitive feeling?

Thank you, Jose. I like controversy.

Dula?

Would you please elaborate, Dula?

What do others think of that idea? Jesse?

Why?

How should we proceed?

And Kelly, roll the ball against it.

[Do nothing]

Are others convinced like Dula? Ashley?

Carlos?

Terry, did you have an idea?

Why do you think that's dumb?

Jo?

Would you demonstrate, Lindsay?

Let me see if I understand what you did, Lindsay.

Good job, Lindsay. Was that what you were going to try, Bryce?

A protractor like this? [Hold one up.]

[Wait]

Pat, did you have a question?

Can you elaborate?

What is that you're saying, Lindsay?

As you know, a math problem isn't finished when we solve it.

[write] Yes, like that. Thank you, Kim.

Everybody please stand up.

I want to take up one of them briefly before starting homework.

Yes. Does anyone remember the name for the angle perpendicular lines form?

Is it obtuse? Mikoo?

What other mathematical ideas have come up as we've solved this problem?

Do you agree that they're equal? Lee?

Would you explain to me what you're doing?

Does your answer make sense to you?

Yes, I think that's what the class agreed to. Is that what you've done?

So here, for example, you traced this incoming angle?

Would you please explain to me what you're doing, Morgan.

Would you show me how you did the measurement here?

Does your answer make sense to you?

Would you please explain to me what you're doing, Dula?

Have you thought of any other questions?

 

 

Bouncing Simulation Questions:

 

When choosing between: ÒAre they equal Lee?Ó and ÒDo you agree that theyÕre equal? Lee?Ó the message conveyed is that the nature of mathematics is a collective project where everyone contributes. Many students think of the nature of doing mathematics is a single thing where you do a problem and will either get it or not get it. This is not correct, we want to teach the students to work together to find many ways of problem solving.

 

At the point when Kim says, ÒIt doesnÕt seem logical though,Ó the meaning of logical is referring to common sense. Her intuition says that if you hit a ball harder it will go a different direction when bouncing than if you hit a ball softer. She first thinks about things with common sense, and then goes on to think about things using mathematical logic.

 

Under some conditions it would be appropriate to ask ÒWhat the heck is that Dula?Ó The tone in your voice must be laughing, but also interested and encouraging Dula to explain what she has done. The statement will also get the attention of the other students and they will want to know what Dula is up to.

 

I think this problem definitely belongs in both a math and physics curriculum. Students need to connect math to the real world and not simply think of it as a list of problems that they have to work out. This problem can help make that connection.

 

Beginning with a problem actually made the score go up overall. This was interesting because it was not my first choice in the simulation, but it can be a type of inquiry based teaching. If you start with the problem students may begin to think of ways to solve it and their brains will start focusing and they will be more creative and able to focus on the math concepts that come out later.

 

I said ÒI like controversyÓ because we need to remember to look at both sides of the problem and solution. Many times students (and teachers) may think they are on the right track and not even consider other possibilities, but with controversy they can look at it from another angle. Having controversy can also strengthen your view of the problem even if you are on the other side of the controversy.

 

The ÒhubbubÓ when the class was gathered on the floor did not show that the class was out of control. It was a good way to get the class moving and refocused while still thinking about the problem at hand. If the classroom has standards and rules about the demonstrations that are followed, then hubbub is not a bad thing and not a sign of an out of control class.

 

The times when I clicked for a long time while not making any choices were good. It allows the students to discover things for themselves and to have an active role in participating in the classroom.

 

The What if questions were definitely worth the effort it took to elicit them. It conveys the message that you canÕt simply stop when you find an answer to a problem, but you also have to continue to think about other questions to fully understand a concept. The question that came out about what happens after the ball bounces off the second cushion is a good example of this and lead the students into the right thinking for their homework.

 

It was a better idea to ask the class about the meaning of obtuse than simply telling them because they have to then figure it out for themselves and will remember the definition better. It will help with their discovery learning and understanding.

 

 

Catchup Simulation Revisited

 

Score: 16000

Choices made:

Who remembers what a table is? Joey?

So do you mean it's a chart with rows and columns?

Thank you.

Dula, can you think of a situation that would give a table with just two columns?

Justin?

I think that has possibilities, Justin.  What would be in the two columns?

Like this?

Thank you, Justin.

Carlos?

What's your reasoning?

How might Carlos be seeing that in the table? Ashley?

Thank you, Ashley.

Do you understand, Jo?

Good. How about you, Mikoo?

If you were going to ask a question, what would it be?

Oh!  Thank you. Do you see what Mikoo is saying, class?

Who would be willing to read it aloud?

Any idea of what to put in the first column, Yiscah?

What's the best thing for the second column, Bryce?

Now how might we find the distance at each time?  Sasha?

OK. Would you please help us fill in the table, Madison?

Can you explain your reasoning?

Papella, how would you fill in the first row of the car's table?

How did you decide on 60 for D, Corrine?

Oh what, Julie?

Why did you change your mind, Julie?

Thumbs up if you agree.

So we have the first row.

Are we done? Mariana?

So what can we conclude? Rory?

What are some good things to do when you're stuck?

Do you think that might help? If so, why?

Could you pick a point and say how far each of them is from that point?

Can you explain it to me, Kim?

What's a good strategy for when you're stuck?

Yes. What should you do with that?

Would a picture help?

Could you pick a point and say how far each of them is from that point?

What are you doing, Kelly?

What's a good strategy for when you're not sure?

 

Week 9 Simulations Questions:

1. mathematical objectives:

            The students will be able to use various tools such as charts and pictures to evaluate and solve a problem.

            The students will be able to connect a math problem to a real world situation.

 

2. This lesson addresses the following MN standards:

            Mathematical Reasoning by the application of mathematical representation, communication, and representation.

            Number Sense because of the use of real numbers represented in various ways to quantify information.

           

This lesson addresses the following NCTM standards:

            Numbers and Operations because students have to understand numbers and relationships among numbers.

            Algebra because students have to understand patterns, relations, and functions.

            Problem Solving because students will be able to solve problems that arise in mathematics and in other contexts.

            Communication because they will communicate their thinking to peers, teachers, and others.

 

 

Additional Simulation Questions:

           

The Catchup problem and the other problem of this simulation are definitely real math problems. They are something in the math class that allows students to discover math concepts and help them better understand. Problems are different than exercises in the fact that exercises are focused on repetition and problems are about discovery and learning.

 

In this simulation the teacher made the connection to an actual time where she drove her car and someone else drove a trailer to another place and she left later. Making connections serves to give students a view of what mathematics can be used for and that it is in fact used other places besides a classroom.

 

This lesson was not taught in the way I was taught mathematics. We rarely ever had discovery lessons. We were simply lectured and then did homework, took tests, and every once in a while did a project. 

 

I think the Catchup lesson would fit into an algebra curriculum. It can be connected with graphing and constructing a function from a table. But, it also could fit into many other math curriculums as a problem of the day or an extra credit problem.

 

I think this lesson was more effective than the lecture based lesson of the Coins class. It got students involved and thinking about the problem at hand. Questions like those in this lesson are often, but not always, more effective than statements when teaching.

 

I do think students should be allowed to use graphing calculators to make tables in this class. It is another way for the students to view the problem and it can also lead to graphing and the writing of equations.

 

Holedigger Simulation Score/Questions:

 

Score: U=13  N=3  I=12  C=7  E=7  composite = 19500

 

Choices made:

Good morning, class.  It's good to see all your cheery faces.

Let's start with some practice at using your WAPS.

As usual, some lucky person in each group gets to read it aloud to the group. Go ahead and start.

[Find a seat or kneel]

Sean, what are your thoughts at this point?

The best mathematicians consult each other for ideas. You might do the same with your group, Sean.

What's the goal of the problem?

And D is the unknown?

It's an idea you might try. [Leave group.]

[Find a seat or kneel]

How did you do that?

What are you thoughts, Sean?

Maybe more thinking can reconcile the two, Sean.

Very nice work! Another group had the idea of using two variables. I wonder if it would be easier if you used a second variable?

Sure.

Would you try that and let me know if you think it's easier?

[Find a seat or kneel]

What's up?

Sometimes it helps to think of a similar situation. Can you think of one?

What does a solution to a single equation look like?

What do you mean?

Sean, what are you thinking?

And H also equals D minus 69.

What in the world could you do with those?

Nice idea. Try making a single equation without H. [Leave group.]

 

Simulation #7 Question (Week 10)

 

I choose the option ÒWould you try that and let me know if you think itÕs easier?Ó This option is communicating the idea that mathematics has many paths to solutions and some are easier for different people. It says that the nature of teaching and learning mathematics is individual to each person.

 

Some problem solving strategies that this group used were to read the problem out-loud, draw a picture and label it, to go back and reread the problem, and to use a graph to help solve the problem.

 

When an equation is being solved, the if-then statement that is being proved is: if there is some equation (the one you are trying to solve) then there is some constant that can replace the variable(s) to make both sides of the equation equal. For example, if you have 2x + 3 = 13, then there is some constant, 5, that can replace x to make both sides equal. An if-then statement for an equation being checked is: if you have some constant or constants and put them into an equation for the variable(s), and if both sides of the equation are equal, then the constant or constants are a solution to that equation.

 

Sitting instead of standing conveys the message to students that you are on their level and working to help them learn instead of being a dictator and forcing them to learn. It helps them to feel more comfortable to ask questions and learn.

 

 

The group generated both a set of equations and also a graph of the equations to represent the situation. Personally I prefer to solve the equations on paper instead of with a graph, but I think it is also a good idea to graph them and know that is another way to solve. I think it is a matter of which you prefer and both ways are good to solve an equation.

 

I think it is pretty likely that students would make comments like a picture at lunch time and not being able to see in the dark. I think students will say some things that donÕt seem to have to do with trying to solve the problem, but they are actually thinking about the task at hand. I know I get distracted when doing anything, so it is allowable for students to comment like that. It will let them relax and think about many different solutions to their problem.

 

Regarding the question ÒShould loners like Sean be forced to work with a group,Ó I donÕt know if I automatically defend my opinion, but I did think about the question. I think it is a common problem in a classroom where not every student likes to work in a group and what to do about students who prefer to work alone. Students know if there is a ÒlonerÓ and they will know if you are purposely trying to get them to include that student, but there are also times where you can get the student to participate in a less intimidating situation.

 

A teacher can create a climate where students automatically show their work by teaching them by example. Every time you give an example problem or teach you can show your work and emphasize that you are showing your work. Also when helping them one on one, you can ask them to tell you how they found their answer and ask them to write the work on their paper.

 

I think it is a good and bad thing that students would say you had nothing to do with them making productive suggestions. It is good in the fact that they made the suggestions and came to good conclusions on their own accord. But, they must also be able to solve similar things on standardized tests and classroom tests, so they need to know if they understand things fully and are able to find a way to do the problem that they are comfortable with on their own.

 

I would authentically assess what I hope students learned from working on this problem by having each group present in front of the class and give them a short rubric before they did this. I would also assess what they had learned from when I went to each group asking questions and helping.

 

 

 

Holedigger Simulation Score/Questions:

 

Score: U=14  N=3  I=12  C=7  E=7  composite = 20500

 

Choices made:

Good morning, class.  It's good to see all your cheery faces.

Let's start with some practice at using your WAPS.

As usual, some lucky person in each group gets to read it aloud to the group. Go ahead and start.

[Find a seat or kneel]

Sean, what are your thoughts at this point?

The best mathematicians consult each other for ideas. You might do the same with your group, Sean.

What's the goal of the problem?

And D is the unknown?

It's an idea you might try. [Leave group.]

[Find a seat or kneel]

How did you do that?

What are you thoughts, Sean?

Maybe more thinking can reconcile the two, Sean.

Very nice work! Another group had the idea of using two variables. I wonder if it would be easier if you used a second variable?

Either that or perhaps the head distance.

Would you try that and let me know if you think it's easier?

[Find a seat or kneel]

What's up?

Sometimes it helps to think of a similar situation. Can you think of one?

What does a solution to a single equation look like?

What do you mean?

Sean, what are you thinking?

And H also equals 2F minus 69.

What in the world could you do with those?

Nice idea. Try making a single equation without H. [Leave group.]

 

 

SeanÕs ÒweirdÓ idea is valid. It is a very creative way to think about the problem that a middle or high school student would have come up with. We need to remember that we as future math teachers have had a lot of mathematics background and practice and will go straight to solving these types of problems with variables, but students often donÕt look at it that way. SeanÕs idea is valid and shows the other students there are many different paths to a solution.

 

I choose the option ÒWould you try that and let me know if you think itÕs easier?Ó I like this option because it lets students know that some methods are easier than others for certain people. It also puts the teacher on that level with the students and lets students know that the teacher has certain learning styles too. This is the same opinion that I had last week that the nature of mathematics changes for each person.

 

If you sit instead of stand you are communicating to the student that you are there to help rather than simply dictate. My attitude for this question has not changed from last week, but it has gotten stronger. I decided to remember to kneel or sit last week when I was helping in a 7th grade math class and I could tell the difference in how students opened up more  and began asking more questions.

 

ÒAs for the best leaders, the people do not notice their existenceÉwhen the best leaderÕs work is done the people say Òwe did it ourselves.Ó- Lao TsuÓ This quote applies to teachers and students because the best learning is where students discover things themselves. Even though students need to know if they truly understand or not, when they believe they discovered a math concept themselves they will remember it forever. My opinion on this has changed slightly from last week because hearing my peers opinions makes me more completely believe it is good for students to think they have discovered on their own.

 

A climate where students automatically show their work first needs to be created when simply discussing a problem. When a student answers a question, prompt them to also explain why they answered in that way. After they are used to this, it will come more naturally to Òshow their workÓ on paper too. I had this attitude last week too, but thought more about showing work when I do examples on the board or paper. This week I also think having students explain their ideas is a way of showing their work.

 

Student # 7

 

For Student # 7 copies of Combining Negatives, Coins, and Ft/Yds are in the paper packet.

 

Simulation #3: Catchup

            At the end of the simulation when I had a chance to ask Kelly, ÒWould the time be the same if the car were going 65 mph and the truck 60?Ó the answer would be no.  If the car and truck were going 65 and 60 mph respectively, the car would catch the truck after three hours, not two hours and forty-five minutes.  I think that when solutions are found, some people consider it ÔmagicÕ because they do not understand the procedure to reach that solution.  Those people think the solution is impossible to find, so when an answer is determined and the answer works, then it is mysterious.  I, on the other hand, do not believe that mathematics is magic.  There is a logical method to reach a solution, thus I consider it a procedure and not magic.

            The catchup problem and the other problems of this simulation are real math problems.  The problems are real in the sense that they involve mathematical steps and reasoning in order to reach a solution.  Also, the problems are real because they consider actual events and situations that people encounter everyday.  These problems are definitely different than exercises because the students are unfamiliar with the solution.  This is clear when the students brainstorm about possible problem solving paths and cannot move right through the steps to a successful answer.  The students began by drawing a diagram, which did not help them solve the problem.  With this, the teacher left the topic of drawing a picture.  Surely, at a different time, the teacher will return to drawing a picture when it has a more positive effect on the problem.  The students were engaged in problem solving.  This was shown when a variety of answers were volunteered, some right and some wrong.  The students were exploring possible methods to solve the problem.

            The car and truck problem had connections to real life.  The connections did not arise from a funny story or a joke, but it concerned a topic that all students are familiar – driving.  This connection made the problem relevant to the students.  If a topic is relevant outside of the math classroom, students will be more interested.  Knowing that the topic will be used in real life, students will understand the importance and tend to remember the problem and the method of solution.  In addition to being relevant, the problem was represented effectively.  The problem was represented by an equation with a variable.  And, equally effective, the problem was represented through a table.  These two methods, if performed correctly, will result in the same solution.  I think it is important to use more than one representation because students learn differently.  One student may understand the equation while another understands the table.  Another representation that could have been used is a graph.  An equation for the car and an equation for the truck could have been built and graphed.  The intersection point would have shown when the car caught up with the truck.

            Through the simulation, it is difficult to judge who was good at math.  We saw that both Kim and Lee offered a suggestion for solving the problem, but that does not mean they are good at math.  Also, just because Julie did not have a suggestion does not mean that she is not good at math.  In order to make a judgment of who is Ôgood at mathÕ much more needs to be considered.  Plus, what makes a student good at math?  This question is pure opinion, thus cannot actually be determined.  I think all students can learn math, but students learn at their own pace with different roadblocks.  Some students learn at a slow pace and need lots of extra help, however; they can still learn math!

            This lesson was taught differently than I remember being taught.  The question allowed students to explore the problem and determine their own solutions, while being guided by the teacherÕs questions.  I remember seeing examples where the teacher worked through the entire problem with limited student input.  My teacher knew how he/she wanted to solve the problem, so he/she used that method and simply explained along the way.  Questions were rarely used to guide the studentsÕ thinking.

            The catchup problem would fit into the math curriculum when working with single variable equations.  These types of equations are commonly seen during algebra.  Also, the problem could be used to encourage different problem solving techniques.  When teaching graphs, the teacher could use this problem to show how the solution is the point where the graphs intersect.  In addition to algebra and graphs, the problem could be used when teaching about tables.  Using a table in this problem was an equally effective approach compared to graphs and equations.

            I think the curriculum is neither the textbook nor the standards.  A curriculum is the content that the teacher will cover throughout the school year.  Even though the curriculum is not a textbook or the standards, those two resources definitely guide the curriculum.  The teacherÕs curriculum must include all of the standards, but is not limited to them.  The textbook may outline a majority of the teacherÕs lessons, but the teacher may pick and choose what sections in the textbook she will teach.  Also, the teacher may add content to her curriculum that is not included in either.  The textbook used in the class will make a difference, because the teacher will either teach mainly lessons from the textbook or if the textbook is not ideal, the teacher may bring in many outside resources and worksheets to build a lesson.  Having a good textbook, I think, helps the students so they have all of their resources together.

            Overall, this lesson was effective.  The questioning guided students in the correct direction, while allowing them space to explore possible paths.  Was this lesson more effective than the lecture-based Coins class?  I donÕt think this lesson was more effective, but it did give variety.  If the teacher used lecture to teach every lesson or if the teacher solely used questioning, then neither would be effective because students need variety to learn.  I think the two lessons were equally effective.  Questioning, however, is effective and forces students to think deeply about a problem.  The students were not simply told which procedure to pursue.  Statements get right to the point and leave no room for students to discover a solution on their own.  If time permits, I would question my students and guide them to a correct solution.     

 

 

Simulation #5: The Monk

            At the beginning of the lesson, the teacher asked for intuition but not reasons.  I think this was an appropriate approach since the students had only briefly considered the scenario.  If the teacher would have asked for reasons, students would not have volunteered answers.  Once the students have worked with a problem for a longer period of time, students would feel more comfortable offering reasons with their intuition.  At the beginning, students are not confident with their response, so there might not be any reasons to back up the initial intuition.  After the initial intuition was gathered, the students had several starting points to consider.  The students could consider a case, and through problem solving, decide if the intuition holds.  Without the intuition, the students would blindly approach the problem.

            Working in pairs allowed the students to discuss their ideas.  The students could bounce their ideas off their partner instead of the whole class.  The students could adjust their thoughts and compromise on a reasonable solution or idea when hearing the other partnerÕs thoughts.  During this time, the teacher observed the groups and listened to their ideas.  This allowed the teacher to hear suggestions that would be offered by the students later.  Also, if a pair of students was confused or stuck on the question, the teacher could offer a starting point.  If students were moving in an incorrect direction, the teacher could guide the students back on track.

            During this lesson, the teacher thought all students were engaged in the problem.  I think that only in a perfect lesson all students will be engaged.  Usually, at least one student in the class will be distracted or uninterested.  Not all students are intrigued by the same concepts or subjects, thus; it is difficult to engage every single student in one lesson.  Engaging a student in a lesson is important to promote learning.  If a student is interested in the topic, that student is more likely to want to learn and understand the concept.  That student will make a connection of the present material to prior knowledge, which will help the student remember it.  If a student is not engaged, he/she will listen but not strive the extra length to make a connection, causing him/her to forget the concept faster.

            When asking a question, the teacher noticed that Jon was not responding.  She asked, ÒJon? Maybe I was looking elsewhere.  Are you OK?Ó  I think the purpose of the middle comment was for the teacher to take some responsibility.  If Jon felt offended by the response, then he would be turned off to the entire lesson.  So, the teacher admitted that that he/she could have made a mistake and missed his answer.  This gave Jon a second chance to respond with his intuition.  Also, if he was confused, he had an opportunity to ask a question or ask for clarification.

            At a point in the lesson, the teacher thought, ÒConfusion is not necessarily bad.Ó  I agree with the thought because, if confused, students will rethink the problem.  Doing this, students may recognize new important information.  Also, students will consider other perspectives and cases when they are not satisfied with the present one.  Confusion in this type of lesson is okay because it involves student exploration and discovery.  However, when doing a direct lesson, I think confusion will cause students to quit trying.  If students are confused with a basic concept or example, teachers should try to clear up that confusion before moving to a new topic.  If students remain confused, they may not be able to master the new concept either.

            When conducting a discussion or conversation, some teachers fear a small group of student dominating.  In the simulation lesson, the teacher did not allow a small group of students to dominate.  The teacher was aware of what students were participating and the teacher made an effort to include students who were not volunteering answers automatically.  If the teacher only called on the students with their hand raised, then it would have been a small dominating group.  This did not happen because the teacher regulated the discussion by requiring students to raise their hands and be called on before they spoke.

            Sometimes, when students hear an answer, they stop thinking about the problem.  I agree with this statement, but whether or not students stop thinking depends on the teacherÕs response to the answer.  If there is only one correct solution, then the students indeed stop thinking.  If the teacher acknowledges the answer and still encourages different equally correct answers, then students will continue thinking.  The teacher has to be conscious of his/her response depending on his/her expectation.  In this class, Sara gave an answer and the other students kept thinking.  This was a result of the teacherÕs response.  She did not praise Sara for finding the one and only answer, instead she asked for responses to SaraÕs point and then continued to ask for other intuitions.

            The lesson was a form of group investigation and discovery.  For this reason, the teacher did not save time and simply tell the students the answer.  The lesson was implemented to improve group skills, communication skills, critical thinking, and evaluating solutions.  If the whole objective of the lesson was to learn the answer, then the teacher may have just told the students how to solve it.  On the contrary, the lesson was designed to teach several skills simultaneously.  Problem solving involves the process to finding a solution and not simply the solution itself.

            During the lesson, the students began communication without the teacherÕs involvement.  This was still good teaching.  I think good teaching can stem from student to student interactions.  When students explain their ideas to other students, then both students in the conversation are learning.  Even though the teacher was not directly involved, the teacher continued to monitor the discussion and could have jumped in when it turned a wrong direction.  As long as the discussion is moving in the desired direction with correct information, it is good teaching to allow students to interact.

            In the lesson, students learned how to think critically and evaluate posed solutions.  Students created visual diagrams to support their conclusions.  In addition to the math concepts learned, students learned how to discuss their solutions and procedures with a partner.  Learning how to verbalize the mathematics and solutions is a large math topic that is mastered through practice.  

 

 

 

Simulation: Bouncing

 

            The simulation, Bouncing, was centered around a question dealing with the angles on a pool table.  The problem does belong in a math curriculum because it is a real life application where students can see the study of congruent angles being applied.  Problems of this type are not only fit for a Physics curriculum.  Mathematics concepts are used in everyday life constantly, and this awareness has to be increasingly raised to students. 

            When a student supported the opposite position, the teacher responded with ÒI like controversy.Ó  I think controversy and considering opposing situations allows the class to analyze both sides of the position.  If all students agreed, then the supporting side would be considered.  Since there as controversy, both sides of the situations were given equal consideration.  A weakness of having a controversy, is that time is spent considering a false position.  Students may remember the class time dealing with the false situation and think it were true.  However, math problems often involve many situations where there is one best situation.  So, considering the controversy aligns with characteristics of actual math problems.  A majority of the time, when a problem is brought up the correct answer is unknown.  In this situation, many sides must be considered to find the correct solution.  In the simulation, the class considered multiple situations instead of simply the correct one.

            Under certain circumstances, studentsÕ actions varied.  In one situation, Papella did not ask her question.  But, when the teacherÕs timing was appropriate, Papella shared her question.  I think it depending on the wait time before the teacher questioned Papella.  If the teacher asked Papella right away when she looked confused, then Papella shrugged off the question and acted okay.  When the teacher discussed the content with another student first and was sure that Papella was stuck, then the teacherÕs timing was appropriate.  Papella shared her confusion and stated her question.  Another situation deals with students who are over-anxious to answer.  In this case, Justin was eager to discuss the angle of incidence and reflection.  The best way to deal with this situation was to ignore Justin for a minute to hear other studentsÕ opinions first.  Once a few suggestions were given, then JustinÕs opinion was asked.  This gave the students several opinions to consider rather than hearing only JustinÕs confident answer.  Sometimes when a student is very confident with his/her answer, the other students assume it is right and stop thinking of alternatives.  Thus, the teacherÕs approach of hearing several opinions before calling on the eager Justin was appropriate.

            When the class was in a ÔhubbubÕ or gathered around a demonstration on the floor, the students were very engaged.  The students were jumping into help the demonstrator.  This was not a sign of an out of control class, instead, the class was fully participating in the lesson.  The students were not in their usual seats but the students were giving input and listening, signs of engagement.  The lesson was moving in the intended direction and staying focused with the pool table problem.  If the lesson were to get off-track, then the lesson may be considered out of control.  In the simulation, though, the class was controlled and the students were learning.

            At times in the simulation, there were long periods where the teacher did not have choices.  In this case, the students were communicating with one another.  The students were discussing the position, fixing actions, and questioning statements.  When this can be accurately done, the teacher can allow students to solely interact with other students.  Effective teaching can occur between students as well as teacher to student.  Having students be the center of attention allows students to improve their explaining and speaking skills.    

            The teacher encouraged the producing of Ôwhat ifÕ questions.  During this time, many students volunteered questions that were not worth considering because they required different factors and math concepts.  However, one student questioned, ÒWhat if the ball was hit directly at the cushion?Ó  This question brought up the topic of congruent angles and right angels.  In the simulation, these concepts were very important and the students had the appropriate background knowledge to discuss the Ôwhat ifÕ question.  The students also had the background knowledge of the three types of angles.  Therefore, I think it was a better idea to ask the class about the meaning of obtuse rather than telling them.  Asking the students about the meaning forced the students to review their knowledge of angles.  If the teacher would have simply stated the definition, students could have overlooked the concept.

            In the simulation, the teacher attempted to review some prior concepts.  One of the concepts, as discussed in the previous paragraph, was an obtuse angle.  Another concept was reflections.  If the teacher would have written ÔreflectionsÕ on the board, the students would have the word but nothing to connect the word to.  By asking, ÒWhat about reflections?Ó the students heard the word multiple times, plus a student explanation accompanied the word.  Asking students to further explain their remarks allows students to explain the procedures. 

            The simulation did not contain a formal summary of the lesson before distributing the homework.  In this lesson, the homework was a direct follow-up of the class discussion.  The class was structured to consider possible situation and the homework was answering the question.  A summary is needed toward the end of a class session if certain skills were learned.  I think it is important to summarize main concepts in several situations.  First, when skills are built up, the summary reinforces the conclusions.  Second, when the teacher observes the students having trouble with a concept, a summary will provide feedback and criticism on how to fix the skill.  Finally, a summary can end a lesson when several lessons should be tied together.  Many individual concepts may be taught, and then the summary will connect the concepts. 

 

My Score:

U = 26

N = 10

I = 9

C = 26

E = 4

Composite = 36700

I will turn in a printed copy of my scores with my Problem Solving 7 the next time

 

Simulation #8:

1. In the simulation Bouncing, identify a question, if possible, at each of the various levels of BloomÕs Taxonomy.

 

Knowledge: Do you mean a protractor?

The question is asking Bryce to recall the correct word for the tool he wanted to use.  It is the remembering of a basic name, nothing more than stating the correctly memorized word.

 

Comprehension: How do balls bounce off of the walls of a pool table?

The question is asking the students to comprehend and explain how the balls bounce.  The students are translating what they remember happening into words.

            Another Comprehension Level question could be, ÒWould you please explain what you mean by angle of incidence and angle of reflections?Ó This question is asking Justin to explain and discuss what he knows about the angles.

 

Application: Would you demonstrate Lindsay?

The teacher asked the student to demonstrate her suggestion.  Lindsay had suggested the path of the ball be drawn using patty paper.  So, in response, the teacher asked the student to demonstrate in order for the class to actually see the suggestion and how it related.

 

Analysis: What do others think of that idea? 

The teacher is asking the class to analyze JustinÕs remark about the angle of incidence and the angle of reflection.  The students were asked to criticize, examine, or question JustinÕs suggestion.

 

Synthesis:  Using a ping pong ball and meter stick, the teacher had the students construct a fake pool table then asked what was happening.  This section of the simulation asked students to assemble the appropriate trial table so they could observe what happened when the ping pong ball hit the wall.  Students had to organize their trials and develop conclusions based on their observations.

 

Evaluation: What if the pool table isnÕt flat?

This question was asking the students to predict the balls path given a different circumstance.  The students had to compare what they knew using a flat table to support their predictions for a not flat table.  Other Evaluation Level questions in the simulation were: How can we find the line that the ball follows?  Do you like that question, Kim?  So, what do other think of that idea?

These questions asked for students to give their solutions and defend their position.  Also, the questions asked students to evaluate what they thought about a particular suggestion.  Students were encouraged to assess and argue using their opinions.

 

2. Write two objectives for the Bouncing lesson.

Given a protractor, the students will be able to measure angles accurately to two degrees nine out of ten times.

 

Given a labeled picture, student will be able to state whether a pool ball will fall into a pocket or miss the pocket eight out of ten times.

 

Given a situation, students will be able to discuss possible solution methods and analyze the effectiveness of the posed procedures.

 

These objectives state what the students will be able to achieve as a result of the lesson.  Each objective describes the circumstances set-up by the teacher, such as, given a labeled picture.  The objectives include a task written in measurable terms.  Also, an evaluation guide is within each objective, for example, nine out of ten times.

 

3. Identify the 9-12 standards from both NCTM and MN Standards that this lesson would address. 

The simulation met many 9-12 NCTM Standards and Process Standards:

Numbers and Operation Standard:

Judge the reasonableness of numerical computations and their results.

Algebra:

Use symbolic algebra to represent and explain mathematical relationships.

Draw reasonable conclusions about a situation being modeled.

Geometry:

Use geometric models to gain insights into, and answer questions in, other areas of mathematics.

Problem Solving:

Apply and adapt a variety of appropriate strategies to solve problems.

Monitor and reflect on the process of mathematical problem solving.

Reasoning and Proof:

Select and use various types of reasoning. 

Communication:

Communicate their mathematical thinking coherently and clearly to peers, teachers, and others.

Analyze and evaluate the mathematical thinking and strategies of others.

Connections:

Recognize and apply mathematics in contexts outside of mathematics.

 

Several 9-11 MN Standards were met through the simulation:

Mathematical Reasoning:

Assess the reasonableness of a solution by comparing the solution to appropriate graphical or numerical estimates or by recognizing the feasibility of solutions in a given context and rejecting extraneous solutions.

Support mathematical results by explaining why the steps in a solution are valid and why a particular solution method is appropriate.

Spatial Sense, Geometry, and Measurement

Use models and visualization to understand and represent three-dimensional objects and their cross sections from different perspectives.

 

4. What were the choices that led to DulaÕs having a Big Idea?  In general, what kinds of choices lead to creative problem solving?

In the simulation, DulaÕs Big Idea was prompted by discovery learning.  When the teacher encouraged students to explore the question, the students tried their ideas and judged if their solution method was appropriate.  The Big Idea surfaced when the teacher asked for intuition, but did not say whether a certain idea was correct.  This left the students open to many options.  I noticed that various solution methods were suggested by the students, and the teacher encouraged further experimentation.  The teacher guided the students through the demonstration with the ping pong ball and meter stick.  After seeing this, Dula felt comfortable to experiment during the individual work time.

In general, allowing students to experiment lead to creative problem solving.  Giving students space to explore instead of giving them a particular solution method promoted student creativity. 

 

5. Was the amount of closure for this lesson appropriate?  Why or why not?  If not, what would you include? 

For the lesson, the closure was included in the homework assignment.  The entire period was dedicated to finding possible solutions and discovering how to work with the posed methods.  Thus, assigning the problem to students to further investigate served as the closure.  In the assignment, the students would work with their suggested method and solve the problem.  Perhaps, the following day the teacher could formally close the assignment by asking volunteers to describe their methods and state their conclusions.

 

Three additional questions:  (My simulation scores print-out is attached to my problem solving.  My scores are: U=27, N=14, I=8, C=30, E=5 composite=39,900)

During the simulation, I had a couple opportunities to ask, ÒDoes your answer make sense to you?Ó  This question reveals that there are multiple solutions to a particular problem.  How the student approached the problem was that studentsÕ individual process, so that solution must make sense to the student.  Often, solution methods are unique to students, so some explanation may be required to describe the solution to others.  Checking if the answer makes sense to students is important, because the solution method is equally as important as the answer.  If the student does not understand the method used to solve the problem, then the problem is not completely understood.

As Dula was working individually on the homework assignment, the teacher could have asked, ÒWhat the heck is that, Dula?Ó  This response is appropriate under specific circumstances only.  This circumstance comes from the teacher-student relationship and how well the teacher knows the student.  If the teacher has an informal relationship with a student whom she knows will accept the comment in a positive manner, then it may be okay.  The teacher must be sure that this student will be confident and be able to explain his/her method.  The comment is never okay if the teacher does not know how the student will react.  The comment could discourage the student and think his/her creativity and exploration is unacceptable.

At lone point Kim says, ÒIt doesnÕt seem logical, though,Ó referring to the conclusion that the angle of the bounce is independent of the speed.  In this case, logical means that it would have matched her predictions.  Many times, students have notions of how the mathematical concept or situation works in real life.  Kim had a predication that the angle depended on the speed, and when the conclusion was different, it did not seem logical.  In these cases, demonstrations would help emphasize the conclusion to help make it more logical for the students.

 

Catchup Revisited

 

Answer six additional questions for the simulation:

            The Catchup problem and the other problems in the simulation were real math problems.  These problems forced students to analyze practical situations by using the relevant math skills.  All of the problems required students to determine the distances based on time, while the students developed knowledge of tables.  The problems encouraged students to analyze the situations in order to use appropriate solution methods.  In all, the situations were indeed problems, and not exercises.  Exercises are questions where students know the correct solution method.  Exercises are routine practice, compared to problems where students have to decide on a solution method.  The catchup problem was solved during the class session.  However, the students could continue to work with the problem to find alternate solution methods.  The homework questions were started during the class.  During the class, the students were engaged in problem solving.  The students studied the question and then gave input on the tables.  The students problem solved to decide what data should be organized in the tables.  Then, once the two tables were complete, the students compared the data to find a solution.

            The simulation was based around an interesting story, where someone was helping his friend move.  This connection helped the students see the relevance of math in everyday life.  Driving and catching up with another car are everyday situations that students have probably experienced or can imaging experiencing.  The simulation included connections between distances and time.  The students utilized the equation D = R * T (distance equals rate times time) without explicitly stating the equation.  Another connection made in the problem was the use of tables to organize data.  Students were solving an algebra problem that was fully organized by using tables.  These connections help students improve a variety of skills through one problem.  In addition, students will remember all of the skills to a higher degree when they can relate back to a relevant problem.

            The Catchup problem was represented effectively.  The problem was represented through tables.  Data about the truck and the car were separately calculated and represented in an individual table.  When doing the homework, tables were also used as effective means of representation.  Also, students used diagrams to represent the problems.  More than one form of representation can be legitimate.  Students could have created linear equations to represent the distances traveled by each vehicle.  Using an equation would have been equally effective compared to using a table.  Further, equations and tables can be used together to calculate data points and to display the data.  Graphs could have been used, as well.  The data points of the truck and the car could have been graphed on the same coordinate plane.  The solution would result when the two graphs intersected.

            In a math curriculum, the Catchup problem would be useful in several places.  One position for the problem is during Algebra.  Students learning linear equations would benefit from the relevant example and the strong use of the equation of D = R*T.  With this equation, students would have the rate and the time and would solve for the distance.  Algebra students are fully capable of inserting values in an equation and solving for one variable.  A second place for the problem would be during a unit on tables.  As in the simulation, the problem could be used to display the productive use of table making.  Further, the problem could be used in an integrated curriculum because it utilizes several different mathematical concepts.  The problem highlights general problem solving and communication, while using tables, equations, graphing, and verifying solutions.

            I do not think students should be allowed to use graphing calculators to make tables for this problem.  This problem was used to teach students about tables.  Students initially learning a concept should do it with paper and pencil before using a calculator.  Creating the table with paper and pencil forces the students to calculate the data, which requires students to understand the origin of the data.  Once students are comfortable with tables, then students can rely on calculators to save them time.  If calculators are used from the beginning, then students will never develop the appropriate knowledge about tables.    

            In the simulation, many questions were asked by the teacher.  When the teacher called on a student before asking a question, more points were received.  I think that calling on a student before asking the question focuses the student.  When the student knows he/she will be answering the question, the student will be sure to concentrate on the question.  When asked before the question is stated, the student has more time to prepare the answer. If the question is asked before the teacher calls on someone to answer, the student may be caught off-guard.  Often, the question needs to be restated so the student is sure of the question at hand.

 

Simulation Score: 26,600

I will attach a print-out of my score to my problem solving. 

 

Simulation: Hole Digger

            During the simulation, the group utilized several problem-solving strategies.  The group began by drawing two pictures and labeling the values with constants and variables.  Distinguishing what each variable represents is an important considerations when drawing diagrams.  After the students drew the diagrams, they created equations with the constants and variables.  Solving the equations gave the students a solution.  The students did not stop once they found one solution, they continued to create more equations and solve those.  Doing this, the students had two equations and used substitution, another problem-solving technique.  Additionally, the students graphed the two equations on a graphing calculator and found the same solution.  Altogether, the students used at least four concrete problem-solving strategies: diagrams with labels, single equations, system of equations, and graphs.  Once the students used the strategies to find a solution, the students reworked the equations with the solutions to check their work.

            This problem communicated that mathematics problems often have more than one equally valid approach to solving problems.  As described in the previous paragraph, the group successfully completed the problem with one approach.  Then, by the teacherÕs encouragement, the students completed the problem using different equations.  Further, the students discovered that graphing the two equations and finding their intersection point provided a third equal solution.  This is the reason that I enjoy mathematics so much – there is more than one way to approach a situation.  Learning how to view a problem in multiple perspectives allows someone to be a good problem solver; something that math has taught me how to do. 

            Math problems often have multiple solution methods, which assists students who think differently.  A solution method that I think is easy may be confusing for a group member, and vice versa.  In the simulation, there was an option asking, ÒWould you try that and let me know if you think itÕs easier?Ó  This option communicates that not everyone will agree that that particular method is easy.  Students will hear that question and understand that not every approach will be received in the say way by all students.  Students learn differently, hence, students will grasp different methods of solution differently. 

            During the simulation, there were choices determining the teacherÕs actions when he/she approached the group.  One choice had the teacher remaining standing, while the other option had the teacher sit or kneel.  Each time, I chose the option where the teacher sat or knelt.  This way, the teacher was among the students instead of above the students.  When the teacher sat among the students, the students continued their conversation and focused their attention on the group members.  If the teacher were to remain standing, the studentsÕ attention would shift to the teacher and communication would be teacher-centered.

            The group created several equations.  When the equation is solved, the ÒifÉthenÓ statement that is being proved is based on the given constants.  The equation is solved using the constants.  On the other hand, when the students are checking the solution, the students are placing a constant on the variable and solving to see if the equations constants hold.  So, when initially solving an equation, the students are saying, Òif the constants areÉ, then the variable equalsÉÓ  When the students are checking the solution, they are saying, Òif the variable isÉ, then the constants provide an equal solution.Ó

            At the beginning of the simulation, the students made comments about the picture at lunch time and not being able to see in the dark.  These comments reflect common remarks stated by students in everyday classrooms.  Because of these comments, I think it is very important for teachers to be specific with what they are asking for in a problem.  Being vague gives students room to consider other situations that may not have clear evidence.

            Each student in the group has a different personality.  Carols was organized.  He began by organizing the group by stating, ÒI guess we should read the problem out loud.Ó  He continued to organize and lead the group by giving input about needing more pictures and by describing the initial variable.  Jo was determined to be involved with the problem.  She read the problem voluntarily and drew the pictures.  Jo was giving input throughout the problem and asking questions.  Dula, opposite Carols and Jo, was the group member who often initiated the off-track side conversations.  She would ask a question and then immediately agree.  For example, she asked, ÒAn equation?  Oh yea.Ó  Dula wanted to complete the problem so she could be done; she was not very interested or intrigued by the multiple solution methods.  Sean was the group member who kept to himself.  Sean began the problem by working alone and throughout the problem, Sean did not give much input to the group.  Even when Sean commented, ÒI donÕt think this will work,Ó the group continued and ignored his comment.

            Leading students to productive suggestions is a good teaching strategy.  Students completed the thinking through the process and made the final conclusion, even if they were guided along the way.  Letting students state the suggestion rather than stating it as a teacher gives the students confidence.  Sometimes, a lack of confidence holds students back from giving suggestions.  If they succeed in a problem, next time they might feel comfortable doing the thinking on their own and giving suggestions.

            I think it is important for all teachers to establish an expectation at the beginning the school year that work is expected to accompany solutions.  If the teacher begins to require work mid-way through the year, the teacher will have low student responses.  Students form routines on how they answer questions, so if the expectations are given at the start, then students will learn to show their work.  Then, students will automatically show their work because they are used to the routine.

            In order to assess the problem, observations during the group work are essential.  Viewing which students understand the problem and the extent to which the students understand will allow the teacher to assess if the students met the objectives.  Also, questions work well.  The teacher could ask specific students questions about the problem and their work to give input about the studentÕs understanding.   

 

My score for the simulation: U=12, N=4, I=12, C=7, E=7, composite=19,000

I will attach a copy of my scores to my problem solving and hand it in on Monday.

 

Simulation: Hole-digger

            During the simulation, Sean said, ÒWe didnÕt need any variables.Ó  I thought his remark was invalid throughout a majority of the simulation because the solution methods used by the students all relied on variables.  However, when Sean explained his ÔweirdÕ idea, he did not need any variables.  The students at the beginning created equations with variables representing the unknown distances, and then they used substitution to solve for the distances.  Once Sean explained his idea, it was apparent that variables were unnecessary for his approach.  He used the idea that the head was one third of the distance and the feet distance was two thirds of the distance.  The hole was a total of four thirds below the ground.  Once Sean knew it was four thirds, he multiplied four by twenty three feet, which was the height of one third.  His result was equivalent to the results found by other methods, so it was equally valid.

            Finding the solution in several ways communicates that there are several possible solution methods leading to an equally valid solution.  In the past week, when we did the simulation the first time, I had not considered the teacherÕs impact on studentsÕ solution processes.  This week, I realized that the teacher must have encouraged several solution methods throughout the class or students would not have automatically attempted different methods.  The teacher must set the climate in the class that several methods can, and should, be used to find a solution.  I will remember to set the climate in my future classroom, so students routinely solve problems various ways.   

            In the simulation, I think it was appropriate to use the substitution method to solve for a variable rather than using the elimination process.  I think this because the variable, H, was already by itself so the substitution could be immediately done.  If the variable was in the form (constant)*H, then the elimination method could be used.  This would skip the division step needed to get H by itself before substituting.

            The group used many non-pictorial representations of the situation.  A majority of the non-pictorial representations were in the form of equations and a graph.  The equations translated between the words of the problem and the diagrams.  A variety of equations could be created with the given information, so the students needed to think hard to create an equation that would result with their desired value.  At first, the students created an equation where their variables cancelled out, so they needed to rethink the problem and create a new equation.  I prefer the use of equations over pictures, however; I think it is important to create both because the values for the equation are pulled from the pictures.  Using equations in conjunction with pictures will be clearer than either equations or pictures without the other.

            A question during the simulation was, ÒShould loners like Sean be forced to work with a group?Ó  When I answered the question, I automatically defended my opinion even though the question did not include instructions to do so.  I answered that, yes, Sean should be forced to work with a group.  I defended my answer by stating that communication skills are improved though group work.  However, I backed up my initial opinion answer with another opinion.  Others may have disagreed and said that Sean should not be forced to work with a group, and maybe that is why I immediately defended my answer.  I felt that reasons were needed to support my opinion.

 

Student # 8

 

Copies for Student # 8 of Combining Negatives and Coins are in the paper packet.  All other simulations are here as they were sent electronically.

 

Simulation #3 - Catchup

 

            At the end of the simulation, the teacher had the chance to ask a student what the time would be if the carÕs and truckÕs speed was increased by 5 miles per hour each.  Looking at this hypothetical situation algebraically, it can be found that it will actually take 15 more minutes in order for the car to catch the truck when speeds are at 60 and 65 mph.  This is true because of the 15 minute head start the truck has at the beginning of the problem.  The faster the truck goes, the larger the deficit the car must make up before the car catches the truck.  Therefore, the bigger the deficit, the longer the time it will take for the car and truck to be at equal mileage.

            Another choice I could have chosen was, ÒAnd now for the magic.Ó  Although a clever comment, I do not believe that mathematics is magic simply because of the firm foundation that it has been set upon since the time of Euclid.  I have had the opportunity to delve into advanced mathematics and, for the most part, have the understanding of how and why mathematics works in general.  Now, it is a slightly different phenomenon when mathematics can describe odd happenings in the environment around us.  In that sense, I do think it is ironic (and maybe magic?) that mathematics has that capability.  Now, I do believe that others with a lesser understanding of the foundation of mathematics do believe that math is magic.  When someone with little knowledge in mathematics gets an answer through a process of algebraic techniques, I do think that they take the answer for granted and donÕt analyze why that will happen every time.  Either way, mathematics is certainly an intriguing branch of life that everyone should enjoy on a daily basis!

            Undoubtedly, the Catchup problem and the other problems of this simulation are real problems.  When a teacher has the capability and time to apply mathematical knowledge to real life events, that problem is a great way to make connections and make the work meaningful to the students.  Now, problems are different from exercises.  I have been taught that problems are those questions that make use of new knowledge not yet mastered; exercises are questions that are asked as a part of rote memorization and/or reviewing previous material.  The problems presented in this class session seem as though they were solved for both teacher and student satisfaction.  The only problem I would question is when the teacher asks if the strategy held when the speeds were increased by 5 miles per hour each.  I donÕt think I saw any conclusions drawn in the few simulations I did.  In addition, I do think that the students were engaged in problem solving because the students were creating and answering engaging questions.  Even after the original problem was solved, extensions were made about other possibilities of speeds of the car and truck.

            During the simulation, a student asked if this hypothetical situation was a real story.  To his or her surprise, the teacher responded that the story was actually true.  The teacher explained that the story was a road trip from two cities in the United States last weekend.  Whether that comment was true or not is a question for debate, but, nonetheless, the comment engages the students because they realize that this problem actually happened.  Teachers make these connections in order to engage the student and to make the students understand that mathematics can actually apply to the world around them instead of on a piece of paper.  It is certainly a good strategy to hook the students and get them involved!

            Looking at the class population, it is definitely not apparent who may or may not be Ôgood at math.Õ  Rather, it is important to recognize that as long as each student is actively engaged in the question at hand, whether it be asking questions, answering problems, or simply listening to the conversation at hand, each one of those students can and will be Ôgood at math.Õ  Not getting the right answer or not asking the ÔcorrectÕ question does not conclude that a student is not Ôgood at math.Õ  It is not only the exceptional students that are Ôgood at mathÕ; instead, it is every student that is willing to make mistakes, learn, and move forward that are Ôgood at math.Õ  Simply said, every student can learn math—it is the teacherÕs responsibility to determine how each student learns in order to give each student the optimal chance to succeed.  In comparison to a similar classroom I was in at that age, this situation is fairly accurate to what I went through.  While the classroom was still teacher-centered, there was constant student dialogue and interaction between other students and the teacher to figure out the problem.  Overall, this simulation was strikingly similar to my experience in a middle school math classroom.

            Now, taking a perspective as to where this Catchup problem would fit into a math curriculum, I am certain that this problem would be placed in an algebra-based course.  I say this because of the criteria the students had to do in order to solve the problem; the students had to find linear equations for each vehicle, put them equal to each other, and finally solve for the unknown variable.  Further, within an algebra curriculum, I would place this problem at the end of the unit on solving system of equations.  Students will be efficiently prepared to do this work, while emphasizing what they just accomplished in the text.  This Catchup problem is a great extension to a unit and brings the math to life—a hook or attention grabber that students need to stay engaged in the curriculum.

            As a teacher is developing a unit, course, and year plans for upcoming academic years, it is important to recognize that a curriculum is not just based from a textbook nor is it based strictly from standards.  Obviously, teachers must incorporate the standards into their curriculum, but the curriculum would be quite dry if it were driven solely by standards.  Within the curriculum plan, there must be a combination of 3 areas:  standards, text, and teacher extensions.  Using the text as a guide, a teacher must develop a curriculum that is efficient but not too rigorous, all while incorporating the standards given and building in enough time for time to slow down if the students struggle.  The text that a teacher uses undoubtedly matters; it is the teacherÕs professional responsibility to make sure that a particular text is appropriate for the grade level, conduct readability tests for appropriateness, as well as, give the students and teachers alike the opportunity to make specified gains (standards) during that year.  Further, a teacher should build in a few extensions and engaging activities in order to keep the class light and not so dependent on the text.  Incorporating variety into a curriculum will not only benefit the studentsÕ learning habits but it will make the year run smoothly for the teacher as well.

            Returning to the simulation itself, as a teacher, I would never ask one student what another might be thinking simply because of the disadvantage I put that student at as soon as I make that comment.  First, that student may have something else to contribute, but now has to put that comment on the back burner in order to explain what another student was trying to say.  Instead, I would move on to other student contributions and tell the student that was struggling with what to say that I would come back to him or her to answer the same question.  This way, the lesson keeps moving, dialogue keeps going, and the student has the opportunity to collect his or her thoughts and contribute them to the whole class.  Now, if a student offers to explain what another student might be thinking, I would accept that.  After the explanation, I would ask the struggling student if it was an accurate explanation, and, if so, move on.  If it was not an accurate explanation, I would then tell the student that I will call again on him or her later.  Building confidence in students is a great way to gain trust along with create an environment where conversation is welcome—asking other students what another student may be thinking is a way to break down that confidence.

            Finally, making choices within the simulation, I did get fewer points for choosing ÒgoodÓ than from choosing Òthank you.Ó  My interpretation of this difference is that when a teacher responds with Ògood,Ó it tells the student that they are on the right track, but nothing innovative was really found on their part; when a teacher responds with Òthank you,Ó the student now believes that he or she has made a significant contribution to that problem and is on the right track.  Saying Òthank youÓ is a stronger message to the students implying that they are being appreciated for the work they are doing—another tremendous way to build confidence in a student.  Moreover, demonstrating positive etiquette in a classroom is a subtle way to let students how to act around peers and adults.  It is a free lesson on PÕs and QÕs!

           

 

 

Student # 8

 

Simulation #4 – Feet and Yards

            In viewing the simulation a number of times, there are many possibilities as to what the objectives are to be accomplished during this lesson.  A first objective the teacher may have set for this lesson is that the student will be able to identify the relationship that is held between a foot and a yard.  Further, a second objective that could have been is that the student would be able to define the different ways these formulas were been used (using different units).  A last objective that could have been set is that the student will be able to change feet and yards to more intriguing units in order to internalize the relationship between similar units.  Looking at these objectives, I think it is necessary for objectives to be set in order to have a goal to achieve in the lesson.  Objectives shouldnÕt be seen as disadvantages—teachers simply need to be flexible and understand that the written objectives in the lesson plan may not be what are accomplished at the end of the period.  Instead, the teacher needs to adjust to what the studentsÕ needs may be and adjust the objectives accordingly.  Objectives are not disadvantages; objectives are flexible goal setters.

            In looking at the equations given within the lesson, I am to determine which one I would say is right.  My answer is dependent upon the units being expressed.  First, the most valid equation given would be that of 3y = f, because when y=1 signifying one yard, then f=3 which holds both algebraically and understanding that there are three feet in a yard.  Now, if one were to talk about inches in a foot and in a yard, the equation of 3f = y is also valid in that, 3*(12 inches in a foot) = 36 inches = 1 yard.  But, looking more closely at this situation, the equation wouldnÕt allow for any other possibilities for f or for y simply because there are no other options for how many inches go into a foot or in a yard.  Therefore, the equation of 3y = f is the correct equation that allows for flexibility and an infinite amount of answers.  On another note, mathematical problems do always have right answers and, sometimes, mathematical problems have multiple right answers.  Even more, math problems will have multiple ways to solve a problem that contain different answers—but there will always be correct solution(s) (explainable or unexplainable) which is a great attribute to the nature of mathematics.

            Ultimately, the emphasis of the problem was in making sense of the two equations.  Making sense of the two equations is a mathematical technique that individuals develop as they learn math.  As the question states, it does makes the same sense as the graph of a linear equation is a straight line.  One must simply experience and be around math in order to create the sense of why certain mathematical concepts appear as they do.  An inexperienced learner will not understand why a linear equation maps to a line on a graph; experience with these concepts helps with studentsÕ overall learning growth.  Even though the yard is a bigger unit, numerically, the yard has a lower number in comparison to the smaller unit of feet.  Therefore, it was necessary to multiply the yard by three in order to achieve how many feet there will be.  As students become more oriented to mathematics, the more ÔsenseÕ odd concepts such as this simulation problem will make!

            When the students were on task (or when the teacher came over to an individual group and inquired as to how each group was doing), the students were involved in problem solving.  Through discussion with their group members and asking relevant questions to either further their progress or to throw out wrong answers, students were actively involved in solving the problem at hand.  A problem is a problem rather than an exercise when recall and rote memorization is not needed—students consider it a problem because the information and/or techniques are new to them.  Moreover, students were engaged in problem solving especially in group work because they were able to benefit from the contributions from their individual group rather than prompts from the teacher in a direct learning environment.  Certainly, there is a place in the classroom for exercises, but the involvement of problems is where new learning takes place.

            The students actively engaged in solving the problem found several solutions to the problem of making sense of both equations.  While most of the solutions given were certainly correct, only one allowed for an infinite domain of values with infinite outputs.  As described earlier, one student was expressing his interest in using the units of inches.  While the equation would have been balanced, the answer was the only answer that would work in that equation because only one quantity of inches goes into either yards or inches.  Having the equation of 3y=f allows for an infinite amount of inputs and outputs as is the correct solution.  Conclusively, while there will be many approaches to solve a legitimate problem, there is usually only one correct solution.  Now, a legitimate problem may require more than one solution, but all must be in the same context (i.e. solving the quadratic function) and do not have other possible solutions other than those given.  Thus, if multiple answers are given to a math problem, students must take a more critical look at their solutions and make sure that it is a legitimate answer—such as the answer to the simulation question with using the units of inches.

            At one point in the simulation, I had the choice of asking, ÒYiscah?Ó or just looking at Yiscah.  This is a great teaching technique to acquire as an experienced teacher because it allows Yiscah the chance to tell the teacher whether he actually knows the answer or not without putting him on display.  I chose to simply look at Yiscah; looking at Yiscah would result in two options of either he will look down or away from me (which indicates he does not know the answer) or look intently at me where I can confidently call him to answer the question.  Now, in just looking at Yiscah, I have found out where he sits in the understanding of the problem as well as given him confidence to answer questions in front of the classroom without a possibility of calling him and have him not know the answer.  Just looking at students, also, is a great technique to let students know what is expected of them in behavior.  If I make eye connection with a unruly student, he or she knows I am aware of that behavior and looking at them is virtually a warning without words.  The student realizes that the teacher is aware of the solution and does not want it to happen.  Communication through body language is a great way to communicate thoughts without disrupting the classroom to verbally communicate them.  For example, having the teacher walk around the room and simply stand next to a student that is not paying attention and/or is disrupting the classroom is a good way to stop the unruliness without reprimanding that student in front of the classroom.  On the other hand, a simple smile to a student lets the student know that what she or he is doing is the correct behavior or action to be taking.  Simple body language is a great way to communicate among students without disrupting the discussion or overall atmosphere of the classroom.

            This simulated lesson would fit nicely into an algebra course while learning how to solve linear equations.  The students could take advantage of graphing techniques along with algebra techniques to visually see how to solve a linear equation and, further, to make sense of that equation.  A curriculum is a thorough plan set up for a course that lays out the semester and/or year in terms of what will be accomplished.  Moreover, a curriculum includes the standards being reached and how the students will achieve those standards as they progress through the semester and/or year.  The curriculum is an important part of the classroom—it is a guide map to where the class should be going and how the students are to get to there both efficiently and thoroughly.

            Moving throughout the groups, I received the most points by encouraging some students to change the problem into something in which they were interested.  I think this is a tremendous idea.  To tap into the interests of the students is a great technique for a teacher to get a student or students genuinely interested in the topic.  By changing the topic from feet and yards to legs and jeans is an authentic problem to the student—the problem would then be meaningful to that individual student.  Now, under the circumstances that the student can correctly move from different ideas without changing the general problem is okay; when the students are changing the premise of the mathematical problem is where concern may lie.  A teacher must be aware of the situation and, while allowing students to change the problem, the teacher must be accessible to come back to the student or students who changed the problem and make sure they are still on the right track.  In all, to make a problem more meaningful to a student will only benefit the student in the long run because the students will actually be genuinely interested in the topic they chose.

            When conversing with Group 3, the group was stuck until they reread the problem.  It is always beneficial to reread the problem, and, even more, read the problem out loud to make sure students hear all of the known facts as well as know what answer needs to be found.  By reading the questions out loud, an auditory learner may be sparked to a new idea and others may hear a new part of the problem they otherwise initially read over.  Further, by rereading the question, a group will be put on the same page and starting all at the same step.  If each student is reading to themselves, students in a group may be at all different points in the steps toward the solution.  Reading the question out loud requires everyone to focus on the reader and actually listen to the problem.  This is undoubtedly a great way for stuck groups to get started!

            Finally, throughout the simulation I was given the option of waiting when talking with groups of students or individuals.  Again, I believe that this is a good technique in the learning process of the students.  As students begin to start to explain their process or their technique to get the solution, it is imperative to let the students either verify that they are correct or to let them find their mistakes.  If a teacher jumps in too early, the thinking process for the student is over—the student now has to listen to you explain the problem over again.  Remaining quiet also encourages students to ask questions about their technique or their process—silence creates critical thinkers without the involvement of the teacher.  Now, if it is obvious that the student is simply confused and no thinking process is happening during the silence, then it is okay to give small hints or reminders to the students to give them a spark.  Regardless, the less words spoken during the learning process is inevitably the better process because it gives students the ability to be critical thinkers and not depend on the teacher to either give them the right answer or always be there to give them hints.

 

 

Student # 8

 

Simulation #5 – The Monk

            In viewing the simulation a number of times, it is apparent that the students had a variety of ways of solving the Monk problem.  First, some students drew a mock hill that recorded the times of the monk going up and down the hill and found that at some point between 12:00 and 12:30 the monk must hit a similar point in both paths.  A second solution was to physically see how two people must cross each other when they walk similar paths.  Even more, a graph was drawn that labeled the increasing and decreasing elevation of the hill and visually showed that both lines must cross at some point.  All of these ways were valid; I think that all of the ways listed were right or best ways simply because each appealed to a different learner.  The more processes presented for a solution will only enhance and solidify the concepts of that particular problem within the classroom.

            The teacher presented the problem in such a choreographed way to get the students engaged in the problem.  Further, the teacher may have presented the problem in this way to get students excited about the problem and making the problem a more realistic problem which is something that most students will want to solve.  In general, a teacher should plan teacher movements into the lesson to a small extent in that it undoubtedly will get the attention of all the students, if only for a few minutes.  Instead of staying at the white board the entire hour, a teacher must engage him or herself, too, to the given situation and portray the feeling that the teacher is genuinely trying to solve the problem as well.  Getting involved in the problem creates excitement in the classroom and will encourage the participation and excitement within the students.

            Before any explanations or proving of the problem was asked, the teacher simply asked for studentsÕ intuition of the problem at hand.  This may be a helpful strategy to start the problem since it requires the students to think about the problem before necessarily handling any mathematical tools that will be used later.  Intuition will allow students to first think if this problem makes sense and, most likely, visualize the monk going up and down the hill and determine whether there may be a point of intersection of the paths.  Unquestionably, intuition plays a great role in mathematics as a learner becomes more experienced in the mathematics arena.  For example, when the base knowledge of a parabola has been formed, intuition tells students the behavior of a parabola while only the details (i.e. negative, shift, transformation) may be a factor that may be needed to look at.  Moreover, intuition tells mathematical students what may happen to a function, graph, etc., as it approaches infinity—there is no way to draw that graph or write that series, but intuition along with previous knowledge allows us to be confident in drawing valid mathematical conclusions.        

            The teacher may have put the students into pairs after shortly discussing the problem for a couple of reasons.  First, the teacher probably wanted the students to discuss their own ideas to their peers and either get peer acceptance or to receive help from their peers on how to figure out the problem.  A second reason for pair groupings is to give confidence to students to share their work out loud.  Most students may have the correct solution and a creative way to solve the problem; the circumstance is that most students will not share their process and solution without knowing it is correct.  Putting students into pairs will give another set of eyes to the processes and, consequently, stimulate conversations with more willingness on how to solve the problem.  The teacher, during the working time, must facilitate throughout the room asking stimulating questions to check for comprehension, make sure students stay on task, and assess at how well the students may be comprehending the problem.  Finally, the teacher can walk around the room and gauge how many students are done in order to move the lesson along.

            As the simulation and lesson progressed, the teacher thought that all students were engaged in the problem.  I find that hard to believe.  While the teacher may have set up a very stimulating and engaging exercise for the students, there will always be some student that is thinking about the other endless factors of teen life.  Knowing this, as a teacher, the lesson must be broken up into chunks (with smooth transitions) so those ÔdaydreamsÕ are kept to a minimum.  Having one hook or catch within a lesson may not be sufficient—a veteran teacher should know how to draw students in throughout the lesson in order to get the most out of the student for that period of time.  In all, having students engaged is very important for learning.  If students are engaged, the students are genuinely attempting at using mathematical knowledge to solve the problem.  If students are not engaged, the process or math work may be present, but the learning opportunity has been lost.  Engaging the students may be the hardest task for the teacher to accomplish during oneÕs time in the classroom; it is the teacherÕs responsibility to encourage every student to maximize the actual engaged learning time within the classroom each and every day.

            The purpose of the teacher saying, ÒJon?  Maybe I was looking elsewhere. Are you OK?Ó was intentional action by the teacher to get the attention of Jon.  Now, the teacher stated Ômaybe I was looking elsewhere,Õ to not bring attention to Jon inattentiveness—the teacher was putting blame on him or herself by saying that he or she wasnÕt looking at all the nods for understanding around the classroom, and, consequently, not putting Jon at fault.  This is a great technique since it results in having the attention of Jon and having the class move forward without embarrassing Jon in front of his classmates.

            When the teacher responded to LoriÕs comments using proper mathematical language, I donÕt think it was necessarily appropriate for the teacher to respond with words like Ôflattered.Õ  Not only does the comment tend to favoritism, but the comment lacks professionalism in the sense of keeping a proper image of teaching.  Some texts even suggest that praise of good work or comments should be direct and short and not overly ÔgushyÕ praise.  With that in mind, it would have been a better choice for the teacher to respond to LoriÕs comments with, ÒItÕs good practice to use the words of the problem.Ó

            During classroom interaction such as one in this simulation, teachers often fear that a small group of students will dominate the conversation.  At first, I would say that a small group of students had opinions and ideas that did dominate the conversation.  But, after the teacher had given them more detailed instruction and pair work, more students had opinions and suggestions on how to prove why or why not the monk crossed paths.  Even better, the teacher was a great facilitator in that he or she was aware of her surroundings and included students who may or may not have had their hand up.  Involving all students in classroom discussion is crucial; a teacher will lose many students to daydreaming if a small group takes over and solves the problem in the minimal amount of possible ways.  This simulation had great classroom interaction.

            As students Kerry and Toni drew there time going backward in their picture on the board, the teacher thought, ÒItÕs better if one of the students catches it.Ó  As I see it, the teacher is mostly correct.  Having fellow peers catch mistakes is a development in learning critical thinking skills.  Letting the students know that it is the teacherÕs expectation that students help one another puts valuable responsibility of classmates in the room.  Now, the teacher must intervene if the mistake goes unnoticed and will affect later calculations and/or conclusions; the art of intervening is asking leading questions that still leads to students correcting the mistake.  If a classroom is comfortable enough for fellow peers to respectively correct peersÕ mistakes as well as have valuable discussion, that teacher has certainly has a gift and has set a solid foundation for the rest of the school year.

            Finally, one of the teacherÕs thoughts was, ÒWhen people hear an answer, they stop thinking about the problem.Ó  As a student, I would have to agree with this thought.  Especially in a math classroom, once a conclusion has been reached on a problem, students usually shutdown and wait for the next instruction.  The difference between what typically might happen and what happened in this simulation is simple; the teacher continued to pursue an answer despite SaraÕs quick answer to the question.  Although Sara had a definitive answer to the problem, the teacher was quick to accept the response, but pursued different suggestions from other students.  The teacherÕs pursuing didnÕt allow for students to shutdown since they knew that the teacher was still looking for alternative answers.  Another reason for the difference in this simulation (that may happen in typical classrooms) is that students are generally oblivious to the student who answers everything.  All of the students expected Sara to answer the question right away; the difference was that they knew the teacher was looking for more depth than just SaraÕs answer.  In all, the teacherÕs quick acceptance and continued questioning was the factor in keeping the students engaged in the problem despite SaraÕs quick conclusion.

 

Student # 8

 

Simulation #6:  Bouncing

            After viewing the simulation a number of times, I had the opportunity to choose the response of, ÒDoes this make sense to you?Ó numerous times.  Each and every time, I chose this response as a way to get the student thinking about the choices he or she was making.  This choice communicates the idea that the nature of mathematics is not simply getting the answer; instead, it is the ability to clearly communicate how one got to that right answer.  A student might be able to solve one type of problem, but if he or she does not understand the underlying mathematical concept that supports problems, that student will have difficulty moving from problem to problem and advancing forward at a consistent pace.  I, as the teacher in this simulation, am communicating to the students that is unquestionably important for them to not only get the correct answer, but to demonstrate the ability to visually show and communicate the process that allowed them to receive the correct answer.

            Under some conditions, Pat asked where he should aim if the ball were to go into the pocket.  Initially, I would ask Pat where he intended on starting the ball since the starting point will affect what the angles may be.  After the lesson was finished and the class was satisfied on concluding that angle of incidence is congruent to the angle of reflection, I would ask Pat what we should do to figure it out.  Obviously, we would get out protractors and measure angles until we found a point on the opposite cushion that ensured that the incoming angle is congruent to outgoing angle.  Working out this solution will show Pat where he should aim in order to hit the ball into the pocket.

            Again, at some point during the simulation, I ask Terry if there may be more than one way to look at this situation.  Unquestionably, this comment is letting the students know that math is not always clear cut with one solution or process.  While there may only be one correct solution, there may be multiple ways to get to that solution—with all of those processes being completely valid.  This message about the nature of mathematics is simple:  mathematics has many different avenues that can be used to achieve correct answers and it is important to look at and verify as many avenues as possible.  Exploring alternative methods only will benefit the students/teacher involved while showing how different thought processes work as opposed to a single idea.  Hopefully, students begin to understand the beauty of mathematics when looking at all of these alternative methods!

            Further into the situation, Dula gets her Big Idea as the teacher begins to come around to see how the students are doing.  One of the responses I could have chosen was, ÒWhat the heck is that, Dula?Ó  Now, because I was not aware of the classroom environment or how Dula typically reacts, I chose not to choose this response.  However, I may choose to pick this response if I was comfortable knowing who Dula was, and that the class understood that I typically can be a clown.  If I knew that Dula would laugh, full well knowing that I am joking, the classroom environment remains light and everyone may get a hearty laugh from the comment.  On the other hand, I would refrain from making that comment if I knew Dula was not into sarcasm or humor.  Doing so would only create tension and discourage Dula from attempting to do the work she was doing.  In all, making this comment is completely dependent upon the personality of the individual and classroom atmosphere.

            Looking at the problem itself, this problem could belong to both the mathematics and physics curriculum.  Ideally, a math and physics classroom could teach lessons in relation to each other so that students could begin to make connections within different content areas.  Since physics includes a lot of mathematical concepts, it would be great if students could learn the mathematical concepts like congruent, obtuse, and acute angles within a math classroom and further the investigation with the game of pool in a physics classroom.  Certainly, this problem does not need to be taught solely in either classroom—exploring problems that overlap disciplines will only benefit the studentsÕ learning.

            In handling JustinÕs eagerness in the classroom, I would suggest that while the teacher cannot ignore him, it is also important to recognize that he may have something useful to say as well.  As a teacher, I would prefer not to call on Justin immediately after posing the question simply because it would reduce the critical thinking done by other students in the classroom after he suggests his solution.  Instead, I would let the problem sink in, note and let Justin know that I see his hand up, and begin to ask prompts to the entire class.  Once a few suggestions have been offered, I would then call on Justin to give his input.  Recognizing his solution as a valid option just like the others allows fellow students to keep critically thinking and not conclude that JustinÕs answer is always right.  Now, if JustinÕs eagerness continues (which usually does) and I know he understands the problem, I will let Justin know publicly that I know he has a correct answer, but I want other participants to suggest their solutions.  Letting Justin know this publicly is important; it lets Justin to know that the teacher understands he knows the problem, as well as, allows Justin to have a little bit of the satisfaction of knowing that his fellow classmates know that he understands the problem.  However, letting other students answer the question gives the opportunity to others that would otherwise be lost if I, as the teacher, would call on Justin every time.  In all, recognizing and controlling JustinÕs eagerness is the key to using his involvement effectively without hindering the whole class.

            During some segments of the simulation, there was a long period of time before I was able to have an option of saying something in the discussion.  I think that it is good for the teacher not to be the center of attention, especially in a problem-based environment that this problem has placed the class.  To have the students discuss ideas and possible solutions with prompts from a teacher is undoubtedly a goal for this type of classroom.  Now, the teacher would have to facilitate in order to keep the class on task, but to have minimal involved in a student-centered discussion is a great learning tool for the students in the classroom.  Even further, when an idea, possible solution, or demonstration was mentioned within the discussion, there was a momentary ÔhubbubÕ within the classroom.  As a teacher, I would consider the ÔhubbubÕ a great reaction to a possible solution.  Students are genuinely involved in the discussion and are determining whether that solution is valid.  Unquestionably, the classroom is under control, legitimately drawn into the discussion, and determined to find the answer to whether the ball will go into the pocket.  To have rich discussion about a problem like this one is a very rich learning environment for the students within the classroom.

            At the end of the whole class discussion on whether the pool ball would go into the pocket, the teacher asked the students to generate some What if? questions in order to enhance the question.  Personally, I think this was a great move for the teacher.  First, asking the students to generate their own higher level thinking questions is one way to internalize the big ideas of the lesson.  Second, the teacher suggests that the students attempt to answer those questions.  So, by having the students make the question and then answer the higher level questions, the teacher is providing an opportunity for the students to genuinely come away with a deeper understanding of how angles, speed, and pool works in general.  This process was a great use of the time it took to produce these answers and I do believe that the students will benefit from those deeper questions.

            Finally, in the description of the different kind of angles, I think it is still a better idea to use effective prompts to explain the meaning of an obtuse angle.  Since the teacher had already gone through acute and right angles, the teacher could have used some rhetorical questioning to speed up the explaining while still having the students involved.  Overall, it is sometimes necessary for teachers to simply help students down the path of defining certain concepts instead of asking the class about the meaning of obtuse.  Asking students to explore the situation can be drawn out into too much time when that time is needed to achieve other objectives.

 

Student # 8

 

Simulation #6:  Bouncing – Week 8 Extension

 

1.  In the simulation Bouncing, identify a question, if possible, at each of the various levels of BloomÕs Taxonomy.  Check the BloomÕs Taxonomy sheet for further examples of terms used at the various levels.

                        The following list of questions addresses each level of BloomÕs taxonomy within the simulation, ÒBouncing.Ó  I found examples of all 6 levels:

Knowledge – Can you state the sides and vertex of each angle, Carlos?

*This is a very basic question that asks for recall of rote memorization and fits nicely at the knowledge level.

 

Comprehension – Would you please elaborate, Dula?

*This question is digging a little deeper than asking for a definition.  Instead, this question is asking Dula to explain her possible solution in order to express her answer thoughtfully.

 

Application – Would you demonstrate, Lindsay?

*Referring to her idea of tracing the angle and flipping over the paper to get reflection, this question is requiring Lindsay to apply the information that she is saying and physically demonstrate it to the class.  This level is unique in that it helps students learn within different learning styles (kinesthetic, oral, listening, etc.).

 

Analysis – Do you think there might be more than one way to look at this situation?

-- What is the difference [between congruent and equal angles], Lee?

*Both of these questions are intended to make the students analyze the information presented to them and apply it to multiple situations.  The first question allows student to analyze if there is only one way to accomplish the task.  Further, the second question requires Lee to use the information about congruent and equal angles and analyze where the difference lies.

 

Synthesis – What would happen if you hit a ball perpendicular to a cushion?

*This is a great question at the Synthesis level simply because it allows students to take old information and create new information of what would happen if the ball it the cushion at a perpendicular angle.  The students must synthesize the material, predict, and draw accurate conclusions.

 

Evaluation – Can anyone defend an opinion about whether or not it goes through the pocket?

*Finally, this question, after significant discussion, has the students make choices based on reasoned argument that was presented during the lesson.  The students must come up with an opinion of what the resulting path of the ball will be and be able to defend that opinion. 

 

2.  Write two objectives for the Bouncing lesson.

At the completion of this lesson, students will be able to:

Utilize prior knowledge and intuition to critically discuss possible outcomes of a real-life problem and make accurate predictions based on the discussion.

Identify, and make use of, congruent angles in order to predict the path of a ball after the ball has been reflected off a hard surface.

Define and distinguish between congruent, perpendicular, acute, and obtuse angles.

 

3.  Identify the standards from both NCTM and MN Standards that this lesson would address.

The following National Council of Teaching Mathematics (NCTM) standards are addressed in this lesson:

Communication Standard:

Organize and consolidate their mathematical thinking through communication.

Communicate their mathematical thinking coherently and clearly to peers, teachers, and others.

Analyze and evaluate the mathematical thinking and strategies of others.

Use the language of mathematics to express mathematical ideas precisely.

Geometry Standard:

Analyze characteristics and properties of two- and three-dimensional geometric shapes and develop mathematical arguments about geometric relationships:

Create and critique inductive and deductive arguments concerning geometric ideas and relationships, such as congruence, similarity, and the Pythagorean relationship.

Use visualization, spatial reasoning, and geometric modeling to solve problems:

Draw geometric objects with specified properties, such as side lengths or angle measures;

Use visual tools such as networks to represent and solve problems; and,

Recognize and apply geometric ideas and relationships in areas outside the mathematics classroom, such as art, science, and everyday life.

 

The following Minnesota Standards are addressed in this lesson:

Spatial Sense, Geometry, and Measurement:

Geometry Standard:  Identify a variety of simple geometric figures by name, calculate various quantities associated with them and use appropriate tools to draw them.

Classify triangles as equilateral, isosceles or scalene, and right, acute or obtuse.

Measure, identify, and draw perpendicular and parallel lines, angles and rectangles by using appropriate tools such as straightedge, ruler, compass, protractor or software.

Mathematical Reasoning Standard

Assess the reasonableness of a solution by comparing the solution to appropriate graphical or numerical estimates or by recognizing the feasibility of a solution in a given context.

Translate a problem described verbally or by tables, diagrams or graphs, into suitable mathematical language, solve the problem mathematically and interpret the result in the original context.

 

4.  What were the choices that led to DulaÕs having a Big Idea?  In general, what kinds of choices lead to creative problem solving?

            After viewing the simulation a number of times, it seems as though choices that encouraged a form of discovery learning led to DulaÕs having a Big Idea.  Throughout the simulation, I chose questions that facilitated discussion along with encouraging everyone to participate in the discussion.  Also, all possible solutions were accepted and students were even encouraged to get out of their seats to demonstrate the path of the ball.  It is also important to note that whenever I got DulaÕs Big Idea, JustinÕs comment about the angle of incidence and angle of reflection did not come up.  It wasnÕt that I ignored Justin, but I gave other students the chance to explore the ideas before Justin.  Also, after some initial discussion about some student solutions, Pat creates an intriguing question that relates to another studentÕs response and sparks more discussion within the classroom.  These are the choices I recognized as important in the leading to DulaÕs Big Idea.

            Having this open dialogue where students are centered (as opposed to teacher-centered) are the kinds of choices that lead to creative problem solving.  If I allow one or two comments, but stay on the same track in order to get through the lesson, each student will become a Ôcookie cutterÕ and learn exactly the same way I proposed it.  Instead, having a classroom atmosphere that encourages participation and discovery learning will undoubtedly lead to creative problem solving.  With the many ideas and possible solutions floating around the room, Dula was obviously encouraged to try it out as well.  In all, to have a classroom where students are excited to participate with little fear of rejection or un-approval is a classroom environment where students will inevitably excel.

 

5.  Was the amount of closure for this lesson appropriate?  Why or why not?  If not, what would you include?

            As I look back at the lesson, I feel that the amount of closure to the lesson was not quite sufficient.  While the teacher made some significant extension questions for students to answer and extend their learning, I did not see any wrap-up on the problem itself.  There were some students that concluded their conjectures, but, even immediately before introducing the ÒWhat if?Ó questions, a student offered another question.  The teacher never really answered whether the ball would go into the pocket or not.  Doing this could potentially confuse students and not necessarily know how to proceed when working individually if they do not know what the end result will be.  In order to extend their learning and explore extension questions, it would be appropriate for that teacher to have wrapped up the initial question for the students.  I may additionally include how I would have solved the problem, just to show some direct instruction on a process that the students could use.  Illustrating the congruent angles and, as a result, determining whether the ball goes into the pocket would be a simple and effective way to wrap-up the general discussion before students continued to work as individuals or small groups.

 

 

Additional Responses:

            With the continued critical look at the simulation, ÒBouncing,Ó I would agree with the idea that the beginning of the problem would be considered inquiry-based teaching.  The teacher introduces the problem and allows the students to construct their own meaning by exploring different possible solutions en route to a valid conclusion.  Ideally, the teacher would only facilitate and the student-centered environment would be self-sufficient in order to solve this authentic problem.  Although the problem is somewhat weak in the sense of authenticity, the problem is a real-life problem, nonetheless, that will engage the interest of the students within the classroom.  With all of these factors put into consideration, I agree that the beginning of this problem could be considered the start of an inquiry-based teaching lesson.

            At one point in the simulation, I make the comment to Jose that, ÒI like controversy.Ó  While there are positives to making this comment, the teacher has to make sure that this controversy is only an academic controversy not a controversy among people.  Strengths of making this comment is that it lets students understand that it is okay to have conflicting ideas—it encourages students to make as many conjectures as possible so that the class can explore many different possibilities.  It also gives students the ability to not fear being wrong; it is okay, and actually encouraged, to make competing claims so the class has the opportunity to analyze both sides of the argument.  Students naturally want to solve a problem that has differing sides of opinion to simply rid the tension that it creates in their minds.  On the other hand, the major weakness of this comment is that students may interpret it the wrong way.  It is not a controversy among students who make competing claims—it is a controversy among the claims among themselves.  Controversy is a strong word and it is important for a teacher to preface that comment with appropriate measures so students are not put down in the problem solving process.

            Finally, at a different point in the simulation, I had the choices of deciding that the class should experiment and asking the class how to proceed.  In trying both directions, I found that while the nature of mathematics, comfort in participating, and engagement were increased in either case, while the level of independent thinking was also increased when I asked the class how we should proceed as opposed to when it was not increased when I chose to say that we should go ahead and experiment.  Taking a step back to look at the whole situation, it makes sense that the level of independent thinking increases when I give the students the choice of how to proceed.  Giving students control of how they will proceed to solve this problem is a critical thinking process that increases independence.  It also makes sense that their level of independent thinking would not increase if I chose how the class will proceed.  Obviously, independent thinking will not be furthered if a teacher automatically tells the students what the next step will be in the problem solving process.  Instead, allowing students to hold responsibility to solve this authentic problem will give students the optimal opportunity to increase independent thinking and, ultimately, to succeed in this discovery

learning environment.

 

Student # 8

 

Simulation #3:  Catchup Revisited

 

1.  For the simulation Catchup, identify the mathematical objectives of the lesson.

As a result of this lesson, students will be able to:

Set up and use tables that relate to a given problem.

Draw information from multiple tables in order to identify equality.

Determine periods of time a vehicle is moving if only given the speed of a car.

Make use of various problem solving techniques in order to find when objects going different speeds will meet.

Discuss the reasonableness of a solution by justifying solutions mathematically.

 

 

2.  Identify the standards from both NCTM and MN Standards that this lesson would address.

The following Minnesota Math Standards are addressed in this lesson:

Mathematical Reasoning Standard:  Apply skills of mathematical representation, communication and reasoning throughout the remaining three content strands.

Translate a problem described verbally or by tables, diagrams or graphs, into suitable mathematical language, solve the problem mathematically and interpret the result in the original context.

Support mathematical results by explaining why the steps in a solution are valid and why a particular solution method is appropriate.

Algebra (Algebraic Thinking) Standard:  Solve simple equations and inequalities numerically, graphically, and symbolically. Use recursion to model and solve real-world and mathematical problems.

Use a variety of models such as equations, inequalities, algebraic formulas, written statements, tables and graphs or spreadsheets to represent functions and patterns in real-world and mathematical problems.

Measurement Standard:  Use the interconnectedness of geometry, algebra and measurement to explore real world and mathematical problems.

 

The following NCTM Standards are also addressed in this lesson:

Number and Operations Standard:  Compute fluently and make reasonable estimates

Judge the reasonableness of numerical computations and their results.

Algebra Standard:  Represent and analyze mathematical situations and structures using algebraic symbols.

Use symbolic algebra to represent and explain mathematical relationships.

Algebra Standard:  Use mathematical models to represent and understand quantitative relationships.

Draw reasonable conclusions about a situation being modeled.

Geometry Standard:  Use visualization, spatial reasoning, and geometric modeling to solve problems.

Use geometric models to gain insights into, and answer questions in, other areas of mathematics.

Problem Solving Standard:

Build new mathematical knowledge through problem solving.

Solve problems that arise in mathematics and in other contexts.

Apply and adapt a variety of appropriate strategies to solve problems.

 

Additional Simulation Responses:

            In doing several of these simulations throughout the duration of this class, it is unquestionable that these problems are real math problems that would be used in a classroom.  Also, these problems are realistic in the sense that students can relate to the problems that are presented.  Making the problems realistic and genuine will only benefit the learning of the students.  Moreover, when the students were on task (or when the teacher came over to an individual and inquired as to how each group was doing), the students were involved in problem solving.  Through discussion with their group members and asking relevant questions to either further their progress or to throw out wrong answers, students were actively involved in solving the problem at hand.  A problem is a problem rather than an exercise when recall and rote memorization is not needed—students consider it a problem because the information and/or techniques are new to them.  Furthermore, students were engaged in problem solving especially in the whole class discussion because they were able to benefit from the contributions from their fellow classmates.  Certainly, there is a place in the classroom for exercises, but the involvement of problems is where new learning takes place.  Particular to this simulation, I believe that all the problems presented were solved both in the whole class discussion as well as when the students had individual time to work on other problems.

            Looking at the class population, it is not apparent who may or may not be Ôgood at math.Õ  Instead, it is important to recognize that as long as each student is actively engaged in the question at hand, whether it be asking questions, answering problems, or simply listening to the conversation at hand, each one of those students can and will be Ôgood at math.Õ  Not communicating the right answer or not asking the ÔcorrectÕ question does not conclude that a student is not Ôgood at math.Õ  It is not only the exceptional students that are Ôgood at mathÕ; instead, it is every student that is willing to make mistakes, learn, and move forward that are Ôgood at math.Õ  It is the teacherÕs responsibility to determine how each student learns in order to give each student the optimal chance to succeed.  In comparison to a similar classroom I was in at that age, this situation is fairly accurate to what I went through.  While the classroom was still teacher-centered, there was constant student dialogue and interaction between other students and the teacher to figure out the problem.  Following, there would be some sort of individual work to ensure that students understood the material before attempting it as homework.  In all, this simulation was strikingly similar to my experience in a middle school math classroom.

            Looking at the problem that the simulation addresses, I would place this problem into an algebra course early in the year.  Because this simulation did little with setting up equations and solving for unknown variables (unlike the prior Catchup simulation), I would place this version of Catchup towards the start of an algebra course as an exercise on how to develop tables simply by using base knowledge of speed and time.  Placing this problem early in the year also gives the class an opportunity to come back to the problem after developing alternative techniques for solving similar problems and see how their math connections have grown and how they could go about solving the same problem again but, for example, using linear equations, setting them equal to each other, and finding the quantity of the unknown variable.  I would consider this problem an algebra problem simply because of the numerical analysis being done in conjunction with drawing information from a table to solve for unknown quantities.

            Throughout the simulation, I had many opportunities to choose either the response ÒgoodÓ or Òthank youÓ when students had made a comment in the class.  In making these choices, I did get fewer points for choosing ÒgoodÓ than from choosing Òthank you.Ó  Where I find the difference is when a teacher responds with Ògood,Ó it tells the student that they are on the right track, but nothing innovative was really found on their part; when a teacher responds with Òthank you,Ó the student now believes that he or she has made a significant contribution to that problem and is on the right track.  Saying Òthank youÓ is a stronger message to the students implying that they are being appreciated for the work they are doing—another tremendous way to build confidence in a student.  Moreover, demonstrating positive etiquette in a classroom is a subtle way to let students how to act around peers and adults.

            Finally, in concern to whether the students should have been allowed to use graphing calculator to make the tables, I think that it wasnÕt necessary in this simulation.  Since the tables were made by full hours (as opposed to fractions of hours), it should be expected that students can find how many miles the car or truck travels when given the speed of the vehicle.  Ultimately, the table is being made by addition and little multiplication; these operations can and should be done relatively quickly, especially with whole numbers.  Now, if the table included fractions of hours or students had to compute mileage that wasnÕt integers, I would have no problem using calculators simply because, if not, students would be using valuable time that can be used otherwise more efficiently on higher level concepts being taught in this lesson.

 

Student # 8

 

Simulation #7:  Holedigger

            During the simulation, I had the opportunity to choose the option, ÒWould you try that and let me know if you think itÕs easier?Ó  In communication about the nature of mathematics, I think that this message is telling the students that there are multiple ways at getting the same answer.  This particular group even found a third way through the graphing calculator simply because the teacher gave them the opportunity to explore deeper into the problem and see if there are alternative routes to get to the same solution.  Furthermore, this comment communicates the complexity of the nature of teaching.  As a teacher, one needs to see that while this particular group may have gotten to the solution more quickly than others, the teacher needs to have ÔwithitnessÕ and keep students challenged.  Giving students another idea to work with on the same problem is a great example of showing how the nature of teaching is keeping students on task and involved in the process.  Finally, this comment simply shows how the nature of learning differs so widely.  While one of the students understood how to set up the single variable solution, it was two other students that led the group throw the two-variable equations.  Both processes were valid and got to the same solution, but both ways communicated the process to different students.

            In relation to what has been already mentioned, finding solution in several ways communicates that the nature of mathematics is not always one direct route.  Instead, multiple strategies to attack a similar problem may be completely valid and will result in the same solution.  That is what is so tremendous about mathematics—different students learn differently and will inevitably use different strategies to solve a problem, and mathematics welcomes diversity in every way.  This is precisely the reason why teachers need to be prepared and flexible when giving students this diverse opportunity; to restrict students to only one possible way (given that there are other ways to solve a problem) would undoubtedly hinder some students in the classroom.  In all, finding solutions in several ways is a great opportunity for students to learn in their own unique way.

            As the group progressed through the story problem, several problem-solving strategies were utilized along the way.  Initially, the students began to naturally draw figures that represented the depth of the hole during both morning and night.  With these pictures, the students were able to label known variables and unknown variables.  With these pictures, a second problem-solving strategy used was the use of single variable equations.  The students set up appropriate equations in order to solve for the unknowns on the picture, and, consequently, solved the problem.  After some of guidance from the teacher, the students also made use of two-variable equations, setting them equal to each other in order to solve for one unknown at a time.  Finally, one of the students used the problem-solving strategy of using a graphing calculator and finding where the two equations intersect each other in the Cartesian plane in order to check their work.  Utilizing all of these problem-solving strategies inevitably helped the students visualize the problem, solve the problem, and check their solutions in multiple ways. 

            When an equation is solved, the Ôif-thenÕ statement that is being proved is if the equation is defined as the given expression, then the unknown variable can be solved to equal some constant.  In the same sense, when the solution to an equation is being checked, the Ôif-thenÕ statement being proved is if the solution equals some constant, then, when that solution is put back into that expression, the expression should hold and either side of the equation should be equal.  Looking at both of these Ôif-thenÕ statements, it is clear that both are opposites of each other.  In this case, putting both Ôif-thenÕ statements together, one could produce an Ôif and only ifÕ which allows the statement to be valid in either direction.

            During the simulation I had multiple opportunities, when approaching the group of students to either remain standing or kneel/sit within the group.  Each time, I chose to kneel/sit so that the teacher is at the same level of the students.  Choosing to remain to stand communicates to the students that the teacher is still above the students and, ultimately, not part of the group and its processes.  In complete contrast, when a teacher sits among the students, the teacher is at the same level and doesnÕt communicate that same kind of dominance or control over the students.  Instead, it communicates that the teacher genuinely wants to help the students and is willing to sit down and work through the roadblocks that is holding back the group.  Even more, sitting/kneeling portrays that the teacher is also trying to figure out the problem at the same point of the students.  This is completely opposite of when the teacher stands since standing communicates that the teacher already knows the answer and is simply overlooking to make sure the students are going in the right direction.  Sitting down and getting oneÕs hands dirty within the problem is a great relationship builder and gives respect to both teacher and student.

            In developing their first picture, the group mentioned a picture at lunch time and not being able to see in the dark.  I think it is very likely that students will make a comment such as this one.  Not only would it come from a comedian in the classroom, but students that are really involved in the process would make a comment like this while trying to talk through possible difficulties.  Being deeply involved in problem solving often makes people neglect other actions (such as a filter on the logic of a comment), which just means that students are putting their concentration in other areas.  In addition, the comment is actually reasonable—the student just may not realize right away that darkness will not serve as a hindrance in order to figure out the problem on a piece of paper.  It is a funny remark that will lighten the mood and will undoubtedly happen on a regular basis.

            One of the questions provided was, ÒShould loners like Sean be forced to work with a group?Ó  When I answered this question, I naturally defended my opinion simply to justify why I made that decision.  Generally, I think it is necessary to defend an opinion so that any person that wishes to read my opinion will have the opportunity to judge the credibility of my opinion based on my argument.  In regards to the question, I still think that Sean should be forced to work in a group.  To not include him is only reinforcing his loner demeanor which is not something that needs to be supported in the classroom.  While I respect those who want to work on their own, there is a difference between those who are loners and those who work individually.  Students like Sean need to be put into groups in order to help develop social skills as well as develop relationships with peers.  Group work is a great way to not only learn mathematical material with the guidance of others, but it also gives a great opportunity to students to appropriately mature.

            To create a classroom climate in which students automatically Ôshow their workÕ without being instructed to is to demonstrate the capability as often as possible and put emphasis on that every time available.  In other words, as a teacher, I must show work when demonstrating a problem along with putting emphasis on showing work on homework, tests, group work, etc.  Putting emphasis on work portrays to the students that the teacher believes it is important, and, ideally, the students will pick up on that habit.  Another technique is to create an expectation at the beginning of the semester/year that showing oneÕs work is necessary in every situation and spend the first few days explaining and demonstrating what exactly is expected on homework/test/etc.  Implementing these strategies should create a classroom climate where students automatically show their work without being instructed to.

            In making various choices throughout the simulation, the students begin to make productive suggestions, though the teacher may have asked the question that planted the seed.  Unquestionably, I would consider it a great success to have students offering productive suggestions rather than me, as teacher, giving away every detail every step of the way.  It is a great atmosphere when the students are fully engaged and have the capability and understanding to make suggestions that may trigger an idea with another student in the group.  Instead of having the teacher orchestrate every movement of the class, the class will be much more successful and productive when student are able to move forward without the guidance or approval of a teacher.

            In respect to authentically asses what I hope the students learned from working on this problem, I would assess the students in a few ways.  First, as the simulation was leading towards, I would assess the groups of students on how well described and correct the presentation of their solution was to the class.  If it was satisfactory, I can assume that the group, as a whole, understands the problem.  If the presentation was not satisfactory, I can use effective questioning techniques in order to pinpoint where the difficulty lies.  Second, as an individual assessment, I would give my students some sort of homework assignment that addresses the same concept of using multiple variables in an authentic story problem.  By doing this, I am able to assess whether each student understands the concepts addressed both in class and in the homework.  I would obviously need to correct that homework as part of my formative assessment and adjust my teaching strategy accordingly.  In all, assessing both the group and the individual should be sufficient in assessing what I hope students learned from working on the problem offered in class.

 

Student # 8

 

Simulation #7:  Holedigger Revisited

            Looking again at the simulation Holedigger, I came across a comment made by Sean saying, ÒWe didnÕt need any variables,Ó after he explained his way of solving the problem.  While his explanation was valid and unique, it certainly was confusing because of all the different heights he referenced.  It is true that he didnÕt use variables in his explanation, but it is also true that his explanation was very unclear.  Even as the teacher, it was difficult to understand what height Sean was talking about at any point, and, if this simulation were regular conversation within a classroom, I donÕt know if I would have picked up his explanation on the first try.  Therefore, while I agree that one doesnÕt need to use variables, I feel that using variables not only makes the work more clear, but the explanation of that work will be more precise and easier to follow.  Looking at SeanÕs method, I still do think that his method was algebraic (mostly used mentally) since he used thirds to explain the height of the worker which then related to the depth of the hole at night.

            Now, although Sean did not use variables to solve this problem, his ÒweirdÓ idea was valid nonetheless.  In my own words (rather then trying to explain his reasoning with no variables), he split up the height of the worker (69 inches) into thirds or 23 inches.  He called that the head distance and concluded that the depth of the hole at night is a head distance plus the height of the worker.  Simply put, 23 inches + 69 inches (height of worker) = 92 inches.  Honestly, looking critically at his reasoning within the simulation is not very clear to me, but, basically, he is saying that the head distance is only one-third of the total height of the worker.  Thus, in the morning, the workerÕs head was 23 inches above ground.  At the end of the day, the workerÕs head was 23 inches below ground which does make sense since if the feet double in depth, the head distance must double as well.  Then, adding in the height of the worker, the depth of the hole can be found.  Conclusively, SeanÕs ÒweirdÓ idea is valid, but hard to follow!

            Watching the small group work throughout the simulation, the students used non-pictorial representations of the problem in conjunction with pictures in order to assist in solving the problem.  The non-pictorial representations used most were the equations drawn from the information given in the problem and from the pictures the students drew.  Initially, the students created one-variable equations to solve for the unknown depth, D, of the hole at night.  After some encouragement from the teacher, the students were also able to come up with two-variable equations describing the head distance and the total depth of the hole at night and, accordingly, solve the system of equations.  These non-pictorial representations of the material are another great problem solving technique that should be encouraged in the classroom.  In terms of preference, I think it is necessary and beneficial to have a combination of both non-pictorial and pictorial representations of the material.  As witnessed in this simulation, the pictures set up the information, and, following, the non-pictorial equations were drawn from the pictures the students drew.  The pictures often depict what is happening in the problem and fulfills the intuition students may have about the problem.  Then, the non-pictorial representations, such as equations, tend to justify those pictures and intuitions.  To me, a combination of both of these representations would serve as a best option within the classroom.

            In reference to Lao-TsuÕs quote on the best leaders, it is undoubtedly true that the same applies among the best teachers.  Very similar to what I responded to last week, it is unquestionably better for students to be able to solve their own problems and come up with their own solutions with little guidance from the teacher.  If the teacher is always over the shoulders of the students, the students will learn to depend on that teacher and never will really learn how to critically problem solve.  Conversely, if a teacher only gives input as a spark to a new idea or a new look on the same problem and the students are able to make the jump and solve the problem, then the students have meaningfully learned how to solve their own unique problems that may occur during the problem solving process.  Lao-Tsu is right; when the best teacherÕs work is done, the students claim that they did it themselves—which is what teachers should strive for in the first place!

            Finally, since I was never able to get SeanÕs ÒweirdÓ idea last week, I will extend my response from last week on finding solutions in several ways communicates several ideas about the nature of mathematics.  While the group itself solved this problem in 3 ways (one-variable equation, two-variable equations, and graphing linear equations), Sean still came up with an alternative route to get at the solution without any symbols at all.  Looking at this situation, it is amazing to me that 4-5 people can solve the problem 3 or 4 different ways.  This undoubtedly communicates the idea that while the solutions to a mathematical problem may be consistent, the processes that are taken to get to that solution are endless.  Moreover, as students get older and develop more mathematical knowledge, the processes taken to get at a solution increase incredibly with new, more difficult mathematical concepts that describe the same mathematical behavior.  The nature of math allows for different learning styles to thrive in the classroom with each of those learning styles being completely valid in the process of getting at common solutions.

            Comparing these responses to last week, my responses stayed fairly consistent.  Note that I tried to answer questions that I did not address in last weekÕs responses.  Besides the addition of SeanÕs ÒweirdÓ idea, I recognized the fact that it is great that math allows for so many different ways to get at a common solution.  Also, the ability for students to come up with answers to their own problems is an ideal situation for the teacher; simply being present among the groups and provide prompts should be the teacherÕs role when doing group work.  In all, this simulation was a great exercise for a teacher to learn how to encourage students to both work together (meaning, learning how to include students like Sean) and to solve the problem in multiple ways.

 

Student # 11

Paper copies of Combining Negatives and Coins are in the packet.

 

EED 331 Middle School Math

Prof. Shager

2/20/06

Simulation Reflection: Catchup

 

            One advantage to doing, exploring, and learning math through problem-solving versus worksheets, is that they more often have real life connections and applications that students can see.  The simulation that we did for this week, ÒcatchupÓ involves one such problem.  In this problem, students are asked to calculate how long it would take a person driving at 60mi/hour to catch up with his friend whoÕs driving a truck at 55mi/hour, but is 15 minutes ahead of him.  This is a very valid question, as it is something that could really happen (as asked by a student in the simulation.) in other many similar situations.  I think it was great that the teacher in this simulation had the option of making it a ÒtrueÓ story, which I think helped the students see the application of math in the real world and it also gives the problem more meaning. 

            Though I think that all math problems are ÒrealÓ problems, I think that when we put them into real life terms (word problems) it gives the students more motivation to solve the problems versus if they were doing a worksheet and number-crunching. 

            Another great thing about the real life connections that this type of problem gives is that the same math and situation can be represented in many different ways, to best suit the needs of the class. For example you could use the situation of a race and ask how long it would take a fallen runner to catch up or pass another runner.  You could also use the situation of machines and one has a malfunction.  There are many different ways to present this problem.

            One question that was a great question I think for students to think about was posed by Kelly, Òwould the time be the same if the car were going 65mi/hour and the truck 60 mi/hour?Ó  If I were teaching this lesson, I would have students calculate it out and see if there is a difference.  This is also because I know that there would be a slightly different answer if the speeds were both changed by 5 mi/hour.  From the time that the car leaves the gas station, it will take fifteen minutes more than if they were traveling at 55 and 60 mi/hour, making it 3 hours, not 2 hours and 45minutes.  

            One thing that I liked about this simulation is that it gives me ideas for how I want to teach-ways that I present material, interact with my students, and answer questions, etcetera. One thing that I noticed in this simulation is that for some choices I got lower points than other choices.  For example, choosing to say ÒgoodÓ instead of Òthank youÓ in response to an answer was given by student, gave me fewer points.  I think that this is because it is trying to make me understand and see that all answers given by students should be validated (whether right or wrong) and welcomed; and that by saying good, IÕm validating the fact that and answer was right, not that s/he actually volunteered an answer.  This is important to take note of because we want a learning environment where students feel safe to participate in class, ask questions, and offer solutions to a problem.  By validating all student responses, we are promoting and encouraging this behavior.

            Along the lines of getting points in the simulation, I also noticed that I got more points if I called on a student before asking a question-because often, they either asked or answered the question I was going to ask and allowed them to think about it on their own. 

            Another way that this simulation helps me think about my own teaching skills is through experience handling mistakes made either by the students or myself.  The great thing about mistakes is that they are perfect opportunities for Òteachable momentsÓ.  I really think that by learning through mistakes, we understand problems and the math better.  It is so important to talk through mistakes-why they were made, and how they can be avoided and not just correct them and move on.

            As a future math teacher, I have to think about how often and when I would let my students use calculators.  In this particular problem, I do not think that a calculator is all that helpful or needed. It is important for students to be able to make tables by hand and see the relationships and patterns for themselves.  The only thing in this problem that I would let my students use a calculator for would be to graph the functions and find their intersection point-but only after they have worked out the answer to the question by hand.  Here, it really only makes sense to use a calculator as an extension tool and should not be used to replace the actual math being learned.

            In terms of curriculum, I think that this problem can fall into many different categories.  In a basic sense of the technical math involved, it is an algebraic problem-where we are solving for some unknown number. However, this type of problem, because of its format can be used in a non-algebraic setting.  That is one of the great things about these types of problems: students are learning the skills without having to weed through the formal mathematical vocabulary and explanation (which can come after the skill has been learned) and lets them just do the math and feel good about successfully completing a problem.

            As a teacher, I think that it is important to examine a textbook carefully to make sure that it fits curriculum and goals for the class.  A good textbook should ask students to think critically for themselves, be able to relate the math they are doing to the real world, and provide problems that is well built on prior knowledge and skills-in other words, starts Òwhere the students areÓ. That is not to say that a curriculum is the same as a textbook.  Instead, I believe that a curriculum is actually made up of and based concepts and skills that we want our students to learn and acquire. 

 

Simulation 4: Yards and Feet

 

            This weekÕs simulation was question which formula was ÒrightÓ: 3f=y or 3y=f where f equals feet, and y equals yards. One of the things about this simulation that was little different from the others that weÕve done is that I really had to think about how IÕd answer the problem given to the students. It really is a difficult problem to make sense of.  After pondering the question for myself, I came up with the same answer that most of the students in the simulation came up with: both work, but in two different situations.  For the first equation, 3f=y only works if the variables, f and y do not actually stand for numbers, but words.  To me, it is not an equation, but a statement, Òthree feet is equivalent to one yardÓ.  It is stating what the conversion rate is.  Because, as we discovered in the simulation, if we input numbers to represent feet and yards, the equation no longer makes sense. (ex: let f=6. 3*6=18, but 3 feet does not equal 18 yards).  However, if we use the equation 3y=f and input numbers instead of words, where f is the number of feet and y is the number of yards, it does make work.  (ex f=6, then y=2 and yes, 2 yards is equivalent to 6 feet).  One great aspect about this problem is that it highlights the fact that math is not one dimensional, and that multiple answers are sometimes possible.  Going along this line, you can also mention to your class or show them that there are other types of math problems that have more than one answer.  Examples of these include equations such as quadratic and cubic functions. 

            Overall, I would say that the objective of this lesson is for students to work on their problem-solving and reasoning skills, while also thinking about the role and importance of placing units on our numbers. There are definitely reasons and advantages for having objectives written into lesson plans.  They give us a purpose and a focus for our lesson and also remind us as teachers of what we want our students to be learning. One of the disadvantages that I can think of is that by listing and focusing on one objective, we forget to give attention to other learning moments and ideas that tie into the lesson. Just because we choose a specific objective does not mean that we should forget about all other skills and teachable moments that are available in a lesson.

            Earlier in my reflection I mentioned that part of this problem involves Òmaking senseÓ of the two equations.  In saying this, I do not mean that in the same way that you would say it makes sense that a graph of a linear equation is a straight line.  This is because the latter only produces one answer.  There is no way that a linear equation makes the graph of a parabola or cubic. By definition, a linear equation is a straight line.  For this particular problem, I would replace the word, ÒsenseÓ with decoding. 

            As always, these simulations make me think about my teaching philosophies and they also help me see how teaching methods affect student responses and learning. One thing that I noticed when doing this simulation is that a couple groups were not as engaged in the problem as other groups. I think it was great how the teacher in this simulation had the option of making the problem relate to the students in a way that makes them more engaged in the problem. The reason why I think this is such a great ideas it that it doesnÕt change the actual math of the problem-just the story behind it.  It is amazing how that by just changing such a small part of the problem it suddenly becomes more interesting and applicable to students.

            Another option that these simulations frequently offer is to wait and see what a student might say next, versus prompting them.  Many times, I find it the better option to wait and see what my students will do next and then if they still donÕt seem headed in the right direction, I give them a little boost.  I think that this is a good teaching strategy as there are often times when kids need more time to think things through in their head before they can articulate what they mean.  By giving them this extra time, we allow them to make advances on their own, which I think gives them a great deal of satisfaction and confidence in their abilities as learners.

            Probably the best feeling you can get as a teacher is what Jesse sometimes says at the end of the simulation and that is, ÒWow!Ó  This response is one of the many joys of teaching.  It shows that the student is interested in what he or she is doing and that theyÕre excited about it.  As a teacher, it also is a sign that my students have learned something and that skill is something that they value and can use out side of the classroom.  There is no greater response to a lesson than one that shows enthusiasm for the subject and for the learning that has just taken place.

 

Simulation 5: The Monk

 

            The simulation that we did this week, I thought was pretty interesting.  Unlike any of the other simulations that weÕve done, we did not get to participate and instead, were given the thoughts of the teacher along with the virtual teaching of the lesson. 

            In addition to the actual simulation, I was intrigued by the actual problem that was posed during the simulation.  The problem posed is that a monk climbs to the top of a mountain starting at sunset of the last day of the month and paces it so that he arrives at the top exactly at sunrise.  He then leaves that mountain the next day at sunrise and arrives at the bottom of the mountain exactly at sunset. Is there a point on the mountain that the monk will cross at the same time he went up and came down the mountain?  Many of the students proposed different solutions. One student reasons that it would be the same as if there were two separate monks and one walked up the mountain while the other was walking down it-there would have to be a point that theyÕd cross at the same time.  Another suggests that the there is a point and itÕs at the midpoint and demonstrates it on the board using the special case that the monkÕs rate is constant going up and down the mountain.  Others suggest that because sunrise and sunset isnÕt always the same, that the rate canÕt be the constant and therefore, there isnÕt necessarily a point.  Of all of the ideas proposed, I think that most of them were trying to reason out their thinking.  There was not right or better answer in my opinion because all of the responses indicated that the students were actively thinking about and reasoning through the problem. 

            Looking at how the teacher taught the lesson, I think that it would have been helpful if she had written the problem on the board, or given each student their own copies of the problem.  I think that this would have been beneficial because like myself, there are students that need to see a problem visually, either in words or in pictures, or a combination of the two.  I think that some of her students may have been confused by the problem simply because they only heard it and couldnÕt make sense of those words. 

            I also noticed in the thought bubbles of the teacher that she often commented on the engagement level of the class.  I think that keeping a class engaged and involved in the learning process is so important.  For one, it shows that the student is interested and actively learning, which in the end will help to ensure and solidify the concepts.  While it is often the case that a few students may not be engaged, I do think that it is possible to have a lesson where everyone is engaged and participating in the lesson.  If planned wisely, I even think that it would be possible to have the entire class engaged more often than not.

            As always, these simulations require me to think about my philosophies regarding math education.  As small as it may seem, wording can have a large effect on the studentsÕ learning.  A slight variation can mean the difference between making sense and not making sense, encouraging a student, or discouraging a student.  In this particular simulation, the teacher responds to a student by saying, ÒIÕm flattered that you used my words.Ó  I think that this is a great response because it positively reinforces the behavior without actually telling the student thatÕs what they should do; itÕs more of an encouragement than a direction.

            Another point that came up during the simulation was the idea of confusion.  Being confused can often be one of the most frustrating states to be in, but from a teaching point of view, it can be a good sign.  While confusion sometimes is a result of poor communication and instruction or direction, it is a sign that students are involved and thinking about what is going on.  Also, a confused student I would say is generally one who wants to make sense of the problem and work through it, instead of sitting back and giving up.  I think that it is a catch-22.  It may show weakness in the teaching method for that lesson, but it can also show strengths in a studentÕs willingness to problem solve.

            ÒTelling is rarely teachingÓ.  I think that this statement is so true.  While some people can remember concepts simply by hearing them, I would say that most need the ability to get involved in what is going on in order to truly learn.  To go with that quote, I would say that Òyou learn by doingÓ.  Students must have the chance to try things for themselves and often that occurs best when they are not told what to do but rather are allowed to explore and discover on their own. This means that as teachers we should not be standing at a board writing down formulas, algorithms, and drill problems for our students.  Instead we should give them the opportunity to see how the relationships work for themselves.

            A second thought of the teacher that made me think was about students who stop thinking once they have an answer in their heads.  I think that this can go both ways.  I know that there are probably some students who would think, Òoh well, itÕs solved I donÕt have to think about the answer anymore.Ó  On the other hand, I do think that there are many students who would continue to think about the problem-and even more so now, to try and figure out how that person got that answer.  So in a way, I think that it can act as a catalyst for student thought.

            An action that the teacher took in this simulation, was to sit down.  I think that when able, this should be done during discussions and lessons.  The reason that I think this is because it puts the teacher at a more student-friendly level.  The students still know that he/she is in control because that hopefully has been established in the classroom.  The benefit is that it makes the discussion atmosphere more open and inviting for all to participate. I think that when a teacher is willing to come down to the student level and talk with them (vs. to them), it makes the students more comfortable to share their thoughts and ideas with the class.

            Another situation that arose during the simulation was the teacher allowing a student to correct another student.  I think that this is a good idea in some cases.  I think that it is important for students to learn from each other.  However, I think that it can often be embarrassing for students to be corrected by a peer who is at the same ÒlevelÓ as him or her.  ItÕs definitely something to be cautious with.

            Overall I think that this was a pretty interesting simulation, because it allows us to participate in a different way.  ItÕs not often that you get to see how a teacher is thinking during a lesson, and itÕs helpful to compare what theyÕre thinking to how IÕm thinking.

Student # 12

Combining Negatives for this student is in the paper packet.  All the rest are here.

 

February 6, 2006

Week 2 Simulation

 

Coins

 

The objective of this lesson for me was to get the students thinking about what they were working on. If this had been a lesson I wrote, I would have made sure that working in groups and problem solving were part of the objectives for the lesson.

I think practice does make perfect, but it must be done in a meaningful way. If it is just repetitive busy work, it will do no good; if it is something like this activity where students have to think about what they are doing and why they are doing it, then I think that having worksheets works. It works for some students, but not all and you have to keep that in mind when you are thinking about doing any repetitive activity in your classroom.

The students were engaged in this problem. They were talking about what they needed to do in order to solve the problem, not just the arithmetic. Figuring out what the problem is and then solving it is what I consider to be problem solving. An exercise is simply giving the problem to the student to solve.

Yes, there is always more than one way to solve a problem. In my classroom, I think it is valid to explore all of the ideas that come up. All are important to the learning process. Just because something doesnÕt work doesnÕt mean that it wasnÕt a good idea. All ideas are welcome in my classroom.

I donÕt think any of the explanations were good. Then again I am not a fan of lectures. While they did explain what was going on, I donÕt think that they would have made sense to a Middle School Student. In order for an explanation to be good, I think that teachers need to make real connections to what the students know. This way they can connect math to the real world, not just the classroom.

 In order for an explanation to be a proof, it must be generalized. While examples are great, they donÕt prove anything. As far as I am concerned, none of the explanations given in the lecture were proofs.

The thought of doing a problem correctly might be enough motivation for Bryce. He may also be motivated by the chance to getting to teach what he knows to a small or large group. Motivating middle school students is never easy, but it is possible. Another thing that can motivate many students is letting them pick from a list of topics to work on. If they are learning about something that they got to choose, then they are responsible for what they have chosen. This usually directs them to choose something that they would enjoy and WANT to learn about.

ValÕs approach to solving the problem was solid in her head. I think we need to allow students to think in a way that makes sense to them, as long as it works. The major advantage to this is that the math will make sense to them in their way. The disadvantage that I would be concerned about would be if Val would be able to apply her way of thinking to a different situation.

I think what makes the difference with Val staying on track is her ability to be able to not worry about the math so much and just focus on the process. I know I was in her place when I was a child as well. I still am in that place sometimes. I sometimes forget where I am if I have to stop and do the algebra and then get back into what I was doing.

I think it is important to let students use calculators if they are getting hung up on calculations. When I was in school I know that we could not have them and that is ridiculous because you will always be able to find one if you need one. I like to compare this to say the periodic table. If you need one, you can look it up. If you donÕt know what 689,372,846,129 times 8,916,983,568,930,935 is, I think it is safe to find a calculator.

 

 

 

 

 

 

 

Week 3 simulation

2/13/06

Catchup Simulation

 

When we asked Kelly about the car going 65 and the truck going 60, the teacher was getting at the point that we are getting closer by 5 miles every hour. The teacher in this case was trying to get the students to think outside of the box. Although it was a good idea, it may be a little confusing. We know that the truck is ahead of the car. Now we need to think about how far that truck got in the first 15minutes that it was moving. It gets farther down the road going 60 mph; therefore the time to catch-up is going to be greater than the original problem.

I donÕt think that Math is magic. I guess that I have studied it enough to know that it really does have a logical pattern. IÕm sure that some people who are not comfortable with math feel that it is magic. When I was in Elementary and Middle school I know that I felt like algebra was magic. A lot of times when I have been working on things for too long I feel like the things I remember to get it done are magic. I guess it all depends on where you are and what is happening around you. When you are first learning something, anything can feel like magic.

This is a real life problem. I think this is a great way to show students that some times it is good to know math. Although in this problem it is not totally necessary to know how long it is going to take, it is fun to be able to show someone how it would work if you wanted to. This was a great problem for engaging students. I particularly liked it when the students had the chance to get up and move around to act out the problem. This gives them a chance to Òdo mathÓ.

There were a few jokes in this simulation. While I think that jokes are important to gain attention of students, I think it also important to not be sarcastic all the time. I am not in this class, so it is hard to say what was really being said and how it was being said, but I think that some of the choices were very sarcastic and many were just plain rude. I hope that teachers arenÕt really like that.

Connections did arise in this problem. The teacher made great connections to driving across the country. These connections are good, but it is also important to stay on task and not totally distract the students from their train of thought.

This problem was represented well. The students were given a problem and then asked to look at it closely. When the teacher felt like they had it down, she then asked them to try something that was similar but had a different answer. This proved to me that there is more than one representation that is legitimate.

The students who were Ògood at mathÓ were the ones who could reason through what they were thinking. It was not necessarily the students who could whip out the formula, but the ones who could explain why they were thinking that was the correct formula. All students have the capability to learn mathematics. It takes some problem solving skills and some reasoning skill, but all students can learn math in their own way at their own pace.

Yes, this lesson was taught in the way that I learned math. The teacher would ask a question and the ÒsmartÓ kids would answer and I would be in the back of the class lost. I would never know what anyone was talking about and when we moved on to a new idea that incorporated the old one that I didnÕt get, I was lost. Even though I got lost, I think that it is possible to make sure that everyone understands what they are doing before you move on to the next topic. Not only is it possible, but it is very important if you want your students to succeed.

The catch-up problem is both a problem solving skill and an algebra problem. It also could be a physics problem. That is the good thing about this problem; it can be used in a variety of different curriculums.

A curriculum is neither a textbook nor standards. A curriculum is what you teach. You donÕt need standards and you donÕt need a textbook. You need good common sense and maybe a little direction to help guide you on what you should be teaching in your classroom. You shouldnÕt have to teach the ÒstandardsÓ if your students are too far behind or too far ahead of what they want you to teach.

This lesson was much more effective than the lecture-based lesson on coins. Whenever you can ask questions rather than tell students something, you are getting them to think for themselves. You can create those problem solving skills with little effort on your part. It is just a matter of Òteaching by askingÓ rather than Òteaching by telling.Ó

I would never ask another student what someone else is thinking. You always want to ask the student who made the comment what they were thinking. Why ask someone who has no idea what was going on in the other personÕs head?

In classrooms, you should ALWAYS discuss mistakes before moving on. If you have one student who brings it up, you probably have more than one who was thinking of it. It is always good to address those issues and fix them rather than ignoring them and moving on.

 

 

 

February 20, 2006

ESE 331

 

Feet/Yards Simulation

 

I think that the mathematical objectives of this lesson would be to explore different ways of thinking about equations and how they work. There would also be something in the objectives about exploring variables and what they really mean in context. I think that there can be negatives to having objectives when teaching. They can be negative when it comes to exploring, but I think that in some cases you may have objectives and even if those objectives are not met directly, there are always cases that straying from the planned objective is not a bad idea. Although objectives can make you focus your teaching too much at time, I think that as a first year teacher, it is important to have objectives so you know where you are going with the lesson. As experience allows, I feel that it is ok to stray from what is planned and think about ideas that are brought up by the class.

 

While I can see the correctness in both equations, I think that 3y=f is something that I see as correct. This will tell you the number of feet you will get when you plug in the number of yards. For example, 2 yards, times 3 will give you 6 feet. There are 6 feet in 2 yards. This can also work in the other direction. For example, if you have 9 feet, you would then plug it in to get 3y=9ft. Then with a little algebra, you get y=9ft/3; 3yards is your answer.

While this is what I see as the right answer, it does not always work this way. Math problems donÕt always have a ÒrightÓ answer. While I hear it all the time from others who are going to teach social studies or science, math does always have some answer, but sometimes it has multiple answers, and it is possible that some problems have multiple ÒrightÓ answers.

 

This is not the same sense making as a linear line on a graph. This is reasoning through a problem. When looking at a graph, we can see that it is a straight line, and we can prove that it is a straight line. With this problem, we can not ÒproveÓ anything. We have to reason through what we know and use examples to figure out what we donÕt know.

 

I think some of the students were engaged in problem solving. The ones who were not initially thinking that feet and yards was fun, soon realized that they can make their own problem and at that point, they became more engaged in their problem solving strategies. This was considered a problem rather than an exercise because there was no ÒrightÓ answer. In an exercise you just go through the steps to get to the answer, this was more than that. It was thinking things though and reasoning through things and being able to explain why you thought one way or another.

 

Many problems do have more than one answer. I remember being very confused by that when I first learned it, but problems like x^2 have more than one value for x to get the same y value. It is possible to get more than one answer to a math problem, and it will happen many more times in the math career of these students.

 

Thinking outside the box to me means getting away from the equation and applying it to something that you know. In order to apply something, you have to step back away from the box that the equation is in and figure out a way to make something else fit into what you have been given. The students who compared their equation to jeans or to football teams were definitely taking that step outside of the box to look at the box from the outside.

 

When you are working in small groups, I think it is ok to take the time to step back and just look at students. You are giving them that wait time that they need, and looking at a student may give them the ÒokÓ to say what they were thinking. Some other ways that teachers communicate with body language is how they act when they are upset at something. You can tell when a teacher crosses their arms or just gives the ÒlookÓ that you know something has not gone the way the teacher wanted it to go. A positive way of body language is something as simple as a smile. In a middle school classroom, a smile can go a long way.

 

IÕm not sure what the criteria were for visiting the groups in the order that I was required to. In this case, I think that tending to the groups that were not working was something that needed to be addressed. Also, I think that when working in groups and you know that there is a student who tends to be a loner, that you make sure that the students(s) feel comfortable.

 

I think changing the problem into something that the students were interested in helped them to focus on the problem better than they were focusing on it when they didnÕt care about feet and yards. I think anytime a group doesnÕt care about a problem or is having trouble making sense of it, it is a good idea to encourage students to change the problem into what they want to understand.

 

Rereading the problem aloud helps many students make sense of what they are to do. I think that the auditory learners do much better when it is read out loud and this helps everyone to catch information that may have been missed the first time. It is always a good idea to read the problem out loud when someone is stuck; it helps to recognize more information or hear it a different way then how they were reading it.

 

February 27, 2006

Week #5

 

SIMULATION #5: The Monk 

 

The students solved the problem in a variety of different ways. Some students drew graphs while some just talked through it and made thoughtful arguments about why they thought the way that they did. When it comes to the solution that was ÒrightÓ or ÒbestÓ I donÕt think that we can determine that. I think that all of the students were making thoughtful ideas of what they were thinking, and there is no best way or right way to think through a problem. All ways of thinking had valid points.

 

Both Sara and Maurice had justifications of the problems. They had proof by example, which turns out to be an inaccurate way of proving. When we look at the way that both Sara and Maurice explained their problems, they really are quite similar, but in many ways they have their own explanations. Both used representation to show how they came to their conclusions, which was very helpful. In this case, for it to be an accurate proof, one would have to prove that the monk never meets in the same place at the same time, which would be very difficult.

 

The teacher should always display a written version of the problem. This way, when someone is stuck, they donÕt have to have it read again, they can read it on their own. Some good communication that was going on in this problem was when the teacher stepped back and let the students talk to each other about the problem. This gave students the chance to talk though their thoughts without feeling intimidated. Some of the worse communication came when the teacher was when the teacher told students to nod their head to stay awake. I thought this was a little strange because this way you are telling students to agree with you, even if they donÕt care or are not listening. I think noticing who is not shaking their head is a good indicator of who is listening and who understands what you are asking.

 

I was wondering the same thing. I donÕt think anything in teaching needs to be that choreographed. Teaching should be something that you just do. As a teacher, you do need to be prepared, but I think it is silly to make up and act and go into a classroom as someone who is just on stage and not a real person. A teacher needs to plan movements as they come up in a classroom. As you are teaching, you realize that not everything goes the way you would have planned and therefore, I find if ridiculous to try to plan every movement of everyday.

 

Many times students need to learn to trust what they are thinking. The teacher asked for intuition to get an idea of what students think shortly after the question is asked. If everyone is coming up with the opposite of what you had hoped, it is maybe time to either re-phrase the question or to try to figure out why students are thinking the way they are. Intuition of the students will often give you an idea of how well or not so well you stated a question.

 

The teacher wanted students to work in pairs so they could share their ideas with a small group and feel comfortable doing so. While the students met in their small group, the teacher more than likely was going around the room and assessing understating of the problem. The teacher could also be answering questions at this point or could simply be listening to what the students are thinking about.

 

There are many times when all students are engaged. While it is often the exception rather than the rule, it does happen that every student is enjoying what they are learning about. Being engaged in a problem is important because those that are engaged have a better chance of remembering what was learned that day. They will remember the problem because they got to be part of it.

 

It was nice that Lori used the words of the problem, but it is possible that you can restate the problem with different words and have the same answer. So, since you can have an answer that is equivalent with different words, it is important to say that you appreciated it, but also allow students the freedom to use their own words.

 

Confusion is not always bad. I actually think that it is sometimes good. For me, the best questions surface when I am confused. Being confused means I have questions and having questions means I want to understand. So at least for me as a learner, confusion is not a bad thing.

 

That is a fear that I have seen in classrooms before, but the reality is that the teacher still has control of when each group presents. (Or each student talks) In this case, the teacher called on a few others to get some ideas rolling before calling on the students that she knew often have the right answer. She allowed other to think before the usually ÒdominatingÓ students got a turn.

 

I donÕt think that sitting down that the teacher gave up control of the class; I think it helped the students feel more comfortable talking to each other. She still had the power to bring the focus back to her, which means she had never given up the control. When you are in control of a class, you know because you are able to let students talk amongst each other and you are still able to bring the focus back when you feel the need to address something.

Student # 13

Paper copies of Combining Neg, Coins, and The Monk are in the packet.

 

SIMULATION #3: Catchup

 

o One choice included ÒAnd now for the magic.Ó Do you believe that mathematics is magic? Do you think others believe that mathematics is magic?

Some student think math is mystical, my job is to dispel the mystic.  That does not mean that I would not say something like that, if I wanted to get their attention or to point out a place where math is really a cool way to get an answer.  Anything that catches a students interest and has them paying attention is great.  But then you need to have them understand how the magic works.

 

o Which of the students in this class are Ògood at mathÓ?  Can all students learn mathematics?

Students who were Ògood at mathÓ were the students who could explain why they are doing what they are doing.  They are raised to a higher level understanding once they are able to explain it to someone else.  All students can learn math, to what extent depends on the individual.  Ideally yes everyone can learn, in reality NCLB is not realistic.  I have seen students who truly gone as far as they can in math, the abstract reasoning skills have net been developed to the same extent as her peers.  It may develop later or may not.  This is an IEP situation.

o Was this lesson taught the way you were taught mathematics?

This lesson was not taught in a way that I would teach mathematics.  I would never call on a student and let them off the hook for coming up with an answer.  I may help them along to reaching an answer but giving up is not an option.

 

o Is a curriculum a textbook or standards?  Does it matter what textbook you use?

Curriculum is the material/concept which are taught over the coarse of the year while a textbook is a resource used to help the student understand the material.  The textbook is important tool to help students learn, in that respect the choice of a textbook should mirror the content standards that your district has deemed necessary to cover.

 

o Was this lesson more effective than the lecture-based lesson of the Coins class? Are questions more effective than statements for teaching?

Effective is relative, this lesson was more effective in that it used questioning more than the previous lesson.  But in both cases I would have gone for deeper questioning and asked better explanations why they chose to do what they did.  The examples are too simplistic or they donÕt go far enough.

 

o What kinds of questions get more points than others?  In general, what was the difference between calling on a student before and after asking a question?  Can you think of exceptions to the general principle?

Questions that have a student explain their reasoning are far more effective than questions that require a yes/no response.  By calling on a student before you ask the questions gives them time to actively listen to the question.  I never call on special ed students without giving them prior warning.  I usually make a deal with them at the beginning of the year that if I am going to call on them I stand in front of them before I start talking.  Also I tend to ask special ed students questions that I am fairly confident they can answer.  I ask them the more in-depth questions privately where they are less intimidated by being wrong in front of the class.

 

o Would you ever ask one student what another might be thinking?

Not usually, I would rather ask the student what they were thinking.  Then open it up to others to either elaborate or help fix the reasoning.  If you ask another student to explain what someone else is thinking, that is a lot peer pressure on the student to make the correction and the person being corrected can close down and never volunteer information again.

 

o Why did you get fewer points from choosing ÒGoodÓ than from ÒThank you?Ó

Good could just be that you have gone far enough in the problem for the teacher to move on where think you implies that they have explained their answer sufficiently.  Thank you is also polite and shows respect for the students thoughts.

 

o Which of these questions had you thought of before seeing them here?  Which if any of these questions are important for teachers to think about?  the most important thing  for math teachers to think about is how to draw information out of students in an way and in an environment that is safe.  How do you ask a question so that a student is willing to risk being wrong or right in front of their peers.

 

o Should students have been allowed to use graphing calculators to make tables in this class?

I think graphing calculators are being more frequently used in the middle school grades.  I have a problem with them being a constant tool(required) because they are simply too expensive for most students.  It is not far to use a tool to teach a concept unless it is available to all.  The next point is the calculator is s tool that short cuts the process, at some point students need to learn the process.  It is one the biggest problems outside of the math classroom where math is an application tool.  They are so dependent on the calculator that they donÕt know how to follow the steps to solve a problem in science.

 

 

 

 

SIMULATION #4: Feet and Yards

 

Are there any disadvantages to having objectives governing your teaching?

I think that any teacher need to understand what they are trying to accomplish in the whole and objectives help define the way.  But, there are drawbacks objectives.  They restrict what can be taught,  my time at workshops are so focused on making sure that the standards are imbedded that you do loose some creative control.  There are so many objectives that you can®t possibly complete them all.  So some of the uniqueness that can be created for individual classes have been lost.

 

Which of the equations 3f = y and of 3y = f would you say is right?

Do mathematical problems always have right answers?

3f=y   Mathematical problems usually have a right answer but 36in =y is also accurate.  There are many approaches to solving a problem.

 

What makes a problem a problem rather than an exercise?

An exercise is lead by the teacher who takes them through the steps, where a problem happens either after they have some background and are apply that knowledge or in a discover activity to learn a new concept.  In either case in a problem situation the students are trying to discover the answer.

 

The students found several solutions to the problem of making sense of both equations. Can a legitimate math problem have more than one solution?

Yes some math problems can have a different answer is they use different variable.

 

Communication

At one point you have a choice between asking, ™Yiscah?š or just looking at Yiscah. Under what conditions, if any, is it good teaching just to look at someone?

Sometimes looking a a students allows them permission to speak up.  Or sometimes just looking at a student calls their attention to the fact that you are not going to let them not participate.  The answer to this question really depends on what is happening prior to the look or the calling their name.

 

What are various ways a teacher might communicate with body language?

If a group of students isn®t on task moving closer to them will let them know that you are aware of what they are doing.  Looking directly at a student can also accomplish this.  On the other hand sometimes a simple look can give the courage to a student that they are on the right track.  You do need to be careful in your looks that students don®t perceive that they ar being trivialized.  You may just be tiered but they think it is something else.

 

 Where might this lesson fit into a curriculum?  What is a curriculum?

Learning and teaching

Why were you required to visit the working groups in the order that you did?  Does it matter what order you visit groups?

Moving around the the different groups let the students know that you were available as a sounding board, that they need to be on task, that they didn®t just have free time, and that they were able to explore the problems in a variety of ways.

 

Group 3 was stuck until they reread the problem aloud.  Why might reading the problem aloud for a second time have helped a group get unstuck?

Rereading a problem aloud lets the students hear the problem instead of just reading it.  The oral follow through can let different subtext of the problem become clearer.  Sometime you just need to hear it again before it makes sense.

 

In Group 4, after talking for a while with Sean (who was working alone), you said, ™Maybe your group has ideas you can use.š Was that a good thing to say to a loner?  Should loners be required to contribute to groups?

There are times when students should work alone.  If they are the ones who always get it and feel like they are always having to explain to others, give them a break occasionally.  This does not mean that I will let them get away with working alone very often, ultimately the best form of learning is to explain it to someone else.

 

In this instance Sean isn®t getting it right away and encouraging him to work with his group will help alleviate his frustrations.

 

A number of times you were given a choice to wait.  Under what conditions, if any, is this a good teaching strategy?

Waiting is one of the hardest things for teachers to learn to do well.  We want to get in there and help, but waiting allows students time to process their own thought and attempt to solve the problem.  In the end the goal is the the students to be able to problem solve.

 

Toward the end of the simulation Jesse might have said ™Wow!š How does that response from a student make you feel?  What assumptions about teaching and learning does that feeling represent?

A wow, if genuine, is that good moment that we don®t always get daily, but when you do, it is why you teach.  It represent that moment in a students development when the concept clicks into place.

 

 

Student # 15

Paper copies of Comb. Neg, and The Monk are in the paper packet.

 

Simulation 2

                                                                                                                        Score: 18800

 

After looking at the coin simulation again, I think a lot of my original answers are appropriate and how I would have answered them anyway.  There are some new details I noticed, however, that help me understand this simulation better, which I will address here.  I will also make sure to answer some more of the questions so I have completely answered at least ten.

            The main thing I noticed when I did the simulation on my own was the difference between the studentsÕ individual work when they worked in groups compared to the individual work when the teacher taught them how to solve the problem.  When the students worked in groups, they understood the material better and had new ideas about how the problem could be solved.  When they were simply taught the process, students did not understand the process, using the wrong numbers to solve their individual work.  There was also a lack of creativity in how to solve the problem and the students did not try to think of new ways it could be done.

            Val was the most obvious example of this.  When she worked with a group to discover the mathematics, she was able to complete the work on her own and thought of a new way that made sense to her and may help other students who did not understand the coin problem.  Giving students the freedom to discover the properties of coins on their own allowed them to understand the process more deeply and they were more comfortable with the math. 

            I also think it was important to let the students use calculators.  It was not as apparent in this simulation as it would be in real life, but if calculators were not available, students may get stuck on the computations rather than the processes and problem solving.  In this situation, those concepts were what the lesson was trying to teach; therefore the students should be allowed to use calculators.  Although it is important for students to be able to do addition, subtraction, multiplication, division, and other computations in their heads or on paper, that was not the intent of this lesson.  Students who were not as strong there may have been discouraged from learning the other skills this lesson provided if they had not had access to that technology. 

            The final thing that I did not address in my original homework was BrunerÕs stages of representation.  In this simulation the students were at the third, or symbolic, stage of representation.  They did not use pictures to represent the coin situation, but used symbols to represent the information in the problem.  For example, the students understood the meaning of N, the number of nickels, even though N is simply a letter of the alphabet.  

                                                                       

                                                                                                                        Simulation 3

                                                                                                                        Score: 16000

 

            In the catchup simulation there were a lot of different types of choices and the students responded in interesting ways.  I enjoyed completing this simulation especially, because the students came up with some ideas of how to solve the problem that I would not have thought of and it was a great example of how students can learn when they are allowed to explore and have some freedom with mathematical content.

            At the end, there was a time when the teacher asked Kelly, ÒWould the time be the same if the car were going 65 and the truck 60?Ó  This was a great question because it extends the math beyond the one problem the students were working on at that time and generalizes the ideas they were exploring.  In this case, the answer would be yes, because the car would still be able to make up five miles every hour, and although the truck would be further ahead after fifteen minutes, the car would also be going faster and would be able to make up the distance in the same amount of time. 

            Questions such as that one are what make great problems.  When students can extend the learning or look at it from several approaches, the math is more meaningful to solve.  In this simulation, the students were engaged in the learning because they were all thinking about the problem in their own ways.  The teacher was able to ask a lot of open ended questions to get the students on the right track with their thinking about the problem without giving them answers.  This approach allowed the students to construct their own understanding of what was going on and used their ideas and thinking to guide the class.

            That is why I do not think math is magic.  Some people may think that because there are certain ways to ÒmagicallyÓ get the correct answers, or because it all fits together, but I disagree.  There are certain mathematical structures and there are ways to make things work, but there are also multiple ways to get to a solution.  There are concrete explanations for everything that can be done mathematically, so while some things may look or seem like magic, I do not think that is the case. 

            That is, however, one thing that teachers could say to get students engaged in the learning.  Stories, jokes, or interesting comments help students connect their learning to meaningful situations in their own lives.  In this simulation, the teacher used a situation that happened, or could have happened, in her own life to get the students interested in the mathematics.  The catchup situation is familiar and when students can make connections to real life, they understand the importance of the math and are more willing and excited to learn it. 

            Making connections such as these, allow all students to be Ògood at math.Ó  I believe anyone can learn math if they have the right situation and support.  Everybody does not learn things in the same way so problem solving is a good strategy to let students discover the math in a way that is personally meaningful.  I remember being taught mostly through lecture and examples and, while that worked for me, a lot of students did not understand.  They learned to believe they were simply not good at math and that some people can do it and some can not. 

            That personalized approach to teaching is also apparent through the types of questions and comments a teacher makes throughout a lesson.  In this simulation, the teacher often asked questions that led to several students giving different answers.  Each of the answers was appropriate in answering the question and the students were able to see there was not one ÒrightÓ answer.  That was also the reason I got more points for choosing Òthank youÓ rather than Ògood.Ó  Telling a student that their answer is good in front of the class implies that was the answer you were looking for.  I also got points from calling on students who did not have their hand raised, or calling on students before answering a question because that required every student to think about the problem and showed them multiple ways of thinking. 

            The main idea I got out of this simulation, was that students need to have opportunities to share their own ideas, rather than simply following the teacherÕs instruction.  While there are concrete rules and answers to problems in mathematics, it is also a very personal subject that can be understood in multiple ways.  Whether or not you, as the teacher, can think of multiple approaches yourself, the students will have new ideas that will not only give you more insight, but may also help the rest of the students understand.                                                                                              

                                                                                                           Simulation 4 Ft/Yards

                                                                                                                        Score: 22,400

 

            In this simulation, the mathematical objective of this lesson was to get students to explore two similar equations and to understand the importance of measurement units to the equations.  The students were asked to figure out which equation, 3f=y or 3y=f, was correct when discussing the idea that there are three feet in one yard.  However, neither answer was correct because both could be explained using different units of measurement to make sense of the equation.  For example, the equation 3f=y made sense to a lot of students because it was written as we talk, Ò3 feet in 1 yard,Ó but did not make sense when numbers were plugged in until the students converted to inches.  Similarly, the teacher allowed students to look at different situations such as shopping for jeans or playing football to illustrate the same concept because the idea of feet and yards was not the objective of this lesson.

            When the teacher allowed students to use their own interests to solve the problem, they became more engaged.  She also gave them choices and left them to explore the concepts rather than giving them answers.  Even when students asked questions, the teacher did not simply tell them what to do, but asked questions to get them thinking about the two equations.  This was a good problem because students used multiple approaches, were actively involved in the mathematics, and came up with numerous solutions that made sense.  Although people tend to think that math problems have one correct solution, that is a misunderstanding that was shown clearly through the simulation.

            The problem not only engaged the students in the math, and led to numerous solutions, but it also required students to think outside the box.  For example, Yiscah understood that the equation 3y=f worked when plugging in a number of yards to find the number of feet in those yards, but he realized that there might be another solution that made sense of the equation 3f=y.  There were also several groups of students who were not interested in this particular problem so they made up their own, similar, situation to solve.  Any good problem requires thinking outside the box because there are multiple ways to solve and, in this case, multiple solutions that require thinking beyond the obvious answers.

            For some of the groups, that was difficult.  The students did not want to think hard about the problem, but simply wanted to choose the answer they thought most obvious, whether or not they could explain why it made sense to them.  That is why the teacher went to the groups in the order that was presented on the simulation.  Groups that were not working or that were working individually needed encouragement to get engaged in the lesson. 

This was not only an important lesson in measurement units and explaining equations, but it also gave the students an opportunity to work together to gain insight into othersÕ thinking.  Therefore, the teacher needed to get everyone involved.  The groups that appeared to be working hard were not always discussing the assignment, but that gave the teacher a chance to apply the situation to their interests.  Similarly, it was good to simply wait and listen to a group when they were discussing the problem because you could then gauge their understanding of the problem and decide what clues or ideas to give them.  

This was a fun simulation because it gave a lot of insight into how students think and I think it had a lot of situations that would arise in a real classroom.  Although these students were not always engaged in the problem, it seemed like they were all thinking about the two equations and making sense of them for themselves, as well as listening to the ideas of their group members.  Ultimately, they came to some pretty sophisticated conclusions and did not need too much prompting from the teacher.