I left my book in California. I had to order a 2nd one, as my sister can't find where I left it! I'll get my book tomorrow. My comments will likely come a little after 8pm. Sorry for an inconvenience...
In each one of the chapters the major theme was writing! As planned by the group, each one of the chapters worked well together. In each there were wonderful quotes involving writing and understanding. These understandings can be brought out through many different writing strategies which "makes teir [students] insights visible not only to themselves, but to others, including teachers and peers." After reading I've come to enjoy the idea of peer review with a rubric. With a rubric, students can see what "good" and "poor" responses look like. If students know what the teacher is looking for, I would assume (I know, assume is a bad word") that students may take more care in answering wuestions on examinations. The peer review allows students to see more than simply their own work. Seeing other students succeed and fail is thought provoking. When student A reads student B's work, student A will be thinking about student B's thinking. Thinking about student B's thinking, sutendt A will hopefully create schemas to help in the future. Perhaps student B solved a problem in a succinct way. This may be something for student A to try and emulate.
Math is subject which involves numbers, pictures, and words. The Multi-Rep chart makes students explain mathematics in these four dimensions including the real-world example. The question that every student asks the teacher will now be the question students need to answer on their own. Students must come up with a scenario to fit the mathematics. <3 it. Why don't math content textbooks suggest reading and writing strategies to help the reader understand the text? Maybe good-bye teachers?
Chapter 6: I really like the multiple representations charts strategy. It gives students the opportunity to view their mathematical thinking in more than one way. In addition, everyone has their own style of learning, and completing the box that is their strength might help them complete another box that they are struggling with. For example, a student might learn better by using visuals, therefore, have no trouble completing the “visual example” box, but might struggle with the “explanation in words” box. However, using their knowledge of the information they put in the “visual example” box might help them complete the “explanation in words” box. Also, what I like about the multiple representations charts strategy is that it allows for differential learning. Students can be on different levels of learning and complete their chart based on their own individual level.
Chapter 7: I thought “You be the Judge” was an interesting strategy, but I am curious if it would be used in a real-life classroom as it involves students reviewing their peers work. From what I have been told by teachers, students are no longer allowed to switch papers and grade their classmates work. I don’t know if this is a district policy or a larger policy, but I wonder if due to that policy if teachers would avoid using this strategy despite the work being anonymous. Also, I wonder what affect receiving negative peer feedback would have on a struggling student. Would it really help them or upset them to the point where they won’t try to correct their mistakes.
Chapter 12: I like the idea of students writing out their reasoning and methods used to solve a problem. Many times in math students learn the steps to complete a problem, but cannot explain why they did it a certain way besides saying, “that is how the teacher showed me.” By writing out the reason for completing each step helps students get a better understanding about the concepts they are working on. I know firsthand this is a good strategy to use because when I was first learning algebra I would have to write out my reason for each step I took to complete a problem. By having to explain each step it forced me to think about the mathematical reasoning that went into each step of the problem.
Chap 6 Comments: I agree with the above observations. I love the math rep charts. Of course, each lesson has new material, so technically, one could use math rep charts in each lesson! So one has to pick and choose carefully. The teacher chose to use this to assess prior knowledge. But thinking larger... the chart showed that teachers need to get students to write in multiple representations. One could use any combination of the 4 representations, either individually, or in combination, at any time within a unit. I also wonder whether one has to regularly share the multirep charts. It is interesting for students to share their ideas with others, and that could help "unblock" a student who's struggling. But it could take a whole period to do a rep chart and go over it. By that time, other concepts the teacher might have been able to cover could get missed. STudents sharing work with each other helps students communicate their knowledge and rethink their understandings... if they are the ones who share. If a student is lost, it isn't clear to me that hearing a concept from fellow students, which might have subtle errors, is better than learning from the teacher. So I think that multi-rep charts could be used more often than shared amongst the whole class.
Chapter 7 Comments - I want to add something to what Ken and Ann said. I think the chapter didn't do a great job of explaining this, especially in their summary. It seems to me that in the vignette, the process of having students apply a rubric helps them learn what makes a good response. If this is done in the process of evaluating a solution to a problem several times, students will learn the process of evaluating their work before "calling it quits". For example, if the rubric says "double check your answer", then students will start double checking their answers as a result! But it is unrealistic to provide a rubric for each and every homework and test problem. But using it, at least several times, to help educate the students HOW to solve problems and evaluate their solutions, then you won't have to keep using a rubric. So I think the point isn't that one should use rubrics and peer review, but rather to get students evaluating their own work for accuracy, completeness, correctness, neatness, etc.
Chapter 12 - Wow... what a lot of ideas. Many seem interesting to try one or two times with certain lessons to see if they work. But it has to be a natural fit. INtroducing extra writing, as a way of improving learning of a topic or the process of learning, is a great idea. But it must be used wisely, and not overused.
A few comments on some of the strategies:
Test Notes: This is how I've studied for tests since High School. Two days before a test, I start reviewing the material, and write down what I think is important. Then the day before, I review that. I never knew I was using a documented strategy! Though of course, they talk about handing them in!
Symbols-Meaning: This is what we do with proofs all the time.
Portfolios/Journals - Interesting to hear that these should be reviewed and critiqued, to make sure students are learning from these. They shouldn't be graded just for content. Interesting. My wife uses journals, but I don't think she does this. She really looks for completeness of content.
Chapter 6: I've heard numerous times that representing math concepts in different ways can be beneficial for student understanding. The multi-rep chart strategy presented in chapter 6 takes this idea of multiple representations and packages it into a neat and comprehensive chart. I was intrigued by this when I began reading the chapter, and by the end I was saw numerous potential uses and gains in implementing this strategy in a classroom. As the author explains, one benefit of the multi-rep chart is gaining insight into student understanding; one can see how much or how little a student understands a particular concept, and where misunderstandings a student has lie. In this chapter the chart was used as a formative assessment and teaching tool, but I wonder if it can be used on a cumulative assessment. I often hear concerns about testing for understanding (and how standardized tests don't really do this) and I think that this chart could be a possibility in testing for this. Also, all though not explicitly stated, the multi-rep chart allows for classroom discourse. Discourse was something that was emphasized in the introduction to this section of chapters (6 being the 3rd in the section). I believe classroom discourse is important because it allows students to communicate and exchange ideas. This is something that can take place through the use of the multi-rep chart, which was demonstrated by the teacher in the vignette. In the vignette discourse takes place on multiple levels, with pairs, groups, and the whole classroom. Even with just these two uses for the multi-rep chart (discourse and gauging student understanding) I begin to see more ways in which this strategy can benefit students and how these benefits are interconnected. One such connection, which is described in the chapter, is that discourse involving an exercise with the chart can help guide student understanding: the idea was that when students shared their charts in groups they could compare ideas to see what others came up with (and broaden each others' horizon of examples). The class discussion at the end of the exercise also demonstrated this connection because the teacher was able to ask students about their work and probe them for further explanation and justification of what they included in their charts. I really like this strategy because it appears as if it can work with any math topic (a thought which turned out to be confirmed at the end of the chapter) and can really give some good insight into student comprehension and thought process.
The strategy described in chapter 7 is completely new to me. I'm still not sure what to make of it. So far I think that when implementing this strategy it is important to protect student identity. Using responses from another class seems like the best way to do this. Though another layer of protection would help, such as the teacher having the students agree not to discuss the work they evaluated outside of class, so that students whose work was deemed not up to par would not end up hearing about it. The purpose of this exercise is for students to learn frames of reference so that they can evaluate their own work. Also, what happens after the students evaluate their own work? Do they hand it back in with an explanation of their scoring? If they decide it's doesn't meet the requirements for a good answer do they also have to redo the problem?
Chapter 12: The premise of chapter 12 is the importance of writing in mathematics; that is it the beginning, middle, and end to learning mathematics. My professors in college always stressed the importance of being able to write a good proof or a good answer to a problem, so that an audience (namely a reader) can understand it. To me, understanding our world through math is only half the story. The other half is being able to communicate our ideas and understanding. I feel that many students probably don't realize that when they do homework, take tests, or engage in other forms of assessment that they are writing to an audience and clear communication is important. Even when they solve a simple problem, like an equation with one variable, an answer doesn't mean anything without the rest of the solution preceding it. Simply writing down an answer doesn't communicate much of anything. It is the complete solution that allows an audience to see the reason behind the answer, and possibly understand the writer's thought process and intentions. Merely writing down a number as an answer is like writing the conclusion sentence to a proof and leaving out the preceding logical steps that led to the conclusion. Doing mathematics isn't putting an answer on paper. It is a communication of thought processes and justification. Therefore, it is absolutely essential that students learn how to effectively write mathematics.
Based on this response it probably appears that I'm more interested in why and how writing in mathematics is important and less so in the strategies presented in chapter 12 that can be used to teach students how to write mathematically. I am very much interested, but didn't have much to say about them now. I started thinking more about the ideas behind the importance of writing rather than how to teach it, so that's what I chose to write about. I do plan to give them a closer look and discuss them in the book club meeting.
Sorry I'm late to post though I've read these posts more than once - thanks to them coming directly to my mailbox! Thanks, Mark. I was up here looking for the summary of the conversation. Though I caught some of it, not all of it so I'm curious ... I so appreciate the depth of the comments here. There seems to be lots of agreement about the purposes of using writing in math (Countryman is a reference for math and writing worth checking out). I appreciate Raphaella's comment re: discussing the why of these strategies rather than the procedural details because the procedures can always change. I agree with Mark's clarification re: the peer review piece. I agree that in practicing a process of review, the idea is to internalize a process of self-review. By making the invisible visible, you eventually are able to make the visible invisible again. That's how good scaffolding works. Thanks for these comments.
Last Thursday our discussion revolved around chapters 6, 7, and 12. For chapter six the discussion revolved around the multiple representations charts strategy. We agreed that math is really the only subject that can be explained four different ways. We thought this strategy promoted discourse and differential learning. This strategy would be good to use to let students assess their progress by doing charts in the beginning and the end of a unit.
For chapter seven we discussed students creating rubrics to assess their work and the work of their peers. We had a conversation about whether it was a good idea to have students assess their classmates’ anonymous work. We concluded that it was a helpful idea as the ultimate goal was for students to form the ability to review their own work without any help. In math, students reviewing their own work is a problem and this strategy can help them develop the tools they need to succeed at that task.
We did not spend that much time discussing chapter twelve, but we did have a conversation about how teachers check corrections students make on tests. We wondered if teachers just check to see if the student finally got the correct final answer or do they check the process as well. We concluded that the focus should be on the process not the final answer.
I left my book in California. I had to order a 2nd one, as my sister can't find where I left it! I'll get my book tomorrow. My comments will likely come a little after 8pm. Sorry for an inconvenience...
ReplyDeleteMark
In each one of the chapters the major theme was writing! As planned by the group, each one of the chapters worked well together. In each there were wonderful quotes involving writing and understanding. These understandings can be brought out through many different writing strategies which "makes teir [students] insights visible not only to themselves, but to others, including teachers and peers." After reading I've come to enjoy the idea of peer review with a rubric. With a rubric, students can see what "good" and "poor" responses look like. If students know what the teacher is looking for, I would assume (I know, assume is a bad word") that students may take more care in answering wuestions on examinations. The peer review allows students to see more than simply their own work.
ReplyDeleteSeeing other students succeed and fail is thought provoking. When student A reads student B's work, student A will be thinking about student B's thinking. Thinking about student B's thinking, sutendt A will hopefully create schemas to help in the future. Perhaps student B solved a problem in a succinct way. This may be something for student A to try and emulate.
Math is subject which involves numbers, pictures, and words. The Multi-Rep chart makes students explain mathematics in these four dimensions including the real-world example. The question that every student asks the teacher will now be the question students need to answer on their own. Students must come up with a scenario to fit the mathematics. <3 it.
Why don't math content textbooks suggest reading and writing strategies to help the reader understand the text? Maybe good-bye teachers?
Chapter 6: I really like the multiple representations charts strategy. It gives students the opportunity to view their mathematical thinking in more than one way. In addition, everyone has their own style of learning, and completing the box that is their strength might help them complete another box that they are struggling with. For example, a student might learn better by using visuals, therefore, have no trouble completing the “visual example” box, but might struggle with the “explanation in words” box. However, using their knowledge of the information they put in the “visual example” box might help them complete the “explanation in words” box. Also, what I like about the multiple representations charts strategy is that it allows for differential learning. Students can be on different levels of learning and complete their chart based on their own individual level.
ReplyDeleteChapter 7: I thought “You be the Judge” was an interesting strategy, but I am curious if it would be used in a real-life classroom as it involves students reviewing their peers work. From what I have been told by teachers, students are no longer allowed to switch papers and grade their classmates work. I don’t know if this is a district policy or a larger policy, but I wonder if due to that policy if teachers would avoid using this strategy despite the work being anonymous. Also, I wonder what affect receiving negative peer feedback would have on a struggling student. Would it really help them or upset them to the point where they won’t try to correct their mistakes.
Chapter 12: I like the idea of students writing out their reasoning and methods used to solve a problem. Many times in math students learn the steps to complete a problem, but cannot explain why they did it a certain way besides saying, “that is how the teacher showed me.” By writing out the reason for completing each step helps students get a better understanding about the concepts they are working on. I know firsthand this is a good strategy to use because when I was first learning algebra I would have to write out my reason for each step I took to complete a problem. By having to explain each step it forced me to think about the mathematical reasoning that went into each step of the problem.
Chap 6 Comments: I agree with the above observations. I love the math rep charts. Of course, each lesson has new material, so technically, one could use math rep charts in each lesson! So one has to pick and choose carefully. The teacher chose to use this to assess prior knowledge. But thinking larger... the chart showed that teachers need to get students to write in multiple representations. One could use any combination of the 4 representations, either individually, or in combination, at any time within a unit. I also wonder whether one has to regularly share the multirep charts. It is interesting for students to share their ideas with others, and that could help "unblock" a student who's struggling. But it could take a whole period to do a rep chart and go over it. By that time, other concepts the teacher might have been able to cover could get missed. STudents sharing work with each other helps students communicate their knowledge and rethink their understandings... if they are the ones who share. If a student is lost, it isn't clear to me that hearing a concept from fellow students, which might have subtle errors, is better than learning from the teacher. So I think that multi-rep charts could be used more often than shared amongst the whole class.
ReplyDeleteChapter 7 Comments - I want to add something to what Ken and Ann said. I think the chapter didn't do a great job of explaining this, especially in their summary. It seems to me that in the vignette, the process of having students apply a rubric helps them learn what makes a good response. If this is done in the process of evaluating a solution to a problem several times, students will learn the process of evaluating their work before "calling it quits". For example, if the rubric says "double check your answer", then students will start double checking their answers as a result! But it is unrealistic to provide a rubric for each and every homework and test problem. But using it, at least several times, to help educate the students HOW to solve problems and evaluate their solutions, then you won't have to keep using a rubric. So I think the point isn't that one should use rubrics and peer review, but rather to get students evaluating their own work for accuracy, completeness, correctness, neatness, etc.
ReplyDeleteChapter 12 - Wow... what a lot of ideas. Many seem interesting to try one or two times with certain lessons to see if they work. But it has to be a natural fit. INtroducing extra writing, as a way of improving learning of a topic or the process of learning, is a great idea. But it must be used wisely, and not overused.
ReplyDeleteA few comments on some of the strategies:
Test Notes: This is how I've studied for tests since High School. Two days before a test, I start reviewing the material, and write down what I think is important. Then the day before, I review that. I never knew I was using a documented strategy! Though of course, they talk about handing them in!
Symbols-Meaning: This is what we do with proofs all the time.
Portfolios/Journals - Interesting to hear that these should be reviewed and critiqued, to make sure students are learning from these. They shouldn't be graded just for content. Interesting. My wife uses journals, but I don't think she does this. She really looks for completeness of content.
Chapter 6: I've heard numerous times that representing math concepts in different ways can be beneficial for student understanding. The multi-rep chart strategy presented in chapter 6 takes this idea of multiple representations and packages it into a neat and comprehensive chart. I was intrigued by this when I began reading the chapter, and by the end I was saw numerous potential uses and gains in implementing this strategy in a classroom. As the author explains, one benefit of the multi-rep chart is gaining insight into student understanding; one can see how much or how little a student understands a particular concept, and where misunderstandings a student has lie. In this chapter the chart was used as a formative assessment and teaching tool, but I wonder if it can be used on a cumulative assessment. I often hear concerns about testing for understanding (and how standardized tests don't really do this) and I think that this chart could be a possibility in testing for this. Also, all though not explicitly stated, the multi-rep chart allows for classroom discourse. Discourse was something that was emphasized in the introduction to this section of chapters (6 being the 3rd in the section). I believe classroom discourse is important because it allows students to communicate and exchange ideas. This is something that can take place through the use of the multi-rep chart, which was demonstrated by the teacher in the vignette. In the vignette discourse takes place on multiple levels, with pairs, groups, and the whole classroom. Even with just these two uses for the multi-rep chart (discourse and gauging student understanding) I begin to see more ways in which this strategy can benefit students and how these benefits are interconnected. One such connection, which is described in the chapter, is that discourse involving an exercise with the chart can help guide student understanding: the idea was that when students shared their charts in groups they could compare ideas to see what others came up with (and broaden each others' horizon of examples). The class discussion at the end of the exercise also demonstrated this connection because the teacher was able to ask students about their work and probe them for further explanation and justification of what they included in their charts. I really like this strategy because it appears as if it can work with any math topic (a thought which turned out to be confirmed at the end of the chapter) and can really give some good insight into student comprehension and thought process.
ReplyDeleteThe strategy described in chapter 7 is completely new to me. I'm still not sure what to make of it. So far I think that when implementing this strategy it is important to protect student identity. Using responses from another class seems like the best way to do this. Though another layer of protection would help, such as the teacher having the students agree not to discuss the work they evaluated outside of class, so that students whose work was deemed not up to par would not end up hearing about it. The purpose of this exercise is for students to learn frames of reference so that they can evaluate their own work. Also, what happens after the students evaluate their own work? Do they hand it back in with an explanation of their scoring? If they decide it's doesn't meet the requirements for a good answer do they also have to redo the problem?
ReplyDeleteChapter 12: The premise of chapter 12 is the importance of writing in mathematics; that is it the beginning, middle, and end to learning mathematics. My professors in college always stressed the importance of being able to write a good proof or a good answer to a problem, so that an audience (namely a reader) can understand it. To me, understanding our world through math is only half the story. The other half is being able to communicate our ideas and understanding. I feel that many students probably don't realize that when they do homework, take tests, or engage in other forms of assessment that they are writing to an audience and clear communication is important. Even when they solve a simple problem, like an equation with one variable, an answer doesn't mean anything without the rest of the solution preceding it. Simply writing down an answer doesn't communicate much of anything. It is the complete solution that allows an audience to see the reason behind the answer, and possibly understand the writer's thought process and intentions. Merely writing down a number as an answer is like writing the conclusion sentence to a proof and leaving out the preceding logical steps that led to the conclusion. Doing mathematics isn't putting an answer on paper. It is a communication of thought processes and justification. Therefore, it is absolutely essential that students learn how to effectively write mathematics.
ReplyDeleteBased on this response it probably appears that I'm more interested in why and how writing in mathematics is important and less so in the strategies presented in chapter 12 that can be used to teach students how to write mathematically. I am very much interested, but didn't have much to say about them now. I started thinking more about the ideas behind the importance of writing rather than how to teach it, so that's what I chose to write about. I do plan to give them a closer look and discuss them in the book club meeting.
Sorry I'm late to post though I've read these posts more than once - thanks to them coming directly to my mailbox! Thanks, Mark. I was up here looking for the summary of the conversation. Though I caught some of it, not all of it so I'm curious ... I so appreciate the depth of the comments here. There seems to be lots of agreement about the purposes of using writing in math (Countryman is a reference for math and writing worth checking out). I appreciate Raphaella's comment re: discussing the why of these strategies rather than the procedural details because the procedures can always change. I agree with Mark's clarification re: the peer review piece. I agree that in practicing a process of review, the idea is to internalize a process of self-review. By making the invisible visible, you eventually are able to make the visible invisible again. That's how good scaffolding works. Thanks for these comments.
ReplyDeleteLast Thursday our discussion revolved around chapters 6, 7, and 12. For chapter six the discussion revolved around the multiple representations charts strategy. We agreed that math is really the only subject that can be explained four different ways. We thought this strategy promoted discourse and differential learning. This strategy would be good to use to let students assess their progress by doing charts in the beginning and the end of a unit.
ReplyDeleteFor chapter seven we discussed students creating rubrics to assess their work and the work of their peers. We had a conversation about whether it was a good idea to have students assess their classmates’ anonymous work. We concluded that it was a helpful idea as the ultimate goal was for students to form the ability to review their own work without any help. In math, students reviewing their own work is a problem and this strategy can help them develop the tools they need to succeed at that task.
We did not spend that much time discussing chapter twelve, but we did have a conversation about how teachers check corrections students make on tests. We wondered if teachers just check to see if the student finally got the correct final answer or do they check the process as well. We concluded that the focus should be on the process not the final answer.