I totally agree with Steven Leinwand's shift that we need to make sure that students understand and know how to explain how they did their homework rather than if they got the correct answer.
Jeanne, I agree. There is too much emphasis on "the right answer" and not enough credit for mistakes. Students tend to learn more when they get the wrong answer because they have to back track and figure out what they did wrong. After confirming the correct answer, the teacher moves on and the students do not reflect on the problem they just figured out.
I completely agree. The explaining is helpful not only for students who don't understand but also for students who did get the correct answer. Students need to get into the habit of writing and explaining how they solved a problem.
I agree Jeanne, I find that most of the time my weaknesses in teaching math occur because I know how to solve a problem but I do not know why I did what I did to solve the problem. I realize that most of my math teachers taught me how to solve a problem but not why we solve the problem that way. Teachers should focus not on correct or incorrect but the process of solving a problem and why we take certain steps in solving a problem.
Accessible Mathematics makes tackling math sound so logical. Steven Leinwand breaks it down and offers sound insight. The idea of encouraging children to picture familiar objects when faced with a problem makes perfect sense. The children were able to understand when they began thinking about money, for another student, the image of the measuring cups worked. Building on the children's strength and giving them the confidence to work on something new are essential.
I agree with Leinwand in that there are parallels between the questions we ask our students in reading and the questions we can ask our students to answer mathematical problems, however this is something I am still struggling with as a teacher. Math has always been one of the easier subjects for me, but I can see how it can overload other students who it doesn't come quite as naturally for. Any thoughts on how to formulate these types of questions for elementary students? I find it difficult for me to get my students to explain their thinking in a way that others understand and I think if I knew some great prompting questions I could tackle this. I also thought it was rather mundane of Leinward to mention that teachers need to give meaningful homework assignments. Maybe because I already had this philosophy, but why give homework if it's not going to be productive?
I agree with you Holly. In all the time I have spent in a classroom and in asking my own children about their experiences, I know that the homework given is meant to reinforce what was taught during the day or week. I don't remember ever reviewing the math homework from the previous day for a teacher. The teachers that I have worked with and those my own children have had use homework to evaluate if the students are grasping the concepts taught that day.
I think you need to formulate questions that are math-related but also are applicable to real life situations rather than rote computation. I think if you give your students, "if....then...." scenarios they will be able to apply simpler math skills to bigger pictures.
I agree with Leinwand that it is important to determine what works in reading and apply those approaches to teaching math. Happily, I think that this is being done in more and more schools. I know my children are never allowed to turn in math work without showing how they reached their answers. In many cases they can get partial credit if they had the correct process but maybe added incorrectly so the ultimately answer was wrong. My daughter came home from her first math class in middle school and told me her teacher was explaining a concept and a student didn't understand it so he explained it in a completely different way so the student could grasp it. She thought that was fabulous. I hope that teachers like this are more and more common.
Lyanne, I too hope that teachers like the one your daughter has are more and more common. I was lucky enough last year to work with many teachers all with different styles and I believe that the teachers the students learn the most from, and like the best, are the ones that teach to their needs. This means we as teachers have to teach in many different ways, to many different types of thinkers. It makes their understanding better, and our jobs more fun!
I also think its a great tool to make students explain and show all of their work. I have heard "I can do it in my head" numerous times but once they stopped getting credit suddenly they changed their attitude! It also helps to show different ways to approach a problem, especially with so many unique learning styles.
This was a great chapter for me. I don't know about any of you but I have definitely had that teacher in high school who gave forty-something problems for an evening's homework assignment then in class the next day just read down the list of answers. I have never been strong in math. I have always tried, but the chances of me getting forty problems correct is slim if I don't have an extremely good grasp on how to work towards my answer. The best math teacher I have ever had was a teacher that seemed to be in with this second shift that Leinwand writes about. At the beginning of class we were to come in and write down a problem number on the board that we had difficulty with or did not understand at all. She would then open to the page, write it on the board, go through the problem, and all the while ask questions to the people who didn't understand, and ask for help getting through it to the people who did. I learned so much more in her class than in any other math class because she made me think about WHY I was taking the steps I was taking, and WHAT the problem was really asking. She asked questions the way my other teachers for English or Social Studies would ask them, trying to make our brains work differently and wonder, rather than just giving the steps and the answer. It made me appreciate and feel confident in math class. That is the type of teacher I would hope to be someday.
I enjoyed this technique because I always try to incorporate language into math lessons. This includes explaining and answering "how" and "why" questions just like in Reading. I have seen many students who can find the right answer, but cannot explain the steps taken to get it. Having students document steps taken as they work on a problem makes it easier to explain what operations they used and why the answer makes sense. This is also useful for students who don't understand math, because it allows them to slow down and work it out step-by-step as opposed to a race to get the right answer.
Creating and encouraging language-rich math classes offers students the opportunity to expand their thinking and ideas of how and what "subjects" are all about. So many people define themselves as either a "math person" or not. Leinwand reminds us to think beyond those labels and incorporate more into the classrooms. The memorization of math techniques did not work for me and I eventually tuned most math lessons out. The idea that there are many ways for children to learn and absorb should not be the exception, but rather the norm.
I agree with Liewand that there are many parallels between reading instruction and math instruction. I work in a school for students with needs that cannot be met in a regular public school setting. I find that their thought process is very different than my own thought process. I can be explaining how to solve a problem in a way that is comprehendable to myself, but then I look at my students and they have no clue what I'm talking about. I have to use different language with my students a lot that may make less sense to me, but makes the problem and explanation much clearer to them. Language is involved in every topic that is covered in the common core, especially math. ELA and Math go hand in hand.
Victoria - I love that your teacher made you come in with a problem you didn't understand. I just today was trying to help my daughter with her math homework and wasn't sure how to help her without giving away the answer. That strategy would have worked perfectly for her. If everyone has to do it there is no stigma on the child who often has difficulty with math too. And if everyone chooses different problems each one is gone over in depth to give greater understanding to all the students.
As part of a PD project, a group of teachers developed a booklet of open response questions that students worked on over the course of the year. They were taught to dissect the questions in the same way they would look at a question after reading a book in class. It was amazing how their ability to explain mathematical concepts improved by incorporating this reading strategy.
I totally agree with Steven Leinwand's shift that we need to make sure that students understand and know how to explain how they did their homework rather than if they got the correct answer.
ReplyDeleteJeanne, I agree. There is too much emphasis on "the right answer" and not enough credit for mistakes. Students tend to learn more when they get the wrong answer because they have to back track and figure out what they did wrong. After confirming the correct answer, the teacher moves on and the students do not reflect on the problem they just figured out.
DeleteI completely agree. The explaining is helpful not only for students who don't understand but also for students who did get the correct answer. Students need to get into the habit of writing and explaining how they solved a problem.
DeleteI agree Jeanne, I find that most of the time my weaknesses in teaching math occur because I know how to solve a problem but I do not know why I did what I did to solve the problem. I realize that most of my math teachers taught me how to solve a problem but not why we solve the problem that way. Teachers should focus not on correct or incorrect but the process of solving a problem and why we take certain steps in solving a problem.
DeleteAccessible Mathematics makes tackling math sound so logical. Steven Leinwand breaks it down and offers sound insight. The idea of encouraging children to picture familiar objects when faced with a problem makes perfect sense. The children were able to understand when they began thinking about money, for another student, the image of the measuring cups worked. Building on the children's strength and giving them the confidence to work on something new are essential.
DeleteI agree with Leinwand in that there are parallels between the questions we ask our students in reading and the questions we can ask our students to answer mathematical problems, however this is something I am still struggling with as a teacher. Math has always been one of the easier subjects for me, but I can see how it can overload other students who it doesn't come quite as naturally for. Any thoughts on how to formulate these types of questions for elementary students? I find it difficult for me to get my students to explain their thinking in a way that others understand and I think if I knew some great prompting questions I could tackle this. I also thought it was rather mundane of Leinward to mention that teachers need to give meaningful homework assignments. Maybe because I already had this philosophy, but why give homework if it's not going to be productive?
ReplyDeleteI agree with you Holly. In all the time I have spent in a classroom and in asking my own children about their experiences, I know that the homework given is meant to reinforce what was taught during the day or week. I don't remember ever reviewing the math homework from the previous day for a teacher. The teachers that I have worked with and those my own children have had use homework to evaluate if the students are grasping the concepts taught that day.
DeleteI think you need to formulate questions that are math-related but also are applicable to real life situations rather than rote computation. I think if you give your students, "if....then...." scenarios they will be able to apply simpler math skills to bigger pictures.
DeleteI agree with Leinwand that it is important to determine what works in reading and apply those approaches to teaching math. Happily, I think that this is being done in more and more schools. I know my children are never allowed to turn in math work without showing how they reached their answers. In many cases they can get partial credit if they had the correct process but maybe added incorrectly so the ultimately answer was wrong. My daughter came home from her first math class in middle school and told me her teacher was explaining a concept and a student didn't understand it so he explained it in a completely different way so the student could grasp it. She thought that was fabulous. I hope that teachers like this are more and more common.
ReplyDeleteLyanne,
DeleteI too hope that teachers like the one your daughter has are more and more common. I was lucky enough last year to work with many teachers all with different styles and I believe that the teachers the students learn the most from, and like the best, are the ones that teach to their needs. This means we as teachers have to teach in many different ways, to many different types of thinkers. It makes their understanding better, and our jobs more fun!
I also think its a great tool to make students explain and show all of their work. I have heard "I can do it in my head" numerous times but once they stopped getting credit suddenly they changed their attitude! It also helps to show different ways to approach a problem, especially with so many unique learning styles.
DeleteThis was a great chapter for me. I don't know about any of you but I have definitely had that teacher in high school who gave forty-something problems for an evening's homework assignment then in class the next day just read down the list of answers. I have never been strong in math. I have always tried, but the chances of me getting forty problems correct is slim if I don't have an extremely good grasp on how to work towards my answer. The best math teacher I have ever had was a teacher that seemed to be in with this second shift that Leinwand writes about. At the beginning of class we were to come in and write down a problem number on the board that we had difficulty with or did not understand at all. She would then open to the page, write it on the board, go through the problem, and all the while ask questions to the people who didn't understand, and ask for help getting through it to the people who did. I learned so much more in her class than in any other math class because she made me think about WHY I was taking the steps I was taking, and WHAT the problem was really asking. She asked questions the way my other teachers for English or Social Studies would ask them, trying to make our brains work differently and wonder, rather than just giving the steps and the answer. It made me appreciate and feel confident in math class. That is the type of teacher I would hope to be someday.
ReplyDeleteI enjoyed this technique because I always try to incorporate language into math lessons. This includes explaining and answering "how" and "why" questions just like in Reading. I have seen many students who can find the right answer, but cannot explain the steps taken to get it. Having students document steps taken as they work on a problem makes it easier to explain what operations they used and why the answer makes sense. This is also useful for students who don't understand math, because it allows them to slow down and work it out step-by-step as opposed to a race to get the right answer.
ReplyDeleteCreating and encouraging language-rich math classes offers students the opportunity to expand their thinking and ideas of how and what "subjects" are all about. So many people define themselves as either a "math person" or not. Leinwand reminds us to think beyond those labels and incorporate more into the classrooms. The memorization of math techniques did not work for me and I eventually tuned most math lessons out. The idea that there are many ways for children to learn and absorb should not be the exception, but rather the norm.
DeleteI agree with Liewand that there are many parallels between reading instruction and math instruction. I work in a school for students with needs that cannot be met in a regular public school setting. I find that their thought process is very different than my own thought process. I can be explaining how to solve a problem in a way that is comprehendable to myself, but then I look at my students and they have no clue what I'm talking about. I have to use different language with my students a lot that may make less sense to me, but makes the problem and explanation much clearer to them. Language is involved in every topic that is covered in the common core, especially math. ELA and Math go hand in hand.
ReplyDeleteVictoria - I love that your teacher made you come in with a problem you didn't understand. I just today was trying to help my daughter with her math homework and wasn't sure how to help her without giving away the answer. That strategy would have worked perfectly for her. If everyone has to do it there is no stigma on the child who often has difficulty with math too. And if everyone chooses different problems each one is gone over in depth to give greater understanding to all the students.
ReplyDeleteAs part of a PD project, a group of teachers developed a booklet of open response questions that students worked on over the course of the year. They were taught to dissect the questions in the same way they would look at a question after reading a book in class. It was amazing how their ability to explain mathematical concepts improved by incorporating this reading strategy.
ReplyDelete