This totally makes sense but in the past, when teaching or learning about math, the focus was always whether or not you or your student got the correct answer....this Shift focuses on the process rather than the product. If students can justify their answers and explain how they got their answers, teachers will have evidence that the student has mastered that particular concept.
Jeanne, I agree that a teacher has evidence that a student has mastered a concept when they can justify and explain their answers. As teachers we really need to know when this is happening with our students, and when to provide extra support. I also like how you brought up what the teaching of math used to look like. I remember when I was in elementary school it seemed as though all the teacher really cared about was whether or not we had the right answer; We never really focused on the reasoning behind it. It's great to see that times are changing, and I think for the better.
I think that times are changing-but just this past week, both of my children had math homework and each child's paper had no room to show "the work"-when I asked them where their work was they both said we do not need to do that. How do the teachers know they are not simply using a calculator to get the answer?
I feel like a lot of teachers are still stuck in what they have been doing. I do agree the process is more important than getting the correct answer, and I even would take off points on homework if students did not show work - even if they got the correct answer. I think this encourages kids to think outside the box but to also show teachers they understand what they are learning.
I agree with Leinwand that most of the time when watching students struggle with math, or even watching teacher struggle to help students get mathematical concepts, I want to immediately go to number lines and manipulatives. I find that manipulatives make a huge difference when working with students. Students really enjoy working with objects such as counter, beads and blocks and get a real picture of what is going on in the problem. I know when introducing fractions I let the students use magnets. In addition, for learning the basics of multiplication students get to use bowls of counters in order to make arrays to solve problems. Even the more advanced students love the opportunity to get hands on and the lowest students or the students who have a hard time understanding the process get to physically solve the problem, which make it easier than computing the answer in their heads.
Meaghan, I am glad you pointed out that there are different levels of students who need the same tools for learning but used differently. We all work with different leveled students but this shift can really be used at any age, and for any type of problem. It then becomes our jobs as teachers to do the modifying for the different students and their needs.
It is truly amazing how much easier math problems are to understand when you use pictures. When Leinwand asked to picture three quarters I immediately came up with coins and a pie but didn't think about a number line or some of the other options he provided. The chapter really made me think about how important it is to have students not only visualize math problems but also ask them to articulate how they visualize them. The use of bar graphs in the example of the suitcase weights really made the problem so much easier for me to grasp and I probably would not have thought about using a bar graph to help solve it. In the past, I have used manipulative unifix cubes and ones, tens and hundreds cubes to help students with basic addition and they have worked well in allowing the students to visualize the problem. In the future, I will definitely use the phrase "How did you see it?" in my teaching and in helping my own children with their math homework. The reminder that the same visual will not work for all children is important to remember as well.
Lyanne, I had the same reaction to the three quarters problem. I was stunned that I only came up with two immediate options, although there were so many possible pictures. It just goes to prove that all minds think differently and that we really have to provide connections with students when it comes to deeper understanding and background knowledge in mathematics! I like that you are going to use "How did you see it?" in your teaching, because that will allow your students to share their voice, and their visual, without feeling that they saw it wrong.
As I picked up my textbook today to do the readings for this weeks class, I found it coincidental that I was just talking about the new math programs we are piloting at the school system I work in with a few coworkers. The new math programs are based on teaching students to understand math topics conceptually and not just learning the procedure. I love how the third instructional shift that Leinwand mentions is specifically geared towards the teaching of mathematics and using a variety of representations. One of the teachers in my school was discussing at lunch today how many children know how to "drill and kill," but when it comes to understanding why a math procedure works, like "borrowing" during subtraction, they have no background knowledge or understanding of their answers. As an undergrad, my second major was mathematics. I truly believe that in order to teach students math concepts, especially in the elementary grade levels, seeing is believing. By using a variety of manipulatives, drawings, and real life connections students can see how the math works and not just repetitively going through the procedure of solving a problem. Leinwand made a great point in this chapter that not all children learn the same. By incorporating a variety of different teaching methods into mathematics, this creates more opportunities for students to have that "ah-ha!" moment. I absolutely agree with everything Leinwand presents in this third instructional shift.
I don't want to seem like I'm Leinwand's biggest fan right now but I am really enjoying this book. I completely agree with everything he writes about in this third shift. As a student I would be completely lost without a visual aid. When a teacher gives me the background I need to know about something, and gives me a picture to then connect with I understand things much better- no matter what subject it is. In math if we, as teachers, provide a solid explanation that includes a picture of what happens in a problem and why, the students will be more likely to soak in the information, create an associate between mathematical terms and the picture and then have success solving a problem correctly. Proof that a picture works wonders: I can still remember my elementary school teacher showing us a visual on how to multiply into groups. When I come across a multiplication problem that I am having trouble doing mental math for, I sometimes write down a number multiple times, and then add. The visual aid of writing the problem down in that formation is something that a teacher taught me to explain multiplication, and I feel that it is a great way for me to check my work, and a more reliable way for me to come up with an answer. Another picture that proved to stick in student's minds when I taught fourth grade was the terrible picture I drew them to get to remember the steps for long division. They made fun of my terrible artistic abilities but months later they could recall how to do long division because they could remember the picture of my steps. These are the associations we want our students to make, and then remember, and these pictures contribute to the deeper understanding of problems which is something very valuable to our students.
Skills like this are important to teach to students. Not all students are paper and pencil learners, many times they need a visual model to help learn. I'm glad that the author encourages teachers to continue to show the process instead of having students copy problems down until they memorize the answers.
During my student teaching, I saw how valuable it was to use visuals to help show how math can make sense. Many students were good at taking the formula and simply finding the correct answer, but when asked to explain how they got the answer it was hard to really describe the process outside of repeating the formula. Using visual models to supplement teaching has helped my students in many ways. Currently, we are doing a unit on adding and subtracting money. Instead of just doing problems on worksheets, we use money and show how different dollar amounts come together. It is helpful for students to see a real-world application to work they do in the classroom. With my class being a life skills class, this is a skill that they will be using more and more as they get older. I completely agree the visual models such as the fraction charts can help students to grasp a concept more than just copying problems down.
Rob - I agree that visuals are so important. I worked in a 1st grade classroom several days last week and the everyday the teacher sang a money song to help the students learn about the value of coins. As they count the days they are in school she has the daily helper put a penny in a chart. The kids were so excited because they were up to 25 pennies and they knew that they could swap them out for a quarter. I think the visuals really help the kids make sense of numbers.
The use of manipulatives is even more important now that one of the practice standards in the Common Core is that students are able to model problems. I wonder where it comes from that students feel it's "not cool" to use manipulatives. Even as an adult, I often take out my models to solve complex problems!
This totally makes sense but in the past, when teaching or learning about math, the focus was always whether or not you or your student got the correct answer....this Shift focuses on the process rather than the product. If students can justify their answers and explain how they got their answers, teachers will have evidence that the student has mastered that particular concept.
ReplyDeleteJeanne, I agree that a teacher has evidence that a student has mastered a concept when they can justify and explain their answers. As teachers we really need to know when this is happening with our students, and when to provide extra support. I also like how you brought up what the teaching of math used to look like. I remember when I was in elementary school it seemed as though all the teacher really cared about was whether or not we had the right answer; We never really focused on the reasoning behind it. It's great to see that times are changing, and I think for the better.
DeleteI think that times are changing-but just this past week, both of my children had math homework and each child's paper had no room to show "the work"-when I asked them where their work was they both said we do not need to do that. How do the teachers know they are not simply using a calculator to get the answer?
DeleteI feel like a lot of teachers are still stuck in what they have been doing. I do agree the process is more important than getting the correct answer, and I even would take off points on homework if students did not show work - even if they got the correct answer. I think this encourages kids to think outside the box but to also show teachers they understand what they are learning.
DeleteI agree with Leinwand that most of the time when watching students struggle with math, or even watching teacher struggle to help students get mathematical concepts, I want to immediately go to number lines and manipulatives. I find that manipulatives make a huge difference when working with students. Students really enjoy working with objects such as counter, beads and blocks and get a real picture of what is going on in the problem. I know when introducing fractions I let the students use magnets. In addition, for learning the basics of multiplication students get to use bowls of counters in order to make arrays to solve problems. Even the more advanced students love the opportunity to get hands on and the lowest students or the students who have a hard time understanding the process get to physically solve the problem, which make it easier than computing the answer in their heads.
ReplyDeleteMeaghan,
DeleteI am glad you pointed out that there are different levels of students who need the same tools for learning but used differently. We all work with different leveled students but this shift can really be used at any age, and for any type of problem. It then becomes our jobs as teachers to do the modifying for the different students and their needs.
It is truly amazing how much easier math problems are to understand when you use pictures. When Leinwand asked to picture three quarters I immediately came up with coins and a pie but didn't think about a number line or some of the other options he provided. The chapter really made me think about how important it is to have students not only visualize math problems but also ask them to articulate how they visualize them. The use of bar graphs in the example of the suitcase weights really made the problem so much easier for me to grasp and I probably would not have thought about using a bar graph to help solve it. In the past, I have used manipulative unifix cubes and ones, tens and hundreds cubes to help students with basic addition and they have worked well in allowing the students to visualize the problem. In the future, I will definitely use the phrase "How did you see it?" in my teaching and in helping my own children with their math homework. The reminder that the same visual will not work for all children is important to remember as well.
ReplyDeleteLyanne,
DeleteI had the same reaction to the three quarters problem. I was stunned that I only came up with two immediate options, although there were so many possible pictures. It just goes to prove that all minds think differently and that we really have to provide connections with students when it comes to deeper understanding and background knowledge in mathematics! I like that you are going to use "How did you see it?" in your teaching, because that will allow your students to share their voice, and their visual, without feeling that they saw it wrong.
As I picked up my textbook today to do the readings for this weeks class, I found it coincidental that I was just talking about the new math programs we are piloting at the school system I work in with a few coworkers. The new math programs are based on teaching students to understand math topics conceptually and not just learning the procedure. I love how the third instructional shift that Leinwand mentions is specifically geared towards the teaching of mathematics and using a variety of representations. One of the teachers in my school was discussing at lunch today how many children know how to "drill and kill," but when it comes to understanding why a math procedure works, like "borrowing" during subtraction, they have no background knowledge or understanding of their answers. As an undergrad, my second major was mathematics. I truly believe that in order to teach students math concepts, especially in the elementary grade levels, seeing is believing. By using a variety of manipulatives, drawings, and real life connections students can see how the math works and not just repetitively going through the procedure of solving a problem. Leinwand made a great point in this chapter that not all children learn the same. By incorporating a variety of different teaching methods into mathematics, this creates more opportunities for students to have that "ah-ha!" moment. I absolutely agree with everything Leinwand presents in this third instructional shift.
ReplyDeleteI don't want to seem like I'm Leinwand's biggest fan right now but I am really enjoying this book. I completely agree with everything he writes about in this third shift. As a student I would be completely lost without a visual aid. When a teacher gives me the background I need to know about something, and gives me a picture to then connect with I understand things much better- no matter what subject it is. In math if we, as teachers, provide a solid explanation that includes a picture of what happens in a problem and why, the students will be more likely to soak in the information, create an associate between mathematical terms and the picture and then have success solving a problem correctly. Proof that a picture works wonders: I can still remember my elementary school teacher showing us a visual on how to multiply into groups. When I come across a multiplication problem that I am having trouble doing mental math for, I sometimes write down a number multiple times, and then add. The visual aid of writing the problem down in that formation is something that a teacher taught me to explain multiplication, and I feel that it is a great way for me to check my work, and a more reliable way for me to come up with an answer. Another picture that proved to stick in student's minds when I taught fourth grade was the terrible picture I drew them to get to remember the steps for long division. They made fun of my terrible artistic abilities but months later they could recall how to do long division because they could remember the picture of my steps. These are the associations we want our students to make, and then remember, and these pictures contribute to the deeper understanding of problems which is something very valuable to our students.
ReplyDeleteSkills like this are important to teach to students. Not all students are paper and pencil learners, many times they need a visual model to help learn. I'm glad that the author encourages teachers to continue to show the process instead of having students copy problems down until they memorize the answers.
DeleteDuring my student teaching, I saw how valuable it was to use visuals to help show how math can make sense. Many students were good at taking the formula and simply finding the correct answer, but when asked to explain how they got the answer it was hard to really describe the process outside of repeating the formula. Using visual models to supplement teaching has helped my students in many ways. Currently, we are doing a unit on adding and subtracting money. Instead of just doing problems on worksheets, we use money and show how different dollar amounts come together. It is helpful for students to see a real-world application to work they do in the classroom. With my class being a life skills class, this is a skill that they will be using more and more as they get older. I completely agree the visual models such as the fraction charts can help students to grasp a concept more than just copying problems down.
ReplyDeleteRob - I agree that visuals are so important. I worked in a 1st grade classroom several days last week and the everyday the teacher sang a money song to help the students learn about the value of coins. As they count the days they are in school she has the daily helper put a penny in a chart. The kids were so excited because they were up to 25 pennies and they knew that they could swap them out for a quarter. I think the visuals really help the kids make sense of numbers.
ReplyDeleteThe use of manipulatives is even more important now that one of the practice standards in the Common Core is that students are able to model problems. I wonder where it comes from that students feel it's "not cool" to use manipulatives. Even as an adult, I often take out my models to solve complex problems!
ReplyDelete