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Modeling Mathematical Thinking

Problem-solving has been defined as knowing what to do when you don’t know what to do. I love that definition, although I don’t know who to credit it to. Last summer, I was privileged to hear Yeap Ban Har speak about Singapore math at the SDE National Conference on Singapore Math. One fact about the Singapore system that stuck with me is that 20% of their standardized tests are items of a kind that students have never seen or worked before. That’s true problem-solving.

Students need to opportunity to work on non-routine problems, but they also need to be taught what it sounds like for a mathematician to work through a tough problem.

Consider this item from the 2013 Texas 4th Grade STAAR test:

This item was coded as a perimeter problem, but because it has process skills embedded, a child could totally know how to find the perimeter of a figure and still miss the problem. Stop for a minute and consider the skills needed to solve this problem.

Here’s where modeling comes in. Students need to hear your thinking about how to work this problem. That’s the way to develop true problem-solving/critical thinking skills. It might sound something like this…

[teacher reads the problem, which is projected on an overhead, document camera, or interactive whiteboard]

Wow!  That’s a pretty complicated problem! I heard several math vocabulary words as I read, so I’d better think carefully about the meaning of each word. Let me break this down and see if I understand each part.  

[re-reading]  Use the ruler provided to measure the side lengths of the figures below to the nearest centimeter. Okay, I need to measure the sides, and I’m using centimeters, not inches. Let me make sure I’m using the correct side of my ruler. [measuring the first side] I’d better be sure I line my ruler up carefully. I know that’s important when I’m measuring length. As I measure, I should write my measurements on each side. That way, I’ll have the information I need to work with. [measures all sides and record the measurements] Hmmm, I notice that all the sides on this hexagon, because it has six sides, are congruent. They’re all the same length.

I’m done with the measuring. I’ll read more and see if I can figure out what to do next. [re-reading] What is the difference between the perimeters of these figures? Okay, it mentions the perimeters of the figures. I know that perimeter is the distance around a figure. I’ll find the perimeter of each figure by adding up the side lengths. Oh! The sides on the hexagon are equal, so I can multiply those. [finds perimeter of each figure, recording the calculations].

Hmmm, now I’ve got the perimeters, but is that my answer? I’d better re-read the question again. [re-reading] What is the difference between the perimeters of these figures? Oh!! I need to find the difference between the perimeters. I remember that difference is like subtraction. I’m comparing the perimeters of the two shapes, so I’ll subtract the smaller perimeter from the larger perimeter. Oh!  I already know that 29 can’t be the answer because that number is too big. I’ll bet that’s adding the perimeters. The perimeter of the triangle is 17 cm and the perimeter of the hexagon is only 12 cm, so 17 – 12 equals 5.

Wow! That wasn’t really so hard after all. It had a lot of steps, but each of the steps was pretty easy once I read it and broke it down.

Notice a couple of things:

  • The teacher does all the talking. This is not the time to have students help you solve the problem–that can come later.
  • It really doesn’t take all that long! Model one problem like this a day, and you’re sure to see your students’ problem-solving abilities grow.
  • Pack as much math into the problem as possible. This was not a geometry problem, but I took the opportunity to use the names of the geometric figures as well as the word congruent.
  • Be sure to model the habits you want your students to develop. By recording all of your work when you model the problem, and commenting on why it’s important to do so, you are showing the students that mathematicians communicate their thinking using words, numbers, and pictures. That’s better than just telling your students, don’t forget to show your work.

Bottom line, our students don’t come to us knowing how to think critically. It’s part of our job to help students develop analytical skills right along with the math content.

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  1. This is a fantastic post! I’ve done think alouds with reading strategies, but never realized that I do this in math! Thank you so much for taking time to share math strategies, resources, and your expertise. I’m sure you are very busy with work and life, but want you to know that your posts mean a lot! I look forward to reading them and always find something useful!

    1. I really appreciate your kind comments! Isn’t it cool when we find connections between the content areas? Good instructional strategies are just that, whether for math or reading. 🙂

  2. I love the detail you provided for what a true think aloud sounds like and looks like.

    Some of my pre-service teachers struggle with the difference between a “think aloud” in math and a “tell aloud” in math. It’s that ever popular “tell aloud” they are so used to seeing because it is mostly how they were taught math. Your posts always hit home for my kiddos. Thank you!

  3. I love thinking aloud through math too! After much training the last few years I do next to no modeling though. I’ve only taught one math lesson this year and my students are doing astonishingly better! When I taught the lesson or modeled it, it was only after the class had earned it by exhausting everything they knew. Problem solving truly is the key as you said.

  4. Thank you for this. I needed to know how to think aloud in Math so it doesn’t sound as if I’m trying too hard and make it sound too obvious to the students.

  5. This is such a huge contrast to the routine drill and practice style. Cannot agree more that students should be able to see & hear how teachers are thinking through while solving Math problems. Thanks for this thoughtful article. One question that does come to my mind is – does the traditional teacher training teach teachers on how to develop this type of thinking in students?

  6. As math teachers we are told Inquiry Based or Discovery Learning develops problem solving. However, some students don’t have the skills, strategies, or confidence to work through problems. I believe modeling mathematical thinking is an integral part of the process. Great article validating modeling thinking is not only appropriate it is necessary.

    1. I agree! And I love how this article and what you posted reminds me what my students need. In addition to modeling, I find that I need to let them know that it is okay to feel like they do not know the answer. With our traditional style math program that we are directed to follow, my higher achieving students are used to getting the answer quickly, even when working with higher numbers in story problems. Modeling the approach as well as “o.k., I am unsure of what to do, so let me use what I know…” will be so helpful! I also plan on giving a copy to our math aide to help facilitate her interactions with these students.

  7. This is a great post! I completely agree think alouds in Mathematics are beneficial. They do not take long at all and the benefits will instill a lifelong change in the way our students approach math problems. It is not about the “do it with me” approach many teachers take. Thank you for you valuable posts.

  8. Very helpful. I teach small groups of Year 3 students who are new arrivals to Australia and speak little English. Typically they’re very good at maths but struggle with word problems and I can see how this will help me teach them. Thank you.

    1. Great question! Think of this as a mini-lesson, of sorts. They are just listening and following your thinking.

  9. It was fabulous to hear the thinking of the teacher. I’m wondering if this thinking aloud could be used with those students who need modeling from the teacher. For some of my students, I’d like to see the productive struggle of initially solving the problem on their own. I think it’s a great strategy for the math teacher’s toolbox.

    1. I totally agree with you about the power of productive struggle. I’ve found, however, that many students don’t know how to attack a problem, so the struggle is not productive because they have no strategies. I use the think aloud process to help prepare the students to struggle with problems on their own.

  10. THANK YOU!!! This is such a wonderful way to help students be able to understand a problem. I love how you provided us detailed information to help our students understand! Thank you! I’m getting pumped about being able to help my students with each post I read from you. Thank you!

  11. While I like this idea, I wonder its value if the teacher is doing the work up front. I completely get the modelling component but what if this problem was presented and you had the students take a look at the problem and break it down – perhaps working with a partner or in a small group. Give them 5 minutes or so to figure it out, talk about their strategies as a whole class and see/hear from your students. If many were confused or didn’t know where to start or how to proceed, I would work with those students or have students ask classmates to explain as it puts the learning on them and thus would be more apt to ask their peers.
    I find there is much greater value to have the students try and perhaps achieve or fail before modelling as they may come up with their own strategies and be able to figure it out. Thoughts?

    1. I am all for productive struggle! The point of the think aloud is to expose students to what mathematical thinking sounds like. The next step is to have students struggle with their own problem.

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