68. Problem Solving & How to Write Out Your Thinking

Ellie 0:00

Writing might not be the first thing that comes to mind when you think about math class or solving problems, but this powerful practice can enhance understanding, communication skills, and metacognitive awareness. I'm Ellie, and I'm here with my buddy Brittany.

Brittany 0:37

Hello.

Ellie 0:38

In today's discussion on The Teaching Toolbox, we'll explore how writing out math thinking can help students in their math classes.

Brittany 0:47

Why should we have students write explanations in math class? It does take time, and I suppose it might even put you behind in the curriculum, but we're here to tell you it's well worth the time that it takes. First off, when students write explanations for how they solve math problems, they have to articulate their thought process and reasoning. This helps them clarify their own understanding of the mathematical concepts involved. By breaking down each step or their process and explaining the reasoning behind it, students can self identify any gaps in their knowledge. It helps students practice using math vocabulary correctly, and this deeper engagement with the material leads to a more comprehensive grasp of the math concepts. Another benefit of writing out math thinking is that it enhances communication skills, and we can apply what we're discussing today to verbal explanations as well. Writing explanations helps students develop their ability to communicate complex ideas clearly and effectively. Math often involves abstract concepts that can be challenging to explain, but by practicing how to explain their solutions, students learn to express their ideas in a structured and logical way. This skill is valuable, not just in math but also in other academic subjects and real life situations. And then, thirdly, encouraging students to write about their problem solving process promotes metacognitive awareness, which is the ability to reflect on your own thinking. This awareness helps students evaluate the strategies they use, recognize successful approaches, and reconsider those that aren't working. As students become more aware of their own cognitive processes, they can develop more effective problem solving strategies and become more independent learners.

Ellie 2:52

And as you're explaining that, I was just kind of thinking about how when we explain something to someone, it starts to make more sense to us. Right? We remember it better.

Brittany 3:06

Correct.

Ellie 3:06

The understanding is deeper, and so it's basically doing the same thing, but they're doing it with their own mind, to start with thinking about it, explaining it to themselves in a way, as they're writing it down. Does that make sense?

Brittany 3:18

Yes. If you think about it, it's a lot like usually in second or third grade, teachers will have kids explain how to make toast, and then they'll go through every kid's explanation,.

Ellie 3:32

Yup.

Brittany 3:32

and bring in a toaster and a loaf of bread and see if it see if it works.

Ellie 3:37

Yeah, yeah, exactly. So how do we actually approach this practice and incorporate it into the classroom? So you'll need to take a couple class periods to introduce this idea of writing, practice it together, and set expectations for this kind of work. When I had an hour for math in elementary school, this was fifth grade math, we spent part of almost every class period working on problem solving and explaining, like writing their explanation of their solutions. And we started this the year that state testing began incorporating open ended questions that required written explanations. So there was kind of a push to get that started. And while we started it for that reason, I continued it for years and years and years, because students became so much better at communicating their math ideas and processes. And I think they became much better thinkers because they weren't just writing down some numbers and being told that the answer was correct or incorrect. They had to justify what math work they had done and basically prove through writing that they understood the steps to solve. And having to justify their work made them slow down and take much more time to try to reach correct answers.

Brittany 4:53

Yeah, I think with kids, it's a lot of - in math, they just want to know, am I right? Am I right? Am I right? And you have to have them slow down and really think about the processes that you're doing and be able to explain what you're doing, and why are you doing it.

Ellie 5:13

Right. Now, when we were in elementary school, we wrote several paragraphs. As time moved on, and I went to middle school and we didn't have as much time for math that got shortened up, but still even a couple sentences like if you start with a few sentences of explanation that's helpful to students. When you start using this practice, I would recommend using problems that are more general or incorporate skills from previous grade levels so students aren't hung up on I don't know this math when they're trying to just understand how to write an explanation. We used problems that address problem solving strategies like guess and check, work backwards, solve a simpler problem, create a table, look for a pattern, make an organized list - those types of strategies. And we were really fortunate where we had a book, it was called the problem solver, I believe, at a couple different grade levels, and it had numerous examples of these types of problems. So we use those in class a lot. As I said, in middle school, I tried to do the same thing as elementary school, but since we only had 40 minutes, we couldn't spend as much time and do it every single day. So eventually we did it once or twice a week. So if you're starting to incorporate this, you might want to do a few days of introduction. This is our process. This is the expectation. This is how we're doing it. And then try to work it in maybe once a week, if you have shorter class periods, if you're lucky, and you have block scheduling or something, or you have longer class periods, then you could spend a lot more time on it.

Brittany 6:40

40 minutes is crazy short.

Ellie 6:43

It is. I think it was 44 but by the that included switching time,

Brittany 6:46

yeah.

Ellie 6:47

So by the time they got in, got settled, yeah. And so I would say, as you know, we kind of mentioned in the beginning, you might worry that this is going to put you behind in your curriculums, but do not be afraid of that. As your students get better and better at being able to write out and verbalize their math thinking, their math performance improves overall and other lessons may go a lot faster than they would have if you had not started using this practice.

Brittany 7:12

So when teaching students how to problem solve and write out their thinking, it's important to focus on a few key principles. First, you want to model the process. Start by showing students how to break down a problem. Walk through your thought process, step by step, thinking aloud as you approach the problem. This helps students see that problem solving isn't about instant answers, but about thoughtful exploration. I like to use, find out, choose a strategy, solve, and check your answer.

Ellie 7:48

Another thing that you want to think about is teaching metacognition, which we mentioned in the benefits part of explaining the thinking. Encourage students to think about their thinking. Ask them reflective questions like, What do I already know about the problem? What strategies have worked in the past? By pausing to think about their approach, students become more aware of how they're solving problems, and they can transfer those strategies to future situations.

Brittany 8:14

Then you want to scaffold the problem solving steps. Understand the problem. Teach students to take time to fully grasp the problem before rushing to solve it. They find out or identify what they need to know to solve the problem. And some of this information in the problem, and some might be background knowledge, so encourage students to rephrase the problem in their own words. Then devise a plan or choose a strategy, and guide them in identifying possible strategies in which to approach the problem, like drawing a diagram, making a list, solving a simpler version first. Then they want to execute the plan or solve it. So help students carry out their plans Step by Step. Encourage them to write down each step, explaining what they're doing and why. Then review and reflect or check your answer. Once a solution is reached, have students evaluate if the answer makes sense. If not, encourage them to retrace their steps and identify where they might have gone wrong.

Ellie 9:25

Once you've guided students in solving then you want to guide them through the examples of written explanations. One way to make written explanations easier for students is to provide some sentence starters. Many students struggle with writing out their thinking simply because they don't know where to start. Providing some sentence starters gives them a structure to build their explanations. And these can be starters, like from the problem I know. And then they pick out things in the problem that were important to solve, and they write them down. What I need to find out is. And they identify the question in the problem, and then rephrase it in their own words, so that they're clarifying that they know what they actually need to find out. To solve the problem, I will draw a diagram, work backwards, solve a simpler problem. I know my answer is correct because so then you know, in between all that they will write out what they actually did and then justifying that they know it's correct because might be, you know, reiterating the important points. You know, I know my answer is correct because it answers this question to the problem. I do have a blog post that has two example problems and example explanations, just like this, so we can link that in the show notes if you want to check out those examples and see something a little bit more concrete.

You can create a supportive environment for mistakes by emphasizing that mistakes are part of the problem solving process. When students feel safe about making mistakes, they're more willing to take risks and think critically about their approach. Encourage them to see mistakes as learning opportunities and to reflect on what they've gained from the process. Take time to have students share their explanations with the class or in small groups, so students can get an even better idea of what a complete explanation might sound like. Point out one or two aspects of the explanation that you really liked.

Some students are super proud to share their explanations, while others are hesitant, and that's okay. They don't have to, but it's usually a fair number of students are willing to share, which is excellent. You should regularly practice reflection after a problem solving activity, have students reflect on their own explanations. They might read the explanation to a partner, as we talked about, you know, sharing, to have that partner follow the directions and see if they can solve the problem given that explanation, to see if they can't do everything, then they know what they've missed, and that it's kind of fun, you know, to just listen and and see if you can figure it out from what somebody else is telling you. You can ask them questions like, What would you do differently next time? What would you add to or take away from your explanation? Or what part of this problem was the hardest to solve, and that will help them become more strategic in the future.

Celebrate different approaches which students use. Show students that there's often more than one way to solve a problem. Share various strategies from their classmates and discuss the pros and cons of each approach. This helps students appreciate diverse ways of thinking and find approaches that work best for them.

I think this is that's so important, because there's so many times that students just think it has to be one way.

Right.

And there are times for me that I think of something in one way, but a student would think of it differently and it was perfectly correct, and I'd be like, Oh, wow, I never thought of it that way. And it's good for them to hear you say that, because then they understand that even, even you might think a different way than them, and it's all correct and okay.

Yeah or they go home and they fight with their parents. That's not the way we did it in class,

right, right, right,

or whatever. And then you look at their paper, and you're like, your dad did it great, you know? I Yeah, it came out wonderfully. It looks good, you know, you've explained it well. And then they're like, Okay, I can trust my dad,

right? Well, by emphasizing these types of strategies for writing about your thinking in math class, you can help students build confidence in their problem solving skills and communicate their thought processes clearly. This not only improves their ability to tackle problems, but also deepens their overall understanding, critical thinking abilities, and communication skills. It's a power tool for your teaching toolbox.

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