Ep 29: Worked Examples in the Elementary Math Classroom

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Sometimes you sit down to write an email or something like that and you have absolutely no idea where you want to start and you're just kind of looking at a blank screen with a cursor.

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Sometimes our students can feel like that when problem solving, if they're just really not sure about the context, not really sure how to approach the problem.

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So using worked examples or worked out examples is an amazing way to help your kids figure out different strategies or discuss different strategies for solving problems in a low pressure way because they don't actually have to solve the problem.

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They're just looking at problems that have already been worked out and trying to analyze and figure out what is going on.

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So in this episode today, I'm going to share with you some different types of worked out examples you might want to use for your next problem solving session.

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Hello and welcome to Making Number Sense Make Sense, a podcast for elementary teachers, specifically early elementary teachers, looking to really make an impact in the number sense of their students.

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So worked examples are step by step solutions or partial solutions to math problems and each step is very clear for students to see.

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So some of the things that you'll find if you're using worked examples is you can show or model different problem solving techniques and strategies.

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It can help break down complex problems into more bite sized pieces.

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It can help students learn specific strategies and how they might apply those strategies to different problems in the future.

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It helps students focus on understanding a concept or strategy without the cognitive load of having to figure out all the steps.

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So with that, there are different types of worked examples that you might want to use.

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One type is the fading example.

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So the fading example might be a series of problems where students might start with most of it worked out and they just have one little part of the problem to figure out.

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And then next time they'll have maybe two steps that are not figured out that they'll have to work on.

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And then next time, maybe three steps. And it goes like that.

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So it provides less and less scaffolding or support as kids get more familiar with that style of problem.

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This type of worked example, it helps students perform better on harder problems.

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It takes less time to achieve the same level of performance.

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And students end up having less unproductive moments of struggle. So a lot of times you'll hear people talk about the productive struggle and how that's good for students.

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There comes a time where it's not productive anymore and they're just struggling.

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So this type of problem can really help alleviate that.

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The next type of worked example is prompting students to explain what is going on in this problem.

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So when you're asking kids to explain what's going on, it forces them to tap into their prior knowledge and integrate that with this new information that they're seeing and helps to make the new learning that they're doing very explicit.

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So not just like by osmosis, they're figuring out how to do this. It really helps them solidify what they're doing.

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In the math setting, it helps kids increase the likelihood that they'll be able to pick out relevant information in a problem.

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It helps with understanding underlying concepts in a problem and how to carry out steps.

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It helps increase their declarative knowledge and conceptual knowledge.

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The next type of worked example is an incorrect worked example.

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And a lot of times, especially when students are struggling, people might be hesitant to show students an incorrect example.

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The one caveat with this kind of worked example is you're going to want to make it very clear that there is a mistake somewhere here, that this is not the way that you're supposed to do it.

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And then from there, lead a discussion on helping students figure out where the error is and how they might be able to correct that error in the future.

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And this type of example is really good for if you're seeing common mistakes in certain kinds of problems, to have a discussion about what that error is and why it's not working and how to change it so that you can figure out the answer.

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And once students are aware of this mistake, it will help them be less likely to repeat that mistake in the future if you've really discussed what's going on.

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It also helps learning by contrast. So if you only ever show something positive, you're not going to know what the negative even looks like.

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So if you only show correct examples, you're not necessarily going to know what is incorrect or where students might be going wrong of all the possible things that might happen when they're solving a problem.

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So by showing that contrast, you can really hone in on where the errors are occurring.

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And just like in the previous one where why explaining something does work can help you understand, also explaining why something does not work can also help you understand.

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This is really helpful for all students to get in some error analysis and it helps fine tune problem solving strategies.

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The next type of worked example is comparing multiple examples.

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So this kind of worked example is great if you're wanting to show students two different ways to solve a problem or if you're wanting to move from a certain type of problem solving strategy that might be less efficient than a different one.

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You can show the two problems, kind of show that they're both correct. They both led to the same result, but really have a discussion about why one strategy might be more efficient than a different strategy and kind of make connections between the two strategies.

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So this is great if you're moving from concrete to more representational, representational to more abstract, but really highlighting those connections can help students.

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So there's lots of things that you can compare. You can compare to, like I said, two examples of the same problem.

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You can compare two correct solutions to two separate type of problems that might have an underlying similar concept and also compare a correct and an incorrect solution to the same problem.

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Again, kind of like the previous one with error analysis to figure out what's going on and where the error might be.

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When comparing two solutions to the same problem, it can help students determine, it can help students kind of think about what strategy is better in the sense that it will work for them.

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It makes the most sense to them. It can lead to flexibility in choosing the best strategy for a specific type of problem.

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So maybe in one kind of problem, this strategy works, but in this kind of problem, I might not want to do the exact same thing I just did. It kind of broadens their repertoire of strategies to choose from when problem solving.

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To get the most out of this kind of worked example, it can be helpful for students to have a certain threshold of prior knowledge to really benefit.

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So if you're comparing a concrete and representational and students are not yet ready for that representational, it might not be the best time to use this kind of strategy.

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But if you feel like your students are successful in solving problems concretely, that might be the time where they're ready to move on to more representational.

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And that's when this kind of strategy can really be helpful. So which type do you choose? It really depends on your goal.

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So from what I've explained, some strategies might work better than other, but if your goal is specifically procedural fluency, then a faded worked example might be best for that because the steps are very similar.

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If you're wondering when to include these or how much to use these, worked examples should be used a lot more than they are typically.

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Students can really benefit from studying why something does or doesn't work versus just doing endless problems and not really sure what they're doing.

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If they're solving problems incorrectly, then they're learning that that's the way to do it when it's not benefiting them because they keep making errors.

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Worked examples can be used up to half of the time because from there, they've talked about, they've had the chance to discuss different problem solving strategies, figure out why something does or doesn't work.

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And with all of that in their head, the time to process, the time to talk about it, the time to reflect, they can use what they've been thinking about and apply that to new problems.

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So definitely something that you can consider adding to your classroom.

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I guess to tie it up with a little bow.

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Using worked examples help students develop a deep understanding of why something works instead of just procedures or algorithms and not really connecting it to prior knowledge or the ability to apply what they've learned to future problems.

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Students can learn a lot from mistakes.

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You know, if you're cooking and you're cooking with oil and it splatters on you, you're going to know next time you cook with oil to step back a little bit because you're not trying to get burned again.

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So again, learning from mistakes, you're like, oh, that didn't work.

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And if you can talk about why it didn't work in a judgment free way, then that's really helpful.

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I would stay away from holding up a certain student's example and say like, this isn't an error that so-and-so made.

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You're not going to want to do that.

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You can even present it as like, can you can we figure out what the mistake is in this problem?

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And if you're framing it that way of can we find the mistake here and figure out why this isn't working and figure out how to correct it, it can help students see mistakes as opportunities for learning instead of something that they're dreading and hope never happens.

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Because at the end of the day, we want our students to be able to jump into any problem and try and solve it. And if they're held back by the fear of making a mistake, they're not going to try new things.

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They're not going to be able to learn from new things and feel confident enough to discuss what's going on.

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And finally, worked examples help improve conceptual understanding and procedural skills when used appropriately.

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I'm going to link in the show notes an article about worked examples if you want to learn a little bit more about it.

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And I will catch you next time.