At CMC-South, I attended a workshop by David Chamberlain on the distributive property. At the end he posed the question: is this a trick? These days that question might be really asking: Is this something I can teach my students responsibly or is this intellectual poison? “Tricks” are recipes that students follow without understanding, and they can even shut down further thinking because the solution is so readily accessible that the process becomes automatic. Calling something a trick is pejorative, and implies that it doesn’t belong in a classroom where real learning is going on… but that is an overly simplistic point of view. I prefer the label mnemonic to distinguish “tricks” from models. Both are useful tools, each with a time and place, and the labels help us recognize and communicate the role each plays in our teaching, which in turn makes it easier to think about when to use them.
So instead let’s ask, is this a model or a mnemonic?
Models have a number of important characteristics. The ones we use in math often make an abstract idea visible. They also are usually something that students can manipulate and experiment with. These two characteristics make the third possible: students can do their own problem-posing and -solving. This is the defining characteristic of a model in math class. A model allows students to think for themselves and make their own discoveries. So the model must allow students to explore uncharted territory, without teacher guidance, and uncover new ideas.
An example of a model is a balance or hanger diagram to represent an algebraic equation. Many of us use these in algebra classes to help students discover or justify for themselves the steps involved in solving an algebraic equation. Using balance diagrams makes it clear for students why we “do the same thing to both sides” and when it is more direct to add or subtract first, or when multiplying or dividing first might be better.
A mnemonic is a tool for recall. PEMDAS, cross product, keep-change-change (for dividing fractions) are all mnemonics, helping students remember the order of operations or the sequence of steps to work with fractions. “Do the same thing to both sides” as a stand alone phrase is a mnemonic for an important part of the solving process. This is not wrong. We assume that understanding something means we will remember it, and vice versa - if we don’t remember something, we must not have understood it. But it turns out this is not the case (Google Robert Bjork and memory or listen to some of Craig Barton’s math podcasts if you want to learn more). Understanding and recall are more loosely connected, and each needs to be cultivated and strengthened. Students build recall strength by recalling, and if they don’t practice remembering, the accessibility (but not necessarily the knowledge) fades.
So what about the wrist tool that David described - is it a mnemonic or a model?
For me, it seems to be more of a mnemonic for remembering that when multiplying two sums, there are four products that need to be calculated and then summed. Using wrists doesn’t provide any justification for that, and there is nothing inherent in the model that tells students what to do next, the same way that there is no way to tell if you should add denominators together or not when you see a fraction addition problem on a page. You have to get that information somewhere else.
That said, the line between a model and a mnemonic is not black or white. Using wrists makes the process more visible, and one can see what is happening better than on paper. The closest it came to being a model for me was when we factored polynomials. Then I could see why shifting from (x+3)(x+4) to (x+2)(x+5) would leave the first and middle terms the same, but change the last term. And sometimes mnemonics communicate some of the underlying concepts. Models themselves can also serve as mnemonics, helping students remember what steps to take. But in general we can ask does the tool mostly help students think or does it mostly help them remember? If we asked them to give an explanation, would they be explaining their reasoning or listing their steps? During the session, when we described what we had done, I felt it was mostly the latter.
Why is the distinction important? We want to cultivate both understanding and memory in our students. Neither one alone is sufficient for success in mathematics. David referred to area models and, as I recall, we talked about using the wrist model after using some other context to explore the concepts behind the distributive property. So we need to be aware of when we are asking the students to think and discuss and justify their reasoning, and when we are asking them to remember and perhaps share their process. We want to make sure we are giving proper time to both.
David quoted one of his students saying “Why did we have to go through all that other stuff [the area model I assume]? Why didn’t you teach us this first?” Robert Bjork also has researched desirable difficulties, and the fact that learning happens when things feel hard - once it feels easy and comfortable, we aren’t learning any more. But managing frustration (for both students and parents) in math class is a real issue, and mnemonics are an important tool for that. We just have to be careful that ease does not surpass thinking and reasoning as the main goal for our math classes. Making intentional choices about when to use models and when to use mnemonics will help.
So instead let’s ask, is this a model or a mnemonic?
Models have a number of important characteristics. The ones we use in math often make an abstract idea visible. They also are usually something that students can manipulate and experiment with. These two characteristics make the third possible: students can do their own problem-posing and -solving. This is the defining characteristic of a model in math class. A model allows students to think for themselves and make their own discoveries. So the model must allow students to explore uncharted territory, without teacher guidance, and uncover new ideas.
An example of a model is a balance or hanger diagram to represent an algebraic equation. Many of us use these in algebra classes to help students discover or justify for themselves the steps involved in solving an algebraic equation. Using balance diagrams makes it clear for students why we “do the same thing to both sides” and when it is more direct to add or subtract first, or when multiplying or dividing first might be better.
A mnemonic is a tool for recall. PEMDAS, cross product, keep-change-change (for dividing fractions) are all mnemonics, helping students remember the order of operations or the sequence of steps to work with fractions. “Do the same thing to both sides” as a stand alone phrase is a mnemonic for an important part of the solving process. This is not wrong. We assume that understanding something means we will remember it, and vice versa - if we don’t remember something, we must not have understood it. But it turns out this is not the case (Google Robert Bjork and memory or listen to some of Craig Barton’s math podcasts if you want to learn more). Understanding and recall are more loosely connected, and each needs to be cultivated and strengthened. Students build recall strength by recalling, and if they don’t practice remembering, the accessibility (but not necessarily the knowledge) fades.
So what about the wrist tool that David described - is it a mnemonic or a model?
For me, it seems to be more of a mnemonic for remembering that when multiplying two sums, there are four products that need to be calculated and then summed. Using wrists doesn’t provide any justification for that, and there is nothing inherent in the model that tells students what to do next, the same way that there is no way to tell if you should add denominators together or not when you see a fraction addition problem on a page. You have to get that information somewhere else.
That said, the line between a model and a mnemonic is not black or white. Using wrists makes the process more visible, and one can see what is happening better than on paper. The closest it came to being a model for me was when we factored polynomials. Then I could see why shifting from (x+3)(x+4) to (x+2)(x+5) would leave the first and middle terms the same, but change the last term. And sometimes mnemonics communicate some of the underlying concepts. Models themselves can also serve as mnemonics, helping students remember what steps to take. But in general we can ask does the tool mostly help students think or does it mostly help them remember? If we asked them to give an explanation, would they be explaining their reasoning or listing their steps? During the session, when we described what we had done, I felt it was mostly the latter.
Why is the distinction important? We want to cultivate both understanding and memory in our students. Neither one alone is sufficient for success in mathematics. David referred to area models and, as I recall, we talked about using the wrist model after using some other context to explore the concepts behind the distributive property. So we need to be aware of when we are asking the students to think and discuss and justify their reasoning, and when we are asking them to remember and perhaps share their process. We want to make sure we are giving proper time to both.
David quoted one of his students saying “Why did we have to go through all that other stuff [the area model I assume]? Why didn’t you teach us this first?” Robert Bjork also has researched desirable difficulties, and the fact that learning happens when things feel hard - once it feels easy and comfortable, we aren’t learning any more. But managing frustration (for both students and parents) in math class is a real issue, and mnemonics are an important tool for that. We just have to be careful that ease does not surpass thinking and reasoning as the main goal for our math classes. Making intentional choices about when to use models and when to use mnemonics will help.
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