Ever used a model in your math class? Maybe you got out some manipulatives to work with a concept, or you used hangers or tape diagrams or algebra tiles to solve equations. How did you use them - were they for modeling or for demoing?
There is an important difference between the teacher using a model to demonstrate and the students using a model to aid their thinking and develop understanding. When a teacher puts a model up in front of the students, demonstrates a problem or more and explains the connection between the model for the students, that is a demo. Even if the teacher asked "What do you notice? What do you wonder? What do you think?" and had the students explain some of the connections, that's a demo. A demo has limited value, because the students don't have an opportunity to develop deeper understanding.
We have all had the experience when someone explains something to us with the utmost clarity, pictures, diagrams, tables, whatever... and we still don't quite get it until we have had time to think it through ourselves. In math class, we want to give students that opportunity to develop their understanding of the concepts. In order to do that, they need time and they need something that they can think about. The purpose of a model is to make the relationships and connections visible so the students can literally see what we want them to understand, and think about it for themselves. With a hanger diagram, for instance, the concepts of balance, doing the same thing to both sides, and simplifying both sides with opposite operations are all inherent in the diagram, and students can reason with those concepts independently with relative ease. Working with algebraic symbols before those concepts are in place is a purely mechanical exercise and students rely on the teacher to do all the thinking for them.
There is an important difference between the teacher using a model to demonstrate and the students using a model to aid their thinking and develop understanding. When a teacher puts a model up in front of the students, demonstrates a problem or more and explains the connection between the model for the students, that is a demo. Even if the teacher asked "What do you notice? What do you wonder? What do you think?" and had the students explain some of the connections, that's a demo. A demo has limited value, because the students don't have an opportunity to develop deeper understanding.
We have all had the experience when someone explains something to us with the utmost clarity, pictures, diagrams, tables, whatever... and we still don't quite get it until we have had time to think it through ourselves. In math class, we want to give students that opportunity to develop their understanding of the concepts. In order to do that, they need time and they need something that they can think about. The purpose of a model is to make the relationships and connections visible so the students can literally see what we want them to understand, and think about it for themselves. With a hanger diagram, for instance, the concepts of balance, doing the same thing to both sides, and simplifying both sides with opposite operations are all inherent in the diagram, and students can reason with those concepts independently with relative ease. Working with algebraic symbols before those concepts are in place is a purely mechanical exercise and students rely on the teacher to do all the thinking for them.
Giving the students a quick demo with a model is not utilizing its full potential. It may clarify things for some students, but for many, it is likely to build few lasting connections or even be more confusing. Here are some things to remember when using a model:
1) Make sure it is a model that the students can manipulate and reason about on their own. Start with simple examples that illustrate the key relationships, then gradually make the problems more complex. If you can pose a problem and the students can solve it on their own or mostly on their own, it is working. If the students are frequently asking how to proceed or why a step comes next, there is something wrong with the model or the problems you are asking them to do.
2) Give the students time to work with the model, use it to solve problems independently, and notice things on their own.
3) Once the concepts seem to be well-developed with the model, you can begin to use the mathematical symbols to represent them. Begin by pairing model representations with symbols and asking the students to provide one or the other. Can they draw a model for these symbols? Can they write the symbols that represent this model. Continue to cross-reference the symbols as long as the students need it.
1) Make sure it is a model that the students can manipulate and reason about on their own. Start with simple examples that illustrate the key relationships, then gradually make the problems more complex. If you can pose a problem and the students can solve it on their own or mostly on their own, it is working. If the students are frequently asking how to proceed or why a step comes next, there is something wrong with the model or the problems you are asking them to do.
2) Give the students time to work with the model, use it to solve problems independently, and notice things on their own.
3) Once the concepts seem to be well-developed with the model, you can begin to use the mathematical symbols to represent them. Begin by pairing model representations with symbols and asking the students to provide one or the other. Can they draw a model for these symbols? Can they write the symbols that represent this model. Continue to cross-reference the symbols as long as the students need it.
If you are using bar models to represent fraction division by whole numbers and a student is stumped by 1/4 ÷ 5, you could ask them to draw the bar model or imagine the bar model.
4) Models and understanding do not replace practice. The algorithms and rules are different from the concepts and are stored in a different place in the brain. The students can understand the concept behind 1/4 ÷ 5 and still forget the "shortcut" for solving it. Understanding concepts through exposure to models and problems is complementary to practicing algorithms. Surprisingly, the research doesn't seem to be clear about whether one has to occur first. If the algorithm is taught first, it can sometimes be challenging to get students to engage properly with the models. But I think it is clear, and often misunderstood, that both have to occur. Neither can replace the other.
4) Models and understanding do not replace practice. The algorithms and rules are different from the concepts and are stored in a different place in the brain. The students can understand the concept behind 1/4 ÷ 5 and still forget the "shortcut" for solving it. Understanding concepts through exposure to models and problems is complementary to practicing algorithms. Surprisingly, the research doesn't seem to be clear about whether one has to occur first. If the algorithm is taught first, it can sometimes be challenging to get students to engage properly with the models. But I think it is clear, and often misunderstood, that both have to occur. Neither can replace the other.
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