I’ve mapped out in my mind the different stages I want to help learners move through as they learn a new math concept. I think sometimes people jump straight to explanation and practice when I think there are a couple key stages that should come before those.
- Inspiration. Before a learner starts in on learning a new concept, I want them to be feeling at least some curiosity about it. Not only will it make the learning more engaging and more fun, but it will set them up for a better understanding. It helps so much to have a good feel for what the question or problem really is about, before coming to understand a solution, method or concept that addresses that problem. Dan Meyer talks about creating mathematical headaches to which mathematical tools are the “aspirin”. Interest can also be kindled by giving some interesting historical context about how the concept was invented, or by finding an interesting problem that the learner can relate to but finds they are stuck without some additional mathematical tools.
- Inquiry. Before learners turn to explanations of the concept presented by others who already have an understanding, I want them to have an opportunity to struggle with the problem themselves. It can be a brief struggle, if there’s limited time, patience or interest, but even if it’s brief I feel like the struggle is important. The feeling of being stumped by a problem can make the explanation of the concept seem magical. Getting stuck gives a feel for what is difficult about the problem, and allows insight into why the solution is so useful. Once in a while this step can even be stretched into the learners inventing their own complete solution and explanation.
- Explanation. After a good struggle, even if learners have invented a complete mathematical tool themselves, it is important to think through some good explanations of a new concept. It’s good to look at a couple different explanations and perhaps invite learners to reflect on which explanations make the most sense to them. Since the concept is new I wouldn’t necessarily expect learners to be able to give a full explanation of the concept itself yet, but they should be able to tell when a concept “clicks” for them.
- Practice. In my enthusiasm for the above three steps it is easy for me to under-emphasize the importance of practice in math learning. It was again Dan Meyer (infinite font of teaching wisdom that he is!) who joked on the Vrainwaves podcast a while back that he wouldn’t want anyone who discounts the importance of memorization to operate a vehicle. It’s a good point. Once a concept has been understood in a way that makes sense, the next step is to get it to where it’s easy so that the learner no longer has to think hard to do it. This allows higher level thinking about the concept, connections with other concepts and makes it easier to apply the concept to challenging problems. Sometimes a little practice is even needed before an explanation can be completely understood.
- Mastery. To complete the learning journey a learner needs to make the concept their own. This can be done by writing down a complete explanation or explaining the concept to someone else. Learners can apply the concept to difficult problems that require a little creative thinking, or extend or generalize the concept. At this stage it is great to ask lots of “Why?” questions and challenge learners to prove that the concept always applies (or analyze when it does or does not apply).
What do you think? How do you view the journey of a learner?
Enjoyed reading through this, very good stuff, thanks. “To be positive To be mistaken at the top of one’s voice.” by Ambrose Bierce.
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