I’ve been thinking recently about what it takes to make a concept “stick” in a student’s mind. When first looking at a topic, it’s tempting to show a student easy examples that get them familiar with the mechanics, before moving on to more difficult problems. However, when this new concept is a new way to see an old idea, it can be difficult to sell the concept to the students if the old idea seems to be just as effective as the new one. After all, why should the student have to learn a new method if the old one still works?
This is the topic that Dan Meyer talks about in his presentation on the needs of students. His main point is that mathematics education is too much like a collection of solutions, and not enough of giving students “headaches” in which mathematics can help. Throughout the talk, he gives several examples in which the audience is asked to solve an easy problem, and then to solve a similar but much more demanding problem which creates the “need” for new mathematical tools. In particular, I was struck by the example at around the 35:00 mark in which he asked two people from the audience to describe the location of a chosen point in a sea of other points. Crucially, there were many points, so it wasn’t a trivial matter to describe with words where the point one chose was located. The first person struggled with this, but when it was the other person’s turn, the dots were underlayed with a Cartesian grid. You can hear the audience at that moment laugh at how easy everything becomes. I highly encourage watching the whole thing, but if you don’t want to spend the full fifty or so minutes, watch that one part.
Related to Mr. Meyer’s point is the idea of showing easy examples. I’m reminded of the fact that many students are given easy examples only. You’ve seen this before in the fact that assignments have “nice numbers” as answers, and that nothing is too difficult. This is good to start, but it can often lull students into not really grasping why certain methods are used versus others. If there are five ways to answer the question, why does a student have to do it in a specific way? If this method really is important, I think one needs to create problems that show-off the merits of a particular method. As Mr. Meyer says, give the students a headache and show them a particular method as the aspirin. Don’t just give them easy example problems to work through when trying to motivate them on a new technique or concept. Skipping this step means students won’t have buy-in on the idea, and will simply wonder why they have to learn this esoteric method.
Now, I’m not saying that easy examples aren’t useful. They can help during the beginning of practice, where students are motivated by the concept, yet still need to work on getting the technique. Then, it makes sense to give easier examples. This can also help students check their work, since they can compare the new method to their old method. However, once the technique is mastered, don’t give them problems which are so easy that a different (and more familiar) method can be used. Make sure that students see the need for this new technique. There’s a reason it’s there.