In my decades as math department chair, I learned that if you want to teach for understanding, you need to approach important topics repeatedly, and from different points of view. This includes different representations of concepts, and different learning tools to get at them. One reason for this is that since students have different backgrounds, experiences, and talents, some approaches will work better for some students, and other approaches will work better for others. Another reason is that extended exposure helps make ideas stick in a way that rushing through cannot. Finally, making connections between multiple points of view is an important part of understanding.
But multiple approaches take time, and whether you like it or not, the only way to make time is to take out or reduce unnecessary parts of your program. Today’s post is about the intricacies of pruning your curriculum. (Even if you do not have the option of taking topics out, you might still change the amounts of time allotted to various items, so read on.)
Almost certainly, many things can go. Any topic that is only there because it’s a personal favorite should at least be considered for deletion. Any topic that is unconnected to things that come before or after can probably be skipped. Any topic that is only there out of habit should go. Any topic that only reaches a handful of students should be taken out as it is wasted time for the others. (Do those super-hard topics in the math club, or move them up the grades to where they have a chance.)
Most controversially, you should consider spending less time on techniques and procedures that have been rendered obsolete by the availability of technology:
– Less time on paper-pencil multi-digit multiplication and division
– Less time on complicated factoring problems and techniques
– Less time on complicated equation-solving
– Less time on complicated manipulation of radicals
All of these things can be done faster and more accurately by machines. Speed and accuracy in computation are no longer legitimate priorities for math education — if they ever were.
You may feel I am reckless in making these suggestions. I am not. I want you to be careful because the concepts behind these manipulations remain essential.
– Multiplication and division will of course always be important, and there are many interesting things you can do to help students make sense of them, for example exploring arrays with Base 10 blocks, estimation, and mental arithmetic.
– The concept of factoring polynomials will always be important, and a student who cannot factor anything does not understand the distributive law. That is pretty much catastrophic from the point of view of developing any sort of symbol sense. The area model can help, this time using Lab Gear or some other algebra manipulatives.
– Building connections between equation-solving, graphs, and tables is a good way to develop sense-making. Solving simple equations mentally is for some reason not a standard activity. It should be!
– An interesting way to talk about radicals is geometric, as developed in Geometry Labs, 8.4-9.4. Simple manipulations of radicals remains important for communication and reasoning. I recently came across an interesting geometry problem which I was only able to solve because I knew that
In any case, I am confident that you will reach more students with deeper understanding of these important concepts if you prune your curriculum. One step at a time, and in consultation with your colleagues, cut back on complicated manipulations, do more intellectually engaging work, and watch your students grow.
[related post: Mapping Out a Course]
PS: Alas, the authors of the Common Core failed to grasp this important idea and ended up stuffing high school math with way too many standards. Read more about this here.