"There is no one way"

Tuesday, April 22, 2014

Reasonable Acceleration

In previous posts, I expressed my concern about hyper-acceleration, the foisting of ever more advanced math topics on ever younger students. After illustrating some of the problems with that trend, I suggested a strategy for teachers to resist it. In this final post in the series, I will unpack one ingredient in that strategy: reasonable acceleration.

A moderate amount of acceleration can indeed be a good thing. Here are some examples:

- Some algebra before 9th grade is an excellent idea. In most countries, including the countries that do well in international comparisons, algebra is integrated into math education starting in middle school. That makes a lot of sense, and is far better than the traditional US approach of rehashing arithmetic in middle school, and then hitting students hard with a gigantic amount of algebra to be learned in a single year. In age-appropriate doses, earlier algebra is a good thing, and I welcome that aspect of the Common Core State Standards for math (CCSSM). (However this needs to be balanced with postponing some of the end-of-book and highly technical Algebra 1 topics, which only become useful in more advanced classes. For more on this, see my CCSSM analysis.)

- New technology and better pedagogy can make some topics accessible to more students than they once were. For example, electronic graphing and an emphasis on "real world" connections have made the concept of exponential growth and decay accessible to ninth graders, even though it was once an 11th grade topic. Interactive geometry software makes it possible to introduce transformational geometry to eighth graders, when this was once an end-of-book tenth grade topic if it was seen at all. Those are additional positive aspects of the CCSSM.

- Having students of different ages in the same class is one way to reduce tracking and enhance equity. For example, a geometry class with strong 9th graders and not-as-strong 10th graders provides both groups with the same opportunities, at slightly different times in their school career. (This is vastly different from assigning some students to an honors class, and others to a "regular" class, because that entails different expectations, and guarantees different outcomes.) But acceleration by more than one year tends to be counter-productive.

- It is possible to teach some interesting and motivational college-level mathematical ideas in high school. I have enjoyed bringing topics such as different-sized infinities, basic group theory, the fourth dimension, dynamical systems, and fractals to 11th and 12th graders. However to do this well, it needs to be done in a form and at a pace that is appropriate to high school, not by using college textbooks and approaches. (See my Space and Infinity classes for one way to do it.)

The main point is to acknowledge that racing students through topics that they are not ready for leads to a rote mastery which does not stick in the long run. Paying attention to students' developmental readiness, and taking the time to teach for understanding enhances learning and retention. Finally, incorporating very challenging problems at grade level is better in the long run than rushing to more advanced topics, as it teaches perseverance, reflection, and collaboration, all of which are more important than any particular topic.

--Henri

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