In a recent series of articles, Professor Keith Devlin of Stanford University outlines some of the issues related to creating video games to enhance math learning. Part 6 of this series – the most recent one – gathered 15 comments at the time of this writing. The comments, while related to video game design, also speak to some more fundamental issues in teaching math. I quote some parts of the discussion, as a way of illustrating what I see are two important pedagogical issues.
(a) “Are you going to approach mathematics as a collection of procedures or as a way of thinking? These are not completely separate classifications; indeed, the latter is in many ways a broader conception than the former. But they do tend to cash out in very different forms of pedagogy.”(Devlin, part 5)
(b) “Even if K-8 math is taught extremely well, how can a video games of this kind teach higher level math (precalculus and calculus) – which is where even those who’ve done well in K-8 math struggle?” (Jeremy Millington, Comments, part 6)
(c) “…we don’t need games (or other new learning methods) merely to teach fractions or to bootstrap kids into algebra. We could really use them at higher levels, too. At those higher levels, the gap between conceptual understanding and symbolic understanding seems bigger to me.” (Anonymouse, Comments, part 6)
(d) “In terms of gameplay not in conjunction with a teacher-led class, this strikes me as harder, since you need to provide the player/student with good feedback on what they have done.” (Devlin, Reply to Jacob Klein, part 6)
Procedures vs. a way of thinking [(a) above] is, in my opinion, a fundamental pedagogical issue. The ways textbooks are written, the way most teachers teach, math is a collection of procedures. Most teachers announce the topic, give some definitions, work out some examples in class, assign homework and then go to the next topic. Even if exams are cumulative, in the minds of the students, the topics are not related – yesterday we did exponents, tomorrow we do rational expressions. There is no attempt (or no time) to present a bigger picture.
As an example, 90% of the California standards for Algebra II deal with functions (the exceptions are series and probability). However, most teachers, in my experience, do not emphasize the commonality that exists between lines, quadratics, exponentials, higher order polynomials, etc. In my experience, most teachers do not show that all of these are functions that represent different models of reality. Neither do most teachers emphasize the common theme of finding the roots of these functions, why we try to find the roots or what do these roots mean.
Because we teach sequentially, often with an eye toward the released questions of the state test, we do not often assign synthesis problems. For example, if in the same problem we would ask students to find the roots of (1) a line, (2) a quadratic, (3) a cubic, that would strengthen the concept that in all cases the x-intercepts are the roots and that the function goes to zero at that point.
In my personal experience, this lack of a broader emphasis on the patters of mathematics is often due to students’ lack of symbolic understanding and difficulty in manipulating these symbols [(b) and (c) in the comments quoted above]. When the students do not understand the procedures very well, it is pretty hard to emphasize the concepts; it is natural for teachers to try and “fix” the basics first, before talking about the broad principles of algebra. I will write about what I think might be some potential solutions to this conundrum in a future post.