The Math Wars:
Asking the Wrong Questions
Having taught at two countries and at multiple levels (elementary to university) I have seen that Math is a universal problem. Many countries struggle with how to teach math, and even those that do well do not argue that it is easy to achieve. Therefore, the conclusion must be that it is not a cultural issue, as many in the US like to believe, although culture and values can lessen or strengthen the problem.
The Math Wars got it right, though neither camp is right. To distill the approaches, we can say that one camp, supported by education researchers, favors discovery mode, where kids discover solution approaches by themselves and learn to explore math, and at its strongest approach relegating algorithms (solution methods) non-existent. The other camp, supported by mathematicians, and sometimes confused with “traditional math”, favors a strong technical foundation. While the first camp has been winning on the curriculum front and most school district in the US are teaching variants of the “New Math” (Discovery Math, etc.), the US test scores on the international test (TIMSS – see www.timss.org) have fell, and as students grow they do worse on the test, lending support to the “algorithmic” camp. The arguments on each side are complex at times and have devolved into what the title implies: a war. It is no longer a discussion but more of a shouting match.
As a mathematician I KNOW that the discovery type of curriculum is bad for kids but at the same time, as a long time teacher, I KNOW that rote learning of algorithms is not enough either. The problem is that each side is answering a different question, and both are the wrong questions!
The “discover/new math” camp asks, “How do kids learn and how can we facilitate such learning?” It is a question that is natural for education researcher but it is the wrong question since ALL of us have flaws in the way we learn. While some of us are better at it than others, without guidance most if not all would end up with some faulty problem solving approaches. As millennia of experience show, learning how to think SYSTMATICALLY requires long and rigorous training. As I have constantly seen with my college students, in some of the top universities in the US, the education system fails in teaching them how to systematically analyze, since it is not taught AT ALL. We can think of the question posed using my favorite metaphor of sports. A coach would not ask, “How do kids shoot and how can we facilitate such?” because the coach knows most kids do not naturally end up with the right method of shooting. While it is a critical and important question to ask, it is not the RIGHT and main question, but rather a supporting question.
The “algorithmic” camp asks, “How do we make sure kids know how to solve the math problems they study in school and college?” While in terms of ensuring performance in Math it is a better question than the “discovery” one, it is also the wrong question. As studies across nations have shown, technical skills are important for understanding math concepts. But, technical skills, while they might ensure good scores on standardized tests, do not ensure deep understanding of concepts. However, it is the wrong question because it focuses on Math skills, and not thinking and learning. Having seen thousands of college students who mostly got high grades in their studies, Math included, I have learned that their general analysis capabilities and skills, as well as understanding of basic mathematical concepts, are severally lacking at best.
To reach the right question we have to examine what the objectives of education (the “thinking” part of it), and not just Math education are, or should be. There are numerous important objectives that have to do with how students approach learning and specific topics (sense of wonder and joy, curiosity, etc.) and even broader ones that has to do with their social and personal life. But to distill the issue at the center of the Math Wars I focus on the intellectual/mental elements only.
There are three elements that education should strive to achieve in students:
Ability to learn.analysis, thinking, and problem solving abilities.specific knowledge and skills.ability to apply the second to the third.
The first objective is general to all topics and subjects and provides students with the fundamental capabilities they need to approach any domain of knowledge, be it Math or Literature. The second is obvious and arrives from the need to master certain knowledge and skills, although which ones can be argued on. The third is but a merger of the first two, but it is not necessarily trivial.
It is clear that the “discovery” camp tries to answer a question related to the first objective, believing that answering the “how kids learn?” question leads to achieving that first objective, which in turn results in achieving objectives 2-4, when used in a specific domain, such as Math. The problem is that facilitating students learning does not necessarily (and that word is critical) results in students achieving even #1. They may, or they might not learn how to learn. Further more, even if #1 is achieved, #2, #3, and #4 might not be, since the link is not automatic.
It is also clear that the “algorithmic” camp focuses on the third objective. As a result, while this approach would result in higher Math capabilities, it would not guide students to develop to their full potential, and might kill their internal curiosity and sense of joy in learning, due to over emphasis on rote and drills. It would also not teach them how to learn, or systematically analyze and solve problem, not just in Math.
It is important to note that when students know how to learn, and know how to approach problems, they enjoy the specific domain in which they have those capabilities. This is why those who are good at Math tend to enjoy it, and why those who are good at sports tend to enjoy sports. Naturally, enjoying an activity creates a strong incentive to learn and practice it, thereby creating a positive cycle. But it is the ability that drives joy, or can kill it if absent.
The right question is therefore simple:
How do we achieve all four objectives?