In this series we are looking at ways that math education can be reformed (Part 1 here). In contrast to some other math reform efforts, we are not trying to fundamentally rewrite what math education is doing but to simply admit that we can do better and see where that takes us. This article, Part 2, will concentrate on improving math education by better identifying where students are when we encounter them.
Mathematics curriculum is generally developed with a goal of “fitting it all in.” That is, educators assume that people learn at a relatively fixed pace. They then pace the lessons so as to fit all of them into the curriculum in the right amount of time.
However, this approach has almost no relationship with how students actually develop in mathematical ability. Starting math too early means that a lot of them will be doing math before they are physically or cognitively ready. This is what causes them to feel that they are “bad at math.” They might be perfectly fine at math in a few years. However, because they weren’t good at math as early as other students, they get permanently labeled (and worse, permanently label themselves) as “bad at math.”
The fact is that before nine years of age, there is no reason to get serious about math. It should all be games and common sense. Developing number sense is important at this time. Looking at basic math facts is good. Being tested on them is not. Under nines just aren’t mentally ready for that. Even if a child “gets ahead” at this stage, it isn’t much benefit, nor is it a problem to be “behind.”
Once children are cognitively ready, math facts will make sense to them almost effortlessly. Before that, you might as well not try. Many generations have seen math as a struggle because we worked too hard to get children who weren’t ready to do math to proceed at the pace we wanted. Learning to read is the goal for this age. Math is secondary.
From age 9 to 12, the focus should be hard-core arithmetic. This is where students need to learn the math facts and learn the rules for addition, subtraction, multiplication, division, etc. Any ground that was “lost” during the previous years can be easily made up here. This is an age where memorizing and following instructions comes easily. It is not an age at which the student should be asked to worry too much about the “why”—it’s about the “what.”
Common Core math oftentimes overemphasizes the “why” aspect of math, creating confusing systems for doing basic arithmetic. Instead, the focus should be on learning an algorithm that is straightforward to follow and leads straight to the answer.
After the age of 12, students are more capable of dealing in abstract ideas. And they will be able to do so more easily if they have the arithmetic down. Lack of proper grounding in arithmetic will bog down the performance of the more abstract skills like algebra and geometry. Every algebra teacher I’ve ever met will tell you that instant recall of math facts is the best predictor of algebra success. If students are to engage their newly-acquired abstract reasoning skills, they can’t be bogged down in puzzles about the arithmetic they should have learned in middle school.
Another common problem students have is dyslexia. Dyslexia affects nearly 20% of students and it can be a major hindrance to learning mathematics. Most mathematics texts aren’t written with dyslexics in mind. That needs to change. Fortunately, some very simple changes can help.
First, differentiate numbers clearly. Always strike through a zero, put a middle line on the seven (to distinguish it from a 1), put a middle line on a 5 (to distinguish it from a 2) and make sure the 9 and the 6 are not upside-down versions of each other.
In algebra, don’t use variables that can be confused with each other in a problem. For instance, don’t have both “m” and “n” as variables in the same problem or “p” ,”b”, and “d” all occur together. These simple changes can dramatically affect the success of a fifth of the students!
Next, recognize that many students simply get stuck somewhere. Math tends to be cumulative, so getting stuck at one point usually means that they won’t be able to move further. When I was in high school, I tutored a kid who was in “developmental” (i.e., well below grade level) mathematics. I spent a total of four hours helping him out. It turned out, he just needed some one-on-one help with fractions. Once he got it, he moved up to the advanced class within a few years.
Finally, sometimes math problems are generational. With large class sizes stretching the teacher’s attention, the person most likely to give a student individual help is a parent. If parents have qualms about doing or teaching math, that attitude can rub off on the students. Having resources available to help parents help their kids would also go a long way to helping students succeed in math.
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Part 1: How can we really fix the way math is taught? First, we must understand why we teach math in the first place. Math teaches students how to think more clearly in all areas of life but it mostly performs this function silently, invisibly. (Jonathan Bartlett)
Part 3: Helping students see how math benefits them in the long run. To keep them motivated, we need to answer the “Why bother?” question honestly and directly. Most mathematics topics teach a specific logical skill that will help students solve problems on any career path.
Part 4: To fix math education, see it as a program that needs an update. As a computer programmer, I’ve seen this problem in my work: The basic idea is still sound but “fixes” have made it too complex. We need to “refactor” math education, the way programmers must sometimes refactor old code that still works in principle but needs simplifying.
Bartlett’s calculus paper reviewed in mathematics magazine. The paper offers fixes for long-standing flaws in the teaching of elementary calculus.