Answer by Alon Amit, Ph.D. in mathematics, math circler:
This is (obviously, I hope) a complex and controversial question. It is also geographically diverse: Whatever problems exist in math education in the U.S. are very different from those in India, Mexico, or Mongolia. I'll focus on the U.S.
My understanding is based on the following ingredients:
- I have been regularly teaching in math circles around the San Francisco Bay Area for the past five years, and through that I've gathered some knowledge of what those children who are really interested in math know and what their unfulfilled needs are.
- I have been regularly speaking at math circles for teachers, where I've worked with middle-school and high-school math teachers who are interested in expanding their understanding of mathematics. I know something about what teachers tend to know and not know.
- I have spent my career working with software engineers, data scientists, economists, and other "knowledge workers" who rely, to varying degrees, on mathematical training and math-like problem-solving skills. (I've been a researcher and engineer myself before succumbing to the irresistible allure of product management.)
- I have a Ph.D. in math, so hopefully I know a little about what mathematics is.
- I've held official positions at the Mathematical Association of America (the Northern California, Nevada, Hawaii and section) and have had some exposure to the challenges facing an organization dedicated to exciting students about mathematics.
- I have two children in elementary school in the U.S. (and another little waif who isn't quite there yet).
- I've read lots about the issue. I care.
Those are my credentials, such as they are. I don't purport to understand the issues completely or even reasonably well; the following are my personal opinions based on my experience and observations.
First, there are two meta-problems: 1) The lack of consensus around the goals of K-12 math education. Many people have strong opinions, but they are often at odds with one another. 2) The lack of consensus around the proper way to define and measure the success of the math education system. Not everyone agrees that there is a problem, and among those who agree, many disagree on its specific form.
Beyond these two higher-level issues, I think the following are true as well:
- The vast majority of the people who teach mathematics in schools know very, very little mathematics. They are not to blame—many are intelligent, caring, wonderful people—but they severely lack training and skills. This includes not just exposure to higher mathematical content; for the most part, they're not even aware what mathematical problem solving is, let alone how to go about solving problems.
- People who do have the skills and even the passion for teaching math are not often excited about the career prospects of being a math teacher. Compensation is part of this; public perception and prestige is another. And I am aware of at least anecdotal evidence that the arduous certification process is not helping either.
- The textbooks are horrendous. They are massive, confusing, uninspiring, incredibly inefficient, and stupefyingly boring. I lack the expressive skills to describe how awful and misguided they are.
- The recent adoption of the Common Core standards is actually a positive and promising move, but as of this writing, both the teachers and the textbooks are unprepared to actually teach students to those guidelines.
As a result, many children are led to hate the subject and lose all confidence in their ability to excel at it. This is an oft repeated cliché but it is, unfortunately, true.
Despite all of that, some people do become software engineers or physicists or mathematicians, having developed a taste for mathematical thinking, acquired problem-solving skills, and dove into deeper training. This is often done outside of the school system, but not entirely—there are certainly some great teachers and some useful resources within schools. Not all mathematical teaching in schools is broken; but much of it is, and many naturally talented and curious minds are turned off by the broken parts and face an uphill battle as they seek to nurture their talents and interests.
Those are some of the problems as I see them. How to solve them is an even harder question (but I do have a few opinions on that, too).
* * *
Answer by Neil Aggarwal, lawyer:
I was a seventh- and eighth-grade math teacher. From my experience, it seems that the basic problem with math ed is the lack of focus on the fundamentals.
Let me give you an example: Most teachers taught 2(x+y) using dolphins. They would draw two dolphins traveling from the 2 to the x and the y. This was meant to indicate that we distribute the 2 to get 2x + 2y.
Teaching this way allows kids to answer that specific construction of problem, integer(variable + variable) = integer * variable + integer * variable, but not much else. What happens when the kid sees (x+y)2 or (x+2)(y+2)? The dolphins can only allow a child to guess at what to do in these new circumstances.
Rather, if the teacher had taught that 2(x+y) = (x+y) + (x+y), therefore giving the students some insight as to how and why the distributive property works, then kids might be more able to approach new circumstances and prevail by force of logic rather than speculation.
A student facing a new construct—say (x+2)(y+2)—has a shot at realizing that the (x+2) can be seen as its own number, thus (x+2)y + (x+2)2. The kid didn't have a shot with the dolphins.
Moreover, this drilling approach that is commonplace in schools is very inefficient. Think about it: The above two examples take up the better part of two quarters over the course of two years in most schools using the drilling method. Is that really necessary? And imagine you are the student. How boring! Three to four months of what?
The drilling approach requires that the teacher drill almost every new circumstance with the same ferocity as the first circumstance. Rather, when teachers focus on fundamentals, new circumstances still need to be taught, but usually not drilled to the same extent.
Perhaps the worst consequence of the drilling paradigm is that students and adults have no ability to use the math they learned in school outside of the boxes within which they were drilled. The value of math is in its predictive powers. You combine math with economics, or math with biology, or math with physics, even with law (see Coase) etc., and suddenly you can predict the future. But these uses of math require extremely strong fundamentals, which most people were never taught.
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