An article in the Baltimore Sun this past week: “A Failing Grade for Maryland Math,” highlighted a problem that I believe is not unique to Maryland. The author, Liz Bowie, explained that the math taught in Maryland high schools is deemed insufficient by many colleges. More and more entering college students are required to take remedial math. In many cases incoming college students cannot do basic arithmetic even after passing all the high school math tests.
The article resonated with me because in recent years I’ve witnessed first hand the disconnect between high school and college math curricula. As a parent of three children with current ages 14, 17, and 20, I’ve done my share of tutoring of middle school and high school math. The problems assigned to my children have become progressively more difficult through the years to the point of being bizarre. My wife keeps shaking her head at how parents without my level of math expertise assist their children.
For example, my eighth-grade daughter asked me one evening how to perform “matrix inversions.” This is a technique I teach in a college sophomore-level mathematical methods course for physics majors. Matrix inversion is difficult for me to do off the top of my head. I needed to refresh my memory by referring to a highly advanced math book. Another night my daughter brought home a word problem that was easy for me to do with my advanced knowledge of differential equations but it took me a lot of thought to arrive at an explanation comprehensible to an eighth-grader.
My other daughter struggled through a high-school trigonometry course filled with problems that I might assign to my upper-class physics majors. I certainly wouldn’t assign problems at such a high level to college freshmen. I kept asking her how she was taught to do the problems. I wondered if the teacher knew special techniques unknown to me that made solving them much easier. Alas no such techniques ever materialized. The problems were as difficult as I judged. At least I could solve the problems, a feat the teacher couldn’t manage in a number of cases.
At the same time I work the summer orientation sessions at Loyola College registering incoming freshmen for classes. Time and again students cannot pass the placement exam for college calculus. Many students cannot pass the exam for pre-calculus and that saddles them with a non-credit remedial math course. Without the ability to take college-level math the choices students have for majors are severely limited. No college-level math course means not majoring in any of the sciences, engineering, computer, business, or social science programs.
A colleague in the engineering department complained to me that many students who wanted to major in engineering could not place into calculus. The engineering program is structured so that no calculus means no physics freshmen year and no physics means no engineering courses until it’s too late to complete the program in four years. For all practical purposes readiness for calculus as an entering freshmen determines choice of major and career. The math placement test given to incoming freshmen at orientation has much higher stakes than any test given in high school. But, the placement test has no course grade or teacher evaluation associated with it. No one but the student has any responsibility for its outcome.
So if eighth graders are taught math at the level of a college sophomore why are graduating seniors struggling? From my knowledge of both curricula I see three problems.
1. Confusing difficulty with rigor. It appears to me that the creators of the grade school math curricula believe that “rigor” means pushing students to do ever more difficult problems at a younger age. It’s like teaching difficult concerti to novice musicians before they master the basics of their instruments. Rigor—defined by the dictionary in the context of mathematics as a “scrupulous or inflexible accuracy”—is best obtained by learning age-appropriate concepts and techniques. Attempting difficult problems without the proper foundation is actually an impediment to developing rigor.
2. Mistaking process for understanding. Just because a student can perform a technique that solves a difficult problem doesn’t mean that he or she understands the problem. There is a delightful story recounted by Nobel-prize winning physicist Richard Feynman in his book Surely You're Joking, Mr.Feynman!: Adventures of a Curious Character
about an arithmetic competition between him and an abacus salesman. (The incident happened in the 1950’s before the invention of calculators.) Here is the link to the full text of the story.
Feynman and the abacus salesman competed on who could do arithmetic faster. Feynman lost when the problems were simple addition. But he was very competitive at multiplication and won easily at the apparently impossible task of finding a cubed root. The salesman was totally bewildered by the outcome. How can Feynman have a comparative advantage at hard problems when he lags far behind at the easy ones? But when Feynman tried to explain his techniques he discovered the salesman had no understanding of arithmetic. All he does is move beads on an abacus. It was not possible for Feynman to teach the salesman additional mathematics because despite appearances he understood absolutely nothing.
This is the problem with teaching eight-graders techniques such as matrix inversion. The arithmetic steps can be memorized but it will be a long time, if ever, before the concept and motivation for the process is understood. That raises the question of what exactly is being accomplished with such a curricula? Learning techniques without understanding them does no good in preparing students for college. At the college level emphasis is on understanding, not memorization and computation prowess.
3. Teaching concepts that are developmentally inappropriate. Teaching advanced algebra in middle school pushes concepts on students that are beyond normal development at that age. Walking is not taught to six-month olds and reading is not taught to two-year olds because children are not developmentally ready at those ages for those skills. It is very difficult to short-cut development. All teachers dream of arriving at a crystal clear explanation of a concept that will cause an immediate “aha” moment for the student. But those flashes of insight cannot happen until the student is developmentally ready. Because math involves knowledge, skill and understanding of symbolic representations for abstract concepts it is extremely difficult to short cut development.
When I tutored my other daughter in seventh grade algebra, in her words she “found it creepy” that I knew how to do every single problem in her rather large textbook. When I related the remark to a fellow physicist he said: “But its algebra. There are only three or four things you have to know.” Yes, but it took me years of development before I understood there were only a few things you had to know to do algebra. I can’t tell my seventh grader or anyone else without the proper developmental background the few things you have to know for algebra and send them off to do every problem in the book.
All three of these problems are the result of the adult obsession with testing and the need to show year-to-year improvement in test scores. Age-appropriate development and understanding of mathematical concepts does not advance at a rate fast enough to please test-obsessed lawmakers. But adults using test scores to reward or punish other adults are doing a disservice to the children they claim to be helping.
It does not matter the exact age that you learned to walk. What matters is that you learned to walk at a developmentally appropriate time. To do my job as a physicist I need to know matrix inversion. It didn’t hurt my career that I learned that technique in college rather than in eighth grade. What mattered was that I understood enough about math when I got to college that I could take calculus. Memorizing a long list of advanced techniques to appease test scorers does not constitute an understanding.
Joseph Ganem is a physicist and author of the award-winning The Two Headed Quarter: How to See Through Deceptive Numbers and Save Money on Everything You Buy