# Everything We Know About Mathematics Achievement is Wrong!

by on October 16, 2009

In his landmark movie, “Annie Hall,” Woody Allen, in the guise of his alter ego, Alvy Singer, posits a theory that everything our parents thought was good for us was, in reality, actually bad. Topping his list was sunshine, milk and college. While the validity of the items listed in this theory could be debated (especially college, considering its current costs), there is no doubt that many of the truths that we have come to accept as “self-evident,” may, in fact, be entirely wrong. Nowhere is this more apparent than in mathematics education.

For example, suppose you were the head of admissions at an elementary school and wanted to select out for admission those students who were most likely succeed in mathematics. What kind of assessments would you administer? It would stand to reason that it would include items like counting, number recognition, and simple arithmetic. After all, if you are proficient in these skills, you should have a head start in your mathematical career.

According to Daniela O’Neill, a Canadian research psychologist at the University of Waterloo, this is not the case at all. O’Neill administered a series of tests to a group of pre-school aged children and then checked their mathematics achievement two years later. She found that the strongest predictor of success in mathematics was a child’s “narrative skills.” That is, the more sophisticated a pre-schooler was able to narrate a story, including switching perspectives in that story, the more likely the child would be excelling in mathematics two years later. What was even more interesting was that this ability did not correlate to later performance in reading, spelling or general knowledge.

For years we’ve assumed that children who showed precocious skills in early mathematical skills were destined to be among our best and brightest mathematics students. O’Neill’s study shows that, like the man searching for his wallet under a streetlight when he lost it in the dark alley, we’ve been looking in all the wrong places when assessing whether a child will succeed in mathematics.

Here’s a second piece of the math puzzle: built into our brain are two numerical systems. The “bestial,” is our approximate sense; that intuitive feeling we have for numbers and quantities. We use this system all the time, whether we’re judging how much time we will need to get to a destination, or how much spaghetti we should take out to feed four people. In contrast, our “celestial” system is responsible for precise computations. It is this system we study for our 12+ years in school, and hopefully right into college. The celestial system is used to calculate the sum of 1 3/8 and 2 3/4 (4 1/8) or the square root of 529 (23.) The bestial is our approximate number system, while the celestial is the one concerned with precision. Which do you suppose correlates most highly with success in elementary and middle school mathematics? If you recall the title of this article, then you know where this is going.

A group of researchers from Johns Hopkins University and the Kennedy Krieger Institute in Baltimore, studied 64 14-year olds to determine their approximate number sense, by asking them to discriminate between ratios of one color dots to another. For example, they would show a picture with 9 yellow dots interspersed with 10 blue dots and ask, “which was more?” Those students who had the more refined approximate number sense also had the stronger scores on a battery of standardized math tests from kindergarten onwards.

While it is too soon to draw a causal relationship one way or another, the study does suggest that children would benefit from more exposure to activities that exercise their approximate number system, as opposed to those that focus on the precise. Unfortunately, many “back to basics” math curricula (and out-of-school math programs like Kumon) are channeling children directly into this mathematical dead end. In theory, both should be leading to higher levels of math achievement. The reality says something different.

Finally, I’d like to examine the effectiveness of test preparation classes. Each year, vast amounts of time and resources are marshaled to help schools and individual students to perform better on the raft of citywide, statewide and nationwide tests that are administered each year. Billions of dollars are spent on test preparation classes and materials, and thousands of hours, which should be used for instruction, are instead diverted to engage in test practice.

Like any other logical idea, the “truthiness” of this practice seems to make sense: the students have a test, therefore, if you practice for the test, the students will do better on the test. Not so fast.

According to a meta-study by the National Association of College Admission Counseling, the average increase for a student taking a test prep class is 30 points on the SAT (out of a possible 1600), and 1 point (out of 36) on the ACT. What is frightening about this study, beyond the fact that it points to a less than a 2% gain in test scores, is that this is among students who are keenly aware of how much they have riding on the outcome of their preparation. What kind of results would we find for a third grader who has little understanding or interest in the results of his/her standardized citywide exams?

The truth is, there is a lot of contradictory evidence about what contributes to mathematics achievement. From these studies, however, I hope you have a better idea of what kinds of questions you should be asking as your child goes through school. As Sy Syms once intoned, “a well informed consumer is our best customer.” To me, a well-informed parent is the best advocate for a math program that is challenging, robust, and, most importantly, incorporates practices that rely on facts, not worn-out assumptions.

About the Contributor: Robert Berkman is a regular contributor to the NYC Firm Schools Blog in the area of mathematics education.

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