By Barry Garelick
Peg Tyre, author of “The Good School” and “The Trouble With Boys”, is well regarded by many, including those who write about education. But I believe she missed the point in her latest article in The Atlantic. The story explores why there has been a surge in the number of teenagers who have excelled in advanced math topics as evidenced by increasing numbers of award winners in prestigious math competitions such as the International Math Olympiad. She describes what those students are doing differently than the students who do not make it to such airy heights. In my opinion, her article amounts to an advertisement that tells parents to get on board or get left behind.
A One-sided View of the Extracurricular Schools: Where Conceptual Understanding Rules the Roost
Tyre focuses on the extracurricular schools that these students attend, one of which is the Russian School. Students start early in such schools—as early as first or second grade—and continue on. (Other venues include the website “Art of Problem Solving” which she mentions.) In discussing the Russian School, Tyre talks with the founder of the school about his experiences with his kids when they were in second grade in Newton, Massachusetts in 1997.
“I’d look over their homework, and what I was seeing, it didn’t look like they were being taught math,” recalls Rifkin, who speaks emphatically, with a heavy Russian accent. “I’d say to my children, ‘Forget the rules! Just think!’ And they’d say, ‘That’s not how they teach it here. That’s not what the teacher wants us to do.’ ” That year, she and Irina Khavinson, a gifted math teacher she knew, founded the Russian School around her dining-room table.
Although this makes for a compelling story, we really don’t know what his kids were being taught. We only know that they were being taught to do something step by step, and Rifkin admonishes them to forget the rules and just think. The average reader, having been told by countless newspaper articles that depict traditionally-taught math as rote memorization with no understanding, may assume that this is what was happening. Many such readers have therefore been led to believe that conceptualizing mathematical problems is something you can do with little instruction or foundational skills and memorization— a pervasive theme of the article that Tyre, as a highly skilled writer, manages to keep understated.
She accomplishes this by first acknowledging that fluency is important and also by describing the prevailing disputes about how best to teach math:
“Fiery battles have been waged for decades over what gets taught, in what order, why, and how. Broadly speaking, there have been two opposing camps. On one side are those who favor conceptual knowledge—understanding how math relates to the world—over rote memorization and what they call “drill and kill.” (Some well-respected math-instruction gurus say that memorizing anything in math is counterproductive and stifles the love of learning.) On the other side are those who say memorization of multiplication tables and the like is necessary for efficient computation. They say teaching students the rules and procedures that govern math forms the bedrock of good instruction and sophisticated mathematical thinking. They bristle at the phrase drill and kill and prefer to call it simply ‘practice’.”
It would have been informative if she had identified the “well-respected math-instruction gurus” who claim that memorization of procedures as well as addition and multiplication facts obscures student understanding. It would also have been extremely valuable and instructive if she had asked the opinions of the heads of these various extracurricular schools where they stood on the matter. As the paragraph stands, however, it is left to the reader to decide which one is correct. In the context of the rest of the article, the reader is led to believe that in the supposed dichotomy between procedure and understanding, understanding always should take precedence. (She makes this point explicitly in her book “The Good School” in the chapter on the teaching of mathematics.)
And the Other Cherished Concept: Problem Solving
Tyre talks also of “problem solving” which, like conceptual understanding, she casts as something that can be taught independent of foundational skills. Her disdain for such skills is embodied in the following statement:
“Sitting in a regular ninth-grade algebra class versus observing a middle-school problem-solving class is like watching kids get lectured on the basics of musical notation versus hearing them sing an aria from Tosca.”
Problem solving is much more complicated than that. One doesn’t learn to sing an aria from Tosca by doing just that. It is based on years and years of training in basic vocal skills. The majority of music lessons are about skills. Musicality is built up from mastery of the basics — and it is the same thing with math problem solving. Students are given instruction in solving basic types of problems such as distance/rate, mixture, work, number, coins, etc. Though frequently derided by reformers as not motivating students to solve them, nor giving any significant problem solving skills, these type of problems are an essential starting point. From there, students are given variants of the problems—well scaffolded so that they ramp up in difficulty—and eventually graduate to non-routine problems.
But for many educational experts and their followers, foundational skills play a minor role in learning how to solve problems. In their view, problem solving is taught by giving students open-ended, multi-answer problems and an insistence on different approaches. There is a belief that continued exposure to difficult problems for which students have had little or no prior knowledge builds up a problem solving schema, and provides them the motivation to learn what is needed to solve such problems on a just-in-time basis.
But math problems are not necessarily useful just because they may require outside-of-the-box insight and/or inspiration. Nor are they likely to result in a problem-solving “habit of mind.” Tyre falls prey to this fantasy, though it is highly unlikely that the Russian School and other schools operate in such manner.
The Real Message: Let’s Help the Gifted
Tyre’s solution to delivering better math education to more students is to get more students into these special schools—by better identifying gifted students and attending to their needs, particularly in low-income areas. She argues that No Child Left Behind, in fulfilling the noble goal of providing help to struggling learners, did so at the expense of those who could benefit from accelerated learning.
What this argument leaves out is that the programs in place to teach math in the lower grades have over the past two and a half decades slowly and steadily been influenced by reform-math agendas and do a poor job of teaching basic facts and skills despite the increased focus on helping the weaker performers. What Tyre proposes is to expand the number of students who go to such special schools to include those from low-income families. But this “let them eat cake” solution fails to help those low-income students who remain stuck in poorly taught math programs because they were “ungifted”.
Furthermore, it is not obvious how one would select gifted kids in the earliest grades for math potential. Math is not an all or nothing proposition. Success breeds interest and love, and success in the earliest stages of anything has more to do with the mechanics than some deep understanding or analysis. Finding and separating these kids is not the role of educators. They should not attempt to separate those with promise versus those who just work hard.
Tyre concludes with:
“Perhaps the moment is right for members of the advanced-math community, who have been so successful in developing young math minds, to step in and show more educators how it could be done.”
I would agree that perhaps the techniques used in these schools could be extremely effective – particularly the mastery of basic skills and facts which serve as the foundation on which students can build conceptual knowledge and problem solving skills. If they offer a proper (STEM-level) curriculum in class, push a little bit, and then provide advanced opportunities after school (in areas like math, robotics, poetry, etc.), then students will get what they need—thus fulfilling what she calls “a noble goal”. Kids in a STEM level curriculum can join the after-school programs and be successful later on. AMC math and the International Olympiad are competitions, not curriculum. The teacher should be the mentor and the one driving (and pushing) the learning process with a group of equal-level students. This should handle all required top-level learning. After-school should only be for “extra.”
If, however, the after-school program provides properly taught lower level instruction that students should have received in the first place, then it’s just a divergent form of tracking. Given the direction math education has been going for the past twenty five years, it is not hard to guess how Tyre’s article will be interpreted.
Barry Garelick has written extensively about math education in various publications including The Atlantic, Education Next, Educational Leadership, and Education News. He recently retired from the U.S. EPA and is teaching middle and high school math in California. He has written a book about his experiences as a long-term substitute in a high school and middle school in California: “Confessions of a 21st Century Math Teacher.”