This article talks about a book titled “Systems for Instructional Improvement” coauthored by the dean of the University of Southern California ed school. It is described as “dedicated to improving math instruction in the U.S.”
Why is it that just about every book, article, tweet, and Linked-In polemic that purports to put math education back on track starts from the following assumption:
“For the past 25 years or so, there’s been a growing recognition that students at the middle-school level, in particular, aren’t developing a deep understanding of mathematics,” said Thomas Smith, dean of UC Riverside’s Graduate School of Education. “A big piece of that is because of the way students in the U.S. are taught; current math instruction tends to be highly procedural — as in ‘use these steps to solve these types of problems’ — instead of allowing students to investigate real-life problems and experiment with different types of solution strategies.”
The same catchwords end up in this catechism of edu-wisdom: “Deep understanding of mathematics”. Just how deep of an understanding of math do you want middle school students to have? Even in freshman calculus courses, students learn a short-cut version of what limits and continuity are in order to get to the powerful applications of same; namely derivatives and integrals and the (yes) procedures for taking derivatives and finding anti-derivatives.
Students who go further in math will master the delta-epsilon definitions of limits and continuity, and learn about least upper bounds, greatest lower bounds, limit points, cluster points, open and closed sets and compactness. Such deep understanding of calculus is largely inappropriate at the freshman level–one has to start somewhere in order to go deeper. And so it is the same with middle school math. One starts with procedures and in subsequent math courses, one goes deeper.
And it isn’t as if teachers and textbooks have not provided the conceptual understanding that underlie the procedures that math reformers seem to find so “rote like”. It’s just that students like to know how to do things and as has been stated in various research studies, procedural fluency and understanding often work in tandem. Sometimes conceptual understanding comes first, and sometimes the procedural understanding. But there is a stated resistance to learning the procedure first as if it is handing it to the student and taking the easy way out. Teachers are made to feel guilty when, seeing that their students are unable to solve the problems assigned, take to breaking down the procedures step by step.
Also, the problems that middle school students have had to solve over the years are really not as far-removed from reality as the spin-meisters would have us believe. K-8 math is largely “applied math” and in earlier times was an essential part of everyday life. A glance at the textbooks used in Singapore (the Primary Math series is a good example) has very straightforward problems that allow students to apply the concepts of things like decimal, fraction and percent operations. An example: “Alex spent 1/3 of his pocket money on a toy airplane and 2/3 of the remainder on a toy robot. He had $20 left. How much did he spend altogether?”
Singapore has boasted high scores on international tests for years, but the problems that students solve there may be held in disdain by math reform types. Perhaps they think they are not relevant to students concerns. If students are not given proper instruction on how to solve such problems, and are expected to discover “strategies” for solving, they will tend to ask “When am I ever going to use this in real life?” If given proper instruction with scaffolded problems that are variations on the initial worked example, students generally will tackle such problems. The “When will I ever use this” question is generally an expression of frustration.
Math reformers want kids to use elaborate techniques to get simple answers and call that “understanding”. Or even “deep understanding”. What they think is understanding is visualization which is not what mathematicians mean by understanding.
But the standard talking points that pervade books about education continue. The article ends with this chestnut:
“When it comes to teaching, people are often wedded to the ways from which they learned,” Smith added. “But the reality is those ways don’t always help all students learn. Our goal is to support teachers and coaches, as well as school and district leaders, to improve teaching and learning for all students.”
Interestingly, over the past 30 years, math has been increasingly taught in the manner these reformers want to see it done (in the lower grades primarily). If you point this out to them, they will generally say one of two statements: “No, that isn’t true, but I wish it were” or “They aren’t doing reform correctly.”
In the meantime, many books like “Systems for Instructional Improvement” have been written and not much has changed. Oh, right. It’s because they aren’t implementing the recommendations. Or they’re doing them wrong. Or (insert your own excuse here _________________).