What’s that you say? You’re sick of all those long-winded education rants that never go anywhere? Me too! However, I can’t pass up this opportunity to commend the California Commission on Teacher Credentialing for so succinctly summarizing everything that’s wrong with elementary-school-level mathematics education today. And in an exam that every prospective teacher in the state is required to write, to boot – now that’s efficient delivery!
Many people believe children will never learn mathematics if allowed to use pocket calculators. Having spent countless hours memorizing multiplication tables and doing long-division problems unaided by any mechanical device, many adults cannot conceive of anyone acquiring this knowledge without similar effort and practice. ______________.
What many people fail to understand is that mathematics is constantly evolving; it is not a fixed body of facts. Students must still learn basic skills, but they do not need to perform the endlessly repetitive exercises that calculators largely eliminate. Youngsters can better use their time—time they would have spent performing long-division problems—to learn mathematical concepts that will enable them to become better problem solvers.
Which sentence, if inserted into the blank line, would best focus attention on the main idea of the passage?
(A) It is true that mathematics is not the easiest subject in the typical elementary school curriculum.
(B) Many of you have doubtless heard about the bitter classroom experiences of students who learned mathematics this way.
(C) There is much to be said for instilling this kind of discipline in students.
(D) Although it was clearly not fun, students trained in this manner rarely forgot what they had learned.
(E) Such views, however, seem to reflect a resistance to change rather than a rational approach to mathematics instruction.
(F) Contrast this instance of common sense with the hallucination that follows.
Just kidding! (F) isn’t an option. The correct answer is (E).
Yes, yes: fish in a barrel are targets for amateur marksmen, I know. But what we have here is a barrelful of tranquilized guppies that been given free reign over early mathematics education, so let’s have a go at it before any more metaphors get mixed:
What many people fail to understand is that mathematics is constantly evolving; it is not a fixed body of facts.
Yes, if there’s one field that has evolved beyond recognition in the past quarter-century, it’s introductory calculus. Why, back when I was a tot, the area of a rectangle was length plus width, the derivative of sin x was 5, and we only had whole numbers, so whenever we needed to compute the area of a circle we had to use pi=3, AND YOU NEVER HEARD US COMPLAIN.
– to be hysterically funny, because it was so over-the-top. Except that…it wasn’t. It was very much beneath the top. It was smack dab in the middle of what those enlightened educators, unlike “many people”, have succeeded in understanding.
Students must still learn basic skills, but they do not need to perform the endlessly repetitive exercises that calculators largely eliminate.
They don’t? Really? I’m not convinced. Let’s see what a cognitive scientist has to say about this!
It is difficult to overstate the value of practice. For a new skill to become automatic or for new knowledge to become long-lasting, sustained practice, beyond the point of mastery, is necessary.
… By sustained practice I mean regular, ongoing review or use of the target material (e.g., regularly using new calculating skills to solve increasingly more complex math problems, reflecting on recently-learned historical material as one studies a subsequent history unit, taking regular quizzes or tests that draw on material learned earlier in the year). This kind of practice past the point of mastery is necessary to meet any of these three important goals of instruction: acquiring facts and knowledge, learning skills, or becoming an expert.
But what does that guy know, anyway? The times (tables), they are a-changin’! Mathematics is evolving! Get with the program!
I’m curious about something, though: is this anti-repetition view unique to mathematics education? Have the musicians among my readers, for instance, noticed a similar trend in music pedagogy? Music is EVOLVING! It is not a FIXED BODY OF FACTS! We have new-fangled technomology that enables students to bypass all that boring stuff, like learning scales! Or…do music students still practice scales, even though scales really aren’t that much fun to practice? I know that when I teach pottery, I spend a fair bit of time focusing on the basic skill of centering the lump of clay, even though it’s more rewarding to throw teapots.
Youngsters can better use their time—time they would have spent performing long-division problems—to learn mathematical concepts that will enable them to become better problem solvers.
I’ve spent countless minutes (that seemed like hours) waiting for students I’ve tutored to multiply 4 times 5 or 3 times 6. Oddly enough, they weren’t adept at understanding mathematical concepts or solving problems.
The idea that students who don’t waste their time learning the boring basics will have freed up valuable time and brain space to become creative problem-solvers is one I’ve seen cited often. It’s an idea that every single mathematics educator with whom I have ever communicated – and I have communicated with hundreds, across several continents, both in person and through this blog – considers to be utter hogwash.
But it’s possible that we’re just outliers. It’s possible that on the whole, students who don’t learn their times tables beyond mastery are brighter, bolder, and more creative problem solvers than their predecessors. And if that’s the case, then there should be plenty of statistics to bear that out. And surely the proponents of calculator-based curricula have plenty of rigorous statistical studies at their fingertips.
Let’s see them. Put up or shut up.
Show me the data.
Show me a single peer-reviewed study that indicates that students who were raised with calculators show greater facility and more creativity in problem-solving than those who were raised without. I want to see test scores. I want to see comparisons of performances in university-level mathematics classes between students who were made to memorize and practice their times tables, and those who weren’t. I want to see evidence that learning the basics and learning advanced mathematics are negatively correlated.
Until then, I’m going to rationally resist change, and teach mathematics in the way that my peers and I actually learned it.