In theory, today’s Constructivist classrooms, inspired as they are by educational progressivism, are supposed to favor child-centered discovery learning. And yet, in many ways, they are less child-centered than ever.
We see this, for example, in Reform Math and Common Core-inspired math classes. Here, children are told not just to solve problems, but how to solve them. And they are often required to solve them several times over in multiple ways. Typically, the standard algorithm is only one of several options, and preferred options are things like counting forwards or backwards from “landmark numbers,” “splitting” numbers via “number bonds,” repeatedly adding, repeatedly subtracting, or (if you’re lucky enough to be using Everyday Math) multiplying via “lattices”. Not to mention explaining, verbosely, how you did what you did and why.
A more child-centered, discovery-oriented approach to math problems—indeed, a more “authentic” and “organic” approach—would be to let the problems themselves, rather than the verbal directions, suggest one particular type of strategy or another.
999 + 77 vs. 956 + 77
The first of these invites a “landmark” numbers approach: solve it by converting it to a similar problem involving 1000 instead of 999. The second problem is much more rapidly solved using the standard addition algorithm, left to right, with regrouping.
1000 – 7 vs. 956 – 77
The first invites a counting back approach (count back 7 from 1000); the second is much more rapidly solved using the standard subtraction algorithm with borrowing.
9 × 1004 vs. 9 × 1234
The first invites a “splitting”/distributive approach (split 1004 into 1000 and 4 and multiple each by 9 separately, then add). The second is much more rapidly solved via that standard algorithm for multiplication.
8032 ÷ 8 vs. 8032 ÷ 7
The first of these invites a “splitting”/distributive approach (split 8032 into 8000 and 32 and divide each by 8 separately, then add). The first is much more rapidly solved vis that standard algorithm for division.
Offering a large number of problems that invite different strategies is the approach taken by so-called tyrannical, teacher and textbook -centered traditional math. Here, it’s much less common for kids to be told how to solve particular problems. Yes, the standard algorithms are “privileged” as the most efficient ways to solve most problems. But students weren’t generally forced to use these—so long as they solved the problems correctly.
But the problems themselves were different. There were many more of them than what kids get today; they involved more digits and fewer “friendly” numbers. The result: many more problems for which the most efficient strategies were the standard algorithms. Also, calculators weren’t—so to speak—part of the equation. But speed often was. Timed math tests and timed problem sets were frequent. As I discussed in my earlier post:
Many people assume that speed tests (especially multiple choice speed tests) measure only rote knowledge. But they’re also a great way to measure conceptual understanding. Performance speed reflects, not just rote recall, but also efficiency, and efficiency, in turn, is a function of reasoning, strategizing, and number sense.In particular, time pressure will inspire you to use the standard algorithms on problems like 956 + 77, and nonstandard shortcuts (landmarks, splitting, etc.) on problems like 999 + 77.
This brings up another difference between traditional math and Reform Math. Traditional math doesn’t belabor the shortcuts—or, indeed, even teach them. After all, if you have enough timed tests involving problems like 999 + 77, you will figure it out on your own—in the spirit of true, child-centered discovery.
It’s only when you drastically lower the number of problems that you assign, allow calculators, and dispense with speed tests, that you find yourself having to start spoon-feeding students the shortcuts and other ad hoc strategies. (At the expense, of course, of the standard algorithms).