If you’re anything like me, math instruction up until middle school is a haze, an impenetrable fog. I couldn’t tell you what we did. I think there was long division in fourth grade, maybe multiplication tables in second. I remember a teacher telling me five minus 10 equaled zero. The instruction was all very diffuse and general, skipping from basic geometry to reading an analogue clock, as if they were rushing to prepare me for the time, post-5th grade, when I would leave school to work on the homestead. There was no overarching principle. The very divisive Common Core Standards — say what you will about them — are making an attempt to concentrate early education and drive it towards some goal. For math, that goal is algebra. But they can still do better.
As researchers have noted, algebra often “marks more or less the end of [students’] mathematical growth” as instruction fails to communicate to students that algebra isn’t just applying “memorized procedures” of symbolic manipulation, but is a way of thinking about problems. To remedy this, researchers advocate introducing letters earlier in math problems.
This is a simple solution, but it’s not being followed by the Common Core. Looking at the 1st grade math standards, one might indeed get the impression that schools are already taking the advice. Solving word problems with “equations with a symbol for the unknown number” sounds promising. But take a look at the table where they give an example of this, and it’s not right at all. “2 + 2 = ?” is not introducing a letter in math problems. It’s a glorified mathematical expression, as reprehensible as the scores of worksheets with fill-in-the-answer arithmetic problems. We need to introduce the concept of the mathematical variable as quickly as possible in American education. Replace all those “2 + 2 = ?” problems with “2 + 2 = C” problems, and then instead of asking what the answer is, we can ask what “C” is. Adding that extra level of abstraction is critical to laying the groundwork for developing the sort of abstract reasoning skills that algebra heavily tests. It’s important we introduce young students to the concept of a variable, even if not rigorously, at the start of their education so they don’t get blindsided by algebraic concepts.
The researchers also advocate broadening kids’ “definition of the equal sign,” meaning they should “be trained to view an equal sign (=) as balancing an equation, not as a command to produce an answer.” This poignant idea uncovers probably the most nefarious byproduct of using expressions or “?” equations to teach students math: it commits a subtle semantical error. Asking children to fill in the spot after the equal sign suggests that by doing the arithmetic, one is actually creating the answer to the problem. The equal sign in this case is an operator, yet this isn’t true in the slightest. An equal sign is just an expression of equality between two sides. The answer already exists, and by doing the arithmetic one simply uncovers it. Understanding that the equal sign is a fluid, non-operating symbol — which was probably the hardest part of pre-algebra for students — is absolutely crucial to early algebra, where one adds, subtracts, multiplies and divides each side to solve for a variable. Thus, placing young students under the impression that the equal sign is doing something in an equation besides expressing equality is directly undermining their ability to adapt to algebra when the time comes.
We’re really wasting an opportunity here to jump-start kids’ mathematical understanding. Solving expressions is an inefficient teaching-method because we could easily introduce the variable, and directly harmful because it incorrectly uses the equal sign. I’m not saying this would make us all prodigies, nor that it would completely fix America’s math curriculum (I’m still not completely sure what Algebra II is, for instance), but it would definitely help. And maybe it would bump us up a few spots in the international math and science rankings. I think we need it.
Brennan Edel is an Opinion columnist for The Cavalier Daily. He can be reached at b.edel@cavalierdaily.com.
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