Short-Circuiting the Understanding

When my son was very young, about 3 or 4 years old, he liked to bring me the Harbor Freight sales ad from the newspaper and sit in my lap. He would point at pictures of different tools and say their names. He could recognize compressors, table saws, drills, chainsaws, and many other tools. I figured since he knew so many tools, it was time to put him to work.

So I bought him a chainsaw.

We went out to the backyard, where I filled the gas tank, put in the oil, and started the chainsaw up for him. As you can imagine, it didn’t go well.

Are you kidding? There’s no way I or any other responsible parent would do that.

But do we actually teach mathematics in a similar way?

There are many first grade teachers out there who regularly think things like: “All my students coming in from kindergarten can write all of their numbers (symbols) and count from 1 through 20, so they’re ready for the algorithm.”


Kindergarten Worksheet
First Grade Worksheet















Turning them loose with the algorithm, without conceptual understanding, is the mathematical equivalent to turning a 4-year-old child loose with a chainsaw. Both have the potential to cause long-term damage. They may memorize their addition facts and demonstrate what is often called “fluency” (really just speed with correct answers), but they aren’t doing math; they are parroting. The math has been short-circuited.

On the other hand, there is a conduit that promotes understanding and helps students construct mathematics. Ernst von Glasersfeld (1991) called it “re-presentation.” He defined mental re-presentation as “the re-generation of a prior experience” (p.4). The hyphen (-) was included in the word to differentiate between “representation” and “re-presentation.” For example, if I say the word “enchilada,” we all bring up different images in our minds that are based upon our own experiences with enchiladas (for more, see my blog from February 7). Mine is a pan of enchiladas my wife makes. Yours is probably different. This mental re-generation is “re-presentation.”

“Representing,” though, is an action. If you were to say some obscure word in Spanish, I could approximate a spelling of it for you (represent it in writing). I know the rules of spelling in Spanish and they are very uniform. I could even throw it into a spoken sentence. I can speak with a fairly decent Spanish accent. Would my sentence make sense? No. I could “represent” the word verbally or in writing, but without being able to “re-present,” I would make no sense of it. This act of output is “representation.” In math, it’s the same with a spoken number, written symbol, an algorithm, etc.

Students can “represent” without “re-presentation.” That is most often the short-circuit.

To truly understand the very abstract concept of number (notice I am not saying “a” number), students must be able to “re-present” connections to number words and/or symbols (for more, see my blog from June 5). They must be able to re-generate experiences that they have had with counting items or patterns, finger patterns and usage, movements, subitizing, and so on. Numerous and varied instances of “re-presenting” those experiences will help the child form an abstract understanding of a number word and avoid short-circuiting the understanding.

Comments? Questions? Leave them below. Let’s start a discussion.

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