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My main source of income is based on the premise that I can help other people to understand math. At least, that’s what I want it to be based on. I’m finding, though, that it’s easy to slip into a crucial trap, which is the idea that my income is based on my ability to help other people get good grades in math, which is related, but not at all the same thing.

The problem is that most parents and nearly all the students who hire me do so with the latter hope. The fact is, it’s much easier to short-term improve grades than to insist on true understanding.

When a student has learned a “shortcut” in math, some algorithm that allows them to get the right answer to a given problem easily even if they don’t understand it, they are pleased. Not only are they pleased, they are highly resistant to revisiting that material in a different way because it’s going to “confuse” them.

Let me give you an example that infuriates me. In one school from which I have many students, the 6th-graders have all learned to deal with adding and subtracting positive and negative numbers via the following 3-step process:

1) If the question is subtraction, change it into addition by adding the opposite of the number that was originally subtracted. (i.e. -7-(-4) becomes -7+(+4). )
2) If the signs of the numbers are the same, “add” the numbers together. If the signs of the numbers are different, “subtract” them from each other. (i.e. -7 and +4 have different signs, so we would subtract 4 from 7 to get 3.)
3) Give your answer the + or – sign of the “bigger” number (meaning, in reality, the sign of the number with the larger absolute value.) (i.e. in our example, -7 is the “bigger” number, so our answer has a negative sign and is therefore -3.)

Now, this algorithm works just fine. But no student I have encountered who uses this algorithm really understands what they’re doing. They’re just following the steps and writing down an answer that pleases their teacher. When I ask them why subtracting a negative number is the same thing as adding a positive number, they stare at me blankly. When I ask them why they’re assigning the answer the sign of the “bigger” number, they say ‘because that’s how you’re supposed to do it’ or ‘because it works.’ When I try to explain that -7 is actually not really “bigger” than +4, they start to doubt me or their teacher.

This algorithm works fine as long as all you’re doing is adding or subtracting a single pair of numbers, in isolation. As these students move on into more complex algebra, this algorithm will serve them not at all, because they will be encountering negative quantities of a variable. When they crank addition and subtraction of ‘x’s through their algorithm, they are losing the intuitive understanding of what it means to have quantities of x, and most of algebra will therefore be reduced to algorithms they need to memorize. And as the math gets more complex, the algorithms pile up and get harder and harder to memorize, until the student fulfills minimum academic requirements and gives up in exhaustion, never having found out that it’s even possible to *have* a deep understanding of math.

The problem, though, is that because they’re getting all the questions right, they think they “understand” it. And when I want to cover this ground with them again, confusion and frustration reign. Why am I asking them all these complicated questions when they already know how to do the problem? Why are we spending more time on this material? ‘I understood it before, but now I’m all confused because you’re telling me something different’.

There are lots of different ways to address this concept. You can use number lines, or “jars of zero” that are full of equal numbers of little pluses and minuses that cancel each other out, or money and the concept of debt. Lately I’ve invented an analogy of piling up dirt to make a hill or digging down to make a hole (height of hill=positive number, depth of hole=negative number) and that works pretty well because you can draw little pictures to illustrate, and the concept of “subtracting hole” comes pretty easily after you’ve talked about adding and subtracting hill.

The point is, every student will click with a different method of understanding, but sometimes it doesn’t happen right away. And if you don’t get it after a single session, students go home feeling a little bit broken—they “understood” it before, but now their tutor said a bunch of weird stuff and it doesn’t make sense any more. So confidence crumbles. 6th-graders are not known for being strong on patience. And the next time they come back, they don’t want to keep working on it because their experience last time was a negative one. And in the meantime, their regular class keeps moving forward, and new topics are piling up that need clarification in a limited amount of time.

So what’s a tutor to do? What the students want, and what the parents want, are good test scores. They have hired me because that’s what they want. They are happy with the nonsensical algorithm because it can be memorized quickly and it works every time. So why should I fight this uphill battle that risks my clients’ satisfaction, when I can simply help the student memorize and apply the algorithm and move on to the next lesson and make everybody happy?

I think that would make me a pretty mediocre tutor. And while the students might not know the difference, I know the difference. But is it a good idea to convince my employer that they’re hiring me for the wrong reason? Is it even my place, as the hired contractor?

Maybe I need to become so good at tutoring that I can flawlessly help any student to have a deep understanding of the concept in 20 minutes or less. But until I achieve that level of sublime teacherhood, I have a puzzle. My soul says, insist on understanding. My pragmatism says, give it a go and let it pass if I’m spending too much time on it. Any thoughts out there?

Love always,
Alissa