‘Simple’ Isn’t ‘Easy’

You are probably aware that 3^{1/2} = \sqrt{3}. Sometimes when I’m tutoring I wind up teaching this to young students. Here is the story I use:

You already know that 3^4*3^2 = 3^6 for a very simple reason.

Forget the reason for a moment, and just focus on the rule. When you multiply exponents with the same base, you can add the powers.

That means

3^{1/2}*3^{1/2} = 3^1 = 3

Evidently, 3^{1/2} is a number such that if you multiply it by itself, you get three. But that is exactly the meaning of the square root! Hence 3^{1/2} = \sqrt{3}.

This is a very simple idea, but when I try it on students, it usually fails.

After going through the story, I ask what 16^{1/2} is. I’m hoping to hear “four”, but that’s not what happens. Sometimes they say it’s eight. Sometimes they say they don’t know. But the most common response is to go through the whole thing again. The student writes down

16^{1/2}*16^{1/2} = 16^1 = 16.

They stare it at for a while. Then they look up at me and say, “Is that right?” We discuss it a bit further to clarify. Circuitously, we stumble upon 16^{1/2}=4. After that we do a few more half-powers and they get it right. Then I ask what 8^{1/3} is. The student will write down

8^{1/3}*8^{1/3} = 8^{2/3}.

“It doesn’t work for that one,” they say. “You just get a 2/3 power, and we can’t do that.” So we talk about it some more, until after some time the student can go between roots and exponents.

Then I ask what 4^{3/2} is, but they struggle with this, too. Once that’s down we try for 6^{-1}, but that is also impenetrable (I usually hear that it’s -6). When I suggest trying to figure it out based on the rule of exponent addition, the student feels frustrated and defeated.

It’s curious that I have such difficulty teaching this idea. It is not too complicated or too difficult, even for a young child. It is far simpler than long division and far less abstract than “set the unknown variable equal to x”. The problem is not the sophistication of the idea, but a more fundamental error in communication. When I give my little presentation, the students simply have no idea what I’m doing.

An analogy: I’m teaching someone how to lift weights (this is very hypothetical). I take a dumbbell and I start doing some bicep curls. It’s only a 5-lb dumbbell, and the motion is very simple, so I figure the guy I’m teaching will get it for sure. I hand him the weight and say, “You try.”

When I hand over the weight and the student starts yanking it up and down. He purposely mimics the way I grunt in exertion and copies my facial expressions. He remembers how I looked over my shoulder to talk to him while I demonstrated the exercise, so he looks over his shoulder when trying it out. The weight ultimately does go up and down, but only with a great deal of extraneous commotion. I straighten him out with some effort, but when we move over to the bench press we’ll repeat the whole confused process.

The problem is that before we began, my student didn’t know what weight-lifting is. He didn’t know the point is to make your muscles stronger, or the counter-intuitive idea that to make your muscles stronger, you first have to tire them out by working them hard.

Similarly, my math students watch me do this strange algebraic exercise with exponents not knowing that the goal is to discover new things. They think, instead, that I was simply teaching a new procedure, as in, “This is how you solve problems where the exponent is one half.”

This is not really a big problem. Students can learn new things; that’s what being a student is about. The problem is that students’ ineptitude at this task frustrates me. At times, when watching a student struggle with a problem, I’ve felt ironic wonder at the student’s remarkable creativity – how do they find so many unexpected ways to get everything totally wrong? I wind up concluding that the student is “stupid”, and the student leaves the lesson with only the impression that they have somehow failed at a task they never even understood.

I make these grievous errors in judgment because I assume that since I’ve seen the student handle far more complicated tasks, they should master this one right away. That is not so. ‘Simple’ isn’t ‘easy’. Computing a determinant of a 4×4 matrix isn’t simple, but my students can blaze through it. Showing that the determinant will be zero by noticing that the last row is equal to first row is very simple, but I’ve never had a student use that method.

The things we’re good at are not what’s simplest, but what’s most familiar. The converse also holds: things that are unfamiliar are difficult, even if they’re simple. I personally find it much easier to solve geometry problems using coordinates, algebra, and calculus than using Euclidean geometry, even when the Euclidean approach may be just a few lines of sketching and finding a similar triangle.

When I first noticed that students were having a hard time with problems because they required unfamiliar thinking, and not because they were too hard or because the students were bad, I tried to remedy the situation with speeches. I would talk about how interesting it is to figure out where a formula comes from. I would say over and over that no, I don’t have all the formulas memorized, because as long as I know most of it, I can figure the rest out. I would prove my point by waiting until they embarked on a difficult calculation, and then solving it quickly in my head using some trick or other, supposedly demonstrating how useful it is to be able to approach a problem many different ways. Then I would describe how it’s done. “You’ll like this thing I’m about to show you,” I would say. “It’ll make your life easier.”

This backfired. It mostly led the students to believe that I either gained some ineffable voodoo skills in college or that I am in possession of an extraordinary native intellect that they could never hope to emulate.

I still don’t know quite how to handle the “simple isn’t easy problem”. I have become far more patient when trying to push students’ boundaries, and far less ambitious. I regret the many times I compromised a student’s chance at learning and my own at equanimity by failing to recognize “simple isn’t easy” in practice. I continue to search for simpler and simpler teaching stories, but I don’t spend enough time searching for ways to make the unfamiliar territory easier to navigate. I don’t know how complicated a task that is – to figure out how to build a stepladder to a new level cognition – but I know it isn’t yet easy.


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4 Responses to “‘Simple’ Isn’t ‘Easy’”

  1. Woods Says:

    One of the things that I’ve found works well for me in getting students to think “properly” about math is to try to teach almost by virtue of only asking questions, but in a very specific way. I’ll let the student go through a problem, and not say anything until they’re done; while they’re working, I’m only watching and cataloging their mistakes in my head. Then, once they’ve finished, I’ll point out a mistake by asking them a question about something they’ve done in a way that makes a prediction about some other, related, simple problem that they can immediately see the answer to. I’ve found that repeated application of this usually both helps them build up intuition and helps get them in the mindset of looking for ways to check the math that they’re unsure of.

    For example, say a student is working on the following problem:

    Solve the equation x^2 + 4^2 = 5^2 for x.

    One of the common mistakes I see is that students immediately take the square root of both sides incorrectly. e.g., they might work out:

    x^2 + 4^2 = 5^2

    => sqrt(x^2 + 4^2) = sqrt(5^2)
    => x + 4 = 5
    => x = 1

    Clearly their mistake is that they don’t understand that sqrt(a^2 + b^2) != a + b, so here’s how the following dialog usually goes:

    me – “Is that your answer?”
    student – “Yeah.”
    me – *Pointing at the line in question* “So you’re telling me that sqrt(x^2 + 4^2) = x + 4?”
    s – “Yeah.”
    me – “So, you would also be telling me that sqrt(2^2 + 2^2) = 2 + 2, right?”
    s – “Yeah!”
    me – “Is it?”
    s – “What do you mean?”
    me – “Is the sqrt(2^2 + 2^2) = 2 + 2? That’s simple enough to figure out by hand.”
    s – “Oh, ok. Uh… nope.”
    me – “Ah hah! So we’ve learned that that’s not how sqrts work! We can’t do that. You better rework the problem.”

    And then they’ll go rework the problem, and if there are any mistakes remaining, we’ll go through the questioning process again, and they’ll rework the problem again, and eventually they get it right.

    Usually after a few weeks of tutoring, I’ll start to see them checking their math with simple examples whenever they’re not sure how an operation works.

    The other specific example where I’ve used this teach-by-questioning tactic often and to great effect is when teaching students what trig functions like sine, cosine, tangent, cosecant, etc look like. That always begins with writing down the unit circle until it’s correct, then I let them draw an entire graph, and there’s lots of questions like:

    me – “So you’re telling me that the sin(0) = 1?”
    s – “Yeah.”
    me – “Sine is the y-part of the coordinates of a point on the unit-circle, right?”
    s – “Yeah.”
    me – *pointing at (1, 0) on the unit circle* “So you’re telling me that the y-part of this point is 1?”
    s – “Yea… oh. Oh! Nnnnooo nonononono. Hang on a minute.” *goes to redraw the graph*

    With the trig functions, getting students to correctly draw the graph of tangent is my favorite, because at that time in their math education most of them are very unfamiliar and uncomfortable with the idea of asymptotes, so they almost always draw a continuous graph to start with, so we have lots of fun discussions that start with “So, you’re telling me that somewhere between *here* and *here*, tan(x) = 0?”

  2. Salomon Trujillo Says:

    So, my mom used to teach algebra at my old high school and it was a small school. (now closed due to the population density gap between the baby boomer’s children and their grandchildren) As a result, her classroom had the full gambit of smart and stupid, lazy and motivated, with every combination of those two spread across five different years of students. All smashed into one class.

    Now, my mom is very smart, but she didn’t always think so. She took a math class (I forget if it was algebra or calculus) a bit too young and watched her older brother do well and decided she wasn’t good at math. A few years later, she took it again, initially dreaded it and discovered at it was really easy. Between her own experiences and watching years and years of students take algebra, she came to the conclusion that almost everyone is capable of doing well in algebra, but your brain doesn’t mature to that point until about the age of 13 or so, the variance is quite wide. A student who’s doing poorly in math one year might benefit greatly by simply waiting a year.

    On another topic, I recall taking CS1 at Caltech. I’ve been programming since I was a little kid, so I figured CS1 would be a repeat. Well, it was taught in Lisp and it was “Introduction to Computer Science” not “Introductory Programming.” About 90% of was simple and straightforward, learning to program primarily recursively was interesting. But a few things threw me: Big-O notation was one of them. I completely missed the point of the concept at the time (perhaps it was because I skipped lecture, but it’s an example of something that’s very clear to me now and was a mystery to me then.) A year later, I TA’d CS1 and I was talking with the professor as to why we were using Lisp and covering some of the “advanced” topics. He said: “Ever see a terrible program? We’re going to rid the world of bad programs by teaching people how to be computer scientists from the beginning.” In my head, I wondered if he had been taught as a computer scientist from the beginning and if he would have understood encapsulation while he was still trying to master the fundamentals of object-oriented programming.

    So, finally, with these two stories in mind, I had a couple questions about your story from above. You ask the kids what 25 ^ 0.5 is, but are they capable of answering “What is the square root of 25” easily? I recall seeing a half in the exponential and being told: “That’s another way of doing the square root” then verifying it with the exponential arithmetic rule. I will fully agree that the deductive path to figuring it out on the fly is superior and if a child uses it, they will remember it better. But the bread-and-butter aptitude needs to be there in place first. If anything, I would use it as a double-check method, similar to teaching very young children to double-check their subtraction and division problems by relying on their presumably superior addition and multiplication skills, respectively.

    Basically, most of these student are just trying to survive their homework and they might not be mature enough. I’ve had some success with getting students to understand the material at a fundamental level, but it has always been under the guise of “this method is easier” and it is easier because it doesn’t require memorization. But it’s always been with material that’s just at the edge of their understanding, and it always used methods that they’re comfortable with.

    I guess what I’m trying to say is: don’t feel bad about not pushing your students, they might not be ready yet. They will be.

  3. Mark Eichenlaub Says:

    Wise advice, Sal. I only have to deal with one student at a time. An entire classroom is an entirely new level of challenge.

    I had a similar experience with linear algebra. Only a vague and confused notion of what they were talking about during core, but when I had to go back and learn it for quantum, I picked it up much more easily.

    I’m not sure whether struggling with it the first time around and half-comprehending set the stage to understand it the second time, or whether it was mostly other supporting knowledge I picked up in the two intervening years that did the trick, but either way what you mentioned is definitely something I want to keep in mind when I’m tutoring.

  4. My Brown Big Spiders « Arcsecond Says:

    […] “Simple” Isn’t “Easy”, I learned not to judge the difficulty of new ideas by how simple they are, but by how familiar to […]

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