…a person in calculus might be led to believe that the following are true?

- The graph of the derivative of f(x) is just the graph of f(x) turned upside down.
- Dividing by x is the same thing as subtracting x, as in: (1 + x
^{2})/x = 1 + x^{2} – x.
**Update**: Dividing by x is the same thing as changing the exponent to -1 and adding, as in: (1 + x^{2})/x = 1 + x^{2} + x^{-1}.
- ln(x) = 1/x.

I’m getting these on the final exams I am grading right now; I got them from some of the students throughout this course and have gotten them before in every calculus class I’ve taught going back for years. I can see how students might get screwed up on the third point, but the first two are just inexplicable — there’s nothing that would cause a person to think this, at any level of math.

Updates will follow as I encounter more “new math”. Hopefully that means no more updates. (**Update**: Oh well.)

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Do you really want to know my theory about how this happens? I have one… and I’ve discussed it with my mathematician-husband (whose name you might know since your work is in similiar areas of math). It makes him no more sympathetic to the mathematically challenged, though, so I’m not sure what good my little theory is….

Any theory would be better than what I’m working with now, so go for it.

I suspect that many students don’t understand the difference between “is equal to” and “has something to do with.”

It’s certainly true that a lot of students use the symbol “=” to represent any sort of transition in a problem — whether it means setting two things equal, saying that two things are equal, indicating that a derivative has been taken (as in ln(x) = 1/x), etc.

Andrea’s theory of mathematical pedagogy follows. I have yet to convince any mathematicians of it😉

I think it’s more than a lack of algebraic skills. High school teachers want to hammer away at rehearsal of technical skills -pretty much context-free. Then students are destroyed when they get to college and are encouraged and expected to think as mathematicians (okay..fledgling mathematicians) think. They don’t know what that means.

The bottom line thing they don’t get it that mathematics is a language -among other things. So those dreaded equations that, in the above cases, contain absurdities…. they don’t SEE that they’re absurdities. They don’t know that an equations is a sentence that one could diagram -assuming they knew how to do THAT, which they don’t.

Another problem…. It’s taken me 25 years of living with a mathematician to grasp that math is beautiful. You can see that, perhaps, because it’s a natural talent for you. Of course your talent has been nurtured and trained and you’ve worked hard. I don’t want to minimize that. But something drove you to do that work in the first place. Trust me on this, the beauty of mathematics is a big black spot for most people. I don’t know what you could do about that, though. I think elementary and secondary education are largely responsible for having ruined that.

I have more….🙂

Andrea

Good stuff, Andrea. Bring it on.

I think that some of these errors–especially some classic algebra blunders like

(a+b)^2 = a^2 + b^2 (when I’m feeling especially snarky, I’ll write “Only in a

ring of characteristic 2!” as a comment to that error) or a/(b+c) = a/b + a/c–

often occur not becuase the student honestly doesn’t know any better, but because

they panic. They may think to themselves “I don’t know how to integrate the

square root of 1-x^2, but I know how to integrate 1-x”…

At least, that’s my impression.

Oh, and Andrea, I think your ideas are well thought out. And I definitely agree

that too much drilling of fundamentals can be a bad thing. However, the current

shift in the K–12 math curriculum is going too far the other way: fundamental

skills are being replaced with button pushing, and both problem-solving skills

and critical thinking are being made less of a priority…which is a damn shame.

Interesting. “Critical thinking” means something very different now than before, particularly when it comes to math ed. “Critical thinking,” or “higher level thinking” now means a problem with multiple solutions, and allowing students to come up with their own methods for discovering the solutions. It’s inefficient, and as far as math ed goes, counter-productive.

What’s really interesting, though, is that I see contradictory “thinking” (not sneer quotes — just the best word I could come up with on only two cups of coffee) among the same students. Maybe it would be better said that students confuse one correct answer with black-and-white, all-or-nothing thinking.

We do two basic types of problem-solving. Absolute, and statistical. Absolute problems are those that have one correct answer. Problems that require statistics to solve, of course, can have more than one answer, because statistics is probability. We have students who are obviously victims of current math ed, and don’t have the math skills to do even pretty simple, straightforward problems — and don’t understand that one answer is correct, and all others are equally incorrect. They often argue that they were almost right, or something to that effect. But we also have students who understand the “one correct answer” concept.

What students really struggle with, though, are the statistical problems. They carry over the “one correct answer” mindset, I guess because there are numbers involved, and really have trouble when we get to statistics, for example, running simulations and interpreting the results. They don’t understand that simulations aren’t REAL, that the output data aren’t REAL, and that there can be more than one best option or answer because the interpretations are based on probabilities.

Panic is certainly a factor, but:

(1) I get that stuff on homework assignments too, where there is essentially no time pressure.

(2) I’m a believer that panic can be minimized by preparation.

I know for a fact from talking with students in the past who’ve done things like change division by x to subtraction that it wasn’t so much panic as it was just not thinking about what they were doing — possibly a result of the mindset that math problems have to be done in the shortest possible time, and so there’s no time to reflect on whether an action in a solution makes sense or is legal. So maybe not so much panic as time pressure, either real (on a test) or imagined (on homework). High-stakes testing has forced students into the “speed” mentality, but that’s another rant.