Words Matter!

Here's a thing that irritates me more and more over time: When a math problem doesn't have any words in yet. Most specifically: when it lacks an action verb on what you're supposed to do with it. For example, here's a problem that comes from a textbook I use (and similar stuff even pops up from time to time on our department-wide final exams):

For all x, 30x⁵ + 32x⁴ + 8x³ = ?

Well... it's equal to all kinds of friggin' stuff. Like: 30x⁵ + 32x⁴ + 8x³ + 1 - 1 and an infinite number of other things. Now, in this particular case, if it's a multiple-choice problem, then you can look at the proposed answers and infer that what's being requested is for the expression to be factored. Although you can still get in trouble if one of the options is only partly factored, but it's still technically equal to the original expression. Stuff like that. But this sample problem is definitely not a fair question, because you could not tell what action to take if it were posed completely alone, outside the context of a multiple-choice test (plus: many of our students' abilities to look at multiple-choice responses and back-infer intent will be shaky at best).

I think that this is a major symptom of a scurrilous disease that lets students get away with the false impression that for any given algebraic expression, there's some implied thing that you always "do" to it -- when that's absolutely, totally not the case. Different use-cases will require different actions to be taken (e.g.: sometimes to factor, and sometimes to simplify, which are opposites).

So once again: It really all comes down to a matter of reading. If students think they can "do" math through rote mechanical processes without reading the words -- at least a requested action to take, a single verb at minimum -- then they are tremendously, grievously in error. The #1 skill that I tell my algebra students they're expected to master is learning new vocabulary, so that we can have an intelligent discussion about math, and so they can follow the instructions on a test from me or anyone else (and more generally: make use of that learn-new-vocabulary skill elsewhere in their lives). Failing to phrase our math questions with clear, well-defined action requests in words is simply an atrocious example to set.

One last example: Take the expression 4(x²-9). There's all kinds of things we might have to do with this at different times, including but not limited to any the following (so: get in the habit of reading & writing the words carefully for any of these):

• Simplify. (Answer: 4x²-36).
  
• Factor. (Answer: 4(x+3)(x-3)).
• Identify the Degree. (Answer: 2nd).
• Determine the Roots. (Answer: +3 and -3).
  

Dead Grandmother/Exam Syndrome

There was a discussion the other day where I mentioned a student that a colleague and I had many years ago. The colleague once said, "I just feel terrible for his mother; every time I give a test she has a heart attack."

So apparently this problem is more widespread than I first thought; someone on Slashdot kindly linked to an article by Mike Adams of the Eastern Connecticut State University Biology Department, on the subject of "The Dead Grandmother/Exam Syndrome and the Potential Downfall of American Society" (Connecticut Review, 1990). A truly stellar piece of work; highly recommended.

Against Factoring Trees

Nowadays, I'm anti-factoring trees. It's funny, because they're not usually part of the classes I teach, but they've come up a few times recently -- including twice just yesterday (as I write this), when they were included in a new book I received, and then within an hour a student came asking about them (because they were part of a YouTube lecture on reducing radicals she'd been trying to watch).

By factoring trees, I mean the method for producing the unique prime factorization of a number that looks like this:
By which we can conclude (re-sorting the leaf nodes) that 48 = 2^4 * 3. Of course, this is pretty customary, and it's how pretty much everyone I know (including myself) learned how to do it.

But now my primary complaint against them is that they're a nonstandard method of writing mathematical relationships, and most of all, they're a lost opportunity to practice writing equality statements. With all the problems that students have using, writing, and understanding the equality sign, why not use this as an opening to reinforce their meaning -- especially so in a context just like this, where we do not intend to simplify (evaluate) on the right hand side? Why not instead write in a more standard format like this:

(Or whatever your preference is for use of parentheses or exact number of steps.) It highlights all these issues with the meaning of equality signs that we struggle with later on students' behalf, and it avoids using a special one-off writing technique for the singular task of factoring a number. It's likely easier to read for some students (who may have trouble identifying where the leaves of the tree are). It even saves on lines of paper, and is easier to type out in an email or website if you have to do that. This is actually what I do in class when it comes up now. The more mental connections we can make to the "correct" way of writing math, the better.

PEMDAS Poetry, Pt. 1

Like a drunk uncle
Stumbling towards the bathroom door
Mistakes will happen

More anti-PEMDAS proselytizing here.

Comments on Decimal Places

Previously observed -- Operations on powers with the same base effectively shift one place down in the order of operations. Examples:

• Exponentiation will multiply powers; e.g., (x^3)^4 = x^12
• Multiplication will add powers; e.g., x^2 * x^3 = x^5
• Addition does a no-op on powers: e.g., 3x^2 + 5x^2 = 8x^2

Note that the same rule generally applies regarding number of places after a decimal point. Examples (working in the other direction):

• Addition does not alter number of places; e.g., 1.2 + 3.4 = 4.6 (1 place)
• Multiplication adds the number of places; e.g., 1.2 * 3.46 = 4.152 (3 places)
• Exponentiation multiplies the number of places; e.g., 1.234^2 = 1.577756 (6 places)

And the reason, of course, is that the decimal places themselves represent exponents to the common base 10. Example:

• 1.2 + 3.4
  = (1*10^0 + 2*10^-1) + (3*10^0 + 4*10^-1)
  = (1*10^0 + 3*10^0) + (2*10^-1 + 4*10^-1)
  = 4*10^0 + 6*10^-1
  = 4.6

Since the number of decimal places is dictated by the lowest power, if the addition operation 2*10^-1 + 4*10^-1 doesn't change the power (as above), then neither will it change the number of decimal places. And so on and so forth.

Calculator Equals

On the subject of students not understanding equals signs -- probably not the first time someone pointed this out, but -- How much of this is caused by usage of the equals sign button on a calculator?

It's really a bit malformed, if you think about it. Mathematically what's really happening when you hit that button is a request to "simplify" the numerical expression that you've typed in so far. So perhaps it would be better if the button were labelled "simplify" or "evaluate" -- or maybe a "total" button like on cash registers, or some abbreviation along those lines.

Possibly the malformed understanding of the equals symbol (thinking that a simplified number always goes on the right side) is due to the hundreds and thousands of times that students have used a calculator "=" button by the time the issue matters in algebra?

Related posts:

• Equals or Not Equals
• More Equals Signs

Concrete P-Value Demonstration

I find that students in my statistics class are almost totally bewildered by the logic of hypothesis testing and P-values (for hypotheses based on a population mean), no matter how carefully I try to explain the concepts. Here's an idea for a super-short and simple, concrete demonstration of hypothesis testing. Tell me if you think this would be worth the class time:

1. Start with a hand of four cards: {A, 2, 3, 4}
2. I'll turn my back and secretly do one of two things:
   H₀: Leave the Ace in, or
   H_A: Take the Ace out
   
3. Now shuffle the hand and deal out 3 cards.

Question: Say I get a draw of {2, 3, 4}. What's the chance of this happening if I did not take out the Ace (H₀)? Note that all possible draws would be {{A,2,3}, {A,2,4}, {A,3,4}, {2,3,4}} so the probability of seeing that would be P = f/N = 1/4 = 0.25.

Conclusion: If I draw {2,3,4} then we have some evidence that I did change the deck (H_A) -- because it's unlikely to see that result if I didn't (P = 0.25).

Now -- You can actually demonstrate this and ask the class if they think I left the Ace in or took it out each time. I'd recommend 3 run-throughs: leave it, leave it, then take it out. (In the latter case, also ask: Is it possible that I left the Ace in?) In reality, you should probably hold the cards against the otherwise full box, so it isn't obvious if your hand becomes empty in the take-it-out case. (And otherwise practice the prestidigitation in advance so your handwork doesn't give it away.)

Open Question: Should I actually reveal to the class which one I did each time (for confirmation), or leave that as a mystery (modeling real-world usage)?