## Saturday, November 23, 2013

### Oh Why Can't My Kids Count?

I have a freshman class, a sort of Humanities hybrid thing where students learn about college and so some "great books" style reading. I usually teach it once every two years and it's sort of fun. I couldn't do it every semester, though.

Yesterday I had Stupefied Simone in my office. She wanted to get her grades for the semester, even though I give all their work back and even though the standard syllabus shows the 100 points available each term, with letter grades corresponding as: A-90+, B-80-89, etc.

Let it begin:

Simone: I wonder what my grade is.
Kimmie: Well, do you have all your assignments with you?
S: Yes!
S: Okay, they come to 45.
K: That can't be right, Simone, let me count: 8, 8, 7, 8, 9, 10, 8, and 7. That's 65.
S: Oh, what grade is that?
K: Well, it's no grade yet, really, but you can see that we've done 8 assignment so far, all worth 10 points. We just have the final left and it's 20 points.
S: I need an A in this class.
K: Okay, well you need 90 for the A for the semester. You only have 65 so far, and there are only 20 left.
S: So can I still get an A?
K: Really? What's 65 + 20?
S: (Brightening): Oh, 95. No, wait, 75.
K: It's 85.
S: Oh no, so I'm going to get a B?
K: Well, if you get at least 15/20 on the final you will. Less than that and you can get a C.
S: I thought I had 95, no, 85.
K: You have 65 now. There are 20 points left. The MOST you can get is 85, and that's a B. If you get LESS than 15, you'll get a C or worse.
S: WORSE!?!?!

1. I know why they can't count. The educational fad lately in elementary math education is discovery-based learning, where you do exercises like count little boxes that you arrange into squares and whatnot. And after you've "discovered" the conceptual basis of arithmetic they hand you a calculator in about first-grade now.

The result is that students do not understand arithmetic as relationships between numbers, they understand it as operations to carry out on a calculator. This is part of why they can't cope with symbolic manipulations; so they can't add fractions by hand and they can't do algebra is more than variable is present. There are studies on this.

One example: you will frequently see students write such things as

2+3 = 5+2 = 7. Neither 7 nor 5+2 are actually equal to 2+3. What they are writing is the sequence of buttons they pushed on the calculator.

Students are actually pretty helpless with calculators too. They need fancy ones that let you type in an expression as it appears in printed material. If you give them a simple scientific calculator like a TI-30 they can't cope with it because they can't do the operations in the order in which they are written down, they have to apply the rules of arithmetic and they don't know them.

We do not teach driving by discovery. We do not teach football or basketball by discovery. We teach these things through drill so that the operations are second nature and can be done without thinking, so you can think about what you need to do next and carry out your intentions in real time. A basketball player who has to concentrate on dribbling has no processing power left for figuring out how to evade defenders, seeing who's open for a pass, or figuring out where they are supposed to be in relation to their teammates.

A calculator is an excellent servant and a terrible master. My students are helpless with any problem that requires any sort of planning ahead because their brains are consumed by the basic symbolic manipulations that they never learned how to do without thinking about them.

1. BURN THE HILL OF CALCULATORS!!

2. This explains why it's so difficult to teach the concept of significant figures. I have just about given up with my general-ed, intro-to-astronomy-for-non-majors students. Even pointing out how fractions of a cent aren't counted draws a blank.

Worse, though, is how often graduate students in physics don't understand significant figures. Traditionally, significant figures are taught on the first day of introductory chemistry, the first college science class for science majors. They should have learned this on their very first day of college. They should have seen it many, many times since then and now. It still bounces off harmlessly.

@Strel: It saddens me to say that if you try it, you'll stand a good chance of being lynched.

3. ''students do not understand arithmetic as relationships between numbers....they can't cope with symbolic manipulations"

That's definitely something I've noticed for some time now. If you give them an equation like PV = nRT, and ask them what happens when the Temperature (T) goes up, they can't see that the Pressure or Volume are going to go up. Without numbers to plug into a calculator, they're helpless.

It's quite foreign to them that math is a way to describe relationships among real world entities.

2. 2+3 = 5+2 = 7. Neither 7 nor 5+2 are actually equal to 2+3

I first encountered this in the mid 1990s when I was tutoring a family friend's daughter. When I explained what was wrong with it, she got it.

But she kept doing it if I wasn't in the room because she was just writing down what she said as she worked through the problem and this had become a habit.

At least he parents were the kind of educated professionals who had made her learn he basic arithmetic. There was only the misunderstanding about how to write math problems to overcome.

3. "It's quite foreign to them that math is a way to describe relationships among real world entities." (Rand/orG) That's exactly right. It's common sense (or visual intuition) made precise, nothing else. And to produce (and convey) logical reasoning precisely, you need to learn how to use symbols correctly. Don't they learn that in high school?

And yet if I were to dock points for incorrect use of notation on test questions, in most cases there would be no points left. So I write a comment, and leave it at that.

My favorite innumeracy story is a Calculus II student who came to my office a few years ago to ask about a particular point in an example done in the text. The "particular point" was:
1/6+1/7=13/42. "I don't understand how they got that". After explaining it to her, I pointed out, as diplomatically as I could, that she was in the wrong class. She agreed, and dropped.

And I also get these queries "what is my current grade"? Even though I write their current HW score each time I return a test. Last week I got one who wanted an answer "since I think the drop deadline is tomorrow". He was wrong, the deadline had been two days earlier.

4. You weren't giving her the answer she wanted. She was re-asking the same thing in different ways in order for you to give her the "right" answer.

5. That's not even math: it's basic second-grade arithmetic, right? I'm unable to do "math," but basic counting? We are so fucked when these kids get jobs.

6. I feel your pain, Kimmie. I have students who cannot understand that it is mathematically impossible to go from an F to an A in two weeks of class ... and these students are in a STEM major.

7. What I've taken to do doing in my online grade sheet is to simply calculate their expected grade, extrapolating numbers from previous tests and assignments onto likely future performance. I find that this predicts students's grades plus-or-minus about 5%, which means only a small fraction of students will change by more than a half-letter-grade.

Heartbreakingly, this turns out to work extremely well, even in student cases where they suddenly decide to apply themselves to studying for the next exam...