Algebra students who forget their thinking caps

Miscellaneous and Personal Stuff I Must Share
1

I’m in the process of grading final exams for the Introductory Algebra I & II courses I teach. All quarter long, when it comes to word problems, I stress that students should think about the answers they come up with in the context of the problem. Of course, I always tell them they should check their answers anyway, but failing that… if the answer doesn’t seem reasonable, perhaps they’ve done something wrong, and should rework the problem. Do you think they take my advice?

One problem I gave is to figure out the speeds of two buses. The students who came up with 90 and 100 mph, and 130 and 140 mph, probably should have thought “Hmm, that seems a little fast. Maybe I should go over this again.” Nope! Those are the answers they went with.

Another problem asks to find how long it will take two people working together, at different rates, to complete a job. The student who wrote “-9 hours” didn’t think twice about her answer.

:rolleyes:

2

I remember back in high school when we started working with quadratic equations and word problems and the teacher telling us the same thing. If you have two answers (as you will with a quadratic) and one of them doesn’t make sense…the answer is the other one.

Sure, technically, the answer to the equation is 5 and -17, but the store probably isn’t going to sell the books for -$17. (Also, if you put -17.0001, you probably used a calculator)

3

I was surprised that a teen I was helping hadn’t been taught to quickly sketch an equation. It helps to have a mental picture of what you’re working with. What quadrant(s), where it crosses the axis etc. Is it a line, circle, ellipse, parabola, the orientation up/down… We were expected to know that based on the equation and a few quick test points.

The teen was a senior taking high school calculus and was looking at me like I was talking greek. :wink:

4

This is because people don’t see math as correlating with real life. And we don’t do a lot of favors in the way we teach math, as it’s mostly abstracted away from real life.

Plus, these kids have probably expended all their mental energy doing the math in the first place, and don’t have any left over to check the answers. Plus most checks take a lot longer and most kids need the entire time to do the test in the first place. I admit, even as the most advanced kid in the glass, I almost never checked my answers.

5

Me, neither, but I’d usually have noticed an improbable or impossible one like the examples.

6

I honestly like playing with numbers. When I’m on hold, I will calculate Pi just to give my hands something to do. My ex-minion thought I was amazing because I could average 3 numbers without doing anything but hearing them. One day, he was talking about a TV show and said something about Schrodinger’s Cat. I was excited, we could talk maths.

As it happened, he had no idea that said cat was a real concept and I tried to explain it to him several times. (It was our unspoken agreement, I was allowed to talk to him about books and geeky stuff and he would listen because he would tell me about his new golf clubs and talk sports to me and I would listen.)

As it happened, I posted a thread about it and someone gave me this link and I sent it to him. He called me and seemed to actually have gotten it. So, my point is that its as much the presentation as it is the content. IMHO, of course.

I used to have a friend in her 20’s who didn’t know how to double a recipe because she couldn’t add fractions. Her solution to this was to just measure everything twice because she really didn’t get the connection between 1/3 + 1/3 = 2/3 on paper and her measuring cup. When I showed her how it worked she was amazed and happy. When I moved on to percentages and how 50% was half a cup or half a dollar, she actually started crying, she was so happy.

This was not a stupid woman, she was just poorly educated.

7

You only have realistic questions? My teacher didn’t hesitate to have a bus go 140.

8

That’s what I was going to say. My teachers’ buses could have set land speed records.

9

After I wrote the OP I came across another good one. A variation problem, the question gives the loudness in decibels at the stage at a concert, then asks to find the loudness 60 feet back. One student answered 6000 dB. I thought “wow, would that be loud enough to shatter the planet?”

I’ll cut her some slack, though. Most people aren’t familiar with decibels in the way they are miles per hour and whatnot. I doubt she would have known what a crazy answer that is, even if she had thought about it.

10

Let’s not forget about professors too!

At the ripe old age of 40-something, I went back to community college.

Anecdote #1: Introductory College Chemistry:

Instructor was great at chemistry and great at teaching except he was lousy at math.
Lecture on “dimensional analysis” or whatever they called it that year: He gave an example of converting some length in nanometers to kilometers. (Presumably, one should predict in advance that the result would be a much much smaller number than the original.)

In the midst of working that out, at one point he moved a factor from the denominator to the numerator, but forgot to change the sign on its exponent. I noticed it but kept my mouth shut. (This was in a huge lecture class of about 200.) Nobody else said anything either.

He ended up with a result something like 23 nanometers = 230000 kilometers or something like that. :smack: ONE student in the front row raised her hand and questioned this result. Then I remarked about his missed exponent sign change. The instructor absolutely would not acknowledge that the might have made a math mistake, nor that the result was highly absurd. The entire class broke into a free-for-all of arguing back and forth across the whole lecture hall over it.

This same instructor also had a tendency to put questions on exams (typically involving some math) that he didn’t know how to solve himself, and then marking wrong any student who actually got the right answer (although I suspected in some cases I was the only one in that boat). One such question involved approximate numbers, where the proper rounding of the numbers was the whole point of the problem, but he didn’t know how to work with round numbers very clearly.

Anecdote #2: Health Education Class:

Taught by a P. E. Instructor who had also majored in P. E. Science or Physiology or something like that. (i.e., he had relevant science background.)

When we studied blood alcohol levels, we learned that if you ever manage to get a B.A.C. as high as 0.5%, you are most likely already dead. One student questioned this, asking: Doesn’t that mean your blood is half alcohol? I mean, how could you blood ever become even remotely that high in alcohol content?

She clearly didn’t understand the meaning of 0.5% – But hey, give her credit for noticing the absurdity of what (she thought) that meant, and give her credit for asking!

The instructor confessed that math wasn’t his strong suit, and he couldn’t explain to the class any answer to her question! :smack:

11

That’s why they say “half of 1%” so exactly that kind of confusion can’t occur.

12

I didn’t mention it above because it wasn’t the purpose of my post to brag. But when the teacher couldn’t explain that, I raised my hand and said exactly that. Actually, I said something like “If one half your blood is alcohol, that’s fifty per cent. Zero point five means one half, and zero point five per cent means one half of one per cent.”

Nobody made any further comment or asked any further question about it. I got the sense that the person who asked might have understood that, although I never found out for sure.

Then there was the Introductory Statistics class I took, after which I tutored during the following semester. They require one semester of algebra as a prerequisite, which seemed technically adequate, but not really. Most people in the class seemed to struggle with it. One guy I tutored was really lost at the elementary algebra level. We worked through one statistics probability problem where the final answer, after a lot of work, came out to .1/.2 (Yes, simply point one over point two.) So I asked him to simplify that, or put it into a more standard form. He reached for his calculator. I said to him: “You don’t need no steenkin calculator to do that!” and he had an instant total panic attack. So I let him use the steenkin calculator.

ETA: BTW, I mentioned these stories in an essay I wrote for my English 1A Composition class, and the teacher wrote a remark on my paper indicating a certain degree of flabbergastion.

13

Sigh. Reasonableness tests were something I could never get students to apply.

14

I don’t think it’s so much reasonableness as a basic difficulty in the way math is (or isn’t) taught. Different students need different approaches in math more than any other subject, and often teachers fail to even cover one approach adequately. IMHO, math requires:

(1) Laying out the overall principles of what today’s lesson actually means.
(2) Explaining what the operation today actually does. I’ve forgotten many a lesson that was simply a rote math exercise because I had no idea why you wanted to do it or use it.
(3) Explaining how it works with order of operations and different oddities added. This isn’t always as obvious as you might like, and it leaves some students confused.
(4) Pure-math practice.
(5) Real-world examples.

I’ve often felt that most mathematics teachers hand out way too much homework, and the wrong kind. There is value in rote learning with some problems at some levels. But often, a week will be spent learning how to do one thing with different variations, and the fiftieth (or hundred-and-fiftieth) problem is no different than the first. Doing all those is boring, frustrating if they’re very long or complicated, and means absolutely nothing to anyone.

When I went back to college, I was most concerned with the Calc courses I needed to take. I utterly bombed the “How-Well-Do-You-Know-Your-Math” test (which didn’t count towards grade), but then did extremely well in the course. Literally all it took was a quick sheet explaining how all the basic operations fit together and suddenly years of math became obvious. Yet in fifteen years of math, nobody had explained simple things in a straightforward and complete manner. It took a single page to do what a dozen teachers had failed at.

Sure, that worked for me and someone else might need another method. But that’s the point, ain’t it?

15

If you’re their Algebra teacher, they’ve had years of teachers who wrote tests and problems where someone gets negative change back from a purchase, someone else farms on a 1’ x 80’ plot of land, and people routinely run 30mph - 50mph over long distances. While you write reasonable problems, it isn’t a universal condition among math teachers. While by finals they should have figured your tests out, the previous years of impossible math results are hard to undo.

Also, sometimes, you know the answer that you’ve arrived at is wrong, but you cannot figure out where the mistake is. In which case, you write down the wrong answer (because a wrong answer is better than no answer), show your work, and hope that the teacher gives partial credit for the part of the problem that you might not have completely screwed up.

16

Well, duh. Everybody knows if the speed of the bus drops below 50 mph, it’ll blow up.

17

These are the sorts of people who don’t understand what’s wrong with prices like .99¢

18

Yeah. In many cases, I suspect that the problem is not that they don’t notice the answer is unrealistic, but that they don’t know any better way of solving the problem than the way they tried, that gave them that unrealistic answer.

19

I’ve had one or two teachers that at least claimed to give (more) partial credit to students who wrote something like ‘This answer is wrong because dogs don’t weigh -1500kg’.

I know that when I tutored math students I’d love to know they understood enough to know when an answer was wrong.

20

Like I said, I give her credit for realizing that, by her understanding, there was something wrong with that number. In any case, people who don’t know 99¢ from .99¢ can always work in customer service at Verizon.