Calculator cripples

On the Irascible Professor’s site, guest commenter William Kohl writes about calculator dependence. He tried without success to persuade an algebra student to analyze a problem rather than graph it on his calculator.

One of the first problems was:

y = x2 + 5, if the constant 5 is changed to 1, the curve

a. does not move
b. shifts 1 unit up
c. shifts 1 unit down
d. shifts 4 units up
e. shifts 4 units down

I said, “What would you do to find the answer?” He said, “I have to get my calculator.” I said, “Why?” He said, “I need it to work the problem.”

I said, “Couldn’t we just think about the problem first? Even though it may seem hard, (as it probably did to him), perhaps we can start by finding a simpler problem inside this difficult problem.”

I was thinking of analytic strategies I had learned in math, one of the basics of which is, look for a simpler problem inside a complex one. He argued with me, and said it was necessary to have his calculator. It seemed as if he had relegated part of his thinking process to this calculator and his math brain would be incomplete with out it.

Kohl, an engineer, is a mentor to a student “on the verge of success,” which is the new esteem-guarding euphemism for “at risk,” itself a euphemism. But he can’t help describing his student as “deficient” in math understanding.

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  1. Um, e is the answer, right? x2 is the slope and 5 (or 1) is where it crosses the Y axis of the graph (in my head, no calculator required for this one).

  2. Using “on the verge of success” for a student who isn’t actually passing (which is what I assume the euphemism means) is kind of like saying “a little bit pregnant” for someone who’s knocked up.

    Then again, if we’re allowed to be “on the verge” of anything we desire, I could say:

    I’m on the verge of slimness.
    I’m on the verge of fame.
    I’m on the verge of being super-hot and attractive.

    (Nope, doesn’t make me feel any better.)

  3. Half Canadian says:

    I believe it’s e. Take it this way:

    x=1 — y = (1)2 +5 — y = 6.
    x=2 — y = (2)2 +5 — y = 9.
    x=3 — y = (3)2 +5 — y = 14.
    Ad nauseum.

    If the formula is changed to y=x2 + 1, then . . .

    x=1 — y = (1)2 +1 — y = 2.
    x=2 — y = (2)2 +1 — y = 5.
    x=3 — y = (3)2 +1 — y = 10.

    No calculator required, and never have used a graphing calculator.

  4. Because the question refers to “the curve,” the student goes right for the calculator. Unless the student knows how to graph said curve or has been taught the relationship between curves and their equations, this question makes no sense unless it is done with a calculator.

    The correct answer is e – any value added to the equation (y-value) of a curve shifts the curve vertically; multipliers stretch or shrink the curve; numbers added to the x-value shift the curve horizontally. Once the student understands this concept, it can be applied to any curve, whether the student has seen it before or not. This concept is introduced in 9th grade Algebra, but hammered home in 11th grade Algebra II. Depending on the age of the student, he or she may have a grasp of this concept, but if it hasn’t been taught as a connector to equations, the student will have no clue. Tutors need to remember that some kids don’t think analytically until they have been taught to think that way – for a lot of them it doesn’t come naturally. Some people are natural born spellers, and some really have to work at it. Some people are born mathematical analyzers, and others have to work at it.

  5. Alex Bensky says:

    It’s even worse. A few years ago I substitute taught part-time in a Detroit suburb, one of the more affluent school districts on the planet…and by the way, generally a good system. I had a seventh grade pre-algebra class. One set of problems was to find between which two integers the square root of a number lay. Note, it wasn’t to find the square root, simply to ascertain between which two integers it would be.

    About half the class was stymied. “Look,” I said, “if you think about it then it’s easy.” I took one of the problems, say the number given was 38.5, which I wrote on the board.

    “See, you know that four times four is sixteen.” I wrote “16” to the left of the number. “Five times five is twenty-five.” Again, I wrote “25” to the left of the number. “Six times six is thirty-six.” I repeated the process. “And seven times seven is forty-nine, so your answer is ‘between six and seven.'”

    And half the class couldn’t follow me, I discovered, because they couldn’t do their times tables in their heads.

  6. ” any value added to the equation (y-value) of a curve shifts the curve vertically; multipliers stretch or shrink the curve; numbers added to the x-value shift the curve horizontally”

    And, to make it much easier to memorize:

    f(x + n) shifts n to the left.
    f(x – n) shifts n to the right
    f(x) + n shifts n up.
    f(x) – n shifts n down.

    Along with “multiply x by integer to make a steeper slope, by a fraction to make a flatter slope”.

    As Jill points out, the tutor isn’t doing a good job.

    He also mentions the kid had to graph both equations separately because they couldn’t be super-imposed. Not with any TI I’ve seen, so I’d be interested in knowing what he was using.

    Calculators are an outstanding tool. Not everyone gets concepts without a strong visual. I have my students start with x^2 and then test all the ways to shift it.

  7. “Calculators are an outstanding tool. Not everyone gets concepts without a strong visual”

    Blackboards serve such purpose as well, but they are generally used as bulletin boards these days so there’s not much room to write. Calculators are overused in beginning math classes and while they can be used effectively, they are often not and very often used as a “thinking aid”.

    I disagree that Kohl was a bad tutor. I think he was very good. He was trying to show the student the fundamentals. Please don’t tell me that calculators are the “new fundamentals” and that mathematic principles are not fixed. I’m in ed school and hear that crap all the time.

    I’m about to take the Praxis II in math today. It REQUIRES a graphing calculator. Taking the sample test, I noted that some problems can be solved by plugging in the answer choices into the calculator to see which one works to solve the problem. For other problems it no longer makes it necessary for one to know how to factor a polynomial equation to find its zeroes.

    I tutored a student once, who when the calculator was turned off, actually was quite good at doing calcs in his head. Once he turned it on, however, his mind shut off. One time, faced with the equation 30x = 210, he reached for his calculator. I stopped him and told him he could do this in his head. I pointed out the 21/3 relationship inherent in 210/20 and then he got it. Similarly, he had a tough time with breaking down radicals, like square root of 48. Break it down into factors like 16 x 3, and then you’ll get 4 times square root of 3. These used to be drills, but drills are said to deaden the mind. In my case, I don’t waste precious time figuring out how to simplify these things, and I can solve equations much more quickly. As a tutor, I try to get students to that point. Using a calculator can deter in such cases. Its use must be judicious.

  8. Not only they not learn to analyze the problem, but there is no sense of the magnitude of the expected result.

    Being an engineer of the age that goes back to the slide rule, you rapidly learned to estimate the result to be expected. It was a necessary part of the process.

    If you don’t know multiplication tables you can’t do this. Calculators make the estimation part of the process seem to be unnecessary, but it still is a key part of the process.

    Failure to teach manual graphich and the multiplication tables is a really good way to produce graduates that can’t do much but be english majors…

  9. Robert Wright says:

    Bill, that would be English majors, not english majors.

    But that’s OK. My superintendent wrote in his May newsletter,

    “….the students who come to their schools have a variety of needs including learning to speak English, poverty, and special education programs.”

    Many of us are on the verge.

  10. “I’m in ed school and hear that crap all the time.”

    Well, anyone foolish enough to go to ed school deserves whatever crap he gets.

    He should have taught the student the formulas for determining the effect on the curve. He could have then explained how to analyze the problem within that context.

  11. “Well, anyone foolish enough to go to ed school deserves whatever crap he gets.”

    Since I wish to teach high school math, please tell me any alternatives you know to ed school. Sounds like you teach; how did you get out of going to ed school?

  12. So, Robert Wright, are you saying that schools don’t teach students to speak “poverty” and to speak “special education programs”?
    Would that you were right.

  13. Actually, Barry, there are a bunch of alternatives in various states – you might want to see if there’s something around where you are, ESPECIALLY for people looking to teach high school math.

    Usually these alternatives are called “alternative route” or perhaps “alternative path.” I’ll look up the info I have; there’s an organization that specializes in alternate route teacher certification, too.

    Usually what these programs do is give qualified people (that would be you) a temporary certification so you can start teaching.

    Then they might have some kind of mentor program along with a night class or two, depending on the state.

    Ed’s assistant, who has a Ph.D. in history, took a job teaching high school in NJ. He has a mentor teacher (I think) and he takes a class one night a week at state expense (I believe).

    I don’t think he had to do that for very long.

    You end up with full certification.

  14. “….the students who come to their schools have a variety of needs including learning to speak English, poverty, and special education programs.”

    We get that stuff constantly.

    It’s just amazing.

    Meanwhile the English teachers are giving our kids’ papers grades of Cs and Ds.