Calculators: Useful or not?

In response to Konstantin Kakaes’ Why Johnny Can’t Learn Without a Calculator, math teacher Paul J. Karafiol argues that Calculators in the classroom are useful.

Teaching math requires actually understanding math, and people who understand math have always been in short supply, in and outside of the teaching profession. So a different, simpler explanation for the failure of students to learn math is that there aren’t a lot of excellent teachers out there teaching math. Technology doesn’t enter into the picture.

Where it does enter the pictures is in a new and completely unexpected change in mathematics education. Excellent teachers who use technology can increase access to higher mathematics for students with poor computational skills, by allowing these students to reason about concepts without getting bogged down in computation. This year, my AB Calculus class included some students who couldn’t reliably add fractions. By the end of the course, almost all of them could explain what the derivative of a function means (in abstract and contextual terms), how it is calculated, and what it could be used for. They could do all this because they used calculators with computer algebra systems—calculators that give algebraic answers, not just numbers—to do the heavy lifting.

Finally, Kakaes never engages what is, to me, the central question that technology poses to the mathematics teacher, namely, what of the traditional pencil-and-paper mathematics is worth teaching?

“The argument should be about when and how often students should be taught to use their calculators,” Karafiol writes.

Kakaes responds here.

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  1. Elizabeth says:

    Somehow, my generation learned to use “technology” without meeting much of it until college. Technology changes, but basic math does not. I believe there is no place for calculators in elementary school, and their use should be limited through high school.

  2. Obi-Wandreas says:

    My general rule of thumb is that a student should not be allowed to use a calculator until they don’t need it.

    If the problem is to find the surface area or volume of a cylinder, for example, I have no problem with the use of a calculator , especially when computing the final step (in which they would use the pi button on the calculator to maximize the number of digits). In cases like that, my students are still required to correctly substitute, and work out the intermediate steps by hand. In fact, we spend some time leaving the answers in terms of pi before they are allowed to touch a calculator.

    For log and trig, also, the calculator is key. The use of tables in prior years was simply a function of necessity – there is no real useful mathematical skill related to hunting and pecking around a table of numbers. I have no problem with allowing trig tables to die an inglorious death.

    I am very much against the use of calculators for graphing, however. The interface is lousy, the display is atrocious, and trying to get any useful information is much more of a headache than its worth. Graphing on a computer or iPad can be useful. Trying to do so on a calculator, however, is an exercise in frustration.

    In general, calculators can be very useful, if used correctly. Just like using the elevator all the time to go just a few floors leads to a weak body, however, using a calculator all he time weakens the mind.

  3. GEORGE LARSON says:

    ” This year, my AB Calculus class included some students who couldn’t reliably add fractions. By the end of the course, almost all of them could explain what the derivative of a function means (in abstract and contextual terms), how it is calculated, and what it could be used for. ”

    I do not understand why a calculater would have aided in this. I would give the credti to the teacher, not the tool.

  4. Mark Roulo says:

    By the end of the course, almost all of them could explain what the derivative of a function means (in abstract and contextual terms), how it is calculated, and what it could be used for.

    I do not know what the author means by “contextual” in the quote above.
    But …
        (a) Explaining what the derivative of a function means,
        (b) Explaining how it is calculated, and
        (c) Explaining what it could be used for

    Should take no more than two or three days.

    The trick to differential calculus is:
        (1) Being able to actually calculate the derivative for lots of different functions, and (maybe)
        (2) Being able to prove why it all works.

    My guess is that very few students actually master (2). Not even before graphing calculators. Most students just trust the teacher that the whole thing works …

    A sufficiently powerful calculator can handle (1).

    Was the Calculus AB class mostly focused on how to use a modern graphing calculator to solve calculus problems? That seems unlikely …

  5. The calculator (and I do own a TI-89 Platinum, which actually gets used perhaps 2-4 times a year) should not be allowed into elementary or middle school (when the student should have learned their basic math skills, etc).

    The fact that students are taking AP calculus and not being able to add fractions consistently is something I as a lay person would have a great deal of concern with. The issue of adding fractions is something I learned to do in elementary school in the late 60’s to early 70’s, and as other posters have commented, when deli clerks have no idea what 1/3, 2/3, 3/4’s of a pound are in digital (which is something everyone should know how to figure out in their head, btw), I can see what calculators have done to the math ability of an entire generation (or more) of persons.

    Btw, when we needed to multiply something by PI, we usually were told to use 3.14 or 22/7 to do the math, so the actual math wasn’t really that complicated.

  6. Elizabeth says:

    I second the use of calculators as a substitute for log and trig tables.

  7. I think that calculators are a bad idea, but I would take it further, along the lines Bill mentioned above. You will have a big advantage in the world if you can do simple calculation in your head. I see this all the time with younger folks.

    It’s very helpful to be in a meeting or hallway conversation and be able to calculate things. To be able to say something like, “That part is six dollars? That’s going to push our COGS (cost of goods sold) up to $24 and knock our gross margins down to under twenty percent, I don’t think the boss is going to go for it.”

    If you can do this and your coworker can’t, you have a distinct advantage. And really, it’s not like it’s hard. It just takes a little practice. You don’t have to be able to figure it super accurately, just get close. Nearly all of my most successful colleagues can do math in their head, it’s a basic business skill.

  8. LSquared says:

    All very tricky. Yes, there are a few people who just can’t seem to learn computational skills, although they can learn some of the meaning-based things (as described), but is that really the same skill? It makes it tricky for the hiring office/admission to college office if “Calculus” on a transcript could mean: “understands what a derivative is and how to use it” or it could mean “understands what a derivative is and can calculate a bunch of stuff”. These really aren’t the same skills, and yes, there’s a need for people who can “calculate a bunch of stuff” as opposed to being able to successfully ask a calculator to “calculate a bunch of stuff”.

  9. One issue I frequently see is that the people who can’t calculate stuff often have a hard time recognizing if their error is wrong, even when its off by an order of magnitude or higher. People who can do those calculations often are better at saying “this doesn’t look right”, trying it again, and fixing their mistakes.

  10. Is it at all significant that the Gregory Euclide uses a pretty primitive technique (omigosh – the man is drawing with a pen!) and gets praised for it. But somehow knowing how to add fractions is not a good thing.

    disclaimer – I was a college Latin major and teach art history…and I can add fractions.

  11. Without a lot of practice, and maybe even hands-on lab time for things like metric units, students don’t develop a good number sense. It seems like it wouldn’t be a big deal, until a nursing student calculates that a patient needs 2000L of medicine and can’t see that the answer has to be wrong. They assured me that it wasn’t an issue, since for calculating dosages ‘there’s an app for that’. Only when we’re not in a crisis that prevents recharging…and have you ever watched the good carpenters on DIY/HGTV shows? Fractions everywhere, and angles, too.

  12. The guy who loads big racks of bread onto trucks at the local distribution center has to be able to add, subtract, and multiply in his head when both hands are managing the racks. He’s an older guy; he says his job is safe because none of the younger people can do that.

  13. Supersub says:

    It’s stories like this that make me somewhat eager for a massive solar flare to scramble our electronics…

    • Sean Mays says:

      Why wait on nature? The same thing can be done through EMP!

      • SuperSub says:

        True…but I don’t have any spare nuclear devices hanging around to detonate in the atmosphere…

  14. I actually know teachers personally who consider not allowing a child to use a calculator in their classes – at ANY age, even Kindergarten – to be child abuse. Seriously!

  15. therese says:

    My son is one of those who cannot memorize his math facts, and has to count even simple adding on his fingers, but has been taught a strategy for this. He is mildly dyslexic and transposes numbers frequently. Math homework goes extremely slowly unless he uses a calculator.THere is a method of teaching for this kind of disability called Making Math Real.
    My son understands the algebra. but he gets bogged down with the arithmetic. He built his own computer at age 14 and wants to be a programmer. The calculator is indespensible to him but he does tend to take advantage of it to do things that he should be learning to do without it.

  16. Writing in math journals also provides an alternative mode of mathematics teaching and learning for those creative students who have not previously been reached by more traditional, structured, or linear teaching strategy. Because a student often knows more than he or she can or is willing to explain verbally, journal writing provides a safe venue for students to express what they know or do not know about specific math concepts or problem solving.