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Calculators are like Bicycles
April 21, 2004
We could go on for days about what we really think of calculators in K-12 math classes (we think they're great for science classes, especially stochiometry in Chemistry), but suffice to say it just might be one of the signs of the Apocalypse.A recent entry by Joanne Jacobs quoted a study demonstrating that students who use calculators do poorly in arithmetic when forced to do without. While School of Education wonks insist that students be "exposed to technology" at every level of K-12 education, we know that putting calculators in the hands of students learning arithmetic is incredibly shortsighted and harmful.
Our analogy relates to a child learning to walk.
Imagine taking a child who's pretty good at crawling, and starting to pull himself up on things, and placing him in a modified tricycle, one with a supportive cradle for a seat so he can't fall out. Soon he'd learn to push the pedals, randomly at first, then with purpose, as with steering the handle bars.
In no time he'd be zooming around on his little three-wheeler, after which he can be moves up to a real tricycle with a bicycle-style seat. From this he could evolve to a bicycle with training wheels, which can be raised incrementally so that more and more of the time he's on two wheels, after which they can be removed completely.
Now, if the child were placed in these three- and two-wheelers, each and every time he was attempting to stand or walk, there's little doubt that he'd quickly be far more skillful in pedaling himself around than two-footing it. Sure, the kid might learn to walk (probably looking like Frankenstein's monster), but he'd probably never learn to run.
Why bother when it's faster (and easier) to bike?
And so it is with calculators.
Personally, we think calculators (and just about any other technology made possible by the transistor) are great, and they sure do save a lot of time.
But that's not the point behind teaching children arithmetic in school.
The point of doing paper-and-pencil arithmetic (for example, finding the product of 47 and 9) isn't to actually find the product (like we didn't know what the answer was, and that's why we have kids do these problems). The whole purpose of this exercise is to practice a skill accurately to the point of automaticity.
Memorizing the times tables (for example, up to 12x12) is a gateway to so much more. Once a child has memorized every single fact contained in this grid of 144 products, then simple division (where both divisor and quotient are numbers between 1 and 12) is only a matter of backwards thinking.
For example, in picturing the number 24 one might think of 4x6, 3x8, or 2x12 (and of course the commutative versions of those three). Thus 24 divided by 8 is a fairly simple matter.
But doing longer division, especially when it involves remainders, is harder, because the dividend often is not a number found in the times tables. And very quickly, if the lesser skills of simple addition, subtraction, multiplication, and division are not mastered to automaticity, it quickly becomes a chore.
One reason why paper-and-pencil long division is going the way of the dodo bird is because it involves too much computation, 100% of which has to be done accurately, or the whole answer is wrong.
Take a simple problem of dividing a three-digit number by a one-digit number, for example dividing 301 by 7.
7 into 3? No.7 into 30? Yes, 4 times.
7x4 = 28
Subtract 8 from 0? No.
Borrow a 1 from 3 (make it 2), write it next to 0 to make 10.
10-8 = 2
2-2 = 0
Bring down the 1.
7 into 21? Yes, 3 times.
7x3 = 21
21-21 = 0
Depending on how you count, it takes about ten steps, most of which are calculations, but even so this problem should be done in under 15 seconds.
Truth be told, it should take fewer steps the more fluent one is with numbers, for example the step of subtracting 28 from 30 can be done as a quick mental math step without all the borrowing. And of course there's no real need to actually subract 21 from 21 at the end.
But the point is that this fluency cannot come if students aren't asked to solve a great many paper-and-pencil problems (yes, this means worksheets and flashcards) over a number of years of studying mathematics.
Unfortunately in a great many places, the handheld calculator is seen as a way of relieving this drudgery. Students may still be given worksheets, but are permitted to use calculators; a pointless exercise.
We mentioned the shortsightedness of this strategy earlier. Many fresh math teachers (especially in our inner cities where calculator use seems to be at pandemic levels) see nothing wrong with the use of calculators to do simple basic arithmetic, for the simple reason that the calculator is fast and accurate. So much time is freed up for higher-order thinking skills!
Unfortunately, middle and high school math desperately requires a certain fluency with numbers which dependence on a calculator simply cannot permit.
In late elementary school and middle school, students learn things like manipulating fractions (where finding the lowest common denominator is a key skill, one that requires being able to instantly look at several numbers and mentally seeing the greatest common factor or least common multiple) and using square roots (also requiring the ability to instantly factor numbers).
And in late middle school or early high school there's algebra, one place where the calculator isn't much help.
While we're sure a calculator exists which can factor the binomial 49x² - 25 such a beast is not in wide circulation, and if it were, most kids wouldn't know how to tell it to solve such a problem. But if a student has some number sense the answer of (7x + 5)(7x - 5) should be completely obvious just by inspection.
Unfortunately most of the folks advocating rampant calculator usage in the early grades probably can't do algebra, let alone trig or calculus.
In having a child master the calculator, that child just might become its hobbled slave.
Agreed 110%! And that is exactly how I learned math, and how we are teaching our HS'ed kids. The differnce is that while I spent my 5th grade year doing multiplication tables, my kids mastered them in the 3rd grade.
Chris O'Donnell April 22, 2004 08:24 AMMy daughter's in 2nd now, and she's got her multiplication tables memorized. She's working on long division now.
Her class? No, she's doing this on her own with homeschool workbooks. Her class is still struggling to learn borrowing for subtraction. But they just recently had a session on division. When did they cover multiplication, you ask? They haven't. Syllabus courtesy of Everyday Math. Makes me nauseous.
Claire April 22, 2004 03:48 PMEven if you teach kids to do arithmetic in their heads, they can lose it if it's not reinforced. When I was in sixth grade or so, my parents decided it was okay for me to use a calculator. (This was in ancient times, when calculators weren't routinely allowed in primary/middle schools.) I gradually became totally dependent on the damn thing, to the point where I couldn't look at an answer and know whether or not it was the right order of magnitude -- whether or not I'd made a mistake pushing the buttons. So I couldn't to arithmetic well with a calculator or without.
This persisted until my sophomore year of college, when I took a surveying course (I was thinking about studying civil engineering at the time). Outside, in the sunshine where LCD displays were useless, I had to do basic arithmetic -- lots of it -- with pencil and paper. It was really, really painful at first, but by the end of the quarter I'd finally regained the skills I'd learned in primary school.
Since then I've made a point of doing arithmetic by hand occasionally, just to keep the skill. I'm also quite accurate with a calculator, because I can do enough arithmetic in my head to know sort of what the answer should be.
Karen April 23, 2004 01:53 PMyeah, pretty much, I'm a victim of this whole calculator crisis... I was ecstatic in the 6th grade when I learned that, being in an advanced math class, I would be able to use a calculator... unfortunately, i'm really slow with basic arithmetic (but really fast at the conceptual stuff). Why? Because you can't think about X's and Y's on a plain calculator, so I had to learn that stuff in my head... somehow, that doesnt manage to carry over mentally with regular numbers and basic things... The worst part is the no-calculators section of the AP Chemistry exam... i'm gunna be slowed down by the arithmetic...
Justine April 24, 2004 02:47 PM"I gradually became totally dependent on the damn thing, to the point where I couldn't look at an answer and know whether or not it was the right order of magnitude..." Like the time I was at the checkout counter in Kmart with two $0.99 items. The clerk punched some buttons and something like $1.61 popped up on the cash register. ($1.49 plus tax.) I told her that couldn't be right. She studied the tape with a blank look and finally took a cheap calculator from the checkout aisle sales display. Yep, when two one-dollar items add up to a dollar and a half, your computerized cash register is broken...
What really struck me about this, though, was that I was just back from a trip to Korea. Street vendors in Seoul can convert dollars to Won and back in their heads, accurate to a penny or two. Some of the stores had cash registers, but the only time I saw a calculator in use was by the owner of one up-scale clothing store - and he'd worked in auto plants in Detroit for a dozen years...
markm April 24, 2004 04:05 PMYou hit two of my soap box issues, calculators and long division.
As a high school math teacher, I see students everyday who have been ruined mathematicaly because of the calculator. They have no concept of number relationships. Need to find half of 10? Grab that magic box. No idea how much 5 squared is? No problem, punch, punch, equals.
Every chance I get I make them work without it. Boy, talk about complaining!
I think long division is the perfect math problem. It involves estimation. (How many times will 17 go into 30?) It involves evaluation. (Did I estimate too high? too low?) It involves following steps. (Bring down the next digit and repeat.)
Funny, NC's objectives include estimation, evaluation, and following procedures.
When this subject has come up on various fora, I have _never_ yet seen a teacher rise to defend the current widespread use of calculators in school (as well they shouldn't.) This leads me to be mystified as to how calculator overuse became so popular in the first place. Does anybody have any insights on that?
Steve LaBonne April 26, 2004 11:17 AMI recently talked to several 5th grade teachers who are ambivalent about calculators in the classroom. They insisted they never used calculators until calculator use was permitted on state assessments. Then they felt it was very important for the students to be comfortable with the tools.
I don't necessarily agree with this perspective (or tests that allow calculators!), but i think i understand it. If "calculator literacy" is expected on tests or in other ways in life, then many teachers probably feel they're doing their children a service by letting them practice with those tools.
Nina D. April 27, 2004 02:01 PMHow long does it take to learn to push buttons on a calculator? Especially if the students have solid math skills _before_ they pick one up?
Steve LaBonne April 27, 2004 02:04 PMSteve,
It's more complex than that.
First, TI calculators are pretty powerful little machines and it does take practice for 10-year olds to use graphing functions, etc.
Second, i am in no way am i endorsing this reasoning, but the teachers' perspective was this:
The students will do better on this particular statewide assessment if they learn to do division of fractions with the calculators. This allowed more time for practice and developing understandings of other mathematical content.
The teachers felt pressured to "cover" an unreasonably large amount of content before this statewide assessment and calculators allowed them the time to develop "solid math skills" in other areas.
It's a poor test (why calculators?) compensated by poor teaching strategies. A worst-case scenario for childrens' learning.
Nina D. April 27, 2004 08:46 PMWhat are fifth graders doing using a calculator for graphing? Everything they need to know about this can and should be done by hand.
So little Johnnie has to learn the difficult, but "solid math skills" of learning the graphing functions on a calculator (following the directions), but is using the calculator to avoid learning the solid math skills of dividing two fractions by hand. Which do you think is more important as a lead-in to algebra?
I have seen this sort of discussion for the last 30 years - with calculators and with computers. There are many things that require a lot of hand work before one should use a calculator or computer. However, I have also seen cases where schools (and colleges) emphasize paper and pencil work far longer than is necessary for proper understanding of what is going on.
There is no exact boundary line. How many digits of long division do you have to do (and how many problems) before you decide that it is OK to use the calculator? Do you really need to do square roots by hand? Unfortunately modern reform math curricula for the lower grades are nowhere near any sort of reasonable boundary. Many don't even see any boundary.
They think that calculators makes math easier. Actually, calculators should make math more difficult and challenging. Calculators should not take away, they should add. Calculators were just coming out when I started college. They actually improved teaching and learning. Tests were changed to require algebraic manipulation rather than number crunching and homework changed from 5 pages of hand calculations to 30 - 40 pages of calculator number crunching using much more rigorous methods that then became reasonable to do. The key point is that we already knew how to add, subtract, multiply and divide. For kids, this isn't true.
The question is not just what you take away, but what you add. Replacing division of fractions by hand with learning graphing functions on a calculator is a very bad choice. The problem is that modern math curricula have very skewed ideas of "solid math skills".
Steve April 27, 2004 10:57 PMYes, but when the math curricula are being designed and taught by people who admit that they hate math and aren't very good at it, it's no wonder the state of our kids' math knowledge is so abysmal.
Claire May 19, 2004 02:41 PM