My 14yo is a highly gifted math student. 4 years ago, if I’d asked him how many ways you could choose 3 people out of a group of 8, if order didn’t matter, he’d have reasoned something like this:

Well, you have 8 choices for the first person, 7 choices for the second, and 6 for the third. So that’s 8x7x6 possible ways. But that counts the same sets of 3 people in several different orders, and since order doesn’t matter, you have to divide by the number of different orders you could have chosen the 3 people. That’s 3x2x1 since there would be 3 choices of which you chose first, 2 for which you chose second, and 1 for which you chose third. So that’s (8x7x6)/6 = 56

Now he’s in honors geometry. They are learning combinatorics and probability first. He asked me a question about a homework problem that was bothering him, so I asked him, “well, what’s the formula for combinations look like?” His reply, “I don’t know, I just know how to type it into my calculator.” ARGH!

I can understand that for large numbers, it is efficient to know how to do such a thing on a calculator, and in one easy step even. But I can’t stand that they are being taught to use calculators to mask the fact that they have NO CLUE what they are really computing! This is a bright kid. He was perfectly capable of understanding the ideas behind permutations and combinations many years ago, and worked with them successfully in many contexts (mainly contest math). I assume that every student in honors geometry is perfectly capable of understanding this topic, which is well within the reach of most middle school students. WHY are these students being taught to be trained monkeys and push the right buttons on their calculators without knowing what the button does?

—–

My first clue that calculators were being over-used by high schools was when I was proctoring a computer science exam at a very selective engineering college. I watched an undergraduate pull out a calculator to multiply a natural number by 10. Really. I was dumbfounded.

—–

Fast forward many years to when I began volunteering to help with math teaching at my kids’ K-8 school. I handed out a mixed set of MATHCOUNTS problems for the older (10-13yo or so) kids to work on. I didn’t specify whether or not calculators could be used, and apparently they were used to being allowed to use them all the time. There was a problem which required the kids to read a bar chart and then figure out the percentage of people who fit into a particular category. Conveniently, 100 people had been surveyed. So, they read the correct bar and found that 8 people were in that category. Then they took out their calculators in order to calculate 8 out of 100 as a percent. Several of them got it wrong! coming up with answers like .8% or 80%. They had recently been taught how to compute a percent using their calculators (and yet had clearly not yet mastered the skill) and it never occurred to them to engage their brains first (or even second) and just write down the obvious answer (or sanity check what their calculator had presented them with).

So… I went on a rampage against calculator use in our school. Their regular math teacher argued that she didn’t want to “torture” them with messy calculations. This is fine. There is a time and a place for calculators, and calculating a messy percent is a perfectly good place for them. Once you’ve shown mastery of long division (and all of these students had done that!) I agree that there is no need to make middle school students regularly work math problems that require messy long division. But the solution is not to just throw a calculator at it. A better solution is to, at least most of the time, engineer the problems so that the numbers work out neatly and no messy calculations are required. Once students are used to doing the problems by hand, if at some point you want to introduce messy ones and allow a calculator, at least the students, with a good feel for what the numbers should look like, have half a chance of noticing if the calculator comes up with something absurd. Fortunately, the teacher agreed with my arguments, and they now use calculators much less frequently.

MATHCOUNTS is interesting in this regard. Their contests have sections where calculators are not allowed, and others where they are. On the “no calculators” sections, they fix it so that the numbers are usually (but not always) nice. On the “calculators allowed” sections, they don’t. I suspect that the reason they design any part of the contest to employ calculators is that MATHCOUNTS is run by engineers, rather than mathematicians. In any case, the free problems they sent to schools to use with their students during the year share this split between the ones with “nice” numbers where calculators should not be needed, and those with messy numbers where they are, frankly, pretty useful. I tend to use mostly the “no calculator” problems to get around the whole issue. I really prefer to make them practice using numbers mentally and by hand as much as possible. I really do think it makes a difference. K-8 students should be working on becoming quick and comfortable with numbers, whatever else they are working on.

There is an argument that calculators are important tools, and students need to learn to use calculators properly and powerfully. This is true, but only to a degree. Yes, calculators are tools that students should learn how to use, BUT, not as a substitute for learning how to work with numbers by hand. There is a skill to working with a calculator, knowing how to use the tools it provides effectively, and absolutely knowing how to sanity check your answer. But when calculators are used without understanding, as a substitute for knowing how to do the problem by hand, they’re horrible tools that lead to innumeracy.

If I had my choice, calculators would not be used at all in K-3, and only very minimally in grades 4-6, in separate “how to use a calculator and estimate if it gave you the right answer or not” type lessons. And then, maybe, pull it out occasionally when working on an interesting real-life application that brings in big, messy numbers. But generally, K-6 students should be using their brains, pencil and paper as their “calculators”.

For pre-algebra students, I’m not opposed to a little more calculator use, but again, only in specially selected situations, when messy numbers are being used either because of the nature of a real-life application, or explicitly for calculator practice. The rest of the time, practice working with the numbers by hand is extremely important and valuable.

1. September 18, 2007 3:32 pm

I would also say that a student should never use a word processor until they’ve learned to write neatly. Penmanship is underrated.

2. September 18, 2007 9:05 pm

Calculators are relied on way too much. I’m reviewing trig in my AP calculus class and the kids are upset because I won’t let them use their calculators. They used them so often in Math B that their trig memory is non existent.

Our first test is tomorrow. It is a no calculator test and includes graphing and transcendental functions. They are not happy. I have only let them use the calculator to check and demonstrate so far. As always, it will be an interesting term.

3. September 18, 2007 9:44 pm

Pissed Off — does you department have any kind of policy on calculator use? Is there any chance that you could appeal to the Math B teachers to use them less? What do you think can and should be done about this?

4. September 19, 2007 8:45 pm

Math B is a NY State test that was designed with the calculator in mind. We can teach multiple approaches to problems, but we would put our own students at a disadvantage if we were to artificially limit their use below what the test demands, which is a lot.

Math B is the reason I teach a (completely separate, and voluntary) once a week elective in, essentially, hand-graphing.

5. September 19, 2007 8:56 pm

Thanks for the clarification, JD. Is Math B essentially Algebra II? Is there a class between Math B and AP calculus (pre-calculus?), for the “calculus-bound”? And if so, could calculator-free practice of the prerequisite skills for calculus be fit in there? I think it’s great that you offer a separate elective in hand-graphing to make up for the calculator-focus of the other course. You seem like a very cool and dedicated teacher. I hope the kids and parents appreciate you!

6. September 19, 2007 9:30 pm

My school makes the kids take a year of pre-calculus before AP. In that class, they do more work without a calculator but, in my opinion, they still use it too much. The calculator is essential to being able to solve more difficult problems. In fact, there are problems that cannot be solved without it. Kids need to do both kinds of work.

7. September 19, 2007 9:34 pm

Very cool. Though my students might disagree. Math B is a mishmash of topics from geometry, algebra 1, algebra II, probability, stats, hmm, something else. NY has gone two miles wide, and about as thick as a TI-83. Good for us, we’re dumping this lousy test. (Bad for us, we wouldn’t trust the people designing the substitute exams, Geometry and Alg II/Trig, to tie their own shoes)

In some schools the sequence might be:

AAAABBBB (one letter per term)
or
AAABBBCC (c for calc)

We do

AAGGTTPP (Alg, Geo, Alg II/Trig, Precalc) or
GGTTPPCC

But we are not exempted from NY State testing.

Not very clear, but at least you get an idea.

8. September 19, 2007 10:01 pm

Thanks for the clarification. I agree that the calculator enables kids to do stuff that they couldn’t do without it, and that makes it a great tool. Finding balance is hard. But when I see kids being taught/encouraged to use a calculator exclusively for things that just aren’t that hard to compute by hand (8C3, for example — when you do those by hand, many of the factors cancel out and they’re quite easy to compute for reasonably sized numbers) it leads me to believe the correct balance hasn’t been found. 😦

9. September 23, 2007 4:48 pm

In his 17th Carnival of Mathematics post, Dave Marain points out that the AP calculus exam (like the MATHCOUNTS contests) is a two-part test, which uses a graphing calculator on one part and on calculator on the other. It does seem like the way to go to make sure that students both learn how to use the tools well, but also understand what they’re doing and how to do it for “cleaner” examples without the calculator.

September 23, 2007 10:35 pm

I recall an old joke: As far as many American schools were concerned, the 3 R’s were Reading, wRiting and Remembering to bring your calculator.

11. October 4, 2007 10:51 pm

JD, I just rescued your Sept 19 comment from my spam queue. Not sure why it landed there since it doesn’t have any links, which seems to be the surest way to get grabbed. Your math sequence looks like a DNA sequence. (Actually, it looks like what we have here — and what I thought was “normal” for US high schools.)

12. October 5, 2007 6:22 pm

Ah, the strings of letters must look spammy.

There are two semi-normal sequences in the US:

(pre-algebra)
algebra
geometry
algebra II with trig
precalculus
calculus

or the same with trig with precalc instead of alg 2 and precalc (gets to calculus faster, with less proof along the way).

In New York State, that sequence is coming back, but for over two decades it has been the exception here.

Jonathan

13. October 5, 2007 6:49 pm

Our Honors sequence is the one you listed above, with algebra (I) taken in middle school. The honors sequence is the only one that includes pre-calc and calculus. The regular academic sequence has 2 algebra II with trig classes, though with a very high score in the first, they can jump to honors pre-calc.

14. October 6, 2007 2:18 am

I see the logic for running to calculus fast (prestige), but I can’t see the mathematical or pedagogical justification. Colleges universally look for 3 or 4 years of college prep math; better colleges and engineering-oriented schools tend to want 4. But very very few ask for calculus (and not necessarily the strongest schools).

Let them learn well, rather than fast.

15. October 6, 2007 7:15 am

Are you saying the normal honors sequence shouldn’t go up to calculus, or that kids not in the honors sequence shouldn’t worry about it?

16. October 6, 2007 9:59 am

The latter. I can see a kid starting algebra in middle school, and reaching calculus. I can’t see a kid going from geometry to one intermediate year, and then calculus, especially since our algebra (and geometry, etc) now also include so much probability and stats, etc. I do not understand how the kid can get a good feel for number, for variable, if they are being rushed forward. Scope trumps depth? Not for me.

17. October 6, 2007 1:00 pm

I’ve heard of mathematically gifted kids skipping pre-calc and going from Geometry, to Alg2 with trig, to Calculus. I’ve heard that in some schools precalculus is mostly a review/consolidation year, and kids who are “naturals” at math don’t really get much out of it, especially if they’ve gone ahead and learned some things on their own for math contests etc. I don’t know if that is the case in our system or not. My oldest is in Geometry at the moment (though he has over 100% so far due to extra credit) :-} But you know that in general I am not in favor of rushing and skipping depth.

18. October 6, 2007 2:57 pm

We also have much repetition from Algebra II to Precalc, but Precalc treats the topics more theoretically, with more depth, and reaches a bunch of topics that do not show in Algebra II, including inverse trig functions, polar coordinates, complex numbers (as more than objects to be manipulated), sequences, series, limits. In fact, the slow, detailed work on limits is crucial, imo. Also, our precalc teachers have been introducing the derivative at the end of the year (right after sequences and limits), giving everyone who is not going into high school calculus a leg up for their college class, and for the kids doing AP with us, giving them a slightly advanced place to start that very difficult September.

We do the extra credit thing, too, but not in a way that raises eyebrows (tests in our school are hard, but we often modify the scoring so that more than 100 points are available, so that the kids are not at a competitive disadvantage when compared to neighboring schools where they are not tested as hard).

Jonathan

19. October 7, 2007 7:53 pm

The extra credit actually hasn’t be extra busy work since the first time I wrote about it. There are sometimes a few bonus points on a test or assignment, etc.

When I was in HS, limits were part of calculus. It’s interesting that they’re done in advance now. But that was in Ontario, and we had an odd math sequence anyhow, and 5 years of high school at the time. Grades 9 through 12 the math classes were just called “grade 9 math” etc. but 10th grade was geometry and I think 11th was trig. I have no idea what was in 12th. Grade 13 then had 3 math classes you could take: “functions and relations”, “algebra” and “calculus”. Some kids would take them all at once, but many kids managed to get a year ahead in math at some point so they could take one in 12th grade (usually functions and relations) and only 2 in grade 13.

20. October 16, 2007 6:29 am

RIght on! Calculators were invented by vampires to suck your brains out! (At least until advanced algebra or trigonometry). Doing math mentally helps develop skills beyond simple arithmetic. It is great all-around mental exercise, and helps you learn about patterns.

Giving a kid a calculator is like giving him/her a gun. Sure, there are uses for guns, but not for kids.

Hotcha!

Brian at http://mathmojo.com/chronicles

21. April 13, 2009 7:21 pm

I know I’m a year and a half late to this conversation but I wanted to add my POV in the whole calculator scandal 🙂

I teach in a Baltimore school with NO honors class, an no “levels” at all – and lots of special education inclusion. Since IEPs say that the special education kids need calculators I just give ALL of my kids calculators since I don’t want to put the spotlight on any one child. This seems to be standard practice at my school.

I’m strongly, strongly pushing for an honors track, where I’d like to rely a whole lot less on calculators. But then again, I’d say that a lot of the math TEACHERS at my school can’t do a lot of the math without calculators, which is a whole separate, probably worse problem.

22. April 14, 2009 11:01 pm

I’m saddened that you feel the need to aim your class at the least common denominator. Just because some kids have IEPs that allow them to use calculators does not mean that no one in your class needs to develop number sense and an ability to do computations by hand or in their heads. And it’s not only honors kids who need this. Every child without a disability preventing it should be expected to do computations without a calculator.

What grade level do you teach?

April 23, 2009 6:10 pm

An important point mathmom I hope you dont loose sight of – calculators are essential propelling you forward into higher level mathematics. I could not imagine if I had to do vector calculus or diff eq by hand. It is ridiculous to do by hand equations at this level… just a different point of view to balance out this discussion.

24. April 23, 2009 6:38 pm

Absolutely, George, calculators (and/or computers) are indispensable and necessary tools for many aspects of higher mathematics. But they shouldn’t be used by elementary and middle school kids who have not yet developed number sense, and they should not be used even in high school or beyond to replace understanding basic processes. Because you have to have the basics and be able to go back to first principles sometimes to make a connection or solve a problem that isn’t a straight computation. And if all you know about a process is how to press the calculator buttons, then you’re screwed if you ever have to do anything more involved than that.

25. April 9, 2013 2:09 pm

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