Kids don't learn arithmetic anymore. Adults are forgetting everything they ever knew about it because calculators will do the computations for them. Fractions, signed numbers, decimals, etc, etc, etc can all be done on these little machines. After all, most of us use a remote to change the station when we are watching television and we certainly use a washer-dryer to launder our clothes instead of beating them with a rock on a river bank, so why not replace our brains with little machines that can be purchased for less than $20?
I don't have a problem with calculators if they are used to enhance learning, not replace it. Kids need to understand how operations with signed numbers work before they just start pressing buttons to hopefully arrive at a correct answer. They need to understand what happens when they find a percent of a number, know whether the answer should come out larger or smaller than the original number. They should know that multiplying by pi will give a number slightly larger than three times the original number. Too many times I have seen kids come up with totally insane answers and write them down because that is what the calculator gave them.
I do believe that arithmetic shouldn't hold anyone back. Not mastering fractions should not be equated with the inability to solve an equation. After years of trying, kids should be able to move ahead with a calculator if that is the only way, and I do mean the only way.
Calculators do have a place in higher mathematics. By freeing the individual from tedious calculations, more difficult conceptual problems can be studied. But, even in the higher level classes we are doing our students a big disservice by allowing a calculator on every exam. For example, kids no longer know trigonometric relations of quandrantal angles or of the special angles, and some of the beauty of math is being left behind. We used to teach them to find sin 15 by using sin(A - B) formula. Now, they just want to get the answer from their calculators.
The AP calculus exam is divided into four parts. Parts II and III allow calculator use. Parts I and IV do not. The calculator parts are the most difficult parts because of the concepts that are being tested. In my class, to prepare for this exam, I give both calculator and non calculator exams. At first, the kids whine about not being able to use the calculator. THEY HATE DOING ARITHMETIC! THEY DO NOT REMEMBER SIMPLE TRIG RATIOS! They soon find that the arithmetic they are being forced to do is the easiest part of the exam. I find that I have to show them silly calculation tricks to make the arithmetic easier, things that they should have learned (and are more than capable of learning) a long time ago.
So, what is the answer to all this? I don't want to do away with calculators. I just think calculators need to be used sensibly. Teaching must include calculator and non calculator questions. Exams must sometimes be calculator exams and other times be calculator free exams.
Mathematics is more than arithmetic, but, an understanding of arithmetic is vital to being successful with it. Technology must be used, but, it must not be used to the exclusion of everything else.
10 comments:
I just came from a message forum in which a math professor was saying that at his university they administer their own placment tests to the incoming freshmen and forbid the use of a calculator and the results are much less favorable than some students anticipate.
Besides, calculators are as old school as slide rules. Any real math geek uses Excel for graphing, number crunching, and checkbook balancing.
PS Is interpolation still taught?
No more interpolation. I'm not sorry to see that bite the dust.
My daughter was a math major and my son an engineering major (at different schools). Both were not allowed to use calculators in introductory calculus classes--a wise idea.
Here in Nevada, students are not allowed to use the calculators on the state tests. I do not allow my students to use the calculators all year, until we have reached the CRT's. Then they must show that they can do the arithmetic and processes on a test without the calculator before they are allowed to use them in class.
What suprizes me so much is that 7th graders can not do multiplication and division! I am having to do multiplication drills with them just so they know up to their 12 times tables.
While it makes teaching easier to just give the kids a calculator, it is only harming them by not learning the basics first.
In NY, kids can use calculators on state exams. My AP pushes them to the point where we teach kids to pass the exams by using the calculators. I got 27 out of 28 to pass the regents last year when they knew nothing. The sad thing about calculators is that they are even making the smart kids dumb! I can't believe how many of my AP kids have forgotten how to work with fractions.
What are you talking about?
The calculator sections are the HARDEST parts of the exam. They take the chance to ask really theoretical and abstract questions where a calculator couldn't possibly be useful anyway.
Like I would have loved those questions if not for the time factor.
(What was that FRQ #3 last year? That was the hardest problem on the exam and it was on the calculator section on the exam, but a calculator wasn't useful there because it was an arbitrarily defined function.)
Oh, not to mention that on #2 I screwed up totally (but still got my 5) because I couldn't remember how to graph piecewise. On the calculator. (I tried using the boolean notation, but it gave me multiple syntax errors -- I spent 8 of my precious 15 minutes trying to work my calculator problems out.)
Especially since you had to integrate across a discontinuous function and I practically had to reconstruct the antiderivative, mentally, in my head, without the use of the calculator.
Thanks a lot, calculator questions. If it weren't for those two pesky little calculator problems I think I would have gotten a 6 (well, if there were such a thing as a 6; getting a 5 isn't as fun when you realise that over 20% of test-takers get a 5 anyway).
The free response non-calculator ones were substantially easier. #6 was the easiest of all, because you could also mentally view the equations too. Having the major parts of the equation cancel out was beautiful, oh so beautiful!
#4 (as I recall, the one with the inflating hot air balloon) was a problem I really liked. It wasn't until I did this problem that I had one of them Eureka moments. We had done lots of related rates problems before, but it wasn't until that I was actually taking the AP exam that I realised that changing the "dx" to a "dt" in a related rates problem wasn't just an aesthetic thing to be done for consistency, and that there was a difference between integrating x with respect to dx as opposed to dt. (It's curious how you tend to discover a lot of things by thinking, "hey, that's strange....", even in a 3 hour exam with a 100 different questions.)
I tell my students that the calculator section is the hardest one also.
I make them do lots of practice both with and without the calculator.
I once observed a strong (B+/A-) precalculus student multiplying 6*7 on her calculator during a test. Later, I asked her "Carrie, if your calculator had answered something other than 42 would you have
a) decided your calculator was wrong and written down 42 anyway? or
b) written down what your calculator said.
She answered a (thankfully), but then quickly realized how silly it was to use the calculator since her answer would have been identical without it.
I don't know the answer to this problem. I do know, though, that the stronger math students are better at discerning when a calculator will help and when it won't. Which means that if we teach weaker students using calculators, and I agree that we have to sometimes, we should be sure to spend a lot of time teaching when it is and isn't an appropriate tool for solving the problem.
My five-year-old daughter is just about to start with a calculator at school. As a friend of mine, with slightly younger kids, said: "I'd refuse."
My theory is that the remedy is to introduce other things that are more visuo-spatial and tactile.
For example, she knows how to use an addiator. She knows how to form numbers on a soroban, but can't add with it yet.
My theory is that the more tactile nature of mechanical calculators gives you a better understanding of what's going on than an electronic one. Maybe there's money to be made in selling adding/subtracting slide rules...
I have another theory about the decline of number skill, as someone who was brought up with calculators, but is numerate nonetheless. Actually I think the main problem is the decline of cash as currency, so that people do not have to think about whether they have enough money in their pocket or if they have received the correct change. With plastic cards and automated bank statments, everything is calculated for us, and even when using public transport, many people buy monthly passes. In supermarkets, too, because cheeses and meats, etc., everything is prepackaged, so we don't learn how to buy in weights and measures and calculate prices. How many people today even know what half a pound of cheese looks like? or how much haddock costs per pound? My grandmother's everyday numeracy was brilliant, even though she had little education, because she needed it. These days, people can get by fine without.
The Calculator Home Page at calculator.org is a collection of resources related to Pocket calculators, online calculators, calculator software and units conversions. You can download the free version of Calc98 now, view the product documentation online, or Pocket PC version screenshots.
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