Why are students permitted to use calculators on AP math?

[sear_squid]

When I was in college, calculators were forbidden on all calculus exams and quizzes. In fact, many universities still forbid the use of calculators. Why are calculators, such as the TI-89, even permitted on such exams as the SAT and AP? I'm a recent college graduate and I was allowed to use these calculators in high school. I quickly weened myself off the use of a graphing calculator and was eventually able to get through problem sets in college using nothing but a pencil and paper. Are we setting students up for failure by placing way too much emphasis on calculator use? What are your thoughts?

 

[Timo]

A few random thoughts here (and since this is the Internet where tone of voice does not carry over: No offense meant by any of them):

You seem to be very much against the use of calculators. Yet, you do not mention a single argument against them (suggesting we set up students for failure is not an argument, neither is the fact that you can do exercises without them). As if the arguments were self-evident. They aren't to me. First gut-feeling argument is that students don't properly learn the math if the computer does it for them. I am not quite sure why that should be the case. Doing the derivative of a polynomial by hand gives me no more insight into how derivatives work than my calculator doing it and plotting the function at the same time. Arguably, the insight or at least the insight/time is even less.

An argument for the use of calculators is that computers are an essential part of modern world. Particularly in the fields where a lot of computation is being done. I do consider learning the efficient use of such tools to be vital to modern education. That does, admittedly, not imply that calculators must be used all the time. But the more you use them the better you get at them. And in real world (outside of school) computers are used a lot. So unless there is a good reason not to use them, or at least sometimes not to use them despite them being helpful, there is no good reason not to use them (*building up my score in the tautology ranking*).

There is one argument one could theoretically make against using calculators - which goes completely against what you probably had in mind. I haven't used one for ten years at least. Reason is: If there is a problem that I cannot conveniently solve on paper I boot up Matlab, Mathematica or their equivalents (or type echo "print 5.1234232*12323/0.13" | gnuplot into the console ;). I believe that -theoretically- a point could be made that students should be taught to use proper computers/software instead of calculators. Not really realistic in practice, of course (costs and the much higher skill-threshold required to start).

Bottom line: I am with you in terms of gut feeling. In fact, I enjoy doing things by hand. But based on rational thought I argue that even extensive use of calculators is a good idea (not meant to imply that people disagreeing with this statement were irrational, of course).

 

[Bystander]

Bottom line: I am with you in terms of gut feeling.

extensive use of calculators is a good idea

You've yet to see the kids given two numbers produce the sum, the product, the results of two subtractions, and of two divisions and their best guess among the six for an answer, I take it. Very few are competent to apply calculators as tools; they're used as answer machines --- incorrectly.

 

[QuantumCurt]

When used properly, calculators are incredibly valuable tools. Calculators shouldn't be allowed when students are learning elementary arithmetic obviously, but I see them as an essential part of algebra and calculus today. Technology is an available tool, and it's silly to not use it due to some yearning for "the good old days." I love math. That's why I'm double majoring in physics and math. But...I'm not such a fan of long strings of numerical computations.

This is more than a black and white issue though. Every math class I've taken in college has allowed calculators, but in very few instances would the numerical solving capabilities of a calculator have actually helped anyone on an exam. It could be used sometimes to check answers and such...but in all of my math classes, a correct answer without clearly displayed supporting work to arrive at the correct answer would still be worth 0 points.

When one is first learning how to graph equations, they should learn how to graph them without using a calculator. This is how an intuitive understanding of what graphs mean and how they work is formed. But is it necessary to keep graphing them? I know how to sketch a graph. I can take the time to find all of my extrema, test for concavity and points of inflection, find intercepts and such...which depending on the function may take 10 minutes. Or I can spend 20 seconds punching it into my calculator and bringing the graph up, get the information I need, and continue on with the things that I actually DO need to spend time working through formally.

In physics we often get large, nasty strings of computations. Why on Earth would anyone choose to evaluate something like $\sqrt{\frac{(8.964)(0.002)(28.6)-8.964(.050)}{(28.6)(.08026)}}$ longhand when they can just punch it into a calculator? I know how to evaluate that. That doesn't mean that I want to when I can have a valuable piece of technology save me the time that would be spent doing it.

I can appreciate both sides of this issue. Calculators with symbolic solving capabilities should not be allowed on multiple choice tests where only the final answer is graded.

 

[momof4]

Former teacher here. I was in college in the early 90s and we were allowed to use simple calculators in exams. I have yet to see a kid who knows nothing about calc be able to solve a problem just because they have a calculator. If they are smart enough to figure out how to use the calculator correctly to solve a problem they've never seen, more power to them. Did you take spell check off when typing reports? Did you run a grammar check? Or did you go old school and use a dictionary? No matter how much you use spell and grammar check, it's not going to make a poor writer a brilliant one, especially if they are not familiar with the subject. In short, a calculator is just a tool.

As teachers, we focus on what we are testing or teaching. If I'm testing math facts or multiple digit division then of course no calculator. If I'm teaching calculus, then I assume they already know the basics. Therefore, I'm not concerned if they need a calculator to determine 2 + 2= 4. I know that's simple but you should get the picture.

And since I was actually an English teacher, despite a science undergrad degree, it's wean not ween.

 

[QuantumCurt]

That's a very good point. A calculator might be able to integrate a function or take a derivative, but it's not going to set the integral up for you.

I've been thinking about this since I commented a little bit ago. I couldn't even imagine doing things like using Newton's Method for finding roots of a polynomial without a calculator. I recall working those problems back in Calculus I, and they typically involved 5-6 iterations of a complicated recursive function involving numbers that were basically 9 digit strings of decimals. If you were to do that by hand, you'd be there for an hour doing arithmetic. What about the Runge-Kutta method for approximating numerical solutions to differential equations? That's an incredibly time consuming and tedious process even -with- a calculator. I shudder at the thought of doing that by hand.

 

[MidgetDwarf]

Depends. If the teacher makes the exam with friendly numbers, then no calculator is needed for calculus and above. However, I would only allow scientific calculators that have no key for symbolic/numerical integration (ti 36 is an example). The calculator can be abused. If a teacher requires no calculator, I expect them to pick nice numbers and not try to find 34623×2313 +232/900023 by hand.

 

[Ophiolite]

I spend a portion of my time teaching technical aspects of the oil and gas drilling business to graduate engineers and Earth scientists. A disturbing characteristic of many of them is their inability to mentally compute a close order of magnitude value for comparatively simple calculations. Consequently, if they make any error in inputting the numbers into a calculator, or spreadsheet, or "black box", they fail to recognise the ludicrous nature of their answer.

I am all for the use of calculators and software and, indeed, anything that can increase the accuracy and speed of calculation, but too often these are used as crutches to cover up weaknesses, not turbo chargers to enhance performance.

 

[Mark44]

I spend a portion of my time teaching technical aspects of the oil and gas drilling business to graduate engineers and Earth scientists. A disturbing characteristic of many of them is their inability to mentally compute a close order of magnitude value for comparatively simple calculations.

Completely agree. This is the often overlooked cost of relying too much on computing devices.

Consequently, if they make any error in inputting the numbers into a calculator, or spreadsheet, or "black box", they fail to recognise the ludicrous nature of their answer.

I am all for the use of calculators and software and, indeed, anything that can increase the accuracy and speed of calculation, but too often these are used as crutches to cover up weaknesses, not turbo chargers to enhance performance.

 

[CWatters]

I was at school when calculators were invented. By the time I took my exams at 18 in 1978 they were allowed in exams. However you had to show all your working and got most of the marks for that not the final answer. That meant there wasn't much point in having one with too many exotic functions.

 

[momof4]

I spend a portion of my time teaching technical aspects of the oil and gas drilling business to graduate engineers and Earth scientists. A disturbing characteristic of many of them is their inability to mentally compute a close order of magnitude value for comparatively simple calculations. Consequently, if they make any error in inputting the numbers into a calculator, or spreadsheet, or "black box", they fail to recognise the ludicrous nature of their answer.

I am all for the use of calculators and software and, indeed, anything that can increase the accuracy and speed of calculation, but too often these are used as crutches to cover up weaknesses, not turbo chargers to enhance performance.

I agree the lack of number sense prevents students from being able to see that their answer makes no sense. At the point they can use a calculator the time for developing this is long gone though.

IMHO, the issue is that in elementary and middle school math is rushed. The amount of information they are expected to learn is insane. There is no longer time to master a concept or develop number sense. And the year long lesson plan means that you must move on to the next concept. Teachers try to use tutoring or small groups to fill in the gaps. The students become even further behind as lessons build upon previous material. It's a vicious cycle. We are too pc to divide kids for reading and math. Put lower kids with one teacher for two hours a day, average and higher kids with another for reading and math. It's bs the reason they don't, "at least they are exposed to it." What good does exposure to long division do if you can't do single digit division, etc. And stop spiraling curriculum and slow it down. Rant over.

 

[symbolipoint]

QuantumCurt, I want to be able to apply THREE "like's" to your post #4; unfortunately this forum system does not allow me to do that.

 

[RacinReaver]

In physics we often get large, nasty strings of computations. Why on Earth would anyone choose to evaluate something like $\sqrt{\frac{(8.964)(0.002)(28.6)-8.964(.050)}{(28.6)(.08026)}}$ longhand when they can just punch it into a calculator? I know how to evaluate that. That doesn't mean that I want to when I can have a valuable piece of technology save me the time that would be spent doing it.

You know that's approximately equal to $\sqrt{\frac{(10)(\frac{1}{500})*30-10*(\frac{1}{20})}{30*0.1}}$ ;)



That said, I think teaching graphing calculator use in high school isn't a bad thing, as it serves as a nice introduction to programs like Matlab or Mathematica. Both of those tools are vital to study in an engineering curriculum.

 

[QuantumCurt]

You know that's approximately equal to $\sqrt{\frac{(10)(\frac{1}{500})*30-10*(\frac{1}{20})}{30*0.1}}$ ;)



That said, I think teaching graphing calculator use in high school isn't a bad thing, as it serves as a nice introduction to programs like Matlab or Mathematica. Both of those tools are vital to study in an engineering curriculum.

lol

Approximately, yes. However, I need it accurate to two significant figures. I'll let you do the work to show the answer. I'll punch it into my calculator. :p

 

[Ophiolite]

But if I calculate it approximately in my head while I punch the numbers into the calculator then if I get a close order of magnitude answer I have higher confidence that I've punched in the right numbers.

 

[QuantumCurt]

This is very true. A calculator doesn't replace number intuition.

 

[Almeisan]

Number intuition is less useful than being able to use technology.

 

[Ophiolite]

Number intuition is less useful than being able to use technology.

I suggest that those with number tuition are more likely to be the ones designing the technology. It just depends whether you want to be a leader or a follower. Learning new technology is straightforward. Understanding when and why to use it is likely more aligned with skills like number intuition.

 

[Almeisan]

Apparently, using technology leads to less number intuition.

 

[QuantumCurt]

Apparently, using technology leads to less number intuition.

I don't think anyone is suggesting that the simple act of using technology leads to less number intuition. What is being suggested is that the lack of the actual teaching of methods that result in number intuition leads to a lack of number intuition. Technology often gets substituted very early on as a means to perform arithmetic operations. Because of this, many students lack a great deal in number intuition.

Is it perhaps important to teach both number intuition and the utilization of technology? Does it have to be one or the other? I don't think it can be substantially argued that number intuition is less important than knowing how to use technology. Technology is a wonderful thing, and it absolutely should be utilized. That's what it's there for. But without having some conceptualization of how a calculation should come out, it's far more likely that one will realize the mistake later. If I calculate something that the result is several orders of magnitude away from where it should be, I can be reasonably sure that I punched something into the calculator incorrectly. I'd rather recognize this mistake immediately, versus 5 steps later when the final numbers won't add to zero as they're supposed to. We've probably all had those fun experiences of following problems back to find arithmetic errors.

 

[Almeisan]

People are. They just want to ban calculators and then that problem is solved.

Instead, they should ask themselves if they are copping out of actually deliberately teaching things like intuition just because it apparently came automatically when solving problems back in the '60s.

 

[vela]

It's complicated because despite the best intentions of teachers, students will often find a way to miss the point and not learn the material. Technology can cut both ways. It can remove the drudgery that obscures the main point, or it can be a crutch that let's students get the right answer without engaging their brain.

 

[symbolipoint]

Apparently, using technology leads to less number intuition.

For some yes, and for others no, and still for some, depends on what KIND of effort they make for understanding.

 

[Ophiolite]

People are. They just want to ban calculators and then that problem is solved.

Instead, they should ask themselves if they are copping out of actually deliberately teaching things like intuition just because it apparently came automatically when solving problems back in the '60s.

I'm slightly confused by "People are." People are what? I realize you are referencing a comment in an earlier post, probably the preceding one, I just can't figure out which. Perhaps that also would clarify who the "they" are that "just want to ban calculators".

I can't speak for "them", but when I teach, I continually emphasise lateral thinking, sound background knowledge, practical experience and other actions that will lead to the development of intuition. I don't expect this to come naturally, but I do expect people to work at acquiring the skill. The alternative is that they should learn to say "Do you want fries with that?".