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I know this subject is inflamatory... but I also believe it is something that requires broader discussion (and hopefully some consensus).

My own position (from personal experience) is that today you rarely (if ever) do calculations (sums, multiplications, ...) by hand. Remembering the hours I spent (not all exactly happy ones) being drilled in multiplication and division, doing so-so in that, but going on to end up here, I'm looking for better use of that time. In college, I see student's (and teacher's) time being wasted on "what clever trick gets an integral for ..." and similar questions.

My question is then, can (and should) we take out this "drill" aspect of teaching, and what should we do with the freed time?

Follow-up question is what this means in formation of future teachers?

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  • $\begingroup$ I agree that this is deserving of broad discussion, but I don't know that the question is formulated well at present. a) I don't think the first paragraph is necessary b) phrases like time being wasted seem to presuppose a conclusion. $\endgroup$ – user37 Mar 16 '14 at 16:15
  • $\begingroup$ @Mike, perhaps my opinion on the matter shows a bit too strongly. Please feel free to edit the question if you believe it should be more neutral. $\endgroup$ – vonbrand Mar 16 '14 at 16:19
  • $\begingroup$ I have my own feelings about calculator use, but when you talk about 'drills,' doesn't that occur early on, 3rd to 5th grade? $\endgroup$ – JTP - Apologise to Monica Mar 16 '14 at 17:18
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    $\begingroup$ I hate calculations and I see them as necessary. Perhaps, in the future, when computers will be able to pick our thoughts and give an answer before we are actually conscious we want it, it might be different. Right now, doing things by hand is faster in many more cases that I would like to admit. Nevertheless, after the theorem (or whatever I was working on) has been proved, I do check the calculations, once again, this time using a machine, I do not trust calculations by humans. $\endgroup$ – dtldarek Mar 17 '14 at 1:41
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    $\begingroup$ Even if a computer can pluck the question from our brain and give us the answer, knowing your multiplication tables makes it easier to understand the concept of division (for example). I observed a 4th grade classroom that was learning division by drilling multiplication factors backwards and after the 3rd or 4th one, the kids started picking up on what was going on and "got it" much easier than if the teacher had simply tried to explain the method. $\endgroup$ – David G Mar 17 '14 at 4:03
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I think some practice in manipulating symbols and number is necessary for many mathematical activities. To do everything "by machine" is first inconvenient and second there is also some insight to be gained from doing it by hand.

I think by solving some systems of linear equations by hand one gains some insight that one would not gain just by typing it into some CAS and getting the solution back out. (Leaving aside the aspect that some need to build and improve such things, as this is a minority.)

The same goes first for basic calculations, then for more complex ones, then for manipulating expression containing indeterminantes and so on.

The above being said, I stress the some. I agree it is nowadays pointless (for most anyone) to develop high proficiency at these tasks. And, yes, I do not understand either why some still let students do so many integrals. Surely, one should know how to do integration by parts and substitution, but a few examples should suffice.

Generally speaking, I think some practice in such tasks is important, but only to the extent that one can do things by hand if one must. Forced "drill" to do such things fast and reliably is besides the point in my opinion (I stressed the forced as, personally, I enjoyed to do calculations a lot).

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    $\begingroup$ I discovered one day when doing calculations for my research that I was being surprisingly accurate and making very few mistakes. I don't know why this is, but I presume the practice I had in computing complex things accurately when in school and when I was an undergrad helped. $\endgroup$ – user1729 Mar 17 '14 at 12:21
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    $\begingroup$ Speaking as a current undergrad doing a lot of tutoring in math, "a few examples" for u-substitution is rarely enough for the calc I or calc II student to really grasp what to do. It takes extensive coaching. It is always (to me) somewhat painful to see the differential equations student failing because they do not understand substitution and chain rule. If a concept needs to be leaned back on in a future class, shouldn't that concept be thoroughly covered? $\endgroup$ – Opal E Aug 1 '14 at 16:06
  • $\begingroup$ @OpalE personally I feel that it is better to do very few examples properly and in depth than to do many. I do appreciate however that some students need (at least in the sense that they are so used to it that they believe they need it) more examples. I admit I struggle to find the right balance there in my teaching. $\endgroup$ – quid Aug 1 '14 at 21:23
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    $\begingroup$ I see what you're saying; I thought you meant that few examples were needed for their practice! (as opposed to demonstration and modeling, which is what you seem to be saying). $\endgroup$ – Opal E Aug 1 '14 at 21:28
  • $\begingroup$ @quid You seem to have shifted in this discussion from the issue of how much student practice "drill" is needed (as in your answer) to how many examples by a lecturer. I agree with Opal that most students do need a reasonable amount of drill to internalize things. $\endgroup$ – guest Oct 24 '18 at 6:57
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I agree that in the current paradigm, there is no need to emphasize the skills of computing things by hand for the sake of it. Having said that, I totally agree with the answer above which says that "there's a valuable insight to be gained in doing things by hand". I think that precisely because it is laborious, the act of writing things out forces you to take notice of them, and in doing this a student takes in more. If I am allowed to be poetic, I would say that the act of writing or computing things forges the first relationship between a student and symbols. Of course this loses its meaning when computing by hand also becomes mechanical. It is the same reason I prefer board talks to slide shows. As a student, I was more awake in the class where the instructor wrote things on the board, talking to us as he did. As an instructor too I am able to engage my audience when I am not feeding them pre-packaged slides. Hence I don't think that asking students to do things by hand is a waste of their time. But again, balance is the key.

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In addition to the other useful answers... there is a general "problem" that computers, given the long-known algorithms, ironically many not really human-executable, can do almost all of the algorithmic parts of k-12 and undergraduate and some once-upon-a-time-grad-level math. So it seems silly to struggle to teach kids to do things that machines can do better (the superiority of calculators at arithmetic was just the beginning...)

The genuine sociological and pedagogical problem is that humans are not getting smarter or more capable to keep ahead of things that can be implemented algorithmically on machines. And, to add insult to injury, the same kids that had trouble getting the algebra right in by-hand-calculus will typically have similar troubles getting the input right to have software do it.

That is, math software mostly helps the most able people.

Returning to some aspect of the question at hand: yes, it is ostensibly silly to teach algorithms that are far-better executed by machines. (I don't know whether anyone's bothered to write Latin or classical-Greek or classical-Hebrew grammar-checkers...) Ideally, yes, it would be better to use the machinery and allow kids to look at higher-level issues.

However, sadly, those higher-level issues are intellectually more taxing than (pointless, yes) execution of algorithms. I am disappointed to report that my experience with trying such things generates such reports.

That is, the kids who'd make sign errors or not be able to do middle-school algebra will also be flummoxed by trivial computer-language syntax, etc.

Nevertheless, it is better to be honest, to be genuine, to have math courses address real issues rather than traditional-fake. But we must brace ourselves for the disappointment that there'll still be a cadre of kids who can't do it... now for just-slightly-different reasons than the reasons for which they couldn't do hand-executed algorithms.

So, yes, one should not insist on hand-execution of algorithms, but giving that up and doing better things will make it harder, overall, for the students. But that doesn't mean we shouldn't move in that direction. A tough situation, quite seriously, in my opinion.

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  • $\begingroup$ I can corroborate; my experience as a tutor (not yet math teacher) suggests that the students who struggle the most with the basic algebra also struggle the most with inputting their answers to the online homework website. In particular, they struggle with order of operations, and so their parentheses get "messed up." These tend to be the same students who complain that they shouldn't have to learn math because "computers and calculators can do it for them." $\endgroup$ – Opal E Aug 1 '14 at 16:11
  • $\begingroup$ As a frequent user of Excel in business analysis, my strong impression is that those who are good at Excel are also strong at algebra and the like. If you stay weak at manipulations because "the computer can do it", you will be much MORE prone to logical errors in making models. $\endgroup$ – guest Oct 24 '18 at 7:11
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I have some documents discussing various issues about the use of Maple in teaching here:

http://neil-strickland.staff.shef.ac.uk/talks/maple2/

(This is actually quite old, but I think that most of it is still valid.)

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    $\begingroup$ Can you summarize or quote some of the key points? Especially if your site gets moved one day; it seems interesting, and I don't want it to get lost. $\endgroup$ – Brian Rushton Mar 17 '14 at 3:02
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For your college example: remove the techniques of integration from the calculus curriculum.

Teach numerical integration instead, e.g.: What is $$\int_0^1 e^{-x^2} dx$$ and how do you calculate it to within 0.001?

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  • $\begingroup$ What does it mean to "teach numerical integration?" Should we teach students to use calculators to numerically integrate? Should we teach algorithms for numerical integration? Should we teach students to implement these algorithms? Should we teach error bounds for numerical integration? $\endgroup$ – petehern Mar 17 '14 at 5:15
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    $\begingroup$ @petehern: Yes to using calculators (or Excel or whatever), yes to teaching algorithms, yes to implementing algorithms, even at the cost of less time for a rigorous definition of the integral. I'd leave rigorous error bounds to an analysis class, and have the students subdivide the interval until the answers seem to converge well enough. $\endgroup$ – user173 Mar 17 '14 at 5:34
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    $\begingroup$ No! Students must have something to practice on, and most learning comes through practice. Most students find purely conceptual problems (with very little calculation) more difficult, not less. $\endgroup$ – kjetil b halvorsen Mar 20 '14 at 10:46
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I have several arguments for having some hand-calculations in the classroom, but with some caveats:

Firstly, as a pure mathematician, whose entire research was doing things by hand entirely in symbols, I can confidently say that it was important to me! Also, when I enter various applied mathematician's or statistician's or scientist's offices at my university I see whiteboards and scraps of paper with symbols scrawled on them every time. I have watched even those whose work mostly involves computers constantly switching back and forth from the computer to hand-caclulations. So clearly hand-calculations are still important for the working scientist.

Often there is a decent amount of algebraic thinking that needs to be done in order to make it possible to turn your problem into something the computer can solve. Without some training in this, you may never be able to even enter the correct input. (Of course, this also means it is probably a good idea to introduce calculators/computers early so that people can learn to use them in conjunction with their algebraic manipulations, rather than leaving it till later to learn that skill.)

Secondly, symbolic thinking is a style of thinking that is actually quite useful, and allows people to do things they could not otherwise do. Sometimes an in-depth study of the symbols and rules allows you to know what is possible and what is not possible and what can be made possible. Deferring all of that to a machine can rob us of coming up with a way that is different and makes more sense to us, rather than the way that makes sense to the person who programmed the computer.

However, sometimes our approach to using symbols and calculations does not highlight these thinking skills. Techniques of integration can be a study of how integrals are connected to each other and can be a study of the difficulties that integration poses compared to differentiation, and it can be a study in how to earn how to choose from a set of strategies in a field where multiple approaches may work or none may work. However, sadly, it usually just feels like a lot of "here's another fancy trick". So in terms of structuring a course, I'd suggest making sure your emphasis is on the higher thinking if you do choose to do calculations (or have no choice).

Finally, something that is never mentioned in these discussions is self-efficacy. There is something highly satisfying about knowing that you can do something yourself without the need of technology or assistance. My daughter learned to crochet for no purpose but to know that she could. I personally love doing integrals simply because I can. The nursing students I help love to be able to calculate their drug caclulations by hand once they know how, because it gives them power over their calculations. This personal power is an important aspect of doing things by hand. Note you don't necessarily need absolute fluency to have this feeling, but it does take some practice.

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