[cydonian_monk]

Cheat-sheets are also far more "real world" than the fake environment many schools cling to. I don't know a single practicing engineer that doesn't have a note scribbled somewhere. My engineering school recognized this decades ago and adjusted accordingly.

Now the math department? They were a different story. No calculators. Period. Ever. Made higher-level calculus... interesting. Thankfully I'd already had it in high school in a tools-based curriculum, so repeating it with just fundamentals was less of a headache than it might've been.

And I agree - the simpler the cheat sheet the better of an understanding of the subject the pupil is likely to have.

[jacobolus]

> Now the math department? They were a different story. No calculators. Period. Ever. Made higher-level calculus... interesting.

There is no reason for any math course at any level to ever need a calculator, period.

The point of math courses is learning to think, not learning to avoid fat-fingering tiny buttons or learning the specific crappy interface of some anachronistic antique machine.

Without an electronic calculator students can’t be expected to do as much mindless number crunching, so instead the problems can be made much more interesting, unique, and conceptually challenging.

Frankly the same goes for science courses. If people need to process data resulting from physical experiments they should use a machine with a full-sized keyboard and a real programming language. If you want something portable a slide rule is entirely sufficient for anything that might come up in high school or intro undergrad level science courses. The students might even learn something about significant figures.

I could plausibly believe that upper-division engineering courses benefit from handheld calculators – I have no experience with those – but foisting $100 calculators on every high school student is a tremendous scam.

[saagarjha]

FWIW, not having to go into the details of how to find the antiderivative of an some mildly complicated function was a huge boon in my physics class.

[eru]

I like mathematics that's about concepts and understanding.

But numbers are a big part of some mathematics too. And getting some help with them can be useful.

Lots of our past mathematical geniuses were great at calculating by hand. And some of them even invented some mechanical calculators (or electronic ones, too).

[madengr]

Couldn’t agree more. Though this was 1990, the professor for the calc classes could care less if you had a calculator, as it “won’t be of any use”.

Even in engineering and physics courses, you had better show all your work. If a numerical calculation was made in error, and you should all your work, you could still get a good amount of partial credit.

30 years later, I still just use a basic RPN calculator.

[baud147258]

For sciences courses, I remember one exam (at least) where calculators were forbidden and instead the exam sheet included a bunch of calculations that might or might not be needed. Of course it helped since if you land on a calculation that's on the sheet, you're likely on the right direction.

[toomanybeersies]

The odd thing about my exams at high school (in New Zealand) is that although you were allowed to use a graphing calculator, you still had to show your working. So you'd plug the numbers into your trusty Casio graphing calculator, get the answer, then still have to work out the answer anyway.

In theory that means that you could use your graphing calculator to verify your answer. But in practice, people ended up spending too long trying to fiddle with their calculator, not knowing how to use it properly, and getting the wrong answer in their calculator but the right answer when they did it by hand, getting flustered that they didn't match, and crossing out their correct answer.

[mkl]

> There is no reason for any math course at any level to ever need a calculator, period.

Numerical methods courses need calculators for questions about practice rather than theory. Basic (non-graphing) cheap ones are usually sufficient, though.

I think slide rules are thoroughly obsolete in almost every context. There's no point in wasting students' time when better tools are available.

[jacobolus]

None of the professional numerical analysts I know work with a handheld calculator.

The point of learning a slide rule is not that it is a particularly important practical tool, but that understanding how it works has independent pedagogical value.

If someone built their own electronic calculator from discrete components, programmed one on an FPGA, or even implemented a bunch of mathematical functions on an existing computer, that would be similarly educational (though teaching different things than the slide rule), but just knowing how to navigate the interface of an electronic calculator doesn’t teach anything.

[mkl]

Professionals don't, but students need to, especially in study groups, tutorials, tests, and exams, when computers are not available. Calculators are also much more approachable than computers for this material for students with no programming knowledge.

The point of using a calculator is to skip over the tedious unimportant details when learning other things e.g. Newton's method or Euler's method. The calculator itself is a tool, not an educational destination.

Learning a slide rule as you say makes sense in a history of maths course, and implementing one's own calculator makes sense in an electronics or computer science course.

[jacobolus]

Computers are plenty available in “study groups and tutorials” (for one thing almost all college students and many high school students now have smartphones, and most college students also have laptops and/or tablets, but if you want a cheap computer just for math class, get a netbook or cheap android tablet and external keyboard or something made from a raspberry pi or ....)

There’s really not much pedagogical value in using a handheld calculator to apply Newton’s method to some root-finding problem or apply Euler’s method (forward differences) to model a differential equation. Both of these are very simple and students can learn enough of some simple programming language to implement them both in a very short amount of time. That time is much better spent than doing 4 or 5 examples of each with a handheld calculator. If a general-purpose programming language seems too much, get them implementing these simple tools in desmos or geogebra.

On a timed in-class exam in an introductory calculus course, there are much better ways of judging someone’s understanding than making them perform a bunch of tedious and error-prone number crunching. (For example you could give the students rulers and printed graphs of a function – without any symbolic expression written down – and ask them to sketch approximately what a solution using Newton’s method would look like).

If you want a nice introductory calculus book organized along more computer-focused and conceptual lines, take a look at http://www.math.smith.edu/~callahan/intromine.html

In a post-introductory-calculus “numerical analysis” course, the exams should consist of writing proofs, not performing algorithms.

The important thing for numerical analysts about different root-finding methods (etc.) is their convergence speed, numerical stability, computational complexity, and so on. In the 1960s and before it might have made sense to get students performing the role of human computer, but nowadays it is anachronistic.

> Learning a slide rule as you say makes sense in a history of maths course

No, learning how to use a slide rule makes sense in an algebra course for ~15-year-old secondary math students who are learning about logarithms, and for 15–17-year-old secondary science students. They’ll end up with a better intuitive understanding of logarithms and significant digits and error bounds after regularly using a slide rule for even a few weeks than any amount of reading about it or doing formal algebraic manipulation.

Electronic calculators give students a very misleading impression that all of the digits printed on its display are meaningful. But in high school chemistry, physics, etc. courses there is pretty much no experiment ever done with better than about 2 digits of precision.

[missosoup]

You can't really do problems of meaningful complexity without either a calculator or forcing your class to do a large amount of arithmetic which is really not what you're testing. And at the same time you can't really twist those problems around to not require arithmetic, without also making them abstract and too different from what's encountered in the real world.

Try doing matrix multiplications for hidden markov chains without a calculator. I dare ya.

[jacobolus]

Nobody should be using a handheld calculator for nontrivial matrix multiplications. In “the real world” literally nobody does this.

Making students do nontrivial matrix calculations on a timed in-class exam just tests their calculator skills. There are a wide variety of alternative types of problems which will better probe their understanding of the course.

If your students are trying to learn about numerical linear algebra, consider getting them implementing the relevant algorithms in computer code.

[missosoup]

> Making students do nontrivial matrix calculations on a timed in-class exam just tests their calculator skills [...]

> consider getting them implementing the relevant algorithms in computer code.

Cool, so we went from a statistics exam with no programming experience required, to a programming language exam with a theme of statistics. Because what? Because calculators are bad mmkay?

Yeah I'm not sure this idea has been thought through.

[jacobolus]

I thought your big concern was “what's encountered in the real world”?

Seems to me like the concern is “what someone decided should be in the curriculum 30 years ago and nobody ever bothered to change even though it is now an anachronism”.

But anyway, I am suggesting writing programs could potentially be part of the homework, not part of exams. (At least, writing computer models was far more useful for learning about statistics than any textbook problem I ever did. YMMV.)

On timed in-class exams, there are many relevant pen and paper exercises that could be posed. Posing problems requiring a handheld calculator is a generally poor way to track whether students understand the content of a course. Then again, personally I think timed in-class exams are terrible. YMMV; some teachers seem to love them.

[tripzilch]

You're right except for when the exam is in numerical methods :)

[whatshisface]

>Now the math department? They were a different story. No calculators. Period. Ever. Made higher-level calculus... interesting.

When I took calculus the professor made sure that every arithmetic problem could be solved quickly. The trick he used was to carefully track factors when setting up the problem, so that the numbers never got very big, and also so that things had a way of cancelling (I can't tell you how many problems had one or zero as an answer.) Calculators weren't necessary and in fact would have been slower.

[spookthesunset]

Yup. One clue you were going off course on a test in a high level math class was when all your numbers were getting rather crazy and unwieldy...

Usually the same rule applied in physics exams too...

[blattimwind]

Except some professors / PhDs deliberately set up exercise in a way that they involve large-ish (3-5 digits) or stupidly large (9+ digits) numbers to turn otherwise trivial exercises mostly into a mental math test. (Another all-time favourite of mine was the guy who always included an exercise in his exams that required you to determine the non-trivial prime factorization of a four digit number).

Good exam questions can be approached in multiple ways and are not a mental math test.

Bad math exam questions are usually those that can only be solved using one specific technique or lemma (ideally one that was only mentioned in passing once or twice) or require a lot of error-prone calculation.

[dagw]

One clue you were going off course on a test in a high level math class was when all your numbers were getting rather crazy and unwieldy...

We had one professor who was rather infamous for being really sloppy when writing exam questions. On more than one occasion did he accidentally transpose a couple of numbers in a problem, turning an equation that should have easily reduced to a trivial linear problem into a 4th degree polynomial with complex roots.

[toomanybeersies]

I only did 100-level mathematics at university (except for discrete mathematics), but for all my tests and exams, the lecturer made sure that all the problems could be solved (or reduced) without a calculator.

As long as you showed the working and the closest you could get without a calculator you'd get full marks (e.g. you could leave the answer as a fraction like 543/42, or as a power like 26^12).

They were quite good at my university for mathematics exams though. You were usually allowed a handwritten cheat sheet in the exam, and if you couldn't get the answer to a question that had follow-on questions they'd still give you marks for the follow-on question if you just made up an answer for the earlier question. If only the lectures weren't usually at 8 AM I might have gotten good grades.

[jimktrains2]

I don't remember a math course I college anything but a ti89 or one of the other few capable of handling symbolic expressions would have even remotely helped with. The arithmetic was usually small or just non-existent.