## Math without calculators

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relative_entropy
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### Math without calculators

I'm a math teacher at a community college, and I teach a course that includes the fundamentals of math, multiplying whole numbers, adding fractions, adding decimals, and so on. In most of the course, my students aren't allowed to use calculators. These rules aren't set by me, they're set by the administrators, but I agree with them.

My students often ask me why they're not allowed to use calculators. I explain it by asking them to think of math as exercise for the mind. When a person studies math, they develop mental skills, logical thinking patterns, an ability to pay attention to detail, and so on. You can make a good analogy between math and weightifting, comparing the development of the mind to the building of muscle.

Continuing this analogy, using a calculator to do math is like using a forklift to do bench press. You would achieve the same thing, lifting a weight up and down ten times, but you wouldn't be getting any exercise.

On the other hand, it's also good to learn how to use calculators. In the "real world," (the corporate world, the business world, etc), if you want to multiply two numbers, you use a calculator. So it's good to use a calculator because a college math course should, in part, prepare you for the world.
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Xenomortis
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### Re: Math without calculators

So what's the question?
When the purpose of the course is to teach basic arithmetic skills, then of course it makes sense to forbid calculators.
When the purpose of the course is to teach something more advanced, like statistics or calculus, then allowing calculators becomes more reasonable (who cares if they can't compute an answer to x digital places?).

Yakk
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### Re: Math without calculators

In the real world, if you where asked to multiply 27 times 33, if you have to use a calculator to generate a decent estimate, you will be slower than if you can generate a pretty accurate guess. (calculating the exact answer isn't much harder -- difference of squares -- but that is mostly due to a bit of symmetry I snuck in.)

Generating said guess when you have always done math with calculators will be difficult.

Even at a low level, I was buying some confectionery at a busy food stand (10+ sellers). Most of the staff could make change without any mechanical aid (and in fact had to, as there wasn't a cash register). Mine needed a calculator, and because she needed the calculator when the calculator spit out nonsense wasn't able to correct it on the fly. It was both slower and less accurate. I'd be surprised if she was retained as an employee much longer, she tried to give me 15\$ change for a 15\$ purchase from a 20\$ bill.

At higher levels, number crunching is leverage. You can be a "business person" who doesn't actually *do* business but rather manages other people who do the real work: or you can be a business person who understands what is going on. Understanding what is going on in even a medium sized organization requires understanding numbers, as humans are not smart enough to do it any other way with any efficiency, and if you have to rely on the crutch of a calculator you aren't going to be able to do it.

Can you imagine having a boss who knows everything that 100 employees are doing at any moment? It isn't possible for any mere mortal. Understanding aggregate information filtered through numbers just requires numerical literacy, and some sanity checks. The other option is to be a really good judge of people and pick people under you who can understand the next level under you: but then you are at the mercy of anyone who is more socially competent than you and who appears competent, despite not being so.
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Last edited by JHVH on Fri Oct 23, 4004 BCE 6:17 pm, edited 6 times in total.

mathmannix
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### Re: Math without calculators

I don't have much to add upon the good replies above, so I'll be brief.

I think both calculator math and non-calculator math should be taught at the high school / college (community college, or math for non-math majors in university) levels. Not necessarily in the same course, as long as the curriculum is well thought through.

Both are checks against the other... using Yakk's example, 27 times 33 is pretty easy to do in your head if you know difference of squares, but even if you don't, or you have something slightly more difficult (and/or realistic) like 27.5 times 32.4, then you should probably use a calculator to get an exact answer * (and check your head work), but also estimate in your head (roughly 30 times 30, so 900-ish) to get a ballpark (and thus check that you used your calculator correctly. Everybody makes transposition errors and fat-fingers keys every once and awhile!)

* - I totally didn't plan that it would be the same answer!
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Wildcard
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### Re: Math without calculators

Working with high school students one on one, I watch how they work through the problem. (Let's say a word problem.) When I see them hit a multiplication problem or subtraction or something that they should be able to do in their head and they reach for the calculator, I grab it and gently move it out of reach and say, "Come on. You can do that in your head."

Encouragement works wonders. But you get them started on it. So they're looking at 12 x 15. You say, "What's 15 x 2?" Show them factoring. 6 x 2 x 15 = 6 x 30 = 18 x 10 = 180.

I agree that calculators are more likely to be used in the real world...but that's for, like, accountants and engineers (and some others, I know). 99% of the math that an average person encounters in day to day life SHOULD be done easily without a calculator. So I completely agree with teaching mental math, actually to a much higher level than most teachers I've encountered. Paper should be used to keep track of the numbers you've already figured out when working a big problem, but I encourage students to do even advanced algebra, trig, calculus, probability—all of it in their head. At least to try to do it that way and go all the way through the problem, then check it on paper if they have to.

But one reason I think most people don't learn mental math, is that they are taught these absurdly backwards algorithms in their grade school math books. 99 x 3 = 27 + 270 = 297. Instead of: 99 x 3 = 300 - 3 = 297. Not only that, but they do the math right to left instead of left to right, leaving them often without any good conceptual grasp of the size of the answer to a math problem. Like 144 x 7. The way it is taught in grade school, following the algorithm you will be writing down these digits in sequence: 8 2 0 3 0 1, giving you 1008. Whereas to do it mentally, you are much better off starting out with 7 x 100 = 700, then adding 280 to get 980, then adding 28 to get 1008.

Here are a few multiplication problems that you can use to test out your skill with various approaches to mental math:
99 x 3
97 x 5
29 x 31
47 x 53
41 x 41
24 x 35
11 x 725

Pseudo-edit: I guess I'm talking about math without paper, not just without calculators. But I think that's better anyways.
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Tyndmyr
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### Re: Math without calculators

relative_entropy wrote:On the other hand, it's also good to learn how to use calculators. In the "real world," (the corporate world, the business world, etc), if you want to multiply two numbers, you use a calculator. So it's good to use a calculator because a college math course should, in part, prepare you for the world.

Meh. Everyone should be able to at least make change to get by in the real world. I will never bother with calculators for two digit math. In fact, I didn't realize that there were people who would have trouble doing fairly simple mental math(say, calculating 10% off a price), until I had to interact with them. So, I screen for that now. If a person can't handle basic math without reaching for pen and paper, let alone a calculator, I won't hire them. For anything.

Why? Because the person who isn't familiar with the math, and is completely relying on the calculator won't notice when they fat finger the numbers, and will happily make huge errors. Automation is great for when you fully understand the process, not as a shortcut to avoid it.

Agreed w Wildcard. There's a number of patterns that, once you get familiar with them, become basically automatic, and you can start doing much, much larger calculations with excellent accuracy, as you're simply managing a stack of simpler equations mentally, which seems to be cognitively much less taxing. Now, you don't HAVE to get to that level of proficiency to just get by, but there's few easier ways to develop a reputation for genius than being able to glance at messy 4 or 5 digit numbers and spit out the answer.

Chaoszerom
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### Re: Math without calculators

I suppose this is a bit of a bump, but anyway. I'm a year 1-10 tutor in Australia (so, first grade to sophomore?), and that being true I encounter a lot of children who have trouble with maths. It's part and parcel of tutoring -- not many people who are good at maths will be showing up. The methods that we teach now (from the Australian curriculum) are no longer the vertical algorithms that I learnt as a kid, but ones that translate better to a mental setting. Partitioning is the most common:

Code: Select all

`236+579= 200 + 500 + 30 + 70 + 6 + 9 = 700 + 100 + 15 = 800 + 10 + 5 = 815`

and Rounding and Compensating is my favourite

Code: Select all

`297 + 368+3    -3= 300 + 365= 665`

I've found a few things. The younger you get them enthusiastic about this stuff, the easier it is for them to pick it up -- if they're not doing lots of this stuff mentally by grade 4, they never will. As for the question of "is it worth teaching mental maths?", I'd have to say yes -- my experience shows that people who don't have an understanding of the numbers tend to struggle a lot more, even when they understand the methods they're employing. Additionally, those who struggle tend to be those who don't understand the mental math, but they're both potentially caused by the same thing, not each other. However, a lot of it is also motivation -- good kids are not always motivated, and vice versa, but the kids who are having fun and are awake tend to progress faster (we give them games every 5-10 minutes to try encourage this state).

If anyone can think of some interesting hypotheses that would be interesting to test with the access to students of all ages that I have, feel free to suggest.

elasto
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### Re: Math without calculators

I would say for a community college course, using calculators should be fine because the skills should have been learnt already. It's primary schools which should be banning calculators.

In the real world, I suppose you need catch up for those students whose primary schools failed them, but that's a pretty frustrating state of affairs.

With my 8yo, who goes to a normal state school in China, she's already so good at arithmetic that when she comes home with yet more pages of arithmetic examples to work through, I let her use a calculator for her homework because she gets 90%+ in tests even without one.

(The schooling system is incredible - the number one reason I want my family to grow up here. At 6yo she had a schoolday of 7am to 5:30pm AND then an hour or so of schoolwork to do every evening.

Personally, as much as Western schools are too lax, I think Chinese schools are too tough in many ways - but then again the day does include exercise every morning and some afternoons, and a nap at her desk at midday.

The school grows its own vegetables and prepares all meals in house too. Nothing pre-processed. It's a very healthy and well-rounded lifestyle in many ways.)
Last edited by elasto on Thu Oct 08, 2015 11:32 am UTC, edited 1 time in total.

LaserGuy
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### Re: Math without calculators

elasto wrote:The schooling system is incredible - the number one reason I want my family to grow up here. At 6yo she had a schoolday of 7am to 5:30pm AND then an hour or so of schoolwork to do every evening.

Poor kid. That's kind of insane.

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### Re: Math without calculators

LaserGuy wrote:
elasto wrote:The schooling system is incredible - the number one reason I want my family to grow up here. At 6yo she had a schoolday of 7am to 5:30pm AND then an hour or so of schoolwork to do every evening.

Poor kid. That's kind of insane.

My thoughts exactly.

Code: Select all

`sed 's/kind of/completely/'`

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### Re: Math without calculators

The quality of a schooling system really shouldn't be judged on the basis of how many hours a day a kid spends in school. Neither for good nor bad. Kid could be having an extremely productive day with supervised free-time that isn't spent on watching TV, or be forced to do stupid tasks all the time ... can't tell.
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### Re: Math without calculators

I've seen a lot of weird stuff happen when people rely too heavily on calculators, especially in a maths-focussed course. Here in South Africa the standard is that you start using calculators in High School, and then if you do a maths course at a Universtiy you don't use it. This seems to work against people who go into maths heavy degrees because they did their entire high school career, so trigonometry and basic calculus, on calculators.

I feel like an ability to do at least basic maths, like the stuff the first post talked about, in your head, makes a lot of sense. That kind of maths crops up a lot in everyday life, and if you can't estimate what 10% of your bill is mentally, you're going to either have to guess or use a calculator. While we now have cellphones everywhere, it's still a bit tedious to use one for simple maths. Similarly, if you're doing a bigger, more complex problem, relying on a slow medium like a calculator will slow down your overall solving speed, which is important in an exam scenario.
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DanD
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### Re: Math without calculators

Just don't use the explanation that most of out teachers did growing up. The "you won't have a calculator with you everywhere you go" thing kind of failed around a decade ago.

Xenomortis
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### Re: Math without calculators

To be fair, I still don't carry one that I'm willing to use.
Fumbling with my phone to calculate an exact answer would be much slower than just thinking something in the right ballpark.

Although difficult arithmetic doesn't come up very often - mostly I just need to be able to calculate the cost of my lunch, or approximate something to the nearest power of two.

Yakk
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### Re: Math without calculators

Well, there is calculating tips. I guess it keeps my logarithms fresh.

And the worst thing ever, doing mod-12 math to work out when X hours later is. My usual error is either applying the off-by-2 twice, or applying it in the wrong direction, or even applying it twice in the wrong direction.
One of the painful things about our time is that those who feel certainty are stupid, and those with any imagination and understanding are filled with doubt and indecision - BR

Last edited by JHVH on Fri Oct 23, 4004 BCE 6:17 pm, edited 6 times in total.

rmsgrey
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### Re: Math without calculators

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### Re: Math without calculators

mathmannix wrote:using Yakk's example, 27 times 33 is pretty easy to do in your head if you know difference of squares, but even if you don't, or you have something slightly more difficult (and/or realistic) like 27.5 times 32.4

[...]

* - I totally didn't plan that it would be the same answer!

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### Re: Math without calculators

Yakk wrote:Well, there is calculating tips. I guess it keeps my logarithms fresh.

And the worst thing ever, doing mod-12 math to work out when X hours later is. My usual error is either applying the off-by-2 twice, or applying it in the wrong direction, or even applying it twice in the wrong direction.

Time math is one of my least favorite kinds of math.
What would be the most effective way to learn time math?
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### Re: Math without calculators

liberonscien wrote:What would be the most effective way to learn time math?
...

...

...hours of practice?

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### Re: Math without calculators

Soupspoon wrote:
liberonscien wrote:What would be the most effective way to learn time math?
...

...

...hours of practice?

I should have expected this.
I meant would one be wise to teach oneself mod 12?
Is there a more effective way to learn how to use what may be a broken system?
I know the Babylonians or someone else divided time into sixty but I don't understand it.
Time how we measure it on the average clock is somewhat like a mod 12 system coupled to a mod 60 system. We are taught at birth to use base ten. What is the reasoning here and what is the best way to learn it.
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### Re: Math without calculators

Again, practice? Just keep calculating times in your head? "My thing is in 15 minutes, that's one quarter of an hour, the time will be whatever. Ten hours from now is..." and so on?
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Soupspoon
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### Re: Math without calculators

Sexagesimal (counting in 60s) is handy for dividing a sixty unit into equal parts of 2, 3, 4, 5, 6, 10, 12, 15, 20 and 30 units. We don't tend (at least in English) to use divisions other than 15 minutes (1/4 hour), 30 minutes (1/2 hour) and 45 minutes (3/4 hour), which is easy to remember, and thirds of an hour (20 minutes, and 40 for 2/3rds or dealing with in negative way for hour-crossing sums) are something I deal with on a semi-regular basis, but fifths (12 minutes) and tenths (6 minutes) might also be useful to some. Not much to remember, through use. (40 minutes plus 40 minutes is 1/3 above the next hour, etc, just remember to 'carry the one'.)

Twelve hours per half-day may have been similarly chosen to subdivide (non-linear) time of daylight/night-time into halves, thirds, quarters, sixths. A day and a night give 24 (these days consistent, not welded to dawn and dusk and angles of celestial bodies and shadows) which subdivides into those and the new double/half handy amounts, e.g. eights (three of them in 24, as there were three fours in twelve; there are now six fours in the 24). But I know different cultures (even with the 24-hours with 60-minutes system) talk about times differently... I believe that "8:35" in Swedish is said (in long-hand) as “five past half to nine”, for some reason they went with "half to X" instead of "half past Y" for X=Y+1, and while I would say "twenty-five past/to", they like going by five (to/past) that half-hour that is invariably 'to' the next hour.

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### Re: Math without calculators

liberonscien wrote:
Soupspoon wrote:
liberonscien wrote:What would be the most effective way to learn time math?
...

...

...hours of practice?

I should have expected this.
I meant would one be wise to teach oneself mod 12?
Is there a more effective way to learn how to use what may be a broken system?
I know the Babylonians or someone else divided time into sixty but I don't understand it.
Time how we measure it on the average clock is somewhat like a mod 12 system coupled to a mod 60 system. We are taught at birth to use base ten. What is the reasoning here and what is the best way to learn it.

Spot research suggests that the ancient Egyptians invented the 12-hour day (with the period between sunrise and sunset divided into 12 equal parts, longer in summer and shorter in winter) based on a counting system where you touch your thumb to any of 12 finger bones on the same hand (rather than counting the digits on both hands) and also a 12-hour night, but the ancient Greeks came up with the idea of having 24 fixed-length hours between noon and noon as a convenient mathematical fiction for astronomy. The idea of standard hour lengths for common time is relatively recent, and came in with mechanical clocks some seven centuries ago.

Meanwhile, minutes and seconds arose from geometry - arc-minutes and arc-seconds - fractions of a degree. Early mechanical clocks didn't get below 12ths of an hour, but by 400 years ago, 60 minute divisions of an hour were standard.

In principle, you could replace the current system of time with a decimal version, with the day divided into 10 parts, each divided into 100 sub-parts, each divided further into 100 sub-sub parts, which would roughly correspond to a heartbeat (running slightly faster than seconds) but 19th century attempts have failed to catch on.

When it comes to getting comfortable working with time arithmetically, the only real option is to practice - look at a bus timetable and consider the intervals between buses, or find out when something or other gets released, and work out how many hours away it is, or make a friend in a different timezone and convert between your times when you chat with them (for example, I know that my friend on central daylight time's local time is approaching quarter past 1 in the afternoon as I post this).

Yes, the mixed-base horology is sometimes inconvenient for calculating with, but it fits human activity patterns reasonably well, and it's pretty rare to be concerned about more than one level at once - if you're working in times to the nearest second, you're unlikely to be dealing with intervals more than 5 minutes; if you're worrying about hours, you're often happy rounding to the nearest 15 minutes. And so on.

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### Re: Math without calculators

I think only Math related major students should be the only ones allowed to use calculators for their math classes.

Everyone else who is taking a math class just because it is required for them to graduate shouldn't be allowed to use calculators. Those mental skills and brain exercises will be more beneficial to them than those that are already math savvy.

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### Re: Math without calculators

heuristically_alone wrote:I think only Math related major students should be the only ones allowed to use calculators for their math classes.

An odd perspective.

My attitude: Learning doesn't ever require a calculator.

Doing math, in practice, may involve much bigger and more tedious calculations than are necessary for learning. So sure, use a calculator then. Since you already know how to do it the long way, it doesn't make the calculator into some mysterious magical oracle.

But if you don't want students to have to do the work...don't use the long equations (long calculations required) in teaching.

Then no one uses a calculator in the classroom. Unless you're building one from scratch, programming it, and testing it.
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### Re: Math without calculators

That's not true. There is definitely some math that requires a calculator because it is easy to use but very hard to actually do by hand. The trigonometric functions came to mind for example. Sure it would be possible to teach how to use sin() and cos() without actually using them, but that makes the task really much harder for no good reason. (Also, yes there are trig tables that one can use instead of a calculator, but that really is just a less precise calculator with a cumbersome user interface.)

tl;dr: To calculate trig functions by hand should not be required of someone just learning how to use them.
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### Re: Math without calculators

lorb wrote:tl;dr: To calculate trig functions by hand should not be required of someone just learning how to use them.

Calculate, no. Simplify, yes. I had several no-calculator trig exams, it's just the expected answers were expressions rather than numbers.
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### Re: Math without calculators

If the angle is pi/3, you don't need a calculator. If it's 0.4, you do. Or you leave the answer as sin(0.4) in which case, you don't again!

Best of all, angle is just alpha. Why would you ever have numbers, this is math.

Though I think it could be rather useful when first learning trig to be forced to evaluate your own numerical answers using a pencil and paper, so that these functions aren't black boxes. Take one you can calculate algebraically, and use the angle sum and addition formulas and small angle approximations and such to get to the desired number. It can be done. And then the function isn't a black box. It is fine to take shortcuts around a function you understand, but you have to get it into your noggin the hard way. If it's important.

Trig functions aren't. Instead, we should be making kiddies calculate the standard deviation of a set of five values by hand.

Which, I repeatedly told my students, "This will be question 1 on your 4 question exam, so please do not get this wrong." Extremely disappointing success rate.
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### Re: Math without calculators

HES wrote:
lorb wrote:tl;dr: To calculate trig functions by hand should not be required of someone just learning how to use them.

Calculate, no. Simplify, yes. I had several no-calculator trig exams, it's just the expected answers were expressions rather than numbers.

This was my experience as well, Both my High School level Trig class, and my college level Calc class were non-calculator classes. Granted, I despised them, but not as much as my Construction Documents class where the teacher thought that doing floorplans and footing details by hand on pen and vellum was the only acceptable way, because "Computer Drafting is a fad."
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### Re: Math without calculators

doogly wrote:Though I think it could be rather useful when first learning trig to be forced to evaluate your own numerical answers using a pencil and paper, so that these functions aren't black boxes. Take one you can calculate algebraically, and use the angle sum and addition formulas and small angle approximations and such to get to the desired number. It can be done. And then the function isn't a black box. It is fine to take shortcuts around a function you understand, but you have to get it into your noggin the hard way. If it's important.

That sounds like a really difficult approach. Isn't it easier to grasp when the students are asked to numerically approximate the angle (if the original expression was symbolical), draw a unit circle, draw a triangle with that angle inside and measure the length of the correct side?

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### Re: Math without calculators

Word, that is good. The ruler and protractor are nature's calculator.
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### Re: Math without calculators

HES wrote:
lorb wrote:tl;dr: To calculate trig functions by hand should not be required of someone just learning how to use them.

Calculate, no. Simplify, yes. I had several no-calculator trig exams, it's just the expected answers were expressions rather than numbers.

Yes, exactly. This.

And then estimate, through any one of a huge number of methods, the numerical answer. With some idea of how exact your estimate is. I wouldn't go so far as to ask about significant figures, but just get enough of an estimate to sanity check a calculator answer.

I can't tell you the number of students I have seen who would (for example) use a calculator to compute the cube root of 2739, press the wrong button, get an answer a little over 50, and not even notice. (Can you guess what they did wrong?)

This is what comes from failure to estimate actual numerical answers: A total inability to sanity check.

A more factual example: Given some trig word problem where the answer obviously (from any rough sketch) must be an angle between 90 and 180 degrees, found by taking an inverse sine of some ratio, what do you think is the most common mistake made? (Again, failure to sanity check.)

doogly wrote:And then the function isn't a black box. It is fine to take shortcuts around a function you understand, but you have to get it into your noggin the hard way. If it's important.

Trig functions aren't.

I totally agree with you about "getting it into your noggin," but I think there's a distinction between algorithms and functions. It's possible to get the function solidly into your head without ever learning some particular algorithm for numerically computing it—AND it's important to do so.

I wouldn't make anyone numerically compute a trig function by hand unless they wanted to. But I would and DO insist that they be able to tell me, at a glance, "roughly" what the answer is.

Specifically, for trig functions, they should be able to tell me instantly (for sine, cosine, tangent, cotangent, secant, cosecant) whether a particular value is positive or negative, whether it is greater than or less than one (or negative one), or whether it is exactly equal to 0, 1 or -1.

For inverse trig functions (or relations)—or more broadly, for trig word problems or diagram based problems where the answer being asked for is an angle—they should be able to tell me immediately, by inspection not only which quadrant the angle is in, but what "octant," i.e. for a first quadrant answer, whether it is greater than or less than 45 degrees (or pi over 4).

I recall once teaching a student how to sanity check 2.73^3.39 and give some kind of range immediately. Of course, you just compute four values: 8, 27, 16, 81. It's somewhere in between 81 and 8 for sure. Probably reasonably close to 27, but not close enough to bank on.

To get algebra students out of the constant figure-figure fiddly work with coordinates and an inability to calculate slopes correctly (wherein they would make dumb mistakes like giving slopes of the wrong sign in their homework) I made a set of flash cards with lines on grids that deliberately didn't cross any grid intersections (so they couldn't find two points' coordinates and compute). I only made 7 cards. For a first pass through the cards I would just have them say whether the slope was positive, negative, or zero. Once they had that (and it takes some surprising patience sometimes on the first pass to have them snap out of fiddly calculations into instant recognition), then I would go through again and I would want to know whether the slope was greater than 1, equal to 1, between 1 and 0, equal to 0, between 0 and -1, -1, or less than -1. Usually just, second pass, "Positive or negative?" "Positive." "Good; greater or less than one, or is it one?"

I made a similar drill for spotting, for particular points on curved lines, positive/negative/zero for the value, the slope and the concavity for calculus students.

This is what I'm REALLY talking about when I say "math without calculators." I mean instant ability to RECOGNIZE basic facts of math. Calculators are totally and entirely worthless without this type of ability to recognize and sanity check your answers.

Computing numerical answers without calculators is good too, but it's much much much less important than recognition.

(Hmmm, maybe I should have a blog....)
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doogly
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### Re: Math without calculators

Yeah that is all legit. I think there is value to learning *some* algorithm for a function. Bu tdefinitly the major points are what you describe, and that does likely take full effort to learn.
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### Re: Math without calculators

Learning algorithms for arithmetic and other such things, sure. I know of no exact simplification for sin(.4). If someone is studying trig they should be able to ballpark it, but I see no benefit in math classes to bringing out a calculator to give a less precise answer. When things start become a real pain to compute (e.g. triple integrals) the real work is usually in getting them setup, anyway. After the first test for ability to do them, stop wasting everyone's time reconfirming and focus on the more thoughtful part.

Even for some simpler stuff this applies. In a popular finite mathematics textbook I see way too many matrices that require doing stupid things like 51/27-43/51 a dozen times to get into reduced REF. Two, three, and five are wonderful enough denominators. (Also a large amount of students using calculators end up with silly things like 51/27-43/51=1.05.)