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Public Forums - October 19, 2010
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I can never understand those people "oh young people these days, back in my day we had no calculators". Mental arithmatic is not mathematics.

"We should ban Calculators in Mathematics" is like saying "We should ban reading in English", "We should ban textbooks in History" or "We should ban instruments/paints and brushes in music/art"

Hmm. I agree that mental arithmetic is not the same as math, but IMHO a better analogy would be:

"Banning calculators in Math is like banning a device (possibly still uninvented but probably coming to the market sooner than I care) that reads a book aloud for you in English."

Watch the literacy rate drop...

I teach undergrad math for now, and the students that need a calculator to add fractions are the ones who have problems manipulating rational expressions. I was once helping a group of students do their vector calculus problems. Towards the end we got the answer down to 1/4+1/6. At this point a student dives into her bag, and starts searching for her calculator... COME ON, is that really the fastest method of adding these two fractions that you know???!!! And she was actually among the best 25% in that course.

But... does banning the calculator help those students, or does it make life harder for them, because they have to do arithmatics as well as the original calculus?

Being able to read text is not literacy. If we had a better way to move information around (I don't think we ever will), then hanging on to text would be stupid. Just like hanging on to long-form division and multiplication skills is stupid.

That is very true, however, with graphics calculators now all you have to do is type in the eqn and you get all the information under the sun without needing to know the slightest thing about what it looks like and its general nature, this is the more difficult problem when people head to uni and are asked to actually think about things as opposed to typing it and reading the numbers.

You will see the point of banning a calculator, like me, when you see students taking a university Calc course relying on the calculator to calculate 10x100 or sin(0).

That may be true but calculus is mostly about algebraic manipulation anyways. What I find happens is that as you move on and see those kinds of things more and more you just remember them and I think that as you advance in mathematics at least at the university level you rely less and less on a calculator because the realization of being able to do things without relying on a tool that if you use incorrectly gives you a wrong answer.

After grade 10, eliminate numbers, and use only symbol manipulation.
In grade 11, require that most basic proofs be worked out from axioms.
In grade 12, teach everything in the most abstract terms possible, so that students are forced to abandon their physical intuition and work only from logic.

http://math-blog.com/2010/09/27/symbolmania/

You might be interested in reading that.

But read critically. Simplifications and half-truths abound leaving an unsupported argument. I suspect the author was writing down to what he perceived as the limitations of his audience.

From experience, the people who suffer most with mathematics are those who don't get a solid grasp of mental arithmetic from the get go, and are moved on to more complicated subjects without a solid foundation. The introduction of calculators really hasn't helped in this regards. Granted, mental arithmetic is far from all that there is in maths, but it's a really important basis to start from.

Agreed.
In school, I was always the worst at mental arithmetic because I never memorized anything. I always worked out every problem using the basic rules of arithmetic.
To this day, I don't know my multiplication tables, and yet here I am, studying math in uni.
The moral of the story? We teach math to people the wrong way. It should be about the method, not the answer.

On the subject of understanding symbolic mathematics without having worked enough on concrete examples, happy Grothendieck prime day! (In my time zone and date format, it's 20/10/2010; 2010 is 57 in ternary. What? It worked for 42 day.)

uhmm, not really.. we never needed a calculator in most of my math courses at the university..or in most of my math classes in hungary even in elementary OR highschool. many types of math isn't about .. crunching numbers, as far as i know. why do you need a calculator then? kids these days (i'm not even 20 XD) don't know what it means to multiply, divide, or maybe even subtract.. because they just plug it into the calculator, and it shows the magical answer!

Calculators these can be used for so much more than basic arithmetic.

*these days

The rightmost guy's suggestion is truly evil.

I'm a little confused. Are you defining basic math as arithmetic? and which one of those guys is supposed to be a mathematician? From a mathematican's perspective, a calculator wouldn't hurt or help someone, I would think. Adjusting coins/greedy aligorithm, sounds more like what a computer scientist would just...ehh... ok maybe an applied mathematican.

So in this context, I take it "Optimal solution" means least total coins?

So, for example, if the nickel was eliminated, 30 cents change would be given as a quarter and 5 pennies under the greedy algorithm, but 3 dimes would be a better solution.

How to improve math amongst the public: Replace every subject, but math, with math. That'd do it.

But as for the last suggestion: I'd really hate to fire up the non-deterministic oracle in my head every time I have to pay for something! Strongly increasing vectors rule :)

Here's a related problem I thought of while watching House yesterday. In the show, House and Wilson were given 68 cents change from the chinese place, but could only account for 58 cents, and instantly concluded the baby had swallowed a dime. It occured to me that if the numbers were, say, 78 and 68 instead, it wouldn't have even been possible to be ten cents short, since you can't make ten cents out of 3 quarters and 3 pennies.

So what I'm wondering is if you think of "giving change" as a function from positive decimal numbers (with at most 2 digits to the right of the decimal) to integer row vectors in the obvious way, can one describe when this function is additive? For instance, making the vectors (quarters, dimes, nickels, pennies), we get f(.58) = (2,0,1,3), f(.10) = (0,1,0,0), and f(.68) = (2,1,1,3) = (2,0,1,3) + (0,1,0,0), so it's additive for .58 and .1

Well, I'm no mathematician, but if f(z) = f(x) + f(y), and f(z) has a component that is 0 and either f(x) or f(y) does not contain a 0 in that same position, then the example is not additive.

Well, supposing x is the number of cents, generally, we'd have something like:
f(x)=(a, b, c, d)
with
a = x modulo 25
b = x-(a) modulo 10
c = x-(a+b) modulo 5
d = x-(a+b+c)

So you could set out conditions for when this function is additive. Here's one example:
f(x)+f(y) =/= f(x+y) if
(x modulo 25) + (y modulo 25) =/= (x+y modulo 25)

the other 3 conditions would be much more complex...
And that's not even considering the cases where x+y>99

I honestly doubt the approach you've taken will make the problem any simpler...

damn it all... I keep confusing my operations...
for my last comment, pretend modulo means "take the integer part of the division" as opposed to the remainder.

My problem with calculators in the classroom is that they are robbing youngsters of the chance to develop number sense, which manifests itself in many other areas of Algebra. Yes, smart kids who have developed number sense should use calculators as a time-saving device when working on more serious stuff. Moreover, they can recognize obvious incorrect answers which probably resulted from bad input. OTOH, students without number sense use calculators as crutches to do work for them, and wouldn't recognize the mistake in saying -3 * 2 = 25. (true story, that)

What's up with Greedy algorithms? Second this week :)

Maybe Mike is greedy for comments?

In my Honors Algebra 3 and Trigonometry high school class, we haven't been allowed to use calculators yet this year. When we get to the trigonometry part of the year we will, but having to do pencil and paper arithmetic (specifically for simplifying radicals) is a real throw-back for some of my classmates. I think it's easy, but just makes tests and quizzes that much longer to take. That may have something to do with a good number of people doing quite poorly... Ha...

Me too, except for some of the questions are evil if you haven't got a calculator. for example, factoring polynomials with 4 digit coefficients...

Hmmmmm. The calculator/no calculator argument. I have students that continue to have problems with addition/subtraction of negative numbers. After working with the calculators (with the instruction to pay attention to what they are calculating) they have become stronger with the procedures necessary for doing the math without a calculator as they are recognizing the pattern that is coming out of the calculator. And many of the students no longer need to use the calculators to do the simple arithmetic. Win-win?

I'm just saying that when you have students that have struggled with a concept/procedure for a long time, perhaps if they can use the calculator to internalize a rule for themselves they could begin to learn to help themselves.

I am not in a "all calculator all the time" camp, but there are some calculations that I don't need to revisit (linear approximations of trigonometric functions between two whole numbered degrees using a table comes to mind).

Absolutely. No point in not using a calculator to compute the values of trig functions at odd angles like 34.6 degrees, or multiplication of multidigit integers. But sin(0) or 4*3 is another matter.

Also either many calculators have some precedence rules wrong or (more likely) the users are not aware of some grey areas. Iterative exponentiation is a case in point. If I had a dime for each student, who thinks that 5^5^5 equals 25^5 "cause the calculator says so!"...

I also fully support the idea of using a calculator to experiment and build rules from the outcomes (demanding for an explanation would be even better!). I used my younger sister as a guinea pig once, and made her do multiplications like 5*5, 4*6, 8*8, 7*9, 10*10, 9*11,..., at which point she figured out the pattern. Next I made her do 9*9, 7*11, 13*13, 11*15, and again she caught on quickly. My job was done.

I'm not sure what calculator the students are using to get 25^5 because my cheap sharp when you enter 5^5^5 gives a number on the order of 10^17

Well. Then your calculator is wrong. 5^5^5=5^3125 which is approximately 1.9*10e2184. Your calculator is off by only 2167 orders of magnitude :-)

That was exactly my point!

Arrgh. I meant that their calculators claim that 5^5^5=3125^5=5^25 that is approximately 2.98*10e^17. Not 25^5 - my bad.

Just to make sure. The correct value of 5^5^5 is approximately 1.9*10^2184. Many carelessly programmed/used calculators come up with an answer appsoximately 2.98*10^17. The reason for the mistake is that the calculator (or user) happily takes the "intermediate value" 5^5=3125, and goes on by raising that to the fifth power. In other words they calculate (5^5)^5=5^25 instead of 5^5^5.

haha yeah, well I'm doing a course right now in numerical analysis. Calculators and the way they calculate things certainly have their limitation. Now I just plugged in 5^5^5 and that is what the calculator spat out, but in reality you do need to put the brackets 5^(5^5) and this would give you an overflow error. And in that sense I always put in brackets all over the place when using a calculator so that I don't make a mistake and it's really explicit what I want the calculator to do. Even if it is not necessarily required.

What I do find interesting is that using modular arithmetic can be used to calculate small bases with very large exponents. That's something I've learned in Cryptography. I really enjoy learning about all the different uses for mathematics.

Enjoy the crypto applications! Hopefully your teacher also gives Diffie-Hellman key exchange as an example. The first time I learned about this deficiency of calculators was when I was giving a midterm on a freshman algebra course, and one of the problems was to compute the remainder of 5^5^5 modulo something. I was very annoyed, because I happened to choose such a modulus that the answer didn't depend on whether you computed 5^5^5 or (5^5)^5. A nightmare to grade - the students have to show the intermediate steps to get more than "pity partial credit".

I admit I thought of this story after reading some of the comments: Asimov'a "A Feeling of Power" http://downlode.org/Etext/power.html

Let's take down wolfram alpha! Or whatever it is called this days. BTW, is it true that you can't do long mental arithmetics using Roman Numerals?

the US Army gives their soldiers advanced tech weapons that makes the job easier, but they also still have to be taught how to work with their hands and all other low forms of tech just in case our high tech is lost or can not be used.. so in the same way i think when student are given tests or quizzes they should hand in their calculators but for everyday run of the mill stuff they should be able to use them so they can be faster and not loose what little interest they might already have in math

I remember when I took calculus and the teacher basically forced us to use calculators to evaluate some integrals... he actually got mad at me once because I used integration by parts instead (about a month before we covered it in class :)

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