This post was inspired by a conversation with Rick, at Exploring Binary.

Rick: I have to learn to go to Wolfram Alpha for mathematical queries such as the one I mention in the article. Typing4, 24, 124, 624, 3124into Wolfram Alpha gives the answer I sought directly: a_{n}= 5^{n}– 1.

Sue: But then you wouldn’t have had as much fun! And I’d say that’s the problem with Wolphram Alpha. Having fun with math is often hard work. It’s so much easier to click your way to an answer. But just not the same.

Rick: For me, it goes beyond Google and Wolfram Alpha — it’s the availability of computers in general. I often find myself slapping together a small program or script or spreadsheet before I sit down and think about a calculation first. Sometimes this leads to serendipitous discoveries; other times it leads to more “Well, Duh!” moments .

Maybe it’s good I started my serious study of math before the computer era really got underway. When I started high school ('70), calculators weren’t yet in common use. When I started college ('74), we didn’t have graphing calculators. I remember drawing hundreds of graphs while I was in calculus. For a long time, I felt like I wasn't a real mathematician because I never learned how to use a slide rule. All the math people I knew could use one, because they'd been doing difficult calculations for a few years more than I had, and hadn't had calculators. There was no internet (in my personal life, anyway) until well after I was done with my formal education (’89).

I saw a game for learning factoring, called Divisor Miser, at the Colorado NCTM conference in ‘82, I think. Someone had programmed it on a Vic-20 or TRS-80, or something like that. But I don’t remember computers being used to solve math problems in the computer course I taught in the mid-8o’s.

I taught junior high for a bit, and one of the math teachers wanted me to help him write a program to solve a probability problem. But he had the calculations all wrong, and the right calculations were simple enough that a computer program would have been silly. I was shocked at his bad understanding of math. He was our department chair. I knew almost no probability at the time, but I learned enough from the student materials (extra credit stuff) to figure out that problem. Yikes!

And yet, I love how technology can help us see. Here are some examples that come to mind:

- Dan wanted to simulate an amusement park ride he calls a scrambler, and pulled up Geometer's Sketchpad. Nice results.
- I used the matrix solution capabilities of my TI-83 when I was solving the regions in a circle problem. (Put n points on a circle, connect each to each with a straight line, how many regions in the circle?) If you already understand the solution to that, you may think this was a totally unnecessary use of technology. But it helped me to see. I'd like to post about that problem soon.
- At last year's Math Circle Teacher Training Institute, this problem was posed: "Can you tile a rectangle completely, if you require that only squares are used and each square must be a different size from all the others? How?" We used Geometer's Sketchpad to play with our sketches of how it might work. I dropped out of the group thinking about this. When they presented their solution the next day, they mentioned having used Mathematica to solve some equations.
- Back to my conversations with Rick. The one above is on his blog. We've been having another conversation at the same time in the comments to my post about the Carnival of mathematics. I wondered about the pattern in decimal fractions, .99 in particular, when they're converted to binary. He gave me .99 in binary, and pointed out that it repeats. I started playing with the conversion by hand, got frustrated, and turned to Wolfram Alpha for assistance. (My first time using it for a serious purpose.) Without any more tedious calculations, I could think about patterns. Turns out .9 is a repeating decimal in base 2 with a four-digit repetition, but .99's representation in binary takes 20 digits. Weird. I haven't seen the light yet on that one.

I come down heavily on the "tool" side.

ReplyDeleteReally, if you have a good problem, the machine is not going to solve it FOR you, right? It's going to automate the annoying, error-prone stuff. And even using technology, there are elegant and inelegant solutions, that you gain the capacity to judge the difference with experience.

And more often as I get older and I like to think more sophisticated with my maths, the answer is not the point! How often at MCSI do we get an answer, but are dissatisfied with it, because we didn't gain an intuitive understanding of why it was so?

I think I agree with Kate. I used to think that calculators were a really bad thing for math students. I have learned that they are bad when a student is learning arithmetic. But for math, they often become a necessity. It's not worth the time for a kid to trudge through the quadratic formula if a=1.3 by hand as long as she has already proven herself proficient in the arithmetic.

ReplyDeleteI think "both" only answer I could give, because everyone uses technology in different ways. I find it can be a great tool, not just for generating data but for leading you to an intuitive understanding of the problem. More than once I've tried to get some answers from a computer, only to have the process of translating the problem into something a computer can understand give me the exact insight I needed. However, it's absolutely true that many of my students use it as a crutch and rely on calculators rather than understanding, to their detriment.

ReplyDeleteIt's a tool if used correctly. Unfortunately there are many teachers in our profession who use it as a crutch or actually as something harmful. This is why as students move up to more difficult levels of math, we end up having to reteach lots of relatively simple things that they were never really taught how to do correctly.

ReplyDeletewhy not just give capital

ReplyDeletecompete control of the budgets?

*plus*!

make each generation of workers

obsolete before they come out the gate!

*and* look good doing it!

school is just more TV!

whee!

(i'm for it as you can see.)

the thing is...

ReplyDelete*i* use *tools at hand*

and this is the gospel i preach.

i had a class where i could put the net

on the screen during lecture.

darn *right* i put the net on the screen

when wanted to look something up...

on the very principle that

"this is the way i actually do it".

i *love* showing a whole class

the "way i actually do" something

when this differs from the *usual* way;

this would appear on a glance

to be most of the value-added

from a given lecture in most cases

if we tried a results-first accounting.

the textbook says *that*

but everybody really does *this*....

well, it's a damn shame that they've

created so *many* of these situations

but... anyhow here's something i can

do right as it were by instinct.

anyhow, somebody has a calculator;

they're working on a problem;

i know how to use the calculator

to throw some light; obviously

i mention it. this used to happen

*all the time*... but only *because*

somebody had designed these

math *courses* to make it so.

it's always "classroom" math for me

in or out of the classroom.

and "technology", if it isn't

*always* about the digital divide

and getting into everybody's pocket...

might as well be. it's no crime to be poor.

I'm trying to think how we, as people who do math ourselves, may use technology as a crutch. Maybe I gave too much evidence for the tool side and not enough for the crutch side? Here's one from me on the crutch side:

ReplyDeleteYou have a circle, you put n points on it and connect each point to every other with a straight line, how many regions are created?

I'd worked on it many times over the years. In January 2008, when I got a formula that wasn't making sense to me, I googled it. And saw how

otherpeople thought about it.Now I'll never know if I could have discovered this result all on my own. (I did forget the details, and work it out on my own a year later, I'm proud to say.)

So I guess I'm asking something like, are the temptations of Google and Wolfram Alpha a problem for you?

Owen, what if we're working toward all open source software (etc), does that take it out of the hands of the capitalist masters? To my mind, textbooks are more a problem for my low-income students than technology. The internet just might help us walk away from those.

As for using search engines -- sure, they take away the satisfaction of discovering a solution yourself. But I find that, after reading a multitude of solutions to the same problem, I come away with a deeper understanding -- an understanding I'm not likely to achieve working the problem in isolation.

ReplyDeleteI've gotten stuck and turned to the Google in desperation - I'm not proud of it! But I REALLY try not to, and even if I do, if all I'm getting is "the answer," I keep working until I understand it myself.

ReplyDeleteBut maybe we should just think of it as scaffolding. If a student was working on a problem that we knew was too far out of reach, we'd ask her leading questions, right? If a problem is hard enough that I need to go hunting around the internet for similar problems, maybe I'm just scaffolding myself.

It's interesting that you mention the connecting points in a circle problem - that's one that I know the function is that weird 4th degree polynomial, but I still don't understand why. And it still seems like there must be a function that involves C(n,2).

but i don't *do* math for real world reasons

ReplyDeleteand never did... only ever to talk to other people

interested in talking about math problems.

(like hardy, i've never done anything useful.

*unlike* him, alas, i've also never done

any significant mathematics...)

so research-as-opposed-to-development...

looking it up if you can instead of working it out...

becomes a question of "what community

should i imagine myself as working with"...

and i just don't seem to imagine myself working

except with fellow students of math per se...

where, if we've got access built in,

i'll look on the web at the first sign of trouble

because it's *more comfortable* having

answers and hints available and ghod knows

trying to solve math problems in quasi-public

is a pretty awkward situation even for

most math students.

the only reason to *ignore* a tool-at-hand

would appear to be a sort of "you're not ready"

this-is-the-way-we-do-it-in-*this*-course

-ism

"prerequisite" fetishism that one would like

to get away from.

the technique i'm advocating here is:

"real" problems are there to be solved

*by any means necessary*...

and we can put in esthetic touches at leisure.

but if i'm supposed to order something

or download something or blahblahblah.

forget it.

@Kate, the solution takes that weird 4th degree polynomial and untangles it a bit so C(n,2) shows up as part of it.

ReplyDeleteIt's both. It all depends on how you use it.

ReplyDeleteIt's much like a car.

For simple tasks like getting to your neighbor's house, you walk, no one drives. Driving in this case is like students using a calculator to find the result of 4x8.

For getting someplace far (where you know you could get there but just takes you a long time) a car is a great tool. Gives you ability to explore places/things with minimal time.

I talk about tricking graphing calculators to graph piecewise defined functions. The process itself is inefficient, but it is an extension to understanding holes in rational functions and domain restriction of functions and multiplicative identity. The learning that takes place to put it together is much more interesting than the final result.

I've also used calculators to introduce transformations of functions. I ask students to sketch a series of graphs (y=x^2, y=x^2+1, y=x^2+2, y=x^2-1) and look for a pattern. I found it much more time-efficient than giving them a worksheet that tells them to fill out a table of x and y values, sketch the functions, and look for pattern. I still give the worksheet, but it's alot shorter and only after they already know what they are looking for to reinforce what they just learned.