Teaching Mathematics With Computers

[Paul Beardsley]

As part of my teaching course, I am doing some research into the role of computers (and ICT in general) in the teaching of maths.

I would be interested in reading anybody's accounts of how they learnt or improved their maths through the use of ICT. Examples might include:

* Doing an on-line course

* Mathematical games such as Dolphin Racing (where you have to choose the biggest fraction to get your dolphin to go faster than the others)

* Nntendo brain trainers

* Serious maths packages such as Mathcad or graph-drawing programs

* As part of another activity, e.g. writing formulas in Excel

Alternatively, I'd be interested in computer-based maths teaching that didn't work, or where there was an overreliance on computers when dice, counters, cardboard boxes and so on would have been far more effective.

Thanks in advance.

[davidlpf]

As long as the time on the computers is balanced with time of actual doing the problems with pen/pencil and paper then it is good idea.

[Paul Beardsley]

As long as the time on the computers is balanced with time of actual doing the problems with pen/pencil and paper then it is good idea.
Interesting thought, David - thanks.

Some questions: By "balanced" do you mean a roughly equal amount of time, or does that depend on the topic? Do you think this is true at all levels (from, say, a child learning how to add up to a trainee engineer learning calculus) or only at certain levels? What if the learning is fairly informal, designed only to get a "taste" of a topic?

[Demigrog]

Heh, online courses have never been a favorite of mine for math, because having somebody watch what you're doing on a problem and point out your mistakes is much easier than staring at an unhelpful screen for ten hours and still not making any progress. :)

Also, online testing was pretty annoying when I was in college, especially when there were bugs in the testing software (ie entering a negative value was always counted correct on my linear algebra tests--which the students noticed immediately while the professors were baffled at the odd grade distribution for most of the semester)

[davidlpf]

Interesting thought, David - thanks.

Some questions: By "balanced" do you mean a roughly equal amount of time, or does that depend on the topic? Do you think this is true at all levels (from, say, a child learning how to add up to a trainee engineer learning calculus) or only at certain levels? What if the learning is fairly informal, designed only to get a "taste" of a topic?
For children using computers as a way of making the math a little more fun to keep their attention levels.
I once took a course were part of the course was a regular calculus course and the other part was to figur how to set an equation for maple to do the intergration. This was excellent because you learned the basic theory then how to apply to problems using a computer.
But at the end any course the student should be able to set down and write an exam to prove they know what they are doing and this should be done without much assistance from electronic aids.

[mike alexander]

Used properly, I think that one can use a computer as a very valuable tool. I like Excel or a similar spreadsheet because I can do experimental math with it. Calculating/ graphing the binomial distribution using random numbers to represent coin flips, showing how the Fibbonaci series number ratios converge to the golden ratio, these can be great fun.

[Paul Beardsley]

Good answers, guys!

Demigrog - your on-line course sounds less than ideal. I'm surprised it wasn't set up to give you instant feedback and help. Did you have to wait for a human teacher to get back to you all the time?

Can you remember when you sat the course and the test? I'd particularly like to know about the test, as I'm guessing the people who programmed it now know to look out for bugs!

davidlpf - from your post, I think you are emphasising the importance of assessment, regardless of how the teaching is delivered.

Mike - I like the idea of experimental math, particularly when you can use it to demonstrate the cool stuff you've mentioned. Of course, it is in theory possible to do those things without a computer, but it would be so incredibly laborious that in practice it would probably never happen unless you had a huge amount of time on your hands, and a worrying degree of single mindedness. It seems to me that there are parallels with chaos theory here - the sheer speed of number crunching enabled people to perform iterative calculations that led to fractals.

[HenrikOlsen]

I'd say once the kids know the basics of math, give them access to MathCad or Maple, then let them use them to solve real-life problems.

The actual skill needed isn't to manipulate random formulae, it's describing the world in mathematical terms in the first place.

All this pen and paper stuff is really not needed beyond getting an idea about what's going on.
Either you have access to a calculator/pc, or you're doing quick sums in your head while shopping.

[davidlpf]

I'd say once the kids know the basics of math, give them access to MathCad or Maple, then let them use them to solve real-life problems.

The actual skill needed isn't to manipulate random formulae, it's describing the world in mathematical terms in the first place.

All this pen and paper stuff is really not needed beyond getting an idea about what's going on.
Either you have access to a calculator/pc, or you're doing quick sums in your head while shopping.
Sometimes it is handy to the calculation by hand first, because you might not right by a pc at that time.

[Noclevername]

As someone who was hamstrung by the American public school system, I'd like to improve my math skills. I have add, subtract, multiply and divide down (with occasional calculator assisitance) but I have only the vaguest knowlegde* of any higher functions. Calculus? Didn't he conquer the Goths?
Trigonomonomony? Tenspeed functions? Al-jabber? No clue.

Any suggestions for online learning in these areas?






*EDIT: Or spelling. D'oh.

[davidlpf]

Honestly I do not Noclevername, at university it took me several times to get past the first half of the first year of calculus, mostly because I had hard time through high school and most math programs in my area are not geared towards advance math at leaast the time.

Calculus is generally divided into to catagories differential which was invented by Newton or Leibniz and is based off of the deriative which is the slope of the line tangent to a curve. It is very useful when finding how fast something is changing such as speed. The are is intergral calculus and has alot of different inventors and generally a way of finding the area under curve, this is helpful in finding things such as how much work is done (at least in a mathematical way).

[jfribrg]

I don't have a specific idea on what works and what doesn't with computers, but I do have an enlightening story. I was taking a class at the local community college (with a bunch of college kids half my age). One of the students remarked that calculus cannot be taught without a graphing calculator. That piqued my interest because "back when I was in college", graphing calculators didn't even exist. I asked the kid for details about why a graphing calculator is so essential, and he asked how one could find the graph for an equation. I responded that you calculate first and second derivatives, intercepts, maxima ,minima, and inflection points, and from that you can usually come up with a good idea of what the graph looks like. The kid responded by saying that he always learned it the opposite way: you start with the graph and from that you can deduce information about the maxima,minima, inflection points and intercepts etc. It never occurred to him that you could do it in the reverse manner. Bear in mind that this wasn't a dumb kid, but the material was presented differently because of the availability of the graphing calculator. In the end ,you sort of have the same knowledge, but you aquire the knowledge in a different way. I don't really have an opinion on whether this is good, bad, or neither, but I thought I'd share it anyway since it sort of relates to the OP question.

[Paul Beardsley]

I thought I'd share it anyway since it sort of relates to the OP question.
It certainly does - thank you very much for that!

Thanks also to Henrik and NCN.

[peter eldergill]

I teach Calculus at the high school level, and I have a love-hate relationship with them.

First off, their resolution is crap. I much prefer using a basic math graphing program.

On the other hand, they are convenient to bring into the classroom and distribute. I much prefer to not use them personally and show the students pen and paper type problems.

Calculus in highshool is about behaviour of functions. To me it's more important to undersand why the maximum occurs where the slope is zero than to use a calculator to find it

Pete

[Larry Jacks]

The kid responded by saying that he always learned it the opposite way: you start with the graph and from that you can deduce information about the maxima,minima, inflection points and intercepts etc. It never occurred to him that you could do it in the reverse manner. Bear in mind that this wasn't a dumb kid, but the material was presented differently because of the availability of the graphing calculator.

One of the side benefits of being able to do math in your mind or manually is that you can estimate what the answer is before using a computer or calculator. Then, if you make an all to common entry error and get an incorrect result, you can detect it. Otherwise, you need to try entering the values at least twice to see if you get a different answer. Even that doesn't work if you make the same mistake. I've seen too many people just push some buttons and accept the result no matter how absurd because they didn't realize they made an entry error.

[Paul Beardsley]

One of the side benefits of being able to do math in your mind or manually is that you can estimate what the answer is before using a computer or calculator.
I don't know if I'm confusing fact with fiction here, but I recently heard something about a student who used Eratosthenes' method of measuring the diameter of the Earth... and got an answer of 1.2 metres! If true, it sounds as if the student had the necessary maths skills, but not the commonsense to check for obvious errors.

And even if it's not true, it's sufficiently reminiscent of the Mars probe where they confused imperial and metric measurements. Or one time I was writing up a school lab experiment, and I said the equipment was delivering water at a rate of 40 litres per second. I wasn't working with a fire hose!

[ngc3314]

I don't know if I'm confusing fact with fiction here, but I recently heard something about a student who used Eratosthenes' method of measuring the diameter of the Earth... and got an answer of 1.2 metres! If true, it sounds as if the student had the necessary maths skills, but not the commonsense to check for obvious errors.


Ditto. I have classes work out the lunar distance using parallax, starting from the location of the lunar limb during a solar eclipse at various times done from our location, and the knowledge that the center of the umbra was at a certain place when we had greatest eclipse. One of the students didn't think anything was odd on finding that Moon was closer than Seattle. I mean, that's what the calculator said, so it must be right...

One aspect which I think operates here is also at work in Another Controversy Which I Will Not Name. For lots of students, they've almost never seen any math but trivial arithmetic impact their daily lives, so these sots of calculations have always been school assignments of the jump-through-hoops-for-a-grade variety. This suggests a certain lack of practice in estimation of physically reasonable outcomes. I start many of these same classes with Fermi's question about the likely number of piano tuners in New York City, to set the tone for estimating.

[Larry Jacks]

And even if it's not true, it's sufficiently reminiscent of the Mars probe where they confused imperial and metric measurements.

This one comes up a lot but I suspect most people don't really know what happened. JPL was flying the Mars Climate Observer (MCO) that was built by Lockheed-Martin. LM was providing JPL with a lot of essential information. One piece of information dealt with the delta-v imparted by thruster firings during momentum wheel unloads. It was this delta-v information that was incorrectly reported to JPL in English units instead of metric. The actual delta-v imparted by these unloads was quite small and the error was on the order of 1 part per million or less (the flight path from Earth to Mars was something like 400 million miles and the accumulated error was approximately 40 miles*. It wasn't as if they got all of the measurements wrong.

*Just goes to show once again that Murphy's Law or something like it works in space. The 40 mile error was in the worst possible direction - the MCO hit the martian atmosphere 40 miles too low and burned up. Had the error been in the other direction (or any other direction) then the vehicle might've ended up in a slightly incorrect orbit but likely would've survived. Unfortunately, JPL didn't catch the navigation error in time to correct it, which is very unlike them.

[Tobin Dax]

One aspect which I think operates here is also at work in Another Controversy Which I Will Not Name. For lots of students, they've almost never seen any math but trivial arithmetic impact their daily lives, so these sots of calculations have always been school assignments of the jump-through-hoops-for-a-grade variety. This suggests a certain lack of practice in estimation of physically reasonable outcomes. I start many of these same classes with Fermi's question about the likely number of piano tuners in New York City, to set the tone for estimating.
Heck, I have to wonder how many people use any math in their everyday lives. To help students solve how far the moon moves in an hour, I give the every-day example of freeway travel. You're traveling at X speed, how far do you go in Y time, etc. I post this for my online classes, and they still can't figure it out. Honestly, that's one of my biggest hurdles as I start teaching these classes online. I can only do so much when I'm on the other side of the state. (Then there are the math tutors who can't do science problems, but that's a different issue.)

[Paul Beardsley]

Heck, I have to wonder how many people use any math in their everyday lives. To help students solve how far the moon moves in an hour, I give the every-day example of freeway travel. You're traveling at X speed, how far do you go in Y time, etc. I post this for my online classes, and they still can't figure it out. Honestly, that's one of my biggest hurdles as I start teaching these classes online. I can only do so much when I'm on the other side of the state. (Then there are the math tutors who can't do science problems, but that's a different issue.)

Now this is very interesting! I am getting the idea that learning maths depends on getting feedback almost immediately, probably at several stages - which cannot be done (at least not as easily) with distance learning. I also suspect that tone of voice, body language and other visual clues are vitally important - and, again, generally absent when you're a hundred miles away from your students.

A lot of my own teaching is concerned with what Henrik described as "quick sums in your head while shopping". Basic Skills, recently renamed Skills for Life, soon to be renamed again to Functional Skills, mainly for adults who had a bad experience of learning maths. (These people are typically intelligent enough in other areas - and I cannot help but admire them for giving up their free time to come and listen to me waffling on about their least favourite subject.) Quite often they need a lot of guidance, patience and reassurance - some are genuinely scared of maths - and there is simply no way I could provide that online. Saying things like, "You're doing fine," "What do you need to do before that?" or "How can we simplify that?" are okay in conversation but don't work so well on the screen.

At higher level, I imagine the reassurance isn't so important - the fact that the students got to the higher level suggests they have a reasonable understanding of the subject - but teaching it online does sound a lot harder.

Incidentally, I recently met a woman called Jo who teaches geology online. Quite a few of her students are from Other Cultures, and do not approve of women teaching; but they also don't realise that Jo is a woman's name, so she doesn't bother to tell them.

[Disinfo Agent]

If teaching mental arithmetic is your goal, you are probably right that close contact is important. (In my opinion some close contact is always important. It's too easy to get lost, be discouraged, and give up, when you're learning math.)

There are probably good (old) books on the topic of mental calculation and estimation, though I wouldn't be able to recommend one.

Games that involve estimating probabilities, like poker, might be interesting.

[Tobin Dax]

At higher level, I imagine the reassurance isn't so important - the fact that the students got to the higher level suggests they have a reasonable understanding of the subject - but teaching it online does sound a lot harder.
(Maybe I'm a little cynical in this response. Oh, well.)
It depends on what you mean by reasonable. In my opinion, you're average college freshman doesn't have a reasonable understanding of math. If you're lucky, they can do arithmetic without a calculator, or they can solve math problem that they're given. But they can't apply what they know. Science is word problems which contain both implicit and explicit information. You have to be able to figure out all the numbers you need to solve that problem. Then you have to find the right equation and put the numbers in the right places. Most students don't understand an equation in pure symbolic form, and they don't know where to put the numbers into the equations. And it's like putting together DIY furniture: they have a symbol left over, and they don't know what to do with it. (This is the implicit value that they should know or find on their own.) And then they hate science because of these math problems they couldn't do.
This applies to both in-classroom and online students.

Maybe I'm only thinking of the below-average students. Maybe this isn't covered well in high school algebra (but it's a pre-requisite, so it should be). Maybe their brains just aren't wired that way. Maybe I'm biased because I've been up to my eyeballs in this stuff for darn near a decade. But simple algebra problems are done in everyday life (as the word problems I said above, even), and I don't see how anyone can function without knowing how long a trip will take or what their gas mileage is, among other examples.

Paul, I apologize if I'm pulling this thread off-course here, but I think it has some application to your topic.

[Paul Beardsley]

Paul, I apologize if I'm pulling this thread off-course here, but I think it has some application to your topic.
No need to apologise, Tobin, this is good insight.

In fact I'm finding all the posts interesting.

[Jeff Root]

One of the students didn't think anything was odd on finding
that Moon was closer than Seattle. I mean, that's what the
calculator said, so it must be right...
A couple of weeks ago I got an e-mail asking:


How many miles is it from earth to the moon?
I'm asking you because my Uncle says the distance is shorter than
the distance from San Francisco to Los Angles. So can you please
e-mail me back with the correct answer?
(My web page on the Moon gives the distance in kilometres only,
and that in a table that someone reading the text might skip over.)


I start many of these same classes with Fermi's question about the likely
number of piano tuners in New York City, to set the tone for estimating.
Do you have a "real" number to use as a check? Or is the "real" number
irrelevant? I would think the number would be impossible to know unless
(as I suspect) piano tuners have to be licensed in NYC, and you can
look up how many licenses are current.

-- Jeff, in Minneapolis

[Jeff Root]

Incidentally, I recently met a woman called Jo who teaches geology online.
Quite a few of her students are from Other Cultures, and do not approve
of women teaching; but they also don't realise that Jo is a woman's name,
so she doesn't bother to tell them.
Ow. My guess is that that prejudice only applies above primary school
level, and is even reversed for primary teachers. Ow.

My personal math history: I did fine in 10th-grade geometry. On one
homework problem the teacher said I was the only person in his three
classes (possibly 90 kids) who got the right answer. It was a very
pretty 3-D construction, but nothing I'd consider difficult. I'd have
done better in the class if I had copied all the propositions and
theorems from the textbook into my notebook as I was supposed to
do, so that I could refer to them during tests, but my perfectionism
made it too much work, so I fell behind. I did badly in 11th-grade
intermediate algebra (I hadn't the slightest inkling what to do with
a polynomial), and failed advanced math (pre-calculus, I guess it was)
in the first semester. First day, really. If we had started on page 1
of the textbook I might have had a fighting chance. We started on
page 12. Polynomials. The teacher did nothing but transcribe
the book to the blackboard. I don't know where the other kids in that
class came from. They were like they had been imported from some
planet where everyone learns everything the first time through and
nobody ever asks any questions, and they were all going to Harvard
and Yale and across the ocean to Oxford the next year.

I later took intermediate algebra again twice more, but was never
able to grasp what to do with a polynomial. To replace the advanced
math class I failed out of, I took a one-semester speech class.
Eighteen kids, the teacher was about 75 years old with a permanent
smile like Ed Wynn's, and we each gave 20 speeches during the
semester! Louis Claeson gave almost all of mine A+, except for a
few where I was really unprepared, and got an A.

Nobody told me that I needed the credit from the speech class to
graduate. The summer before I took classes at the Children's Theater,
and found out later that I got credit for them, and that if I hadn't I
would have had to take an additional class. Everyone needs guidence.
I barely got any, and for me it really wasn't enough.

-- Jeff, in Minneapolis

[Van Rijn]

Do you have a "real" number to use as a check? Or is the "real" number
irrelevant? I would think the number would be impossible to know unless
(as I suspect) piano tuners have to be licensed in NYC, and you can
look up how many licenses are current.

-- Jeff, in Minneapolis

Well, you know that the number can't be greater than the population, and only a fraction of the populuation has pianos, and you don't need a tuner for each piano. As a first cut, I'd say perhaps 1 in 10,000 would be a piano tuner. So, say under a thousand. I don't know much about pianos, but I can think of things to check to refine that. Then again, I could look in the phone book.

[Jeff Root]

... and you don't need a tuner for each piano.
Agreed. One for every 1.6 pianos?


I could look in the phone book.
I was wondering how many are not listed in the yellow pages.

-- Jeff, apparently on an off-topic binge in Minneapolis

[HenrikOlsen]

If you need a piano tuned on average once a year, and it takes half a day to tune a piano, it's one tuner per about 700-800 pianoes.

[Paul Beardsley]

I am nearly ready to write my essay on this topic. There might be occasions when I would like to quote some of the comments made here.

So... Would anyone have any objections to being quoted? Would anyone particularly like to be quoted? As I've already said, there is some very good material here.

[Tobin Dax]

I have no problem with you quoting me, Paul, if I'm on your list. IIRC, you may even have my real name somewhere if you want to use that.

[HenrikOlsen]

Feel free to quote me.

[Paul Beardsley]

Thank you, Tobin and Henrik.

Even if I don't actually quote you, I shall certainly give consideration to what you (and the others) have said.

[Theunknownbook]

Demigrog - your on-line course sounds less than ideal. I'm surprised it wasn't set up to give you instant feedback and help.

True, I've seen some programs for submitting homework that actually provide instant feedback or help you along with clues as you progress through the mathematical problem.