Technology and Mathematics: Is There a Downside?Author(s): Chris LittleSource: Mathematics in School, Vol. 24, No. 4 (Sep., 1995), pp. 36-37Published by: The Mathematical AssociationStable URL: http://www.jstor.org/stable/30215201 .

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lOwnWSI by Chris Little, St Vincent Sixth Form College, Gosport

Is it not time we had a serious, balanced debate about using technology in mathematics education? The zealots are usually engaged in teacher education (you need more time to think than "real" teaching allows to be a zealot) and can be identified by incipient symptoms of RSI, and a born-again faith that revolution is about to happen. Backwoodspersons think that arcs are biblical, macs come from north of the border, and maths is best taught using blackboards and pencil and paper, accepting the use of calculators only under sufferance.

Most teachers of mathematics drift with the tide. Occasionally they are persuaded by the zealots to invest in computers, which look good in the school brochure, but which are then rarely used in maths lessons, and therefore become a source of nagging guilt easily exploited by advisors, inspectors, etc. Usually they are locked into exam syllabuses which test maths in the time-honoured way, and certainly do not require much computing power.

Where do I fit in here? As a paid up zealot, I own a Mac, love and use Cabri, use a graphics calculator a lot (I teach in a sixth form college), I have put my departmental records on a spreadsheet, and I word-process most of my supplementary teaching material. In the back-woods, I find Derive and symbol manipulation software bring out the traditionalist in me (why can't "they" do algebra any more, etc), and I despair for 16-19 year old students who do not know what a percentage is, need a calculator to work out 10 x i2.40, and do not know their multiplication tables. (Zealots will say this is not the fault of technology - I will return to this later.)

I am completely convinced that technology can and has benefited mathematics learning, but I am equally sure that there is a down-side to technology, which we must accept and be aware of.

These thoughts were recently prompted by an INSET session where we looked at MathsCad. This software enables you to write mathematics, such as a system of simultaneous equations:

x+2y=8 3x-4y= 9

but then saves you the trouble of solving them by supplying the values of x and y on request:

x:=5 y:=1.5

You can then edit the original equations and the solutions are updated accordingly, as with a spreadsheet.

In the discussion which followed the workshop, I surprised even myself by the vehemence of my defence of traditional algebra. If Jekyll is the zealot, and Hyde the backwoodsperson, Hyde was winning hands down. Had he been there, John Patten would have been proud of me, and signed me up for "Back to basics" on the spot! I began to worry - is this the first sign of a male menopause?

The point at issue is whether the existence of systems which not only write the mathamatics but do the calcu- lational "legwork" undermines, or at least weakens, the necessity for pupils to go through the algebraic treadmill themselves. Why bother to teach algebraic manipulation at all if the computer or calculator will do it for you?

Many mathematics teachers will find this argument seductive. All agree that algebra is hard to teach and to learn. Students struggle to make sense of formal algebraic language, which presents a formidible obstacle to progress in mathematics. If we can now bypass this obstacle using technology, would it not make algebra more accessible and enjoyable?

36 Mathematics in School, September 1995

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This is probably where some of the zealots and I part company: I refuse to accept that there is no penalty to pay here. Returning to the case of simultaneous equations, it is quite true that the computer encourages the "modelling" aspects - setting up the equations, defining the variables. It also adds immeasurably to the problem-solving power at our disposal, by enabling any pair (or system) of simul- taneous equations to be solved instantaneously. But by abdicating the process of analytical solution to the com- puter, the student's understanding of the underlying mathematics must be weakened.

Here, there is a crucial distinction to be made between delegation and abdication. I am perfectly happy to delegate the task of solving these equations to the computer, confident in the knowledge that I could do it myself, given the time and the computational staying power. But abdication: "you don't need to know how to do this, just leave it to the computer", is another matter. Delegating tasks which are routine can enhance our freedom to engage in higher-level activities; but abdication to machines is de-skilling and limiting. Compare the market stall with the supermarket checkout. I could give further examples from many other areas of mathematics, and many different levels. For instance:

Fractions The zealots - "These are hard going to many stud-

ents. Why bother to teach cancelling fractions if a calculator can do it for you? Indeed, why bother to teach fractions at all if we use calculators with decimals? The utility of fractions in measurement, money, etc is weak- ened by decimal currency, metric units. We now rarely need to calculate with fractions."

Me - "On the other hand, the concept of a fraction (1 means a -b) is fundamental to the develop- ment of mathematics. Mental manipulation of fractions is essential to our understanding of what a fraction is, how fractions behave. By all means delegate, but to abdicate responsi- bility for the fraction to the calculator or computer must weaken the basis of this understanding. "

Statistics The zealots - "Why bother to teach the formula for

standard deviation, for example, if pressing one calculator key will do it for us? Why not just establish the properties of standard deviation in a qualitative fashion, and leave the for- mula out. They always get it wrong anyway! It's the same with all that coding of data we used to do ... all quite unnecessary now!

Me - The formula for standard deviation is an algorithmic expression of what standard devi- ation actually means. Unless students under- stand the formula, they do not understand standard deviation. I am happy to delegate the calculation once students have used it, but not to allow the calculator to short-circuit this step. As for coding, as a technique in itself, its utility is admittedly reduced by the calculator. But how do students understand the math- ematical properties of mean, variance, etc without experiencing the effect of coding on the basic calculations?

Calculus The zealots "Why bother to teach integration by

parts, substitution, etc. Students rarely understand these processes, but blindly copy down the lines of the argument. Derive can do all your integrals for you, and you can concen- trate on serious modelling."

Me - I accept the need to review the relative emphasis we place on analytical methods of integration in a general A level mathematics course. But you must acknowledge that there is a loss involved here. I agree that students seldom understand the logical argument behind integration by substitution (I am not sure if I do, fully), but, certainly for those who wish to progress with their mathematics, they need to be introduced to this logical structure, the patterns and the symbols, in order that their understanding can deepen later.

The issue extends beyond mathematics education. For example, why bother to teach students to spell, now computers have spell checkers? At the level of utilitarian communication, one could argue that correct spelling is unimportant (as long as we know what she means, who cares if she spells cat "kat"?). At a higher level, surely accurate spelling manifests a feel for language, a love of words which is essential for good writing?

I do not know of research evidence to prove this, but it would be surprising if skills which are made partially redundant by technologies did not atrophe. Is it not likely that spell checkers erode spelling skills, and calculators erode pencil-and-paper calculation skills? Technology may have made the lack of such skills less crippling, but dependancy on machines is also disabling.

This not the same as saying that calculators cannot be used as a highly effective method of teaching basic number work, or spell-checkers cannot help to teach spelling, but simply that mental calculation and spelling improves with practice. Neither does it imply a "back to basics" approach where we lock all these dangerous calculators and com- puters away and force students to practice hundreds of column additions, or hundreds of spellings for that matter. In embracing new technologies, however, we need to acknowledge the potential for de-skilling which they carry, and place proper emphasis on using our own brains rather than someone else's. We need a balance.

To become confident, effective mathematicians, students must master some "basics" of mental calculation, of reasoning, for themselves. Pencil and paper can serve a purpose here. So can computers. For example, constructing a square in Cabri forces the student to think about what a square is. It is interesting that using the computer here makes the task not easier, but harder, for the student, although the pay-off is a much more sophisticated and better-adapted operational understanding of what square- ness means.

By all means delegate some tasks to the computer and the calculator (thank goodness we've buried bar numbers for logarithms!), but wholesale abdication produces an unhealthy dependency. It may make teaching easier, but it will not ultimately serve our students. Before dismissing from our classrooms analytical algebraic methods, or arithmetic algorithms, there is obligation to think through clearly what the effect on mathematical understanding will be of removing these weapons from the student's mental mathematical armoury. Unless we understand and acknowl- edge the de-skilling potential of technology, there is a danger that it will undermine conceptual understanding rather than enhance it. B

Mathematics in School, September 1995 37

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