We don’t need to spend much time with calculation. Technology can do that for us. We can use the extra time saved for engaging in ‘higher-level’ thinking. Mr. Conrad Wolfram in his TEd Education Talk makes a lot of sense in this video. But, I have my reservations but not because I’m for teaching a lot of calculations.

We also said the same during the era of calculator. Did it improve the math education of our youth? So, what makes us think it will happen in the age of Mathematica, GeoGebra, Sketchpad? There is something in our math classes or math education that’s stuck in the middle ages, that’s not keeping with time. Unless we find and address this, no amount of technology can help us.

I taught elementary grades – mostly 3rd and 4th grade – during the time when calculators were introduced into the classroom. I considered this to be a godsend. Until this time the standard textbooks spent 90% of the time on teaching algorithms; i.e., computation. Even ‘story problems’ were at most thinly disguised calculations; there was no time for a deeper understanding of number or mathematical thinking. Calculators allowed students to save time, and allow more for higher level processing.

In time I realized the importance of number sense and mental math, and shortchanging the ability to learn basic facts or understanding the relationships and patterns of numbers (e.g, multiplying by 4 is the same as multiplying by 2, then the product by 2 again.) So yes, students are ’empowered’ by learning the *basics* (good question – we assume we know what these are!) but why would anyone do long division with a 2 digit divisor if there was a calculator handy. Is it important to know how to do long division? Oh, once I thought it was important to be able to find a square root. Of course it is important to know when division is called for, but long division is rather quaint. On the other hand, we should absolutely be able to arrive at a close estimate, so a degree of number skills and analysis is quite important.

I wince when a teacher says what they teach is important because they’ll need to know it for the work in the next grade. Or that we need to know the properties of numbers because they are used in Algebra. This begs the question. Do we really need algebra? Do we need calculus? (ha – show me one person out of a thousand who does). We are too vested in protecting and overvaluing our turf at times. Much more important is to help students understand the world makes sense; ok, at least the mathematical universe does. Providing the concrete and conceptual foundation of why we do what we do, and why it works, is essential. Tedious or excessive manipulation of numbers following a rote algorithm is obsessive and often distorts the value of math.

I think Wolfram is right in the three reasons to learn math. From time to time it is valuable to step back and take a more overarching view of what we are doing, and why.

I believe that one MUST have a SOLID UNDERSTANDING of NUMBER and their relationships otherwise HOW can you possibly make any sense of what the calculator OR computer software is displaying after you push some buttons or keys?

Just memorizing button sequences on a calculator to get a specific result is not higher order thinking; it is an even lower level of rote memory.

I agree that calculator and computer software programs take the drudgery out of repetitive computations to get a particular result.

For example to study parabolic functions I find it helpful for students to graph the parent function manually so that they can get a ‘FEEL’ for what they are seeing develop. To keep on doing this for the complete study of this function is TEDIOUS, TIME-CONSUMING, and becomes BORING. However, then shifting to using a calculator to have students investigate What do I have to do to the equation of the parent function to make the parabola be upside down, move up, move down, etc. Of course extend this to a complete study of this function. Same goes for other functions.

Very well said, Wally. Thank your for sharing your thoughts. To get a ‘feel’ of the function, one must have indeed have an experience of doing its graph manually first.

What Wolfram is doing over here, like a good salesperson, is promoting his product. What he says is not totally wrong, but its neither totally correct.

Computers can be helpful to make maths interesting. Thus it has a supplementary role. What Wolfram is talking is to entirely replace current curriculum.

Students ought to know the BASICS right. At least the basics. There is no doubt that computers can speed up the calculation, but it would increase our dependence on the machines.

I once had a small debate with my students as to why calculators are not allowed to school children. They were of the opinion that since everyone in the real world use calculator, calculators should be allowed to be used in the exams. And once calculators are allowed, the sum is solved. I gave them my calculator and gave them a sum. What should be the list price (printed price/marked price) of a pair of shoes if you wish to sell them at Rs. 2250 (Indian currency) after allowing a 10 percent discount. They used the calculators can came up with Rs. 2475. The actual answer of course is Rs. 2500. For if the had a list price of Rs. 2475 the discount would be Rs. 247.50 and the selling price would be Rs. 2227.50 (a loss of Rs. 22.50).

Calculator did nothing wrong. With wrong buttons pressed, you can only expect wrong answers. So basic concepts are of very high importance. Or should we be totally be dependent on apps. No apps no answer. Or should we sit and write a program of every single problem we face.

Wolfram conveniently forgets that to make a math app, one requires sound knowledge of maths. And that he too was trained in maths traditionally.

Knowing the basics is like to know what can be done in case of our car breaks down in the middle of the road. So along with driving a working knowledge of engines and other parts is a must. Ya, we need not be a mechanical engineer to drive a car, but working knowledge of the car machinery is a MUST. Even our computers break down every now or then, we do not rush to a computer engineering for petty matters.

So I conclude that sound knowledge of Math basics is always advantageous than not knowing about it at all. And more over having knowledge of Math never stops us from using the ready-made apps.

Couldn’t agree with you more. Thank you.