Mathematics of need

Anand Krishnaswamy
Jul 23, 2017 · 7 min read

I often hear Maths teachers exclaim that a lot of mathematics (arithmetic included) follows the pattern of “theory first, application later” by which they mean that the student has probably not encountered the circumstances to apply the topic (e.g. surface area of a cylinder or quadratic equations) but must still learn the theory because s/he is likely to encounter it in the future. And if they are in a rather rattled mood, they will throw at you the 3 C’s of a deluded mathematics curriculum (calculus, complex numbers & conics) to make their case “stronger”. I agree that the 3 C’s are an extreme case of topics which 99% of the population will never encounter as a vital issue & to expose children to that rare possibility while, say, excluding estimation & back-of-envelope calculations in their curriculum is an indication of a clueless club of curriculum (3 C’s back atcha!) designers.

Recently, a class of 7th graders were in the computer lab hacking their way through Scratch. While discussing the characteristics of motion, I realised that they looked more puzzled than I was prepared to admit as normal. I had segued into describing objects in 2D space & was referring to X & Y coordinates only to face rather blank looks.

“Have you studied the Cartesian system?”

Everyone shakes their head.

“Or as the Cartesian coordinate system? X-Y coordinate system?”

Everyone shakes their head.

Swiftly on my shoulders alight the “Teach them quickly”-me & “Hey! Here’s an opportunity for learning”-me ready to argue about how to deal with this situation. The former usually loses & did again.

I moved to the lone whiteboard, wiped it clean & marked a dot on one corner.

“If you had to go home & tell your mom where I placed the mark on the white board, what would you say?”

“Corner!” (in chorus)

“There are 4”

“Right corner”

“There are 2”

“Top right corner”

“And this one?” - I place another mark.

“Bottom right corner”

“Good! Here is another” — I place a mark near the 1st one.

“Middle”, “No, top right… corner?”, “No, centre!”

“It clearly isn’t the centre. How would you describe this point?”

1 child says “Top portion”

“Fair enough. How do you recognise top & bottom of this board?”

“Half above is top & below is bottom”

I draw a line dividing the board vertically.

“Does this feel right?”

“Yes, sir.”

“Ok, what about this point?” — I place another mark to the left of the most recent one.

“Top portion”

“How do I then differentiate between these 2 points to someone?”

“Top left side portion”

“Ok” — I draw a vertical line — “Does this split the board into right & left?”

“Yes, sir.”

“What about this point?” — I add one more in the 2nd quadrant.

This gets them thinking & one kid says “We can draw more lines.” I begin adding vertical horizontal lines till we have a partial grid.

“How will you describe the points now?”

“We can number the horizontal & vertical lines & then describe the point depending on its proximity to a line.”

I add letters of the alphabet to the vertical lines & numbers to the horizontal lines & soon we have a chess board system of notation (called algebraic notation). The children look happy.

“So that mark is…?”

“E6”

“Not top left?”

“No!”

“But is E6, English?”

“No!”

“Will your mother understand, then?”

They start discussing their thoughts & ideas of how they will educate their moms. I didn’t want to hear what their moms would think of that idea! If I am found hanging from a cross, you know who did it to me!

“So, we have moved from using simple English in describing a mark’s location on the white board to creating a new language. Why did we need that?”

“Because 2 points can both be top left portion”

“And hence, English is insufficient in describing accurately the location of a point as the board gets crowded. Maybe that is why English isn’t our native tongue in India.”

Kids burst out laughing! They need to work those muscles, too.

“But can you do it in Hindi?”

After some thought, “No!”

“But you can still keep Hindi as your native tongue! Every language has its power & limitation. What we have here is a language for describing points. It is powerful but cannot be used to describe why you forgot to do your homework!”

Laughter again!

“You could use the new language to tell someone where you are itching on your back!”

“Now, this board is messy. Too many lines. And as I draw more points, you will need finer lines & soon this whole board will simply be lines. I want a better scheme. Help me!”

Many variations of the Algebraic Notation follow. It seems natural for us to mill around a winning idea & convince ourselves that the minor variation is a new idea. More on that later.

“Let’s say I wipe out all the lines except the central horizontal line. How can you use it to describe a point on this board?”

After some thought a slightly reserved boy says “Maybe we can say how far it is from the line?”

“Good! Build on it.”

Others jump in. “We can say how much above or below it is from the line.”

“What about these 2 points?” — And I draw 2 points equidistant from the line & above it.

The reserved boy again responds with “Maybe we can have one more line… or maybe that won’t work… Sorry!”

“Why do you say it won’t work?”

But he was drowned in the enthusiasm of the others. They decided to add a vertical line through the centre.

“One point is on the right of this line & the other is on the left.”

“Good, but what about this one?” — And I mark a point at the same level & side as one of the earlier points.

I shall stop recreating the dialogue & move on to say that gradually & systematically, they constructed the mechanism of measuring from one line & another to the point & describing it as a tuple of distances. While they were referring to the points as “top”, “right” & “left” still, I introduced a convention which they were ok to adopt — “Let’s call all left as minus from the vertical line & all bottom as minus from the horizontal line”. Thus, they were able to describe any point on the board.

I was overjoyed to see that these 7th graders were able to create the Cartesian coordinate system from scratch well in advance of the “prescribed age” when it is supposed to be introduced. I am not sure they realised the depth of their feat. They probably were amused at seeing me so happy. Later when I read about how Descartes arrived at the coordinate system (or as legend has it), it only made me reflect on how that tale was missed out in my education.

I quickly introduced them to conventions & units of reference. It is merely a convention to consider points on the left as -ve. We could well adopt all points on the right as -ve. Simply because many people are right-handed, left doesn’t imply -ve & it is merely a convention. We could adopt the opposite as long as we remain consistent throughout our measurement. Similarly, we could measure the distance in inches, cm or mm. Based on that, the coordinates will vary. We could also measure it in thumb lengths (although it suffers non-standardisation) & the point can be described in thumb lengths (which is what you see in the picture above) as long as we keep the units consistent throughout.

I shall wrap this account of my class with 2 other events. Firstly, while this was a class in programming in Scratch & learning about algorithmic thinking, not one child complained about switching to mathematics.

I hold that as long as an activity of learning stems from their need, it is not required to find ways to make the topic (artificially) interesting or invoke their discipline (as Dewey says in his book).

When some students declared that they didn’t understand, the ones who did, got an opportunity to explain it to them. They were, thus, compelled to polish their vocabulary while elucidating (another need based learning). Throughout this, not one child moaned about how boring maths is & how they’d rather be coding in Scratch.

Secondly, one boy didn’t stop at the “winning” idea (of the Cartesian system) but proposed another one (and I am unaware of whether this system has any official name. If it doesn’t then it should be called Ramzan coordinate system, after the boy). He suggested we measure the distance from the intersection of the 2 axes & use a D to measure the angle. I asked him what a D was. He gesticulated with his hands.

“Oh! Protractor!”

“Yes, sir. Then every point can be a tuple of distance from origin & angle made.”

I loved the idea & nearly hugged him. It would definitely work. What I loved in his idea was his willingness to explore unconventional or unpopular lines of thinking. I hope that encouraged his friends to try the same in other excursions of learning.

In summary, I think there are multiple ways to create a learning experience & one that arises from need feels more organic & has greater sticking power.

Anand Krishnaswamy

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Focused on community driven creative education & eco-consciousness. Integrated curious teacher, computer scientist, photographer, traveler, cook, writer

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