A boy wonder from the 1780s shows us where school maths gets it wrong

The genius of Gauss and the lessons for educators

In the 1780s, a group of eight-year-olds were asked by their teacher to sum the integers from 1 to 100. The teacher needed a way of occupying his students for half an hour as he got on with other tasks. On the surface, it was the perfect problem: the students would likely add the integers in turn (1+2=3, 1+2+3=6, 1+2+3+4=10 and so on), requiring no fewer than 99 precise calculations. That would keep them busy for a good while. Yet within moments, one student piped up to declare his answer: 5050. He was correct.

His name was Carl Friedrich Gauss and he would go on to become one of the most celebrated mathematicians of all time. Here was an early sign of his genius (there were earlier signs still; he was rumoured to have taken on his father’s accounts at the tender age of five).

The brilliance of Gauss’s solution is that it bypasses the need for brute force computation. Instead, he had the insight to ‘fold’ the integers provided. Rather than adding the integers in sequence, Gauss paired them together to obtain fifty copies of 101. That gives a final total of 50*101, or 5050. Gauss probed the structures of numbers and arithmetic to reduce the intended scope of the problem to a few simple calculations.

Gauss’s folding method — not bad going for an eight-year-old

Most students today encounter the summation problem in high school. It is presented in the context of arithmetic progressions; strings of numbers that differ by a fixed amount. You may recall the formula for summing such strings — it goes something like this (for completeness, a is the first term, n the number of terms, d the common difference).

The dark side of arithmetic progressions

So the sum of the first 100 integers? Easy — just plug in a=1, n=100, d=1 and voila, you arrive at 5050 once more.

Does that satisfy you? Yeah, me neither. This syntactic bastardisation hardly makes the heart flutter. The formula drip feeds us a narrow learning objective: know how to calculate the sum of an arithmetic progression. There is scarce value in applying algorithms and formulas that are not well understood — let’s leave that to computers, shall we?

Rote learning robs us of our own Gaussian moment — of the freedom to play as he did, to plumb the depths of arithmetic and to dare to tear up the script by creating our own solution paths. We are deprived of richer learning outcomes: the heuristics of problem solving, deeper conceptual understanding, the confidence to debate ideas and take ownership of a problem and, most important of all, the joy of seeing mathematical beauty reveal itself. There is immense power in engaging with mathematics as child’s play.