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.
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).
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.
It is tragic that most students never experience mathematics as play.
School maths lets students frisk around the edges of problem solving, but keeps them firmly rooted in the rote acquisition of disjointed knowledge.
Students are constantly fed mistruths about mathematics; they are instructed to learn facts and procedures that are apparently important for solving problems ‘later in life’. Well, when? Where is that dividing line between acquiring knowledge and applying it to solve problems? There isn’t one; they go hand-in-hand. The research is clear on the interplay between mathematical knowledge, skills and understanding.
Beyond the basics of whole number addition, what factual or procedural knowledge did Gauss need to produce his solution? His solution was ingenious because it wove together procedure with understanding. It derived the maximum amount output from minimum input, and delivered the generalised formula as a by-product.
Working within constraints is the art and science of mathematical reasoning. When problem solving is squeezed into the margins, mathematics loses its chief virtues.
We should apply caution when framing pedagogy around cases of individual genius. We cannot expect students to reproduce the magic of Gauss. If we believe the popular telling of Gauss’s story, we can rebuke his teacher for abdicating his responsibilities by leaving his students to it. While Gauss prospered from unaided problem solving, his peers no doubt flailed. Problem solving must be a carefully guided activity, inclusive to all.
The ‘teacher as facilitator’ role is often looked upon with contempt, supposedly disarming teachers of their authority in the classroom. But what could be more liberating — and more challenging — than nudging students towards genius?
The delicate craft of maths instruction is in offering students the right amount of information. Not so little that they will feel helpless, but not so much that they are deprived of the opportunity to discover mathematical truths for themselves.
The best teachers will adapt to students’ cues, steer the discussion of new ideas and approaches, and spread problems in all directions. The summation problem alone can take on various extended forms — what is the most elegant way of summing all the even numbers from 1 to 100? Define elegance!
Gauss stunned his teacher because the folding method was novel for its time, unknown to most. Educators today will be just as stunned by what their students can do, if only they give them the space to explore and solve problems — to be mathematicians.
