“Ontogeny recapitulates phylogeny” in math education

The phrase “ontogeny recapitulates phylogeny” is one of those phrases, like $E = m c^2$, that many people have encountered in school, fewer remember, and fewer still have a clear idea of what it means.

The phrase “ontogeny recapitulates phylogeny” comes from biology, evolutionary biology, though it precedes Darwin by decades.  What it intents to convey is that there is a rough parallel between the way an individual embryo develops and the way the species evolved.  A human embryo, at some early point, has a tail, which is later absorbed.  The parallel is very rough and not to be taken literally.

A similar parallel was noted by Herbert Spencer about education:

If there be an order in which the human race has mastered its various kinds of knowledge, there will arise in every child an aptitude to acquire these kinds of knowledge in the same order…. Education is a repetition of civilization in little.

– Herbert Spencer, Education (1861)

The idea here is that children learn things in the educational system roughly in the order in which the civilization learned it.  Clearly, not to be taken literally.

But it could be a useful perspective to look at the history of mathematics, the history of mathematical ideas, and see if it sheds any light on how we now view the job of  teaching kids mathematics.

My interest and research in this topic started when learning that the concept of zero took such a very long time to develop into its current form, which at least suggests that there is something inherently difficult and tricky and counter-intuitive about the idea of a something that stands for nothing.  And yet, the kids I encountered in various math classes seemed to take the idea of zero pretty much in stride.  Sure, there were plenty of confusions between the digit zero and the number zero – but the number zero seemed not to be a particularly serious roadblock for kids.

In contrast, the idea of negative numbers is not an easy one for most kids: what does it mean for a number to be less than zero, how come subtracting a negative number is the same as adding its positive counterpart, and how come multiplying two negative numbers gives rise to a positive number – this gives kids plenty of challenge.  If all this material is introduced in seventh grade, you find plenty of kids who still trip over this in ninth grade, suggesting that they often passed their tests in seventh grade by memorizing the rules long enough to last through the test.  Often, right after they’ve learned that multiplying two negative numbers gives a positive number, you can easily confuse a student by ‘suddenly’ asking them to add two negative numbers.

In the history of mathematics, it took many centuries for negative numbers to be seen as full-fledged numbers.  Famous mathematicians – mathematicians that school children have heard of – would warn against negative numbers.  Descartes was one of them, in his book Géometrie from 1637, he called negative roots of equations “false”.   John Wallis, also a famous mathematician though not generally known by school children, argued in a book of 1685 that “it is also Impossible, that any Quantity… can be Negative.  Since that it is not possible that any Magnitude can be Less Than Nothing, or any Number Fewer than None.”  Wallis, in fact, came up with an ingenious theory that negative numbers weren’t less than nothing, they were greater than infinity!  He arrived at that remarkable conclusion by noting that 100 divided by 5 is less than 100 divided by 4, and that is less than 100 divided by 3, less than 100 divided by 2, less than 100 divided by 1, so if you trust that pattern to continue, you would get that 100 divided by 0 be bigger still (infinity), and 100 divided by -1 bigger yet.
In the nineteenth century, a formidable mathematician like Hamilton would contrast the certainty he saw in the foundations of geometry (two thousand years old, systematized by Euclid) with the uncertainty in the foundations of algebra.  This is from a paper he published in 1837: “But it requires no peculiar skepticism to doubt, or even to disbelieve, the doctrine of Negatives, when set forth (as it has commonly been) with principles like these: that a greater magnitude may be subtracted from a less, and that the remainder is less than nothing; that two negative numbers, or numbers denoting magnitudes each less than nothing, may be multiplied the one by the other, and that the product will be a positive number, or a number denoting a magnitude greater than nothing. […] It must be hard to found a SCIENCE on such grounds as these…”

Not a bad description of the difficulty kids today still have with negative numbers.

note: the remarks about negative numbers in the history of mathematics are adapted from the book Negative Math by Alberto Martínez.