One way of understanding the subject disciplines of an academic curriculum is to see them as representing different ways of thinking; different and powerful ways. In this schema, mathematics represents deductive logic. It is iron-clad. If x=3 then 2x=6. No question. Mathematics has none of the fuzziness of the inductive logic characteristic of science and, to a certain extent, history. And it has none of the raging ambiguity afflicting the interpretist arts such as English, where you can just make up words if you want to and where I am informed by domain experts that although the word ‘literally’ literally means ‘literally’, we should be relaxed about people using it to mean ‘really’ because those two contradictory meanings can exist in some kind of lexical superposition.
Mathematics is a refuge of certainty and I am sure that this is part of the appeal for many who fall in love with the subject.
No matter, there is no object in the world so beautiful that someone out there won’t want to deface it and there are folks who are set on defacing mathematics.
Apparently, there is a move afoot to ‘humanise’ mathematics. You may wonder what is so inhuman about deductive logic, but the term seems to originate in the narrower sense of the humanities disciplines. In other words, we are talking about a movement seeking to make maths more like history.
This clearly misses the point that mathematics is so powerful precisely because it represents a different way of thinking than other disciplines.
Writing on his blog, maths education pundit Dan Meyer proclaims that:
“Math is only objective, inarguable, and abstract for questions defined so narrowly they’re almost useless to students, teachers, and the world itself.”
Instead, we need to focus on questions such as how many bricks there are in a pile of bricks. These remind me of those occasional fundraisers where you are asked to guess how many jellybeans are in a jar; a mild distraction that would rapidly become tedious as a curriculum.
According to Meyer, we should:
“Ask students to make claims that demand to be argued and interpreted rather than evaluated by an authority for correctness… If our students leave our classes this year without understanding that they have had made unique and original contributions to how humans think mathematically, we have defined “mathematics” too narrowly.”
No. We already have humanities subjects. We don’t need to dress mathematics up as something that it is not. It serves no purpose and it misunderstands precisely what it is about mathematics that makes it valuable. Moreover, mathematics won’t work very well as a humanity, kids will see through it and check out. This is precisely what happened in the U.S. when history was redefined according to Deweyan doctrine to become an ugly form of social studies that everyone dislikes.
Sometimes people approach me via social media and plea for nuance. They seek to build bridges. They ask why I cannot make my peace with the proponents of fuzzy maths. What’s wrong with a few rich tasks here and there? I cannot see a clearer and more definitive divide than between those who want to respect the discipline of mathematics and pass this on to future generations, and those who want to pulp it into something else that involves endless windbaggery.
There is no room for compromise here.
32 thoughts on “The beauty of maths is that it’s right or wrong”
There’s a certain paradox here–for all that maths derives its power from being purely abstract, civilisation could never have emerged without it.
I often wonder if the appeal of progressive education is to people who just don’t like the subject they are meant to be teaching.
I remember a conversation back when I was training to be a teacher. We had been grouped by subject so I was sat at a table with a number of Maths trainees. We were discussing teaching I think expanding brackets and I remember one of the trainees saying something along the lines of “put a problem up on the board which they will need to expand brackets for and get them to discuss in pairs how they would do it” whilst another suggested some weird treasure hunt/carousel. I said that’s all well and good but you haven’t taught them to expand brackets. They asked me how I would do it, and I said that I’d explain what the brackets mean, how to do it, do a couple of examples, get them to do some on mini-whiteboards, do some more examples with algebraic terms outside the brackets, then give them a worksheet to do. The response from one the trainees was “but that’s so boring, I’d have been so bored if I’d been in that lesson”. At the time I found it strange but bit my tongue.
Post script: I continue teaching expanding brackets (indeed most topics) in exactly that way. My students may be bored at times (although I try to minimise this by keeping the pace high and having some challenging questions later on – e.g. for the expanding brackets I would include some expanding and collecting, and maybe some double brackets (which I would make sure I taught properly later on)), but I am pretty certain that they can do what I want them to do. They also enjoy Maths. Go figure.
Or tl;dr: I think you have a point.
I instinctively started looking for the “Like” button after I read your comment. I think you’re on to something.
I love the two quotes you chose there. It really makes the point – guess how many bricks are in a pallet and make an original contribution to mathematics. I am not sure Dan saw the problem with asking for both of these but it seems while in art there is the impressionist and cubist movements in education we have the stupidist movement.
An attempt to put the ideas together in the dumbest combination and find people that will applaud you for standing out with original thinking.
It is worth asking what we would call what Dan is trying to teach with his final questions – how are you thinking about the number of bricks how are they thinking about it. It seems like some part of psychology or cognitive science. This is a disciple with its own name that has little overlap in the lives of actual mathematicians. That is most math majors don’t go on to become cognitive scientists. They may or may not take an interest in it but that is not what they would consider their discipline to be about.
It takes some massive level of hubris to ask those working in a disciple with such a rich history to give up their name to another one which except for Dan is not asking anyone to do this.
I have no problem with someone teaching some relevant cognitive science as part of any other subject. I’d expect people to figure out what is most useful and how that stacks up with the actual subject content.
Dan Meyer has a tag-line on his site: “less helpful”.
Sometimes I wonder if he considers the implications of that. He is actually asking us to be less helpful — but that cuts both ways.
He’s a clever man, but I think sometimes too clever. And this is one of them.
We are teachers. If we aren’t being helpful, we are in the wrong job.
I think your right but your argument dosn’t require maths to be perfectly objective it merely needs to be pretty close to that state (approaching the limit to steal a term). If you make your case on that pillar it only needs one example to topple it over and philosophers of maths and science have plenty of candidates.
I think you are wrong on two fronts. Firstly, I am not convinced there is much in mathematics that is not deductive logic. I may be wrong on this because my background is physics and not pure maths, but I find it hard to imagine non-deductive maths. I can certainly imagine there may be holes in our knowledge or paradoxes, but if we do resolve these then they will be via deductive logic. If we cannot resolve them then they are unknowns.
However, if we accept the premise that there are some rare examples of nondeductive mathematical reasoning, then does that refute my point? Only in a pedantic way. My statement that ‘mathematics represents deductive logic’ is a bit like saying ‘a horse is a four-legged animal’. Most people would accept the truth of the latter statement, even when presented with the rare example of a three-legged horse.
Some bits of Statistics are as much art as science. In the case of Bayesian statistics, it is very definitely subjective.
Still, even then there are still definitely wrong techniques, even if exact answers are hard to pin down.
How is Bayes subjective? The inferences drawn from it might be, but then we’re into the realm of science.
Actually, the act of creating mathematics is much more than simply linear-deductive logic(induction, conjecture, use of heuristics,…) And I believe that’s the point of Dan Meyer, even when I don’t agree with his post, at all.
You are right that creating mathematics – inventing new axioms – is not linear deductive logic. But if it’s mathematics then anything you then do with those axioms involves deductive logic. Meyer does not focus on the creation of new axioms. See, for example, his pile of bricks question.
When you select your priors in Bayesian statistics there is often considerable room for discussion. A search for “Subjective Bayesian” turns up plenty of cases.
Selecting the priors isn’t maths though, is it?
Greg that was exactly the point I was making. The arguments around the subjectivity are part of a legitimate field of study that most of us (myself included) don’t really understand. I think your on solid ground by dismissing them as essentially irelavent but that doesn’t require maths to be perfectly objective, just for it to be very close to that state or alternatively as close to a perfect example of a objective subject that we can find in the real world. If you insist on an absolute then your argument is defeated by a single three legged horse. This is easily avoided by saying most people consider maths as an examplar of an objective subject. People can ignore that premise but they will likely look foolish to those your actually aiming your argument at. Your dropping your guard and exposing your chin.
P.s I agree with Chester. Bayesian reasoning is heavily subjective that’s the usual counterargument to it and why it hasn’t won its war with the frequentists. It’s subjective with its initial assumptions and therefore subjective in its predictions and often it’s arguments are as opaque as any postmodernist. This is likely why it remains a fringe idea (though a fascinating one).
Isn’t Bayesian reasoning a method of taking something subjective – assumed priors and using something entirely objective to improve on the subjective assumptions in a way that offers objectively better conclusions.
That this still has some dependency on the subjective inputs doesn’t change the objective rigour of the mathematical part.
I think you would have to go further to find something mathematical and subjective.
Again the simpler argument is that what Dan wants to talk about is not math it is meta thinking or cognitive science. A fine subject that is clearly not about learning algebra.
Imagine that in another subject – instead of learning French grammar or vocabulary you spend all day thinking about how you think about one sentence in French.
Well quite. Consider the following syllogism
All waiters are evil
This person is a waiter
Therefore, this person is evil
Clearly, the initial axiom is a debatable point. However, that is not the maths part. The logical structure of the syllogism is the maths part. Just because you can do maths based upon questionable assumptions, it doesn’t make the maths subjective. The syllogism is either logically sound in its structure or it is not.
although not directly related to your point about different disciplines, could another reading of Dan Meyer’s post be that he is describing a “primary knowledge” first approach to teaching maths where the “primary knowledge” is drawing on folk concepts of the physical world, in this case a pallet of bricks?
your syllogism commentary reminded me of a study that used Geary’s primary/secondary concepts to teach syllogisms to secondary school kids – Using Primary Knowledge: an Efficient Way To Motivate Students and Promote the Learning of Formal Reasoning (https://www.researchgate.net/publication/332685749_Using_Primary_Knowledge_an_Efficient_Way_To_Motivate_Students_and_Promote_the_Learning_of_Formal_Reasoning)
anyway thought it might be of interest to you (a Geary fan) to comment about this
Yeas Harry. There are correct ways of doing Bayesian statistics and incorrect ones. That doesn’t affect that the starting point involves subjectivity, even when the following steps are objective. The syllogism analogy isn’t really valid, because the starting point is either true or false (all waiters are evil or not), and the logic will follow from that. A Bayesian analysis will take a prior that is true, but they might have a choice of true (and defensible) priors — and that choice decides the end result.
It’s not like frequentist statistics doesn’t have such issues though. What constitutes an “outlier” and what should be done about one? There’s no agreed answer to such questions in either theory or practice. I teach my students to do the analysis with a supposed outlier and without, and so present two different answers to an supposedly objective problem (say, making a prediction from a trend line).
All math statements are provably true or false? Have you read my work? 🙂
I haven’t made this claim. You could make a mathematical statement that is not provably true or false, but if it is a mathematical statement then neither will it be a matter of opinion. Neither have I posited the existence of a single formal system for proving all mathematical statements.
Headline: “…it’s right or wrong.”
There are infinitely many mathematical statements that are neither right nor wrong, because we demand proof. And it can be proven that (in these cases) there is no proof.
“…neither will it be a matter of opinion.”
If I understand the latest findings, the Continuum Hypothesis is a matter of ‘opinion’, in the sense that it can be proven true under some axioms, and it can be proven false under other axioms. Selection of the axioms is a matter of choice, which may follow from your opinion.
The history of Maths is littered with ideas that were initially rejected. It would be absurd to assume that the situation has changed, and now we do not have a hidden infestation of widespread opinions that need to be changed over the coming years.
To deny the influence of the human element (e.g. opinion) is to deny the lessons of history.
If it’s any consolation, it’s much better in math than (for example, picking the worst case) the forever self-correcting field of dietary health advice.
In some versions of indoor soccer, it is against the rules to kick the ball above head height but in other versions of soccer it’s allowed. This does not make it a matter of opinion whether it is OK to kick the ball above head height.
Harry that’s naughy!
To remind you I think Greg is right. I think Dan is talking twaddle. However we should hold our own arguments to account. Shifting the position to whe are objective where it really counts is a different argument. For starters a lot of academic disciplines would be objective under that argument. History follows an objective process as does economics and psychology (most disciplines do) and it produces practical outcomes. The difference is that we more readily aknowledge their subjective states.
On a fundamental level mathematics has the same epistemological issues as all knowledge with Bayesian reasoning openly acknowledging this with its concept of priors.
Of course we can simply refuse to follow Alice down the rabbit hole. When we say maths (or science) is objective we are making a comparative statement. This is more then sufficient to dismiss Meyers nonsense.
Hope this clarifies.
No, the difference between mathematics and history is the difference between the use of deductive versus inductive logic.
An interesting lens but hideously simplistic.
Someone should inform the folks over at the Stanford Encyclopedia of Philosophy, although they are strictly discussing the ‘markedly’ different methods used by mathematics and the natural sciences (link via David F):
No the difference is concrete. In history we attempt to be objective but any conclusion is subject to new facts or new interpretations of old facts.
In math Pythagoras theorem is true if we agree on some assumptions. No one is ever going to find fault with this or show there is a case with the same assumptions and a different result. In math we have a closed system – our assumptions (axioms) and our theorems.
The only difference is that are axioms, assumptions (prime movers) have different likelihoods of being correct. If you assume subjective statements are absolutes then other systems can become as closed as maths. Complex arguments simply require more assumptions and an avoidence of non-sequiters. I
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Meyer clearly hasn’t been around when an investigative model based on complex database queries produces information that saves $27m (did it at work…with a couple of ‘data scientists’ as we like to call them these days).
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