When I was a math tutor in school, everyone would complain about lack of why. This was my approach to address the issue. I would describe a problem relevant to the class, say something like find the angle between two vectors, and ask them to think of a general way to do it. Usually this require some help from me but the solution was always "theirs". Then we would play with their solution: find corner cases, figure out which operations broke, develop a couple of theorems with it. Then I would show them the dot product, and a couple of its tricks (a.k.a. theorems), and they would get a good appreciation for why the dot product works the ways it does. And that's the why and the reason it stuck. It's useful way to solve a common problem. There is no magic or deep philosophical truth.
The problem is this teaching style works with a dialog to a few number of students. I am not sure how well it would work in a lecture setting. It was also not so easy on me as I would be the person who had to spot the problems in their definition, basically on the spot, and lead them to it on their own without revealing the answer. With basic things like a dot product it's pretty easy to know where to look (i.e. is their definition commutative, pretty much never) and finding issues and nudging them too it wasn't too hard. But it's likely hard to scale things like this on many metrics.
'Why' is shorthand for a description of the context of the operation or method being taught. My senior high-school experience was that my maths teacher just plowed through the curriculum from limits to derivatives to integrals and beyond without explaining what each was used for or why. I managed to scrape a pass in high-school calculus but never grew beyond the most rudimentary of understanding until I was taught by a university lecturer who invested time in explaining the why.
The backbone of my mental schema is narrative, so just throwing equations and processes at me does not cause the knowledge to "stick".
Teaching me that Gauss had a problem X that he tried to solve by Y, then showing me what he discovered process N, is invaluable to me because it aids my recall.
My mother was a public school teacher.
She passed her test on the first try, which was extremely rare at her school. She was taught in India, and became an American teacher as an older woman. She was an outlier. She graduated from a major public teaching program in our state of California (CSU Fullerton).
Most teachers fail their tests on the first try. This is not advanced calculus... this is elementary mathematics and reading.
The issue is that accolades in the education department do not translate to results. The state creates a monopoly on teaching via its certification process, with means education departments are guaranteed funding regardless of their results, given that once you have the credential, you're considered 'equal' to other teachers.
Despite my mother doing very well on her tests, she was fired and not granted tenure because she didn't follow her principal's teaching methodology. Even though her students were passing their own exams at higher rates and doing better (And of course my mother taught two boys who went on to make careers in science and mathematics), she was fired for not following the failing methodology.
This is what is wrong with American schooling. It's a race to the bottom and the teachers have no idea what's actual achievement because they've never achieved basic elementary schooling, much less seen it in others.
Here are example questions: https://www.mometrix.com/academy/praxis-math-practice-test/
These are not difficult, at all.
EDIT: in some ways america is doomed by its own success, in that, for those who can actually do math, reading, and writing, there are significantly more lucrative fields than teaching. Unfortunately, since teachers can't teach those skills, it's really a new aristocracy formed by parents who do know those skills passing them on to children.
Maths DO have a why - We needed way to describe and model the world around us, and math was a requirement to do that.
Now - once the model and rules are put in place - Fine, you can bugger off and be as self-referential and contained as you'd like - "There is no why!"...
But lets be clear - there ABSOLUTELY is a why, and the second the world around you no longer matches the model of your maths, we start debating whether or not to throw the thing in the trash and make new rules (see set theory as the classic example...)
I would consider it hostile to your students if you were a maths teacher and withheld practical applications from your students on purpose.
Consider the natural numbers. There is no 'why' behind them. There are axioms behind them, and -- given those axioms -- there are statements about the natural numbers that can be logically reduced to the axioms, but the axioms have no why.
Moreover, it is provable the axioms have no why, because they cannot have a why, because the axioms cannot be proven except in relation to themselves. If you're so convinced the axioms have a why, please prove me and Godel wrong.
The 'why' behind natural numbers is a social one, and one of convenience. The natural numbers make it easy to solve and communicate about certain problems, but they are not the only way to solve those problem nor are they the only way to communicate about these problems.
For example, another way to deal with basic arithmetic, is to talk about numbers as sets. Now you can define certain operations on them, and completely ignore the axioms of the natural numbers. This model is way better than others for certain problems. However, you now have a new set of axioms.. and oh yeah, actually the most obvious ones are completely self-contradictory, so you'll need to choose Zermelo-Frankel or something else.
Or if you want to be even more general, you can simply talk about the lambda calculus, but good luck trying to 'prove' the lamba calculus theorems in itself, because you'll quickly hit the halting problem.
Of course you can then say... well let's get rid of that and use the typed lambda calculus, but then oh yeah you can't do anything interesting. Why are these choices made? Can the choices be justified in the systems themselves? No of course not. The idea that you can use 'logic' to derive these systems is also ridiculous because formal logic is itself a system with axiom (and a very controversial system at that).
But if you look at the lambda calculus, ZF set theory, and the natural numbers as simply models and systems that are sometimes useful, then it makes sense as to 'why'. But the 'why' exists independent of them and is not provable in them and is social and cultural in nature. It is certainly not mathematical as in order to 'do mathematics' (symbolic manipulations) you first need axioms.
Mathematics education in this country has been replaced by rote dogmatism which is why many Americans cannot handle this ambiguity.
What must be explained is that mathematics is a language and in order to communicate with other educated humans about these abstract concepts it behooves everyone to speak the same language. It is the same reason the word for 'dog' in English is taught as being spelled D-O-G. There is no why behind it. It's just the result of thousands of years of culture. Except mathematics is a more global language and more useful for different kinds of manipulations.
In a universe with more than one object, cardinality exists. Natural numbers are how we can discuss cardinality.
Natural numbers are also how we discuss ordinality, because ordinality exists in any universe having at least one dimension.
Axioms are how we discuss natural numbers rigorously. But natural numbers exist independent of any axioms. That's why they're called natural numbers
> It is the same reason the word for 'dog' in English is taught as being spelled D-O-G. There is no why behind it.
I agree with you completely, but I think you're guilty of speaking the wrong language in response to the question (and it would behoove you to consider it from the perspective of someone outside the field).
The question "Why" in maths almost always gets asked by someone new to the field, and they are not asking from a mathematical perspective - They are not asking you for a formal/provable "why", they're asking you what is the utility of learning this thing.
So lets go back to D-O-G. The utility is clear - I have this hairy, 4 legged animal that keeps licking me that I'd like to discuss with you. We can agree that D-O-G (or perro, or 개) refers to it.
But with math, SO MANY PEOPLE (especially those established in the field) jump right into the "Here are the rules of this system of math" without ever taking the time to talk about why someone might give a flying fuck.
It would be like me going and making up my own language and forcing you to learn it. No one else speaks it, it's got no books/literature/history, there are no works of art that reference it - it's literally the language this random person made up that serves ZERO purpose except for talking to that person.
No wonder so many kids don't like math!
Instead you need to explicitly start with the utility of math - ideally in ways that are entertaining and fun. Once a person has an application for some of the rules, they become SO MUCH MORE INTERESTING! Suddenly I care about why this rule might impact that rule over there, or why A and Z are related, or what sin/cos/tan mean.
Basically - sell me on the value proposition of your fucked up whacky language - That's what "why" is asking. Once you know those rules do something useful, it becomes a much more engaging field of study.
We must motivate math, absolutely. And to do so in my opinion starts with socratic questioning. You must convince the student that such an inquiry is even worthwhile.
One thing I'll point out is that we're not just seeing this in math. We see it in every field. More and more kids every year are insisting ( and their teachers are agreeing) that we can do away with inquiries into the English language and the humanities as well. There is a small, but continuing, effort to remove the knowledge of English masters like shakespeare and classic philosophers and treatises from the curriculum.
As a whole, American schooling fails to motivate learning of any kind. Math was the first victim, but the other subjects are also failing.
The whole discussion is from the perspective of the cultural and sociological.
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As an aside, I generally agree with you about american schooling. I think it's less a concerted effort, and more a sad reality of the fact that modern schools have essentially become federally funded child care in the US.
Indeed... As my mother was told by her principal in her inner city school for poor minority kids... "We're just here to watch them until they go to prison".