Exactly, and that comment was from the evolution of the discussion. People cannot say math is black and white, because it isn't. It is to those who only understand basic mathematics, but innumeracy is so high even for the basics that we do need to consider teaching kids that it isn't as rigid as people make it out to be, then maybe they wouldn't be so scared of it.
I'll give the group a euro centric example since it will make them feel more comfortable, my dad's family immigrated from Brasil to Portugal, when my dad does long form division it is completely backwards from what I was taught and confused the hell out of me until he showed me how he does it. Knowing another way to do long form division only made me better at math because it forced me to think of numbers slightly differently.
I was ALWAYS naturally good at math, from grade 2 I was beating all my classmates in flash card games and those 60s drills. But come secondary school where I was being forced into a math box I started butting heads with my teacher since I wasn't following all their exact steps on paper as my brain did most of the work just by looking at the equation. My teachers consistently tried to dock me points even though my final answer was correct and my parents were in there consistently arguing with them that writing all the steps is semantics and what matters is that I learn the concepts and get the correct final answer.
Things like this is why students struggle, the concept of mathematics is a construct, your grade school math level might not have taught you this, but it is. Western society early on agreed on a set of mathematical rules that if ever disrupted would make math impossible - an easy example of this is BEDMAS, we HAVE to write equations to follow this, if we don't no one will get the right answer. BEDMAS is not a natural concept, it is a construct, and there are many more of them in math that we had to agree to to make math as we know it work.
So the irony is, those crying about losing critical thinking in this thread are those who lack it because they were unable to realize that math is not simply first principles, those first principles lay on a bed of assumptions that if ever disturbed would cause havoc in our generation just as it has in previous generations when mathematical "discoveries" came to light.
I'm going to (sort of) disagree with some aspects here. I'm drawing on some general experience , some of which may not apply to you specifically, so please forgive me if I appear to inaccurately attribute certain details to your particular experience.
While there are many different ways to do things in math, I also think it's entirely reasonable to require students to follow particular methods. There are a lot of reasons for this.
Linear systems presents a good example. As the curriculum exists, by grade 10 students are exposed to three methods for solving linear systems of two variables: graphing, substitution and elimination. Now, for the algebraic methods, a lot of students tend towards one or the other. Each method has situations in which it's more convenient, though you can actually solve any question you're likely to get at this level using either, so those students often try to only use their preference and balk at the requirement to use the other. But what if, in future study, you are exposed to nonlinear systems? There are times when elimination is the only practical approach and substitution is effectively useless, and there are times the reverse is true. So in practice you would need to have developed both methods. This kind of thing happens all the time in math. We learn different methods for doing things not only to give us options for the immediate task, but because those methods develop different ways of understanding and can be extended in different ways. Very often, students (and teachers for that matter) get caught in the trap of approaching math with only the short term in mind. I believe that's one of the primary reasons many students say "I was really good at math until grade X, and then I hit a wall."
BEDMAS is actually a really good example to delve into for discussing convention. Because it's absolutely true that it's a construct, and it could have been constructed differently. Indeed, taking brackets as a tool to communicate the desired order of operations, we wouldn't technically need any further convention at all, though this would make for some horribly unwieldy expressions. While BEDMAS is a construct, there are also very good reasons it's constructed the way it is. There is a natural progression from addition to multiplication to exponentiation, and operations are partnered with their inverses. In most cases with which we are concerned, it leads to the most efficient way to communicate expressions. And as you said, in practice a knowledge of order of operations is essential to function in algebra.
I absolutely believe a consideration in the assessment of a math student's work is the way they present that work. In higher level math, no one really cares if you got the right answer if you can't clearly communicate why it's the right answer. In most cases, using the established language of mathematics is going to be part of that.
(Aside: I anticipate someone reading this is thinking "who cares about higher level math. Most people never use that anyway. I just want my kids to get the right answer." I think it's valuable to address this. Firstly, a basic principle in education is that you generally teach a subject on the basis that students will progress further in that subject. Consider what the alternative would lead to. And much of the time the value of education doesn't actually come from the specific skills being taught, since it's impossible to anticipate the specific skills everyone is going to need. Rather, it is the development of metaskills that have value no matter what the future holds. Math is an excellent subject for many of those, including the development
and communication of convincing logical arguments. A
lot of students will get value out of this no matter what they do, just as many will need to follow industry conventions in communicating information in the same way a math student can be expected to use established mathematical language.)
More generally, this issue of the language of mathematics needs much greater attention than it gets. I knew several people who studied math in French immersion until grade 11, at which point they switched to studying math in English. They received no particular support for this transition and it was a real problem for them. Math is a highly technical subject with a great deal of terminology, and they didn't know what their teacher was saying half the time. This is a somewhat extreme example but it is part of a much larger problem. For immigrants, if they come from a place that uses very different conventions, going into a math class can be practically as jarring as going into an English class if they don't speak English. Even for students who always studied in Ontario, sometimes they hit a point they are expected to use conventions no one ever taught them. Frankly, a lot of the time this happens because teachers at the elementary level didn't know those conventions themselves! Which is one of those systemic issues I alluded to in my previous post: there are a lot of people teaching math who don't really know enough about it. That usually isn't their fault; it's the way the system works.
