That's the problem we treat math like it is some innate talent, it's not it just requires certain ways and forms of thinking. Example: I was decent at math in HS, I didn't do very well because I didn't like doing the homework and quite frankly most people good at math aren't going to end up being public school teachers, so the explanations were always lacking.

Well I went on learning programming on my own got a job as an SWE, then a couple of years later ran into a situation where I had to learn some new math. It was world's easier this time, because of my work with programming it all jelled immediately and suddenly clicked it all became very logical. I don't believe I suddenly became better at math, but rather I had developed certain thought patterns and ways of thinking that could be applied to mathematics.

Anyway, this all led to me doing very poorly in math at school, and it’s entirely because my brain just isn’t good at it. Luckily these days the educators are smarter and my son has always been allowed a calculator for math class.

But all of the students who came to the help desk in college were practically addicted to their overpriced graphing calculators. A decent fraction of the time I would look over their work, they'd be on their last try in WebAssign (may it be cast into the fire) with a scared look on their face, and I would say "you entered this part into the calculator wrong". I knew immediately because I learned to estimate these things. You don't have to even know the multiplication algorithm; you can stick to multiplying the first digit, adding exponents in scientific notation, etc. Learning to work with the calculator instead of behind the calculator will make you use it more effectively.

The multiplication table has 45 entries that aren't duplicates or multiples of 1 or 0. Kids memorize longer things all the time. The brain is definitely not the limiting factor.

>Textbooks still focus things like von Neumann's ability ability to multiply large digit numbers in his head as a proxy for his math proficiency.

I don't recall ever seeing that in a textbook. There is a famous story about JvN summing an infinite series in his head, but very few people can do that or would be expected to. He also estimated the yield of Trinity within a factor of four by just looking at it. One should never compare themself to von Neumann.

I honestly believe the biggest factor in math anxiety is osmosis. Their parents are afraid of math, their teachers (often!) are afraid of math, and there's a robust body of evidence in developmental psychology that children inherit the worries of their guardians.

Math is different from most other primary school subjects in that it builds on itself year after year. There is a danger to letting computing devices take over before the students fully grasp a math concept. If they fail to learn a concept, moving on with a calculator as a crutch may inhibit them from understanding the next math concept.

A second degree polynomial with leading coefficient 3 has zeros of -1 and 2. Find all the terms of the polynomial.

Most students can’t do this. Even most calculus students can’t do it.

We teach algorithms like long division and the quadratic formula because they are relatively easy computations to learn but they don’t in any way lead students to fully grasping a concept. It’s only with a certain level of mathematical maturity that one is able to understand the full import of even basic concepts.

I can walk into pretty much any first semester calculus class and ask students to write down an example of an equation with no solution. A large majority will fail to do so. It doesn’t occur to them that 0=1 is such an example. They’ll play around with x’s in various complicated looking expressions. Even something as basic and fundamental as the meaning of an equation eludes people at this level even though they have been dealing with equations for years.

Well, such is my experience teaching math at a community college for over 20 years.

It's interesting that you picked 0=1 as your example, because I'd argue it stretches the definitions of "equation" and "solution" into semantic triviality. It's more of a falsehood than an "equation", since the two sides are trivially defined as not equal, and there's no variables to "solve". Using that as example exists somewhere between sophistry and pointing out the absurdity that mathematical definitions for terms technically hold even in trivially untrue situations. That's not how normal human communication works, and not recognizing that divide probably goes a long way in explaining the "inability" you see in students.

In other words, maybe you should have just used "0x=1" as your example :P

Suppose I said solve

x = x+1

You then subtract x from both sides and end up with

0 = 1

Then you conclude that the original equation has no solution. I’m guessing that you wouldn’t realize that the reason we conclude that the original equation has no solution is because the two equations

x = x + 1

and

0 = 1

have the same solution set since adding the opposite of x to both sides is a solution set preserving operation. It transforms a given equation into a new equation with the same solutions and clearly 0=1 has no solution. That is, 0=1 is a perfectly valid equation.

The larger point, that is missed by people, is that an equation in essence is asking for one to find the instances when two expressions are equal. To find an example of an equation with no solution just find two expressions that are never equal to each other.

Respectfully, you've got this backwards. An equation, by definition, is an assertion that two expressions are equal. 0=1 is a logically consistent assertion, but it happens to be false. Most students will intuitively have trouble with the idea that you want them to make a false statement, even if they don't realize that, because their whole schooling has taught them the opposite.

The issue is precisely that we are teaching those students that that "an equation in essence is asking for one to find the instances when two expressions are equal". Mathematical statements don't "ask" anything, they simply are. That's a pedagogical definition, not a mathematical one, and by teaching students that, you're teaching them how to pass a math test rather than teaching them math. And there's no blame on you for that, since you're paid to teach students to pass math tests. But framing it that way doesn't teach them math, it teaches them how to guess the teacher's password[1]. It's a focus on getting an answer rather than understanding the actual axioms.

So of course students don't come up with an equation with nothing to solve, because you've taught them equations are things that only exist as things with unknowns to solve.

It might be obvious to someone who already is extremely well versed in mathematics that 0=1 is "an equation without a solution". But it's unfair to expect students who don't already have that answer to derive it, because they're working off of the wrong axioms. It's a communication failure, not a mathematical one.

[1]https://www.lesswrong.com/posts/NMoLJuDJEms7Ku9XS/guessing-t...

x^2 + x + 1 = 0

And say solve it we are definitely not asserting that the two expressions are the same. Indeed they are not the same polynomials and if your view were correct we wouldn’t spend time teaching how to solve the equation. There are values for which the two polynomials evaluate to the same number. Those are the solutions.

EDIT: In mathematical logic class one talks about predicates and you learn to think of equations as assertions that two expressions are the same. However, as people typically use and think about math they don’t think in these terms. Indeed, the graphical interpretation of an equation in one variable lends itself to the idea that solving an equation, in essence, is finding values of x that make two functions have the same value.

It is also equally clear that you haven’t taught basic mathematics to innumerate students. When students are taught to solve basic linear equations we include in our instruction that they can encounter situations like:

x+1 = x

And that they can see there is no solution because they reduce the equation to solving 0=1 and that equation has no solution.

You are in an absurd position when you think

0x = 1

is an equation but that

0=1

is not. I doubt that when you simplify:

x^2-2x - (x^2 -2x)

You write 0x^2 + 0x. What I wrote about solving equations has an important word in it. Namely “essence”. In essence…. I was not providing a mathematically rigorous definition. Indeed, the rigorous definition is far beyond the scope of students of basic mathematics. So we have to teach them the essence of things.

> we are definitely not asserting that the two expressions are the same.

Correct. Not the same, equal. Because that's definitionally what the equals sign means. "A=B" is a symbolic representation of "'The expression A' equals 'The expression B'". I hope we can agree on that?

>if your view were correct we wouldn’t spend time teaching how to solve the equation.

What I actually said implies the exact opposite. You teach how to solve equations because that's the use case for equations as tools. That's not a bad thing, it's an extremely useful thing to teach.

But teaching how a tool is used is not the same as teaching the fundamentals of what a tool is. It can help in that goal, certainly, (and might even be required as a prerequisite) but it's not the same. It's exactly like you said:

>We teach algorithms like long division and the quadratic formula because they are relatively easy computations to learn but they don’t in any way lead students to fully grasping a concept.

It's not fair to blame students' "innumeracy" for not being able to derive "0=1" as "an equation without a solution", because they've successfully learned the thing that they were actually taught, that equations are "things with unknowns that we have to solve for". Of course generating a solution that has neither unknowns nor a solution is foreign, because everything they've learned about it as a tool goes against that.

(It's worth noting that there's another reason that the teachers teach this, one that's perhaps even more important for the school system; it's an easy thing to evaluate student understanding of. You can easily test whether a student can "solve" an equation, and return the correct answer. It's something you can get immediate, iterative feedback on. You can't really test if they actually grok a definition, because they can just parrot a definition with no understanding.)

Fundamentally, my argument is about language, not mathematics. You're saying that students aren't able to derive answers based on the definitions of terms, but not only are those definitions wildly divergent from their English meaning, they're divergent from the actual definitions the student are learning via practice.

Take, for example:

>There are values for which the two polynomials evaluate to the same number. Those are the solutions.

Values of x, you mean but didn't say. Because it's so heavily implied in the existence of an "equation to solve" that unknown quantities you are solving for are the ones written in the equation itself, that it's not even worth mentioning. But it's precisely this linguistic assumption that obscures what an equation actually is to students.

I was taught that using the quadratic formula to find roots is "cheating" when I tried using it before we had covered it in class. What exactly does "completing the square" test for other than your skills at mental arithmetic?

The reason why many student believe that polynomial doesn't factor is because teachers do a lot of hand waving when it comes to explaining what it means to have no rational roots of a polynomial. Few teachers will take the time to teach the foundations of the cartesian coordinate system and how complex solutions don't map easily on the typical plane of rational numbers. All students learn is if there's an "i" the answer is "no solution".

Zeros being the solutions of functions is a question on the finding roots of a polynomial chapter in basically every single high school algebra 2 class. It's a prerequisite to learning how to graph second order polynomial functions. Many students learn and forget how to do it before reaching calculus, let alone college.

I genuinely haven't done long division in the last decade. I struggled to help my younger cousin with it recently and had to relearn myself because it's such a useless algorithm in the age of computers. Certain multiples and powers I remember, because of how often I come across the numbers, but in general I will choose a calculator every time. I would even choose a calculator to double check my own work with a paper and pencil. At this point what is the value in doing the work by hand? In many cases a decimal to the hundredths is required as well.

When I hear the question, write down an example of an equation with no solution, my intuition and experience doesn't lead me to writing an incorrect equation. It leads me to think about writing a polynomial function that I know will have complex roots because I was taught the answer to that is "no solution", or writing a system of equation where x is a specific number while at the same time having an equation where that specific number can't be a part of the domain. More fundamentally, a system of equation with no solution is one where the two lines that are graphed are parallel.

I have to admit, my experience is a little biased as I was placed in accelerated math since elementary school. It wasn't difficult for me or my peers. My math class senior year was fundamentals of multivariable calc and linear algebra as a senior in high school, having finished AP calc bc the year before. I was far from the only one in that situation, there were at least 60 of us that year, some seniors and some juniors. I can't say I have experience teaching a full class but I have been tutoring high schoolers in math for over 8 years. Many of my students have also been in accelerated math, but not all of them. I don't think anyone tested out of multivariable calculus, but I did have a friend who tested out of linear algebra at reputable universities.

I know that I have some time and experience left before I feel confident in my own mathematical maturity, but I'd like to imagine I'm somewhat good at math. At the very least I wouldn't consider myself bad at math, even though I still feel like I am at the early stages of learning in specific branches of math.

By the time one reaches calculus they have been taught that complex solutions are valid solutions. They just aren’t real solutions. Therein lies one of the problems teachers of mathematics have. Conveying the concept of the answer depending on what the current algebraic object one is working on. We have to hand wave do some brain washing because the nuances involved are far too complicated for the students to understand at this level.

I agree with your general point that math curriculum builds on itself and if students fall behind it can make things worse and worse for future classes. But in general I think, at least when I was in school, there was probably too much focus on algorithms and not enough on conceptual understanding. I think part of the reason is that algorithms historically were necessary to know before calculators were ubiquitous, and also I think algorithms are much easier to teach and evaluate so there's probably a tendency to focus on them for that reason.

The point is not for students to learn how to follow an algorithm (or master doing it quickly), but that they learn why the algorithm works and how to interpret the result.

The algorithm matters and I argue that once students can naturally derive the algorithm on their own, there's no place for human calculations. Computers and calculators should be used. We're no longer living in a society where students will have to worry about "what if you don't have a calculator on you". It seems as weird as memorizing unit conversions to me.

What's interesting is that the purpose of implementing the common core standard was exactly what I am arguing for. A shift away from algorithmic memorization to a number sense way of doing math. The failure lies with the teachers not being able to teach well much more than students not learning.

Really? Which textbooks are these? Done high school, uni. Never heard that, or really much about von Neumann outside a few passing mentions in advanced topics. Sounds like some really strange math textbooks.

My husband and I went to uni together. I dragged him through the mandatory maths units in the computer science degree by reframing the equations we were working on as "a bunch of steps, just like code" and he got through the unit, but our lecturer was a horrible, impatient person and managed to reinforce his fear of the subject.

A lot of people I know have similar stories. Maths and physics seem to attract teachers who think (out loud) that students who don't intuitively "get it" are beneath them.