[syzar]

Students don’t fully grasp concepts at the early stages and they never will. Very few students who master the quadratic formula understand that this formula allows you to factor second degree polynomials. Many students will correctly solve x^2+x+1 = 0 while simultaneously believing that this polynomial doesn’t factor. Here’s a simple problem that tests understanding,

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.

[LordDragonfang]

>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.

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

[syzar]

0x is the same thing as 0 so it appears my example is a good one in that you yourself don’t fully understand the concepts involved. This isn’t pejorative.

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.

[LordDragonfang]

>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.

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...

[syzar]

It is clear you are not a mathematician. When we write something like:

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.

[knaik94]

Given the first polynomial, when asked to solve it, there's an implied "for x" attached to the question. Even in higher level math you assume you're solving for a variable. When writing an equation without a solution, you don't naturally think about not including any variables. While 0 = 1 is an equation, it's not an equation you "solve". The meaning of equation is not in question, just the association of the terminology of equation to something without variables. Context is important, if the expression had a third order term and I had to use synthetic division, I would absolutely write include the zero terms.

[LordDragonfang]

I feel like you're talking past what I'm saying to continue teaching the same math lesson you've taught hundreds of times before, which is exactly the kind of discontinuity in communication that I'm trying to highlight (and, evidently, failing). It's difficult to articulate, and I already feel like this reply is rambling quite a bit, but hear goes:

> 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.

[knaik94]

Actually the quadratic formula is a great example of exactly what I mean. I was taught it as a way to solve for roots of a second degree polynomial when "completing the square" didn't work. The terms in the quadratic formula are defined as the coefficients in the polynomial and the answer is written in the form "x = +-___". Looking back at that chapter in math, that along with the rational roots test for finding zeros in a higher order polynomial were genuine wastes of time. I don't mean the learning the intuition, I mean a question asking to list every single possible root given a second or higher order polynomial.

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.

[syzar]

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",

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.