When you first start teaching math you quickly learn that some things that seemed simple or inconsequential to you are massive stumbling blocks for large numbers of students.
Factoring quadratics is one of these things.
Most people I encountered while majoring in math did not have a "method" to factor quadratics. You look at the thing, see if you can think of some factors, if you can't use the quadratic formula.
Students HATE this "process" it took me a long time to understand why. 1/
@futurebird I damn-near failed Algebra 1 because being presented with an NP-hard problem and told 'try whatever factoring method we learned this chapter and start guessing numbers, they're mostly going to be wrong' was just so profoundly arbitrary and frustrating. I hated both the guessing numbers and the guessing approach. It wasn't until years later when I'd just had enough practice to do it without thinking that it was fine.
@futurebird It just so obviously doesn't scale, it only works with the tiny fraction of cherry-picked examples that math teachers give you. You just know that any real world problem isn't going to have pretty integer roots.
The problem with "real world" problems is they are much more difficult than examples selected to illustrate a particular point. Often incorporating real-world data into a course makes students more frustrated not less. (And the "real world" examples end up even more contrived in their own way.)
I think the real problem is the point of learning these processes isn't to find the answers. Its to understand the nature of polynomials.
@futurebird yes! Math teachers are always trying to walk a line between
'this is useful in real life'
and
'this is a beautiful and important thing in its own right'
Because the hardest sell is the most common truth:
'This will be critical in like two years for more math that is either beautiful or useful or both'
I do a lot of econometrics contrived examples with cooked-up data, because they're the only clean way to definitely see things, and they are indeed a pain in the neck to make good ones!
And lord know we math teachers are bad at making that clear.
I just need to push back on the idea that there is something bad about "contrived examples" the very best examples are deeply contrived. Selected to show some wrinkle in the process, some feature of the algebra.
It takes a long time to design a really really really good math problem.
The best problems are like those toys where you keep unwrapping them and finding new layers and they are crafted and refined.
"What do I do now?" Asks the student. And we tell them to write some parentheses like this:
(x ?)(x ?)=0
Hopefully the concept of multiplying binomials is a familiar one before you get to this. And the idea of the "zero product property" (I think the zero product property is so cool but have not found a way to get students as excited about it as I am... it's a pretty neat algebra trick... anyway)
It's very natural to anyone steeped in math, but I have come to recognize that it may be one of those quirks of math culture that introduces roadblocks that keep more people from joining in with us in the fun.
Not everyone can keep track of the guessing process without a LOT of failures. This leads to frustration and kids give up.
That's not what we want. Not over something so silly.
You see up to this point algebra has been very linear. You can tell what number to use next to solve an equation just by looking at it. Students might even feel a little joy and mastery and we throw this this curve ball.
2x = 8
divide by 2 the opposite of multiplying
x+2=8
subtract 2 the opposite of adding
x^5=32
take the fifth root the opposite of raising to a fifth power
x²+4x-60=0
(x ?)(x ?)=0
Guess.
I think to them it feels like we are going backwards, you know? 3/
@futurebird I remember being annoyed every time math required me to guess. First with long division. Then factoring quadratics. Later, Stoichiometry. It felt unfair, even after I developed the necessary intuition to do it
@futurebird i'm on my second or third read of this thread, after changing my sitting position and opening a window in case the problem had been lack of oxygen in this room
i still have no concept whatsoever as to how you go from
> x²+4x-60=0
to
> (x ?)(x ?)=0
so far i've got
> x(x+4)=60
> 60/x=x+4
and then... guess
at least i can guess something that's slightly less than the square root of 60, and given the context i'm expecting this should be a fairly simple-to-write answer - i.e., a whole number - so i know it's got to be less than 8 (square root of 64) and 7 doesn't work and 6 does (60/6=10=6+4, or (66)+(46)-60=0)
but i can't even begin to understand the reasoning that would create all that quadra-whatever... i know i did this in high school and managed to pass that class, but literally my only memories are sitting in class trying not to doze off while getting increasingly annoyed with the teacher's accent and failing to copy down things on the blackboard fast enough before they're erased
@futurebird i think the terminology is also a huge stumbling block... by the time it even occurs to me to ask what a "binomial" is i've already forgotten to in the midst of an elaborate daydream about tiny bisexual Noldor setting up church on a lawn praying in Latin
And that's why students are so desperate for a set of steps. Guessing and checking is how they solved linear equations before they learned algebraic manipulation.
But, I think we don't impress upon them enough that:
x²+4x-60=0
Isn't a linear equation. It's a roadblock. The linear toolbox fails. We need to find a way to make it linear again.
But even once I understood why they wanted a "method" I still found the idea of methods ... annoying. 4/
This was part being stubborn. But also methods, such as the AC method, are presented like magic trick. "Multiply these numbers and add that number and now you've solved it" might as well use the quadratic formula.
But I've come around to find "the AC method" to be very cool, though the way I develop it is very different from what I've seen in textbooks.
The AC method is a lot like completing the square. And just like completing the square is the right way to develop the quadratic formula-- 5/
We could use "complete the square" I've attached an image of what that is like. Either you will find it "satisfying and elegant" or the fractions will make it feel like not much fun. The quadratic formula is much the same.
So, back to FACTORING we want two binomials that multiply to:
6x²-13x-5=0
So we need the first terms in each to multiply to 6x² the last terms in each to multiply to -5. And the factors of 6*(-5) =-30 to add to the middle term.
@futurebird In my poking around about slide rules I learned about a nifty method they used with slide rules, which kind of depends on the way you can use an analog scale to generate a continuum of "guesses" all at once.
It does involve getting the polynomial into a=1 form: x^2 + bx + c = 0. You set up a couple of the scales (D and CI, I think) so that every pair of numbers on them will multiply to c, and then slide the cursor along there until you get to the pair that sum to b. And now you've factored the quadratic. Sounds like it could be generalized to the ac method without too much trouble.
@futurebird
this is the method I was taught first, but I do not recall it ever being called "the AC method". Instead had a very long name nobody remembered, something like "Factoring the Product of First and Last Terms".
@futurebird I know of this quadratic factoring method by an alternative name!!
It's the "tic-tac-toe" method, to me!
Draw up a tic-tac-toe board, and put the three coefficients in the top row.
The next two rows are where you start playing with factors of the quadratic and constant coefficients in the left and right columns respectively, to find two pairs of numbers that cross-multiply into two numbers in the middle that add to make the linear coefficient.
Oh god. I've seen students do this and always been scared to ask what they were doing.
I still really just like to think about what the factors are... the key words that made it make sense were "it's so you can do factoring by grouping"
I'm going to have the next student who makes one of those explain it to me in more detail. As I don't totally follow where everything goes from your description.
@futurebird I’ve been helping my niece with pre-calculus math and am reteaching myself all of these methods. I think this is one I reverse engineered to figure out why it works - just for my own satisfaction, mind you. My tutee isn’t as enthusiastic about math 😄
I think one issue students have at this point is that they're making a leap to algorithms that require more steps than they are used to, and they need to take a more expansive view instead of hoping to leap to an answer in one or two steps. Also, for me, the chance that a given quadratic wasn't actually factorable was a barrier. The possibility that I was wasting time with a method made me want to give up on it before I started.
I'm hoping to help with this by spending a bit more time on the determinant and letting them use a program they wrote for the quadratic formula to investigate the roots... then still asking for the factors.
I don't really care if they can do guess and check over and over without getting bored. I really need them to be able to decompose the functions when it's possible.
@dangrsmind@MichaelPorter@futurebird
Just continuing the fine tradition, when students were told you couldn't subtract the big number from the small number.
Though I do get a peverae pleasure in telling students their math teachers lied to them.
@dendari@dangrsmind@futurebird As a teacher (retired), I’d prefer that you didn’t put it this way. We have enough trouble getting students to come along on the journey without this framing.
How about “Your teacher gave you a simple model/method/idea that worked for you at the time. You’re ready for the next-level stuff, now.”
When a student arrives in my class with messed up ideas about numbers it's hard to know if they had a bad teacher, a teacher with no resources, or if they just didn't listen. (If I had a dime for every time a student claimed "but Mr. X said " and I know Mr. X and he DID NOT say that. LOL.)
I make no assumptions.
I want to teach my students how to squeeze learning out of even the worst teachers they will face.
@MichaelPorter@futurebird@dendari@dangrsmind But it is lying. I first learned about negative numbers sitting in study hall (which was a desk in an active classroom, which happened to be math a few years ahead). When my teacher insisted that you couldn’t subtract a big number from a small one, I called it out, but they insisted I was wrong. So as far as I was concerned, I was being accused of lying. And that was the beginning of a lifelong dislike of math.
You can tell kids about things that are possible without explaining the details. It inspires curiosity and interest.
@nazgul@futurebird@dendari@dangrsmind That doesn’t sound like lying, it sounds like the teacher didn’t know their stuff. Bound to happen when teachers at the younger grades have to teach every subject. There will be gaps in their knowledge. But they shouldn’t have doubled down like that, for sure. That’s always a mistake.
I’m just tired of the general statements that paint everyone with the same brush.
I was listening to an audio book the other day, where the author began a section with one of those “Your teacher taught this wrong/never taught this” statements. First of all, no, I did teach that, and correctly. They then proceeded to either grossly oversimplify, or get a concept wrong (depending on which church in physics you attend). So fuck that guy. Sorry for the swearing, but I passionately mean that. If you want to be a science popularizer, you don’t need to alienate your best allies.
To your last point, yes, leave the door open. (It helps if the teacher knows about the door.) One thing to consider is that some students are already intimidated by the material. Keeping things simple for the time being, without mentioning the myriad possibilities to follow, keeps some of them from freaking out. If an advanced/gifted kid comes seeking more, then by all means feed them.
I don't have this right here in front of me now, but I can show examples from textbooks where the book literally says a problem has no solution.
It doesn't say no solution in the real numbers, it doesn't say there is a solution but we will learn how to find that in the future. It says "no solution" or "the solution doesn't exist".
This statement is false and it is purposefully placed in the textbook.Students read the book and believe what it says.
A purposeful falsehood is also called a lie.
I always try to tell my students the truth and it seems to work. Even with academically challenged students.
@dangrsmind@nazgul@futurebird@dendari I’ve come across that lately while working with my niece. It’s a problem, for sure, but I wouldn’t overlook the possibility that the error was not purposeful. I’ve seen a number of mistakes in texts that were just repetitions of commonly held misconceptions, errors by graphic artists (probably working from messy sketches by the author), and wrong answers in the back because that task was handed to an uninterested grad student for a few bucks and no-one checked their work.
Attached is one we came across last night (No textbook, just online resources and worksheets).
@MichaelPorter@nazgul@futurebird@dendari oh there are definitely mistakes and errors or problems with "version control" but also books that specifically state there are no solutions to solvable problems.
I don't know whether that is because the authors do not know what they are talking about (doubtful IMO) or they purposefully decided to tell the student there is "no solution" to a problem when there really is one.
The issue is when you tell a student there is no solution then a few months later they find out there actually is a solution. But also there are other problems that don't have a any solutions and really they don't.
Students do not understand what is true and what is not true, find this super confusing at least the ones I have worked with are confused by this.
I used to write "No real solution" for imaginary solutions... but quickly realized my students thought that meant that I didn't think there was any realistic way to solve it... which almost works, but imaginary numbers are very practical and not that creepy or strange since they have geometry and are a proper field and all.
Now write "solution contains sqrt(-1)" then point out they aren't real number solutions and so not what we are studying yet.
@dangrsmind@nazgul@futurebird@dendari We might be talking about different levels of textbooks, and thus different backgrounds of authors and editors. I suspect some of the stuff at the elementary and high school level isn’t vetted as carefully. But really I’m just guessing. I find it easier to believe that an author would make a mistake than intentionally introduce a statement that wasn’t true, when clarifying that “there are no real solutions but wait until next year” would be just as easy.
@MichaelPorter@nazgul@futurebird@dendari the confusion is some problems actually do not have any solutions, some problems may have solutions but no one knows what they are, and some problems have solutions but we tell students to write "no solution" as the answer. Sure some teachers/texts are more careful but some are not.
Examples:
A system of linear equations where the lines are parallel has no solution (x,y) because the lines never cross.
The Collatz conjecture might have a solution but no one knows for sure. Do we always end up on one? It's fun to think about but we don't know for sure what the answer is.
A quadratic equation with a negative discriminant. We tell students this has no solution but it really has two solutions.
@futurebird I WISH I could understand any of this but after the first few lines this thread devovles into what seems to be complete gibberish to me. It is so frustrating to me that anything beyond the simplest of early high school maths is just completely beyond my grasp. Has been this way my whole life , and it damn near cost me the career in science I so dearly wanted (fortunately a chink in the admission system got me into uni even though I failed maths outright).
@futurebird Then they get the same shock later when passing from differential to integral calculus. The derivative of any function you encounter in high school is just turning the crank once you learn the rules. Turn it around in reverse and suddenly you have to think, hard, guess which of your sprawling and amorphous bag of tricks might apply, and many innocent-looking functions don't even have a closed-form solution.
@futurebird I really struggled through algebra in high school, and I remember lots of that being because of factoring quadratics. I say this as someone who ultimately wound up in a math PhD program (but left before finishing).
Factoring took whatever systems I had away from me. When I learned CTS and the quadratic formula, it was a relief for me. I taught a bunch of intro algebra classes in grad school, and tried to bring that understanding, but I still didn't know how to help bridge that gap.
@futurebird I wonder how "important" teaching factoring really is. I mean, it's our traditional way of developing algebraic skills, and I'm certainly not opposed to that, but... I've always felt that factoring is relatively "slow" to connect to higher mathematics relative to a number of simpler, more approachable ideas.
Or stated another way, it feels like factoring doesn't build on itself in the mathematical curriculum in quite the same way as say, Pascal's Triangle.
But as an approach to getting to the quadratic formula, yeah I agree.
I think it's very important for understanding what polynomials are, how many roots they can have and the expanding complexity as you move to equations with higher powers.
And knowing the polynomials helps the transcendental functions to stand out and have meaning.
Factoring is important for primes as well, there are parallels between factoring integers and factoring polynomials to find rational roots, and polynomials that can't be factored without real roots...and the primes.
That's the quadratic formula ... or rather it is the ancestor of the quadratic formula the process we call "completing the square."
But you can always write a quadratic as a product of binomials where the solution to each binomial solved as an independent equation is one of the roots.
@futurebird What I’m learning as my child goes through kindergarten is, it goes even further back than this to more fundamental concepts.
He is in public school. He went to a good preschool and had some math education going into kindergarten. Not all of his classmates did, so they are starting their math education with stuff he has already covered.
@futurebird He was complaining to me “we aren’t doing real math in math lessons!” and it turns out they were doing exercises like “sort these things into categories” and “how is this different from that,” so, like, basic logic exercises.
My son could do these easily, but some of his classmates struggled with it and really needed the instruction. They would have been in trouble later without mastering these fundamentals.
@futurebird@MisuseCase This reminds me of a Prof of Education around 1980. One of his kids teachers disliked the new math because students were asking her questions that she couldn't answer.
My prof thought that kids asking their teachers questions was wonderful. And that exploring these questions should have become part of class rather than being scary to the teacher.
@futurebird This is so confirming for me. Every step I made would be logical and mathematically correct, but it wouldn't go anywhere in terms of solving for x. I was just flailing.
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