Comment on What would you do?
definitemaybe@lemmy.ca 1 day agoThat kinda breaks down in practice, though. Math is hard for a lot of students. Adding an extra layer of domain-specific application on top of an already confusing topic just makes it worse.
Like, we need polynomials for huge swathes of higher-level math. My favourite application of polynomials is that most continuous functions can be approximated by a Taylor series, which makes some functions that are otherwise impossible to calculate a derivative or integral trivially easy. It’s elegant, beautiful, and deeply practical.
And completely useless for a grade 8 student learning about polynomials for the first time.
Sure, there’s lower-hanging fruit for practical uses for polynomials, but they’re either similarly abstract (albeit simpler) or contrived. Ain’t nobody making a sandbox with length (3x + 5) and width (2x – 7), eh?
I could go on. At length.
Point being, yes, practical applications are better. BUT (and this is a big but) only when there are simple practical applications.
Instead, recent math education research supports teaching fluency through playing with math concepts and exploring things in many ways: symbolically, graphically, forwards and backwards, extending iteratively with increasing complexity, etc. This helps students develop intuition for math concepts and deeper understanding. Then, and only then, teach the standard algorithms and methods, as students will appreciate the efficiency of the tool and understand what they’re doing and why they’re doing it.
Thank you for listening to my TED Talk.
gandalf_der_12te@discuss.tchncs.de 1 day ago
Polynomials:
They exist because they are efficient to compute. Computers do well with basic arithmetic operations like addition (+) and multiplication (*). The polynomial functions are simply those that you can construct from those two operations, and constant numbers.
Like consider a polynomial like f(x) = 5x^3 + 3x^2 + 2x + 7
What it really says is
f(x) = 5*x*x*x + 3*x*x + 2*x + 7and here you can see how it’s all built from + and *.This is why polynomials are useful. Because computers have an easy time calculating them. And all modern mathematics is done on computers. All the engineering uses computer simulations, and we want these simulations to run fast on computer hardware, so we make it easy for computer hardware to do. That is why we’re using polynomials wherever we can.
That is how you explain polynomials to 8th graders. No taylor series needed.
definitemaybe@lemmy.ca 1 day ago
Yes, examples like that are good, of course. But, frankly, abstract examples like that won’t do much to motivate the students who need the most help to get motivated learning math.
I like to interject little anecdotes like that, too. One of my “go tos” to “why are quadratics useful” goes something like “Well, they come up a fair bit, so I could give you some examples—and I will, as we with through the unit, but the real reason we teach quadratics is because they’re the simplest non-linear function. This is the first steps into looking at functions that aren’t a straight line. And the tools you use to work with quadratics are super important for understanding all the really cool functions you get to learn on the next couple of years…”
That’s basically your example, but one step lower and more directly applicable to students, imho. The Taylor Series thing I usually only drop in grade 11/12 (pre)calculus classes, mostly as a hook for the math nerds that they have really cool things to look forward to learning in post secondary. It’s a terrible application to use to try to motivate learning about polynomials for a student who couldn’t care less, lol.
Really, we need to intermix all approaches, depending on the students in the class. At private prep schools, leaning into academic needs works well. In a non-academic math stream, both your example and my examples will go over like a lead balloon.
But, regardless, motivating students to be excited for math, and the excitement of finally figuring out a tricky concept/problem? That’s what we need more of.