I’m always wary of the idea learning should be “practical”. You never know when something will matter and there is an intrinsic value in learning for learnings sake.
Learning needs to be tangible, but I’m not sure it necessitates practicality.
That 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 aresimplepractical 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.
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 + 7 and 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.
The hidden Factor here is coercion, if you don’t go to school the cops will literally show up at your door eventually. In light of that it’s completely reasonable for the people who have no choice but to be there to ask what purpose it serves
jacksilver@lemmy.world 4 weeks ago
I’m always wary of the idea learning should be “practical”. You never know when something will matter and there is an intrinsic value in learning for learnings sake.
Learning needs to be tangible, but I’m not sure it necessitates practicality.
Contramuffin@lemmy.world 4 weeks ago
Sure, but learning tends to be easier when there’s a practical application to the things you’re learning
definitemaybe@lemmy.ca 4 weeks ago
That 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 4 weeks 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.
fluffykittycat@slrpnk.net 4 weeks ago
The hidden Factor here is coercion, if you don’t go to school the cops will literally show up at your door eventually. In light of that it’s completely reasonable for the people who have no choice but to be there to ask what purpose it serves
mnemonicmonkeys@sh.itjust.works 4 weeks ago
Just look at Boolean mathematics. It was developed 100 years before it had a use (digital logic)