[simplotek]

> A high school "data science" course, if designed properly, will be far more useful to students and beneficial to society than calculus.

How do you expect students to understand what they are doing with "data science" without learning probability and statistics, and how do you expect students to get probability and statistics without learning calculus?

I mean, Bayes' theorem. How do you get people to get it if they don't know calculus?

[tzs]

I don't recall Bayes' theorem involving calculus. Are you sure you aren't thinking of some other theorem?

Bayes' theorem follows straightforwardly from P(A & B) = P(A|B) P(B) and P(A & B) = P(B & A). The latter tells us that we can swap A and B in the former without changing the value, giving us P(A|B) P(B) = P(B|A) P(A).

Rearranging gives P(A|B) = P(B|A) P(A) / P(B), which is Bayes' theorem.

[mturmon]

You can sidestep calculus by just using the discrete setting rather than a continuous one.

If you want to introduce continuous distributions like the Gaussian one, you can just say "area under the curve" if you need to connect the density to a numerical probability. They don't have to know how to do the integral, in the case of a Gaussian, it's just tabulated anyway.

I'd argue that you could teach a perfectly reasonable high school stats class using this kind of approach.

A "calculus-free" method is mostly what is done for high school physics, with occasional nods in that direction to set the students up later. And like physics, the obvious connection to of continuous probability to calculus will be a nice motivation later on.

One analogy is how we teach probability to sophisticated engineering undergraduates. I'm not aware of undergrad engineering curricula that use measure theory. This results in awkwardness around delta "functions" and probabilities of certain sets of measure zero (sets that cannot be integrated without the Lebesgue integral).

And sure, some of those undergrads don't ever take that measure theory class, so they escape to the wild without knowing the answers to awkward questions.

[simplotek]

> If you want to introduce continuous distributions like the Gaussian one, you can just say "area under the curve" if you need to connect the density to a numerical probability.

What name do you give to this "area under the curve", or the "rate of change" of this area? They are pretty fundamental concepts with important and basic properties, which affect things like local optima and minimization, and expected value and covariance, etc. I mean, you can't cover linear models and least squares without this stuff, and if you don't then I wouldn't really call it learning.

[comte7092]

You call “area under the curve”… area under the curve. Expected values, least squares, linear model, etc can all be explained in the discrete case without calculus.

High school math isn’t and doesn’t need to be rigorously proofed based, if you lack some do the tooling necessary to demonstrate a proof, you can tell a student, “the proof requires calculus” and boom, you’ve given them a reason to take an interest in the subject.

[mturmon]

You don’t need integration to define expected value or covariance in the discrete case. TBH I’m not sure if you can get around integration in the general continuous case or not.

If not, you could use some limiting argument to handle the moments of a continuous uniform RV, at least, in terms of the discrete analog.

You don’t need calculus to derive least squares estimators. You can follow the logic in this quora answer [1] to show that (e.g.) the mean is the minimum MSE estimator among constant functions, and that the conditional mean is the minimum MSE estimator among “general” (measurable L2) functions.

This derivation is familiar to many who have studied these concepts. It’s clever, it does not need differentiation, just expectation and logic.

It could be that your studies in probability were done using a certain pedagogical path, and that’s blinding you to the fact that other paths are possible.

[1] https://www.quora.com/Why-is-minimum-mean-square-error-estim...

[comte7092]

High schools often teach physics and without calculus as a prerequisite. It definitely makes it more challenging, but you can still communicate the concepts at a different level of detail.

[simplotek]

> High schools often teach physics and without calculus as a prerequisite.

Does it though? For example, you simply cannot teach Newton's laws of motion without knowing what a derivative is.

[comte7092]

> For example, you simply cannot teach Newton's laws of motion without knowing what a derivative is.

You absolutely can do that. You might now want to, but you can, and people do.

[mturmon]

It does, my kid finished calculus-free high school physics last year.

[roddylindsay]

You can definitely explain Bayes' theorem without calculus. I just asked ChatGPT to do it and it came up with a great example using a deck of cards and some fraction math.