# Tag Info

7

If you are teaching this at an introductory level, then the algorithm that calculators use today is going to go far over their heads. (It might go over MY head!) The story of how we developed increasingly accurate trig tables over the course of history would be an interesting topic of inquiry for advanced algebra / pre-calculus / calculus, but at the ...

6

I would suggest inviting, or "recruiting" the teachers you are leading to help you with working out the direction in which you hope to move from an emphasis on "procedural efficacy" toward an emphasis on "conceptual understanding." Try to frame it as a "team project", explicitly recognizing their experience and first-hand perspectives in the class-room. Don'...

5

Set theory has no important implications for 99% of normal mathematics, and the difference between, say, ZFC and NF has even less. Mathematics isn't generally written in the language of ZFC, it's written using more basic ideas and notation that are the same in other foundational systems. If I'm teaching freshman calculus and I talk about $\{x|x>0\}$, ...

5

This is probably too abstract for students just learning about integration by substitution. But convolution is a "real-world application of integration," finding applications in such things as image processing. One defines $$(f*g)(t)=\int_{-\infty}^{\infty} f(\tau)g(t-\tau)\, d\tau$$ and then to show $f*g=g*f$ you use integration by substitution.

4

I don't see any good way to talk about this without explicitly addressing the issue of domains and the fact that the same square root symbol is used for two different functions depending on context. In the first context, the square root symbol refers to the single-valued, nonnegative real square root. In the second context, the square root symbol refers to ...

4

I presume you're not asking for research references, because Googling "procedural fluency vs conceptual understanding" will get you to a page of Google Scholar before you could finish typing out that entire search. I suppose I would lean into how effective pure procedural fluency has been in the past. I'm in my early 50's, and I was taught to do long ...

3

(This entire post is speculation.) I think your teachers think that they are teaching conceptually already and that the students already have a deep understanding. You need to burst that bubble not with logic or reasoning, not with evidence or statistics, but with a sharp and emotional surprise. One way to do that: Take out a video recorder and do some ...

3

This question is mixed with a lot of speculation and commentary, so I'm going to try to focus on the questions. Should the limits of one system of set theory be the limits of a student's mathematical world? No. The idea that a system of set theory should be a limit to mathematical ideas seems like poor pedagogy (as arbitrary restrictions usually are), ...

3

I would consider to add two items to the list, both from a systems slant: Predator prey relations. The behavior can be graphically investigated, but the actual solution function is not analytically soluble. Any solid ODE book will cover this. E.g. Speigel's https://www.amazon.com/Applied-Differential-Equations-Murray-Spiegel/dp/0130400971#...

2

The common way to introduce how trig functions are calculated numerically is via Taylor/Maclaurin series. These are used extensively in a lot of engineering and physics based applications. However, this requires a knowledge of calculus, and isn't very useful in the context of basic trigonometry. It also isn't useful for giving students a "feel" of how these ...

2

Indefinite integrals are easy to avoid, it only takes a will to do so. Basic antiderivatives: $$x^2 =\frac{d}{dx}\frac{x^3}{3},\quad \sin x =\frac{d}{dx}(-\cos x),\quad \frac{1}{x} =\frac{d}{dx}\ln|x|.$$ Antidifferentiation by change of variable: $$e^x\sin(e^x) =\sin(e^x)\frac{d}{dx}e^x =\frac{d}{dx}\sin(e^x),$$ Antidifferentiation by parts: $$x e^x =... 1 You could look at separable differential equations: Given a differential equation of the form$$g(y)y'(x) = f(x),$$integrating between x_0 and x on both side and integration by substitution with u = y(x) gives$$\int_{x_0}^xg(y)y'(x) dx = \int_{x_0}^xf(x)dx \int_{y(x_0)}^{y(x)}g(u) du = \int_{x_0}^xf(x)dx, which allows to solve the ...

1

If they are going over the same material, cross-help by students is valuable: it shores up confidence, and can lead to useful discussion on approaches that do (or not) work.

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