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 $\{x|x>0\}$, ...


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

Can we grant that the students think roots of unity are worthwhile? If so, point out that one of the ways to understand roots of polynomials (like roots of $x^n-1$) is to understand the lowest degree polynomial with those roots. The $n$ roots of unity do not all behave in the same way algebraically, but the primitive $n$th roots of unity (in $\mathbf C$) do. ...


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#...


3

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.


2

On the topic of Sicherman dice: "Cyclotomic polynomials play an important role in analyzing dice labelings of six-sided dice as well as other sizes of dice." http://buzzard.ups.edu/courses/2010spring/projects/jenkins-sicherman-dice-ups-434-2010.pdf


2

I would think you could do this if you are giving the people work and just checking it occasionally. I.e. not holding their hand constantly. This would allow you to timeshare more than one student. Obviously one on one training is the best (for the student) but it is expensive, per capita. This is why it is common that athletic training is done in ...


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

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.


1

I see many good points here, addressing issues going forward. But I have a lower-level reason. I teach point slope form in preference to y=mx+b because it has better flow. When they use y=mx+b, I have seen students plug in a point to get b, and then stop. They forget to give the equation of the line, which is what they were trying to find. When they use ...


1

Surely you teach complex numbers in the curriculum. One thing to do with matrices is to use them to create other mathematical objects, or at least a model of the object. In particular, we can use $2 \times 2$ real matrices of the form $\left[ \begin{array}{cc} x & -y \\ y & x \end{array}\right]$ to model the complex number $x+iy$. Matrix addition ...


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