user52817
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Expansion of mathematical knowledge does not unfold in Bourbaki progression. This is true at the level of both societal and individual knowledge. Just as the invention and significant applications of ...

Ask your pedantic colleagues to reconcile their formatting expectations with the context "solve $F=m\cdot a$ for $a$". Would they prefer to see just $\frac{F}{m}$, or maybe this expression jazzed up ...

The fact that there is a 3-4-5 triangle that is a right triangle is unique to the Euclidean plane. There is no such triangle in the spherical or hyperbolic planes. Since the Pythagorean theorem is ...

One observation is that (sum of numerators) divided by (sum of denominators) is not well defined. For example, let's work with the two ratios $a=\frac01$ and $b=\frac11$. The ratio of the sum of ...

In my job, I evaluate university math courses for transfer equivalency on a regular basis. In the US, "Calculus 1" typically refers to single variable differential calculus up to the fundamental ...

Learning to think about functions abstractly should be one goal in precalculus, and function symmetry helps. Also suppose we carefully protected a student from knowing anything about function symmetry....

I think an important learning outcome of this part of high school Algebra 2 (theory of polynomials) is developing mathematical capacity to juggle several disparate abstract tools all at once, similar ...

This might be a good opportunity to inject some history into your course. Aristotle thought that cannonball trajectories were line segments, up at an angle then straight down. Look in Chapter 5 of "...

I like to include synthetic division as a topic in a college algebra or precalculus course. It is an opportunity to take a 20 minute digression to talk about Horner's method, which is used in several ...

There are two formulations for definite integrals: $$\int_{\phi(a)}^{\phi(b)} f(x)\, dx=\int_a^b f(\phi(t))\phi'(t)\, dt$$ and the one you state: \int_{\phi([a,b]}f(x)\,dx=\int_{[a,b]} f(\phi(t))...

I highly recommend "Flatland The Movie." Your institution should be able to purchase it. You can find a free trailer on the internet. When I was young, I read the book "Flatland: A Romance of Many ...

In many respects this is what is done in European universities, where the degree is often a three year program. First year students reading maths take analysis. Of course they probably learned ...

One approach is to start with the fact that a 90-degree rotation results from a reflection across the $x$-axis followed by a reflection across the line $y=x$. (In general, reflecting across two ...

I think you are overlooking the fact that proof by contradiction must invoke the tautology $(P\ \hbox{or}\ \neg P)$, called the law of excluded middle. To prove $P\Rightarrow Q$ by contradiction, we ...

Some abstraction is perfect for children. Mitsumasa Anno was a Japanese writer of children's books and he had many innovative approaches to introduction functions. This is from Anno's Math Games II. ...

It might be fun to have your students pretend that the only functions they know are sums of monomials $cx^n$ where $n\in{\bf Z}$, and in particular, play like they know nothing about the function $f(x)... View answer 9 votes As others who answered have pointed out, this issue is not the countable choice involved in defining the sequence that makes this challenging for learners. Rather, the difficulty is the semantic ... View answer 9 votes There is a model of how people progress towards abstract reasoning through the subject of geometry called the Van Hiele model. The model describes five levels: visualization, analysis, abstraction, ... View answer 9 votes The reason why learning mathematical induction is difficult is developmental. Your audience of under-prepared college students probably accept "proof by verifying the first few cases" as more ... View answer 8 votes After teaching Calculus 3 for many semesters, your question certainly resonates. I think it is empowering to coach students in how to sketch surfaces in 3D. Their tendency from curves in 2D is to ... View answer 8 votes Your approach is in fact geometrical, if we see all this happening on the Riemann sphere, i.e., the one-point compactification of${\bf C}={\bf R}^2$. Briefly, we have the usual coordinate$z$on${\...

You can also use Wolfram Alpha to look up definitions of words. So by reductive reasoning, there is no need for infants to learn vocabulary because when they are old enough to use a smartphone, people ...

I think the best "real" application of the mean value theorem for integrals is to make a rigorous proof of the fundamental theorem of calculus.

This is among the oldest questions about learning that has been asked. When a young person started learning geometry with Euclid and asked him why he should learn geometry, Euclid replied, "Give ...

It seems unlikely that the Cardano formula has even been of serious analytic use, i.e., used to approximate roots of a cubic. At least since the inception of calculus, Newton's method can be used to ...

This is a very good question. The issue comes up frequently. I explain this using a toy model: throw two regular six-sided die. What is the probability that the sum is 3? With some physical modeling, ...

One idea that comes to mind is the fact that a connected graph has a Eulerian circuit if and only if every vertex has even degree. Another idea that comes to mind is just the Fundamental Theorem of ...

I think a key tool here is tables of values. My approach has always been to give students functions like $f(x)=x^2$ and $g(x)=f(x-1)$ and ask them to fill out tables of values for some range of $x$, ...