user52817
  • Member for 7 years, 7 months
  • Last seen this week
How much credit to give a short exam question with one error?
7 votes

I like to grade questions like this on a five-point rubric. The main aspect of this problem is distributing the negative sign into the second group. Presumably students had practice problems similar ...

View answer
Double Integral: Area or Volume?
7 votes

You are integrating a function $z=f(x,y)$ but the units of $z$ do not have to be for length. The units could be for mass density, in which case the units of the double integral would be for mass. Or $...

View answer
Why does the widespread erroneous definition of "irrational number" persist without being taught?
7 votes

The definition of an irrational number as a "number which is not rational" is not without its own difficulties. It presumes that we have a clear definition of a real number. The audience you refer to ...

View answer
How can I motivate the formal definition of continuity?
7 votes

The prototypical way for a function to not be continuous is that of a jump discontinuity. Imagine a jump discontinuity on the order of a few micrometers, like the width of a hair. If you are tracing ...

View answer
Big list of "interesting" abstract vector spaces
6 votes

The spin states of an electron form a two-dimensional vector space over the complex numbers. Designate "spin up" and "spin down" for a basis. The vector space structure is a ...

View answer
Are there any applications of $x^x$?
6 votes

For the expression $x^x$ we could focus on finding occurrences of $x\ln(x)$. One direction is Stirling's approximation $\ln(N!)\sim N\ln(N)$ so $N!$ is like $N^N$. Another direction is that the prime ...

View answer
Rings in parallel with groups in abstract algebra
6 votes

I think this is an interesting question. In the US undergraduate mathematics curriculum, one often finds a sequence of courses "Abstract Algebra I" and "Abstract Algebra II." I ...

View answer
Question about function notation
6 votes

You are not being pedantic. The name of the process that slices is $g$, and the result of slicing $x$ is $g(x)$. On the other hand, the textbook presentation seems to be for students who are just ...

View answer
How to teach sketching a parametric curve?
6 votes

When I introduce parametric curves in Calculus 3, I like to bring my Etch A Sketch to class. It is a drawing toy from about 1960. The knob on the left encodes $x(t)$, and the knob on the right $y(t)$. ...

View answer
Quote to show students don't have to fear making mistakes
6 votes

If you know what you are doing, then you are wasting your time. Anonymous

View answer
Determining the first digit of the Quotient using hand long division efficiently?
Accepted answer
6 votes

The awkwardness of "guessing" in the division algorithm is an artifact of the base-ten representation of numbers. If you represent in binary, then your only possible "guess" is 1. In binary, your ...

View answer
"Function" vs "Function of ...": how much does it contribute to students difficulties?
6 votes

I really appreciate your question. It gets to something I think about often when teaching calculus. You should read about Ed Dubinsky's notion of reflective abstraction, rooted in Piaget. At one ...

View answer
When higher grades are more common than lower grades
6 votes

Although I have serious issues with "outcomes based" education, its philosophy does combat the pernicious notion that grades should have a normal distribution centered on "average." It is not ...

View answer
Is there a simple explanation for calculus classes of why partial fractions work?
6 votes

I have always found it instructive and fun to address this by thinking graphically. Students in Calculus II will know the graphs of the rational functions $f(x)=\frac{ax+b}{(x-c)(x-d)}$ and $g(x)=\...

View answer
Is there a measurable learning goal related to understanding proofs of important theorems?
5 votes

Your quest for measurable learning outcomes suggests a framework rooted in Bloom’s Taxonomy and the higher-education assessment industry that has been built around it. As I think about your question, ...

View answer
What is the English word for the French "repère"?
Accepted answer
5 votes

Elie Cartan wrote a paper in 1935 that was foundational for differential geometry. It is titled La Méthode de Repère Mobile, La Théorie des Groupes Continus, et Les Espaces Généralisés. In English, we ...

View answer
Tension between the most intuitive definition vs. the most common definition of a concept
5 votes

Another example of a crystalized definition is Dedekind's approach to defining finite. First we define a set $S$ be infinite if it is equivalent to a proper subset, i.e., if there is an injection $f:S\...

View answer
Intuition explanation about Lebesgue measure zero of the rational numbers
Accepted answer
5 votes

Let me build on the idea of Steven Gubkin in his comments. One way to visualize this scenario is to use Ford circles. The standard picture is to plot a circle tangent to the $x$-axis at $\frac{p}{q}$ ...

View answer
Are there direct practical applications of differentiating natural logarithms?
5 votes

Boltzmann's equation for entropy is $S=k\ln W$, and the second law of thermodynamics is all about change in entropy. Maybe this is a place to start with your quest for a practical application of the ...

View answer
Undergraduate Vector Calculus Notation Mess
5 votes

I think you are being harsh in your criticism of the classical notation. Of course, at the mathematician's end of the spectrum, the notation you promote towards the end of your question has merit. But ...

View answer
Practical applications of integration by substitution where integrand is unknown
Accepted answer
5 votes

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 ...

View answer
Is this homework problem on counting triangles within a 4x4 grid too vague?
5 votes

Think of the pedagogical goal of the problem as being to generate discussion about assumptions being made, just as you are doing. The goal is not to come up with a pre-determined correct number. ...

View answer
Logarithm Tables - How were the values reached?
5 votes

Gauss said "You have no idea how much poetry there is in a table of logarithms." The first paragraph of this paper might get you pointed in the right direction ON THE DISTRIBUTION OF PRIMES—GAUSS’ ...

View answer
Simple "real world" l'Hôpital's rule problem?
5 votes

I have a colleague whose hobby is to always figure out how to a evaluate any given limit without L'Hospital's rule. Every week in the lunch room, there is yet another example. I think a very good ...

View answer
Hypothesis Tests in Students' Lives
5 votes

Students in the class could google their birth dates before class and come to class knowing the day of the week on which they were born (MTWRF). In class you then test the hypothesis that days of ...

View answer
What is the rationale for the absent (+) in mixed fractions?
5 votes

The university math department where I work runs a math competition for high school students each year. One year there was a problem with answer $7/2$. May students put $3\frac{1}{2}$ down as the ...

View answer
Early vs. late transcendentals
5 votes

Many places have Calculus I as a co-requisite for University Physics I. The physics instructors like students in thier class to be familiar with derivatives of exponential functions before the end of ...

View answer
Is there an agreed upon difference between how we represent $\frac{a}{b}$ and $a \cdot \frac{1}{b}$?
4 votes

Straightedge and compass construction is an interesting way to thing about this as an educator. This perspective would not be something you actually take into an elementary school classroom. Rather it ...

View answer
What can (and should) an educator do about ambiguous terms like "triangle", "square", etc?
4 votes

A triangle is born from three non-collinear points and the axiom that two points determine a line. In the context of neutral geometry, a triangle has no structure other than three lines and three ...

View answer
Example of why proof by exhaustion is inelegant
4 votes

A classical example of proof by exhaustion is to establish an integer $N$ is prime by trial divisions. One uses trial division by candidates for divisors up to $\sqrt{N}$. This can be made slightly ...

View answer