# College Math Teaching

## August 28, 2018

### Commentary: what does it mean to “graduate from college”?

Filed under: editorial — collegemathteaching @ 1:21 am

Recently, an Oregon university touted graduating someone with Down’s syndrome:

Walking across the stage at graduation was more than just a personal accomplishment for Cody Sullivan as he became Oregon’s first student with Down syndrome to complete four years of college.

Sullivan, 22, received his certificate of achievement at the Concordia University graduation ceremony last month, declaring that while assignments and curriculum were modified for his learning abilities, Sullivan completed all the relevant coursework to make him an official college graduate.

It is every interestingly worded: “certificate of achievement” and “assignments and curriculum were modified for his learning abilities”.

This represents a different point of view than I have.

When a teach a course, getting a certain grade in a course requires that the person getting grade to master certain concepts and skills at a certain level. Those requirements are NOT modified for someone’s learning ability. And getting a degree in a certain subject means (or should mean) that one has established a certain competency in that said subject.

But, well, I wonder if we are moving toward a “meeting a certain competency level isn’t relevant” anymore and just giving “you were here and did stuff” certificates.

There was a time when I thought “aptitude matters” but, well?

### Conditional Probability in the news..

Filed under: probability — Tags: , — collegemathteaching @ 1:11 am

I am going to stay in my lane here and not weigh in on a social science issue. But I will comment on this article, which I was alerted to here. This is from the Atlantic article:

When the ACLU report came out in 2017, Dyer told the Fresno Bee the findings of racial disparities were “without merit” but also said that the disproportionate use of force corresponds with high crime populations. At the end of our conversation, Dyer pointed to a printout he brought with him, a list of the department’s “most wanted” people. “We can’t plug in a bunch of white guys,” he said. “You know who’s shooting black people? Black people. It’s black-on-black crime.”

But so-called “black-on-black crime” as an explanation for heightened policing of black communities has been widely debunked. A recent study by the U.S. Department of Justice found that, overwhelmingly, violent crimes are committed by people who are the same race as their victims. “Black-on-black” crime rates, the study found, are comparable to “white-on-white” crime rates.

So, just what did that “recent study” find? I put a link to it, but basically, it said that most white crime victims were the victim of a white criminal and that most black victims were the victim of a black criminal. THAT is their “debunking”. That is a conditional probability: GIVEN that you were a crime victim to begin with, then the perpetrator was probably of the same race.

That says nothing about how likely a white or a black person was to be a crime victim to being with. From the blog post critiquing the Atlantic article:

What the rest of us mean by “black-on-black crime rate” is the overall rate at which blacks victimize others or the rate at which they are victimized themselves––which, for homicide, has ranged from 6 to 8 times higher than for whites in recent decades. Homicide is the leading cause of death for black boys/men aged 15-19, 20-24, and 25-34, according to the CDC. That fact cannot be said about any other ethnicity/age combination. Blacks only make up 14% of the population. But about half of the murdered bodies that turn up in this country are black bodies (to use a phrase in vogue on the identitarian Left), year in and year out.

In short, blacks are far more often to be the crime victim too. Even the study that the Atlantic article linked to shows this.

Anyhow, that is a nice example of conditional probability.

## August 27, 2018

### On teaching limits poorly

Filed under: calculus, pedagogy — Tags: — collegemathteaching @ 4:52 pm

I will be talking about teaching limits in a first year calculus class.

The textbook our department is using does the typical:

It APPEARS to be making the claim that the limit of the given function is 4 as $x$ approaches 2 because, well, 4 is between $f(2.001)$ and $f(1.999)$. But, there are an uncountable number of numbers between those two values; one really needs that the function in question “preserves integers” in order to give a good reason to “guess” that the limit is indeed 4.

I think that the important thing here is that the range is being squeezed as the domain gets squeezed, and, in my honest opinion, THAT is the point of limits: the limit exists when one can tighten the range tolerance by sufficiently tightening the domain tolerance.

But, in general, it is impossible to guess the limit without extra information about the function (e. g. maps integers to integers, etc.)

## August 20, 2018

### Algebra for Calculus I: equations and inequalities

Filed under: basic algebra, calculus, pedagogy — collegemathteaching @ 9:24 pm

It seems simple enough: solve $3x+ 4 = 7$ or $\frac{2}{x-5} \leq 3$.

So what do we tell our students to do? We might say things like “with an equation we must do the same thing to both sides of the equation (other than multiply both sides by zero)” and with an inequality, “we have to remember to reverse the inequality if we, say, multiply both sides by a negative number or if we take the reciprocal”.

And, of course, we need to check afterwards to see if we haven’t improperly expanded the solution set.

But what is really going on? A moment’s thought will reveal that what we are doing is applying the appropriate function to both sides of the equation/inequality.

And, depending on what we are doing, we want to ensure that the function that we are applying is one-to-one and taking note if the function is increasing or decreasing in the event we are solving an inequality.

Example: $x + \sqrt{x+2} = 4$ Now the standard way is to subtract $x$ from both sides (which is a one to one function..subtract constant number) which yields $\sqrt{x+2} = 4-x$. Now we might say “square both sides” to obtain $x+2 = 16-8x+x^2 \rightarrow x^2-9x+ 14 = 0 \rightarrow (x-7)(x-2) = 0$ but only $x = 2$ works. But the function that does that, the “squaring function”, is NOT one to one. Think of it this way: if we have $x = y$ and we then square both sides we now have $x^2 = y^2$ which has the original solution $x = y$ and $x = -y$. So in our example, the extraneous solution occurs because $(\sqrt{7+2})^2 = (4-7)^2$ but $\sqrt{7+2} \neq -3$.

If you want to have more fun, try a function that isn’t even close to being one to one; e. g. solve $x + \frac{1}{4} =\frac{1}{2}$ by taking the sine of both sides. 🙂

(yes, I know, NO ONE would want to do that).

As far as inequalities: the idea is to remember that if one applies a one-to-one function on both sides, one should note if the function is increasing or decreasing.

Example: $2 \geq e^{-x} \rightarrow ln(2) \geq -x \rightarrow ln(\frac{1}{2}) \leq x$. We did the switch when the function that we applied ($f(x) = -x$ was decreasing.)

Example: solving $|x+9| \geq 8$ requires that we use the conditional definition for absolute value and reconcile our two answers: $x+ 9 \geq 8$ and $-x-9 \geq 8$ which leads to the union of $x \geq -1$ or $x \leq -17$

The fun starts when the function that we apply is neither decreasing nor increasing. Example: $sin(x) \geq \frac{1}{2}$ Needless to say, the $arcsin(x)$ function, by itself, is inadequate without adjusting for periodicity.

## August 3, 2018

Filed under: academia, editorial, research — collegemathteaching @ 12:52 am

Ok, it is nearing the end of the summer and I feel as if I am nearing the end of a paper that I have been working on for some time. Yes, I am confident that it will get accepted somewhere, though I will submit it to my “first choice” journal when it is ready to go. I have 6 diagrams to draw up, put in, and then to do yet another grammar/spelling/consistent usage check.

Part of this “comes with the territory” of trying to stay active when teaching at a non-research intensive school; one tends to tackle such projects in “modules” and then try to put them together in a seamless fashion.

But that isn’t my rant.

My rant (which might seem strange to younger faculty):

A long time ago, one would work on a paper and write it longhand and ..if you were a professor, have the technical secretary type it up. Or one would just use a word processor of some sort and make up your Greek characters by hand. You’d submit it, and if it were accepted, the publisher would have it typeset.

Now: YOU are expected to do the typesetting and that can be very time consuming. YOU are expected to make camera ready diagrams.

And guess what: you aren’t paid for your article. The editor isn’t paid. The referee(s) isn’t (aren’t) paid. But the journal still charges subscription fees, sometimes outrageously high fees. And these are the standard journals, not the “fly by night” predatory journals.

This is another case where the professor’s workload went up, someone else’s expense went down, and the professor received no extra benefit.

Yes, I know, “cry me a river”, blah, blah, blah. But in this respect, academia HAS changed and not for the better.