# College Math Teaching

## September 20, 2013

### Only a narrow view of the students on a campus

Filed under: basic algebra, editorial — Tags: — collegemathteaching @ 4:52 pm

My recent experiences on teaching college mathematics has shielded me from a significant segment of the student population. While I have taught across the curriculum, mostly I’ve taught courses designed for science and engineering majors.

Today, I sat in a so-called “college algebra” course (remedial) to evaluate a new faculty member. This faculty member was getting excellent class participation; I was favorably impressed.

He was teaching them how to graph a polynomial that has been factored; example: $p(x) = (x+1)(x-1)(x-2)$. He wanted them too see if the graph of the polynomial was above or below the $x$ axis; in particular he was interested in the graph of $p$ between $x = 1$ and $x = 2$. So he chose the test point $x = \frac{3}{2}$ and asked the question “is $\frac{3}{2} - 1$ positive or negative?

Many sang out “positive”; a few said “negative” (seriously) but……one student said “$\frac{3}{2} - 1$ can be positive or negative.”

I started to laugh out loud but had to work to stifle it.

Later, when talking to this faculty member, I asked if the person who said “it can be either positive or negative” was making a joke. The faculty member looked down and said “uh….no.”.

So, what is going through the mind of a student who says such a thing? I don’t know for sure, but I think that it might be something like this:

At Cal, he was among the hardest workers in the dorm, but he could barely keep afloat.

Seeking help, he went at least once a week to the office of his writing instructor, Verda Delp.

The more she saw him, the more she worried. His writing often didn’t make sense. He struggled to comprehend the readings for her class and think critically about the text.

“It took awhile for him to understand there was a problem,” Delp said. “He could not believe that he needed more skills. He would revise his papers and each time he would turn his work back in having complicated it. The paper would be full of words he thought were academic, writing the way he thought a college student should write, using big words he didn’t have command of.

Sometimes students are so lost, they don’t realize that they are lost; they don’t understand that the material WOULD be clear to them if they understood it. They don’t see that there IS something to understand; it is almost as if the responses that they hear the other students given are random phrases with certain key words and key phrases in them. That these key words and key phrases actually have meaning is lost on them.

As far as whether these students should even be admitted to begin with is beyond the scope of this blog; personally, I am a fan of the “prep school” approach that the service academies use (a year to address a student’s academic deficiencies prior to being admitted to the main campus) but I haven’t studied the data there.

The issue to me: what does one do with these students? Some MIGHT reach the point where they realize that there is a point to it all, but many won’t.

## September 19, 2013

### What we mean about poor algebra skills…

Filed under: basic algebra, calculus, student learning — collegemathteaching @ 4:47 pm

Yes, mathematics professors have been complaining about their students lack of algebra skills as long as there have been calculus courses.

No, we aren’t talking about a student who, in a moment of panic, decided to write $\int \sqrt{x^2+1} dx = \int \sqrt{x^2} + \sqrt{1} dx$ because they were stuck on an exam. And yes, I once saw a professor walk into an analysis class, write $\sqrt{x^2+1} dx \ne \sqrt{x^2} + \sqrt{1}$ on the board (while grinding the chalk into the board) while saying “the next person who makes this mistake will get an F for this class, ON THE SPOT! 🙂

But the weakness is more of the following: in class today, I wrote
$\int (sec^2(x) - 1)tan(x) dx = \int (sec^2(x)tan(x) -tan(x)) dx$ $= \frac{1}{2}tan^2(x) - ln(|sec(x)|+C$

The student actually understood the integration, but didn’t understand where the first equality came from! I said “it is just algebra” and he STILL didn’t get it.

I have a hard time believing that this student doesn’t understand the distributive axiom of algebra; what I think is going on is that they don’t have the concept as a regular working part of their math/science/engineering mind.

## March 29, 2013

### The Quadratic Formula: case study in misunderstanding its meaning (and a moral)

Filed under: basic algebra, editorial, elementary mathematics, mathematical ability — collegemathteaching @ 8:29 pm

I admit that I never dreamed that something as innocent as this picture (a friend tagged me on Facebook) would lead to a sort-of heated argument.

Of course this is the famous quadratic formula; it gives the roots to the following equation: $ax^2+bx+c = 0$ with $a, b, c$ complex numbers and $\sqrt{w}$ interpreted as the principle solution to $(\sqrt{w})^2 = w$. In fact this works in any field in which the square root is defined.
This formula is just a trivial consequence of completing the square: assume that $a \ne 0$ then
$a (x^2 + \frac{b}{a} +\frac{c}{a}) = 0$ which implies $a (x^2 + \frac{b}{a} + \frac{b^2}{4a^2} +\frac{c}{a}-\frac{b^2}{4a^2}) = 0$ which implies $(x+\frac{b}{2a})^2 = \frac{b^2}{4a^2}-\frac{c}{a}$ which implies $x + \frac{b}{2a} = \pm \sqrt{\frac{b^2}{4a^2}-\frac{c}{a}}$ which implies $x = -\frac{b}{2a} \pm \sqrt{\frac{b^2 - 4ac}{4a^2}}$ which is the formula.

But one of my friend’s “Facebook Friends” said:

I’ve never actually used the quadratic equation, i just relied on algebraic reasoning

That is a curious statement; my friend remarked that the quadratic formula WAS “algebraic reasoning.” I was curious as to what the comment meant so I posted

Ok, solve 2x^2 + 13x – 17 = 0 without using the quadratic formula OR completing the square (the two are actually the same thing).”

Then came the following response:

when you say 2x^2, do you mean (2x)^2? if so then it is one. it really is easy. you are talking to someone who took ap classes from an MIT grad without using a calculator. and jason, the quadratic formula is an example of algebraic reasoning, which is to say that there are other methods. I was not trying to imply that the quadratic formula is different from algebraic reasoning. math is the language of logic, so i usually relied on my own means to find the answer, although my means can be convoluted at times.

Evidently this individual didn’t understand the significance of my response. This is evident later:

“You really should reserve that for yourself. If it is a quadratic equation, it will almost always have more than one answer, which is outlined by the +/- part of the quadratic equation. Is 1 a possible answer? yes. Is it the only answer? no. I am referring to the the equation i provided btw. Did I assert that 1 was the ONLY answer? no.”
[…]
“I know, that is why i said it wasnt the one I was referring to the whole time. Why else would I ask to clarify? I already know that if I were to go with 2x^2, it would produce an answer with a decimal. That is because if you follow the quadratic equation you will notice that the number in the square root is 305, and the root of the 305 is pretty messy.”

See where the confusion is? Evidently he (yes, the friend and his Facebook friend is a male) did not understand that, while the quadratic formula (or the completing the square process) yields ALL possible solutions for every true quadratic ($a \ne 0$) that in no way means that one can’t, at times, guess a solution or, at times, find an easy factorization. So if you want to solve the general quadratic and find all solutions, you need this formula or the completing the square process.

Of course, in the complex coefficients case, the answer is frequently ugly.

Side notes: there is a formula for the solution to a cubic (very messy) and for the degree 4 polynomial. However, it is impossible to find a general formula to solve the degree 5 polynomial; this is a reason to learn some Galois Theory from abstract algebra!

The other fields: in general the quadratic cannot be solved if the field is, say, an integer of odd prime order, unless one extends the field by adjoining $\sqrt{p-1}$ where $p$ is the prime in question. This is a good reason to learn some number theory.

Moral
Often, students will put the time and effort into understanding a concept if they know WHY it is important. However, they don’t always appreciate what a formula like the quadratic does. One doesn’t always have to use it, but it
1. Provides a method of obtaining ALL solutions that is guaranteed to work in every case (where $a \ne 0$)
2. Proves that, in fact, the solutions always exist and what kind they are (real or complex).

These points are not obvious to every beginner, even some who consider themselves to be “bright” and talented. Such self perceptions are the topic of a different post.

## February 10, 2013

### Just for fun: no professors allowed!

Filed under: basic algebra, calculus, Fourier Series, media — collegemathteaching @ 2:32 pm

Ok, here is the quiz:

Aluminum medal:

What are these formulas?

Bronze medal:

Derive one of these formulas

Silver medal:

Derive two of these formulas (hint: one way to derive one of these involves a change of variables and polar coordinates.

Gold medal:

Assuming you have a piecewise continuous function (ok, make it piecewise smooth if you wish) and is periodic over $[-l,l]$ derive the middle formula.

You win: all of the money I made writing this note. 😉

## February 8, 2013

### Issues in the News…

First of all, I’d like to make it clear that I am unqualified to talk about teaching mathematics at the junior high and high school level. I am qualified to make comments on what sorts of skills the students bring with them to college.

But I am interested in issues affecting mathematics education and so will mention a couple of them.

1. California is moving away from having all 8’th graders take “algebra 1”. Note: I was in 8’th grade from 1972-1973. Our school was undergoing an experiment to see if 8’th graders could learn algebra 1. Being new to the school, I was put into the regular math class, but was quickly switched into the lone section of algebra 1. The point: it wasn’t considered “standard for everyone.”

My “off the cuff” remarks: I know that students mature at different rates and wonder if most are ready for the challenge by the 8’th grade. I also wonder about “regression to the mean” effects of having everyone take algebra 1; does that force the teacher to water down the course?

By Drew Appleby

I read Epstein School head Stan Beiner’s guest column on what kids really need to know for college with great interest because one of the main goals of my 40-years as a college professor was to help my students make a successful transition from high school to college.

I taught thousands of freshmen in Introductory Psychology classes and Freshman Learning Communities, and I was constantly amazed by how many of them suffered from a severe case of “culture shock” when they moved from high school to college.

I used one of my assignments to identify these cultural differences by asking my students to create suggestions they would like to give their former high school teachers to help them better prepare their students for college. A content analysis of the results produced the following six suggestion summaries.

The underlying theme in all these suggestions is that my students firmly believed they would have been better prepared for college if their high school teachers had provided them with more opportunities to behave in the responsible ways that are required for success in higher education […]

You can surf to the article to read the suggestions. They are not surprising; they boil down to “be harder on us and hold us accountable.” (duh). But what is more interesting, to me, is some of the comments left by the high school teachers:

“I have tried to hold students accountable, give them an assignment with a due date and expect it turned in. When I gave them failing grades, I was told my teaching was flawed and needed professional development. The idea that the students were the problem is/was anathema to the administration.”

“hahahaha!! Hold the kids responsible and you will get into trouble! I worked at one school where we had to submit a written “game plan” of what WE were going to do to help failing students. Most teachers just passed them…it was easier. See what SGA teacher wrote earlier….that is the reality of most high school teachers.”

“Pressure on taechers from parents and administrators to “cut the kid a break” is intense! Go along to get along. That’s the philosophy of public education in Georgia.”

“It was the same when I was in college during the 80’s. Hindsight makes you wished you would have pushed yourself harder. Students and parents need to look at themselves for making excuses while in high school. One thing you forget. College is a choice, high school is not. the College mindset is do what is asked or find yourself another career path. High school, do it or not, there is a seat in the class for you tomorrow. It is harder to commit to anything, student or adult, if the rewards or consequences are superficial. Making you attend school has it advantages for society and it disadvantages.”

My two cents: it appears to me that too many of the high schools are adopting “the customer is always right” attitude with the student and their parents being “the customer”. I think that is the wrong approach. The “customer” is society, as a whole. After all, public schools are funded by everyone’s tax dollars, and not just the tax dollars of those who have kids attending the school. Sometimes, educating the student means telling them things that they don’t want to hear, making them do things that they don’t want to do, and standing up to the helicopter parents. But, who will stand up for the teachers when they do this?

Note: if you google “education then and now” (search for images) you’ll find the above cartoons translated into different languages. Evidently, the US isn’t alone.

Statistics Education
Attaining statistical literacy can be hard work. But this is work that has a large pay off.
Here is an editorial by David Brooks about how statistics can help you “unlearn” the stuff that “you know is true”, but isn’t.

This New England Journal of Medicine article takes a look at well known “factoids” about obesity, and how many of them don’t stand up to statistical scrutiny. (note: the article is behind a paywall, but if you are university faculty, you probably have access to the article via your library.

And of course, there was the 2012 general election. The pundits just “knew” that the election was going to be close; those who were statistically literate knew otherwise.

## July 30, 2012

### Good writing is difficult! So let’s drop writing requirements from the curriculum….

Filed under: basic algebra, editorial, elementary mathematics — collegemathteaching @ 8:02 pm

Ok, no one I know is advocating for this. But when it comes to basic algebra (the “solve for x” $x^2 + 8x + 12 = 0$ kind of algebra), well:

A TYPICAL American school day finds some six million high school students and two million college freshmen struggling with algebra. In both high school and college, all too many students are expected to fail. Why do we subject American students to this ordeal? I’ve found myself moving toward the strong view that we shouldn’t.

Go ahead and read the rest of the article; it is full of the usual “well people aren’t going to use the quadratic formula in real life”, blah, blah, blah.

Fortunately, the reader recommended comments are good.

So, why should algebra remain part of the curriculum? Here is my opinion:

while algebra is a huge human intellectual achievement, it is also a gateway to things like calculus, economics, statistics, chemistry, physical science and other subjects. NOT requiring algebra will intellectually cripple thousands of students right off the bat.

Of course, I disagree with algebra being used as a “capstone” type course in its current form; its current form makes it into a gateway type of course.

One could devise some sort of capstone type algebra course if one wanted to, but it would be different than the preparatory course.

Interestingly enough, when this subject comes up, I usually hear comments of the following type: “well, *I* am very, very smart but I struggled with it, therefore it is unnecessary”….and usually the sole credential for the person’s intellectual ability is, well, their own opinion of said ability. 🙂

## February 16, 2012

### The “equals” sign: identities, equations to be solved and all that…

Here is the sort of thing that got me thinking about this topic: a colleague had a student complain about how one of her quiz problems was scored. The problem stated: “show that $\sqrt{2} + \sqrt{3} \neq \sqrt{5}$“. She was offended that her saying “$\sqrt{x} + \sqrt{y} \neq \sqrt{x+y}$” wasn’t enough to receive credit and would NOT take his word for it. In fact, she took this to the student ombudsman!!!

But that raised the question: “what do we mean when we tell our students “$\sqrt{x} + \sqrt{y} \neq \sqrt{x+y}$“?

Of course, there are some central issues here. The first issues is that our “sure of herself” student thought that “$\sqrt{x} + \sqrt{y} \neq \sqrt{x+y}$” meant that this relation is NEVER true for any choice of $x, y$, which of course, is false (e. g. let $y = 0$ and $x \ge 0$.) In fact, $\sqrt{x} + \sqrt{y} \neq \sqrt{x+y}$ is the logical negation of the statement $\sqrt{x} + \sqrt{y} = \sqrt{x+y}$; the latter means that “this statement is true for ALL $x, y$ and its negation means “there is at least one choice of $x, y$ for which the statement is not true. “Equal” and “not equal” are not symmetric states when it comes identities, which can be thought of as elements in the vector space of functions.

So, $\sqrt{x} + \sqrt{y} \neq \sqrt{x+y}$ means that $\sqrt{x} + \sqrt{y}$ and $\sqrt{x+y}$ are not equal in function space, though they might evaluate to the same number for certain choices in the domain.

So, what is the big deal?

Well, what about equations such as $x^2 + 3x + 2 = 0$ or $y^{\prime \prime} + y = 0$?
These are NOT equalities in the space of functions; the first means “what values in the domain does $f^{-1}(0)$ take given $f(x)=x^2 + 3x + 2$ and the second asks one to find the inverse image of 0 for the operator $D^2+1$ where the domain is the set of all, say, twice differentiable functions.

But, but…would the average undergraduate student understand ANY of this? My experience tells me “no”; hence I intentionally allow for this vagueness and only address it as I need to.

## February 7, 2012

### Forgotten Basic Algebra: or why we shouldn’t rely on the “conjugate trick”

Filed under: basic algebra, calculus, derivatives, elementary mathematics, how to learn calculus, pedagogy — collegemathteaching @ 7:01 pm

I’ll admit that, after 20 years of teaching at the university level, I sometimes get lazy. But…as I age, I must resist that temptation even though at times I find myself muttering “I don’t have 30 extra f*cking minutes to figure out how to do this…”

But often if I stick with it, it doesn’t take 30 “f*cking” minutes. 🙂

Here is an example: I was trying to remember how to calculate $lim_{z \rightarrow w} \frac{z^{1/3} - w^{1/3}}{z - w}$ and was trying to remember instead of think. I looked at an old calculus book…no avail…then I was shamed into thinking. About 2-3 minutes later it struck me:
“you know how to simplify $\frac{u - v}{u^3 - v^3}$ don’t you?”

Problem solved…shame WIN.

of course things like $lim_{z \rightarrow w} \frac{z^{7/8} - w^{7/8}}{z - w}$ are easily converted to things like $\frac{u^7 - v^7}{u^8 - v^8}$, etc.

This leads to another point. Often when we teach $lim_{h \rightarrow 0} \frac{\sqrt{x + h} - \sqrt{x}}{h}$ we use the “conjugate trick” which only works for square roots. The above method works for the other fractional powers.