# Beyond the Chalkboard: The Job of a Math Professor

About 15 years ago, when I was starting my career as an assistant professor, I was asked to write an article for Imagine magazine, which is targeted toward gifted students in grades 7-12, about what it’s like to be a math professor. While I would probably write something slightly different today (since my job responsibilities have shifted toward administration, academic advising, and the preparation of future secondary mathematics teachers), I think much of what I wrote still applies today.

Source: J. Quintanilla, “Beyond the Chalkboard: The Job of a Math Professor,” Imagine, Vol. 5, No. 4, p. 10 (March/April 1998).

One part of my job is deceptively simple to explain: I teach math to college students. Some people think I’ve got the easiest job in the world. I teach only two classes a semester for just six hours a week, and I have a flexible schedule with summers off. How cozy!

This view of my job is, of course, misleading. The hours I spend actually lecturing are only the tip of the iceberg. Delivering lectures that make sense and maintain students’ interest for a full hour takes considerable practice and effort. Meanwhile, I am constantly fine-tuning the curriculum, writing exams, and of course grading homework assignments — tasks which keep me working late on many nights. My most time-consuming project in recent months, however, has been to write eighty–and counting–letters of recommendation for former students.

My work with students outside the classroom includes one-on-one tutoring, guiding student research projects, advising students about possible majors and careers, and sometimes just lending an ear when someone’s had a rough day. As a professor I am a public figure on campus, and my current and former students come to me for counsel on a wide range of issues, many of which are only tangentially related to mathematics. I hope that through my words and counsel I am contributing to my students’ development as people as well as scholars.

In addition to lecturing, writing recommendations and counseling, I also have to produce original research. At my university, the quality of my teaching and my research will be weighted equally when I am evaluated for tenure in five years. The relative importance of teaching and research, however, varies from college to college. In general, small liberal arts colleges tend to emphasize teaching, while major universities want their professors to be primarily researchers.

When I started graduate school, I was introduced to my current field of research: applying ideas from probability theory to study theoretical problems in materials science. I have found that my research evolves over four stages: months of frustration, several days of sheer ecstasy when I’m overflowing with ideas, weeks of double-checking that my ideas actually make sense, and finally months of writing up my results for publication in scientific journals. I purposely work on three or four research projects
simultaneously, hoping that the cycle of each is slightly out of phase with the others. Though I work on my research all year, it gets my undivided attention during the summer when I’m not teaching.

Of the many aspects of this job, teaching is for me the most satisfying. I know that most of my students will not become professional mathematicians, so I incorporate “fun lectures” into the curriculum. These lectures illustrate how the mathematics we’re studying can be applied to fields of science. In my “Hunt for Red October” lecture, for example, I talk about applying trigonometry to linguistics, opera, and submarine detection; in my “Voyager 2” lecture, I describe how conic sections are used in planetary exploration. For my favorite fun lecture, I dress up in knickers, carry my golf clubs into class, and use calculus
to analyze the trajectory of golf balls. These lectures have become quite popular with my students, and I love to watch their eyes
light up when they’re excited about learning new things, such as how mathematics can be applied to real life.

Does this career sound appealing to you? If so, heed these words of warning: To become a successful professor, you have to really, really want this career. I am not a math professor for its financial rewards; friends of mine in industry earn salaries that are triple what I make. I don’t mind, and I’m not envious of them–I get to do what I love for a living, and I’m not starving. But this job isn’t for everyone. There are innumerable distractions and frustrations along the way that will derail aspiring professors who are not entirely focused on the goal.

For example, I always thought that I would be assured a job after graduation. In 1987, the National Science Foundation projected a shortfall of 675,000 scientists and engineers over the coming two decades. I was a high school senior in 1987, so I assumed I would be able to write my own ticket after earning a doctorate.

Time would show that this NSF projection, now derisively labeled “The Myth,” was amazingly inaccurate. There is currently an overproduction of Ph.D.s in mathematics, and the job market for aspiring math professors is tight. The unemployment rate for freshly-minted Ph.D.s in mathematics has hovered around 10% throughout the 1990s. A new Ph.D. can expect a nomadic
life — bouncing all over the country from one postdoctoral appointment to another — before finally landing a tenure-track position.

Faced with such daunting employment prospects, I braced myself for an unstable life in pursuit of my dream of becoming a professor. Even so, I must admit that getting avalanched by more than a hundred rejection letters was extremely disheartening. In the end, though, I was blessed with a tenure-track position straight out of grad school.

Like many jobs, the job of a math professor is frustrating at times and can feel overwhelming. But when it does, I think about the excited, curious students at a recent fun lecture and remind myself: I love this job!

# Engaging students: Solving one-step and two-step inequalities

In my capstone class for future secondary math teachers, I ask my students to come up with ideas for engaging their students with different topics in the secondary mathematics curriculum. In other words, the point of the assignment was not to devise a full-blown lesson plan on this topic. Instead, I asked my students to think about three different ways of getting their students interested in the topic in the first place.

I plan to share some of the best of these ideas on this blog (after asking my students’ permission, of course).

This first student submission comes from my former student Jesse Faltys (who, by the way, was the instigator for me starting this blog in the first place). Her topic: how to engage students when teaching one-step and two-step inequalities.

A. Applications – How could you as a teacher create an activity or project that involves your topic?

1. Index Card Game: Make two sets of cards. The first should consist of different inequalities. The second should consist of the matching graph. Put your students in pairs and distribute both sets of cards.  The students will then practice solving their inequalities and determine which graph illustrates which inequality.
2. Inequality Friends: Distribute index cards with simple inequalities to a handful of your students (four or five different inequalities) and to the rest of the students pass of cards that only contain numbers. Have your students rotate around the room and determine if their numbers and inequalities are compatible or not. If they know that their number belongs with that inequality then the students should become “members” and form a group. Once all the students have formed their groups, they should present to the class how they solved their inequality and why all their numbers are “members” of that group.

Both applications allow for a quick assessment by the teacher.  Having the students initially work in pairs to explore the inequality and its matching graph allows for discover on their own.  While ending the class with a group activity allows the teacher to make individual assessments on each student.

B. Curriculum: How does this topic extend what your students should have learned in previous courses?

In a previous course, students learned to solve one- and two-step linear equations.  The process for solving one-step equality is similar to the process of solving a one-step inequality.  Properties of Inequalities are used to isolate the variable on one side of the inequality.  These properties are listed below.  The students should have knowledge of these from the previous course; therefore not overwhelmed with new rules.

Properties of Inequality

1. When you add or subtract the same number from each side of an inequality, the inequality remains true. (Same as previous knowledge with solving one-step equations)

2. When you multiply or divide each side of an inequality by a positive number, the inequality remains true. (Same as previous knowledge with solving one-step equations)

3. When you multiply or divide each side of an inequality by a negative number, the direction of the inequality symbol must be reversed for the inequality to remain true. (THIS IS DIFFERENT)

There is one obvious difference when working with inequalities and multiply/dividing by a negative number there is a change in the inequality symbol.  By pointing out to the student, that they are using what they already know with just one adjustment to the rules could help ease their mind on a new subject matter.

C. CultureHow has this topic appeared in pop culture?

Amusement Parks – If you have ever been to an amusement park, you are familiar with the height requirements on many of the rides.  The provide chart below shows the rides at Disney that require 35 inches or taller to be able to ride. What rides will you ride?

(Height of Student $\ge$  Height restriction)

 Blizzard Beach Summit Plummet 48″ Magic Kingdom Barnstormer at Goofy’s Wiseacres Farm 35″ Animal Kingdom Primeval Whirl 48″ Blizzard Beach Downhill Double Dipper 48″ DisneyQuest Mighty Ducks Pinball Slam 48″ Typhoon Lagoon Bay Slide 52″ Animal Kingdom Kali River Rapids 38″ DisneyQuest Buzz Lightyear’s AstroBlaster 51″ DisneyQuest Cyberspace Mountain 51″ Epcot Test Track 40″ Epcot Soarin’ 40″ Hollywood Studios Star Tours: The Adventures Continue 40″ Magic Kingdom Space Mountain 44″ Magic Kingdom Stitch’s Great Escape 40″ Typhoon Lagoon Humunga Kowabunga 48″ Animal Kingdom Expedition Everest 44″ Blizzard Beach Cross Country Creek 48″ Epcot Mission Space 44″ Hollywood Studios The Twilight Zone Tower of Terror 40″ Hollywood Studios Rock ‘n’ Roller Coaster Starring Aerosmith 48″ Magic Kingdom Splash Mountain 40″ Magic Kingdom Big Thunder Mountain Railroad 40″ Animal Kingdom Dinosaur 40″ Epcot Wonders of Life / Body Wars 40″ Blizzard Beach Summit Plummet 48″ Magic Kingdom Barnstormer at Goofy’s Wiseacres Farm 35″ Animal Kingdom Primeval Whirl 48″ Blizzard Beach Downhill Double Dipper 48″ DisneyQuest Mighty Ducks Pinball Slam 48″ Typhoon Lagoon Bay Slide 52″

Sports – Zdeno Chara is the tallest person who has ever played in the NHL. He is 206 cm tall and is allowed to use a stick that is longer than the NHL’s maximum allowable length. The official rulebook of the NHL state limits for the equipment players can use.  One of these rules states that no hockey stick can exceed160 cm.  (Hockey stick $\le$ 160 cm) The world’s largest hockey stick and puck are in Duncan, British Columbia. The stick is over 62 m in length and weighs almost 28,000 kg.  Is your equipment legal?

Weather – Every time the news is on our culture references inequalities by the range in the temperature throughout the day.  For example, the most extreme change in temperature in Canada took place in January 1962 in Pincher Creek, Alberta. A warm, dry wind, known as a chinook, raised the temperature from -19 °C to 22 °C in one hour. Represent the temperature during this hour using a double inequality. (-19 < the temperature < 22) What Inequality is today from the weather in 1962?

# Video pranks in class

Dr. Matthew Weathers is an assistant professor of Mathematics and Computer Science at Biola University and also a world-class showman. Here are some of his biggest hits (often for Halloween or April Fools’ Day). Enjoy. (If you’re interested, you can find more at his YouTube page, http://www.youtube.com/user/MDWeathers/videos?sort=p&view=0&flow=grid.

# Geometric magic trick

This is a magic trick that my math teacher taught me when I was about 13 or 14. I’ve found that it’s a big hit when performed for grade-school children.

Magician: Tell me a number between 3 and 10.

Child: (gives a number, call it $x$)

Magician: On a piece of paper, draw a shape with $x$ corners.

Child: (draws a figure; an example for $x=6$ is shown)

Important Note: For this trick to work, the original shape has to be convex… something shaped like an L or M won’t work. Also, I chose a maximum of 10 mostly for ease of drawing and counting (and, for later, calculating).

Magician: Tell me another number between 3 and 10.

Child: (gives a number, call it $y$)

Magician: Now draw that many dots inside of your shape.

Child: (starts drawing $y$ dots inside the figure; an example for $y = 7$While the child does this, the Magician calculates $2y + x - 2$, writes the answer on a piece of paper, and turns the answer face down.

Magician: Now connect the dots with lines until you get all triangles. Just be sure that no two lines cross each other.

Child: (connects the dots until the shape is divided into triangles; an example is shown)

Magician: Now count the number of triangles.

Child: (counts the triangles)

The reason this magic trick works so well is that it’s so counter-intuitive. No matter what convex $x-$gon is drawn, no matter where the $y$ points are located, and no matter how lines are drawn to create triangles, there will always be $2y + x - 2$ triangles. For the example above, $2y+x-2 = 2\times 7 + 6 - 2 = 18$, and there are indeed $18$ triangles in the figure.

Why does this magic trick work? I offer a thought bubble if you’d like to think about it before scrolling down to see the answer.

This trick works by counting the measures of all the angles in two different ways.

Method #1: If there are $T$ triangles created, then the sum of the measures of the angles in each triangle is $180$ degrees. So the sum of the measures of all of the angles must be $180 T$ degrees.

Method #2: The sum of the measures of the angles around each interior point is $360$ degrees. Since there are $y$ interior points, the sum of these angles is $360y$ degrees.

The measures of the remaining angles add up to the sum of the measures of the interior angles of a convex polygon with $x$ sides. So the sum of these measures is $180(x-2)$ degrees.

In other words, it must be the case that

$180T = 360y + 180(x-2)$, or $T = 2y + x - 2$.

# Welch’s formula

When conducting an hypothesis test or computing a confidence interval for the difference $\overline{X}_1 - \overline{X}_2$ of two means, where at least one mean does not arise from a small sample, the Student t distribution must be employed. In particular, the number of degrees of freedom for the Student t distribution must be computed. Many textbooks suggest using Welch’s formula:

$df = \frac{\displaystyle (SE_1^2 + SE_2^2)^2}{\displaystyle \frac{SE_1^4}{n_1-1} + \frac{SE_2^4}{n_2-1}},$

rounded down to the nearest integer. In this formula, $SE_1 = \displaystyle \frac{\sigma_1}{\sqrt{n_1}}$ is the standard error associated with the first average $\overline{X}_1$, where $\sigma_1$ (if known) is the population standard deviation for $X$ and $n_1$ is the number of samples that are averaged to find $\overline{X}_1$. In practice, $\sigma_1$ is not known, and so the bootstrap estimate $\sigma_1 \approx s_1$ is employed.

The terms $SE_2$ and $n_2$ are similarly defined for the average $\overline{X}_2$.

In Welch’s formula, the term $SE_1^2 + SE_2^2$ in the numerator is equal to $\displaystyle \frac{\sigma_1^2}{n_1} + \frac{\sigma_2^2}{n_2}$. This is the square of the standard error $SE_D$ associated with the difference $\overline{X}_1 - \overline{X}_2$, since

$SE_D = \displaystyle \sqrt{\frac{\sigma_1^2}{n_1} + \frac{\sigma_2^2}{n_2}}$.

This leads to the “Pythagorean” relationship

$SE_1^2 + SE_2^2 = SE_D^2$,

which (in my experience) is a reasonable aid to help students remember the formula for $SE_D$.

Naturally, a big problem that students encounter when using Welch’s formula is that the formula is really, really complicated, and it’s easy to make a mistake when entering information into their calculators. (Indeed, it might be that the pre-programmed calculator function simply gives the wrong answer.) Also, since the formula is complicated, students don’t have a lot of psychological reassurance that, when they come out the other end, their answer is actually correct. So, when teaching this topic, I tell my students the following rule of thumb so that they can at least check if their final answer is plausible:

$\min(n_1,n_2)-1 \le df \le n_1 + n_2 -2$.

To my surprise, I have never seen this formula in a statistics textbook, even though it’s quite simple to state and not too difficult to prove using techniques from first-semester calculus.

Let’s rewrite Welch’s formula as

$df = \left( \displaystyle \frac{1}{n_1-1} \left[ \frac{SE_1^2}{SE_1^2 + SE_2^2}\right]^2 + \frac{1}{n_2-1} \left[ \frac{SE_2^2}{SE_1^2 + SE_2^2} \right]^2 \right)^{-1}$

For the sake of simplicity, let $m_1 = n_1 - 1$ and $m_2 = n_2 -1$, so that

$df = \left( \displaystyle \frac{1}{m_1} \left[ \frac{SE_1^2}{SE_1^2 + SE_2^2}\right]^2 + \frac{1}{m_2} \left[ \frac{SE_2^2}{SE_1^2 + SE_2^2} \right]^2 \right)^{-1}$

Now let $x = \displaystyle \frac{SE_1^2}{SE_1^2 + SE_2^2}$. All of these terms are nonnegative (and, in practice, they’re all positive), so that $x \ge 0$. Also, the numerator is no larger than the denominator, so that $x \le 1$. Finally, we notice that

$1-x = 1 - \displaystyle \frac{SE_1^2}{SE_1^2 + SE_2^2} = \frac{SE_2^2}{SE_1^2 + SE_2^2}$.

Using these observations, Welch’s formula reduces to the function

$f(x) = \left( \displaystyle \frac{x^2}{m_1} + \frac{(1-x)^2}{m_2} \right)^{-1}$,

and the central problem is to find the maximum and minimum values of $f(x)$ on the interval $0 \le x \le 1$. Since $f(x)$ is differentiable on $[0,1]$, the absolute extrema can be found by checking the endpoints and the critical point(s).

First, the endpoints. If $x=0$, then $f(0) = \left( \displaystyle \frac{1}{m_2} \right)^{-1} = m_2$. On the other hand, if $x=1$, then $f(1) = \left( \displaystyle \frac{1}{m_1} \right)^{-1} = m_1$.

Next, the critical point(s). These are found by solving the equation $f'(x) = 0$:

$f'(x) = -\left( \displaystyle \frac{x^2}{m_1} + \frac{(1-x)^2}{m_2} \right)^{-2} \left[ \displaystyle \frac{2x}{m_1} - \frac{2(1-x)}{m_2} \right] = 0$

$\displaystyle \frac{2x}{m_1} - \frac{2(1-x)}{m_2} = 0$

$\displaystyle \frac{2x}{m_1} = \frac{2(1-x)}{m_2}$

$xm_2= (1-x)m_1$

$xm_2 = m_1 - xm_1$

$x(m_1 + m_2) = m_1$

$x = \displaystyle \frac{m_1}{m_1 + m_2}$

Plugging back into the original equation, we find the local extremum

$f \left( \displaystyle \frac{m_1}{m_1+m_2} \right) = \left( \displaystyle \frac{1}{m_1} \frac{m_1^2}{(m_1+m_2)^2} + \frac{1}{m_2} \left[1-\frac{m_1}{m_1+m_2}\right]^2 \right)^{-1}$

$f \left( \displaystyle \frac{m_1}{m_1+m_2} \right) = \left( \displaystyle \frac{1}{m_1} \frac{m_1^2}{(m_1+m_2)^2} + \frac{1}{m_2} \left[\frac{m_2}{m_1+m_2}\right]^2 \right)^{-1}$

$f \left( \displaystyle \frac{m_1}{m_1+m_2} \right) = \left( \displaystyle \frac{m_1}{(m_1+m_2)^2} + \frac{m_2}{(m_1+m_2)^2} \right)^{-1}$

$f \left( \displaystyle \frac{m_1}{m_1+m_2} \right) = \left( \displaystyle \frac{m_1+m_2}{(m_1+m_2)^2} \right)^{-1}$

$f \left( \displaystyle \frac{m_1}{m_1+m_2} \right) = \left( \displaystyle \frac{1}{m_1+m_2} \right)^{-1}$

$f \left( \displaystyle \frac{m_1}{m_1+m_2} \right) = m_1+m_2$

Based on the three local extrema that we’ve found, it’s clear that the absolute minimum of $f(x)$ on $[0,1]$ is the smaller of $m_1$ and $m_2$, while the absolute maximum is equal to $m_1 + m_2$.

$\hbox{QED}$

In conclusion, I suggest offering the following guidelines to students to encourage their intuition about the plausibility of their answers:

• If $SE_1$ is much smaller than $SE_2$ (i.e., $x \approx 0$), then $df$ will be close to $m_2 = n_2 - 1$.
• If $SE_1$ is much larger than $SE_2$ (i.e., $x \approx 1$), then $df$ will be close to $m_1 = n_1 - 1$.
• Otherwise, $df$ could be as large as $m_1 + m_2 = n_1 + n_2 - 2$, but no larger.

# Statistical significance

When teaching my Applied Statistics class, I’ll often use the following xkcd comic to reinforce the meaning of statistical significance.

The idea that’s being communicated is that, when performing an hypothesis test, the observed significance level $P$ is the probability that the null hypothesis is correct due to dumb luck as opposed to a real effect (the alternative hypothesis). So if the significance level is really about $0.05$ and the experiment is repeated about 20 times, it wouldn’t be surprising for one of those experiments to falsely reject the null hypothesis.

In practice, statisticians use the Bonferroni correction when performing multiple simultaneous tests to avoid the erroneous conclusion displayed in the comic.

Source: http://www.xkcd.com/882/

# Engaging students in a different discipline

I have no expertise about how to teach any other subject besides mathematics. But this article from the May/June 2013 issue of the Stanford alumni magazine made a lot of sense to me about how to teach history to middle- and high-school students. The basic principle appears to be the same that governs my classes: figure out a way to make students want to come to class each day. A sample quote:

I easily could have told them in one minute that the Dust Bowl was the result of overgrazing and over-farming and World War I overproduction, combined with droughts that had been plaguing that area forever, but they wouldn’t remember it.” By reading these challenging documents and discovering history for themselves, he says, “not only will they remember the content, they’ll develop skills for life.

# Dimensions

As described by the March 2013 issue of the American Mathematical Monthly, the (free!) two-hour movie Dimensions  is “an impressive computer-generated video of almost 2 hours that describes geometry in two, three and four dimensions. The video assumes an elementary geometry background possessed by most viewers and leads up to an interesting geometric structure, the Hopf fibration of the unit sphere in four-dimensional space.”

The website of this project can be found at http://www.dimensions-math.org.

Here’s the 4-minute trailer for the movie:

This full two-hour movie was uploaded to YouTube in several chapters. The full YouTube playlist is given here.

The links to the 9 separate chapters are below.

Indeed.

# What makes my blood boil

A friend of mine forwarded the following image to me on Facebook:

My response was quick and simple: Anybody who’s able to use a computer or a mobile device to use Facebook to connect with their friends has directly benefited from algebra today, whether or not they realize it.