Analog clocks

What time is it?

From the perspective of an elementary student — even a good student who generally can read the hands of a clock correctly — the answer probably is 2:50. After all, the hour hand is pointing much closer to the 2 than to the 1.

To address this misconception, perhaps the best suggestion that I’ve heard is using the analogy of waiting for your birthday. A child doesn’t turn 7 until his (or her) birthday comes. On the day before his birthday, he still has to say that he’s 6. Likewise, if the minute hand is just a few minutes before 7:00 and the hour hand is not all the way at seven, we still have to say that the time is six-something.

Sorting and interpretative dance

Algorithms for sorting a list, like Bubble Sort, Quick Sort, etc., often require some sort of visual interpretation to make dry algorithms more accessible to students. Here are some YouTube videos that brilliant do this via the medium of folk dancing.

Quick Sort

Merge Sort

Insert Sort

Select Sort

Shell Sort

Bubble Sort

Infraction

While I can’t take credit for this one-liner, I’m more than happy to share it.

A colleague was explaining his expectations for simplifying expressions such as

$\displaystyle \frac{\displaystyle ~~~\frac{2x}{x^2+1}~~~}{\displaystyle ~~~\frac{x}{x^2-1}~~~}$

Of course, this isn’t yet simplified, but his students were balking about doing the required work. So, on the spur of the moment, he laid down a simple rule:

Not simplifying a fraction in a fraction is an infraction.

Utterly brilliant.

Ontologically

Case Study #465 of how not to teach a class.

Me: Ontologically, what kind of object is this?

Student: It might help if I knew what “ontologically” meant.

Math education: an international comparison

A quote from Prof Brian Butterworth, an Emeritus professor from the Centre of Educational Neuroscience at the University College London:

The UK is not very good at maths. We are about average looking at all [Organization for Economic Cooperation] countries. So, we are significantly worse than Canada and Australia and much worse than China and Japan although we are a bit better than Germany and significantly better than the United States.

Credit: Learn maths to boost the economy, scientist advises, The Guardian, November 18, 2010.

Independence

Once in my probability class, a student asked a reasonable question — could I intuitively explain the difference between “uncorrelated” and “independent”? My expert answer: it’s like the difference between “mostly dead” and “all dead.”

Greek letters

One evening, I was watching the Philadelphia Eagles play the Chicago Bears on NBC’s Sunday Night Football telecast. The bottom of the screen showed the score: “PHI 14 CHI 7.” As my wife walked by, she innocently asked, “Why are there Greek letters on the screen?”

We’ve been fans of $\phi$ and $\chi$ ever since.

Student misconceptions about PEMDAS

Simplify $6/2*(1+2)$ .

A Common Incorrect Answer. According to PEMDAS, we should handle the parentheses first. So $6/2*(1+2) = 6/2*3$ . Next, there are no exponents, so we should proceed to multiplication. So $6/2*3 = 6/(2*3) = 6/6$ . Finally, we move to division, and we obtain the answer $6/6 =1$ .

The above answer is incorrect and (even worse) arises from a natural but unfortunate misconception of the way that children are commonly taught order of operations. If you don’t see the misconception, please give it some thought before continuing.

The mnemonic PEMDAS, commonly taught in the United States, stands for

Parentheses

Exponents

Multiplication

Division

Addition

Subtraction

I personally never learned this memorization trick when I was in school. What I do remember, from learning BASIC computer programming around 1980, was the mnemonic My Dear Aunt Sally. I’m told that in the United Kingdom (and perhaps elsewhere in the English-speaking world) schoolchildren are taught BIMDAS, where B stands for Brackets and I stands for Indices.

Unfortunately, all of these memorization devices suffer from a common flaw: they do not indicate that multiplication and divison have equal precedence, and that addition and subtraction have equal precedence. In other words, the order of operations really are

Parentheses

Exponents

Multiplication and Divison (left to right)

Addition and Subtraction (left to right)

Therefore, the correct answer to the above problem is

$6/2*(1+2) = 6/2*3 = (6/2)*3 = 3*3 = 9$ .

In brief, though not intended by teachers, PEMDAS and BIMDAS perhaps promote the misconception that multiplication takes precedence over division and addition takes precedence over subtraction. To avoid this misconception, one of my colleagues suggests that PEMDAS be taught more visually as

P
E
MD
AS

so that students will have a better chance of remembering that MD and AS should have equal precedence.

Fun with Dimensional Analysis

The principle of diminishing return states that as you continue to increase the amount of stress in your training, you get less benefit from the increase. This is why beginning runners make vast improvements in their fitness and elite runners don’t.

J. Daniels, Daniels’ Running Formula (second edition), p. 13

In February 2013, I began a serious (for me) exercise program so that I could start running 5K races. On March 19, I was able to cover 5K for the first time by alternating a minute of jogging with a minute of walking. My time was 36 minutes flat. Three days later, on March 22, my time was 34:38 by jogging a little more and walking a little less. During that March 22 run, I started thinking about how I could quantify this improvement.

On March 19, my rate of speed was

$\displaystyle \frac{5000 \hbox{~m}}{36 \hbox{~min}} = \frac{5000 \hbox{~m}}{36 \hbox{~min}} \times \frac{1 \hbox{~min}}{60 \hbox{~sec}} \approx 2.3148 \hbox{~m/s}$ .

On March 22, my rate of speed was

$\displaystyle \frac{5000 \hbox{~m}}{34 \times 60 + 38 \hbox{~sec}} \approx 2.4062\hbox{~m/s}$ .

That’s a change of $2.4062 - 2.3148 \approx 0.0913 \hbox{~m/s}$ over 3 days (accounting for roundoff error in the last decimal place), and so the average rate of change is

$\displaystyle \frac{0.0913 \hbox{~m/s}}{3 \hbox{~d}} = \frac{0.0913 \hbox{~m/s}}{3 \hbox{~d}} \times \frac{1 \hbox{~d}}{24 \times 60 \times 60 \hbox{~sec}} \approx 3.524 \times 10^{-7} \hbox{~m/s}^2$ .

By way of comparison, imagine a keg of beer floating in space. The specifications of beer kegs vary from country to country, but I’ll use the U.S. convention that the mass is 72.8 kg and its height is 23.3 inches = 59.182 cm. Also, for ease of calculation, let’s assume that the keg of beer is a uniformly dense sphere with radius 59.182/2 = 29.591 cm. Under this assumption, the acceleration due to gravity near the surface of the sphere is the same as the acceleration 29.591 cm away from a point-mass of 72.8 kg. Using Newton’s Second Law and the Law of Universal Gravitation, we can solve for the acceleration:

$ma = \displaystyle \frac{GMm}{r^2}$ ,

where $G \approx 6.67384 \times 10^{-11} \hbox{~m}^3/\hbox{kg} \cdot \hbox{s}^2$ is the gravitational constant, $M = 72.8 \hbox{~kg}$ is the mass of the beer keg, $r = 0.29591 \hbox{~m}$ is the distance of a particle from the center of the beer keg, $m$ is the mass of the particle, and $a$ is the acceleration of the particle. Solving for $a$ , we find

$a = \displaystyle \frac{6.67384 \times 10^{-11} \times 72.8}{0.29591^2} \approx 5.5487 \times 10^{-8} \hbox{~m/s}^2$ .

Since this is only an approximation based on a hypothetical spherical keg of beer, let’s round off and define 1 beerkeg of acceleration to be equal to $5.5 \times 10^{-8} \hbox{~m/s}^2$ .

With this new unit, my improvement in speed from March 19 to March 22 can be quantified as

$\displaystyle 3.524 \times 10^{-7} \hbox{~m/s}^2 \times \frac{1 \hbox{~beerkeg}}{5.5 \times 10^{-8} \hbox{~m/s}^2} \approx 6.35 \hbox{~beerkegs}$ .

I love physics: improvements in physical fitness can be measured in kegs of beer.

I chose the beerkeg as the unit of measurement mostly for comedic effect (I’m personally a teetotaler). If the reader desires to present a non-alcoholic version of this calculation to students, I’m sure that coolers of Gatorade would fit the bill quite nicely.

For what it’s worth, at the time of this writing (June 7), my personal record for a 5K is 26:58, and I’m trying hard to get down to 25 minutes. Alas, my current improvements in fitness have definitely witnessed the law of diminishing return and is probably best measured in millibeerkegs.

Entrance exam at MIT

Here’s a story that I’ll tell my students when, for the first time in a semester, I’m about to use a lemma to make a major step in proving a theorem. (I think I was 13 when I first heard this one, and obviously it’s stuck with me over the years.)

At MIT, there’s a two-part entrance exam to determine who will be the engineers and who will be the mathematicians. For the first part of the exam, students are led one at a time into a kitchen. There’s an empty pot on the floor, a sink, and a stove. The assignment is to boil water. Everyone does exactly the same thing: they fill the pot with water, place it on the stove, and then turn the stove on. Everyone passes.

For the second part of the exam, students are led one at a time again into the kitchen. This time, there’s a pot full of water sitting on the stove. The assignment, once again, is to boil water. Nearly everyone simply turns on the stove. These students are led off to become engineers. The mathematicians are ones who take the pot off the stove, dump the water into the sink, and place the empty pot on the floor… thereby reducing to the original problem, which had already been solved.