Friday, December 5, 2014

Dividing vectors is possible

I was strolling through the neighborhood physics department recently, and I came upon an admonition on a bulletin board. Do not divide vectors the posting said, somewhat sternly. If A and B are vectors, than A/B is undefined.

Unfortunately this is not quite correct. To be fair, the context of the posting is the algebra one encounters in undergraduate physics courses. In the vanilla vector spaces in introductory mechanics or in a first course in linear algebra, e.g., vector division is verboten. But math is a mature and a rich field, and many very smart people have been investigating vector spaces and abstract algebras for more than a century. Some humility is always in order when thinking about absolute statements in algebra.

In a vector space over the field of real numbers R, it is possible to define an algebra in which a given vector has a multiplicative inverse. It's called geometric algebra, which is a kind of Clifford algebra. Dividing one vector by another is a resultant quantity called a quaternion. Quaternions were invented by William Rowan Hamilton in 1843. There's an introduction to geometric algebra book one can download for free. There is at least one physics book that uses geometric algebra, so this is not just mathematical abstraction.

The axioms of geometric algebra are quite simple:
[I apologize for the ugly indentation here. Why isn't LaTeX supported by blogger?]

These axioms lead directly to new quantities, imaginary unit vectors that are analogous to the square root of -1 that is introduced when extending the real numbers to complex numbers. So, to summarize, if we're strictly limited to a vector space in R3, then yes, vector division is not allowed. But it is possible to extend your vector space and introduce new quantities that do allow vector division, where dividing by a vector is defined as multiplying by its inverse. For further details, see this page in the book by Hestenes.

Wednesday, July 9, 2014

Password Generation with R

Submitted for your amusement. This script will generate ten passwords (N) of length 25. These are strong passwords. The program draws from numerals 0-9 and upper/lower case letters of the alphabet. Tune the parameters to your liking.

#!/usr/bin/Rscript --no-restore --quiet
N = 10
for (n in 1:N) {
  pw = sample(choices,length,replace=TRUE)

This is a sample output. Don't use these passwords posted here, generate your own with your own copy of R. It only makes sense to use passwords like this if you're using a password manager such as LastPass or 1Password. You are using a password manager, right?


Tuesday, June 10, 2014


Alfred Wegener is one of my heroes. He first postulated the theory of continental drift, now known as plate tectonics. For this he was ridiculed, scorned, and abused during his lifetime, which ended tragically. He never relented, however. The wikipedia article does not give precise dates, but it took until the 1960s before the theory of continental drift was widely accepted. Someone remarked once that all the old scientists who were opposed to the new, correct theory had to die off before it could be accepted. That's how science progresses sometimes, by the death of the intolerant.

Here's the "Gedenktafel" (commemorative or memorial plaque) on the wall of the gymnasium (think high school if you're American) he attended in Berlin. Gotta love "Kontinentalverschiebungstheorie" for continental drift theory. An mp3 of the pronunciation is at

"The polar researcher Prof. Dr. Alfred Wegener b. 1880 in Berlin, d. 1930 in the midst of the Greenland ice sheet. He was a student of the former Cologne Gymnasium, earned his PhD in 1905 from the University of Berlin, and in 1912 laid the groundwork for the modern theory of continental drift."

As one can read in the wikipedia article linked to above, Wegener perished on the ice in Greenland in late 1930. One can only imagine how brutal the weather conditions must have been. His body was buried with care by a 23-yr old name Rasmus Villumsen, who marked the site with a pair of skis that Wegener had used. Villumsen continued on and is presumed to have perished. His body was never found.

Monday, June 9, 2014

Why college tuition has outstripped inflation, in one graph

The American Association of University Professors has published an interesting graph [pdf]. I would embed it here, but that's not easy with blogger, since it's a pdf. The graph shows the percentage change in the number of employees in higher education institutions, by category of employee, from 75-76 to 2011. The graph is figure 1 in their annual salary report.

To get to the point, it's the administrators. The "Full-time Nonfaculty Professional" category has increased by 369% in the last 35 years. During the same time period, faculty positions have increased a measly 23%. There are now 3.7 times as many administrators bloating the payrolls of our universities as there were 30-some years ago. So that's why college tuition has grown by about 1200% in three decades. We're paying twelve times as much for tuition to finance a bunch of bullshit jobs.

It doesn't have to be like this. As Rebecca Schuman has noted in a post, there is one (one!) university in the USA that has been hiring more faculty and getting rid of bloated salaries on the administration payroll. That school is Iowa State University. What is their secret? ISU provides an existence proof that sanity can be restored. This one school is increasing faculty, while others are eliminating majors.

How do we fix this problem? I suppose the state legislators could do it. They approve the university budgets, right? The professors could go on strike. Eventually, this bubble will burst. This is some kind of deep irony, or a very cruel joke on college students, who are graduating in droves with debt that will take decades to pay off and lousy job prospects.

The TL;DR on this topic is Thomas Frank's post at

Thursday, May 1, 2014

Grading Software fooled by BABEL

BABEL, the Basic Automatic B.S. Essay Language generator, is software created by Les Perelman and others [] at MIT. Perelman was a "Director of Undergraduate Writing" at MIT. He has used BABEL to generate nonsense essays and feed them to automated essay grading software. The BABEL output gets high marks with sentences like "Privateness has not been and undoubtedly never will be lauded, precarious, and decent." I think Perelman has a point here. Clever students will no doubt learn to game any such grading system to their benefit. Teachers must question what, if any, benefit an automated essay grading system has.

Back in 2005, Perelman discovered an excellent predictor of score [] on an SAT essay test. It was the length of the essay. No other variable he examined correlated nearly as well with the score. The top scoring essays had many factual errors, too. No matter, according to SAT. The writing quality depends not on the correctness of any facts, according to SAT. Perelman's advice for scoring well is to practice writing fast and make up facts. Perhaps the high scorers can land a job with Fox News?

A paper by Perelman in the Journal of Writing Assessment critiques automatic scoring of essays.

Tuesday, April 22, 2014

Racial, Gender Bias in Mentoring

Heard on NPR Morning Edition today: the depressing results of a study done by Katherine Milkman of the Wharton School of Business and two others. The researchers emailed 6,548 faculty mentors at  258 schools pretending to be students aspiring to earn a PhD. All potential advisors received the same message; only the name of the sender was changed. NPR says the names were all different. The messages said in part "I really admire your work, would you have some time to meet?" Names were purposely chosen to distribute across racial and ethnic identities. All that was measured was how often the profs wrote back agreeing to meet with the students.

Women and minorities were less likely to get responses, relative to caucasian males, and less likely to get positive responses. Caucasian males obtained access 26% more often. Remember, the text of the messages was identical. NPR says the letters were "impeccably written."

Perhaps the most distressing results were that that the gender of the professor had no effect, according to NPR, and that the business professors discriminated the most in favor of white male names like "Brad Anderson."

Evidence Of Racial, Gender Biases Found In Faculty Mentoring [ audio].

Monday, April 21, 2014

Hybrid Pedagogy

Earlier today I stumbled onto hybrid pedagogy, a website for the eponymous open access journal, while reading a post from Rebecca Schuman, a blogger at Slate who covers education. The director, Jesse Stommel, is an assistant professor at UW-Madison. Give their site a whirl if you're into hybrid learning and pedagogy.

Text Set for NSA Mass Surveillance

Text Sets are a tool teachers can use to scaffold instruction for struggling learners. They generally focus on one topic or theme and are a very useful tool for improving comprehension and literacy. Text sets popped into my head when I found this article about the NSA and its mass surveillance. The author does not use the term "text set," but that is what he's proposing to provide context for the journalistic coverage of the NSA following the revelations by Edward Snowden last summer.

Anyone familiar with the Snowden/NSA story has no doubt read an article in which a journalist has compared the NSA to Big Brother in George Orwell's novel "1984." It's a metaphor with legs (is that a meta-metaphor?). Mr. Berlatsky points out that 1984 is not a book that paints the most relevant picture of the current government-sanctioned surveillance of the citizenry. He argues that other works do. He even goes so far as to say that 1984 can "enslave thought." I think he may be referring to the phenomenon of journalists who repeat what other journalists have already tweeted or written, even after a statement has been found to be unsupported by the facts. On The Media is all over this case.

On to the text set, culled from the article. A caveat: this text set might not be appropriate for scaffolding.

to which I would add these nonfiction works:

Viola, Wisconsin!

Sunday, April 13, 2014

Your Inner Fish

It begins in the city of Chicago, with a room full of human cadavers.

That's how the story begins.

We are descended from fish. It's a scientific fact supported by mountains of evidence. Much of the evidence is written in our bodies. I'm talking about evolution, of course. There's a fascinating look at your inner fish hosted on It's a three-part video series hosted by Dr. Neil Shubin. I'm watching it via the PBS app on an apple TV. You can watch it on your computer in the web browser, or with the PBS app on a tablet.

Dr. Shubin discovered tiktaalik rosea on Ellesmere Island. Tiktaalik is a creationist's nightmare. He wrote a book about it, and now he's doing this great series on PBS. I can't wait to see episodes 2 and 3.

added 4/20/2014: Your Inner Fish passes the Bechdel Test. There are female scientists in this documentary, in one of the scenes we have two women who talk to each other about something other than a man. Also, there are some great visualizations, like when Dr. Shubin is standing on some devonian river sediments on Ellesmere Island and we watch as it is transformed into a stream bed teeming with life some 380 million years in the past. I could see how video like this would be great in a classroom teaching evolution or geology. The website for Your Inner Fish even has a classroom guide. This is a great resource for science teaching.

Saturday, April 12, 2014

Gratuitous Food Blogging

The pizza has landed. I make the dough with a 50/50 mix of bread flour and whole wheat (Pillsbury, usually).

Friday, April 4, 2014

Flipped Learning Guidelines

The Flipped Learning Network (FLN) has announced [pdf] a formal definition of the term "flipped learning" on March 12, 2014, Two things immediately catch my attention here.  First, they do not say "flipped classroom." The other attention-getter is who is the FLN? It is interesting that they do not use the more common "flipped classroom" term. They take care to draw a distinction between flipped learning and the flipped classroom. Until I chanced upon this announcement, I was unaware of any controversy or distinction that involved a definition of the concept.

If you are unfamiliar with the flipped classroom concept, head over to youtube and do a search. Their are many videos on this topic.

The FLN bill themselves as a group of experienced flipped learning educators. Their website is attractive and well-organized. Aaron Sams is at the top of their board members page. I believe he is the creator of the flipped learning model of instruction.

The flipped learning definition is published under a Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International License, from the non-profit Creative Commons organization. Whoever the FLN is, they are clearly interested in openly sharing their ideas and standards. This is commendable. They do not want others creating derivative works of their standard, so they want to keep control of it. They also do not want others deriving works and selling them for profit.  Having looked and worked in the education field, I cannot help but notice all the for-profit enterprises (ETS, MetaMetrics, textbook publishers, Pearson Education, Charter Schools, ad infinitum). But, as I scrutinize the FLN website, I notice ads at the bottom of the page. There's Pearson, Sophia Learning, Cisco, Adobe, etc. This non-profit organization has some deep-pocketed corporations backing them.

A tip of the hat to Casting out Nines, a blog that posted on this new definition.

Monday, March 31, 2014

NCAA bracket odds

We're down to the "Elite 8" in the 2014 NCAA tournament. I've been hearing some discussion of the odds of filling a "perfect bracket." It has been discussed at On Point Radio, which I listen to frequently. The program aired on March 26, 2014. USA Today has a youtube video up of a DePaul math professor estimating the odds. He comes up with an estimate that a savvy bracket filler has about 1 in 128 billion odds of filling in a perfect bracket. It's March Madness.

Then I thought to myself, how did the professor get there? USA today does not go into the details. I wonder what the odds are of finding real math in USA today on a given today? Probably slimmer than 1 in 1 million. I'll point out that precision does not matter when we're finding astronomically long odds. I'll call this a ballpark estimate. With probabilities this small, the upshot is that you'll never win. It's worse than playing lotto.

Let's do some math. Sixty-four teams compete in the NCAA tournament. There are six rounds, including the final game. So we have 32+16+8+4+2+1 = 63 games played. In order to fill out a perfect bracket, I have to choose 63 winners correctly. If we were flipping a coin, there are

possible sequences of 63 coin flips. A bracket filler would have odds of about 1 in 9 quintillion of filling out the bracket correctly. There is no hope of winning here.

But we are not flipping a coin. Suppose you're a savvy odds maker. You know which teams are ranked highest in the tournament. You study basketball statistics, etc. I will quantify your savvy thusly: you have a 90% probability of picking a winner of 16 games in the first round. You have a 90% probability of picking a winner of 16 games in the second round. Your probability of picking all the other winners is 50/50, the same as flipping a fair coin. What is your probability of filling in the bracket correctly? You have to correctly choose 63 mutually exclusive events with a probability of .9 for 32 of them, and .5 for the other 31. Since the events are mutually independent, the individual probabilities multiply. That gives us a final probability of

or odds of

This is roughly twice as much probability as the DePaul math professor estimated. So, my hypothetical savvy guesser had roughly twice as much savvy, but he is still hopeless when it comes to filling in the bracket correctly.

Here is some R code to calculate and print the numbers here. We're right on the edge of R's basic precision, where p could be rounded down to zero. If I change the numbers slightly, I would need to use the Rmpfr package (or logarithms) to get a non-zero printout.

> p=c(0.9,0.5)
> games=c(32,31)
> prob=prod(p^games)
> prob
[1] 1.598934e-11
> 1/prob
[1] 62541682939
> format(1/prob,dig=3,sci=T)
[1] "6.25e+10"

Friday, March 28, 2014

Gratuitous Friday Food Blogging

I used to occasionally make an omelet for breakfast, but now I make fritattas instead. This one had bell pepper, mushroom, onion, and mozzarella cheese in it. I bake it in a 400° oven for about 14 min. It was pretty good. De gustibus non est disputandum, of course.

Friday, March 14, 2014

Calculating pi with a random number generator

It's π day (3/14), so let's estimate π, the ratio of circle's diameter to its circumference, with a random number generator. This post is a blatant ripoff of Rhett Allain's post on the occasion of π day 2010. We're doing this just for fun, of course. The method used here is terribly inefficient. No doubt anyone reading this knows that google can give you a slice of pi. It's also easy to find a page with 100,000 digits of pi. Most calculators and calculator apps have a π key, of course. If you can't find a few digits of π, you're not looking very hard. There is a very nice page at Mathworld that is loaded with approximations for π. Check it out.

Ok. Let's get down to estimating the first few digits of π with a random number generator. I'll be using R and the default random number generator in R, which is the "Mersenne twister." It's used in R, python, php, ruby, and many other software systems. It has a period of $2^{19937} -1$; its "seed" is a vector of 626 integers. To be technically accurate, we should refer to it as a pseudorandom number generator (PRNG), since it is an algorithm. It is not truly random since its output is determined by initial conditions.

To calculate π, we'll focus on the first quadrant of the x,y plane within the square (0,0),  (1,0),  (1,1), and (0,1). We'll use our PRNG to generate $N = 2^{17}$ random pairs and then graph them. We'll be using a uniform random number generator, of course. Here is the R code used to generate the graph:

r = function(z) { return(sqrt(z[1]^2 + z[2]^2)) }
x = runif(N) ; y = runif(N)
color = rep(c1,N)
dists = apply(cbind(x,y),1,r)
color[dists > 1] = 1
plot(y ~ x, pch=".", col = color)
ins = length(color[color == c1])
est = 4 * ins / N
cat(sprintf("estimate for pi = %.4f",est))

The apply() function is used here to calculate the distance from each point to the origin. Our interface to the PRNG is runif(), which returns a random uniform deviate. If the point lies within the unit circle, it is colored. If it lies outside, it will be black. When run, the following graph is generated:

Now, since the points were chosen with a uniform distribution, we expect a fraction π/4 of them to be colored, and 1 - π/4 to be black. The program simply counts the number of colored points and estimates π as 4 times this number divided by N. When I ran it just now, the program printed

estimate for pi = 3.1428.

So that's our Monte Carlo estimate for π.

Sunday, March 2, 2014

Hyperreal Numbers and Calculus without Limits

Once upon a time, I stumbled into a hyperreal number field on the internet when searching for something related to calculus. Maybe I was calculating the volume of a Steinmetz Solid? The so-called hyperreals can be used to develop the calculus without limits. The claim is made that, perhaps, the use of "infinitesimals" is less confusing than the concept of the limit for struggling calculus students. In fact, there is a calculus text using this approach that one can download for free from this page. The author, H. Jerome Keisler, is a professor emeritus of mathematics at UW-Madison.

What are the hyperreals? They are an extension R* of the field of real numbers, which is usually denoted by RR is a subset of R*. Without going in to too much detail, R* is obtained from R by adding in infinitesimals and infinitely large numbers. An infinitesimal number is infinitely small. What does this mean? For every real number a, a number ε > 0 is infinitesimal if -a < ε < a. That's it. In contrast, 1/ε is infinitely large (a "hyperinteger"). Conceptually, hyperreals are similar to complex numbers. However, unlike the complex numbers, they are depicted in one linear dimension, not a two-dimensional plane. The nitty gritty is all in Keisler's book. With the concept of the infinitesimal available, we can have the hyperreal quantity x + ε that is infinitely near x. This hyperreal x + ε can then be used to formulate the derivative, which won't be surprising to anyone who has studied calculus.

Here's figure 1.4.3 from Keisler's book. An "infinitesimal microscope" has been used to zoom in on the hyperreal number line:

The infinitesimal microscope is a captivating pedagogical concept. I claim that it's easier to grasp than the concept of a limit. Children learn about microscopes in grade school. Maybe they learn about it nowadays by zooming in on their iPads. In any case, R* is a useful mathematical concept that can be used to formulate the calculus without that ungainly Σ thing. All you need is a little ε guy. :-)  Keisler's book provides more than 800 pages of evidence.

The history of infinitesimals is interesting. The epilogue [pdf] to the aforementioned calculus text lays out the story. Archimedes anticipated both infinitesimals and the ε, δ approach of limits in some of his proofs. In the 17th century, the modern calculus was developed independently by Newton and Leibniz. In his reasoning, Newton used both infinitesimals and the concept of a limit as well as the so-called velocity method. Leibniz used infinitesimals. So there were three competing methods for doing calculus: infinitesimals, limits, and the velocity method. Jumping ahead (see Keisler's epilogue for whole story), the first truly rigorous treatment of calculus was formulated by Karl Weierstrass in the 1870's. I believe it was Weierstrass who introduced the  ε, δ notation that we still use today.

There can be little doubt that the limit approach is firmly entrenched in the current pedagogy. Other than Keisler's book, I am unaware of any other textbook that uses this approach. I never heard it in Altgeld Hall or Van Vleck. Mathematics teachers should be aware of this alternative approach and should consider using it to introduce students to derivatives and integrals.

Thursday, February 20, 2014

Raising the Minimum Wage

President Obama, in his recent State of the Union (SOTU) speech, called on congress to raise the federal minimum wage and to start reducing the enormous inequality that has arisen in this country in the last 30 years. The people at the top of the economic food chain in this country are doing fantastically well right now, in fact they are better off financially than any similar cohort has been in the history of the world. Congressman George Miller (D-CA) and Senator Tom Harkin (D-IA) have introduced the Fair Minimum Wage Act of 2013, which would raise the minimum wage to $10.10/hr.

As Robert Reich has pointed out, this is a no-brainer. Republicans are claiming that raising the minimum wage will cost jobs. This is untrue. As Reich points out in the 2:29 video embedded below, minimum wage jobs are service sector jobs such as flipping burgers or changing the linen as a hotel maid. Business owners can easily absorb the cost of an increased minimum wage for pennies on the dollar, which they would pass on to customers. Furthermore, workers on the bottom rungs of the economic ladder spend everything they earn. The dollars spent on increasing their wages have an immediate stimulus effect.

Economists agree that raising the minimum wage would reduce poverty. Arindrajit Dube, an economist at UMass Amherst, has calculated that raising the federal minimum wage to $10.10/hr could lift 4.6 million people out of poverty. Of course, republicans are currently howling that any raising of the minimum wage will wreck the economy. There is no evidence to support this doomsaying. What we aren't told is that the real value of the minimum wage has been falling now for decades. This is illustrated by the following graph:

The blue points are the federal minimum wage by year in 2009 dollars. The maximum value occurred more than 45 years ago in 1968. The value was $9.86. This was a time of great economic prosperity in the US. The dollar values were computed by dividing wage values by the consumer price index (CPI) for a given year. The CPI was normalized to 2009. The blue curve is a lowess smooth computed in R with a "smoother span" of 0.4. Note that it has leveled off. The red data points are federal minimum wage data before the CPI adjustment. The first data point in 1938 was $0.25 after the minimum wage was instituted by Roosevelt. The 1938 value is the minimum value on either curve.

Looking at the graph above, one notices that the smoothed fit from 1938 – 1968 is quite linear. It's a simple matter to fit a straight line to these data using R and then extrapolate. Here is the result in graphical form. R-squared was 0.85 for the linear fit with lm.
Minimum wage in real dollars with linear fit 1938 – 1968

Where does this take us? If we had stayed on that linear growth curve, the minimum wage in 2013 would have risen to $18.10!

Reich has been hammering on economic inequality for years. He has put his arguments into movie form, Inequality for All. Watch the official trailer or youtube if you dare. Raising the federal minimum wage to $10.10/hr would put it back to where it was back in the 60s before it started falling. It's the right thing to do for the poorest Americans.

For the first graph above, the data and the R code used to generate it.

[Apr 7, 2014. Edited to remove an html display bug]

Wednesday, February 19, 2014

Spreading misinformation with a bladder

Ray-finned fish are know as actinopterygii. They are the dominant class of vertebrates; most fish species are ray-finned fish, if we're counting species. If we are counting individual fish, I suspect the actinopterygii would not be so dominant. This is one of my favorite ray-finned, fish, the smallmouth bass:

Ray-finned fish have an organ known as a swim bladder or an air bladder. Here is an image from wikimedia of an air bladder:

This organ in a ray-finned fish is homologous to the lungs of higher vertebrates such as apes and horses. So, why bother posting about the swim bladder? Because it is an object used to spread misinformation. It showed up in a class I participated in recently. The swim bladder has been used in a teaching method known as the discrepant event. Here is the discrepant event I became aware of:

True or false?
Sink or Swim Scenario
When a largemouth bass (Micropteras salmoides) [sic] takes air into its swim bladder from the gills, the fish rises in the water. When it releases air from the swim bladder, it sinks.
Students will likely answer that this is true; however, it is actually false because the opposite occurs. When air is taken in, a largemouth bass sinks; when it releases air, it rises.
The appropriate equation for this question is: D = M/V
(D = density, M = mass, V = volume)
When the fish takes air into its swim bladder, the fish’s density, or specific gravity, increases to above 1. The air weighs more than the vacuum created when it is released. Since the specific gravity of fresh water is about 1, the fish sinks. Thus, the fish is able to sink, rise, or suspend itself by changing its density.
This is unphysical nonsense. Unfortunately, it is easily found with a google search. The author has abused Archimedes' principle to reach the wrong conclusion. The first alarm bell is set off by the misspelling. More to the point, we have the statement that "air weighs more than the vacuum created when it is released." Really? A bass has an internal vacuum pump? The author is telling us that a bass can increase it corporeal volume and density using the swim bladder, that this bladder does double duty as a vacuum tank. I doubt it. Physically, a swim bladder is like an extensible bag similar to a ballon.  The "air" (usually oxygen) in the swim bladder is created by a fascinating complex of arteries and veins known as the rete mirabile ("wonderful net") using countercurrent ion exchange. This is an excellent example of adaptation.  I believe it also protects fish from "the bends" when they rise from the depths.

So, when the swim bladder expands, gas molecules are brought out of solution in the blood of the fish, where they occupy essentialy zero volume, into the bladder, where they now constitute a volume with a density roughly 1000 times less than the density of water. The density d = M/V of the fish will decrease, since we have increased V with no corresponding change in M. The buoyant force on the fish will increase, causing it to rise. If the fish swims downward, where the external pressure on its body is increased, the rete mirabile can passively absorb gas molecules from the swim bladder back into the bloodstream to reduce the buoyant force.

Whoever wrote the nonsensical item I quoted above is not understanding the physiology of actinopterygii. Physiology fail, dude! I would question whether any living metazoan can create a significant vacuum within its body. Please comment on this if you are aware of an example.

I will close with one fascinating fact: swim bladders can also receive and generate sound. In many fishes, they can produce a sound like a grunt or a bark.

Tuesday, February 18, 2014

Randomness and literacy

The generation of random numbers is too important to be left to chance. Robert Coveyou

I banged my shin the other day. The bruise on my left tibia just below the knee is a painful reminder of this event. A distraction was provided by the phone ringing. I stood up, thinking about answering, and I smacked my leg right into the coffee table. The ringing of a phone is a perfect example of a random event. If it had not rung, would there be a bruise on my shin? Probably not.

Random numbers, randomness, and the generation of random numbers are important topics. Randomness is particularly relevant to current events because of its essential use in modern cryptography, which has been in the news lately with articles about Edward Snowden and the NSA. Embedded within a larger frame of mathematics, science, and current events, these topics can provide plenty of impetus for interesting conversation and mathematical diversion.

More to the point, I intend to discuss literacy in the mathematics and science classroom. What can we do to motivate students to learn mathematics? One technique I have used and will continue to use is critical literacy. The teacher can display, e.g., the text of a newspaper or magazine article that gets the math wrong, that provides an example of innumeracy. This text can then serve as a jumping-off point for a discussion of a proper mathematical analysis.

Another technique I find intriguing is the discrepant event. A discrepant event is a demonstration or a question with a surprising or startling conclusion.  An attention-grabbing event can be used to initiate the process. The discrepancy creates a cognitive springboard and forces the students to think about the subject matter. An example of a discrepant event was provided to me recently. A question was posed about fish bladders, an organ possessed by ray-finned fish such as the largemouth bass, Micropterus salmoides. I intend to expand on this topic in a later post.

A closely related technique is the thought-provoking question. How many years is one billion seconds? How many cells do you have in your body? These questions can provide a nice stimulus for a lesson and get the gears turning in the student's heads. Each of which has hundreds of thousands of hair follicles, of course.

Literacy can be used in the classroom to motivate, to captivate, and to initiate discussions. Mathematics is a complex topic, and motivating young students to learn can be challenging. It behooves us as teachers to have many arrows in our educational quivers.