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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.