Punished for True Math

Reading some Mathoverflow the other day, I ran into some truly blood-boiling recollections in a discussion of "Examples of common false beliefs in mathematics" (mostly at the research level, but the comments diverged), such as that "Many students believe that 1 plus the product of the first n primes is always a prime number". Recollections such as these (link):

When I was 11 y.o. I was screamed at by a teacher and thrown out of class for pointing this out when he claimed the false belief stated (it wasn't class material, but the teacher wanted to show he was smart). I found the counterexample later at home. I didn't let the matter drop either... I knew I was right and he was wrong, and really had a major fallout with that math teacher and the school; and flunked math that year. – Daniel Moskovich May 5 2010 at 1:19   

@Daniel: Sorry to hear that. When my daughter Meena was the same age (11), her teacher asserted that 0.999... was not equal to 1. Meena supplied one or two proofs that they were equal, but her teacher would not budge. Maybe this is another example of a common false belief. – Ravi Boppana May 5 2010 at 2:59   

@Daniel: I've heard a worse story. A college instructor claimed in Number Theory class that there are only finitely many primes. When confronted by a student, her reply was: "If you think there are infinitely many, write them all down". She was on tenure track, but need I add, didn't get tenure. – Victor Protsak May 5 2010 at 5:38   

This false belief leads to a proof of the Twin Prime conjecture: For every n, (p1p2⋯pn−1,p1p2⋯pn+1) are twin primes, right? – David Speyer May 6 2010 at 15:50   

Daniel, about the same age, I was asked to leave class for claiming that pi is not 22/7. The math teacher said that 3.14 is an approximation and while some people falsly believe that pi=3.14 but the true answer is 22/7. Years later an Israeli newspaper published a story about a person who can memorize the first 2000 digits of pi and the article contained the first 200 digits. A week later the newspaper published a correction: "Some of our readers pointed out that pi=22/7". Then the "corrected" (periodic) 200 digits were included. Memorizing digits of pi is a whole different matter if pi=22/7. – Gil Kalai May 11 2010 at 5:45

I guess I had the good luck to not ever have such a completely horrible math instructor, because I think any of these cases would have made me completely lose my mind. (As an aside, I will say that it's routine in my classes that I'll have to disabuse people of the idea that pi = 22/7... perhaps this is more common in Israel, as Prof. Kalai above is, and many of my students are from.) Have you ever seen someone punished, yelled at, or thrown out of class for actually expressing true basic math facts?

Parity of Zero

Did you know: As much as half the population doesn't know that zero is an even number? And that this can cause problems in cases like gas-rationing based on license plates' last digit (as happened here in NYC last fall after Hurricane Sandy).

http://en.wikipedia.org/wiki/Parity_of_zero

Village Voice on CUNY

Here's a very nice cover story from NYC's alternative newspaper, the Village Voice, basically on the subject of my math teaching job at a CUNY community college (and more generally, community colleges across the country):

http://www.villagevoice.com/2013-04-03/news/system-failure-the-collapse-of-public-education/

Some highlights:

• Enrollment at CUNY community colleges is up 33% in the past 5 years
• CUNY has seen a 40% drop in per-student funding from the state in the last 20 years.
• 80% of NYC public school grads who enroll in CUNY need remedial-level instruction
• Just 14% of public school grads pass the CUNY algebra placement exam
• Only 20% of remedially-placed students have advanced to a for-credit class 2 years later
• Only 1 in 4 remedially-placed earn any degree after 6 years.

Regarding NYC public high school statistics: "The numbers are 'better'—there are more graduates—and yet, in an endless loop of absurdity, these students get to college only to be told they haven't finished high school."

Regarding NYC's Harry Truman High School: "Truman currently boasts an A grade from the city. Yet only 10 percent of its graduates are able to enter CUNY without remediation."

Regarding a new pre-matriculation START program which takes small classes and gives detailed basic math instruction: "That process sounds an awful lot like what we used to think of as 'teaching.'"

Everyday LCMs

Here's an exercise that I'm planning to use in my remedial arithmetic class in the near future. The question is: For each of the following number ranges, state (i) the LCM (least common multiple), (ii) some everyday examples that use that LCM, and (iii) an explanation of why that number is convenient.

(a) {1, 2, 3}.
LCM is 6. Examples: Six-pack of soda, beer, donuts, etc. Convenient because you can divide them evenly whether you have one, two, or three people.

(b) {1, 2, 3, 4}.
LCM is 12 (a dozen). Examples: 12-pack of beer, dozen eggs, hours on a clock, etc. Convenient because you can divide them evenly among either one, two, three, or four people (or dishes or periods).

(c) {1, 2, 3, 4, 5}.
LCM is 60. (And see next exercise.)

(d) {1, 2, 3, 4, 5, 6}
LCM is also 60. Examples: 60 seconds in a minute, 60 minutes in an hour, 360 degrees in a circle (6×60), etc. Convenient because you can divide them evenly into one, two, three, four, five, or six periods, as desired. (See also: Babylonian numerals.)

Explaining Proportions

A common basic math exercise is to set up and solve a proportion (equivalence of two ratios, i.e., fractions), often in the context of some word problem. The funny thing I recently discovered (updating lecture notes for the spring term) is how there's usually a complete absence of explanation on why you're doing this, or justification for why it makes sense to do so. In fact, I flat-out couldn't find any explanation for the procedure in any of the resources that I have available to me at the moment. Here are some examples:

Writing proportions is a powerful tool for solving problems in almost every field, including business, chemistry, biology, health sciences, and engineering, as well as in daily life. Given a specified ratio (or rate) of two quantities, a proportion can be used to determine an unknown quantity. [Elayn Martin-Gay, Prealgebra & Introductory Algebra, 3rd Edition]

We can use proportions to solve applied problems by expressing a ratio in two ways, as shown below. For example, suppose that it takes 8 gal of gas to drive for 120 mi, and we want to determine how much will be required to drive for 550 mi. If we assume that the car uses gas at the same rate throughout the trip, the ratios are the same, and we can write a proportion. [Marvin Bittinger, Intermediate Algebra, 9th Edition]

Proportions are typically used when you want to solve for an unknown. Let's look back to our car example. In the last section we found we could drive 120 miles on 4 gallons of gas. We want to find out how many miles we could drive on 10 gallons of gas. This information is displayed in the table below. The value we want to determine is represented by an x in the table above. We can find this value by setting up a proportion. This is shown on the right. [Syracuse University Mathematics Tutorial, retrieved 2/13/13 -- the first of several Google searches I looked at]

In each case (and there were numerous others), that is the entirety of the explanation of why you'd want to set things up in a proportion. To my mind, each of them are extremely sketchy. And like my own lecture notes up until recently, they have a tendency to start off with a sample problem first; they say something like, "take this and set up a proportion like so", then go through the solving steps. But I've become highly sensitized to that fact that if I can't start out with a simple explanation as to why the mechanics of a certain procedure make sense (in this case, setting up the equation a certain way), then that's an indication that I don't fully understand what I need to answer questions on the subject, and need to rectify the situation.

Here's how I put it in my brief lecture notes nowadays -- Problems involving a constant rate can be set up as a proportion. For example: If 2 boxes of cereal cost $10, then how much do 6 boxes cost? One way of looking at it is this: The cost of one box is 10/2 = 5 dollars, so the cost of six boxes must be 5∙6 = 30 dollars. But another way of putting it is that, if we turn both of these into divisions, then the result is the same; i.e., 10/2 = 5 and 30/6 = 5. Therefore we could set up the original problem as a proportion, being careful to line up like units, e.g.: 10/2 = x/6 [dollars/boxes] → 60 = 2x [cross-multiply] → 30 = x [divide by 2]. And again we see that the total cost is $30. 

Observations: The "constant rate" here is specifically the price point of $5 per box of cereal, which is a reasonable and common-sense assumption we're making in the solution, that the price-per-box is the same for the two transactions (barring some kind of bulk discount, say). But note that the proportion method is not the only way we could solve this problem, and in fact it has some very notable disadvantages: (a) we don't ever see the actual "constant rate" itself in the calculation ($5 per box in the example above), and (b) in the intermediary step it produces a much larger number than anything that existed in the original problem (the 60).

So for me, I frankly find the proportions method pretty unintuitive, and in my own work I rarely turn to it as a first choice in solving strategy. Particularly if I have a calculator or computer available, then I find it easier to do the divide-first-and-multiply-second method, as given initially above (and this strikes my students as far more understandable, i.e., actually seeing the constant rate price-point). Or alternatively, you could divide the box numbers first (6/2 = 3), and then intuit that the dollar amounts would have to be increased by the same factor, i.e., a product of 10∙3 = 30 in the given example (again, dealing with smaller and mentally-manipulable numbers along the way).

That said, the proportions method does indeed have some specific advantages. Ones I can think of immediately are: (a) it encapsulates the entire problem into a neat, concise, and attractively symmetric piece of equation writing; and (b) if you're working by hand, and there's going to be decimals in the final answer, then the decimal work is minimized and only appears in the very last division step (as opposed to dealing with it twice, in my divide-and-then-multiply method). This latter feature is of course devalued the more that cheap computation devices become ubiquitous, and is similar in that regard to a lot of other methods which trade off a large intermediary value so as to delay working with cranky divisions, fractions, and decimals (for example: the "calculating formula" for standard deviation, etc.). Perhaps, then, the proportions method is already something of a legacy dinosaur in that regard; I know that for my own work, I find more utility in actually seeing the constant rate I'm dealing with itself identified in the middle of the workflow.

Can you think of any other advantages to the proportions method for these types of elementary problems?

Graphing Mistake

You would not believe how often I see this mistake in a basic algebra class:

Follow-Up on Elections

Tomorrow, of course, are the elections here in the USA for President and other elected positions. One month ago I posted "Bungled Election Probability", where I griped about the common test-question gaffe of thinking that the preference ratio among voters will be the same as the probability of winning an election.

An excellent case study on that: Nate Silver's been getting some major attention recently with his very nice "Five Thirty Eight" blog at the New York Times, where he uses sophisticated statistical analyses to track the likely election outcome. There are numerous graphs and charts which nicely highlight the difference between the two measurements (accessed today, Nov-5):