Tuesday, November 1, 2011

Statistics à la Mode

Meg Dillon—Southern Polytechnic State University

The last time I taught introductory probability and statistics, I turned in my grades and asked my department chair to take me off the course permanently. I’d spent some time working on a committee to update the course and we’d modernized it roughly to my taste, so my chair was puzzled. The best I could offer by way of explanation was, “I just hate it.” Then I went to France and taught their version of the same course.

My stint in France lasted three weeks. Essentially, I was substitute teaching and not looking for more than an excuse to be in the country for a while. My students were second-year engineering students, pretty much like my students at home. And like my students, the French students were a few notches below elite. While the similarities between my home university and my French university were comforting, the contrasts in the probability/statistics courses could not have been more jarring.

Anyone who has taught or learned in a U.S. mathematics department recently knows the typical introductory probability and statistics course. It involves an expensive, gassy textbook with lots of color pictures, word problems involving industrial applications, and charts to help students navigate problems. American students purchase the textbook and far too often, the ancillaries the bookstore peddles alongside the text.

At my home university, the chair has some difficulty finding mathematics faculty willing to teach the course. While I can’t speak for my colleagues, to me the course seems oddly estranged from mathematics. There is a section on probability, and we love that: the probability laws, the counting. It’s possible to trick out that section and get a chewier piece of mathematics into the act, but, by and large, the course is a hodgepodge of recipes, motivated by problems involving IQ testing, rhesus monkeys, salamanders, and the like. Regardless of the text, there is almost invariably a peculiar pair of caveats presented as from on high: Never accept the alternative hypothesis, and never say the probability is 0.95 that the mean lies in a 95% confidence interval for the mean. I dreaded teaching it in France.

The French course, though, was a different kettle of fish. No one expects French students to shell out money for books, so the course was based on notes produced by the instructor of record. The notes were spare and lacked attribution. They started with simple examples involving coins, dice, and lifetimes of electronic gadgets, what one would expect. The definition of sample space appeared on page one. (That was fast.) The definitions of sigma-algebra (Gasp! Are they joking?) and probability space (Is this a grad course?) appeared on page two. The course spooled out from there. Yes, it assumed more calculus than we do but mostly in the more interesting problems, and it treated testing and interval estimates in much the same way we do. No one was joking, and this was not a grad course: it was introductory prob/stats, in an unapologetically mathematical setting.

Statistics is possibly the most important course we teach in mathematics: for life and for cultural literacy, a basic understanding of it is essential. The high schools teach it, yet I’ve heard excellent high school math teachers express fear, if not loathing, of the subject.

An introductory probability and statistics course based on mathematics is missing, not just from the math education curricula, but from American soil altogether, as far as I can tell. While we teach these courses from bloated texts that avoid mathematics, we might seize the opportunity to teach a critical life skill—understanding statistics—through an exposition that glorifies its foundation in mathematics.

A big chunk of statistics courses in the United States are taught by non-mathematicians, outside math and statistics departments. By the looks of things, students can often get by on facility with software and a foggy understanding of principles. We still see many of these students in the introductory course, though. Could we do better there? Could we rope these students in with mathematical ideas, and could this happen anytime soon?

I don’t know, but I’m hoping to go back to teach in France next year.

About the author: Meg Dillon is a professor of mathematics at Southern Polytechnic State University in Marietta, Georgia.

Aftermath essays are intended to be editorials and do not necessarily reflect the views of the MAA.

Thursday, September 1, 2011

If You Think You Know It, Try to Teach It

Maggie Cummings—University of Utah

I am involved with a Math for America (MfA) project at the University of Utah that helps individuals with strong backgrounds in mathematics (typically a baccalaureate degree in mathematics) become secondary math teachers in high-needs schools. What has been extraordinary to me in this work is the gap between general mathematical knowledge and mathematical knowledge for teaching (MKT). This disparity has received significant attention in teacher education circles. (See, for example, Ball, Thames, and Phelps’s article, “Content Knowledge for Teaching: What Makes It Special?” in the Journal of Teacher Education 59[5], 2008.) The general theme of research in this area is that there is a difference between “doing” and “teaching” mathematics and that while teacher content knowledge is necessary for pedagogical knowledge and skill, the former does not guarantee the latter.

At the University of Utah, we are trying to develop a conceptual understanding of MKT at the secondary level and a means by which we might measure it. In particular, we are interested in identifying knowledge and skills that secondary mathematics teachers need but that are not necessarily possessed by those with degrees in mathematics. It may seem ridiculous to think that individuals with degrees in mathematics don’t know all the math they need for teaching secondary school students, but here are some concrete examples of where we see a gap:

How can you help a seventh-grade student mentally compute 35% of 80?

Why is the area of a trapezoid ½h(b1+b2)?

Is there a geometric reason that the slopes of perpendicular lines are negative reciprocals of each other?

How might you explain why 5 minus -7 equals 12, or why the product of two negative numbers is positive?

Why is anything to the 0 power equal to 1?

When our MfA fellows begin the program, most can provide the algorithms or rules related to the above topics, but when pressed, they generally are not able to give student-friendly explanations that connect a tangible model to the algorithmic procedure. For example, fellows usually set up a proportion to solve the problem of finding 35% of 80 (x/80 = 35/100), but do not know of any way to make this problem simple enough for students to compute the answer in their head. (One approach is to understand that 35% is three and a half groups of 10% portions. Ten percent of 80 is 8, so three of them would be 24 and another half [4] would give 28.)

Indeed, once they examine the conceptualization, the models are not just intuitive—they actually enhance prior understandings of our fellows. The issue is that (a) these conceptualizations are vital to the work of teaching mathematics and (b) they do not seem to be developed in conjunction with typical preparation in mathematics.

It is not enough for a secondary teacher to say a negative times a negative is a positive—she must also be able to engage students in understanding why this is the case and then how this logic can be applied to other situations. In a similar vein, it is not enough that a teacher knows that a student made a mistake in simplifying an algebraic fraction; he must also be able to identify what the student was thinking in the erroneous simplification process. That way, the teacher has a better chance of helping the student connect his or her understanding of numeric fractions to algebraic fractions.

As we prepare individuals with strong backgrounds in math to become teachers, what we have learned is that advanced content knowledge in mathematics must be deliberately linked to content-specific pedagogical knowledge and skills. If that linkage is not made, advanced content knowledge stays “siloed” in the instructor, where it doesn’t do the instructor or the students much good.

Individuals wishing to teach mathematics at the secondary level need more than a strong background in advanced mathematics; they need a strong foundation in the mathematics they are going to teach. So, while it is essential that secondary math teachers understand abstract algebra, it doesn’t necessarily translate into the ability to teach basic algebra. If the truest test of understanding an idea is being able to teach it to someone else, then even some of the strongest graduating mathematics majors still have much to learn about the foundations of their chosen subject. The more they are willing to learn, the more their future students will be likely to follow suit.

About the Author: Maggie Cummings is an instructor with the Center of Science and Mathematics Education at the University of Utah. Email: margaritacummings@gmail.com

Aftermath essays are intended to be editorials and do not necessarily reflect the views of the MAA.

Friday, April 1, 2011

The Problem with Problem Solving

Andy LiuUniversity of Alberta

There are two stages in solving a problem. The first stage is to convince yourself that you have solved the problem. The second stage is to convince other people that you have solved the problem. The first stage is the creative one and is indicative of how mathematics is actually done. The second stage is more formal and often has little in common with the first stage, but ironically it is how mathematics is usually communicated and taught.

Let us illustrate with a simple geometry problem.


P is any point inside an equilateral triangle ABC. Perpendiculars are dropped from P to BC at D, CA at E, and AB at F. Which has the greater total area: triangles PAF, PBD, and PCE, or triangles PAE, PBF, and PCD?

The symmetry of this problem compels us to jump in with both feet and say, “They are the same!” However, the proposer of the problem may be having fun with us, so let’s test our hypothesis with some special positions for P. Putting P at the center of ABC, then at the midpoint of BC, and then coincident with A, we see that in each case our hypothesis holds true. (See figure 1.) So we are confident that our conclusion is correct.

Figure 1

A good way to solve problems is to make use of special cases. The first two attempts at incorporating the special positions into the general diagram are not particularly fruitful. The shaded regions do not correspond exactly. (See figure 2.)

Figure 2

The third special position—the simplest one—delivers! The shaded regions correspond exactly. Moreover, instead of having P coincide with A, we can have it coincide with B and C as well, yielding the diagram in figure 3.

Figure 3

Since PUV, PWX, and PYZ are all equilateral triangles, PZ and PW are both parallel to BC, so that P lies on ZW. Similarly, P lies on UX and VY. Moreover, PXAY, PZBU, and PVCW are parallelograms, and a diagonal of a parallelogram bisects its area. Everything falls into place!

We are now at the end of the first stage—which is where the trouble begins. If you read the solution to this problem in some problem book, you are likely to get the following:

Construct lines ZW, UX, and VY through P, parallel to BC, CA, and AB respectively. Then PUV, PWX, and PYZ are equilateral triangles while PXAY, PZBU, and PVCW are parallelograms. Denoting the area of a polygon Q by [Q], we have

This is very neat and very impressive. The key step is in the first line—the construction of ZW, UX, and VY. But notice that in our work on the first stage, this construction emerges as practically the final step. The two stages run in essentially opposite directions!

Emphasizing the second stage over the first has significantly negative consequences. Performing an excess of exercises may lead students to look at a problem and say, “This one I can do,” and start their write-up before thinking through their own reasoning. Alternatively, they may say, “I have never seen anything like this before,” and move on for lack of any ideas.

The most difficult thing about solving a problem is getting started. Where do ideas come from? There are many techniques, but no sure-fire method because problem solving is not a science. It is an art. Progress comes only with regular practice and sustained effort on the first stage of the process. Unfortunately, only reading solutions by other people is ineffective because other people tend to write only about the second stage of problem solving.

Perhaps this is to be expected. In the world of research mathematics, only the second stage is required to get a paper published. The paradoxical result of this is that we disseminate new theorems of mathematics in a fashion that offers little insight into how they were created.

Bringing this back to the world of the classroom, the upshot is that more attention needs to be paid to the first stage of problem solving. Perfecting the expository skills is certainly important; however, there will not be anything to write about if students cannot conduct informal explorations. They must learn to make something out of nothing, which is the essence of research.

About the author: Andy Liu teaches at the University of Alberta, from which he obtained a Ph.D. in mathematics and a graduate diploma in elementary education, thus becoming perhaps the only person officially qualified to teach mathematics from kindergarten to graduate school. Email: aliumath@telus.net

Aftermath essays are intended to be editorials and do not necessarily reflect the views of the MAA. To respond, go to Aftermath at www.maa.org/mathhorizons.

Friday, February 11, 2011

The Lure of the Dark Side

Doug Szajda—University of Richmond

In the interest of full disclosure, I must confess that though I was once a mathematician, I have since gone over to the Dark Side—computer science. And like any self-respecting Dark Sider, part of my job is to entice others to follow my path. For an undergraduate math major, this translates sim­ply—if you truly want to experience the power of mathemat­ics, then, while there’s still time, take as many applied mathe­matics, computer science, and statistics courses as you can.

I know, your major doesn’t require you to take any statistics or computer science courses. Sadly, such programs still exist. And I understand that it’s comforting to live in the cocoon of pure math. Theory is clean. It is elegant. Yes, pure mathematics can be beautiful in the same way that great literature, art, and music are beautiful. Real-world math, on the other hand, is messy. Hypotheses are not always clear. Boundary conditions and transition phases complicate analysis. Models have to be carefully balanced between being simple enough to be tractable, yet sufficiently detailed that they accurately model phenomena. Dealing with this can be nasty business. But it’s what is required if you want to really use mathematics.

And there are at least two good reasons why you should explore real-world applications of math. First, you like math, and applied areas are where you’ll get to see some really amazing mathematics. In the corridors of my department (a combined math and computer science department), there are 45 AMS “Mathematical Moments” posters. These fliers, which in some math departments should be considered false advertising, depict problems or research areas where math plays a fundamental role. Topics covered include robotics, speech recognition, cell biology, protein folding, and even crime solving. Of this (admittedly unscientific) sample, only three posters discuss problems that might be worked on by a pure mathematician—and one of these is solving sudoku. On the other hand, the topics mentioned on the other 42 posters are most likely examined by experts in the techniques of applied mathematics, statistics, or computer science.

The mathematical techniques most often mentioned on these posters include statistics, dynamical systems, graph theory, mathematical models, pattern recognition, image analysis, differential and partial differential equations, linear algebra, combinatorics, and optimization. As a mathematics major, you’re not likely to see most of these techniques, even if you pursue a math Ph.D., although ironically, your non-math friends might very well be introduced to the basics of very useful topics in linear programming, graph theory, probability, combinatorics, and game theory in the non-major courses they take to fulfill their math requirements for graduation.

The second reason you should take more applied courses is that you likely have an interest in technology, and you live in a technological society. You use a computer and cell phone, probably own an iPod (if not several), and are surrounded by devices that are controlled by microprocessors. And let’s be honest: you probably couldn’t exist without them. Do you want to graduate without having even a basic understanding of how these work? Moreover, you live in a world in which you are bombarded by statistics. It thus behooves you, as a more technologically inclined citizen, to understand enough statistics to be able to see what statistical results really tell us—and also how they can be used in misleading ways.

In case you are inclined to dismiss the opinions of a Dark Sider, then perhaps you will listen to the Mathematical Association of America Committee on the Undergraduate Program in Mathematics, which recommends in its 2004 curriculum guide that mathematics programs should promote learning that helps students better understand the uses of mathematics. This is a refreshing change from the historical norm where applied mathematics was often viewed as a debasement of the Platonic ideals of pure math, and undergraduate programs were designed for the less than 10 percent of students who might have the desire and talent to continue their studies at the graduate level.

So, if you are fortunate enough to be a part of a program that has opportunities for engaging the applied side of math, you’d do well to take advantage. I can assure you, it’s more fun on the Dark Side.

The money isn’t bad either.

About the author: Doug Szajda is an associate professor of computer science at the University of Richmond. He is currently general chair of the Internet Society Network and Distributed System Security Symposium.

Aftermath essays are intended to be editorials and do not necessarily reflect the views of the MAA.