Pressing Toward the Prize

Posts Tagged ‘education

I recently ran across an article that Donald G. Saari (author of Basic Geometry of Voting) wrote in 1996 about that year’s elections, highlighting the ease with which an unwanted outcome can occur. As he explains, when an election result is not what one thinks it should be, that is, the preferred candidate or issue does not win, it is called a voting paradox. This is generally not due to the voters, but rather the voting procedure used. In plurality voting, the process we use in which each voter votes for one candidate and the candidate with the most votes wins, multiple candidates can “split the vote” causing an “inferior” (at least according to the will of the people) candidate to be elected.

Saari demonstrates his point with a very entertaining story about his encounter with a group of precocious 4th graders in 1991. He was attempting to present a counting problem caused by a hypothetical voting example, when the students recognized a flaw in his assessment of the winner in a three-way vote. According to plurality voting, candidate A was preferred to B who was preferred to C in a 6:5:4 vote, when considering only the first choice of each voter. But the students immediately protested, because he did not factor in the “next best” preference, which alters the outcome. In fact, when all rankings were considered, not just the top-ranked candidate, the winner A under plurality voting was actually the least preferred by the majority of the voters, and C was in fact the favorite.

When he asked the students what they thought was the “correct” voting procedure to use, one of the students suggested voters assign 3 points to their favorite candidate, 2 points to the next best, and 1 point to their least desirable candidate. What this student described is the Borda Count method, named for the French mathematician JC Borda, who developed this method in 1770. When the Borda Count method was applied to the voting example, it showed that C was preferred to B who was preferred to A by a 34:29:27 vote – consistent with the students’ earlier assessment. Then Saari presented a version of Marquis de Condorcet’s puzzling example from the 1780’s that shows it is possible to have no winner, because there is a way to count the votes so that every candidate has the same number of votes. The students saw right through this example, as well.

The amazing thing about these students is how quickly they recognized a flaw and were able to suggest reasonable solutions, based solely on their examination of the problem. As Saari points out, they have not yet been conditioned to blindly accept the way things are, but rather used critical thinking skills coupled with their value of fair play. He is concerned that too often our educational system stifles the creativity of our students rather than nurture their inventiveness and innate desire to explore the world around them. He suggests that educators consider changing their classroom approach in order to foster creativity and develop problem-solving skills in their students. As these 4th graders demonstrate, children can accomplish amazing things when given the right environment.

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In Capstone Seminar today we were introduced to the faculty members of the math department at PLU. Each professor gave a brief presentation of his or her areas of expertise, as well as some ideas for interesting Capstone projects. I was impressed by the wide array of mathematical fields represented by the nine professors, as well as the variation among the professors themselves. One works almost exclusively with probability and statistics, and another prefers topology and geometry, having declared statistics “boring.” There was so much information given that it was hard to take it all in, but many wonderful ideas were presented. With so much variety, I cannot imagine any of my fellow students not finding at least one topic that sparked his or her interest.

I am not very spatial in my thinking, so topology and geometry may not be the best for me, although I find them both quite interesting. I struggled a bit with multivariate calculus for that reason: I could do the math with no problem, but I sometimes had trouble “seeing” what was going on. At first blush, there were two topics that caught my attention. One had to do with educational assessments, involving both theory and development, and the other was number theory. I checked out some websites to investigate exactly what number theory entails, and I discovered that it is a very large branch of mathematics. Something that caught my eye, however, is the study of Diophantine equations, or equations that have only integer solutions, of which Fermat’s last theorem is one. I am not sure if either one of these topics will lead to my Capstone project, but they are possibilities to explore.

I was recently discussing with my math professor and a fellow student the relevance, or lack thereof, of current math courses to high school students. When I commented I wasn’t sure if it was the relevance of the courses, or the marketing of those courses, that was the problem, my professor directed me to a video of a talk given by another math professor, Arthur Benjamin. In it he explains why he feels that making calculus the pinnacle of the high school math curriculum is not the best, and that students (and society) would be better served if learning probability and statistics was the zenith. He does not minimize the value of calculus for certain students, but he feels it is not relevant to our everyday lives in the way that probability and statistics are.

After taking my first statistics course, I was so impressed by the relevance of the material that I began to believe, and still do, that everyone in America should know statistics. If followed, Professor Benjamin’s “formula for changing math education” would make that possible. We are so inundated in our culture with facts, figures, and persuasive techniques involving statistics (think sports, politics, lotteries, advertising), that one needs to be able to sift through the rhetoric in order to recognize the reality. If one more fully understands the data being presented, one can make more informed decisions. It would seem, then, that I have had to re-think my position on high school math in favor of changing the current course offerings and emphasis to bring more relevance to our students. And as Professor Benjamin points out, probability and statistics involve uncertainty and risk, which are clearly relevant to our everyday lives.



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  • gramsonjanessa: I can't wait to listen to your capstone presentation in the spring! Your proposal was really interesting and I'm interested to see how the linear alge
  • dewittda: This is impressive! I thought I was good because I solved a rubik‚Äôs cube once in an hour. I served with a guy in the Air Force who could solve a r
  • ZeroSum Ruler: The Euclidean algorithm should me the mainstream way we teach students how to find the GCF. Why isn't it? A mystery.

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