From CNN’s article 6 things you need to know about STEM:
The New York Times consistently provides the best coverage of mathematics and science by a traditional news outlet. Today, I’d like to feature their article The Odds, Updated Continually, which gives a nice synopsis of the growth of Bayesian statistics in recent years and how Bayesian statistics differs from the frequentist interpretation of statistics. For example:
Statistics may not sound like the most heroic of pursuits. But if not for statisticians, a Long Island fisherman might have died in the Atlantic Ocean after falling off his boat early one morning last summer.
The man owes his life to a once obscure field known as Bayesian statistics — a set of mathematical rules for using new data to continuously update beliefs or existing knowledge…
The essence of the frequentist technique is to apply probability to data. If you suspect your friend has a weighted coin, for example, and you observe that it came up heads nine times out of 10, a frequentist would calculate the probability of getting such a result with an unweighted coin. The answer (about 1 percent) is not a direct measure of the probability that the coin is weighted; it’s a measure of how improbable the nine-in-10 result is — a piece of information that can be useful in investigating your suspicion.
By contrast, Bayesian calculations go straight for the probability of the hypothesis, factoring in not just the data from the coin-toss experiment but any other relevant information — including whether you’ve previously seen your friend use a weighted coin.
Scientists who have learned Bayesian statistics often marvel that it propels them through a different kind of scientific reasoning than they’d experienced using classical methods.
“Statistics sounds like this dry, technical subject, but it draws on deep philosophical debates about the nature of reality,” said the Princeton University astrophysicist Edwin Turner, who has witnessed a widespread conversion to Bayesian thinking in his field over the last 15 years…
The Coast Guard has been using Bayesian analysis since the 1970s. The approach lends itself well to problems like searches, which involve a single incident and many different kinds of relevant data, said Lawrence Stone, a statistician for Metron, a scientific consulting firm in Reston, Va., that works with the Coast Guard.
At first, all the Coast Guard knew about the fisherman was that he fell off his boat sometime from 9 p.m. on July 24 to 6 the next morning. The sparse information went into a program called Sarops, for Search and Rescue Optimal Planning System. Over the next few hours, searchers added new information — on prevailing currents, places the search helicopters had already flown and some additional clues found by the boat’s captain.
The system couldn’t deduce exactly where Mr. Aldridge was drifting, but with more information, it continued to narrow down the most promising places to search.
Just before turning back to refuel, a searcher in a helicopter spotted a man clinging to two buoys he had tied together. He had been in the water for 12 hours; he was hypothermic and sunburned but alive.
Even in the jaded 21st century, it was considered something of a miracle.
Set is an enjoyable—even addictive—card game that challenges players to identifycertain visual patterns. A mathematically rich game, it provides ample opportunity forstudents and teachers to ponder combinatorial, algebraic, and geometric questions. Partof Set’s appeal is that once the fundamentals of the game are understood, it is nearlyimpossible to resist investigating its structure, whatever one’s background. We con-centrate on the geometry, introducing interesting objects we call planets and comets,which lead to an elegant variation on the game.
In case you need a subtle hint, here’s another version:
This excellent and engaging video describes how sine and cosine functions can be applied to music.
Every so often, I’ll publicize through this blog an interesting article that I’ve found in the mathematics or mathematics education literature that can be freely distributed to the general public. Today, I’d like to highlight Jo Clay Olson , Sandy Cooper & Tom Lougheed (2011) Influences of Teaching Approaches and Class Size on Undergraduate Mathematical Learning, PRIMUS: Problems, Resources, and Issues in Mathematics Undergraduate Studies, 21:8, 732-751, DOI: 10.1080/10511971003699694
Here’s the abstract:
An issue for many mathematics departments is the success rate of precalculus students. In an effort to increase the success rate, this quantitative study investigated how class size and teaching approach influenced student achievement and students’ attitudes towards learning mathematics. Students’ achievement and their attitudes toward learning mathematics were compared across four treatments of a precalculus course. The four treatments were (a) traditional lecture-based structure, (b) traditional lecture-based structure with a reduced class size, (c) instruction that engaged students in problem solving, and (d) instruction that included opportunities for small collaborative groups. The achievement of students engaged in problem-based learning (PBL) was significantly higher than the other treatments. These findings suggest that undergraduates benefit from instruction that encourages reflection on prior knowledge while developing new ideas through problem solving. Surprisingly, students in the PBE treatment did not continue to outperform students in the other treatments in calculus. These findings suggest the need for longitudinal studies that investigate the long-term effect of teaching approach and small class size on student learning and student success in advanced mathematics courses.
The full article can be found here: http://dx.doi.org/10.1080/10511971003699694
This is a story that I like to tell my probability and statistics students when we cover the law of averages.
One of my favorite sports is golf, and one spring afternoon in my senior year I went out to play a round. I was assigned a tee time with two other students (that I didn’t know), and off we went.
Unfortunately, the group in front of us were, as I like to say, getting their money’s worth out of the round. Somebody would be stuck in a sand trap and then blast the ball into the sand trap on the other side of the green. Then he’d go to blast the ball out of that sand trap, and the ball would go back to the original one.
Golf etiquette dictates that slow-playing groups should let faster groups play through. However, this group never offered to let us pass them. And so, hole after hole, we would wait and wait and wait.
On hole #9, a player walking by himself came up from behind us. I’m not sure how that happened — perhaps the foursome that had been immediately behind us was even slower than the foursome in front of us — and he courteously asked if he could play through. I told him that we’d be happy to let him play through, but that the group in front of us hadn’t let us through, and so we were all stuck.
As a compromise, he asked if he could join our group. Naturally, we agreed.
This solo golfer did not introduce himself, but I recognized him because his picture had been in the student newspaper a few weeks earlier. He was Notah Begay III, then a hot-shot freshman on the Stanford men’s golf team. Though I didn’t know it then, he would later become a three-time All-American and, with Tiger Woods as a teammate, would win the NCAA championship. As a professional, he would win on the PGA Tour four times and was a member of the 2000 President’s Cup team.
Of course, all that lay in the future. At the time, all I knew was that I was about to play with someone who was really, really good.
We ended up playing five holes together… numbers 10 through 14. After playing 14, it started to get dark and I decided to call it quits (as the 14th green was fairly close to the course’s entrance).
So Notah tees off on #10. BOOM! I had never been so close to anyone who hit a golf ball so far. The guys I was paired with started talking about which body parts they would willingly sever if only they could hit a tee shot like that.
And I thought to myself, Game on.
I quietly kept score of how I did versus how Notah did. And for five holes, I shot 1-over par, while he shot 2-over par. And for five holes, I beat a guy who would eventually earn over $5 million on the PGA Tour.
How did the 9-handicap amateur beat the future professional? Simple: we only played five holes.
Back then, if I shot 1-over par over a stretch of five holes, I would be pretty pleased with my play, but it wouldn’t be as if I had never done it before. And I’m sure Notah was annoyed that he was 2-over par for those five holes (he chili-dipped a couple of chip shots; I imagine that he was experimenting with a new chipping technique), but even the best golfers in the world will occasionally have a five-hole stretch where they go 2-over par or more.
Of course, a golf course doesn’t have just five holes; it has 18.
My all-time best score for a round of golf was a four-over par 76.; I can count on one hand the number of times that I’ve broken 80. That would be a lousy score for a Division I golfer. So, to beat Notah for a complete round of golf, it would take one of my absolute best days happening simultaneously with one of his worst.
Furthermore, a stroke-play golf tournament is not typically decided in only one round of golf. A typical professional golf tournament, for those who make the cut, lasts four rounds. So, to beat Notah at a real golf tournament, I would have to have my absolute best day four days in a row at the same time that Notah had four of worst days.
That’s simply not going to happen.
So I share this anecdote with my students to illustrate the law of averages. (I also use a spreadsheet simulating flipping a coin thousands of times to make the same point.) If you do something enough times, what ought to happen does happen. However, if instead you do something only a few times, then unexpected results can happen. A 9-handicap golfer can beat a much better player if they only play 5 holes.
To give a more conventional illustration, a gambler can make a few dozen bets at a casino and still come out ahead. However, if the gambler stays at the casino long enough, he is guaranteed to lose money.
In my capstone class for future secondary math teachers, I ask my students to come up with ideas for engaging their students with different topics in the secondary mathematics curriculum. In other words, the point of the assignment was not to devise a full-blown lesson plan on this topic. Instead, I asked my students to think about three different ways of getting their students interested in the topic in the first place.
I plan to share some of the best of these ideas on this blog (after asking my students’ permission, of course).
This student submission comes from my former student Tracy Leeper. Her topic, from Pre-Algebra: finding points on the coordinate plane.
How could you as a teacher create an activity or project that involves your topic?
After introducing the topic to the students, I will inform the students that we will be playing a game on the computer. After pulling up the game on the screen and demonstrating how it works, I will then issue a challenge using the maze game. The challenge will be to see how many mines they can avoid while using the least number of moves. Before class, I will play to get my best score, to show the students what I am looking for, and then I will see who can beat my score. To encourage the students to try their best, I will offer extra credit to anyone who can get through the same number of mines, with fewer moves. Multiple attempts are possible, and I will allow students to turn in their best game by the end of the week. By offering extra credit, it will encourage the students to play the game at home as well as in the classroom. This game will be fun for the students, as well as support the topic of finding points on the coordinate plane. A common struggle is confusing the x and y axis, so by playing the game it will reinforce the proper name for the corresponding axis, and which coordinate goes first in the ordered pair.
How can this topic be used in your students’ future courses in mathematics or science?
Finding points on the coordinate plane is used in a variety of disciplines. Any type of graph used to represent data, with the exception of a pie chart, uses at least one quadrant of the coordinate plane. Typically, it is quadrant 1, since both numbers are positive. The graph is just labeled to reflect the data shown, instead of using x and y. Scientist use graphs to represent data that has been collected from either observation or experimentation, usually labeled as time and the correlating measurement. In math the coordinate plane is used to represent any function, with x as the input and y as the output, as well as helping to graph things that are not functions, such as circles, and other polygons. As well as adding a third dimension, and including a z axis for graphing 3D objects, such as spheres and cubes. The coordinate plane is also used in other disciplines, such as geography, for determining map coordinates.
How has this topic appeared in pop culture (movies, TV, current music, video games, etc.)?
Video games have changed tremendously since the days of Pong. The graphics, storylines, characters, and amount of programming required has become much more intricate. One aspect of the games that appeals to players is the moving background that changes and shifts according to where the character is in the game, and how the camera angle is changed by the player. This enables different scenery and perspectives throughout the game. This is done by using points on a 3D graph, and as the character moves, the reference changes according to their position. The fundamental skill for being able to build the game this way, is to first learn how to plot points on a 2D graph. Since most teenagers like video games, and the graphics involved, this would be a good point to make, so the students could see the connection between the math they are learning, and something they really enjoy doing. This same skill is used for calculating GPS coordinates on our phones and computers.
References:
http://www.shodor.org/interactivate/activities/MazeGame/
