from this South American country.
Sunday, February 28, 2010
If You Are Interested...
The Teachers Teaching with Technology International Conference is this coming weekend in Atlanta. This is one of my favorite conferences...it is relatively small, and I always seem to learn something new about mathematics, the technology (even if handheld technology may be on its last leg), and how to use it.
I am co-presenting with Todd Edwards. You will find information about the topic HERE and HERE. I am also giving one myself. You can find information about that topic HERE.
The gist of my presentation is a unifying theme for transforming functions I found a couple of years ago in Keith Kendig's book Conics. In particular, in Chapter 9, his "Action Reaction Principle" is pretty darned cool...if you search for that phrase below, you will find it.
Saturday, February 27, 2010
My Favorite Math Movie
I could watch this movie every day and never grow tired of it. The low-quality video does not do it justice. Play it with your sound turned up. There is a link to better quality video HERE.
Have You Seen Prezi?
The following is a Prezi presentation given by my good friend Todd Edwards and I when we had the distinct honor of sharing our work with Metropolitan Mathematics Club of Chicago last autumn. By all accounts, I think it was well received!
Prezi is just another one of those free things that should make powerpoint presentations obsolete. Indeed, Prezi has replaced powerpoints in many or the student presentations at my school.
Creating an Applet
This is weird...a movie of me creating the applet for this post. Pat Ballew asked me how to do this.
The ability to do this easily is what attracted me to GeoGebra almost four years ago. I started out using Cabri on the old TI Voyage 200, then moved to Sketchpad because my school had a site license (which meant my students could actually buy the software foe something like $25). One man's opinion, but GeoGebra is just better. There is no site license to purchase (which means my students can use the software for something like FREE). I look at the new features in the current version of GSP, and it is a good program, but they are playing catch-up. GeoGebra has set the bar HIGH with respect to dynamic mathematics software.
The ability to do this easily is what attracted me to GeoGebra almost four years ago. I started out using Cabri on the old TI Voyage 200, then moved to Sketchpad because my school had a site license (which meant my students could actually buy the software foe something like $25). One man's opinion, but GeoGebra is just better. There is no site license to purchase (which means my students can use the software for something like FREE). I look at the new features in the current version of GSP, and it is a good program, but they are playing catch-up. GeoGebra has set the bar HIGH with respect to dynamic mathematics software.
Tuesday, February 16, 2010
Olympic Analytics or What Would My Sled Weigh?
Japanese Luger Aya Yasuda was disqualified from her first Olympics due to her sled being overweight. This error is apparently attributed to "accidentally miscalculating a somewhat complex formula," I was curious just how complicated this formula was.
RULES
Pre-Race Checks (Singles)
At the start of all four singles runs, each sled is weighed, each athlete is weighed, and temperatures of sled runners are compared to an official "control" runner.
- In singles, the sled cannot weigh more than 23kg (50.6 lbs).
- There is no maximum weight for athletes. But men who weigh less than 90kg (198 lbs) may add up to 13kg (28.6 lbs) of additional weight to act as ballast. Women who weigh less than 75kg (165 lbs) are allowed to add up to 10kg (22 lbs) of extra weight. The formulas below determine how much weight may be added:
MEN: ( 198 lbs - body weight ) x .75
WOMEN: ( 165 lbs - body weight) x .75
- Also, officials check the temperature of the steel runners on an athlete's sled to make sure they haven't been heated. (Warm runners are not allowed because the heat would decrease the friction between the runners and the ice, making the sled faster.) A "control" runner, located by the start house and shaded from the sun, is used as a basis for the measurement. The competitor's runners must be within 5 degrees Celsius of the control runner's temperature.
I got it...you can add 75% of the difference in your body weight and a predetermined minimum amount in ballast, up to a predetermined maximum amount.
Yeah...that formula sucks. Thank goodness that formula wasn't something like this formula for a Player Efficiency Rating used by those who study basketball analytics:
uPER = 1/MIN * (3PM + [(2/3)*AST] + [(2 – factor * (tmAST/tmFG)) * FG] + [FT * 0.5 * (1 + (1 – (tmAST/tmFG)) + (2/3)*(tmAST/tmFG))] – [VOP * TO] – [VOP * DRBP * (FGA – FG)] – [VOP * 0.44 * (0.44 + (0.56 * DRBP)) * (FTA – FT) + [VOP * (1 – DRBP) * (TRB – ORB)] + [VOP * DRBP * ORB] + [VOP * STL] + [VOP * DRBP * BLK] – [PF * ((lgFT/lgPF) – 0.44 * (lgFTA/lgPF) * VOP))])
where
factor = (2/3) – [(0.5 * (lgAST/lgFG))/(2 * (lgFG/lgFT))]VOP = lgPTS/(lgFGA – lgORB + lgTO + 0.44 * lgFTA)DRBP = (lgTRB – lgORB)/lgTRB
Once uPER is calculated, it is adjusted for pace and scalar multiplied so that the league average player-minute is a 15 PER:
PER = uPER * (lgPace/tmPace) * (15/lguPER)
Monday, February 15, 2010
Formative Assessing to the Extreme (A Follow-up)
I just wanted to take a moment and give an update to an earlier post regarding my attempts to check everything my students do as a means of formative assessment.
There is good and bad.
The good? I know almost immediately what I need to address with the entire class or, by looking at a student's work on three or four problems, what I need to address with that particular student. This alone far outweighs any of the "bads" that follow.
The bad? Some students still show no work, or do no work, regardless of all the "atta-boys" and "atta-girls" I can give. For some of these students, yeah...I know they can do it. For others? I am not so sure, and I need to KNOW.
The good? I keep a tally of how many things each student showed me they could do each day. Show me you can do it, you get a check. Make a mistake, I go over the problem with you, ask you to rework it, and either bring the problem back up to me, or shout out the answer to me, in order to get your check.
The bad? Some students just want a check next to their name. I do not like this aspect of it. "How many checks do I have?" I guess I can't really expect them to get too darned excited about calculating the volume of a square pyramid when you are given an edge of the base and the slant height. This is one area where I would gladly take some suggestions as to how I might handle this better.
The good? Seems to work great for the kids who always do what you ask, and for about half of the kids you normally can't get to do anything.
The bad? Some kids just won't do anything. And for some of the things I must ask the kids to do, I honestly don't blame them. The only job I can think of where you will need to calculate the volume of a square pyramid when you are given an edge of the base and the slant height is in the math teaching industry, and I doubt any of these kids will go in that direction.
The good? Lately, I am checking anywhere from 2 to 6 things a day, maybe 10 to 15 things during a week. It really is taking less time than I thought.
The bad? I am still messing with the number things I am checking for each day. This past week, I looked for things like, "Can you calculate the volume of ______?" or "Can you calculate the surface area of _____?" How many do you need each day? Do I include some of the same types of problems each day, this way, you are showing me you can do this over a number of different times? I do know...
The good? I find it is easy to check over the "synthesis" types of problems, the problems where they need to put it all together. I am also finding that my students are improving at presenting and organizing their work, which adds to the easier part.
The bad? It takes longer than the other problems to check over, so I am tending to get lines. I am thinking of going to each group for these problems.
Will I keep doing it? Without a doubt. I feel I am able to give more specific feedback, better feedback, than I ever could before. I also feel I have a better handle of each kid's tendencies...when I go to grade their summative assessment at the end of the week, I can look for those things. I think the jury is still out on my IEP and 504 kids. Overall, their first attempts at the summative assessments have improved, so that is good. However, they generally do not complete everything for whatever reason (mainly, distractions). I think this is helping my ESL students, too, but that data is kind of sketchy. With my ESL students, I am more aware that ever of language difficulties (slant height vs height, for example). Nevertheless, this opportunity for one-on-one time with them is valuable.
Creativity in the Classroom
THIS short article made me think.
Our current teaching method is based on the idea that we can teach students to solve problems. It presupposes that we can define the problems in advance, which in turn presupposes that we know the answers. We give the problems to our students and grade them based on the correctness of the answer. But while creativity may help in problem-solving, it’s a fundamentally different activity.
The difference is this: Problem-solving is a repair activity. Inherent in the concept of a problem is the removal of an obstacle or difficulty.
Creativity is not about fixing things that are broken but about bringing new things into being.
Take the example of a broken car. “Fixing the problem” will get you to a working car. It will never get you to a fundamentally different kind of transportation, such as, for example, an airplane, boat or motorcycle.
Problem-solving asks the question “What is wrong?” or “What is broken?” It assumes there is a right answer, and it’s in the teacher’s edition of the textbook. But creativity asks the question “What do you want?” or “What is possible?” It presumes that there are not one but an infinite field of possible solutions.
It is the difference an industrial model of education and an information model of education...
Someone who is trained by our system to be a good industrial worker will only be confused and disoriented by an information-oriented workplace.
I think three recent posts and subsequent discussions on dy/dan best illustrate theses differences between problem-solving and creativity, between an industrial model of education and an information model of education...
Sunday, February 14, 2010
Comune Di Trentola (Part 2)
I used this Comune Di Trentola panorama (as well as some others) earlier in the year when we studied transformations. One construction popular with my students was repeated translations (as shown below).
Saturday, February 13, 2010
Comune Di Trentola
Take a look around this panoramic view. Pay particular attention to the semi-circular cobblestone design in the street. Try to recreate a drag-testable construction of this street using the GeoGebra applet below.
Comune Di Trentola in Naples
Comune Di Trentola in Naples
Wednesday, February 10, 2010
More Sports and Statistics
What would be cooler than to attend a conference that exists at the intersection of statistics and sports?
How about the 2010 MIT Sloan Sports Analytics Conference?
Here is the panel discussion on Basketball Analytics from last year's conference.
How about the 2010 MIT Sloan Sports Analytics Conference?
Here is the panel discussion on Basketball Analytics from last year's conference.
Tuesday, February 9, 2010
200 Visitors!
My blog has reached the 200 visitor threshold. I first reached 100 visitors on January 15, a little over one month after I started this blog. The next hundred in a little less.
Thanks to those who have visited and have left comments!
When Will I Ever Use Abstract Algebra?
What can you do with the symmetry group of an equilateral triangle? I saw this episode of How It's Made a couple of weeks ago. In it, they focused on how they make highlighters. At about the 3 minute mark of this clip, they show how they assemble 3-in-1 highlighters. At about the 4:30 minute mark, they show an application of Abstract Algebra. In particular, an application of the symmetry group of an equilateral triangle, D3.
Recall, the symmetry group of the equilateral triangle contains six elements: The identity transformation, a 120-degree rotation, a 240-degree rotation (they are both either clockwise or counter-clockwise), and three reflections, one in the perpendicular bisector of each side. The result of the composition of any two of these transformations is equivalent to one of the others.
Dri Mark Highlighters - How It's Made - Watch more Videos at Vodpod.
A Few More Super Bowl Stats From A Former Coach
In light of an earlier post about the role of situational statistics in football, there seems to be a statistical interest in three of New Orleans Saints Coach Sean Payton's decisions during the game: His decision to go for a touchdown on fourth and goal late in the first half, his decision onside kick at the start of the second half, and going for a two-point conversion after a touchdown late in the game. An interesting analysis of all three are found in the post Who 'Dat Gonna Kick Onside to Start the Second Half? at Brian Burke's Advanced NFL Stats.
Monday, February 8, 2010
Three Points and an Orthocenter
Pat Ballew ran with my earlier WCYDWTRP post over on his blog, so I thought I would return the favor.
Pat's take was an old theorem about the vertices of a triangle lying on a rectangular hyperbola. It turns out the orthocenter of these three points will also lie on that hyperbola, as the dynamic applet below illustrates.
Note: Double clicking on any of the applets that follow will open it in a separate GeoGebra window
All of these following applets were made with the FREE Dynamic Mathematics Software GeoGebra.
An interesting property of three points and their orthocenter is this: Select any one of these four points. The selected point is the orthocenter of the remaining three. These four points are sometimes called an orthocentric quartet. Drag the slider in the applet below to see what I mean.
Anyway, I began to wonder, what would happen if the vertices of a triangle were restricted to some other conic? An ellipse, perhaps? Drag the red points in the applet below around. Right-click (ctrl-click) in the green orthocenter to turn the trace of the point on.
What about restricting the vertices of the triangle to a parabola?
Pat's take was an old theorem about the vertices of a triangle lying on a rectangular hyperbola. It turns out the orthocenter of these three points will also lie on that hyperbola, as the dynamic applet below illustrates.
Note: Double clicking on any of the applets that follow will open it in a separate GeoGebra window
All of these following applets were made with the FREE Dynamic Mathematics Software GeoGebra.
An interesting property of three points and their orthocenter is this: Select any one of these four points. The selected point is the orthocenter of the remaining three. These four points are sometimes called an orthocentric quartet. Drag the slider in the applet below to see what I mean.
Anyway, I began to wonder, what would happen if the vertices of a triangle were restricted to some other conic? An ellipse, perhaps? Drag the red points in the applet below around. Right-click (ctrl-click) in the green orthocenter to turn the trace of the point on.
What about restricting the vertices of the triangle to a parabola?
Sunday, February 7, 2010
The Goal Should Be Statistics
A year old, but do you agree?
| The Colbert Report | Mon - Thurs 11:30pm / 10:30c | |||
| Arthur Benjamin | ||||
| ||||
It's Super Bowl Sunday, And I Am Just An Old Football Coach At Heart
Back when I coached, I was a big believer in using statistics to inform my decisions. So, my interest was piqued when I saw THIS article, even though I am not sure what it is getting at. Sounds like some sort of a fantasy league or Super Bowl pool. Who knows.
THIS article begins to get at the heart of the matter, trying to convince the reader that mathematics does matter in football play calling. I was reminded of THIS article, briefly discussing the mathematics behind one of the most controversial coaching calls of this past NFL season, when the Patriots' Bill Belichick decided to go for the first down on forth-and-two from deep in his own territory with a lead late in the game. This lead me to recall THIS article about Pulaski (HS) Academy's Kevin Kelly and his use of statistics to inform his football decisions. Consider Coach Kelly's take on the onside kick:
The onside kicks? According to Kelley's figures, after a kickoff the receiving team, on average, takes over at its own 33-yard line. After a failed onside kick the team assumes possession at its 48. Through the years Pulaski has recovered about a quarter of its onside kicks. "So you're giving up 15 yards for a one-in-four chance to get the ball back," says Kelley. "I'll take that every time!"
However, the article I have enjoyed the most was by the University of California at Berkley professor David Romer, It's Fourth Down and What Does the Bellman Equation Say? A Dynamic Programming Analysis of Football Strategy. In a nutshell, Romer concluded that NFL coaches are far too conservative with their decisions, often going against what statistics suggest is prudent. I actually printed out all twenty-some pages and attached the article to one of our practice plans.
Another interesting article is Stephen Abbott's interview with former Washington Redskins Offensive Coordinator Chris Meidt, an undergraduate mathematics major, in the September 2009 edition of Math Horizons. Meidt's take on statistics:
...in the NFL the data for everything is available. There are a number of statistical firm that capture everything so it’s all there. What’s our run/pass ratio for second and three? What’s our success percentage when we run or when we pass? I used to have to do this myself. I computed it for every down distance in the field. I did it for the red-zone [inside the 20 yard-line] ad nauseam. You’ve got two choices—run or pass—and you have three downs. Once you create your tree, it just becomes a game theory problem—what is the most likely outcome and what has the greatest expected value?
If I wasn't teaching, I think running numbers for a professional or college football team would be an intersting thing to do.
Saturday, February 6, 2010
WCYDWTRP
My writing has been a bit slow the past two weeks, so I pose the question: What can you do with three random points?
Double-click on the applet to open it in a separate GeoGebra window. From there, you can save it.
Double-click on the applet to open it in a separate GeoGebra window. From there, you can save it.
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