Saturday, January 21, 2012

Why we really do need scantrons!

Most schools buy Scantron sheets by the thousands, and the cost per year must be over $1 000 as each pack of 500 are approximately $100. I was thinking that there must be other uses of scantrons other than their traditional use for this amount of annual costs.  After doing some thinking I am now a believer that Scantron sheets do have a place in schools and below is a list, divided by subjects, for which you can use scantron sheets:  I do apologize they are broken up by subject not grade.


·         Provide each student with a scantron sheet and ask them to guess which would be the correct answer to the first question, if you were unable to see the question..  Talk about the percentage of the students in the class which would have guessed correctly.  Extend this to two questions and so on.  You can then talk about the math behind probability.

·          Turn the scantron side ways and draw graphs.  A constant graph would be were all the answers were A, an oscillating graph would go ABCDCBA….  I bet there are many different graphs which could be drawn.  You could then talk about slope, absence of concavity and so on.

·         Using the scantron, you could create a ratio of Surface Area to Volume, or perimeter to Surface area, and so on.

·         You could talk about the costs of buying scantrons and create a Cost graph and calculate the slope and y-intercept and explain what do these values mean.


·         Have younger students read the instructions on the scantron sheet and have them explain to a classmate what the instructions mean.

·          For students learning the alphabet, have them create the different letters by connecting the dots on a scantron in different formations.  (“U” might be a tricky letter!)

Other languages (French, Spanish, etc):

·         Have students translate the meaning of the instructions on the scantron sheet into the desired language.


·         Push a marble down the floor and ask the class to estimate the number of scantron sheets, vertically, it would take to stop the marble.  Increase the speed of the marble and estimate again.  Extend this to the problem “How many scantrons would stop a car going 30 km/h?”  (I would like to know this answer?)

·          Talk about the air resistance as you drop a scantron from a desk to the floor, and then ask “Would the same action occur if this was a vacuum?”


·         Determine how much water one scantron can hold by weighing 20 of them dry and then soaking them in water and comparing the weight difference.

·         Take one scantron and light in on fire.  Talk about the combustion of paper.  You can extend this by dipping the scantron in different liquids and talk about the difference in speed of ignition.


·         Instead of using construction paper, have students cut up Scantron sheets and art with them.  They can be colored on, and easily cut!

·         Have them draw a picture only by connecting the dots on the Scantron.

The best part of these activities is that you don’t need a scantron machine ($5000) to do them…oh don’t even need the scantrons!!

However, if you have other activities, I invite you to share below!

Happy Marking!

Thursday, January 19, 2012

Learning can occur on a final exam

Historically, I would say that the least amount of learning occurs during exam week.  This year, however, I have tried to make this statement false through giving my students a new final exam.  When the first group presented, not only did I see, hear and was taught about unique examples of how to use calculus, I can actually say the group still LEARNED through the assessment. 
Below is a video of their PowerPoint presentation, which they used as well as supplemented with dialogue. 

After the presentation they opened the floor to me by giving me 10 minutes of questioning.  Most of the questions they answered swift and correctly, until I asked “Do all functions, on a closed interval, have an absolute maximum?” . Their answer was “Yes”.  Of course this is incorrect.
Now here is how the learning occurred… First I will address the traditional way of assessment:
If this was a traditional final exam, I would have marked this question wrong and moved on to the next question and continued marking.  These students would never have received any feedback, as in the past, I have yet to see many students come back to see WHAT they did incorrectly on a final exam.  These students would have then gone on to university/college with this false knowledge.
How it has changed this year…
I didn’t let this false information continue.  I then asked, “What if I told you, I could draw a function on a closed interval without an absolute maximum?”  The girls looked at each other with confused eyes, and pondered the idea.  After some passing minutes, one replied accusing me of a liar.  I wanted to ensure they didn't continue on with this false information, so I then asked, “Is there any kind of function that continues upward on forever?”.  One quickly answered, “WAIT! A function could have a vertical asymptote and therefore have no absolute maximum.  I guess you weren’t lying”, the other girl smiled and agreed. 
After this, I wanted to ensure they had a true understanding and therefore I asked, “Can a function, without any asymptotes, on a closed interval, not have an absolute maximum?”  The enlightenment has occurred!  Both girls whispered quietly, and then turned and replied “If the curve has an open point at the highest point, then it would not have an absolute maximum”.
Learning had occurred, and yet it was a final exam.
I continued with my questioning, which they answered correctly, and I am happy to say that this experience has been a success with the first group!

Wednesday, January 18, 2012

15 Geeks who outsmarted the system

Below is a post from, and specifically Tim Handorf.  Cool read about math geeks!

15 Brilliant Math Geeks Who Outsmarted the System

Geeks rule the world these days, with tech geniuses and statistics-savvy entrepreneurs owning some of the biggest and highest-grossing businesses anywhere. Yet while some use their know-how to make a fortune the old-fashioned way, through hard work, others try to outsmart the system and get rich in much less legitimate or sustainable ways.

Here, we’ve compiled a list of math geeks who’ve used their computational skills to make big bucks in Vegas, playing the lottery, or on other forms of gambling. Some did nothing illegal and simply used their knowledge to their own advantage, others used their math skills to cheat and lie their way into millions, and some figured out a system just for the fun of it. While some may not have chosen the high road, these geeks will show you that your math courses really can be useful for more than getting college credit – though we caution emulating some of the methods these enterprising math nerds used.

  1. Mathematics professor Edward Thorp was a pioneer in modern applications of probability theory, which he harnessed in a variety of ways for his own, legal, financial gain. As he documented in his book Beat the Dealer, Thorp was able to show that the house advantage in blackjack could be overcome by card counting, a strategy he used to win tens of thousands of dollars at casinos during the 1960′s – practice which they now guard against, as it produced many copycats. Thorp didn’t just use his gift for math to win in Vegas, however. Applying his knowledge of probability and statistics to the stock market helped him make a fortune off of pricing anomalies in the securities markets, earning an average of a 20% rate of return over the 30 years he was investing.
MIT Blackjack Team
  1. One of the most famous stories of math geeks outsmarting casinos centers around a group of students at MIT and Harvard during the late 1970s into the late 1990s, that was made into a recent movie called 21. Using card counting techniques and other sophisticated strategies, the students were able to win large amounts of money, doubling the initial investment in the venture in the first ten weeks. Of course, with their successes came increased attention from casino management, leading to many players being barred from casinos around the nation. These bans and other career opportunities led to most of the team disbanding by 1993, though some members were to go on to form their own blackjack teams that continued to play up until 2000.
Mohan Srivastava
  1. Unlike many of the others on this list, Srivastava has never used his statistical skills to get rich. In fact, he only set out to crack the code on lottery scratch off cards out of curiosity, not a desire to make money. But crack it he did. Srivastava discovered that there were clear patterns in the numbers on tic-tac-toe game cards that allowed him to predict which cards would be winners with a striking degree of accuracy. He also discovered that these same rules, or adaptations of them, could be applied to other lottery games. While Srivastava could have used these methods to make money, he didn’t feel it was worth the time or the trouble (scratch off cards don’t usually have big pay outs), and he even reported the problem to the lottery authorities. While the tic-tac-toe game was pulled, his warnings were largely ignored and his findings can still be applied to scratch-off tickets in a wide range of locations.
Joan Ginther
  1. There is no proof that Joan Ginther has used her math skills to help her win her four (yes, four) million-dollar-plus jackpots from the Texas Lottery, but there is a whole lot of speculation. Ginther has a PhD in math from Stanford, leading many to believe that this former statistics professor used her math knowledge to help her crack the code for the lottery’s algorithm. Why all the speculation? The odds of someone winning four jackpots is one in eighteen septillion – possible but highly improbable. The Lottery Commission believes Ginther won fair and square, but it’s not likely to stop people from believing that she had a little help from that math degree anytime soon.
Syndicate of British professors and tutors
  1. Winning the lottery isn’t just luck of the draw when you’re a math genius. A syndicate of professors and tutors at Bradford University and College used their knowledge of mathematics to apply probability to playing the lottery, even developing a formula that helped them to pick winning lottery numbers. Using a computer to check lottery numbers each week, an investment of $8,700, and waiting over four years, the gamble paid off and the 17 staff members won a $13 million dollar prize.

David Phillips
  1. David Phillips is perhaps better known as "The Pudding Guy" for the loads of pudding he bought to take advantage of a promotional program offered by Healthy Choice in 1999. Phillips, a professor at the University of California, Davis, used simple math to figure out that the value of the promotion exceeded the cost of the entree which it was offered on. Phillips found that the promotion was even more valuable, as it could be used on pudding packages, which were a mere 25 cents each. Phillips bought over $3,140 worth of pudding, taking the UPCs and donating the food to the local Salvation Army. An additional bonus for the mail-ins held in May of 1999, allowed Phillips to stockpile 1,253,000 in AAdvantage miles, a number which he continues to grow today through other promotions. Phillips may never have to pay for a flight again.
Michael Larson
  1. Michael Larson was a contestant on the 1984 game show Press Your Luck. Larson had figured out how the patterns in the board move and, using probability, could determine which squares would be more likely to contain money. While Larson started off slow on his run through the game show, by the third round he was hitting bonuses on every turn, racking up a total of $110,237 (adjusted for inflation, that’s about $233,000 in today’s dollars). Larson’s successful run would land him a jackpot, but disqualified him from participating in future shows and resulted in changes to the patterns the game show used.

Gonzalo Garcia-Pelayo
  1. Roulette is by and large a game of chance if there ever was one, but one man managed to figure out that certain wheels have a bias for certain numbers. Working with his family, he collected data and analyzed numbers from Casinos all over Madrid, eventually realizing that the numbers on roulette wheels weren’t, in fact, perfectly random. In fact, the statistical data pointed to a distinct pattern, which Garcia-Pelayo would learn to take advantage of. Using probability, he was able to clean up at casinos around the world, winning over $1.5 million over the course of a couple of years. By betting on the numbers he had identified as "hot," he would turn a 5% house edge into a 15% player edge. Casinos weren’t a big fan of his methods, however, and sued to get their money back. The courts sided with Garcia-Pelayo, however, and said it was the casino’s responsibility to fix their wheels. Nonetheless, Garcia-Pelayo, and many members of his family, were banned from casinos.
Keith Taft
  1. Keith Taft is a legend in the world of blackjack. Taft entered his first blackjack game on a whim in 1969 and quickly became hooked. He read numerous books on card counting, but could never really master the strategy on his own. Instead, he invented a computer to do it for him. This is a lot more impressive when you consider that in 1969, most computers were seriously huge, taking up the better part of a room, and ran on punch cards. Taft’s computer would fit on his body so that it could be concealed when he entered the casino and was operated with his toes. After two years of work, Taft would complete the cheating system (which he named George), which used mathematics and statistics to determine the house advantage for a particular hands, count cards, figure out a smart wager, and tell Taft was move to make. The computer was a failure, but it would serve as a model for later systems which would win Taft and a team of pro players $100,000. Taft’s inventions were legal at the time, but all are banned in casinos today.
The Four Horsemen of Aberdeen
  1. The "Four Horsemen", Roger Baldwin, Wilbert Cantey, Herbert Maisel and James McDermott, were a group of Army officers who devised one of the first blackjack strategy guides when they weren’t doing mathematical work for the military. Using probability theory, the group were able to develop strategies that could help a player improve his or her odds of winning substantially, eventually releasing the findings in a book called Playing Blackjack to Win. In fact, this book held the calculations that Ed Thorp would later use to write his own blackjack guide. Using only basic calculators, the group was able to develop a card counting system that gives the player a .1% advantage, though it’s unclear whether they ever used it to their own advantage or just enjoyed playing with the numbers to pass the time.
James Grosjean
  1. Harvard grad and gambling expert James Grojean is the author of Beyond Counting: Exploiting Casino Games from Blackjack to Video Poker, a mathematical treatment of various forms of legal advantage play gamblers can use to increase their odds of winning – and Grosjean should know. He became a professional player after becoming enamored with casino play while a graduate student in economics at the University of Chicago. Often working with partners and a number of computers that are constantly analyzing data, Grosjean has used his mathematically-based strategies to win himself a small fortune, though casinos have fought him throughout it all. Grosjean has won numerous legal battles against casinos, and in 2005 scored a nearly $600,000 judgment in his case against Imperial Palace.
Cash WinFall Group
  1. Gerald and Marjorie Selbee, owners of a gambling company, bought over $614,000 worth of tickets for the relatively obscure game Cash WinFall in July of 2011. With a payout of just $2 million and only one winner in the game’s seven year history, one would think this was a foolhardy way to use that kind of cash, but the Selbees had a plan. For a few days every three months, provided no one wins a jackpot, payoffs for smaller prizes swell dramatically, with pretty much assures anyone who buys at least $100,000 worth of tickets a payoff. The Selbees, along with betting groups from MIT and Northwestern, purchase about half of the tickets for the games, as with a little math know-how the game isn’t really one of chance at all. As of this year, the Selbees have claimed over $1 million in prizes and their final take is likely to be much larger.
Bruce Bukiet
  1. For most, wagering on a baseball team to win the World Series or even to end up there, with the help of stats, of course, would be a pretty big gamble. But Professor Bruce Bukiet of the New Jersey Institute of Technology may have figured out a system that takes a lot of that chance out of the picture. Bukiet has created a mathematical model that computes the probability of one team winning a game against another team, with some surprising results. He’s beaten the odds six of the eight years he’s been using the model. Of course, his system isn’t foolproof. He predicted the Phillies, Dodgers, Cardinals, and Braves would all be division winners in the National League this year. The real winners? The Phillies, Brewers, Cardinals, and the Braves. Bukiet doesn’t gamble on his predictions, however, and states he just does them to show young people that math is relevant, even to things like sports.
John Kelly, Jr.
  1. John Kelly, Jr. used his PhD in physics from the University of Texas to get a job working at Bell Labs in the early 60s. It was there that he developed the famous Kelly criterion (though his discoveries in computer synthesized speech were also pretty amazing). The Kelly criterion is a formula used to determine the optimal size of a series of bets, whether in a casino or on the stock market. The theory was put into practice by gamblers like Thorp and Kelly’s associate Claude Shannon, though Kelly himself never used the theory to profit himself – though perhaps he never got the chance. Kelly died in 1965 at only 41 years old, leaving behind a formula that has been used to make the fortunes of big business moguls like Warren Buffet, Bill Gross, and Jim Simons as a legacy.
Steven Skiena
  1. Think betting on sports is a random gamble? With some sports that might be true, but when it comes to the complex betting systems of jai alai, there is a system and it works, as was proved by computer science professor Steven Skiena in 2004. Skiena began predicting sports results in 1977, correctly predicting the outcome of NFL games with 65% accuracy. He’s refined his methods a bit since then, and turned an initial investment of $250 into $1851. Skiena devised a highly complex mathematical system, and using a computer he could more accurately predict which team or player to bet on. For Skiena, the project was a mathematical exercise, not a money-making scheme, however, and he donated all of the winnings to charity.

Friday, January 13, 2012

Improving the test of provincial exams!

The Alberta Government is about to do something that is going to improve learning in schools.
It is about time the Frasier institute can no longer compare schools, or allow the results of schools to be public.  Finally, Lukaszuk is about to show that there is more to learning than statistics, numbers, test scores, and rankings.  Peter Cowley, from the Frasier institute, says “Teaching teams can directly use the PAT results to the benefit of their students”, and then follows with “It is hard to see how improving teaching effectiveness can be considered a misuse of the PAT data”.  These two statements are begging the question that PATS improve learning, which they do not.

I ask you, have you ever written a test or an exam and became smarter because of it?  In Alberta, I have yet to see a farmer weigh his/her animals more to increase their weight.  Just as weighing an animal won’t increase its weight, testing students won’t increase learning.  Our education has always been test score driven but I wonder if this belief is actually justified?  Has this norm been created out of research and studies?  Is this actually authenticated by pedagogical practice?  Does testing create lifelong learners?  Lastly, does it improve education?  I believe the answer to all of these questions is NO!

I understand that many people outside education, such as Peter Cowley, believe that only a standardized provincial exam can reveal that truth about what is actually occurring in Albertan schools, and is fighting for the idea that three years of education can be reduced to a single number on a multiple choice exam.  However, this idea is completely absurd.  If you find yourself disagreeing, then I strongly urge you to learn more about a standardized exam.    

 Just for starters, before a standardized test is given there are psychometricians who know how many students will pass and how many students will fail, and this occurs before the students ever write the exam.  Why would we support the sharing of grades on an exam that has a pre-determined failure rate?  Next, Peter will have us believe that these tests are an effective way of measuring education in our province, but he fails to realize these tests have a pre-selected material which is unknown to the teacher.  In terms of reliability, most educators resent the idea of a confined focus of testing as it measures only a portion of the domain and ultimately distorts the depth, complexity and dedication of a student’s ability.

 Lastly, Peter will try to convince people that it is these tests scores that will improve education. He fails to understand that it is not the test scores which need to increase to enhance education, but instead the trust that a teacher is a professional, dedicated to providing the best education possible, and the idea that learning cannot be reduced to a mark on a single exam.