Monday, August 22, 2011

Exploring Negative Bases

I love it when math works as it should. Such was the case last night when I was looking for a problem to explore. I began by checking my favourite blogs for a quick puzzle before bed. Nothing really stuck, so I tried some searches on twitter. (#math #mathchat #puzzle usually get the job done). I found a puzzle that intrigued me and began to work.

Before I mention the puzzle, I want to say a quick word about school maths. I have a great personal interest in mathematics, and am able to move on past problems that don't pique my curiosity; we do not afford students the same opportunity. I wonder, if students were given more freedom, if they too would find puzzles and topics that interest them. This is a not the topic for this post, but is an interesting debate in mathematics education. I digress...

Back to the puzzle. It is as follows:

Prove the following subtraction cannot be true:

                                                 E  L  E  V  E  N
                                              -     T  H  R  E  E
                                                     E  I  G  H  T

The intriguing part of this problem is how I initially read it. Because of the spacing and my mathematical mind, I did not see the words initially. When I presented the problem to others, they immediately read "11-3=8", but all I saw was a collection of variables. The second, and more fulfilling, outcome of this problem could not be predicted from the start. I ended up learning far more than the designer could have ever intended. Again, this personal driven learning is lacking in school mathematics. The more I experiment with maths, the more apparent this injustice becomes. 

This is where I began. I assumed that each letter stood for a unique number. I figured if I showed that two variables must represent the same integer, then it was impossible. I counted 9 letters (E, L, V, N, T, H, R, I, G) which meant that there was not much room for ambiguous cases. If there are 10 digits in play (0-9) then there would only be 1 left out of the problem. Again, this is an if, because no base was indicated. Already I was on a trail that was my own, but had no idea where it would take me. 

I began on the far left-hand side. I knew that with the subtraction, we lost a digit. Regardless of base, this meant that it had to be "carried". If it disappeared when carried, then we know that E=1. We also know that L<T to enable the carry to make sense. Even if it was borrowed from previously (to help the E to its right), L would still have to be less than T because if L-1 is less than T but L wasn't, then L would have to equal T. This is impossible by my assumptions. Such is the logic work the puzzle intended; I was going a much different direction. 

Now I knew L<T and then the subtraction gave me this:

L+10 - T = E
but E=1 so...
L + 10 - T = 1
(assuming base 10 on the carry) more generally...
let b = base
L + b - T = 1
L-T = 1-b
T = L + (b-1)

Now this last conclusion fits perfectly with L<T because we must add a positive number to get to T. In fact, because T must be a digit [0, 9] (more generally [0, (b-1)]) we know that L must be 0 and T must equal b-1. It was here where my thoughts veered off track.

These two statements only work together if b>0. If 'b' was a negative number, then adding (b-1) to L would not yield a larger number. Therefore L<T. There seemed to be an incongruence. I had never encountered negative bases; my mind had never gone there. I had seen binary in grade school, and experimented with alternate (positive) bases at leisure, but never negative. Was this possible? What did base -10 look like? 

I abandoned the problem and began to play with its precipitate. Could all numbers be written in Base -10? How could we count in this base? It turns out, it is possible, and creates some very revealing patterns. Instead of a 1s, 10s, 100s place etc., the sign alternates as the negative is exponentiated. We are left with a 1s, -10s, 100s, -1000s place. etc. For example:

472 in base ten is expanded as

4x(10)^2 + 7x(10)^1 + 2x(10)^0
4x100 + 7x10 + 2x1

but 472 in base negative ten is expanded as

4x(-10)^2 + 7x(-10)^1 + 2x(-10)^0
4x100 + 7x(-10) + 2x1
400 -70 + 2

The result is very different. In fact, the difference between the two numbers always seemed to be twice the absolute value of every negative component. Try counting to 20 in base negative 10. What pattern occurs? Take it further. Try an addition. Any quirks? Try anything you like and attempt to make sense of it. I discovered patterns with exploration, but I can imagine how school math would approach this. More to come on that point later.

After playing for a little while, I was developing a list of rules that allowed me to become more comfortable with the base. Along with the absolute value pattern, I noticed that every number with an even number of digits was negative, and all positive numbers had an odd number of digits. What made the rules so powerful? They had meaning because they were developed! My odd/even rule made sense because I had dissected the facts and devised the rule from a logical pattern. After I had grasped the new number system, I sat back and mused on how schools would probably transfer this knowledge to students. It would probably look something like this:

Rule #1: Every odd digited number is positive
Rule #2: Every even digited number is negative
Rule #3: To convert to base-10, the ones digit remains constant, every pair of subsequent digits (ab), becomes 10a-b in the base-10.
Rule#4: If there is no pair, the answer is negative. Subtract what you have from the place value of the remaining digit. 

Crystal clear. Now do problems 1-50 odd. Hopefully these rules sound like garbage to you; they would to me except I wrote them. Actually, I did one better--I created them. 

Now, in my curriculum, there is no space for negative bases. In fact, alternate bases only make an appearance in the form of binary. (If you don't count the bases in logarithms). I would love to take the idea of an alternate base arithmetic system into a math club and explore. Such environments would greatly benefit from discussions and tinkerings with binary, hexadecimal, and neg-ecimal... (to coin a phrase). 

It is tough to incorporate true inquiry in high school these days. Guided inquiry with carefully constructed tasks keeps students on the pathway to the outcomes. The more I experience the freedom I never had, the more I realize that being "good" at math, is much deeper than I ever thought possible.

Nat Banting

PS. I solved the problem the next day. The notation makes it tough to blog, but maybe I will try someday. I encourage you to solve it as well. Also, I used my odd/even rule to show that the base in this problem cannot be negative. ELEVEN has 6 digits, so it is negative. THREE has 5 so it is positive. The result (EIGHT) has 5 digits, so it is positive. A positive subtracted from a negative can never be positive; this explains why my L and T notation didn't work out if b<0. Don't you love how the math comes full circle!

Friday, August 19, 2011

First Day Tasks

You don't get a second chance to make a first impression. 

This is naturally true, and although I don't believe that you can build or destroy a successful semester in one class, it is definitely important to put your money where your mouth is on the first day. I have spent the past few days digging around my materials for the best possible starter activities.

I had some very helpful responses from the twitter-verse, and it prompted me to somehow sort out the information being provided to me. In the past, I have had very productive lessons on the first day. Not productive in the "coverage"sense, but rather in the "intriguing" sense. My goal was to define a set of characteristics that, in my opinion, create a suitable opening day problem.

It should be noted that I have somewhat of a loose cannon reputation in my school. I like to keep students off guard (mathematically) in my class. Most returning students come waiting for the next surprise, and I do not like to disappoint.

The following is a list of four (4) attributes that I believe constitute an effective first-day task:

1) The problem should be rooted in real mathematics
2) The problem should be solved in a cooperative spirit
3) The problem should require authentic problem solving and not algorithm execution
4) The problem should be presented in an intriguing setting

It wasn't until I had drafted these 4, that I realized that these criteria could possibly apply to each lesson. I do believe, however, that the importance of numbers 2 and 4 is heightened on the first day. These build the strongest communal bonds between students and teachers. After I had my target attributes, I set out to review my options. Of course, time and materials became limiting factors for me as well. (when are they not?) I was given 45 minutes total for each class on day 1. 

I began my search by reviewing the activity that I did for the past 2 semesters. I introduced myself to the class and immediately asked students to put everything away. I handed out a test. The tension in the room rose very obviously once I began to circulate with the papers. On the test, there were questions similar to the one below:

A man bought a pair of shoes that cost $75 and gave the merchant a $100 bill. After the man left with his shoes and his change, the merchant discovered that the $100 bill was counterfeit. What was the total loss to the merchant?

Students completed the exam individually, but I did not discourage "cheating". Without announcing an official end to the test, we began to discuss different answers. As is plain to see, students come up with a variety of answers and rationale. What is your answer? Can you back it up? As the teacher, I kept my answer well hidden and agreed with every argument that made sense. Not only did this cause mass mayhem, students began to vehemently defend their position. I was able to step back and allow the students "do" the math. When I look at the criteria, I see this task meeting numbers 2, 3, and 4. The problem became a group task, the contention was a result of the intrigue, and there were definitely no algorithms to solve the problems. (Although some students created them!).

The issue for me is the fact that the problems weren't really based in math skills but rather in the deft wordplay of the questions. Usually the arguments were not math related but rather "real life" related. Students were amazing at finding the loopholes. Like the merchant didn't lose $100 and the shoes, he lost $100 and the wholesale value of the shoes. Such is a splitting of hairs that I didn't want; I wanted the contention to come over mathematical ideas. I decided to scrap this idea, and try to find a task or tasks that fit all 4 criteria. 

The activities come from various places including @jamestanton, A Number Story, by Peter Higgins, and past experiences. I haven't decided which to use yet, but now have a clearer vision.

1) Magic Square Problem
I first saw this problem from a professor. Draw a 5x5 matrix of numbers beginning with 1 at the top left, and continuing across horizontally until 25 is in the bottom right hand corner. The teacher writes a prediction on a piece of paper, and gives it to a trustworthy student (based on first day impressions). Then he asks the first student to pick a number in the first row. Circle the selection and then cross out all other numbers in it's row and column. Continue until 5 numbers are selected, sum them up, and reveal your guess. If they ask you to repeat, do so with a 6x6 matrix. Without fail, some student will say it always works. This is perfect, because now the question is: Why? How does the result relate to the length of the side? Why is the result always the same? Like most of my problems, the secret is in the base-10 number system and the number theory behind it. Encourage students to rationalize and explain to classmates.

2) Box of Toothpicks
I have 39 toothpicks in a box and ask a student to pick out a number of toothpicks and leave any number they wish in the box. To eliminate cheating, I then ask him to take that number he placed in the box, add the digits, and take that sum out of the box. I then guess the number of toothpicks in the box by simply shaking and listening. This can be iterated, but once a repeat comes up, the students become suspicious. Once your cover is blown, ask them to explain why the answer must always be the same? Or does it have to be? A similar example can be found at where a psychic reads the students' minds. After enough iterations, predictions will be made. Always ask for follow up explanations. 

3) Cats and Dogs
This problem has been solved using formal methods on my blog previously. The students are given a question that can be solved using sophisticated methods, but they are not apparent. Get students to work in pairs or groups until they guess their way to the answer. You, as the teacher, should elicit strategy as they go. A very interesting mice strategy emerges. Would you ever add 3 mice to the equation? Why or why not? What special quality do the cats have? Is your solution the only one? How can you tell? Can we add an animal at a different price to switch the solution? The problem has many layers, each of which is rooted in real mathematics. 

4) Iterated Sharing
This problem comes from James Tanton's book, Solve This. It is the second problem in the book and involves a group of people passing candies around a table. I have not personally tried the experiment, but am intrigued by the problem. I will not disclose the entire problem here to respect James' ingenuity. His book is an excellent resource for those teachers (and learners) interested in moving students (or themselves) outside the box. 

You may notice that these problems don't have explicit curricular ties, and that was not one of my goals. Each is rich in mathematics and accomplishes the first day goal of establishing a mathematical ecology in my room. It is a far bigger challenge to design tasks that meet both explicit curricular goals and the four criteria. That is a much larger burden which math teachers collectively bear. Creating an effective environment is very important in mathematics education. A carefully designed first-day task can work wonders. 


Tuesday, August 16, 2011

A Declaration of Independence

I used to be roommates with a magician. He kept all of his materials locked up in a trunk in our hall closet. Although he had devoted himself to the study of human psychology, I still convinced him to crack open the trunk and show me a trick from time to time. This experience was one of the most frustrating yet intellectually stimulating experiences of my life. I was a mathematics undergrad immersed in a stressful environment of number theory, numerical analysis, and abstract algebra. I was being trained to reason effectively, and his antics refreshed my perspective on reality. Life often muddies mathematics; such is the unfortunate reality. 

I was again reminded of this fact this morning when I followed a twitter conversation between @davidwees and @mathfour. The conversation was based on the nature of irrational numbers. Measuring them in the plane yields a rational number, and this could place them in the same, unknowable category of imaginary numbers. In mathematics theory, irrational numbers are very knowable, but when they are transferred into a concrete, classroom environment, they are forged. 

Another intriguing example of this phenomenon is the classification of independent events. We claim that two actions (or events) are independent if one has no bearing on the other. The events could happen in succession or concurrently and it would remain immaterial. When we toss a coin 10 times, the events are independent, but when we ask a student to predict the outcome of tossing a coin 10 times in succession, their guesses may not behave in this way. The involvement of the human psyche tampers with independence.

Testing this influence is a great way to introduce elementary probability to students. I would begin with showing this intriguing video from Derren Brown:

A typical response is immediately available in the comments:

"If he answered the same thing every time, he would have won eventually" - Guitarhero0904

This is a common reaction to such ploys, as evidenced by the 19 "likes" on this individual's comment. (at time of writing) It only makes Brown's statement ring true:

"Our tendency to think that we're not predictable is probably one of our more predictable traits."

This is an excellent conversation to have with your students. If Steve Merchant were to choose randomly, what are the odds that he would get the word correct? How can you represent your answer? Why did you choose this answer? Can you broaden your definition? Before long you are conversing about the sample space, outcomes, and favourable outcomes to a particular experiment. This opens the door to the fundamental counting principle. What are the odds that Merchant would get all 4 wrong? 

You suggest to the students that there must be a way to explain the large difference. Extending this to more trials makes the multiplication of intersection second nature before moving into the next phase. Allow the students to pair up and attempt the game on one another. As the trials continue, take the data on which students are guessing wrong and which are guessing right. This data collection can be done in many different ways, and it is best to allow students to decide which stats to keep. The goal should be to determine which students are the most predictable. You essentially are using the deviance from the theoretical probability to measure predictability of students. 

The power of the experiment is its underpinning in theoretical probability. Students begin to understand probability as an estimate of the norm. If you are lucky, students may question if their results are significant enough. Maybe getting 3 out of 4 wrong isn't that unlikely? This builds a convenient bridge into the union operator. 

Another great way to introduce human impact on probability is through the familiar game of Rock-Paper-Scissors. (RPS). Take a couple minutes to calculate the probabilities of a win, loss, or draw if all trials are deemed independent. Become immersed in the expected value before introducing the human element. Ask the students if they have any strategies to win in a standard game. You will get several responses varying in complexity. A great place to introduce RPS strategy is the strategy guide from the World RPS Society entitled "How to beat anyone at Rock Paper Scissors". The link is found below:

The effectiveness of human influence can then be tested with two online resources. The first game is built on a random number generator. It essentially uses no human influence to calculate its moves. If the students play enough trials, they should notice an even amount of wins, losses and draws. The link for this version is found below:

The next version debuted in the New York Times science section online. The goal of the program is to develop an artificial intelligence based around the game. The computer tracks your trends and makes selections based on prior behaviour. You can choose a novice or expert setting. In the novice, only your decisions are used against you. Students may find that it is harder to win as the trials continue. The expert setting uses thousands of iterations by developers to devise global patterns. It essentially is trying to play like a human--this includes human influence on the probability. The link for this version if found below:

Students may begin to develop strategies to beat the artificial intelligence. The teacher should encourage them to verbalize. Most often, they involve using randomness to trick the computer. The goal of these exercises is to understand what it means to be independent. Students work on the theoretical calculation of probabilities along the way and learn terms like sample space, event, and may even dive into union and intersection. These learning outcomes fall out of the investigation. Probability is mathematical fortune telling, and some of us are more predictable than others.


Thursday, August 11, 2011

The Blue Jays Defense

Baseball is mathematically based. It is the best link between the generally nerdy domain of mathematics and generally manly domain of professional sports. The one downside to this statistically driven machine is that the stats can be selected and used to benefit almost any argument proposed. Major League Baseball keeps such extensive records of stats, that there is always an obscure one in support of your argument. Right now the league is in an uproar over the Toronto Blue Jays sign stealing controversy. Some anonymous players claimed that the Jays were stealing signs with a 3rd party and that effected the amount of home runs they hit at home. In an article by ESPN, they use the personal witnesses' accounts to bring up the topic, but claim that:

"Colin Wyers, a contributor to ESPN Insider who writes for Baseball Prospectus, provided independent analysis that showed statistical deviations in Toronto's hitting stats that he considered too great to be random chance."

If you want the full story visit the original source. This post will make much more sense if you do. Take note on the statistics used, and how convincing they are. You will be hard-pressed not to believe the story. When you are done, come back here and read this; hopefully the same effect occurs. The goal of this post is to selectively use statistics (much like Mr. Wyers) to make a very convincing case, and to show the dangers of a statistically driven world. There is a public myth that mathematics is completely objective--numbers don't lie. In this case, the numbers themselves don't lie, but they may or may not contribute to one. 

I am going to begin my "independent analysis" by detailing the four sections of my argument. It should be known that I am not privy to the virtual cornucopia of stats that the nice folks at ESPN are, but the internet (specifically provided all I needed. I will address the following points through my statistical lens:

Jose Bautista's Numbers
Vernon Wells' Trends
The Trade of Yunel Escobar
AL East Analysis

1) Jose Bautista's Numbers
Anyone who follows baseball knows of the emergence of @JoeyBats19. The reigning home-run king and highest vote-getter for this year's all-star game is continuing that pace again. The article claims that there is a significant difference between his on base plus slugging percentage (OPS) rating. The article claims that this discrepancy is too wide. This begs the first question, "How wide is, mathematically, too wide?". The answer for this (coming from statistics) is: "As wide as you need it to be." Switching your sensitivity (alpha value) is a key ploy for statisticians to get their result. But politics aside, what stat can counter the seemingly overwhelming evidence proposed by ESPN?

Let's look at the home runs. Bautista hit 33 homers at home and 21 on the road. That is a difference of 12. Is that large? Or does it prove that he has serious power. I say the latter. If we eliminate the home field home-runs (and the supposed sign stealing), and pretend that JoeyBats19 played every game on the road, he would have hit 42 home-runs--that is the statistical projection. Even with those stats, he still would have finished first in the home-run race league wide. This shows Jose's power in all ball parks and renders that 22% increase due to home field as insignificant. Twenty-two percent could be explained by many factors: more rest at home, comfortability, knowledge of the ball park, or even hitting in the bottom of the inning. Sure, ESPN's stats truthfully show the discrepancy in homers, but fail to illuminate the insignificance of the gap.

2) Vernon Wells' Trends
The article claims that the descent of Vernon Wells' numbers is proof that he is no longer getting help from the sign stealing at home. Although his average has dropped since the trade to Los Angeles, there are more statistics in play here than ESPN wants to divulge. They cite his OPS and home runs as evidence of his immediate decline, but fail to inform readers of many other statistical factors. 
First, Vernon turned 32 when he was traded to the Angels. I am sure MLB has some pretty juicy statistics comparing players' ages and their slugging percentage, but I will have to stick to what I have access to. ESPN doesn't mention the sample size for their stats. They take Well's Blue Jay stats from a city where he was a 3-time all star and had 3 gold gloves. He also finished as high as 8th in MVP voting. If you chart his trends in Toronto, they go steadily down in both categories. ESPN wants to give the impression that the Angels inherited a top-notch player, but the stats (conveniently left out) show otherwise; Wells was on the down-slope of his career. 
Second, it didn't help that the Angels are also on the fall. Wells' stats were hurt by the overall make-up of the team. In the season before Well's arrival, the Angels got steadily worse. They lost Matsui (21 HRs in 2010), Napoli (Led team with 26 HR in 2010) and Morales (11 HR in 2010). This was only the continuance of the trend which saw them lose Vladimir Guerrero the season before. Guerrero still holds several significant Angels records including highest average slugging percentage, batting average, and OPS. Such significant losses clearly show that Wells wasn't hurting from the lack of signs from the outfield, but rather from the lack of a solid team around him. A waning star on a waning team cannot be held as evidence. 

3) The Trade of Yunel Escobar
Yunel Escobar came to the jays mid-season. He provides the neatest sample of evidence because he almost split the season in half with two teams. If there was significant changes, we  would see them. ESPN only mentions his rise in OPS, but nothing else. Why would that be? Maybe because if you look at his stat line during the season of the switch (available here), the 2010 season contains no major statistical differences. So what about the 2011 increase mentioned by ESPN? Look back at his OPS in 2007 and 2009 with Atlanta. His .837 and .812 averages out to .825. ESPN conveniently chose to compare Yunel's worst year (where he had off-field troubles in Atlanta) with his best year. Of course the stats will show improvement when you do that. Why not compare his best year to his current year? The full story would paint a different picture than the one ESPN is authoring. 

4) AL East Analysis
Finally, let's look at two teams mentioned in the article: Red Sox and Yankees. The two proverbial beasts of the AL-East. Both were seen giving multiple signs at Rogers Centre, so let's assume they kept that up starting in 2010. This would theoretically defeat the Blue Jays' advantage as hitters, so they should have done worse against these opponents in 2010. Let's look at the stats. In 2009, (before the teams were aware) the Sox were 5-4 in Toronto. and the Yankees were 6-3. Pretty good for teams who are being cheated. In 2010, (after they figured it out) the Sox were 7-2 and the Yankees were 3-6. When you take the sums of the records:

Playing Against Cheating Blue Jays:
Playing Against Fair Play Blue Jays

The two teams actually did worse once they began guarding themselves against the Jays' "tactics". This stat doesn't even include the steady Jay improvement over that year. Their combined road records against the same two teams in 2009 was 6-12 and that improved to 8-10 in 2010. These stats show that the Yankees and Sox actually did worse when they played the Jays without the alleged cheating. Blue Jays didn't cheat at home, their wins tell us that, and any sign stealing that was occurring didn't significantly alter what matters most in professional sports--wins. 

When all is said and done, the mathematical argument put forward by ESPN is shoddy at best. For such an esteemed broadcaster to claim statistical differences is a cruel trick on the in-numerate public. Sports' fans and mathematicians both have to be aware of the dangers of statistics. Often times an "independent analysis" showing significant "statistical deviations" is not telling you the whole truth.


Monday, August 8, 2011

Un-Locking Prior Knowledge

I enjoy mathematics in the morning. It wakes my brain up, and makes my coffee that much more comforting. Much of the deliberate mathematics learning that I do takes place in the morning. I say deliberate, because mathematics always finds ways to sneak itself into all parts of my day. Morning is just when I open the door and embrace the learning with open arms. 

Today's dose came courtesy of @republicofmath via @jamesgrime. The problem took longer than I expected, but the result was quite eloquent. I ended up using a method that I had no intention of ever using again. It was the use of this prior knowledge that made the experience valuable. 

The problem is paraphrased as follows:

A lock consists of three reels. Each one is numbered 1-3. You can move reels 1 and 2 together, and you can move reels 2 and 3 together. No reel can move independently. If the combination begins at 333, what series of moves will result in the code 221?

For example, if you chose to move the left-hand reels one spot forward, the combination would then read 113, because both reel 1 and 2 would reset to 1. 

Initially, this is how I began to familiarize myself with the problem. I tried a couple of moves just to see if the solution was obvious. After the trial-and-error approach failed, I took to a more systematized approach. Sticking to Polya's iconic approach, I had to first understand the problem before I could devise a plan. The trial and error allowed me to understand the patterns and environment involved. 

I then had to devise a plan. I did so, but it did not work straight away. I thought that I could work backwards to the solution. If I began with the goal in mind (221) and then performed all of the possible moves (Turn Left or Turn Right), I would eventually land on 333. I would then re-trace my steps back up the path I created. Because of the cyclical nature of the lock, I assumed all moves were forwards. So if I turned right, 213 would become 221. This strategy is shown in the picture below.

I soon found that values were repeating, so I pruned their branches so not to become redundant. After about 10 minutes, I began to get more redundancies than new pathways, and I abandoned the approach. The next attempt was very similar, but with a subtle twist.

I placed 221 in the middle and recognized all four moves (Left +, Left -, Right +, and Right -) as valid. The only change was that I allowed forward and backward turns to be valid. It only took one iteration to realize that this new approach was going to be as unfruitful as the last, and it was abandoned. The first step is shown below:

At this point, I was seriously doubting if there was a solution, and knew the only way to prove the absence of a solution would be the abstract case. I labeled each reel A, B, and C, and began to list a set of possible moves that one could turn A, B, and C.

In invented a notation to go along with this procedure. If a reel moved forward 1 spot (like from 2 to 3) it was given a +1. If it moved 2 spots forward, it was given a +2. Etc.

From this I began a list of moves:

(1) +1A +1B  C -->Was moving the left reel one spot
(2) A +1B +1C --> Was moving the right reel one spot

From here I created the only 5 viable moves:

(3) +1A +2B +1C --> Moving left and right
(4) +1A B +2C --> Moving right forward and left backward
(5) +2A B +1C --> Moving left forward and right backward

Because of the cyclical nature of the lock, a -1 would be equal to a 2. Because if you started at a neutral position and moved -1, you would land at the same position as if you moved +2. Essentially, the moves were all taken modulus 3. This notation was far too complicated to continue. I had to eliminate the letters to create this list of 5 possible combinations of moves:

(1) 110
(2) 011
(3) 121
(4) 102
(5) 201

I then investigated all possible combinations of the moves and the solution came to life. I began by applying every move twice in succession. So applying (1) twice resulted in:

110 + 110 = (6) 220

This was a new result, and had to be added as number (6) to the list. After the original 5 were tested, 3 more created unique entries.

(7) 022
(8) 212

The next part took the most time. I tested every two move combination to see whether it created a unique move. As evidenced by the diagram, each one failed to create a unique pattern. (the attempt is marked with an ‘x’ if is was a repeat.)

It was then when the prior knowledge became unlocked:

I had created a mathematical group of 9 elements. There was no way of applying the elements in sequence that would break the group.

I honestly never thought I would use Abstract Algebra again, but it was a pleasant surprise to have to utilize it. The new group, G, consisted of 9 elements. The aforementioned (8) and an identity element (000) which represented not moving the reels at all. I quickly tested the requirements of a group:

There must be an identity
Operations have to be associative
There must be closure
Each element must have an identity

The operation is piecewise addition modulo 3. This is associative. The identity was mentioned earlier, and the combination calculations showed closure. This was a group whose members represented turns of a wheel.

So the question became, if you start at 333, can you get to 221? 333 is our neutral position. In order to get to 221, we need to turn reel 1, two places, reel 2, two places, and reel 3, one place. I am looking for the element 221 to be included in the group in order to unlock the lock. It, however, is not, and so this problem is impossible.

The only possible combinations that are viable are those included in the group. (pictured above).

Now the point of the problem was not the solution. (although I enjoyed the finality after about an hour of work). The joy came when I unlocked prior knowledge to solve the problem. It encouraged me to know that along my mathematical journey, I have encountered various methods. The real powerful math comes when I can apply a method because of a deep understanding of it. The repetitive, one-dimensional problems given to me in university did not result in satisfaction; I had to use it flexibly in order for Groups to become mathematically potent. I think all teachers of mathematics should have to struggle with a problem for a while. It gives valuable insight to the students’ viewpoint. Sometimes teachers forget how difficult it is to struggle.


Saturday, August 6, 2011

Large Number Numeracy

Gigantic numbers are all around us. This has never been more apparent since the US Debt ceiling became a major issue. The facts and figures are thrown around by the news, and joked about on Late Night television to the point where their potency is diluted. Not many Americans seriously understand what a trillion dollars is. That statement can be broadened to include all earthlings. The comprehension of large numbers is a very interesting task, especially given the role that the media plays in our students' lives. 

Take game shows for example. I remember watching the first ever episode of "Who Wants to Be a Millionaire?" with a much younger and more vibrant Regis. The idea of a million dollars fascinated the public. How could such a large amount of money be given away? Back then one million was at the forefront of the world's consciousness. Everyone thought that their life would be easy if they were "millionaires". Not many understood the actual breadth of that wealth. Nowadays, we see game shows like "Deal or No Deal" have special $10,000,000 shows, "Wheel of Fortune" puts a million wedge on the wheel, and recording artists sing about how they want to be billionaires "so friggin bad". (Radio Edit Included). The world's recognition of large quantities has diluted, and that leaves the generation in today's schools--who never marveled at a million--unimpressed.

Maybe teachers should drop the catchy handles "one million", "one billion", and "one trillion". We could talk in terms of what students understand. When I was 16, I thought $1000 was a large amount of money. Now put a million in that context. A million is one-thousand one-thousands. That is a dollar for every dot on this diagram:

Have students circle the amount of dots that represent a new computer or car; the tiny circle will provide perspective. 

If that is a million, then a trillion would mean that we would have to replace every atomic dot in this image with a duplicate of the whole. We essentially would have a million, millions. Such a large quantity begins to warp our minds. Once quantities get so large, people begin to lump them all together as "big". In fact, the US government is probably lucky the debt is over 1 Trillion, because people would probably view 999 billion as larger. Nine hundred ninety-nine carries more weight that one. 

I was watching a game show where the contestant was asked to estimate how many people watched the 2010 Superbowl on T.V. She guessed 2 million. When asked to detail her thought process, she said that there were 300 billion people in the US, and only about 2 would watch it. She said that she meant to put 2 billion, but mixed up the terms. This tells me two things: 1) This lady has no idea how much a billion is because there is no way 300 billion people would fit on earth comfortably. And 2) She has lumped million and billion into the category of "Big" numbers. Besides her horrible estimation skills, she showed  horrendous large number numeracy. 

So how can we, as teachers, support the understanding of large numbers? I would start with small, lab-like activities. They fit great when there is a little extra space before the school bell. I once brought my class to the library and asked them to estimate, by sight, the number of books in the library. Then they had to come up with a way to justify. I used the librarian's records to check answers; all solutions held some merit. Once I ended class by asking the class how long it would take to type out one-million semi-colons into a single document file. We timed 10, 20, and 50 trial runs, and then extrapolated the data. Such an activity would be excellent for large number numeracy, linear data, scatter plots, line of best fit, interpolation and extrapolation, linear functions, etc. I will leave this up to you, and your students, to solve. 

Allow your students to spend significant time with large numbers. This will only ease the transition into the infinite when that eventually occurs. Use an up-to-the-minute calculator of the US National Debt, and attempt to create a visualization of the monstrosity of the number. How high would a trillion page stack of paper be? How long would a trillion centimeters be? Allow students to pose their own problems and seek solutions and representations. They will be creative in their representations, and impressed with their own ability to comprehend large numbers.