Tuesday, May 31, 2011

Must it Always be True?

This morning on twitter, there was a problem that I just had to solve before going out the door. It is safe to say that these types of problems are my vice. Number Theory has always held a special interest to me despite, according to G.H, Hardy, having "absolutely no practical use." (A Mathematician's Apology, 2001). This has all changed with the inception of encryption.

I wish just to present the problem and then muse on its educational significance both for my personal learning of mathematics, and for that of my students.

N is the 4-digit integer 6_9_. If these two digits are reversed, explain why the resulting number must be 2970 more?
(posted by @dmarain to @cuttheknotmath)

I immediately look toward the base system when digits are switching around. When a digit moves from one place to another, it takes on a new meaning. I expect this new meaning will give me the difference I desire.

First number can be represented as (with a, and b in the set of base 10 digits):

6X1000 + aX100 + 9X10 + bx1

When we switch the digits, the new number becomes:

9X1000 + ax100 + 6x10 + bx1

The second is always larger (which is an interesting discussion to have with students) so we subtract to keep difference positive. We se that because the a and b did not switch orientation in the place value system, their value remains constant. Therefore, they will cancel out upon subtraction and have no bearing on the final solution. This is why it is a constant regardless of the two digits.

9000 +60 - (6000 + 90)
9060 - (6090)

In this way, we show that for the general case (there are 100 cases in total--2 spots, 10 digits) this fact is always true.

The idea of proof to students is very elitist. In high school, a list of examples where it holds is often sufficient. Once the list gets long enough, the proof is concluded. In this case, it would be easy enough to show the students there are 100 cases; this may discourage a plug-and-chug method. Instead, number tricks like this help students realize two things:

1. The basic qualities of our base 10 number system
2. The many interesting patterns that numbers create

In my curriculum, both of these topics are mandated. Posing this problem to a class of grade 10s gives them opportunity to create hypotheses, test them out, and then dive deeper into the number system to look around. I would imagine that such an activity would pair nicely with one on scientific notation or binary numbers. If they are really keen, a proof like the one above may be deciphered. Then we drag algebra into the mix as well.

There are many areas of useful mathematics that are left out of textbooks. As teachers, our pursuit of learning can greatly effect our teaching.


Monday, May 30, 2011

Attaching a "Why" to the "How"

There has been plenty of recent twitter talk about the process of moving the focus of mathematics education away from the "how" and toward the "why". Traditionally, students have been trained to approach a question--usually given to them by an outside source like a teacher, textbook, or test--with the express intent to show the grader "how" it is answered. Such responses often include the use of algorithms, formulae, or memorized facts we know to be true. (These facts are in no way axiomatic, but constant repetition reduces them to that state. Students have answered them so often, the process loses meaning. Take 2x2 for example.)

The focus on the how encourages a race to the finish. Thomas C. O'Brien calls this phenomenon "Parrot Math". (Phi Delta Kappan, Feb, 1999) More specifically, it is the process that every student goes through when answering the question "how"--they attempt to repeat or imitate the process that has been shown to them. These carbon-copy answers are created without the knowledge of the mathematics that ensures their success, just as a parrot can possess a large vocabulary without understanding the intricacies of language.

In my class, I attempt to elicit the "why" as much as possible. The solution to "why" is much harder to come by. There are times in class when I challenge students on their mathematical statements--just to keep them honest. A student was working on sketching the graph of a rational function when she called me over. She explained to me that she could not get part c:

c) Find the equation of the Vertical Asymptote(s)

She was having trouble with a "how". She could describe what a Vertical Asymptote was and how it effected the graph, by had simply forgotten the neat and tidy process to generate them. I explained to her that you set the denominator to zero and solve. Her face instantly renewed with vigor because her question had been solved. I then continued to challenge her with "why"s. I asked why the denominator can't be equal to zero, but got a very standard, and hollow, response:

"Because you can't divide by zero."

Acting as confused as I possibly could, I asked her, "Why can't you divide by zero?" Her joy instantly drained. It took a 5 minute conversation about piles and sticks before we decided that you could divide zero sticks into 10 piles, but couldn't divide 10 sticks into zero piles. Division had become a "how".

Moving on from the distinction, it is plain to see communication as a central element in "why". This includes student-student and student-teacher communication. The Standards published in 2000 by the NCTM call communication an "essential part of mathematics and mathematical education." It is these skills that build "meaning and permanence for ideas" in math. (NCTM, 2000) I think most teachers agree that both meaning and permanence are lacking. Just come back from a summer holiday with a group of grade 9s.

The simplest, and most useful, way I open the avenues of communication in my class is through yellow paper. That's my secret. I grab a couple yellow, lined pads of paper from the office and hand each student a page. We complete the daily task on it, and they are to provide a reflection or explanation when they are through. The students know I read everything handed in on a yellow piece of paper. It took 3-4 practices before the students began to communicate their methods effectively in writing, but my persistence has paid off. I thought it would be interesting to include some student thought from my Mean, Median, and Mode task posted earlier on this blog. (It will be helpful to read that post if you have not already)

The writing created 4 distinct categories of learners, but I would have never made this distinction without the pursuit of "why" and the open communication of the yellow paper. The students were required to fix the set {1,2,3,3,3,4,5,5,6} so the mean, median, and mode are all 3. The mean was the issue.

Some students took out numbers from the high end until they met a target sum. These students knew that the average was dependent on the sum and the number of entries; they altered entries until the mean worked.

"There are 9 numbers in total. Not wanting to change this # that I know will divide by 3. 3 (the # I want to get) timesed by 9 will give me 27. I take the difference of 27 and 32 (32 because thats the original total of these 9 #'s) and I want to get 27 instead of 32 because 27 / 9=3. So I take away 5 (the difference) from 6. 1,2,3,3,3,4,5,5,1 = 27/9 = 3. Mean = 3."

This explanation above (repeated verbatim) shows excellent mathematical communication. She has shown me the "why" in her process. She has also shown me that she understands how the total sum effects the mean. She uses mathematically rich words like sum, difference, and mean. She has also used less eloquent terms like "timesed" and "take away". This shows me that she persevered through lapses in thought. She hit a wall in her explanation, and powered through it by sheer determination. This is how I know she can now answer "how" to find the mean, and "why" it is that way.

Other students tried lowering the entries one by one until they got the mean correct. The logic behind this dilution of sorts was they wanted to keep all entries in order so the median was not altered. Genius! I may have seen this process as primitive if they had not communicated their motives. Other students added entries so the large ones took less effect. Others talked of "balancing" the entries. I had never thought of mean as a balance before.

Creating meaning in mathematics is not about throwing the "how" out the window. Mathematicians have worked hundreds of years to establish hows. True meaning comes when students are given the opportunity to both attach a "why" to the "how" and communicate that connection to others.


Sunday, May 29, 2011

Merit to Mathematics Labs

There is widespread turmoil among teachers and students when it comes to the practicality of mathematics. School mathematics, at the middle and high school levels, has moved out of the elementary niche of rudimentary skills, but has yet to make it into the realm of complexity necessary to apply it back into the world. Our happy compromise, as teachers, is to go with a two-pronged attack:

1. Tell the students that the practicality comes later
2. Create word problems about trains leaving stations or people tossing balls off cliffs

Every teacher of mathematics (from the wide-eyed rookie to the well-weathered veteran) has encountered "the question" numerous times. We hear it so often, that I would imagine many teachers have a well-rehearsed response to the query. I know I do. Even though we are prepared for this onslaught, the thought of having to employ our answer triggers chills down our spines and sends us retreating into the staff room for cover:

"When are we going to use this?"

There are sects of mathematics that present obvious answers to "the question". Measurement has cooking, geometry has the trades, and probability has gambling; each to their own segment of society. I have already begun to envy the science teachers who bottle themselves up in their back room only to emerge with a concrete and engaging example of when people actually use their discipline. This takes the form of a specimen jar, an interesting chemical reaction, or a proposed perpetual motion machine. I still remember anxiously awaiting "Explosion Fridays" in my Chemistry 30 class in high school. It seems as though these subjects have endless applications, and they harness their practicality in a laboratory.

A lab is where the students get a hands-on, experimental crash-course in their discipline. It is my belief, born partially out of my aforementioned jealousy, that there are situations in the mathematics class where students need the opportunity to play around with mathematics in a laboratory setting.

Before we get too much further, I must give a word of caution. You may have noticed that I always use the term "mathematics lab" over "math lab". Although the latter may be easier to say, speaking too quickly may give your students the impression that they will be creating methamphetamines. Awkward explanation for the administration; I digress.

There have been 3 deliberate labs in my math class this year. My hope is that by briefly detailing the setting and task, you too will begin to see the value in a practical, exploratory, and hypothesizing environment.

1. Monty Hall Problem Mathematics Lab

Students were presented the wildly popular problem verbatim from the game show. (I have found that the newer rendition found in the movie "21" works better as an introduction.) We spent a few minutes hypothesizing and gathering into our lab groups of 3-4 students. The majority of the class was spent re-enacting the situation and creating data. Not only did this build skills in the scientific method, it is, in my view, the only way to fully understand probability. Students must understand that the theoretical calculations we create are actually mimicked in their experiments. The second day, the class reported their data, and it was compiled. Most of the class was spent discussing what the data was telling us, which hypotheses were correct, and how we could alter the experiment to get differing results. (Finding the problem is a simple as googling "Monty Hall Problem". I would encourage you not to research an answer until you have tried the very same lab.)

2. Route to the Commons Mathematics Lab

I gave the lab groups a miniature map of the school that is usually given to substitute teachers. I asked them to resolve the problem of hallway congestion by measuring out the fastest way to the commons and back to class. The problem was left intentionally vague, and the students were given a meter stick, a pencil, a calculator, and a centimeter ruler. Students began suggesting the route they took and locating it on the map. A scale factor was devised and tested on various lengths throughout the school. As students became sure of their results, I challenged them to create a map for their most efficient school day. They began charting their path from class to class. Students were encouraged to test this theoretical shortest distance during lunch in the hallway traffic; soon more crowded hallways needed to be weighted differently. Although I never got this far, a general "congestion constant" could have been developed for each corridor. In a main hall, one meter may be equal to 2 meters on an abandoned one.

3. Fermi Estimation Mathematics Lab

I have used this format a number of times this year to build basic numeracy skills. Students are asked to calculate large or small quantities in a general sense. It is entirely about numeracy-based estimation. How many hairs are on your head? How many hairs are on the floor of the school? Students enjoy creating outrageous questions and tackling them empirically. How many hairs does the average student get caught in their mouth in a given school day? The problems expand to include basic arithmetic, probability, surface area, etc. For ideas of Fermi estimations, see "Guesstimation" by Lawrence Weinstein and John A. Adam.

There are a lot of practical topics in mathematics. Creating disenchanted questions about choosing coloured tiles from a bag does not give math its full due. Setting up mathematics labs creates room for curiosity in math; students take direction of the content. Not to mention that giving students a practical platform to do mathematics will reduce the odds of having to field "the question"


Friday, May 27, 2011

Playing With Mean, Median & Mode

Teachers in Saskatchewan, Canada have had a lot to deal with lately in the classroom. The ongoing political battle has effected hours of direct instruction in a very real way. I quickly noticed my classes becoming disjointed with large amounts of time between each encounter with the mathematics. Needless to say, I entered today's lesson in Math 9 with a little apprehension. A Friday morning after 2 days of job action and a long weekend didn't sound like the most nurturing of environments. I decided that the time was ripe to attempt a lesson that has been in my mind for a couple of months; the following account is the story of the task, presentation, student reaction, and important learnings.

The students were introduced to the concepts of Mean, Median, and Mode earlier this week. It had been 3 days, so I quickly refreshed their memories with a standard problem. I gave them a list of data, and had them compute (in pairs) the three measures of central tendency. After some painful re-hashing and peer tutoring, the class was then alerted that we were going to take a major shift. With a quick swipe of the eraser, I eliminated the data set and left only 4 facts on the board:

n = 19
mean = 4.47
med = 4
mode = 2

I asked them if they could re-construct the data set using these facts. It should be noted that I unfairly took advantage of my students' laziness. When asked if they should write down the set of numbers to solve the original problem, I told them not to bother. Do I feel guilty? Meh.

I expected the class to complain about this crazy task. Some began to rack their working memories for the last remaining traces of the numbers--but with little luck. After these efforts fizzled out (2 mins max), I posed the problem I actually intended them to solve:

Find the data set with the following attributes:
n = 1
mean = 3
med = 3
mode = 3

Soon the fear of wrong answers wore off and partners were conversing with other groups. After about 3 minutes, we decided that the only data set that fit all four was :


I asked very quickly for an explanation and verification of the facts, and then erased the "n=1" from the board. I turned toward the class (with a dramatic pause) and repeated the exact same question:

Find the data set with the following attributes:
mean = 3
med = 3
mode = 3

Group work began immediately. An electric hush fell over the room as they worked for 10 solid minutes discovering numerous data sets that fit the description. As I circulated, I overheard phrases like, "what about the median" and "won't that change the mode". At this point, I began to ask questions that required students to search their numeracy skills and metacognition. "How do you know you need to add a large number?" and "Why did you decide to start your set with 1?". In due course, we generated a class list of sets, including one that could go on "forever and ever".

{1,2,3,3,3,4,5,5,6} *

The list itself reveled a lot about how each group thinks about numbers. Certain groups have been trained to think that lists start with one. This seems natural; we start school in grade 1; we begin counting with 1--why not begin listing with 1? Other students took the ingenious route of understanding what a list of 3's does. When we had the list, I asked for observations about it. I often do this with my class to begin the processes of pattern recognition, problem solving, and problem posing. The list of observations was excellent!

"Number 1 is wrong"
"None of the patterns include a zero"
"All use only whole numbers"
"There are no negatives"
"They are all in ascending order"

Although I would love to detail the conversation along each of these points, I would rather use them to illustrate the growing nature of such a problem. These student inquiries can be used to explore the concept of Mean, Median, and Mode in a much deeper way. What began as mathematical play, quickly turned into serious mathematics. I would have loved to set up think-tanks to explore how negatives can be used in sets of numbers. Maybe a particular ambitious student would take on the trouble of finding an integer mean with fractions as data points. (I imagine this would lead quickly to paris of fractions that sum to 1, but I can never be sure where student thought leads). I chose to go into an interesting conversation about zero. I touched quickly on the history of zero, what adding zero to a data set would do to each of the three measures of central tendency, and constructing sets with a mean of 0. The last topic ran naturally into that of negative numbers.

I assigned the class to "fix" the first entry on our list, and describe, in words, their thought process. The topic of writing in mathematics is one for another day.

My point is, changing the focus of a very routine mathematical exercise changed the way the students saw the topic. It began to grow right before them. It is quite artificial for me to separate the extensions; it has already been shown that the topics intertwine. Posing questions in a playful atmosphere unlocks student drive. For an approach often coined, "Fuzzy Math", it sure led me and my students into very serious (and curriculum supported) mathematics.


Thursday, May 26, 2011

Fractions From Digits

This week marked my baptism by fire into the twitter world. It was not long until I was neck deep in tweets, favorites, re-tweets, and followers. The eternal nerd awoke inside me when I was confronted with my first NCTM "Problem of the Day". A simple, yet dangerously deep, question was posed. Wanting to cement my reputation as a responsible twit, I sat down and began to tinker with the theory.

The question was as follows:

How many different fractions can you write using only the digits 1,2,3 & 4?
Be sure to include fractions greater than 1.

Immediately, I changed the word "fractions" into "rational numbers". That way there would be no debate whether a number with a denominator of 1 is a fraction. As I began to experiment with the obvious nature of the problem, certain problem solving strategies emerged. I began to list observations and questions. Every digit, when placed over itself, will create a fraction equal to 1/1. I then became concerned with the reducibility of the rationals involved. What operations (+,-,x,/) are allowed on the digits? Can we repeat digits? It is not hard to see that the problem was becoming very large, very fast.

I decided that ordinary operations were not possible, because there was no way to know if a multiplication of two numbers in the set X={1,2,3,4} would yield a number whose digits were also in X. I then began to explore any operations that I could think of that may be acceptable on this very unique set. After addition, subtraction, multiplication, division, and exponentiation failed, I decided to take a step outside of the box. I decided to show that there are an infinite amount of rational numbers that can be created with the digits by "smushing" digits together.

I called this new operation "composition". If you compose 1/2 and 3/4 you simply get 13/24. This way I was ensured that every subsequent "composed" fraction would have only digits in the original set X. From there I created a list of 11 rationals that had a single digit in the numerator and denominator. (1/1, 2/1, 3/1, 4/1, 1/2, 1/3, 1/4, 2/3, 3/2, 3/4, 4/3). I used the fact that these 11 were unique to begin the proof of this new infinite set of numbers.

From this point on the "@" symbol will represent composition. I now had the fact that:

a/b @ c/d = (10a + c)/(10b + d)

The first thing to be noticed that both a/b and c/d are in lowest terms and unique as rationals. So the newly composed fraction is also in lowest terms. The new fraction would be reducible if a common factor was available from all 4 constituents (a,b,c,and d). If such were the case, c and d would have a common factor, and wouldn't have been in lowest terms to begin with. This contradicts the construction of the new fraction.

The second thing to notice is the uniqueness of the newly composed fraction. We know no a/b = c/d = e/f. Let's assume that for some reason, we find two composed fractions where:

a/b @ c/d = a/b @ e/f


(10a + c)/(10b + d) = (10a + e)/(10b + d)

so breaking apart the fractions we get:

10a/10b + c/d = 10a/10b + e/f

From here it is easier to see that this can only be true if c/d = e/f. We know this is not the case. So from the original 11 fractions, we can now create a set of two-digit fractions that are all unique and irreducible. This list continues to grow when we consider "composing" these two-digit numbers together to get 4-digit ones that follow the same rules. Only now the 2-digit numbers p, q, r, and s follow the form:

p/q @ r/s = (100p + r)/(100q + s)

In general, let 'x' be the number of digits of {p,q,r,s}, then:

p/q @ r/s = [(10^x)p + q]/[(10^x)r + s]

This pattern continues to create an infinite set of rational numbers from the digits 1,2,3, and 4. Of course, this list is not exhaustive--like most lists of infinite length. In fact, using this method, every fraction must have 2^n digits, where 'n' is a natural number.

Today's problem has been yet another example of how an innocent problem, can lead in various directions. How many possible 4 digit combinations are possible in this set of numbers? 8-digit? Such are problems for another day.