Sunday, October 30, 2011

Destroying Functions

I have spent the better part of 2 weeks going over various mathematical relationships in my Grade 10 class. They have been represented as tables of values, arrow diagrams, and sets of ordered pairs. Relationships, both qualitative and quantitative, have been defined, analyzed, and graphed. My focus on graphical literacy has been previously detailed on the blog. See the link below for details:

Numerous relationships were handled. Students we required to create a family tree and then represent its branches as a table of values and set of ordered pairs. Throughout the various exercises, the words "input", "output", "domain", and "range" were consistently used. My family tree mapped the connection between a Domain of "Names" to a Range of "Familial Relationship". Some of my ordered pairs then became:

(Ben, Brother)
(Kevin, Uncle)
(Marla, Cousin)

Qualitative relationships eventually gave way to numerical ones. Students spent time looking at discrete data such as temperatures, test scores, and mileage ratings. It was through these exercises that the terms "independent variable" and "dependent variable" became part of the classroom jargon. On the whole, the topic of relations was well received; I knew that the next step was to define a function--a process that has gone poorly for me before. 

I have seen several analogies that are designed to explain functions. Each is usually paired with the textbook definition of function-- "a special type of relation where each element in the domain is associated with exactly one element in the range" (Foundations and Pre-calculus 10, Pearson, 2010).

1) The Function Machine
This is the overwhelming favourite of math teachers. In the function machine model, the input value (domain element) is placed in some kind of illustrated hopper of the machine. Different portions of the machine contain different operations and the result is an output value (range element) that drops out of the machine. The idea being that if the output was two different numbers, the machine would no longer be functioning. Teachers then stack the "Multiply by 2" machine against the "Add 1" machine to create the "2x + 1" function machine. The proper output values are then reduced to a matching puzzle of the possible machine parts. 

This model may create a visual crutch for the computation of ordered pairs, but does not help students decipher between functions and non-functions. I wanted to use a method that would make the students actively create and modify functions. 

2) The Promiscuous Range
I have never used this analogy, but a student brought it to me from another math class in my building. In a function, the domain and range are given personalities. The domain elements are faithful to their mates. They are portrayed as hard-working, blue collar citizens. If a relation is a function, each domain value would never dream of having more than one range mate--that is out of character. The range values, however, are more promiscuous. A function still allows a range element to be paired with more than one domain element. Students find all sorts of off-colour words to describe the range element; in the end, a function has faithful domain elements but possibly promiscuous range ones. 

3) The Split Decision / Uncertainty Model
This is the model I chose to employ; I felt it could be used as a diagnostic tool for students when presented with a relation. I presented the definition to the students and we analyzed several relations in tables of values. The input values always lead to output values; each ordered pair is presented as a decision. If a certain domain value presents a split decision or uncertainty in the answer, the relation is not a function. If the entry "2" appears twice in the relation, but one time it is mapped to "3" and the other to "4", the domain entry has created uncertainty. How is one to answer definitely if there are two options? Students took to this immediately. 

I created this image to further represent the "decision" model.

Students are asked if the vending machine is "functioning". (You'll have to excuse the pun). It takes a quick second for students to realize that the input "A7" creates uncertainty. Choosing that input may yield 2 possible outputs--there is no way of knowing. 

Functions are practiced for a couple days until I feel it is time to introduce the Vertical Line Test. I do not give them the rule, but rather tell them a story...

The first computer game I ever bought was Sim City 2000. It was somewhat of a cultural phenomenon where the task was to create a functioning city. It gave me my first taste of politics and economics. The game was addictive; hours would melt away as I sat in front of the screen debating whether or not to raise taxes on my figurative citizens. Despite its appeal, there came a point in every game where it was no longer fun to fix run-down neighbourhoods. As seen in the screen shot below, there were several disasters that could be used on your own city. Tornados, hurricanes, and even space monsters were possible. I told my students that we were going to take the same mentality into our study of graphing functions. 

I brought up a blank Geogebra file on the IWB and created a blank Table of Values on the adjoining white board. I filled in points one at a time while graphing them as I went. After six or seven, I told the class that the monotony was too much; we were going to begin destroying the function. By themselves, each student was required to create an ordered pair that would destroy the function. After about two minutes I elicited the responses from the class.

As options came, I graphed them on the IWB. After about five or six, I stopped and asked one question:

What do all of these destructive points have in common?

Discussion ensued. By the end of it, we had coined the Vertical Line Test. Students tested their new theorem on sets of points and continuous functions. They now had linked the fact that two ordered pairs lying directly on top of one another created a split decision. They had linked the algebraic and graphical tests for "functionality". 

Functions are the base of higher mathematics in high school. Grade 11 and Grade 12 courses focus almost exclusively on them. I employed a framework where testing if a relation is a function is an active activity. Creating the space for students to actively create and destroy relationships builds a thorough understanding of the topic at hand. 


Saturday, October 22, 2011

Graphing Literacy

My school division has been pushing literacy for a few years now. The division priority has filtered its way down into many programs at the school level. As a basic premise, if students are exposed to literate people and perform literate activities, their skills will grow. 

As the term is dissected, it seems that every stakeholder can find a way to skew the term to mean that their discipline is a crucial part of being literate. Reading and writing skills are an obvious avenue, but the ideas of technological and social literacy have emerged as important parts of every student's school experience. Riding shotgun to these ideas is the idea of Mathematical literacy--Numeracy. 

Most students carry the misconception that mathematics is a unique commodity that is unlike everything else they encounter in schools. Somehow, math educators have managed to make a system created to understand the world seem completely disconnected from it. I believe that the burden of innumeracy is born from this disconnection. Mathematical frameworks can be applied to situations that were otherwise thought of completely innumerate. 

Students learn the Cartesian system of rectangular coordinates fairly early in their High School career. By grade 10, it is used exclusively for the study of functions--algebraic functions. A system of relating variables has been completely transformed into one that is only associated with "x" and "y" and some kind of "slope". Before, and throughout, the unit on relations and functions, my students are encouraged to graph their thoughts--give me a graphical representation of their actions, emotions, justifications, etc.

Students see that changing the dependent and independent variables affects the story that the coordinates can tell. A graph can guide student decisions. Students are encouraged to graph my tendencies as a teacher. If we put "Time into Class" on the x-axis and "Chances Mr. Banting will let you use the Bathroom" on the y-axis, there is a relationship that exists graphically. Students bring a unique experience into the situation. It provides a rich discussion on the shape of the data, because every student is an expert in the field. Examples from textbooks often talk about things so distant from their lives; homework littered with distant topics only further distances mathematics from them. 

Open up a discussion or critique on the topic. As the teacher, I lobby to fix the graph to make me look "nicer" or "more fair". Maybe we graph a "fair teacher" on the same grid. How far is my line from theirs? What does that represent? What if we switched the variables? What if the variable became "Chances of getting a Drink"? Would the line be lower? Higher? What would each represent? 

Students intuitively graph; I open a unit up with this problem, and most jump right on board with strong opinions. They can begin to apply the mathematical relationships because they are familiar with the data. Discussions of continuous and discrete data are easily pulled from an activity like this. We could have two graphs:

1) Chances of going to the Bathroom after a Maple Leafs Loss vs. Time into class
2) Chances of going to the Bathroom after a Maple Leafs Win vs. Time into class

The vertical translation of these side-by-side graphs can be very telling. Soon the class has a "Mr. Banting Mood Graph". What is the Domain? What is the Range? These will begin as qualitative entities. Maybe we apply a scale of 1-10. Do we include fractions? Maybe the data needs to be discrete? Maybe groups break off and graph the situation themselves and them return with a narrative argument for their solution. When the class is ready to move into function notation, a line of best fit could be developed. Could we extrapolate the data if a class was 2 hours instead of 1 hour? How accurate would the data be? How would the graph shift? 

All these questions fit into the curriculum, but emerge much spontaneously. Students feel the independence of changing variables and graphs. Even as I am writing, my script has taken on a tone of freedom and exploration. A simple concept, linked to some powerful emotions, creates a rich mathematical discussion.

I give my students a personal graph assignment every year. After a discussion much like the above, they are given a handout with a Cartesian Plane on it (only quadrant 1). They are asked to pick any two variables and graph the relationship. There is a space for a description of their reasoning below. This is, by far, my favourite assignment of the year. Every inside joke and class dynamic comes out in one form or another.

This year, the staff won intramural volleyball. Two students we beat in the finals are in my class. Needless to say, our volleyball skills come up in many conversations. One student graphed the "Skill level of a volleyball player" vs. "How valuable they are to their team". I made an unflattering appearance on the graph--their ordered pair was very generous. I make sure to re-draw the graph in my interpretation. The assignment became a form of graphical satire.

This activity is very free-flowing, and builds great numeracy skills. It re-inserts a mathematical framework into student consciousness. I get students graphing freelance relationships for me for months after the topic has faded or been taken over by slope y-intercept form. Many mathematical implications emerge by simply playing with graphs. 


Saturday, October 8, 2011

Proper Workspace for Workplace

My province is in the midst of a major overhaul on its curriculum. This puts me in a very interesting situation. I am a new teacher in a large division filled with veteran teachers that all feel as overwhelmed as myself. I can't decide if this is a curse or a blessing; I simply continue to roll with all the punches that curriculum renewal brings. On top of the nuts-and-bolts of each new course (5 of which I teach for the first time this year), the division heaps on division, school, department, and personal learning priorities. To make matters even more confusing, each initiative comes with about 35 acronyms. I can't tell the difference between AFL, PLO, PLP, PPP, SLI, PBL... you get my drift. Amidst the chaos of red tape, I believe I have found something to hang my hat on.

Our department goal is to find creative ways to develop and foster a growth mindset in our students. Our school has a very large proportion of unsuccessful students. Many students feel as though math is too difficult, or genetics has blocked their possible success. This is not a phenomenon specific to Tommy Douglas Collegiate. I read Dr. Carol Dweck's book Mindset last year as a part of school book clubs, and found its message intriguing. I coupled this with research I have been doing into Problem Based Learning in Mathematics. The creation of the new "Workplace and Apprenticeship Mathematics" pathway in my province completed the perfect storm of personal learning. All these factors contributed to my brainchild.

A little more explanation needs to go into the pathways in Saskatchewan. Grade 9 math is taken by every student. In Grade 10, there are 2 options--Foundations and Pre-calculus 10 and Workplace and Apprenticeship 10. In Grade 11, FPC10 splits further into 2 pathways--Foundations 20 and Pre-calculus 20. Workplace and Apprenticeship Mathematics 20 is also offered. In Grade 12, all 3 strands are offered at the 30 level along with Calculus 30. The whole idea is for students to take the mathematics that is suited for them and their future. 

The Workplace and Apprenticeship pathway (affectionately coined "A&W Math" by my department head) is designed to build skills pertinent in a work setting. The textbooks are designed around unit projects, and the lessons focus on application. Links to the Saskatchewan Curriculum are found below:

I strongly feel that the spirit of the courses require that they be taught in an authentic setting. This feeling, combined with the department initiative and some work with PBL, has begat my vision for how my 3 sections of A&W math ideally will look next semester. My hope is that the reader of this post will contemplate my thoughts and comment on their feasibility. Any feedback from prior experience or similar experience is greatly appreciated. 

I want to set up a completely project based environment for the course. This begins with the set-up of the room. Desks are removed and replaced with small, circular tables with 3-4 chairs around each. I have chosen 3 as my optimum group size, but will leave room for special circumstances. My classroom is equipped with a SMART board and I am working on access to a half-pod of netbooks. Unbridled access to internet resources will only fuel the independent feeling I want in the room. 

The format of the course will look roughly as follows:
  • Every student/group is given a binder with a print out of the curricular objectives in it. The binder includes rubrics, a list of sample projects, and a stock of daily log sheets.
  • Students/groups will design/brainstorm projects that fit specific curricular objectives. They will create a project proposal and get it approved by me before they start.
  • Each project works on highlighted areas of the curriculum. Their outcomes will be physically highlighted in their binders.
  • Every day ends by filling out a daily log sheet which details who was present, what was discussed, decisions that were made, progress to report, and a plan for the future. 
  • Every project ends with a product. A short presentation (business style) is made to me or the class at the conclusion. 
  • Students are able to overlap objectives in successive projects to attain a higher grade for that outcome. The grade for that outcome is never fixed; students may always grow in their understanding of a topic. 
  • Evaluation of each project is based on a rubric (developed in tandem with the group). Self and group evaluation will also play a major role. 
This is an early stab at my vision. I have rough outlines of the group log sheets and rubrics, but nothing is final yet. My hope is to present this framework as my attempt at developing a growth mindset in my students. It also satisfies my curiosity for PBL and creates an authentic workspace for the the workplace mathematics to unfold. There are a few problems that I have to address (and if you can think of more, please comment!)
  • What do I do for students that do not show up for class?
  • What about students that complete all goals before the semester ends?
  • What percentage of the goals completed can be considered a pass?
  • Do I simply check off each outcome, or give a letter grade for how well it was met? Possibly a rubric 0-4?
  • How can I possibly control the freedom granted with group netbooks?
  • Do I allow very similar projects?
  • How large of a scope can a single project encompass?
  • How should I handle an in-class topic that needs to be widely clarified?
Ironically, the design of this class pathway is a perfect example of Problem Based Learning for myself. My short career has not contained an idea quite like it. It requires tremendous administrative and departmental support (which I feel I could get), but also needs wide-scale fine tuning. I am hoping the twitterverse can lend me some all important critical feedback.

I look forward to your remarks. 


Saturday, October 1, 2011

Mathematics for Bros

Before I begin, I would like to make sure that the title of this post was not misleading. If you are reading because you are fuming at the gender inequality reference in the title, please relax. I am in no way advocating that Mathematics is for Bros; the following post is a collection of the mathematical quips garnered from the "New York Times" bestseller, The Bro Code. It is a sacred cannon passed down from generation to generation of Bros designed to guide the lives and decisions of Bros worldwide. The book takes a humourous look at the superstitions of man, and uses mathematics to explain many "manly" behaviours. Hidden within its covers are facts, figures, and formulae that aim to describe even our most irrational behaviours. The link to the popular CBS television show How I Met Your Mother, makes the little activities within motivating and, dare I speculate, relevant. The reader should know that many of the articles mentioned in this post have been altered to fit into a PG setting. As you may have expected, the appropriateness of the Bro lifestyle is not always up to school board standards. 

I will divide the post by strand. The book itself is divided into articles that are designed to govern all Bros. I will reference the pertinent article with each example. I should mention that all math teachers out there that subscribe to "bro-dom" should be incredibly careful in divulging the contents of the Bro Code to others. I, however, am writing this post despite the immense, personal risk. 

Before the mathematics even begins, we see the first jab at the gender stereotypes. Article 4 describes the risks involved with sharing the Bro Code with a Woman. In the notes that follow, the author writes a paragraph to any woman that may happen to stumble upon the book. He begins his address as follows:

"If you are a woman reading this, first, let me apologize: it was never my intention for this book to contain so much math."

It seems as though the author included so much math to encode the rules for Bros only. As a math teacher of numerous excellent, female students, this claim automatically registers as a joke. It is too bad that the notion of gender inequality is so rampant in society that it finds its way into mainstream television. I digress on this point from now on to focus on the mathematics of The Bro Code

Exponents in the Code
Article 48 states that a Bro should never publicly reveal how many chicks he has... dated. (edited for obvious reasons). The author goes on to define a formula for the acceptable response to the question of his past dating history. 

n = number of chicks
a = Bro's age
s = estimation of the chick's past boyfriends (edited once again)
{1 <= s <= 10}

n = (a/10 + s)^0 + 5

I look at this equation as a golden opportunity to look at the concepts of Domain and Exponents. Bounding the possible values of 's' from 1 to 10 gives a very introductory look at  the concept of possible values and Non-permissible values. The formula also serves as a humourous look at the zero exponent. Any expression, save 0, to the power of zero, is 1. Keeping this fact in mind, the acceptable number would always amount to 6. I am not expecting this activity to provide a deep learning experience, but it will create a strong memory cue. 

Inequalities in the Code
Article 59 states that a Bro must always post bail for another Bro, unless it's out of state or, like, crazy expensive. Below the statement is a quick inequality detailing how to determine when the amount is "crazy expensive". 

Crazy Expensive Bail > (Years You've Been Bros) x $100

If two variables were to be inserted for the qualitative description, there are many learning opportunities. Create a table of values. Graph the inequality. Domain and Range of the variables. The other nice piece about this article is it doesn't have to be modified for inappropriateness. 

Article 113 describes the acceptable age-difference formula. The formula is designed to keep "crafty old-timers from scooping up all the younger hotties". It places a floor on the acceptable age of a girl depending on the age of the Bro. Ironically, the book itself uses a less than or equal to sign when it should use a greater than or equal to. That is an excellent conversation to have with your students. What does switching the sign do to the formula? Does this make sense? etc.

x <= y/2 + 7
x = chick's age; y= Bro's age

This is basically stating that every chick must be less than the set line. Switching the inequality sign provides a more accurate article. How do the two graphs compare? This error actually sets the stage for an even richer mathematical experience. The formula is already in slope-intercept form as well; this provides another convenient bridge to curricular mathematics. Below the formula is a table of values for quick-reference. It would be a valuable exercise to have the student create their own table. A graph would also fit nicely into the activity. 

Graphing in the Code
One of the strongest lessons has already been detailed by Dan Meyer (@ddmeyer) on his blog. The link can be found here. Its content comes from Article 86. Giving the students creative license to switch variables and continue graphing builds a deep understanding of the inter-workings of the Cartesian system. It also brings much needed meaning to the ideas of coordinate geometry as a whole. 

Other examples of graphing opportunities can be found throughout the book. Article 56 shows a graph comparing the Bro/Chick ratio at a party and the percent chance of getting noticed. (edited again for content). The image, once edited, can be shown to a class. Reading graphical data is an incredibly important numeracy skill. 

A discussion of the various stages in the graph can impress a deeper understanding of the coordinate system. What happens when the graph gets flatter? Why might some portions be steeper than others? How would you re-arrange it if you don't agree? If you knew there were 300 people at the party, replace ratios with numerical values. There may even be a possibility to discuss piecewise functions.

Article 137 states that a host Bro will always order enough pizza for all his Bros. The equation that follows is a step function. It provides an opportunity to talk about the appropriateness of graphical data. When can the line be solid? When must it be dotted? Why can't this function be continuous? 

The Pizza Equation:

p = 3b/8
p = number of pizzas (rounded up to the nearest integer)
b = number of Bros (including yourself)

Creating a table of values and graphing it makes for an excellent group activity. Rationalizing the resulting pattern makes for a great discussion. I would present the equation to students who have had experience with graphing, and allow them to explore. Some will take (3/8) to be the slope of the linear function, and get a continuous graph. Others will try a table of values and come up with a much different picture. Comparing the two, based on an analysis of the variables, would be an interesting exercise. 

After all is said and done, The Bro Code does an excellent job at poking fun at the ridiculous gender stereotypes in mathematics. The various articles provide engaging activities for teachers to grab on to and modify. Encourage the students to disagree, alter, and create their own mathematical laws.