## Tuesday, November 25, 2014

Every year, my students study the general characteristics of polynomial functions. We investigate the various shapes of various functions and slowly shift parameters to watch changes in the graphs. Eventually, we deduce the roles of the constant term, leading coefficient, and degree.

It should be noted that Desmos makes this process much easier than years previous. Just set up the generic polynomial, add sliders, set specific ones to play (depending on what you want to investigate), and have students discuss in groups.

See sample here. (Sliding "a" to "0" invites an excellent conversation; same with "b" etc.)

After we work with the transition from function to graph, we go the opposite direction. (makes sense, right?)

My favorite types of problems, however, ask students to play with parameters to influence results while leaving some characteristics consistent. For example, they might be asked to write a polynomial function that has an identical Range but different y-intercept. Or an identical end behavior but a different number of x-intercepts.

We play with these choices for a while. (I have them come up with lists of characteristics that are impossible...this is a great conversation)

The student work below comes right before the exam is written. They are asked to write a personal ad for a polynomial of their choice as if it were joining an online dating service. They cannot state their degree, leading coefficient, or constant explicitly. The result is an interesting exercise in encoding and decoding sets of possibly parameters in polynomial functions.

I am a polynomial function, super fun and curvy with two turning points. I am currently on a down slope in my life, but I don't want to sound negative. I am an infinite range of y-values and infinite domain of x-values. It will never be a dull moment. I am looking for a polynomial that is more calm than me. Someone who is basic but positive and going up in life. They need to have 1 y-intercept and 1 x-intercept. I don't want anyone who will throw o curve at me. I hope to have domain and range in common--be something special we share.

I am a very negative and odd function. I have been working my way from quadrant 2 to quadrant 4. I have no curvy parts and I like to rest right at the origin. I am looking for a function to put a little more life in me. Two turning points is a must. I'm, looking for a positive influence in my life.

I am mostly laid back and enjoy to stick close to home. Ever since I was born I have never changed. I prefer to not cross paths with my enemy, but am willing to take the chance to see my friends on the other side. I don't have much of a range. I can't tell which quadrant I start and end which makes me mysterious. I am looking for a soul mate which will help me take risk in life. Someone who leaves the x-axis regularly; three times would be the perfect number. I would like someone who picks me up regularly and doesn't mind hanging out at our common y-intercept. Be curvy and outgoing, but willing to stay close to home as well.

Once students have a grasp on the abstractions, they can begin to play.

NatBanting

## Tuesday, November 11, 2014

### CCSS: Support from the North

I can't--for the life of me--understand why someone would argue to eliminate high level mathematical reasoning in favour of memorized tricks, but that seems to be the case with those arguing against the Common Core State Standards. I cannot fathom how this can be the case except to chalk it up to a case of "he-said-she-said". Change (especially in something as resistant to it as mathematics education) breeds ignorance. And Ignorance breeds fear.

Let's face it: The public are scared of reform efforts and most teachers aren't far behind.

There are multiple (legitimate) reasons.
• Time
• Mathematical Proficiency
• Control / Power
• Outside Testing Pressures
• Personal Histories
the list goes on and on.

All of these things considered, I still find it impossible to figure out why a teacher would want to condemn the methods aimed at deep understanding. (Other than, of course, they haven't taken the time to see what it is all about).

Some would say that the methods do not actually achieve the deep understanding they claim to, but I would caution them (alongside Dewey, 1938) to tread lightly if they believe they can control what anyone learns at any time. Learning is a complex process that is not prescribed; claiming that it is optimal to internalize a 'canon' of 'truth' exactly as it is presented is one assertion, but assuming that such a feat is possible is something much larger. (In my mind, ignorant).

With this incredibly provocative preamble behind us, let's take a look at what sparked this thought. A series of four responses (many more were worthy of exposure) to a typical question on surface area in my Grade 9 regular stream classroom. Each student "received" the same educational experiences, but came up with different ways of conceptualizing the task. People may dismiss this as fluke or ignore it all together, that is your prerogative.

Just know:
• This is real. It is not fabricated. It is not doctored. The scans were simply cropped by myself to ensure complete anonymity.
• This unit didn't take more time than usual. I didn't spend a thousand hours allowing for "aimless" exploration.
• I didn't prescribe what method to use, but encouraged discussions on efficiency. It wasn't an "everything goes" culture; strategies were analyzed.
• The students generated each one of these in consultation with their peers, histories, and classroom experiences.
If you choose to ignore the possibilities (albeit from a very concrete, accessible topic), fine. But if you care to do so by intelligent and sophisticated means, please tell me why students using these strategies is a detriment to them, their future, and the education system.

One last appeasement. Even if one believes that right answers and streamlined edges are the purpose of mathematics (**cough** which they are certainly not **cough**), each of these arrives at the correct answer. This is a nice precipitate come exam time, but certainly wasn't the case during concept development. The mistakes were culture forming; they encouraged reflection and recursion--not something to be shunned. Some of the strategies even looked somewhat "standard" **gasp**

Without further ado, the question and four responses:

Nat Banting

References:

Dewey, J. (1938). Logic: A theory of inquiry. New York, NY: H. Holt & Co.