Sunday, January 29, 2012

VFC: Quadratic Expressions, Equations and Functions

Here is a Virtual Filing Cabinet for Quadratic Expressions, Equations and Functions. This is part of an ongoing experiment in how to better share online teaching resources. If you like this post, then make your own post for a particular topic.

What am I missing here? Point me to your favorite quadratics resource in the comments.

[Last Updated: 1/29/2012]

The Hard Parts
One of the hard parts about teaching quadratics is the complicated formulas that often appear. At its worst, a quadratics unit can get mired down in a lot of meaningless a, b, and c's. A lot of folks seem to approach this unit by asking themselves how they can avoid the formulas for as long as possible.

Your bigger sequencing decisions also matter here. If you have covered GCF factoring before touching quadratics, then you have a tool that can be used by your students for understanding things such as the x-intercepts of quadratics or the equation of the axis of symmetry. Will you use quadratics as an application of trinomial factoring, or as a motivation for trinomial factoring? Have you covered multiplying exponents yet?

Quadratic Expressions
There's a good way of deepening and making factoring problems more open over here.

Quadratic Equations

For a higher level math class, PCMI has a series of problems that drives at the connection between the area and perimeter of a rectangle and the solutions to a quadratic equation.

This puzzle requires students to solve lots and lots of quadratic equations by factoring.

Quadratic Functions
James Tanton is just phenomenal here. He starts with transformations, and has a great conceptual procedure for finding the vertex and the axis of symmetry from an equation.

The Exeter Academy problem set begin this topic on page 62. They've introduced factoring early, so that makes it easy to talk about where the x-intercept are for equations where c is zero. This is a nice set-up for a Tanton-style approach for finding the axis of symmetry and the vertex.

Here's a Malcolm Swan domino game for matching graphs with equations. It also has a set for finding the roots.

Sunday, January 22, 2012

SBG kills motivation.

I've been testing out a new grading system this year in my classroom, and I'm really excited by it.  I used to pay kids $1 per percentage point on a test (e.g. 75% earns you 75 bucks) but I realized that this was encouraging kids to pay attention to the wrong information. I didn't want them to see that they only earned $60 and think, "Well, I bombed that test." I wanted them to dig into the feedback that I was giving them and figure out what their strengths and weaknesses were, what it would take for them to get better.

This year I've changed the way kids get paid in my class. I give them 35 skills, and they get paid $20 for each skill that they can show me that they've mastered. This has radically changed the way that my students think about grades. Instead of focusing on indistinguishable blobs of assessment they are able to get laser-focused feedback about what they need to work on. I can customize assessments for each student. It's great.

Enough coyness. Here's what I'm getting at: let's say you're you've read Dan Meyer's version of Standard Based Grading and you're about to adapt it to your classroom. You might think some version of this:

Mistaken Hypothesis #1: "I'm going to see way more motivated kids in the classroom. After all, their incentive is now to do things that lead to more learning because they're aligned with points."

You're not going to see more motivation, at least not in the ways that you want. Consider the slanty-word classroom at the top of the page. Would those kids be thinking about money or learning? Will some kids try to cheat the system for more money? Will kids check their bank accounts constantly? Don't kids see good grades as rewards?

Are points significantly different from payment?

Mistaken Hypotheses #2"That's why you shouldn't use grades to motivate kids. Instead you should create an environment where students are intrinsically motivated to do the learning. I can adapt the grading system so that it's meaningful as a feedback system, and that will transform grades from rewards into feedback. Yeah, kids are going to be addicted to points, but I can break them by teaching them this new way of thinking about assessment."

If you think that this you can break your student's points addiction while still rewarding them with points, you should read Drive, by Dan Pink. The second chapter is called "Seven Reasons Carrots and Sticks (Often) Don't Work." Here are seven reasons, culled from psychological studies, why extrinsic motivators backfire:

  1. They can extinguish intrinsic motivation.
  2. They can diminish performance.
  3. They can crush creativity.
  4. They can crowd out good behavior.
  5. They can encourage cheating, shortcuts, and unethical behavior.
  6. They can become addictive.
  7. They can foster short-term thinking.
Points aren't just a distraction -- they can actively undermine our students' intrinsic motivation. In "Drive," Pink reports on psychological research that identifies the most problematic types of rewards as "if-then" rewards. These sorts of rewards are exactly what gets proliferated in an SBG classroom. Should we be worried that SBG kills intrinsic motivation?

Hypothesis #3: "But SBG is a huge improvement on the old system." 

Maybe. Kids definitely like it more. I definitely like that it helps me guide student learning. But there are still tons of extrinsic motivators, and they're way more in-your-face than before. The extrinsic motivators are way better aligned than in the old system, but the proliferation of "if-then" rewards can be corrosive to intrinsic motivation. On the other hand, maybe the old system was killing intrinsic motivation as well.

SBG is probably still better, but points are bad.

Let's sum things up:
  • Extrinsic rewards kill intrinsic motivation. Fact.
  • In an SBG classroom that grades students there is a proliferation of "if-then" extrinsic rewards. This should make us really freaking worried.
  • The old points-based system might be way worse.


Let's assume that Dan Pink's recommendations for businesses can be easily adapted to schools. Let's assume that kids : points  ::  grown-ups: money, and that you're at a school that uses grades. What should you do?

"People have to earn a living...The best use of money as a motivator is to pay people enough to take the issue of money off the table." 

So, if you're interested in motivating your kids give them a good grade at the start of the semester, tell them that you won't think about grades unless you're forced to, and then don't talk about grades ever again.

In 95% of classrooms this isn't feasible for a million different reasons, but, hey, here it is.