## Sunday, January 30, 2011

### Slope

I'm still not sure how to use this blog. Frankly, I would like lots of readers to read about my problems and have them be so gripping that a lot of learning--on both sides of the keyboard--is happening around this place. But my struggles are so routine and rookie-ish that I feel strange assuming that there's any insight to be worked out from them. But, you know, forget it, it's my blog and I'll be boring on it.

OK, here's how I'm introducing slope to my geometry class. I'm trying to get better at easing students into concepts, though this is a concept that my students have seen before.

Core idea: Slope measures how steep a line is.

1. Start with drawing two lines with a stick figure standing on them. Ask them, which line is the guy more likely to fall off of? (Draw a floor with a dotted line to give some orientation.)
2. How can we make this idea more precise? (Some students will remember the formula for lines, and just push them into concepts at this stage.)
3. Draw a unit forward, and ask, "how much higher up is the guy when he walks one foot forward?" for each line. Ask for guesses of numbers.
4. (Re-)Introduce slope as the ratio of your height up when you walk forward.
5. But how do we describe walking forward and going up? Introduce the (familiar?) formula.
6. Time for practice calculating slope from two points. Then, at the end of that set, draw a line given a point and the slope.
7. At this stage, students are able to draw lines given a point and the slope. Now they're off to practice given a bunch of problems containing parallel lines and perpendicular lines. They need about 15-20 minutes to work on these.
8. We come back together to discuss the relationship between parallel, perpendicular lines and slope.

## Wednesday, January 26, 2011

### Full contact math

Did I mention that this is my first year and I'm straight out of college and don't really know anything about how to help people learn things? OK, just getting that out of the way.

There are better and worse ways to help people understand things. Here is some of what I've learned about that over the past few months.

1) Verbs are important. When you see 4 x's in the numerator and 2 x's in the denominator are you inclined to cancel, simplify or unmultiply the fraction? One of these verbs is noxious, the other annoying and one tells a student exactly what they should think about doing. This stuff matters. Man, if I have to tell another Algebra 2 student not to cross out a summand in the numerator even though I thought we cancel stuff when it's on the top and bottom of a fraction I swear to God that larynxes will be torn out of children's...happy place, happy place...OK, I'm in control. I'm just going to be careful when I talk to my freshmen, that's all. They're not going to hear the word "cancel" once. But they will hear me talk about unmultiplying fractions, and they'll know exactly what I want them to do.

2) Some procedures are better than others. So, how do you solve a rational inequality, or an absolute value inequality, or a quadratic inequality? You want to give them a procedure that will yield them the correct answer. But you also want them to understand why a procedure works. So you choose, as your procedure, to teach them to solve the inequality like an equation, plot those points on a number line, and then to test each region between those points to see if it satisfies the inequality. And, since you want them to understand all of this, you don't give them the procedure before you explain to them how it works. Right?

Blech. Students quickly forget your explanation, and just rely on the procedure, which is a brainless algorithm that doesn't put the mind in contact with the relevant concepts? But what else can you do?

Well, I've learned that I can design my own procedures, and that with subtle changes I can design them so that they force a student to come in contact with actual math. So instead of the above procedure for solving inequalities, I now teach students to split up the inequality into two functions, to graph both of them, solve the system of equations to find where the two curves intersect and then graphically intuit which regions satisfy the inequality. That's the same procedure as above, in case you're following, just rewritten from code into math for humans.

I suppose that, in sum, my basic moral from my first semester of teaching is: put students in contact with the right ideas early and often.

## Tuesday, January 25, 2011

### Mandelbrot Set for Algebra 2

For my Algebra 2 final exam I wanted to give students a chance to experience putting ideas together to learn something new. I also didn't want them to freak out. So I gave them a short intro to the Mandelbrot set (that was basically ripped off from this article).

A bunch of students thanked me for it, which was nice. They thought it was cool, and not boring. I sent them the link to the Jonathan Coulton song after the final:

Good news: Lots of them were able to tell if a complex number is in the Mandelbrot set and used previous knowledge (function notation, composition of functions and multiplying complex numbers) to learn something new. It also made for a more interesting final.

I wonder how this would work as a full fleshed-out lesson next year. I'm not sure that the Mandelbrot Set is really helpful for leading students to create the math that they ought to be learning. Meaning, I'm not sure how much would be gained from showing them the graph of the set and seeing what questions interest them. Anyway, here's the doc:

Mandelbrot Set