Math in the City

Testing Blogging for English Teachers

2007-02-16T07:47:00.000-08:00

What if, in 9th grade English, you ask each student to separate each sentence as a new paragraph?

That would let you edit each sentence separately.

But how can you make the corrections?

The standard English editing symbols aren't available on a keyboard.

When everything you do seems like a waste

2007-01-16T13:31:00.000-08:00

As years go by, the government-mandated standards just keep rising, and we are increasingly asked to force students into learning experiences they're not ready for. It's a high-stakes, high-pressure field. I read an article recently about students in California who drop out of high school because they've taken Algebra 5-6 times and can't pass it (it's required for a diploma). Among African-American males, not receiving a high school diploma is a strong predictor of future prison sentences. But legislators in California and elsewhere didn't create this situation: they responded to the evidence that students who don't succeed in high school math can no longer graduate from college. It's just that way in the 21st century.

How can we keep strenuous effort without feeling like a cog in an abusive machine? The truth is that 99% of students can master and retain algebra. We have to keep the faith: there are ways to do this work as it needs to be done for almost every student who walks through that door. When you truly own the experience, "Oh, this can be done!", then part of your brain rests from the need to blame everyone when things seem to be a big mess. When it's not being done, "Oh, that's because that essential ingredient is missing. This is like this because that is like that." Whether to roll up your sleeves and dive in with a vision of how to transform things depends on the situation--what will work in some situations is just beating your head against the wall in others. But my experience was that, with this gradual realization, my mind didn't chase every comment it heard about teaching success with its own defensiveness: "Well, they can achieve that, but I can't because blame, blame, etc." Or, "Hell, I could probably do that, too, but I'm not going to give up all my friendships, family, and sleep to meet some standard." Or, my own deeply-ingrained pattern: "It's not my fault! It's not my fault!"

Which is not to say that things settled emotionally for me because everything started to go right in the classroom. Maybe it's just that now I believe in the system: we're not just pretending; we actually can do what we say we're going to do (even if, in some cases, we're nowhere close yet). If tests after a big unit show little learning: "OK, I know there is some way to make this work (that's where the strenuous part comes in). Right now, this is like this because that is like that. What sort of thing would actually work?"

The answer is out there somewhere. And when you find it, you'll know.

Best practice: software to tutor problem-solving

2007-01-07T18:30:00.000-08:00

My biggest discovery so far this year has been the value of computer-aided instruction. I used to think that most computerized learning programs were crap. Actually, I think most of them probably still are. One that's amazingly good is Carnegie Learning's Cognitive Tutor Algebra I curriculum. Key features of the Cognitive Tutor software:

The entire curriculum is mostly problem-based, rather than drill-based. It scaffolds learning precisely in most students and teachers biggest blind spot: modeling real-life situations with algebra, and solving problems out of those models. In other words: a functions approach, presented in the context of situations that are meaningful in the students' world.
It's not "fuzzy math". It's problem-based like lots of reform curricula, but it's structured toward the essential goals: modeling linear situations with equations, tables and graphs; solving equations; graphing lines; solving systems, etc.
Students receive immediate feedback to students on what they got right and wrong. The software doesn't just check students' final answers, it checks their whole work process. And anywhere in the process of solving a problem, it can provide hints (they're actually well-written and useful to students) about how to proceed if they're stuck.
Students can't just SIT THERE and pretend to be paying attention to your class. They have to work. If they don't type something on the keyboard, nothing appears on the screen. There's no way to just "blend in" with the rest of the class anymore. Many students say that part of my class where they work the hardest is on the computer.
The software is designed to be used 2 days/week. The other 3 days/week, you're in your regular classroom working out of the textbook (which is designed to go hand-in-hand with the software...I don't think the software would work well if you weren't also using the textbook). I think it's essential that learning software not take over your class. Personally, I had to rewrite a lot of the material in Carnegie Learning's book, and hopefully I'll be able to post my stuff up here.
Students absolutely love to work on the software. The only limitation on how long they'll work on it is what time I decide to go home. It's literally common for students to stay after last-hour class for an extra 1.5 hours to work on it. And it's not a game! It's just solving word problems and equations, graphing lines, etc. But they can see their progress filling in on a bar graph that shows their skill levels in each of a bunch of different categories, and they get instant feedback, which they ABSOLUTELY LOVE.

To look at the software (and the whole curriculum), go to http://www.carnegielearning.com/products_algebraI.cfm and click on the screenshots for the software and text to see a quick preview.

So, if you've never heard of it and you teach Algebra I, email Carnegie Learning and ask them to send you a demo copy of their Algebra I software. The demo will convince you. Maybe next time I'll discuss Aleks or something (which I don't really like).

Equations for All Learners (a manipulative that works)

2007-01-07T13:15:00.000-08:00

The single best manipulative for high school I've ever found is the Hands-on Equations kits, created and sold by Dr. Henry Borenson. I urge you to take a look at them. To use them well in a high school course (and they can be AMAZING there), you have to adapt the materials. I've adapted them, and hopefully I'll be able to upload those files for everybody's use.

Here is a link to the company, so you can see the kit:

http://www.borenson.com/

You can see a brief video intro if you look at the demo video on their site, or you can get the general idea just looking at the website. (The video takes a while to load.)

The kits were designed to get young elementary students playing with equations, and consequently don't emphasize moving from the concrete/manipulative level to the symbolic/algebraic level. They teach you all the important skills, but only on the manipulative level. What I've done is spiral that learning with the parallel algebraic skills. The result is that students find solving equations to be one of their favorite topics, and the learning is RETAINED year after year. Part of the reason the learning is retained is that the activities hit so many learning modalities: kinesthetic, pictorial, and symbolic.

To adapt them for high school use, you first have to teach the basic concept of the kits. The basic concept is that that you have to take away the same thing from both sides, either by removing the chess pawns that represent x's or by decreasing the numbers shown on the dice, which represent constants.

As students become proficient at that, you introduce them to the pictorial and symbolic methods. Here are screenshots of powerpoint slides I use at this point:

You can take the same types of powerpoints and turn them into worksheets by erasing everything except the original equations.

The Hands-on Equations Kits are amazing in that students effortlessly internalize the rule that you undo adding and subtracting before undoing multiplication or division. Combining like terms before solving is internalized easily, too. Students tend to think of using the kits as a game, and because of their emotional buy-in and the combination of kinesthetic/visual/abstract experiences, they retain the understanding permanently.

I use the initial lessons (which teach studetns how to use the manipulatives to solve equations) straight out of the teachers' manual without changing them at all. Once you get to Lesson 6 or so, if you're in high school, you want to start weaving in the pictorial and algebraic methods of solving. That's when you want to use something like the powerpoint slides I showed above. Don't save the transition from manipulative to symbolic for the END of your study of equations (as the Teachers' Manual suggests). Instead, interweave it constantly with the use of the kits, beginning at about Lesson 6.

The first lessons use only positive constants and positive coefficients for the variables. That's a good thing. Teach this unit immediately before covering signed numbers. That way, while students are learning to perform operations with positive and negative numbers, you can constantly be reviewing what they know about equations. Then after they know the rules for signed numbers, go back and teach equations that contain negative or subtracted terms. This spreads out the teaching of equations, so you have more time after school or in-class to bring those last few stragglers up to speed.

I'd like to point out to Borenson people (if you're out there reading this) and all creators of manipulatives for math that your manipulative-based lessons have to be moving students in the direction of the ultimate goal: to be able to do math without the manipulatives. If your manipulatives give students a concrete understanding of a concept, but they don't lead students towards fluency in working without the manipulatives, they're missing their true potential. (Unless you're teaching young elementary students, who don't have to know the symbolic method yet). So Borenson folks, get with it an publish some materials like I made to interweave the symbolic approach, so high school teachers can use this stuff.

I'll try to make my own materials available for the general public, but first I have to sort out technical and copywrite issues (how/where to host the site, and whether I can distribute my adaptations of other people's copywrited material). If anyone out there knows the legality of the copywrite issues, I'd appreciate your posting a comment to this entry.

Hope everyone out there in cyberspace had a great weekend, by the way.

Last night I think I heard somebody get shot

2007-01-07T12:42:00.000-08:00

(...or shot at). It was about 11pm. I heard the shots (which are not unusual for this neighborhood), but this time they were pretty close, probably somewhere on our block or the next one. Then I heard somebody shouting F***********CK over and over, and then a car speeding away. This is why we teach.

Some statistics. Only 45% of African-Americans entering a 4-year college will graduate. Many drop out of college, and the single biggest obstacle is the math requirement. Students who leave high school prepared to fulfill their college math requirements are far more likely to earn their college diploma. According to a report from the American Association of State Colleges and Universities says, "Finishing a course beyond the level of Algebra II [in high school] more than doubles the chance that an entering college student will graduate."

http://www.ndus.nodak.edu/uploads/document-library/705/6E--HIGH-SCHOOL-COURSEWORK---AASCU.PDF

In contrast, students who get to college and get placed into a remedial math class have something like a 60% smaller chance of EVER receiving a college diploma.

It is the difference between legitimate economic options for our students and the life on our streets and prisons, which according to the sounds outside, is violent.

Hello World!

2007-01-02T13:02:00.000-08:00

Just read an article in NY Times Magazine about how the intelligence community in the U.S. has decided it needs blogs and wikipedias so spies can put their heads together and tease out the meaning of all their clues...they have too many sources of information and not enough time to read and digest it all.

Which sort of sounds like being a teacher, doesn't it? Especially a teacher in an urban setting. How many books/conferences/journal articles have you seen recently that describe the best ways to differentiate instruction, promote student-centered inquiry, etc. Too many. But where are the actually useful parts of each? Let's face it, there's lots of impractical stuff out there. So, whether it helps a lot of people or not, I'm going to declare myself a "veteran in progress" (4th year teaching math in an urban Detroit setting) and blog about my discoveries. Hopefully, you'll join in and we can all learn from each other.