Equations for All Learners (a manipulative that works)

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.