OR The One Where I Use Too Many Quotes OR Making Lame Friends References OR Too Many Alternate Titles OR HELP ME I CAN’T STOP

Today was Open House and so I stayed late at the school, until about 7:15. I spoke with quite a few parents and it was nice to put faces to name and to get to know my students a little bit more. But I am very tired and it doesn’t help that I put off writing today’s post until now.

As I mentioned in my bonus blog post yesterday, I gave Glenn Waddell’s Three Essential Rules of Mathematics and Meg Craig’s Algebraic Epiphany a shot in class. Overall I was very pleased with the result.

First, the 3 essential rules. I changed them to “Three Essential Rules of Algebra”, which felt more appropriate to me. I am in love with teaching to think about equation solving this way. I was already trying to teach the concepts of “making 1’s” and “making 0’s” to my students, but this allowed me something more concrete to give to my students to latch on to. I had already been stressing 0 and 1 as identities for operations and having students practice finding additive and multiplicative inverses, but Glenn’s rule were more clearly written. I’ve already put them up in my classroom and I will always refer to them from now on.

I am also absolutely in love with Meg’s Algebraic Epiphany even if I do have some reservations. The first thing that I love about what I was calling the “block method” (to build on Meg’s verbiage of “building the equation”: building blocks) is that it really gave a visual reinforcement of the “shoes and socks” principle that I had been teaching the students. Before, when I had an equation for students like

7x + 10 = 17

I would have them identify what was happening to the variable x and in what order. Sometimes they get confused but generally they get it right and tell me x is being multiplied by 7 and then added by 10. And I had been telling them that to undo these operations it is best to undo them in the opposite order that they happen to the variable. So first they should undo addition by 10 and then undo the multiplication by 7. I would tell them, “You put on socks and then you put on your shoes. How do you undo this? To take off your socks you must first take off your shoes.” This was working decently well, but this block method gave me a way to draw it. After “building” the equation and then undoing the operations, you can very clearly see what in order the operations happen to the variable and the inverse operations are applied in the reverse order. I explicitly pointed this out in class today and related it to shoes and socks.

The second thing I love about this building block method of solving equations is that I realized it is actually very much how mathematicians think of operations: functions and maps. Drawing a box around the variable and then an arrow with the indicated operation is exactly the kind of diagram I would draw when I was working on far more advanced math in abstract algebra or topology. I am hoping that as students get more comfortable with this method it will provide a good foundation when we return to function notation and inverse functions. I made a reference to the movie Se7en with Brad Pitt and Morgan Freeman because I would ask them “What’s In The Box” and then ask them what the arrow was doing to what’s in the box. I made sure to ask this very loudly. Most didn’t get the reference, but that’s okay. They still thought me yelling was funny. This seemed to click for the lot of them and I was really excited about it.

Some of the students loved the building blocks method for solving multi-step equations just as much as I did, but others were very reluctant because they felt as though the method was “extra steps” or “more work”. And they are right. You have to do the “building up” of the equation before you can tear down to the value for the variable. But I told these students that the extra work was valuable and reminded them that we value process as much as or more than an answer in mathematics. They were various degrees of satisfied with this response. Oh well. They still have to do it anyway. Too bad for them.

I did feel a little silly in my first block because I forgot that half of the reason that I had introduced the block notation was to help students with grappling with “literal equations”, which I learned just yesterday from my Holt McDougal textbook are equations with more than one variable. (Like d = rt, and so on.) I swear that I have never heard that terminology before in my life, but apparently it’s common vocab for “school math” and I must have missed it when I was in high school myself. I don’t know.

Anyway I felt silly because after I introduced the method and had first block practice I let them try the literal equations without using the block method. A mistake I realized and managed to fix before the end of the day, but some of the blocks missed out. But no matter. We will simply do some more literal equations using the block method later this week.

The only other issue I ran into is that I was struggling to help the students see the connection between the two new strategies. I think that Glenn’s Three Essential Rules of Algebra lend themselves more to the “usual” method of solving because it is easier to see the making of zeroes and ones by writing each next line with the appropriate inverse. I still asked and stressed “Are we making a zero or a one” when we wrote the next inverse arrow while using the block method, but I don’t think that connection was as clear.

I will keep emphasizing both and work on making them work better together. Today went really well, even if it was long.

Taylor,

I am very happy to have inspired you! I know this approach made so much difference in my classroom because it eliminated the idea of memorizing and reduced the mental load to one of 3 questions. Can I make a 0, 1, and am I being consistent to the equation?

Rock on! update us on how it is useful to your learners, please!