I am teaching freshman algebra classes for kids who struggle with math, and one geometry class. I am running a PBL style trimester but got sidetracked by a math problem from illuminations. The idea is that you take a square with area = A, where A is not a perfect square and try to find stacks of smaller squares that are the same height as your large square. I gave the problem to my students and watched them go. It is a pure math problem with no real relationship to the project we are doing related to public art funding (I had planned on folding it into some kind of art assignment with the golden ratio but couldn’t really make it work). So these students who hate math tore into this thing like they were wolves on a fresh kill. They did slow down when it got harder. Finding stacks for the square with area = 72 wasn’t too hard until they realized there were a lot of solutions. Then they were asked to describe all squares with no equal stacks and a general rule. They put what they had on a poster for a gallery walk but there weren’t too many able to get to the rule. In the gallery walk they were excited again about the problem because they saw solutions from other students that were different from theirs. None of their solutions involved radicals, only boxes and grids.
The next day I threw up the equivalent question, “How many equations can you make using the square root of 450?” I showed them some examples using their poster solutions and then they started furiously creating radical equations for about 50 minutes, with diagrams of more square stacks. Much better than doing problems 1-31 (odd of course, so the teacher can grade “the process”).
I’m using a version of standards based grading and gave them a quiz. Today, I passed it back and gave them the chance to look in the book to practice for another one tomorrow where they can replace their grades with higher ones. The book gave procedures and properties. No opportunity for thinking but that isn’t what they need right now. They’ve done the hard work and need to tie it up. It wasn’t the best lesson I’ve ever done, but it did convince me again that the worst place to learn math is in a textbook.