We've been working on equations slowly since sometime in September in my Math 8 class. To be successful with equations, we took a detour through integers. Now, we need to refresh our rules for fractions and decimals as well so we have the whole arsenal of rationals at our disposal since most kids are currently screeching to a halt if they see anything that's not an integer. My initial plan was to give the kids foldables, fill in the examples, and move on.
Last night, however, I read Tina Cardone's Nix the Tricks cover-to-cover. Stop now, go download the FREE book, and read it now or file it for reading before January 1. It's important. Essentially, it's a compilation of things that teachers sometimes teach with a procedure instead of conceptual understanding and suggestions for teaching the concept from the get-go. The thing that teachers, students, and parents need to realize is that once you build good conceptual understanding, you don't have to worry about learning a particular procedure because they steps will naturally make a lot of sense.
With this reading freshly in my mind, I couldn't handle it when my students couldn't explain why they convert a mixed number to an improper fraction by multiplying the whole by the denominator, adding that to the numerator, and putting the sum over the denominator. We stopped. We modeled three and one-third. We showed how the whole number 3 represented 9 thirds and that together with the other one-third, we had ten-thirds.
Then next thing students wanted to do was get a common denominator. They couldn't explain why 12 would be an appropriate common denominator in 10/3 + 3/4. We stopped. I projected some fraction strips and we looked for equivalencies. Since thirds and fourths can both be written as twelfths, that is a logical common denominator. Many kids said they'd never seen fraction strips before, so we spent a few minutes exploring how they work. When I asked if they wanted a copy for their notebooks, I had several enthusiastic answers of, "Yes!"
I have to tell you that we didn't even finish that one problem in about 25 minutes. We got so into the modeling that we didn't finish even a small part of what I wanted to get done, but our conversation was rich and dug deeper into the meaning behind the fraction rules they've been trying to memorize for years.
Here's what I see happening in classrooms that I'd like to change. A teacher has a set number of days to "cover" a topic. The class spends a little time on tasks that get to the heart of the concept, but not all of the students "see" it early on. The teacher attempts to build deep conceptual understanding but time is her enemy and she resorts to tricks to help students get through the material in time for the assessment. Later on, the students are weak in those skills because they can't remember the rote procedures and don't understand the concept well enough to develop the process on their own. Besides the time factor, we have to realize that not all students are going to pick up on skills at the same time. I try to build in "did you know..." moments into my lessons in which I provide the background to simple math embedded in the math we're really working on. Most often, these are "aha!" moments I had several years after first learning the material myself.
One of these tips that I regularly share concerns graphing horizontal and vertical lines. As a middle school student, I was forever mixing up whether x = 6 was a horizontal or vertical line. I think that I was taught to find the x-axis and draw a line through 6, but I just remember that being quite confusing. At some point later in high school, I finally realized that x = 6 was a way of telling me that x is 6, meaning the ordered pairs on the line have an x-value of 6. If I could list off a couple of ordered pairs and plot them, I could decide whether the line was horizontal or vertical. This works so well because it's the basis for graphing lines; choose ordered pairs that satisfy the equation and plot them. This is how I teach horizontal and vertical lines now.
Have you nixed a trick? Let me know and I'll make sure to pass along any comments to Tina, who is still working to add to her book.