Here are some useful take-aways for teaching Algebra from the EDUC115 class I mentioned earlier.
First, some definitions are helpful.
Procedural Algebra: Using variables to represent a specific number that you solve for.
Structural Algebra: Using variables to describe patterns/growth in a general sense.
1. Teach structural algebra first. This taps into students’ natural pattern seeking and helps to set norms for classroom discussion. Students can learn procedural algebra later. The shift from procedural to structural is much more confusing than the shift from structural to procedural.
2. Do “number talk” problems, like solving 18*22 in your head. Give students time to work it out. Once most signal that they have an answer, then ask students to share their strategies. Record their strategies on the board and translate them into block pictures and expressions.
3. Consider the following problem: Write an expression for the total cost of apples and pears if apples cost 10 cents each and pears cost 12 cents each. Do NOT teach this as c=10a+12p. This confuses the students to think that there are 10 apples and 12 pears. Here, a does not represent apples, but rather the number of apples. And 10 does not represent the number of apples, but rather the number of cents for each apple. Instead, teach c=10w+12z or something analogous and ask students to think about what the letters actually represent.
4. Start with pattern growth tasks to foster understanding of structural algebra. Always use at least 4 of the following representations–words, pictures, graphs, tables, and symbols. Students should be able to explain what different parts of the expression mean in terms of the figure they are looking at. Do NOT have them extrapolate patterns from the table first.
This one’s not linear so I wouldn’t start with it, but what’s important is relating x to the figure. For me, I see x as representing the number of rows of triangles. For figure 3, there are three rows. The bottom row has 3 triangles, the middle row has 2 triangles, and the top row has one triangle. Each triangle has 3 toothpicks, so the number of toothpicks would be 3(3+2+1). A general formula using Gauss’s sum formula would be t=3x(x+1)/2, where t is the total number of toothpicks and x is the number of rows of triangles.
That’s all I got for now.