Comparison: Comparison sometimes comes before definition, but we are exploring it second. While I admit that we tend to compare things before we truly understand what they are, I also acknowledge that the only way to compare things thoroughly is to first define them. We compare things instinctively, which is why you won't be shocked to learn that in mathematics the most blatant example of comparison is the equation. Whether we realize it or not, we expose our children to truth statements, equations, and comparisons before they are able to fully understand what they are doing. When our first grader completes the mathematical sentence "2 + 3 = " with the number 5, he is staking a claim. As Aristotle says, he is "saying what is, is". This comparison of the sum of 2 and 3 and the number 5 is simple yet necessary.
When students enter Algebra they will be faced with equations that are missing pieces of information. Instead of completing a simple sentence, they will be given a sentence like "2 + X = 5" and told to "solve" for the equation. This comparison is the same as the one from 1st grade, only it includes a very important abstract step. In this equation, we first begin by assuming that the equation is true. We assume that the value expressed by 2 and an unknown number is the same as the value of 5. Don't gloss over this. This may be a simple comparison, but it begins the complex process of thinking.
How do we explore this complex connection? We ask good questions. How is 2 like the number 5? How is it different? What does 5 have that 2 does not? What kinds of numbers are 2 and 5? If the sum is odd and 2 is even, then does X have to be odd? even?
Don't simply fill in those completed equations this week in mathematics, but take some time to wrestle with the similarities and differences on each side of the equation.