Here's how CME introduces the properties of arithmetic to 9th graders:
* This table is actually from Day 4 of their sequence. Day 1 observes patterns in the addition and multiplication tables, restricted to non-negatives. Day 2 reviews arithmetic with negative numbers. Day 3 extends us to the full addition table.
It's not until the next day that we have any sort of general expression of the properties of integers, and at this stage we still don't use any algebraic notation at all. Instead we articulate the "Any order" and "Any grouping" properties of arithmetic. Then we extend these principles to cover rational numbers and then the reals.
Finally, students look at various algorithms and try to explain why they're true. But that's really hard, using English, and so after students struggle to justify the reliability of these algorithms using our common language and vocabulary, the students are presented with variables. Variables are ways of talking about arithmetic with generality. And, boom, then comes the rest of the Algebra 1.
Now, here's how Park Math introduces the properties of arithmetic to their kids:
From there they move to finding counterexamples to claims about this rule, motivating the importance of mathematical justification. From there they quickly move to the introduction of new rules and, then comes this problem:
In a sentence, CME eases kids into variable expressions via the need to justify arithmetic shortcuts. Park Math assumes that kids understand variable expressions (and equations) and asks them to justify algebraic rules.
The personal puzzle is how to hack these two approaches apart and come up with something coherent and useful for my classroom. I'm assuming the pace of CME is too slow for my kids but the abstraction of Park Math is too much to start with. My current plan is to start with CME and move on to Park.
But the more interesting question is why these two curricula assume such different things about the students who are beginning Algebra. Does anyone know why that is? Is there something in the water in Massachusetts/Maryland?