The Common Core State Standards that came out in June 2010 include standards for each of grades K through 8, then one set of standards for grades 9–12. Not surprisingly, many adopters wanted guidance on how to implement the 9–12 standards. Achieve put together a team to map out suggested “traditional” and “integrated” pathways (3-year courses) through the 9–12 standards. The working group delivered this statement about the pathways:
[We believe that these Model Course Pathways] provide states, districts and schools with one possible model of courses and sequences that can be completed in three years. These courses and sequences are worth considering as states, districts and schools contemplate how to organize and implement the Common Core State Standards.
Surely many state and district leaders breathed a sigh of relief that this guidance was provided; it wouldn’t be efficient for every adopting state to develop their own pathways! However, there are a couple of elements of the pathways that are surprising, and I believe adopters ought to question. I’ll just focus on one in this discussion.
The traditional Geometry pathway has much of the content that one would expect, but then ends with a substantial 6th unit on “Applications of Probability.” There are 11 standards listed in this unit (of 52 total in the course), covering topics like:
- understanding independent events and how to calculate their probability;
- understanding conditional probability and that independence implies that the probability of A given B is the same as the probability of B given A;
- constructing and interpreting two-way frequency tables and using them to determine if events are independent and to approximate conditional probabilities;
- applying the Addition Rule, the Multiplication Rule, and permutations and combinations to compute probabilities; and
- analyzing decisions and strategies using probability concepts.
Statistics and probability standards have been added to an increasing number of states’ standards over the last decade or so, possibly starting with NCTM’s 1989 standards. But I’ve never seen these standards explicitly located in a geometry course—only geometric models of probability typically fall there. Usually some basic probability concepts (drawing marbles from a bag) and very basic statistics topics (mean, median, mode) make it into Algebra 1, and Algebra 2 or sometimes precalculus or analysis courses typically deliver the bulk of the more advanced statistics and probability content (to the extent that it’s taught at all).
Can and should this material be taught in a Geometry course? Will adding this content make for a jarring and incoherent sequence of topics? Is there room in a Geometry course for a whole new bank of topics? I’ve thought about this some, and don’t yet see a way to incorporate these topics so that they actually feel like they’re much related to geometry. And, I believe that math should always be presented as a coherent body of knowledge, not isolated and disconnected facts and skills.
Another issue of incorporating this content into geometry courses is teacher preparation. Many highly-qualified math teachers have never taken a statistics and probability course. It was not the norm when I was in high school, nor when I pursued a math major in college, nor were those topics tested when I took the Praxis—the subject matter proficiency test required in California. As more advanced probability and statistics have been added to states’ standards (and AP Statistics courses have become more widespread), some number of teachers have had to ramp up on these topics. But with these proposed pathways, virtually all high school teachers would be responsible for delivering far more statistics and probability content than was ever a part of their own education. And ideally, they’d learning it not only well enough to teach it, and teach it well, but they’d understand it well enough to be able to connect it coherently to other topics (geometry?).
Do you have thoughts one way or another? Is your district planning to implement these pathways? I’d love to hear other educators’ thoughts on this.