Within the big tent that is problem-solving, there are some really different approaches to teaching. In this post I'm going to draw out one of those differences, offer an interpretation of their significance, and then invite you to disagree with me in the comments. Deal?
There are two free, complete, and well-regarded high school curricula available that I know of: the Park School's and Exeter's. (For some quick reading on these schools, click

here,

here,

here, for Heaven's sake don't click

here, and

here.)

Let's start by looking at two problems, the first from Park Math, and the second from Exeter.

These problems look pretty similar, and I think that the generalization is fair: there isn't a great deal of difference between the types of questions that students are asked to solve in the two curricula. But these fairly similar problems are situated in entirely different contexts. In the Park Math curriculum this problem serves as a concrete hook to hang the rest of a series of linear functions problems on. In the Exeter problem set, this problem is preceded by a rates problem and followed by a problem that asks students to practice a guess-and-check technique.

That's a huge difference between Park Math and Exeter. Actually, I think that it's two huge differences.

- In Park Math problems are organized by topic, as in a traditional curriculum. Linear functions is followed by geometry of lines, which is followed by coordinate geometry, etc. Things work entirely differently for Exeter. In Exeter's problem sets there are a cluster of densely related topics that drop in and out of the curriculum as the year goes on. There is no linear functions unit in Exeter; questions relating to linear functions appear on the second page and appear on nearly every page for a month or so. In contrast, there is a well-defined chapter on linear functions in Park Math.

- Because topics are not organized by topic in Exeter, the curriculum does not offer students much context at all for any of the problems that appear. In Park Math problems are very carefully scaffolded so as to allow students to discover solutions to more difficult problems without the help of a teacher. Questions continuously build, taking you deeper and deeper into a subject.

This difference signifies a big difference in the pedagogical assumptions of each school. The Exeter curriculum seems premised on the idea that deep learning happens when students make connections, and solving difficult problems divorced from context forces one to make connections. Students will learn best when forced to situate a new problem among the rest of mathematics. Categorizing problems for students as they're learning them is like organizing questions by topic on an exam. The context of similar problems makes retrieval way too easy on students, and they learn less from it. At the end of the day, it's all about making connections between different mathematical topics.

Park Math, on the other hand, seems premised on different assumptions. Their curriculum is consistent with a vision of learning that values narrative above making connections. Their curriculum is designed to offer concrete hooks and guiding problems for students to dive into. Then the problems develop and complicate the introductory problems and then, after the development, students are offered an array of difficult problems that use the mathematics that was developed. They're betting that it's sufficient to allow students to make connections in this section, after the story of (say) linear functions has been told.

So, that's the framework I'm offering for the differences between Park Math and Exeter's approaches. Park Math favors narrative in learning, whereas Exeter prefers making connections.

Every problem set that I've ever made follows the Park Math format. Exeter is the weird one here for me -- I've seen very little that resembles their approach. But, for a while now, I've been a bit worried that my scaffolded problem sets are offering too much support for my students. I'm thinking that I focus too much on narrative, and not enough on helping my students make connections.

**Homework, for the comments:**

- Do you agree with Michael's analysis? Why or why not?
- What other important differences between the two curricula do you see?
- When designing a lesson, do you aim for narrative or connection-making? Does it depend on the lesson?
- How could you add more connection-making to your classroom, without fully implementing the approach of Exeter Academy?