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?

I think this is generally right...and my problem sets always follow the narrative path, as well--I want students to develop the logic underlying the math, why we're doing a certain method, exploring variations on that method...

ReplyDeleteIt seems to me that the Exeter approach would be valuable if carried on over several years, but would be hard to develop within just one class or one year: your entire department would have to buy into it for students to develop the longer term connection building and, lets face it, patience for not having ideas crystallized every day.

As an aside, I also think CPM follows the spiraling trend of Exeter (also designed by teachers), but

I'm so glad you asked this question. I'd like to hear from teachers at those two schools. (I wonder if anyone has taught at both.) I'd also like to see their success rates, and find out about their students' mathematical proficiency before starting at the school.

ReplyDeleteExeter's format seems substantially more difficult, and I'd expect only the best math students to succeed with it, unless the culture managed to convince the others that success is possible, even with the most difficult tasks.

I'm currently a student in Exeter. After coming from a school where the education was based on teacher lectures, Exeter math seemed very challenging. After a while though, I got used to it. Because every day the class discusses each question, I slowly adjusted to the way the curriculum worked. After a year, it's not intimidating anymore.

DeleteI hope you continue with the blogging. I enjoy your well balanced and intelligent take. I looked through fairly quickly & the Park only had 3 or 4 lessons, so I may be way off on some of this.

ReplyDeleteNeither seems to do a very good job with the visual side of math. I recently stumbled across James Tanton who is spectacularly good at incorporating visual ideas, so that may be influencing me a bit.

Exeter is very formal. Parc seems to have a bit of the reform flavor to it in parts (your lemonade example, the log lesson), but not so much in others. The Park problem sets have too much repetition and not enough range in difficulty, depth, and style for my taste. I think Exeter just took the harder problems from some out of print text books they found laying around.

I don’t really see Exeter as forcing connections. But, the lack of categorizing certainly forces students to understand some key ideas and to read and think about the problems. I think it is all but impossible to determine if students understand concepts when you categorize the problems.

I fairly regularly (every week or two) use at least half a period to work on a problem(s) that is not directly related to the current unit. There are a lot of great problems out there. What I would really prefer, though, is connecting the the current unit more frequently and more directly to past important concepts. By that I mean what does the current topic have in common with a past concept or idea, not the more sequential thinking of how do you use past topic A to figure out how to solve/prove the problem in current topic B.

Yes, I do have a lot of uneasiness when I am giving an overly strong narrative or heavy scaffolding. How do I know whether the students really understands anything, or are just riffing off the common pattern to the problems? That seems like a big issue in the Park problems and just about any textbook I have seen recently.

The more I think about it, the more convinced I am that we over-sequentialize math. We do not need to constantly poke and prod and all but write it on the paper for them, so that a student can discover whatever it is that logically follows in the next lesson of our orderly, sequenced math program. Give them some interesting problems once in a while that are not meant to lead them in a particular direction, and let them actually have a chance to make some insights and connections on their own.

There's a lot to think about in your comment, but I just want to quickly jump in. I don't want to be the guy who equates narrative with scaffolding. I don't think that those are the same thing. For example, in Dan Meyer's Three Act problems there is a great deal of narrative, and not that much scaffolding.

DeleteWhat I was trying to figure out is why someone would choose a problem-solving approach that provides more scaffolding and organizing the curriculum by topic. That's something that we usually take for granted, because everything is organized by topic. But what's te pedagogical advantage of organizing something by topic? What I came up with was that it has to do with the value of narrative in learning.

We could come up with other answers, though, and I'd be interested to hear people explain where I've gone wrong.

A little clarification on connections: Exeter does a good job of forcing students to connect the problem to the math, at least in book 1, I didn’t see a lot of connections or relationships shown between different math topics.

ReplyDeleteNarrative can mean a lot of different things to me.

Dan Meyer’s will the ball go in the hoop is really just a nice clean way to introduce the day’s problem. I see very few draw backs and many benefits for this type of narrative – I wish I had a good one for every lesson. Also, you can use this type of narrative even if your content is not rigidly organized by topic.

There are also extended narratives. Connected Math for example will continue the narrative about a bicycle ride or whatever through several lessons. I gather Park takes this approach at times as well. In this instance, I would argue that the narrative, if affective, is also providing quite a lot of scaffolding. I don’t think this is necessarily bad, but I also think that a large number of students are unsuccessful in applying the very same math to a different context. I would argue that this sort of extended narrative is valuable to an extent, and at some point becomes destructive.

Finally, organizing by topic is forming a mathematical narrative – a sequential “story” about a particular branch of math. Maybe these mathematical narratives just go on a little bit too long and give way to many details.

The advantage of organizing by topic, with lots of little sub-topics, is that it is easier to produce results that appear to show learning. Organizing by topic is a form of scaffolding that is too rarely removed.

As I think about it more I'm liking less and less the way that I analyzed stuff in this post. I need to understand all of this better.

ReplyDeleteI have had a lot of experience with Exeter in the past few years and very little with Park. In fact, just yesterday I finished a 1 week workshop with Exeter faculty where we took a look at some threads, and earlier this summer I attended their conference on campus. Since I'm not familiar with Park I can't comment too much on a comparison, but I will say that it is important to see the Exeter problem sets as a whole. I am really floored every time I read and do problems over the course of several pages because the problems really do make incredible connections to each other even if they are separated by days or weeks. To say that there is no context or narrative is not exactly true. They actually provide excellent cues for the students to connect to previous problems, they just have to remember a little farther back.

ReplyDeleteIn speaking with Exeter faculty, they contend that their students represent a diversity of math ability and pre-knowledge, though they are universally highly-motivated. Some students will work from Math 1 to Math 3 (more or less Algebra 1 to PreCalc) and some will make it though Math 4 (Calculus).