This week I discovered “The Exeter Series” by Glenn Waddell. For me, this was a “You’ve got a friend” moment (not my generation, but I definitely caught the allusion, Lee). If there’s one thing in the field of education that gets me fired up, it is good thinking on the problems and questions that we pose to our students. And, if that’s what you’re into, it doesn’t get much better than Glenn’s “Exeter Series” and the object of that series, Exeter’s math curriculum.
I am not an Exeter expert, but for the sake of a starting point (and to expose any misinterpretations), here’s my understanding of their approach:
- Exeter’s “curriculum” is organized as a sequence of 4 year-long, 100-page probems sets. The problems are written and revised by Exeter faculty, and each problem includes a “comment” with common student misunderstandings and recommended teaching methods.
- There are no chapters or sections. Similarly, there are no “subjects.” Instead, Exeter offers “Math 1,” “Math 2,” “Math 3,” and “Math 4.” Math 1 integrates topics conventionally segregated under Algebra, Geometry and Trigonometry. The curriculum is “problem-centered,” not “topic-centered.”
- Mathematical concepts are introduced in the problems, and are often formally defined after the student has already gained substantial practice. Exeter says, “Techniques and theorems will become apparent as you work through the problems . . . Definitions, highlighted in italics, are routinely inserted into the problem texts.”
- Teachers assign 8-10 problems per night, almost all word problems. Each assignment is designed to require a maximum of 50 minutes of work.
- Students present and discuss most or all of the homework problems at the start of the next class. Students are not expected to have every problem correct, and a small number, if any, of the homework problems are graded.
- In the absence of a conventional textbook, students take their own notes, and all tests are open note.
- The Exeter faculty reviews all of the problems and comments every year and makes appropriate revisions.
I think that most teachers are appropriately skeptical of “silver bullets,” so let me start with a few critical points as evidence that I haven’t totally tossed back the Kool-Aid:
- Some of the problems are mediocre. This one, for example, typifies the well-known “contrived application:” “Dan is 20% taller than Lucy, and Lucy’s height is 30% less than Andy’s. If Andy’s height is A cm, write algebraic expressions for Lucy’s and Dan’s height in terms of A.” Only math teachers have ever asked this question, and even the math teachers didn’t mean it.
- Exeter’s problem sets are paper-based, which probably limits their engagement potential. In Dan-Meyer-speak, they could use some media-rich“Act 1’s.” (Note: For all I know, Exeter uses “Three Act Math” in every class period.)
- Exeter’s curriculum is designed for high schoolers, and I think that important adaptations would have to be made to bring the “spirit” of Exeter’s method to younger students (e.g. less homework, more guidance and structure).
Caveats aside, I think that Exeter’s curriculum is a masterpiece. And, as with many masterpieces, it sets itself apart by looking at a situation in a different way.
If you asked a friend to define curriculum, they might say, “it’s the courses you take” or “it’s the content of your courses.” This agrees fairly well with the standard definition. Exeter’s curriculum reflects a different definition (and one that is closer to the original). Per Wikipedia: “Curriculum came from the Latin word for race course,” and it refers to “the course of deeds and experiences through which children grow to become mature adults.” For Exeter, curriculum isn’t about courses and content. It is about “a course of deeds” that builds toward adulthood. In math, these deeds are problems, and Exeter ‘gets it right’ by investing in an excellent sequence of problems.
Let’s return to the summary points I made above and consider some of the benefits that might flow from an Exeter-like approach:
- Year-Long Problem SetBenefit 1: Focusing on problems communicates that math is about engaging with problems, not memorizing procedures.Benefit 2: Assigning a fixed amount of material for the term (and sharing it up-front )communicates that each problem is part of a cohesive and thoughtfully planned whole. There are no “filler” or “busy-work” assignments.
Benefit 3: Eliminating distinctions between classwork and homework prevents “opt-out” and rewards perseverance. Students cannot avoid an individual assignment, because classwork isn’t over when you leave class, and homework isn’t over when you arrive. Problem-solving carries over, and students keep working until they succeed.
- No Chapters. No Subjects.Benefit 1: Eliminating chapters and subjects communicates that math is not a collection of isolated problems, but a cohesive whole – a discipline – the core of which is quantiative problem-solving.
- Problems Before ConceptsBenefit 1: Introducing problems before concepts decreases the amount of time students spend listening and increases the amount of time they spend engaged in authentic problem-solving.Benefit 2: Introducing problems before concepts ensures that students can attach a name to something they already know.
Benefit 3: Introducing problems before concepts communicates that math is a creative human discipline in which an individual practitioner can solve a problem and discover a “concept” using his own approach.
- 8-10 Wor(l)d Problems Benefit 1: Introducing mathematical concepts through word problems communicates that math is not a separate language applying to a separate world but a way of thinking that can be applied in any language and in any world.Benefit 2: Introducing mathematical concepts through word problems ensures that students become comfortable solving authentic mathematical problems.
Benefit 3: Keeping assignments short (by high school standards) encourages students to fully invest in the problems, and allows for a manageable review of all of the problems in class.
- Present and DiscussBenefit 1: Presenting and discussing work gives students an opportunity to speak about and take credit for their mathematics. It also holds them accountable to the people that they most care about – their peers.Benefit 2: Spending class time reviewing work that students have already attempted (rather than previewing work they haven’t attempted) gives them a clear incentive to attend and an opportunity to appreciate mathematical technique.*
*In most classrooms, students are exhorted to listen to the teacher’s “preview” to avoid a future cost (struggling to finish the homework). Unfortunately, humans discount future costs, which means that students often skip the preview. In a review-based classroom, students are listening to get out of a present struggle, and, having already struggled themsevles, they are much more likely to appreciate the assistance of conventional technique.
- Own Notes, Open NotesBenefit 1: Having students take their own notes communicates that it is their responsibility to internalize the material and provides them with a powerful mnemonic tool – writing and structuring the information.Benefit 2: Empowering students to use their notes communicates that math is not about memorizing formulas and procedures; it is about understanding and solving problems, and keeping good records can help you to do that (even in the digital age).
- Annual RevisionsA. Investing time and energy in good problems means that teachers are more likely to present with confidence and students are more likely to engage with curiosity. In the long run, this means less time and energy spent on external systems of “management.”
I’ll return to Exeter soon. In the meantime, we can all look forward to the final installment in Glenn’s “Exeter Series.”