So, in what sense is the math program at Phillips Exeter Academy that I blogged about below a reform vs. a traditional curriculum?

One source is the math curriculum packets on this website. These packets consist mostly of math problems--hundreds of them--but they also contain a revealing introduction and a by-students-for-students guide. Supporting the idea that this is a "reform math" curriculum are the following excerpts describing the overall curriculum from the introduction:

algebra, geometry, and trigonometry have been integrated into a mathematical whole. There is no Chapter 5, nor is there a section on tangents to circles. The curriculum is problem-centered, rather than topic-centered. Techniques and theorems will become apparent as you work through the problems, and you will need to keep appropriate notes for your records — there are no boxes containing important theorems.

Many of the problems in this book require the use of technology (graphing calculators or computer software) in order to solve them.

...be prepared to explain your method to your classmates.Then there's this excerpt from the by-students-for-students guide:

The methods that you use to solve a problem, the corrections that you make in your approach, the means by which you test the validity of your solutions, and your ability to communicate ideas are just as important as getting the correct answer.Other reform math-linked characteristics are that the curriculum was written by the teachers and that the problems are mostly or entirely word problems.

What about the pedagogy? Student testimonials show further reform-linked traits:

"When you discover or formulate a concept yourself, you remember it better and understand the concept better than if we memorized it or the teacher just told us that the formula was ‘xyz’."

Student testimonials also describe the classroom routine:"One learns many ways to do different problems. Since each problem is different, you are forced to use all aspects of math."

"Usually, every homework problem is put up on the board at the beginning of class, and then they are discussed in class. If you regularly put problems up on the board, your teacher will have a good feel of where you stand in the class; a confident student will most likely be more active in participating in the class."

"The class goes over each problem; everyone shares their method and even difficulties that they ran into while solving it."

"We reviewed the different problems, and everyone was successful. I explained my work and awaited the class’ response. My classmates agreed with the bulk of my work, though there was a question on one part. They suggested different ways to find the answer and we were able to work through the problem, together."Sounds suspiciously student-centered, doesn't it?

There are, however, a few things to keep in mind. For one thing, there is nothing indicating that students are working in groups. Rather, what predominates seems to be a combination of independent work (the homework) followed by whole-class discussion. This protocol more closely resembles the Japanese model (as described by Alan Siegel) than that of American Reform Math.

Second, while it looks like the classroom teacher is holding back and letting students do much of the discussing and explaining during class time, there is abundant opportunity, at this small boarding school, with its low student-to-teacher ratio and its longer hours of teacher and tutor availability, to get direct, one-on-one instruction. From the by-students-for-students guide:

Without any explanations showing you exactly how to do your homework, how are you supposed to do a problem that you have absolutely no clue about? (This WILL happen!) Ask somebody in your dorm. Another person in your dorm might be in the same class, or the same level, and it is always helpful to seek the assistance of someone in a higher level of math.

...

The very first place to turn for help should be your teacher. Since teachers at Exeter have many fewer students than teachers at other schools, they are never less than eager to help you succeed in any way they can. There is actually one designated time slot a week for students to meet with teachers, which is meetings period on Saturday. You can always call or ask a teacher for help. If there is no time during the day, it is always possible to check out of the dorm after your check-in time, to meet with your teacher at their apartment, or house. It is easiest to do this on the nights that your teacher is on duty in his/her dorm. Getting help from your teacher is the first and most reliable source to turn to, for extra help.

…

Along with help from your teacher, there are several other places to get help. From 7-9 PM every night, except Saturday, there is a Math and Science help group in the Science Center. Each evening, the lab is filled with students in a broad range of math levels, which should be able to help you with problems you have. Also, remember that your homework is not graded everyday, and your teacher will usually tell you when he/she will be grading a particular assignment. This means that you can always find someone in your dorm that will help you catch up or simply help you with a tough problem.Incidentally, this is yet another way in which the Exeter System mimics the Japanese system, as we see in this link from Anonymous (thanks, Anonymous!): on top of the more conceptual approach taken in the official school classrooms/assignments, there is a "shadow system" of extra-curricular schooling that involves much more direct instruction.

A second factor is the strong background in basic mathematics, which the American Reform Math pipeline doesn't foster, but which a highly selective high school can guarantee via stringent admissions standards. Here, the exception proves the rule:

"My background in math was a little weaker than most people’s, therefore I was unsure how to do many of the problems."Finally, we learn about the importance of tests and the strictness of grades, which reek of traditional prep school expectations:

"The tests are the hardest part between terms to adapt to, but if you prepare well, there shouldn’t be a problem."

"My first math test at Exeter was horrible. I had never seen a D− on a math test."Those who extol the Exeter curriculum for its reform-linked traits should also check out how well these serve the students. As the by-student-for-student guide explains:

During the fall of 2000, the new students avidly voiced a concern about the math curriculum.One result of this was a survey of new students in the spring of 2001, from which these quotes were taken, as well as a few more:

"It takes longer for new concepts to sink in ...you understand, but because it didn’t sink in, it’s very hard to expand with that concept."

"Hard to separate different math concepts. Not sure what kind of math it is I’m learning. More difficult to review."

“Solutions to certain problems by other students are sometimes not the fastest or easiest. Some students might know tricks and special techniques that aren’t covered.”Some students found the following things helpful:

“You could meet with the teacher for extra help anytime.”

"Extra help session one-on-one with the teacher. My old math text."

## 6 comments:

Wow. You start with a hand-picked group of bright, motivated, hard-working students, and then you throw a curriculum at them that they can't possibly handle themselves, so they must rely on help from each other and their over-available teachers (do they have any private time?) This seems like a great way to produce young adults with no sense of independent agency.

Wouldn't it be better to give them clear problem sets which might be occasionally challenging but that they can still solve on their own? Shouldn't school be about giving kids tools that they can use themselves?

OK, I took my rant to my own blog:

It Takes a Small City

The availability of and encouragement for teacher help amounts to direct instruction--which raises the question of whether that should have been done in the first place. No, better to sustain an illusion that the kids are doing it all by themselves. Sounds like a self-sustaining juku.

My son had an honors geometry class at his boarding/prep school that used the Exeter curriculum. He and more than half of his classmates ended up dropping the class and going down to the regular traditional geometry class. Even with super-available teachers, there is only so much extra help that is reasonable for students to seek, especially for introverts like my kids. It was practically impossible for him to study for a test with only his notes to use. If he missed something there was no chance he was going to learn it from his notes. A textbook would not have been helpful as the first few units did not seem to be about geometry at all. The kids who did not drop the class had Algebra II as their prior class instead of Algebra I, so there are real questions about how much kids were learning and how much they actually were applying prior learning.

We had a similar bad experience with a Harkness method precalculus class for my older son at a different boarding school. He struggled all year with the "figure it out yourself" methodology. He is very strong in math and went on to all 5s on the Calc AB, Calc BC, and Stats AP exams after classes with traditional instruction.

It sounds like the way math used to be taught. Help didnt consist of direct instruction in any thing but problem solving skills. Even the 1990 Dolciani Algebra 1 book on my shelf retains problems of this difficulty. Of the problems you posted #5 would give anyone who wasnt on the football field, using a pedometer, or an outdoors person pause, but they would soon find out from a classmate what a reasonable walking stride was. A marching band member would say '8 steps to the yard line' and explain that meant he was expected to use 8 steps to go 5 yards, and convert that to inches. A woodsman, farmer, runner/walker would know his personal pace from measuring his step via pacing off a known distance. And 9th grade is awful let to be telling students 5280 ft to a mile...that was mastered by the end of jr high in the past....we did a lot of map reading in gr. 4 to 7, which included scale conversions. I think you are seeing the effects of Poverty, not bad teaching. The teaching I had is what Stigler and Hiebert describe as German in their book 'The Teaching Gap' and prepared me well for an engineering undergrad. My peers who had been shown how to solve everything step by step in high school floundered until they learned some problem solving skills

Many of my peers came from city schools, which apparently were dumbed down before rural schools. The approach they used was to memorize solution books and old tests.

My teachers were a bit older, normal school grads, who are in their 90s now. Critical thinking skills were taught.

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