Before I get into some examples of what we did for the 4 days of Exeter training, I want to discuss what the overall philosophy of pedagogy that was modeled exhaustively. Never once did the instructor actually SAY, “This is how we designed and planned the problems.” Instead, he set up the situation where we worked problems, and through my experience at #TMC12 and paying attention to how he phrased things, and how he moved from one problem to the next I was able to work it out.

Then I asked him point blank if I was right. I was. We had a short (very short) discussion on it, and then we did more math.

In describing the system / plan / organizational structure below,* I want to make it clear these are my words, my descriptions, and my labels about their essential methods*.

I am going to use the following descriptors for my understanding of their system; the** setup & modeling, naming**, and **extension**. I will work through each one of these separately, and connect them to some problems found in the Exeter problem sets as examples. When I am talking about problems found in the sets, I will use the following notation: M1:1:3 means Math 1, Page 1, Problem 3. All problems can be found on Exeter’s site here.

**The set up & modeling:**M1:15:6, M1:17:3,4,5,6, M1:18:1,3,4 M1:19:2,3

All of the problems listed above have one thing in common; they all are slope problems and yet NEVER ONCE mention the word slope. They use “rise” and “run” or some other variant. They discuss the change of one thing and the change in another, they talk about stairs or setting up a table of values or walking at a continuous rate or … you get the picture.

All these problem (and more, I just pulled out a sample) model the idea of slope of a line in a REAL WORLD basis. It has the learner calculate slope 5 different ways, from 20 different types of constant change. It clearly equates the idea of slope and rate of change, and puts emphasis on units and context. Never once is slope mentioned. No definition, no definition of a line, no y=mx+b, nothing. Just; here is a situation, figure out the answer. And the figuring isn’t all that difficult. It just asks the learner to understand what the rate of change is for each real world problem.

AND THEN they spring the definition of slope on the learner. It is kind of off handed, “hey, you know that thing you have figured out how to calculate in all those different situations, it has a name, it is slope btw. How cool is that, we now have a name for the idea we have been working with the last two weeks.”

I asked our instructor if he would ever say the word slope before they worked this problem, and the answer was no. Let’s think about that for a second. This problem is on page 19, which essentially means (but not necessarily) the 19th day of class. The learners in this class have been working with linear equations, problem solving with linear situations, and slope problems for about 10 days of the last 19 and the teacher JUST NOW UTTERED THE WORD SLOPE!

This is essentially 100% backwards from how we are taught to teach, and completely and diametrically opposite the textbook approach. We use the word, give the word a meaning, try to get the learner to memorize the meaning, create foldables to help them memorize the meaning, and then are frustrated when the learner forgets the meaning.

Exeter has the learner work with the meaning, solve problems with the meaning, completely understand the meaning, demonstrate they know the meaning through 15 different applications of the meaning AND then they say, “Oh, btw, that meaning has a name.”

Which method do YOU think is better? I know the answer for myself. Having tried and failed the last 5 years at getting learners to memorize meanings for words, it is a hell of a lot easier to get them to memorize a word when it is being attached to a meaning that is well understood.

**The extension**: M1:19:5,6,8; M1:20:9; M1:21:3,4,11

So now that the learner knows the word, the world opens up because instead of long problems that model real world situations the questions can become much more abstract. But they don’t! That is the point!

The problems just step up the level of thought required to a new level. Now the instructor can create activities that challenge the learner to think about slope and y-intercepts in a more thoughtful way. For instance, one hands on activity has the learners working with geoboards and thinking in depth on what it means to attach certain adjectives to the slope, or certain numbers to a slope.

This extension piece can not be stressed enough. It is not working problems 2-30 even from section 3.2. It is; explain what the slope of a line looks like as you take the value of the slope from 1/5 to 5/1 through all integer steps of both numerator and denominator. Then explain WHY your explanation in the first part makes sense.

This is the pedagogical pattern used at every level, and for every topic. **Setup** the topic simply, show how the topic **models** a real world situation and work with that topic for several days in several different ways, then **define** the topic in a very straightforward manner, and finally **extend** the topic to new, novel, and more complex situations.

It is important to understand this progression of understanding for the next post, which will take one very straightforward idea, and end up in a place where we can calculate the volume of a 3 dimensional derive the formula for calculating the volume of any 3 dimensional parallelogram. Yea, it really can be that easy.

Thank you! This is exciting. I’ll see how much I can do this sort of building from problems to definitions as I teach calculus this semester. It looks like the Math4 problems address calculus (along with probability and polar coordinates). I didn’t see a link above. I found the problems here: http://www.exeter.edu/academics/72_6539.aspx

Sue, Sorry I need to put in a link for the problems. I will fix that now. You may also want to look at some of the later materials in Math3 for introductory questions that will lead you to rich discussions at the higher level. Honestly, we took some Math1 questions and did things like graph tangent lines to a parabola w/o derivatives. Their questions are built to be extended.

Do you have any sort of list of the calculus-related questions in Math1 to Math3? (And is it possible to get a modifiable version of the problems?) I’m eager to use as much of this as I can – if it works with my students.

Sue, that is the $10,000,00 question. The answer is no, no one does. The question most asked of the Exeter teacher was “is there a stranding or concordance” and the answer is always no. They take on the huge burden of writing and revising the questions each year, along with writing commentaries on the questions (that is going to be a future post). A stranding or concordance does not exist. (that is the topic of a future post as well).

I just sat down and planned out 5 more posts on the Exeter issue. That is just from this week’s work. I think I will have many more in the future.

Wow! Thanks for writing all this up, Glenn. I appreciate the way you’ve broken down and explained/named their patterning. Your analysis is really thorough.

I need to think about this cycle of Set-up — Modeling — Define — Extend some more. Even though we public school math teachers may be screwed, this patterning could be introduced into any curricular sequence. It seems like once you really internalize this way of teaching, it could be applied in many different ways.

Thank you for writing these posts up!

– Elizabeth (aka @cheesemonkeysf on Twitter)

You are most welcome Elizabeth. I think that the level of how bad we are off is up to us. Just wait, there is more coming, and when it all comes out I think I am going to have a recommendation (but not a solution).

Personally, I see the Exeter model and process working VERY well in the public school setting.