I got some good feedback on the first 2 blogs from Peter Meyer and Kay Merseth. I'll share their comments and my responses soon.

Moreover, 3 old friends commented yesterday. I'll add #3.5 out of 6 as a separate blog so we get the thoughts of Ryan Kelly, Sara Schnitzer, and Max Tuefferd. Plus I think the computer ate some comments by Paul Friedmann so if he tries again, I'll share those too.

For today, remember, this is #3 in a 6-part blog thinking about the New York Times Magazine article about math teaching.

0. Yesterday the Cardinals traded away a very good outfielder, Allen Craig. One reason: they had a young talent waiting in the wings, Oscar Taveras.

Some years ago, Chris Dupuis -- a star calculus, physics, and chem teacher at Match High School -- moved to Amsterdam. Chris was really good. About 10 years ago, Teach For America's then-COO (Jerry), happened to visit and said "Wow, I think that's the best math teacher I've ever seen."

Anyway, Chris's departure was a blow. It turned out there was a young talent waiting in the wings. Eddie Jou.

Kids love and respect Eddie. They learn a ton from him. For example, 19 out of 20 of his Calc seniors passed the AP Exam. To put into context, most of these kids arrived to Match High School 4 years ago below the 50th percentile in math on the state exams. At a typical high school, they'd be forbidden from taking Calc....many would be locked in the "low group." So it's quite an achievement. There's not a data set I know of that pulls how many of the thousands of American kids who pass calculus were below the 50th percentile on basic state math tests earlier in their lives, but it's gotta be really small; this is Escalante stuff.

Eddie is pretty unassuming. I don't know what he'll say this time around, but last summer, when I praised him for the 2013 AP results kids had achieved, Eddie was quick to deflect. "It was a great team effort -- being given extra time on Fridays and having a great Teaching Assistant, etc." Then he spoke about the tutoring the kids had gotten, the teachers before him who taught algebra, trig, geometry; the teacher training he'd received....heck I thought he was gonna take it all the way back to thanking his mom.

Eddie was a solid tutor from his first days in the Match Teacher Residency. Like with all rookies (whether teacher or tutor), usually the first things kids do is size you and up and decide "Am I going to try hard for his person, or mostly goof around?" I found an old email from Alia in 2010, where she wrote, a week after Eddie had started as a tutor:

I just sat in on Eddie’s tutorial with Malik, Luis R., and Randell and saw so many impressive things that I was itching for a pen and pad so I could write them down. Included:

· Great use of Cold-Calling (with the students’ names almost consistently after the question so that all three were doing the mental work for each question…nice management of Ratio) in walking through the Algebra problems. It kept the tutorial pace quick, and kept the students on their toes (and me, too!).

· Seized the opportunity for a Break It Down moment when Malik gave the wrong answer to a question. Eddie first responded with a rollback (“Revenue and Expenses?”), which immediately made Malik think about it. He knew he’d given the wrong answer, but then didn’t know how to find the right one – he guessed “Profit?”, after which Eddie broke it down even further and directed Malik back to the question – in a pure execution of Right is Right.

Here we are 4 years later; after Eddie's residency year and 3 years of math teaching under his belt.

Eddie writes (I edited a bit for clarity, so if you have a beef with any of Eddie's thoughts, first blame my translation):

1. I wholeheartedly agree that you can be an excellent teacher with 80-20 tilt towards I-We-You or and an 80-20 tilt towards You-Y'all-We. Either can succeed.

I think teacher preference, teacher experience level (discovery learning is harder on rookie teachers), and the students who arrive to your classroom (are they already used to thinking a lot on their own from last year, the year before, etc....or would you be the first teacher giving them the You-Y'all-We format) -- all of these should decide the "how" of teaching format.

Where I disagree most with Paul is his belief that I-we-you cannot grow students conceptually except in rare situations. The vast majority of my lessons for AP Calc are I-we-you.

The story I create for students involves giving a large overview of what we are doing conceptually; burying heads in the nuts and bolts of a problem type; then resurfacing to again take a look at the bigger picture. I see no reason why students couldn't learn big picture first, then work on nuts and bolts second -- if a teacher prefers that. What I think that omits is the level of students when they arrive to class. Self discovery is very difficult when you struggle with the basics and all your working memory is being used to handle the basics; many students benefit from grasping concrete, actionable steps first. Practice goes much easier. I cannot reiterate enough how much I think people do not give practice enough credit. With a working base, one's working memory is freed to think conceptually. Then working memory can be freed up for more abstract thinking.

Yes, with the "I" part, it does take time before the teacher is able to see the student deficits in I-we-you, and to plug in those holes with conceptual pushing. This is where I agree with you that new teachers need meat and potatoes practice first, and also where I agree with Paul that experience grows teachers in such a way that they are able to recognize the potatoes need gravy.

2. The article talks about students in Brazil that sell nuts and can do math in their heads, but then when asked to do it on paper, failed. I wondered about that anecdote. Many, including our own Match students, are very adept at solving a fixed problem with slight variations. I would like to see the peanut vendors and milk deliverers switch jobs to determine if they would be able to solve each other's situations immediately. That would show if they are capable of "Complex Math," or if they are simply used to slight variations on a single algorithm.

(Here is the NYT Magazine passage Eddie refers to):

The unschooled may have been more capable of complex math than people who were specifically taught it, but in the context of school, they were stymied by math they already knew. Studies of children in Brazil, who helped support their families by roaming the streets selling roasted peanuts and coconuts, showed that the children routinely solved complex problems in their heads to calculate a bill or make change. When cognitive scientists presented the children with the very same problem, however, this time with pen and paper, they stumbled. A 12-year-old boy who accurately computed the price of four coconuts at 35 cruzeiros each was later given the problem on paper. Incorrectly using the multiplication method he was taught in school, he came up with the wrong answer. Similarly, when Scribner gave her dairy workers tests using the language of math class, their scores averaged around 64 percent. The cognitive-science research suggested a startling cause of Americans’ innumeracy: school.

3. Eddie's final thoughts:

*Agree with much of what the article talks about in terms of the need for teacher collaboration around math topics -- i.e., choosing the right examples, knowing the frequent ways students get confused. (Whatever the merits of a small school in terms of building culture, it makes it harder to do this....example Eddie is Match High School's only calculus teacher).

*Worry that we as a country don't have highly qualified teachers that understand concepts themselves. If teachers need to be taught "rectangular arrays and/or area models," our teachers are just not of the caliber we need.