On the face of it, a math class for teachers sounds like a very much needed and good idea, which is what I thought when I started reading this article:
“You need mathematical content expertise to teach mathematics effectively,” said PMI Director Andrew Baxter, a Penn State math lecturer, noting that the workshops focus not on the mechanics of elementary math but the logic behind concepts. “It’s not just sometimes you do this and sometimes you do that, but there is a cohesive understructure to how everything connects.”
I agree that math teachers need to know the math they’re teaching. I started to get suspicious when I saw the words “cohesive understructure to how everything connects”. The words had a sense of foreshadowing, like movies in which the shot of an otherwise affable friendly character is held on camera for just a little bit at the end of a scene–a scene in which that character does something nice for someone. The character’s eyes shift just slightly–and maybe there’s even an ominous oboe solo on the music score–and you know this friendly character isn’t what he or she seems.
“One of the things to the problem-first approach is we’re a student-centric model,” Baxter said. “How do we know what to say next? Well, it depends on what the student says. It’s not just because of what the script says. It’s because the students are saying this; they seem to be at this point of their understanding. How do we get them to the next point?”
And so now the music in the movie has a foreboding minor key understructure. “Student-centric model”? Yes, I agree, it’s good when a student has an observation that gets at a deeper level of mathematics and teachers have been taking advantage of such moments for years. They’ve also been on the alert for when a student is going off track, and you need to get the lesson back or you’ll run out of time. Such scenarios are downplayed and cast in a negative light in schools of education as “teacher-centric”. The teachers role is to facilitate and if you run out of time, so what? If it was a “deep discussion”, then “deep learning” has taken place. Yes, well some of us do want to cover the material which I know is old fashioned and out of step. But then again, how many of these programs take a look at how successful students in math have been taught, and what these students have done to gain mastery. In fact, procedural mastery is something that rarely comes up in such discussions. It’s all about “understanding”. Or “deep understanding.” Sometimes “deeper understanding.”
Now, she said, she allocates more time for students to share their thinking about strategies, to justify their answers, and to compare ideas with those of classmates.
I do that with warm-ups at the beginning of class. Another thing left out of this discussion is that many students just want to know how to do it. Sure, we teach for understanding. But what teaching for understanding means in the context of PMI and other similar programs is delaying the teaching of the standard algorithm, and using pictures or other techniques that supposedly provide the “understructure”. By the time they get to teaching the standard algorithm, it may seem like an afterthought to some students who are grappling with “what method do I use?” from the various approaches they have been shown. Such approaches, seen through the lens of adults who have been educated in the manner held in disdain, make sense because they know the standard procedure. They think they have seen the light with these “deep learning” approaches and often say “I wish I had been taught this way.” No, you really don’t.
“You can present one problem, but it has to be a rich problem where there’s a lot of thinking and adjusting as you’re going along so that the students are engaged,” she said. Back in the classroom, she employs the PMI approach with problems such as figuring out permutations for an 18-foot chicken coop perimeter. She continues to draw on eight mathematical principles that PMI emphasizes, including making sense of problems and persisting in solving them, reasoning abstractly and quantitatively, and using appropriate tools strategically.
“That’s about thinking like a mathematician, and PMI stresses that a lot,” Romig said. “And I stress that with my students.”
And there you have it. Instead of presenting the standard old fashioned problems of “Given a perimeter of 20 feet, with one side of the rectangle being 5 ft, what is the area of the rectangle?” which requires students do understand what perimeter is and how to work with it, we have “The perimeter of a chicken coop is 18 feet; what are it’s dimensions?” This is considered rich, and in keeping with the eight “Standards of Mathematical Practice” contained in the Common Core Math Standards. It’s all about thinking like mathematicians. But in the words of one mathematician I happen to know, his view of the SMP and “thinking like a mathematician” is as follows:
Mathematical practices grow by the practice of mathematics; not by replacing critical content with lessons targeting platitudes.
This is something that has yet to become the next “shiny new thing” but I look forward to the day when it does.