Wednesday, December 8, 2010

More front-page accolades for hands-on classrooms, II

A front page article in Monday's Local News section of the Philadelphia Inquirer profiles a math class at Philadelphia's Microsoft-funded High School of the Future, whose teacher, Thomas Gaffey, placed second in Microsoft's U.S. Innovative Education Forum and was a semi-finalist in its Worldwide Innovative Education Forum. In Gaffey's ninth-grade algebra class there are:
No textbooks, no paper, no chalk, no desks, and no assigned seats.

Instead, students use laptops while sitting in rolling chairs at trapezoidal tables spaced out in hexagonal classrooms.
Just how newsworthy this sounds to you depends on whether you think chair mobility and table shape have a big influence on learning, on whether you've been following current trends in education over the last 50 years, and on how unusual you think it is for a teacher to "encourage his students to find answers to their own questions" and engage with them in exchanges like these:
"Is this an obtuse triangle?" one student asks.

"Well, what can you tell me about an obtuse triangle?" Gaffey replies.

"One of the angles has to be more than 90 degrees," the student answers.

"Are any of the angles here like that?"

"Yeah. Oh, I get it now!"
As the Inquirer explains:
This snippet of student-driven discussion is a glimpse of the style and approach that have earned Gaffey national and international recognition. 
Student-driven? Who's asking most of the questions? But I'm splitting hairs here. What I should be asking is: Why does this kind of exchange warrant international recognition?

To fair, it wasn't this, specifically, that earned Gaffey his honors. Rather:
Les Foltos, one of the judges who reviewed Gaffey's work, was impressed by his emphasis on "actively engaging students in solving real-world problems." As Gaffey puts it, "If we want to teach math to learners, we should teach math how it is actually used. It doesn't matter how much you know. It matters what you can do."
Ah yes, "real world problems."  Again, only if you've been out of touch with the last half century of educational reform, and with today's Reform Math in particular, will this strike you as revolutionary.  Here is Gaffy's version of real world math:
In his classroom on a recent Tuesday, Gaffey's challenge to his "learners" - as students in the Parkside public school are called - was to estimate Earth's land area.

To solve the problem, the class first covered basic concepts about area and polygons - shapes with three or more straight sides.

Gaffey then asked, "If a shape has four sides, is it always a polygon?"

Learners who answered yes (the wrong answer) were asked to redefine what a polygon is, while those who answered no were asked to draw a four-sided shape that was not a polygon on the class "smart board."

Gaffey drew a shape with three straight sides and one curved side.

"Is this a polygon?" he asked.

"No," the class responded.
The class drew lines through each of the continents, chopping them up into complex polygons, then simple polygons.

The final phase was to derive formulas for the areas of the simple polygons, and add up the areas.
This sort of problem is not particularly new, as a quick survey through now-standard textbooks like Everyday Math and the Interactive Math Program makes clear. And it's been around long enough to have garnered some serious criticism--specifically in what Barry Garelick calls its "just in time" approach to teaching.  

Among other things, "just in time" often means serious delay. For example, one would hope that students would already know the formulas for the areas of simple polygons, and how to derive them, well before they hit 9th grade.

But because so many students are so far behind where they should be, there is one thing in which I and Gaffey are in whole-hearted agreement. In Gaffey's words, as cited by the Inquirer:

"Math education, more than any other subject, is in need of drastic reform." 


Anonymous said...

"It doesn't matter how much you know; it matters what you can do." And how much can you do, if you don't know anything? Also, teachers asking leading questions is as old as the hills. All good teachers do that.

Joanne said...

In what way is estimating the Earth's land area "real world" math? If "real world" means what people use day-to-day, then how many people do this in their day-to-day lives or work?

Also, isn't the purpose of education to prepare students for the "real world?" It doesn't seem to me that the purpose of education is to mimic the "real world" because that would be impossible.

There are too many different kinds of jobs with too many different kinds of needs. Education should be providing a solid foundation that students can build on later on when they choose their field of study and work.

Barry Garelick said...

A friend of mine who teaches algebra tells his students "This problem is fake--meant only to teach you HOW to use algebra. You can't do real stuff until later." The kids appreciate the honesty.

Problem solving skills come about from solving many problems, whose form and solution generalize to solve many other types they will see later.

Multi-step problems that rely on previously mastered material is a time-honored way to achieve the problem solving skills that are so sought after. For example, the following problem requires students to apply their prior knowledge about slopes and to develop a sequence of steps to answer the question:

"A line passes through the points (1,4) and (-2, -8). Find the equation of a line perpendicular to this line and which passes through the point (1,4)"

And it is often necessary to have to do this kind of analysis in science and engineering.

Joanne said...


What exactly is meant by "real world" math? That is what I am confused about. I learned math with a very traditional drill and practice approach. I can use those math skills in the real world. So, aren't these traditional approaches also "real world" if the definition is "This problem is fake--meant only to teach you HOW to use algebra. You can't do real stuff until later."

It seems to me that many educators today are saying that more traditional approaches to teaching math are not relevant in the "real world" and the new fangled stuff we're doing now is. I disagree. I think the traditional approaches were far more "real world" than a lot of what is being done today. When educators talk about "real world," for math, science or any other subject, what exactly do they mean? How do they define "real world?" Aren't they really trying to say that lectures and practice drills don't prepare students for the kind of math that is used in the work world?

Anonymous said...

Really, I don't think there is any such thing as math at the K-12 level that isn't "real world." What teachers and others mean when they say it isn't "real world" is either that it focuses on the concept/practice level instead of the applicaiton level, OR that it involves applications that they personally don't find engaging.

Barry Garelick said...

I learned math in the same way as you (Joanne) and the teacher I cited would probably agree with you. The teacher was anticipating the type of question that goes "When will I ever have to solve this type of problem?" He was also defending the use of traditional problems since he's just as irritated as you are with the criticism of such problems that you bring up.

An example of the type of problem that he was calling "fake" is "John will be twice as old as his sister will be in three years. How old are John and his sister."

Now, I can see the utility of learning to solve such problems because it teaches how to translate situations into mathematical symbols and equations and obviously so does this teacher. But there has been criticism levied against the traditional method that you and I had, by reformers who claim that kids need to see the "relevance" of math to the so-called "real world".

Thus, reformers bend over backwards coming up with things that sound like they relate to the real world and do not appear contrived. (Anonymous' description is very accurate). But they are typically very poor problems that don't generalize well. One example, from my daughter's algebra book is "The trajectory of a kangaroo's jump is given by the equation -18x^2 + 24x +15. Find the maximum height the kangaroo can jump and the length of the jump." I made up the equation but you get the point. The problem doesn't require that the student develop an equation to represent the situation, and is simply an exercise in working with a quadratic to find the vertex and the zeroes of the equation.

Additionally, there is also a belief that providing open-ended and unusual problems (usually for which students have had little to know prior knowledge in order to be able to solve it) are necessary to develop the so-called "habits of mind" required to solve problems. What is really needed are the traditional problems that teach students techniques that apply to any number of real-world (and even open-ended) situations. I wrote about this here.

Interestingly, the traditional problem I cite in the article brought criticism even from people who are defenders of traditional math, claiming it was contrived. I say so what if it's contrived?

Barry Garelick said...


I wrote a comment but it isn't showing up. I agree with you and I believe the teacher I cited would agree with you as well. I think anonymous summarized the issue fairly well.

Yes, some educators (too many, in fact) are saying that practice drills don't prepare students for real world applications. This is nonsense. I wrote about this

FMA said...

Thanks Anonymous. I have also been really confused about what is meant by real world. That makes perfect sense.