# PBL: A guide to the hype

Edutopia has advice for those math teachers who believe that Problem (Project) Based Learning (PBL) actually has something of value to offer.

I offer a brief commentary on their suggestions.

Address math myths: “Some teachers worry that PBL will take away time needed to practice math skills. Others insist that they need to “front-load” concepts before students can apply them, or worry about students encountering concepts out of the order outlined in their math curriculum.”

Those are my concerns as well. What the author calls “front-loading” concepts is what the rest of us call teaching. Many of us teach using direct and explicit instruction with worked examples and pratice problems.  We do this so that students can then put to use what they learn–with guidance.  The alternative is what I call  “just in time teaching”.  This is similar to throwing a kid in the deep end of the pool and instructing him/her to swim to the other side. The belief is that  he/she will be ready to absorb the instructions the teacher is shouting from the side of the pool on how to swim.

Fancher and Norfar rely on research from the National Council of Teachers of Mathematics (NCTM), among other sources, to overcome common concerns about PBL.

That’s their first mistake.

For example, developing what NCTM calls procedural fluency does not require having students labor over worksheets. Fancher explains, “It’s better to have students do four or five rich problems and explain how they solved them.”

Rich problems: What are they? Generally, they are very wordy, tedious one-off type problems that do not generalize. Often they are open-ended and ill-posed (like, the area of a rectangle is 48 sq ft; what are its dimensions?) and require students to learn procedures on a “just in time” basis.

Explain how they solved them:  Back to the kid who was tossed in the deep end of the pool. If by some miracle the kid makes it to the other side without drowning, he/she will likely say: “I don’t know how I did it, but I never want to do that again.”

“Don’t stop doing PBL if your project doesn’t go the way you dreamed,” Norfar cautions. Instead, reflect on what worked well and what didn’t, and consider how you can improve the project next time around. “PBL isn’t a cure-all,” she adds, “but it’s too powerful to give up on.”

My advice: Don’t waste your time, but if you do, your reflection should be on how you could have instead provided direct instruction on mathematical procedures. This could have been coupled with providing adequate practice with scaffolded problems. Finally, you could have provided instruction (and practice) on how to solve specific types of word problems. I believe that such tried and true methods are the ones too powerful to give up on. They are also the ones the education industry has discarded and from which the tutoring industry has greatly benefited.

# The Prevailing Caricature of Traditional Math, Dept.

In a recent article about math education that ballyhoos the “latest approach” in how to teach math, this statement was made:

“If the teachers are telling students how to solve a problem, and then that problem isn’t exactly what’s on the test, it creates this disequilibrium for a student,” said Beverly Velloff, the math and science curriculum coordinator for the University City School District.

There is nothing new about the so-called breakthrough ideas the article discusses. Moreover, this quote is representative of how traditionally taught math is mischaracterized.  The notion is that students are taught by rote, given a set of problems that are all exactly alike, and thus leaves them flummoxed when presented with a problem that is even slightly different. Such a caricature may be true for traditionally taught math done poorly, but it makes no allowance for it being done well.

Looking at an example from algebra: there are many varieties of distance/rate problems. There are problems in which two objects are going in opposite directions, going in the same direction (i.e., playing catch-up), round-trip problems, objects being influenced by wind or current, and so on.  At some point students are given basic instructions for solving these various types.  In opposite direction problems, students should be taught that we are dealing with two distances that are equal.  That is, if two people are driving towards each other, then the distance each travels before they meet up is equal to the initial distance of separation between them.

In well written textbooks, such problems are scaffolded so that initial problems are solved by following the worked example. But subsequent problems might have some small variation.  Instead of two cars coming towards each other at respective speeds of 60 and 40 miles per hour, we might be told the speed of one car, the distance between them initially, and the time it takes for them to meet. For example, two cars are 200 miles apart, and one car goes 60 mph. It takes 2 hours for the two cars to meet. What is the speed of the other car?  We know that in two hours the 60 mph car has travelled 120 miles. The distance of the second car added to 120 miles equals 200 miles. That’s the “distance = distance” relationship so that if x equals the speed of the other car, then 2x+120 =200. The speed of the second car is 40 mph.

Students will need some guidance in going through these problems, but after given practice with these types, they learn what to look for.

Math reformers may look at this as spoon feeding and rote. They would rather give students problems for which they have not been given specific instructions, and need to synthesize prior knowledge. Alternatively, they are expected to learn in a “just in time” manner what is needed to solve problems.  Thus, problems are given in a top down form in the belief that over time, students will develop a problem solving “schema”.

An article by Sweller et al (2011)  states that such notion is mistaken:

Recent “reform” curricula both ignore the absence of supporting data and completely misunderstand the role of problem solving in cognition. If, the argument goes, we are not really teaching people mathematics but are teaching them some form of general problem solving then mathematical content can be reduced in importance. According to this argument, we can teach students how to solve problems in general and that will make them good mathematicians able to discover novel solutions irrespective of the content.

We believe this argument ignores all the empirical evidence about mathematics learning. Although some mathematicians, in the absence of adequate instruction, may have learned to solve mathematics problems by discovering solutions without explicit guidance, this approach was never the most effective or efficient way to learn mathematics.

Nevertheless, reform/progressive math ideas rule the roost in education. Articles such as Sweller’s are thought of as fluff, not proven, no evidence to back it up, or dismissed in light of arguments such as “It has worked in my classrooms”.

Students need instruction, worked examples, scaffolding, ramp-ups in difficulty of problems, guidance, and much practice. Reformers view such steps as “inauthentic math” that produce “math zombies” who do not have “deeper understanding”.  Ignored in all this is the fact that the so-called math zombies are the ones in college who by and large are not in need of remedial math classes.