In an Edutopia article the difference between the two PBL’s: Project- and Problem-Based Learning was addressed. It included a nice definitional list of what is involved in Problem-Based Learning:
Problem-based learning typically follow prescribed steps:
Presentation of an “ill-structured” (open-ended, “messy”) problem
Problem definition or formulation (the problem statement)
Generation of a “knowledge inventory” (a list of “what we know about the problem” and “what we need to know”)
Generation of possible solutions
Formulation of learning issues for self-directed and coached learning
Sharing of findings and solutions
Nothing about this is new. Much has been written about the debate on how best to teach math to students in the K-12grades—a debate often referred to as the “math wars”. I have written much about it myself, and since the debate shows no signs of easing, I continue to have reasons to keep writing about it.
While the debate is complex, the following two math problems provide a glimpse of two
Problem 1: How many boxes would be needed to pack and ship one million books collected ina school-based book drive? In this problem the size of the books is unknown and varied, and the size of the boxes is not stated.
Problem 2: Two boys are canoeing on a lake and they reach a point where the lake joins a river. They are paddling to get to a little cabin upriver that happens to be the same distance from where they are, as the distance across the lake they just crossed. They have an argument. One boy says it takes longer to go upriver to the cabin and back to where they are now, than it would to go across the lake and back, assuming they row
at the same constant rate of speed in both cases. The other boy says it would take the same amount of time. Who is right?
The first problem is representative of a thought-world inhabited by education schools and much of the education establishment. The second problem is held in disdain by the same, but favored by a group of educators and math oriented people who for lack of a better term are called “traditionalists”.
In the spirit of full disclosure, I fit in with the latter group.
I have heard various education school professors distinguish between “exercises” and “solving problems”. Textbook problems are thought of as “exercises” rather than “problems” because they are not real world and therefore, in their view, not relevant to most students. Opponents of such problems contend that they contain the quantities students need to solve the problems and therefore do not require students to make value judgments. Such criticisms of traditional approaches in mathematics have led to the “problem-based learning” approach.
The first problem presented above is an example of a PBL type problem which
appeared in a paper called “Teacher as designer: A framework for teacher analysis of
mathematical model-eliciting activities”, by Hjalmarson and Dufies Dux. The problem is called “The Million Book Challenge”. While it may be engaging, students will generally lack the skills required to solve such a problem, such as knowledge of proper experimental approaches, systematic and random errors, organizational skills, and validation and verification.
Problems such as the million book challenge are predicated on the idea that by repeatedly confronting students with new situations as well as problems with which they have little to no experience, they will develop problem solving schema. The open-endedness of the problem is seen as a means to engage students in the “process” of critical thinking. In my opinion, however, such approach is like learning German by practicing particular sentence constructions in English (e.g., “I know that he the book read has”) in the hopes of building up a structure that then only needs vocabulary to complete the learning process.
Let’s turn now to the second problem—a problem appropriate for an honors algebra 1, or regular algebra 2 class. Unlike the million book challenge, it allows students to rely on prior knowledge, it is well-defined, and has specific mathematical goals. I think of problems such as the canoe problem as going down a well traveled road—you know where it is taking you even though it may have a few twists and turns and detours in unfamiliar territory.
Proponents of PBL tout problems such as the million book challenge as demonstrating that the real world provides more meaningful and useful mathematics than traditional problems. But problems like the canoe problem demonstrate that mathematics is needed to describe what is going on—and can lead us to a conclusion that may contradict what one intuitively believes. Many students will assume that in the canoe problem either route will take the same amount of time. They reason that if the boy travels upstream and downstream for the same distance at the same rate of speed, the amount the canoe is slowed by the current when travelling upstream is cancelled by the additional speed the current imparts when travelling downstream. But the mathematics shows otherwise.
It is also interesting that despite the criticism of math reforms that traditional problems lend themselves to formulaic solutions, the canoe problem does not lend itself to a “plug-in” solution. The proof that the time it takes to go across the lake is less than going up and downriver the same distance is not easy for beginning algebra students. Proofs of inequalities do not easily lend themselves to algorithmic solutions–in fact, they require the critical thinking and analytical skills that many believe problems like the Million Book Challenge develop.
According to Vern Williams, a middle school math teacher, and who served as a member of the President’s National Mathematics Advisory Panel, “By taking math that has been taught to them and attempting to solve difficult problems, they will discover relationships between content and methods that they already have in their arsenal even if they don’t solve the problem or arrive at the correct answer.”
The issue of PBL versus traditional type problems has particular significance
in light of the recent interest in developing assessments for math that are considered to measure “authentic” reasoning and skills. Critics of traditional type problems believe that assessments should evaluate the “critical thinking” skills of students rather than having students solve “exercises” that lend themselves to applications of previously learned problem-solving procedures. Many such critics also believe that students in
other countries that surpass the US in math on international tests are being taught only how to take tests.
Ironically, the problem with a test that emphasizes the ill-posed “rich” type of problems is that it accommodates students’ learning how to answer open-ended questions. Also ironically, the U.S. is not achieving as high scores on such tests as Asian nations. (The PISA exam given every three years contains some of these type of problems). Which begs the question of what mathematics are U.S. students exposed to PBL type approaches really learning? In the end the problems which students in Singapore, Hong Kong and Japan excel at solving are still likely to be off the script for many US students.
In a world in which problems that have a unique answer obtained through systematic application of mathematical skills and principles are deemed “mere exercises”, students are heading down a PBL approach to learning that leads more to math appreciation than math proficiency. If, however, students are taught the skills and concepts necessary to solve well-defined and challenging problems, they will learn to surmount what a disheartening number of U.S. students now consider to be insurmountable.