In an Edutopia article the difference between the two PBL’s: Project and ProblemBased Learning was addressed. It included a nice definitional list of what is involved in ProblemBased Learning:
Problembased learning typically follow prescribed steps:

Presentation of an “illstructured” (openended, “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 selfdirected 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 K12grades—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
opposing sides:
Problem 1: How many boxes would be needed to pack and ship one million books collected ina schoolbased 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 thoughtworld 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 “problembased 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 modeleliciting 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 openendedness 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 welldefined, 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 “plugin” 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 problemsolving 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 illposed “rich” type of problems is that it accommodates students’ learning how to answer openended 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 welldefined and challenging problems, they will learn to surmount what a disheartening number of U.S. students now consider to be insurmountable.
I abhor the poorly defined nature of problems like the million book problem.
But, leaving that aside, there is another reason it is a terrible problem from the perspective of a math teacher.
That is that the level of the material needed to solve the problem is incommensurate with the maturity needed for the student to tackle it.
Now as you suggest one might get into sampling methodology, statistics etc but no student I know will be motivated to research that path — or to even know there is a wellestablished literature in the field. Even if they did it’s irresponsible for teachers to expect students to go on a major reading exploration of their own choosing to learn about such things. If learning statistics is the goal then there ought to be a welldeveloped course of studies leading up to it. You don’t say “go and learn an entire course in statistics to solve this one problem”.
It’s unclear from the start that an experimental approach is even needed. So on the one hand if this is given to a, say, Grade 8 class they have a huge uphill climb just to learn (without assistance apparently) the fundamentals for doing this so it’s well above their level …
Or … they do as I would do with such a problem and decide it’s a “fantasy problem” so one makes up a fantasy solution in which all boxes are the same size and all books are the same size. Then one declares that a box holds 50 books (or some such figure). Then one divides 1,000,000 by 50 getting 20,000 boxes as the answer.
The “figgering” part about boxes and books in this solution is not really of any value in the learning of math. So it comes down to the use of division — by long division, or a simple “strategy” (divide by 100 by knocking off two zeros and then double the answer) or worse using a calculator. This tool is below grade 8 level and so it should not be the key mathematical learning content in the problem.
Either way it is hard to see how any approach is likely to hit the “sweet spot” one wants of a good problem for a particular grade level. If you want them to divide a million by 50 in grade 6 this problem might be a bad way to do so because I expect the illdefined nature of the problem will not sit well with the maturity level of grade 6 students.
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Thanks; good points. Trivia fact: one of the authors of the paper I referenced, was my advisor at ed school.
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