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Most students never learn how to think about solving problems. Throughout their education, they are constrained apply the material from each chapter to solve a few problems given at the end of each chapter (why else would a problem be at the end of the chapter?). With this type of approach to “problem solving,” it is not surprising that students are ill prepared for framing and addressing real-world with instructions or textbooks.
Although many educators are interested in teaching “thinking skills” rather than “teaching independent thinking (or problem-solving skills) regardless of the nature of a problem. As Alex Fisher wrote in his book, Critical Thinking: “… though many teachers would claim to teach their students ‘how to think’, most would say that they do this indirectly or implicitly in the course of teaching the content which belongs to their special subject. Increasingly, educators have come to doubt the effectiveness of teaching ‘thinking skills’ in this way, because most students simply do not pick up the thinking skills in question.” This approach has dominated the educational arena – whether in history, physics, geography, or any other subject – almost ensuring that students never learn how to think about solving problems in general.
Over the past few decades, various people and organizations have attempted to address this educational gap by teaching “thinking skills” based on some structure (e.g., critical thinking, constructive thinking, creative thinking, parallel thinking, vertical thinking, lateral thinking, confrontational and adversarial thinking). However, all these approaches are characterized by a departure from mathematics as they concentrate more on “talking about problems” rather than “solving problems.” It is our view that the lack of problem-solving skills in general is the consequence of decreasing levels of mathematical sophistication in modern societies.
Hence, we believe that a different approach is needed. To address this gap in the educational curriculum, we have created a new course (based on this book) that focuses on getting students to think about framing and solving unstructured problems (those that are not encountered at the end of some textbook chapter …). The idea is to increase the student’s mathematical awareness and problem-solving skills by discussing a variety of puzzles. In other words, we believe that the course should be based on the best traditions introduced by Gyorgy Polya1 and Martin Gardner2 during the last 60 years. In one of our favorite books, Entertaining Mathematical Puzzles, Martin Gardner wrote:
Many other mathematicians have expressed similar views. For example, Peter Winkler in his book Mathematical Puzzles: A Connoisseur’s Collection wrote: “I have a feeling that understanding and appreciating puzzles, even those with one-of-a-kind solutions, is good for you.” As a matter of fact, the puzzle-based learning approach has a much longer tradition than just 60 BC! Yet the best evidence of the puzzle-based learning approach can be found in the works of Alcuin, an English scholar born around AD 732 whose main work was Problems to Sharpen the Young – a text which included over 50 puzzles.
Many other mathematicians have expressed similar views. For example, Peter Winkler in his book Mathematical Puzzles: A Connoisseur’s Collection wrote: “I have a feeling that understanding and appreciating puzzles, even those with one-of-a-kind solutions, is good for you.”
As a matter of fact, the puzzle-based learning approach has a much longer tradition than just 60 years. The first mathematical puzzles were found in Sumerian texts that date back to around 2,500 BC! Yet the best evidence of the puzzle-based learning approach can be found in the works of Alcuin, an English scholar born around AD 732 whose main work was Problems to Sharpen the Young – a text which included over 50 puzzles. Some twelve hundred years later, one of his puzzles is still used in countless artificial intelligence textbooks!
Of course, it is difficult to give a universal definition as sometimes the difference is not clear between a puzzle and a real problem. However, in this text we concentrate on educational puzzles that support problem-solving skills and creative thinking. These educational puzzles satisfy most of the following criteria (also see the preface in Peter Winkler’s book Mathematical Puzzles: A Connoisseur’s Collection):
Of course, we do not need to satisfy all of these criteria. For example, the zebra puzzle (puzzle 5.4), – not to mention the monkey and the rope puzzle (puzzle 12.28) – are impossible to remember as they contain too many details. Some puzzles (e.g., puzzle 6.2 on the traveling salesman problem) have no entertainment value, but there is no question they are educational! And a few puzzles, such as the 7-Eleven problem (puzzle 12.6) or some versions of Nim games (puzzle 11.6), do not have elementary solutions. Thus, in this book we have focused on educational puzzles using our own intuition and many years of teaching experience.
Besides being a lot of fun, the puzzle-based learning approach does a remarkable job of convincing students that (a) science is useful and interesting, (b) the basic courses they are taking are relevant, (c) mathematics is not that scary (there is no need to hate it!), and (d) it is worthwhile to stay in school, get a degree, and move into the real world which is loaded with interesting problems (problems perceived as real-world puzzles). These points are important, as most students are unclear about the significance of the topics covered during their studies. Often times, they do not see a connection between the topics taught (e.g., linear algebra) and real-world problems, and they lose interest with predictable outcomes.
There are other well-established learning methodologies that address some of the above issues; these include problem-based learning and project-based learning (e.g., Blumenfeld et al. 1991, Bransford et al. 1986). Note, however, that the problem- and project-based approaches deal with quite complex situations where there is usually no single clear, unique, or correct way of proceeding. For example, projects may include assignments such as: Where is the best location for a new airport in our city? Or: How can we run an efficient marketing campaign for a new product with a limited budget? There may not be a single “best” solution to these problems or projects.
The emphasis in these approaches is usually on how to deal with the complexity of the problem and how to integrate the use of a wide range of techniques. Furthermore, project-based learning may involve teams of people with perhaps different specialist knowledge. With both problem- and project-based learning, a major piece of work is conducted under the supervision of an experienced facilitator acting in a mentoring role.
In a complementary contrast to problem-based learning, puzzles tend to be at the other end of the spectrum. They appear to be deceptively simple and usually have a single correct answer. An important part of completing a puzzle is to understand what we have learned by solving the puzzle and how we can apply this knowledge to other problems.
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