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Spring 2002
The goal is to arrive at the number 6. Here are the limited resources you may use to get to 6. You may use the numerals 1, 2, 3 and 4, but only once per solution. And you may use the following mathematical operations in each solution, but only the number of times specified: one x (multiplication) operation, two - - (subtractions) and two // (divisions). Parentheses can be used as often as you like, e.g. (4x3)/2. You may use a given resource-operation only once; that is, 2 x 3 = 6 and 3 x 2 = 6 do not count as different solutions. (4x3)/2 x 1 is unacceptable, because it requires two x operations. The resources yield exactly 12 solutions-12 different ways to arrive at 6. What are they? (See below.) If you find this problem difficult, don't be discouraged. That's the idea. Get out pencil and paper and keep plugging away at it and you'll sharpen your wits. That's what U-M Law School Prof. Layman Allen discovered in the early 1960s when he was a young law professor at Yale, trying to motivate his 12-year-old Sunday school students. He made up a logic game for them to play after they had completed their regular lessons and was surprised at how quickly they became adept at tackling his puzzles. Their skill surpassed the reigning psychological theories regarding the reasoning capabilities of young minds. Allen, seeing how powerful a motivator gaming was in the acquisition of difficult reasoning skills, began exploring its use in the classroom. Working with his brother Bob, he developed several nationally marketed instructional games, including Wff 'n Proof, a game of symbolic logic, and Equations, a creative math game. By the mid-1960s, instructional gaming had caught on in the education field, and a Florida school district brought in Allen, who was still teaching law at Yale, as an instructional game consultant. Soon, educators launched the Academic League of Games, and students began competing in local, state and national academic contests. Today, in addition to his recognition as a research scientist and law professor, Allen is regarded as a pioneer in the mathematics sub-field of instructional games. The publication The Mathematics Teacher said of his Wff 'n Proof game: "Through playing these games, a child can learn propositional logic, how to make proofs in it and how the content of a deductive science depends on the assumptions and rules. The games can be learned as early as kindergarten, and can be of value even to college students." Many studies have shown that mind games like those Allen developed can increase the problem-solving abilities of the players, raise their math scores and boost their scores on IQ tests. A 1972 study compared the Stanford Achievement Test in Mathematics scores of two groups of urban junior high school students. For a nine-week period, two classes received instructions using the Equations game, with team learning and tournaments each week. The other two classes received regular math instruction. All of the classes had comparable scores on the pre-test. However, students in the experimental class that used Equations gained more than double the gains of the conventional class on the post-test scores. A fundamental component of the puzzles is that they are too hard to do in one's head, Allen notes. If the player views the puzzle as difficult, yet is able to solve most or all of it, success provides immediate reinforcement and increases the player's confidence, encouraging further attempts to master new ideas. "No one has come up with anything that fully explains the extraordinary results," Allen says. "I think the kids just become more confident. They are willing to tackle things they don't know and tinker with them." In addition to a confidence boost, players show another attitude improvement-reduced absenteeism. A study of nine junior high math classes in Detroit compared absenteeism of students as a measure of their attitudes about the learning environment. The classes that used the instructional games had three times less absenteeism than classes that used conventional construction. Gloria Jackson, a Detroit teacher in the early 1970s, commented on the effects of games in an issue of Education USA, a weekly newspaper put out by the National School Public Relations Association. "Using Equations allows students to discover ideas for themselves," Jackson commented. "And it leaves the teacher free to be a consultant, explaining concepts like negative numbers as they come up." Last fall, Allen introduced the Math-Science Quest for Solutions, a program that integrates mathematics and science and is available on the Internet for teachers to use in the classroom. Math-Science Quest encourages students to use experimental science to solve problems, using sets of puzzles derived from the game Equations. Designed to take no more than five minutes out of class time, one problem is posed each week and then the proposed solutions and experiments by team members are uploaded. The puzzles can supplement a math or science class, and at whatever level the teacher finds appropriate. "If I were to label the principal thing we've discovered over 40 years, it is just how difficult it is to apply ideas to practical situations; it is enormously more difficult than people are aware of," says Allen, who regards the games and puzzles as "resource-allocation endeavors." "Often people have a goal they are trying to achieve," he explains, "and there are limitations, even active opposition, putting constraints on the resources available to try and achieve that goal. In effect, the games are designed to have students deal with that kind of application, like practical problems in the world." So the next time you are in a quandary, don't be daunted-refine your problem-solving skills. Tackle a logic puzzle, if you feel your brain is rusty. You just might achieve more than reaching a goal of 6. Freelancer Rachel Ehrenberg received her MA in biology from Michigan in 2001. She is also a communications intern at the School of Natural Resources and the Environment.
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