A R T H U R J. B A R O O D Y

AND

J E S S E L. M. W I L K I N S

Inverting a Triangular Array: Involving Students in Mathematical Inquiry

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MATHEMATICS TEACHING IN THE MIDDLE SCHOOL Copyright © 2004 The National Council of Teachers of Mathematics, Inc. www.nctm.org. All rights reserved. This material may not be copied or distributed electronically or in any other format without written permission from NCTM.

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OSTERING STUDENTS’ MATHEMATICAL

thinking requires regularly presenting them with challenging problems and involving them in the processes of mathematical inquiry (e.g., representing and solving problems; making, testing, and justifying conjectures). The Inverting a 36 Penny Triangular Array problem (to be called the Inverting problem, for short) is shown in figure 1. It is a challenging problem that can immerse middle school, secondary school, college students, and in-service teachers in the processes of mathematical inquiry. The Inverting problem is based on an item that appeared at least as early as the 1970s, was used by the University of Wisconsin Middle School Project, and was eventually published in the Mathematics in Context series (Kindt, Roodhardt, Spence, Simon, and Pligge 1998). It can be used to illustrate the value of problem-solving heuristics and the following three points about the processes of mathematical inquiry: (1) each type of mathematical reasoning has its pitfalls; (2) looking for counterexamples can be a valuable method of checking on, and improving, conjectures; and (3) using multiple solution methods and finding converging evidence can strengthen the case for, although never prove, a conjecture. The goal of this article is to share what we have learned from our teaching experience with pre- and in-service teachers and nearly 180 middle school students about using the Inverting problem as an instructional tool.

Getting Started: The Role of, and Creating a Need for, Problem-Solving Heuristics IDEALLY, INSTRUCTION ON PROBLEM-SOLVING

heuristics, like any content instruction, should be done purposefully. It can be achieved by presenting genuinely challenging problems such as the Inverting problem, allowing students to use their own ART BAROODY, [email protected], is a professor of cur-

riculum and instruction at the University of Illinois at Urbana—Champaign. His research focuses on the development of number and arithmetic concepts and skills among young children and those with learning difficulties. JAY WILKINS, [email protected], is an associate professor at Virginia Polytechnic Institute and State University, Blacksburg, Virginia, where he teaches mathematics education courses. His areas of research include educational opportunity and quantitative literacy. The authors are grateful to Jeanne Campione (MahometSeymour Junior High School in Mahomet, Illinois) and Jennifer Wall (Blacksburg Middle School in Blacksburg, Virginia) for allowing them to try out the problem with their classes and providing detailed feedback on the results.

Inverting a 36 Penny Triangular Array What is the fewest number of moves it would take to turn the triangle of 36 pennies (shown at right) upside down? Constraint: Only one penny may be slid or otherwise moved per move (Baroody, with Coslick 1998, pp. 2–14). Fig. 1 The Inverting problem

strategies and become “blocked,” and then suggesting a heuristic as a way to overcome the block (Baroody, with Coslick 1998). By creating a “felt need” for heuristics, students may more likely appreciate these problem-solving aids and use them spontaneously in the future.

Useful Heuristics for the Inverting Problem A NUMBER OF PÓLYA’S HEURISTICS CAN BE USE-

ful in solving the Inverting problem. One is using a model. A teacher can either suggest this heuristic after a class has had a chance to struggle with the problem or, if some students spontaneously use modeling, encourage them to share this strategy and discuss why it might be helpful. Students, though, often find that modeling with 36 pennies, blocks, or other objects is difficult or even confusing. Indeed, the intention in specifying so many pennies in the problem was to create an impasse—a situation in which finding the solution directly by modeling would not be successful—and, thus, create a need for additional problem-solving heuristics. Some include considering simpler cases, organizing data in a table and looking for a pattern (inductive reasoning), and using logical (deductive) reasoning.

False Starts: Pitfalls, Patterns, and the Role of Counterexamples CONCRETELY MODELING SIMPLER CASES, CREAT-

ing a table, and using inductive and deductive reasoning do not guarantee that students will find a correct solution to the Inverting problem. Even so, they can learn some valuable lessons about the processes of mathematical inquiry. Furthermore, even “false starts” (e.g., conjectures that are disproved) can lead to the discovery of important ideas either within or beyond the scope of the problem at hand. An example of the latter is that non-Euclidean geometry was discovered in the search for a proof of Euclid’s parallel postulate. In the following section, we describe several false starts and the students’ thinking that led to them. V O L . 9 , N O . 6 . FEBRUARY 2004

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A centering solution strategy Several preservice teachers proposed a “centering” strategy, which involves choosing the center or a centermost penny in the base of the original triangle to become the top penny of the inverted triangle. As figure 2 shows, this strategy leads to the observation that it takes one, two, and four moves to invert the simplest cases of a 3-penny triangle, a 6-penny triangle, and a 10-penny triangle, respectively. By modeling the centering strategy with the next three larger triangles, students have created the first three rows of table 1. By analyzing row 3 in table 1, some students noticed a growing pattern (the among-the-number-ofmoves pattern summarized in row 4 of the table). A common conclusion is that, except for the +1, the increases occur in pairs: +2, +2, +3, +3, +4, +4, and so forth. Using this induction as their premise, the stu-

A. 3 pennies ➝1 move

B. 6 pennies ➝2 moves

C. 10 pennies ➝4 moves

Legend: A dashed-line triangle indicates the original triangle. A solid-line triangle represents the inverted triangle. An open circle illustrates an unmoved penny. A shaded circle shows the original position of a moved penny. A solid circle indicates the new position of a moved penny (within the inverted triangle).

dents deduced that the seventh (36-penny) triangle can be inverted in 12 + 4, or 16, moves. One preservice student noticed that this solution has an interesting between-row pattern. Specifically, he observed that the difference between the number of pennies (see row 2 in table 1) and the number of moves (row 3) resulted in the number of moves for the next larger triangle. For example, the difference between the number of pennies for triangle 2 and the number of moves for this triangle, namely, 6 – 2, is the number of moves for the third triangle, namely, 4. He thus predicted 16 moves for the seventh triangle. Conflicting solutions Observant students have noticed, however, that other growing patterns are possible and that these patterns lead to a different prediction. For example, the pattern could be (a) +1, +2, +2, +3, +3, +3, . . . (the increases between successive values form an arithmetic sequence: one +1, two +2s, three +3s . . .) or (b) +1, +2, +2, +3, +3, +3, +3, . . . (the increases form a geometric sequence: one +1, two +2s, four +3s . . .), both of which logically lead to a solution of 12 + 3, or 15, moves for the seventh triangle. Clearly, the solution one deduces (15 or 16 moves) depends on one’s premise (the pattern induced). A discussion can help students appreciate that although the solutions of 16 and 15 each necessarily follows from its premise, both cannot be correct. In general terms, if a premise is incorrect, a conclusion can be logical but not necessarily reasonable. Refutation

For triangles in which more than one move occurs, the order of the moves and the new locations of the pennies are irrelevant. For example, for B above, either the penny in the lower-left corner or the lower-right corner of the original triangle could be moved first to either the left or the right corner of the inverted triangle’s base.

To evaluate their conjectured solutions and resolve which conflicting solution, if either, is correct, students can be encouraged to look for counterexamples. Typically, further exploration with models and use of a strategy more flexible than the centering strategy leads to the conclusion that

Fig. 2 A centering strategy

TABLE 1 The Data Generated by Using the Centering Strategy and a More Flexible Strategy

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Row 1

Triangle number

1

2

3

4

5

6

7

Row 2

Number of pennies

3

6

10

15

21

28

36

Row 3

Number of moves using the centering strategy

1

2

4

6

9

12

?

Row 4

Among-the-number-of-moves pattern in row 3

+1

+2

+2

+3

+3

?

Row 5

Number of moves using a more flexible strategy

1

2

3

5

?

?

MATHEMATICS TEACHING IN THE MIDDLE SCHOOL

?

a 10-penny triangle and a 15-penny triangle can be inverted in fewer moves, 3 and 5 moves, respectively (see fig. 3). These results are summarized in row 5 of table 1.

The “Fibonacci Solution” An interesting observation Using a model involving 21 pennies, several groups of preservice teachers found that the fifth triangle could be inverted in a minimum of eight moves and conjectured that the number of moves for successive triangles formed a Fibonacci-like sequence (i.e., 1, 2, 3, 5, 8, . . .). It logically followed from this assumption that the sixth (28-penny) and the seventh (36-penny) triangle could be transformed in a minimum of 13 and 21 moves, respectively. Refutation A class had just found that a 21-penny array could be inverted in 8 moves, when one student claimed he could do so in 7 moves. Despite the skepticism of the

instructor and the rest of the class, the student was given an opportunity to make his case. He proceeded to demonstrate the solution in figure 4 (a solution that can also be found in Kindt et al. [1998]). This counterexample to the Fibonacci solution launched the class into finding shortcuts for the sixth triangle (28 pennies). After considerable exploration, several students found that a 28-penny triangle could be inverted in 9, not 13, moves. The new findings for the 21- and 28-penny triangles suggested that the solution for the 36-penny triangle required fewer than 21 moves—probably far fewer.

Multiple Solution Methods DETERMINING THE MINIMUM NUMBER OF moves for the seventh (36-penny) triangle proved to

be challenging. Over the next several years, successive pre- and in-service classes found a variety of patterns in their data tables or devised a strategy for generating the minimum number of moves, all of which pointed to the same solution of 12 moves. The among-relation pattern The first of these efforts was based on the firstorder relation among the number of moves, which constitute a growing pattern. See figure 5. The argument was that this “among relation” formed a pattern: two +1s and three +2s, presumably followed by four +3s, then five +4s, and so forth. A counter-

A. 3 pennies ➝1 move

B. 6 pennies ➝2 moves

Legend:

C. 10 pennies ➝4 moves

D. 15 pennies ➝5 moves

A dashed-line triangle indicates the original triangle. A solid-line triangle represents the inverted triangle. An open circle illustrates an unmoved penny. A shaded circle shows the original position of a moved penny. A solid circle indicates the new position of a moved penny (within the inverted triangle).

Legend: A dashed-line triangle indicates the original triangle. A solid-line triangle represents the inverted triangle. An open circle illustrates an unmoved penny. A shaded circle shows the original position of a moved penny. A solid circle indicates the new position of a moved penny (within the inverted triangle).

Fig. 3 Solutions that result from a more flexible strategy

Fig. 4 A solution for inverting a triangular array of 21 pennies

The among-the-number-of-moves pattern: 1 +1

2

3 +1

5 +2

7 +2

9 +2

12 +3

15 +3

18 +3

Fig. 5 A growing pattern V O L . 9 , N O . 6 . FEBRUARY 2004

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argument was that maybe the relations among the number of moves formed the following pattern: two +1s, four +2s, and perhaps eight +3s, and so forth. This alternative (geometric growth) conjecture meant that a 36-penny triangle could be inverted in 11 moves, not 12, as proposed. These conflicting claims prompted further efforts to support or disprove 12 as the solution to the Inverting problem.

Reconciling the evidence Readers may have noticed that the increasingnumber-of-increments conjecture (the first amongrelations pattern) is not entirely consistent with the subsequent patterns or the minimum-maximum solution described above. If the first conjecture is corA. 16 moves

The between-relation pattern Several students independently discovered the relation between the number of pennies and the number of moves: divide the former by 3 and ignore the remainder; the resulting integer is equal to the number of moves, as shown in figure 6 (see, also, Kindt et al. [1998]). The between-relation pattern was the first independent evidence that the seventh (a 36-penny) triangle could perhaps be inverted in 12 moves, not 11—that the increasing-number-of-increments conjecture might be correct and that the alternative geometric-growth conjecture was incorrect.

B. 13 moves

C. 12 moves

A minimum-maximum solution strategy A group of preservice teachers demonstrated yet another way of inverting a 36-penny triangle in only 12 moves. They overlaid a frame of an inverted triangle on the original triangle and systematically positioned the former’s base so that it was (a) initially just above the first row of the original triangle (see frame A in fig. 7), (b) then between the first and second row of the original triangle (see frame B in fig. 7), and so forth (see frames C, D, and E in fig. 7). By doing so, they discovered that the number of moves required to invert the triangle dropped from 16 for the first case (frame A), to 13 for the second case (frame B), to 12 for the third and fourth cases (again, see frames C and D), then it went back up to 14 for the fifth case (frame E). This movement suggested that 12 was the minimum number of moves it would take to invert a 36-penny triangle (cf. Kindt et al. [1998]) although they did not venture to test it for different triangles. Further analysis suggested that finding the minimum number of moves is equivalent to overlying the frame of the inverted triangle in such a way as to include as many pennies from the original triangle as possible. That is, it is equivalent to maximizing the number of pennies that do not have to be moved.

3 ÷3 1

6 ÷3 2

10 ÷ 3 15 ÷ 3… 36 ÷ 3 3 r1 5 12

Fig. 6 The integer is equal to the number of moves. 310

MATHEMATICS TEACHING IN THE MIDDLE SCHOOL

D. 12 moves

E. 14 moves

Legend: A dashed-line triangle indicates the original triangle. A solid-line triangle represents the inverted triangle. An open circle illustrates an unmoved penny. A shaded circle shows the original position of a moved penny. A solid circle indicates the new position of a moved penny (within the inverted triangle).

Fig. 7 The minimum-maximum solution for a 36-penny triangle

PHOTOGRAPH BY JESSE L. M. WILKINS; ALL RIGHTS RESERVED

rect (i.e., if the number of moves increases by +1 two times, +2 three times, +3 four times, and so forth), then the tenth triangle consisting of 66 pennies should require 18 + 3, or 21, moves. However, the between-relation pattern, for instance, leads to the prediction of 66 ÷ 3, or 22, moves, which means that the increase in the number of moves between the ninth and tenth triangles is 22 – 18, or 4, not 3. Consistent with the between-relation pattern is the fact that the triangular numbers actually start with 1 (penny), not 3 (pennies). As the number of moves required to invert this theoretical “triangle” is 0, the increase in the number of moves between it and the next larger (the 3-penny) triangle is +1. Thus, the pattern in the increment of moves among successively larger triangles forms an increasing triples pattern: +1, +1, +1, +2, +2, +2, +3, +3, +3, +4, +4, +4, and so on.

Levels of Sophistication in Pattern Use

pattern, but they use the same cases that were used to formulate it. Relatively sophisticated users recognize that a pattern description is merely a conjecture that must be tested using new cases. Highly sophisticated inquirers recognize that, although testing additional cases may strengthen the case for a conjectured pattern or formula, it may not apply to unexamined cases, and a formal proof is necessary to validate the description in general.

We tested the problem with nine eighthgrade classes involving nearly 180 students

Implications for Middle School Instruction TO GAUGE WHETHER OUR EXPERIENCES WITH

pre- and in-service teachers were applicable to middle school instruction, we tested the problem with nine eighth-grade classes involving nearly 180 students.

STUDENTS USE PATTERNS WITH DIFFERENT LEVELS

of sophistication as they engage in mathematical inquiry (Pat Collier, personal communication, February 7, 2001). Many students do not even consider the power of looking for a pattern as a way of solving problems. Somewhat more sophisticated inquirers look for a pattern but believe that the problem is solved once they find one. For example, the preservice teachers who presented the minimum-maximum strategy seemed convinced that they had found the solution and felt that there was no need to check other examples. Even more sophisticated pattern users test their

General results Nearly all the middle school students readily understood the Inverting problem, and the vast majority were interested in solving it. One algebra class, for example, continued to work on the problem during a study hall following their mathematics class. Although many false starts were typical and most students had to be led to examine simpler cases, middle school students—like pre- and in-service teachers— used multiple methods in their efforts to solve the V O L . 9 , N O . 6 . FEBRUARY 2004

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Triangle number:

1

2

3

4

5

6

7

Number of pennies:

3

6

10

15

21

28

36

Number of moves:

1

2

3

5

7

9

?

dents who do not look for patterns or who believe a problem is solved by finding any pattern, a teacher can underscore that arriving at a solution for the 36-penny case does not guarantee that it is correct. Challenge them with a question, such as “How can you be sure that n is the fewest moves?” For all but highly sophisticated inquirers, students can be encouraged to consider other cases to test their conjectures. They can also be prompted to check it by solving the problem in a different manner to provide converging evidence. For students at all levels, sharing solution methods and solutions with a class can promote reflection and development. For example, prompted to justify their solution by looking for a pattern for a variety of cases, several students in one group hinted at the “divide by 3” pattern. However, the group did not accept this justification because one member noted that, in the case of 55 pennies, this number divided by 3 was not 18. By sharing the conjecture and counterargument with the whole class, there is a good chance that someone would have recognized that the divide by 3 pattern works if the remainder is not considered. Although challenging, middle school teachers can use the Inverting problem to promote strategic mathematical thinking.

Fig. 8 A between-relations pattern

Inverting problem, and many concluded that its solution was 12. The among-relation pattern (the number of moves increases by +1 three times, +2 three times, +3 three times, and so forth) was discovered by at least one class. The between-relation (“divide by 3”) pattern was independently discovered in nearly all classes. One prealgebra student noticed the complex between-relation pattern depicted in figure 8. Students in two classes used overlapping drawings or transparencies to devise what was called by one class “The Star of David” solution strategy (see fig. 3). Different levels of sophistication in pattern use Students or even whole classes, however, used patterns with different levels of sophistication as they tried to solve the Inverting problem. Eighth-grade prealgebra students either did not appreciate the value of heuristics, such as looking for a pattern, or uncritically accepted as the solution the first pattern that they discovered. Although the 36-penny case was posed to create an impasse, most tended to work on it, even when examining simpler cases and looking for a pattern were suggested by their instructor. Even those who made an effort to look for a pattern were typically content with their solution until someone else found a better one. In brief, these students did little in the way of conjecturing (e.g., offering alternative patterns or solution strategies) by evaluating alternative patterns or solutions. Eighth-grade algebra students’ level of patternuse sophistication was broader. Some students continued to rely on direct modeling, but most implemented the suggestion to work on simpler cases and look for a pattern. Students typically had to be asked how they could convince themselves and others that 12 moves was the fewest.

Promoting the mathematical thinking of students operating at different levels of sophistication requires flexibility. A teacher could allow students to solve the 36-penny case as they choose; wait until they are blocked; then suggest that they try simpler cases, make a table, and look for a pattern. If they persist with direct modeling, challenge them to solve a larger case, such as inverting a 66-penny triangle, in which direct modeling is not practical. Especially with stu312

MATHEMATICS TEACHING IN THE MIDDLE SCHOOL

HELPING STUDENTS TO BECOME BETTER PROBLEM

solvers must go beyond introducing and practicing problem-solving heuristics, such as looking for a pattern or using logical reasoning. Teachers need to provide ample opportunity for solving challenging problems and for reflecting on the pitfalls or limitations of these methods. This is particularly true when students

PHOTOGRAPH BY JESSE L. M. WILKINS; ALL RIGHTS RESERVED

Instructional adaptations

Conclusions about Fostering Inquiry Development

PHOTOGRAPH BY JESSE L. M. WILKINS; ALL RIGHTS RESERVED

use the look-for-a-pattern heuristic, because many believe that providing several or even one example proves an induced conclusion. A goal of teachers, then, should be to help students better understand the process of mathematical inquiry by prompting them to use patterns in an increasingly sophisticated manner. As the vignettes described earlier illustrate, solving challenging problems like the Inverting problem can help students recognize the importance of looking for patterns, evaluating conjectures generated by intuitive or inductive reasoning about the patterns, considering alternative conjectures, checking additional examples, and searching for counterexamples. Students may also recognize that logical (deductive) reasoning is only as reliable as the premises that it is based on. They may further recognize that even false starts can play an important role in mathematical inquiry. The vignettes illustrate the importance of encouraging students to find supporting evidence for a conjectured pattern or strategy by, for instance, finding alternative methods (patterns or strategies) that independently lead to the same solution. As the vignettes further show, students will probably need help recognizing that finding a pattern is not the same as proving it correct or that finding any number of examples for a pattern is not a substitute for a formal proof. Finally, the vignettes illustrate how important it is that teachers at all levels model the social aspects

of mathematical inquiry, including that of listening carefully to others, even when they challenge a cherished belief.

References Baroody, Arthur J., with Ronald T. Coslick. Fostering Children’s Mathematical Power: An Investigative Approach to K–8 Mathematics Instruction. Mahwah, N.J.: Erlbaum Associates. 1998. Kindt, Martin, Anton Roodhardt, Mary S. Spence, Aaron N. Simon, and Margaret Pligge. “Patterns and Figures.” In Mathematics in Context. Chicago, Ill.: National Center for Research in Mathematical Sciences Education and Freudenthal Institute, 1997–1998; Encyclopaedia Britannica, 1998.


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