JSGE Vol. XIV, No. 3, Spring 2003, pp. 151–165. Copyright ©2003 Prufrock Press, P.O.

opyright ©2003 Prufrock Press, P.O. Box 8813, Waco, TX 76714

The Journal of Secondary
Gifted Education

Mathematical Giftedness,
Problem Solving, and the Ability
to Formulate Generalizations:
The Problem-Solving Experiences of Four Gifted Students

Bharath Sriraman
University of Montana

Complex mathematical tasks such as problem solving are an ideal way to provide students opportunities to develop higher order math-
ematical processes such as representation, abstraction, and generalization. In this study, 9 freshmen in a ninth-grade accelerated alge-
bra class were asked to solve five nonroutine combinatorial problems in their journals. The problems were assigned over the course of
3 months at increasing levels of complexity. The generality that characterized the solutions of the 5 problems was the pigeonhole
(Dirichlet) principle. The 4 mathematically gifted students were successful in discovering and verbalizing the generality that character-
ized the solutions of the 5 problems, whereas the 5 nongifted students were unable to discover the hidden generality. This validates the
hypothesis that there exists a relationship between mathematical giftedness, problem-solving ability, and the ability to generalize. This
paper describes the problem-solving experiences of the mathematically gifted students and how they formulated abstractions and gen-
eralizations, with implications for acceleration and the need for differentiation in the secondary mathematics classroom.

fascinating aspect of human thought is the ability to Psychologists have also been interested in the phenomenon

A generalize from specific experiences and to form new,

more abstract concepts. The Principles and Standards
for School Mathematics (National Council of Teachers of
of generalization and have attempted to link the ability to gener-
alize to measures of intelligence (Sternberg, 1979) and to complex
problem-solving abilities (Frensch & Sternberg, 1992). Greenes
Mathematics, 2000) calls for instructional programs that (1981) claimed that mathematically gifted students differed from
emphasize problem solving with the goal of helping students the general group in their abilities to formulate problems spon-
develop sophistication with mathematical processes such as taneously, their flexibility in data management, and their ability
representation, mathematical reasoning, abstraction, and gen- to abstract and generalize. There is also empirical evidence of
eralization. It goes on to proclaim that students should develop differences in generalization in gifted and nongifted learners at the
increased sophistication with mathematical processes, espe- preschool level (Kanevsky, 1990). At the secondary level, there are
cially problem solving, representation, and reasoning, and their very few studies that document and describe how gifted students
increased ability to reflect on and monitor their work should approach problem solving, abstract, and generalize mathematical
lead to greater abstraction and a capability for generalization. concepts. This leads to the following questions:
Thus, the ability to generalize is the result of certain mathe- 1. What are the problem-solving behaviors in which high
matical experiences and is an important component of mathe- school students engage?
matical ability. The development of this ability is an objective 2. What are the differences in the problem-solving
of mathematics teaching and learning (NCTM). behaviors of gifted and nongifted students?

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 Sriraman

3. How do gifted students abstract and generalize math- tion, respectively. Orientation refers to strategic behavior to
ematical concepts? assess and understand a problem. It includes comprehension
strategies, analysis of information, initial and subsequent rep-
resentation, and assessment of level of difficulty and chance
Definitions of success. Organization refers to identification of goals, global
planning, and local planning. Execution refers to regulation
Problem-solving situation: a situation involving: of behavior to conform to plans. It includes performance of
a. a conceptual task local actions, monitoring progress and consistency of local
b. the nature of which the subject is able to understand plans, and trade-off decisions (speed vs. accuracy). Finally, ver-
by previous learning (Brownell, 1942; Kilpatrick, ification consists of evaluating decisions made and evaluating
1985), by organization of the task (English, 1992), or the outcomes of the executed plans. It includes evaluation of
by originality (Birkhoff, 1969; Ervynck, 1991); actions carried out in the orientation, organization, and exe-
c. the subject knows no direct means of satisfaction; cution categories.
d. the subject experiences perplexity in the problem sit- The metacognitive component of Lester’s (1985) model is
uation, but does not experience utter confusion; and comprised of three classes of variables: person variables, task
e. an intermediate territory in the continuum that variables, and strategy variables. Person variables refer to an
stretches from a puzzle at one extreme to the com- individual’s belief system and affective characteristics that may
pletely familiar and understandable situation at the influence performance. Task variables refer to features of a task,
other (Kilpatrick, 1985). such as the content, context, structure, syntax, and process. For
Generalization: the process by which one derives or induces example, an individual’s awareness of features of a task influ-
from particular cases. It includes abstracting properties (Davis ences performance. Finally, strategy variables refer to an indi-
& Hersh, 1981), identifying commonalties (Dreyfus, 1991) vidual’s awareness of strategies that help in comprehension,
and expanding domains of validity (Davydov, 1990; Dienes, organizing, executing plans, and checking and evaluation.
1961; Polya, 1954). These metacognitive behaviors are associated with the four
Problem-solving strategies: the actions and/or methods cognitive categories. The aim of Lester’s conceptual model is to
employed by students to understand and solve the problem sit- describe the behaviors in the four cognitive stages in terms of
uation. In this study, student strategies are classified according “points” where metacognitive actions occur during problem
to Lester’s (1985) conceptual model on problem-solving behav- solving.
ior described in the literature review. Schoenfeld (1985, 1992) suggested that problem solving
must be studied within the broader context of what learning
to “think mathematically” means. He described thinking math-
Literature Review ematically as developing a mathematical point of view, valu-
ing the processes of representation and abstraction, and having
A well-known problem-solving model is that of the emi- the predisposition to generalize them.
nent mathematician George Polya (1945), which consists of “Generalization is inseparably linked to the operation of
four phases: understanding, planning, implementing, and abstracting” (Davydov, 1990, p. 13). According to Davydov,
looking back. One of the shortcomings of Polya’s model is that the process of delineating a certain quality as a common one
it is algorithmic in nature, and research generated by it has and separating it from other qualities allows the child to con-
focused purely on heuristics. Lester (1985) attributed the fail- vert the general quality into an independent and particular
ure of most instructional efforts to improve students’ prob- object of subsequent actions. Abstraction is a process that
lem-solving performance to an overemphasis of heuristic skills occurs when the subject focuses attention on specific properties
while ignoring “managerial skills necessary to regulate one’s of a given object and then considers these properties in isola-
activity (metacognitive skills)” (p.62). It was suggested that tion from the original. This might be done to understand the
metacognitive activity or knowledge of one’s thought processes essence of a certain phenomena and to later apply the same the-
or self-regulation underlay the application of heuristics and ory to other applicable cases.
algorithms (Lester; Schoenfeld 1985, 1992). As a result, Lester Early research on generalization focused on elementary
modified Polya’s model to include a cognitive and a metacog- school children’s abilities to generalize number concepts
nitive component. (Davydov, 1990; Dienes, 1961; Shapiro, 1965). There was also
In the cognitive component, the four phases of under- considerable interest in the process of generalization among
standing, planning, implementing, and looking back were mathematics education researchers in the former Soviet Union
relabeled as orientation, organization, execution, and verifica- (Davydov; Krutetskii, 1976; Shapiro).

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 Mathematical Giftedness

Shapiro (1965) wrote that among mathematically gifted students in the class during data collection and data analysis.
students, However, after the data were collected and analyses were con-
structed, the researcher accessed their testing profiles to find that
the development of generalizations occurs from the 4 of the 9 students had been identified as mathematically gifted
first examples, at the initial stages of learning. Transfer in their elementary school. Identification at the elementary
in the general form is almost merged in time with gen- school was based on a variety of factors, including IQ scores
eralization and is accomplished immediately for a (over 124), the Stanford Achievement Test (95 percentile),
whole class of problems of a single type . . . among teacher recommendations, and counselor recommendations.
less capable students generalizations ripen gradually Table 1 shows the achievement profiles of the 9 students.
and are manifested at later stages or they do not Journal writing was an integral part of the accelerated alge-
develop at all. (p. 95) bra course. The teacher routinely assigned one nonroutine
problem or puzzle every other week, which the students solved
Krutetskii (1976) analyzed the generalization ability of in their journals. The researcher asked the students to record
both “normal” and “capable” (gifted) students in a series of everything they tried, including scratch work. Students were
experiments. He hypothesized that “students with different given three cues from the researcher:
abilities are characterized by differences in degree of develop- 1. Restate the problem in your own words. In other
ment of both the ability to generalize mathematical material words, what is the problem asking?
and the ability to remember generalizations” (p. 84). Krutetskii 2. How would you begin solving the problem?
studied 19 students with varying mathematical abilities. Based 3. Solve the problem and write a summary of what
on his experiments with the 19 students, Krutetskii concluded worked and what did not work.
that more “capable” (gifted) students were able to form math- Full credit was given to all students for completing the three
ematical generalizations rapidly and broadly. He noted that cues and including all their work.
these “capable” students were able to discern the general struc- The journal writings over the course of the school year
ture of the problems before they solved them. The “average” revealed that most students were articulate in their description
students were not always able to perceive common elements of their solution strategies and were capable of tackling math-
in problems, and the “incapable” students faired poorly in this ematical problems that were not covered by the school cur-
task. In order for students to formulate generalizations cor- riculum. In order to keep the setting of the study as natural and
rectly, they had to abstract from the specific content and sin- unobtrusive as possible and to be consistent with established
gle out similarities, structures, and relationships. classroom practices, the researcher assigned the five combina-
Most of the existent literature on the process of generaliza- torial problems (see Appendix A) for the study as journal
tion is in the context of number concepts, arithmetic, and alge- assignments, starting with the problem of lowest complexity.
bra. There is lack of research on generalization in the context These five problems were assigned over a period of 3 months.
of higher order mathematical processes such as problem solv- The journal problems were chosen with great care and rep-
ing at the high school level. In particular, there is lack of research resented situations that would facilitate representation, rea-
on the differences in problem-solving behaviors between gifted soning, abstraction, and eventually the formulation of
and non-gifted students. Such research would be valuable to generalizations. The following criteria were applied to deter-
practitioners who need to differentiate the curriculum in class- mine the journal problems.
rooms comprising of both gifted and nongifted students. 1. The problems had to be “mathematically” rich in the
sense that they were nonroutine and solving them
would require both perseverance and creativity on the
Methodology student’s part.
2. The chosen problems were combinatorial in nature
The researcher in this study was a full-time teacher at a because mathematics education research with elemen-
rural midwestern high school. The participants in this study tary children has consistently indicated that children
were 9 freshmen (4 males and 5 females) enrolled in accelerated have intuitive abilities in tackling combinatorial prob-
Algebra I taught by the researcher. The participants were lems (English, 1992).
White, with middle-class socioeconomic backgrounds. 3. The problems represented diverse situations and
Enrollment in accelerated algebra at this high school required increased in complexity. The plan of gradually increas-
recommendation from eighth-grade teachers, as well as above- ing the complexity of the problems was consistent
average performance in prealgebra. with earlier research on the process of generalization
The researcher did not access the testing profiles of the 9 (Davydov, 1990; Dienes, 1961; Krutetskii, 1976).

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 Sriraman

Table 1

Achievement Profiles of the Nine Students

Name IQ1 Stanford Stanford OLSAT

Achievement Test Achievement Test

Math Score2 Math Applications3 Nonverbal4

Raw Score Raw Score Raw Score
(out of 90) (out of 30) (out of 36)

Subset A: Mathematically gifted students who formed generalizations

Amy 162 89 30 36
John 124 85 29 32
Matt 140 87 30 33
Hanna 126 85 28 32

Subset B: Nongifted students who formed false generalizations

Bart 100 68 19 22
Jim 120 74 21 25
Isabel 105 70 20 22

Subset C: Nongifted students who did not form generalizations

Jamie 98 60 15 20
Heidi 102 62 16 21

Note.
1 Stanford-Binet (4th ed.). Mean = 100 ; Standard deviation = 16.
Data in columns 2–3 is extracted from the Stanford Achievement Test Series (administered to students in the 1st grade)
2 The math portion consisted of 90 items on number concepts (34), computation (26), and applications (30).
3 The mathematics applications portion consisted of 30 items on problem solving (12), graphs (3), geometry (6), and measurement (9).
4 The Otis-Lennon School Ability Test (7th ed.) administered to students in grade 6. Nonverbal portion of test consists of items on figural reasoning (18) and quantitative reasoning (18) involving mathematical concepts.

4. Both the problems and methods of solutions were initiate the four phases of problem solving (Lester, 1985).
generalizable. In addition, the solutions to a class of Students were given 7–10 days to solve each problem. The
seemingly different problems was characterized by an researcher collected the journals weekly in order to read the
overarching common generality, namely the pigeon- solutions developed by the students. The researcher then
hole principle, which states that if m pigeons are put in recorded in his journal possible questions to ask the students in
n pigeonholes and if m > n, then at least one pigeon- the interview.
hole will have more than one pigeon. Students were interviewed before or after school during the
The researcher conjectured that the strategies developed by week after the journals were turned in. The researcher followed
the students would evolve with the complexity of the prob- the clinical interview technique attributed to and pioneered
lem, depending on the mathematical sophistication of the stu- by Piaget (1975) to study the thinking processes of the stu-
dents, and eventually lead some students into discovering the dents. The interviews were open-ended with the purpose of
general principle that could be applied to all of them. getting students to verbalize their thought processes while solv-
Data were collected in the second semester of the school ing a given problem. Five rounds of interviews were conducted
year through students’ journal writings, clinical interviews, and with the 9 students over the course of 3 months. Students were
the teachers’ journal writings. The students were assigned the asked questions along the following lines:
five combinatorial problems as journal problems, starting with 1. How did you start the problem?
problem 1. The rationale for providing the three cues was to 2. How long did you spend on this problem?

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 Mathematical Giftedness

3. How does this problem compare to the algebra prob- In order to meet concerns of validity, the researcher had
lems we are doing in class right now? triangulation of data sources, namely data from students’ jour-
4. How can you be sure that your solution is correct? nal writings, interview protocols, and the researcher’s journal
5. How would you explain your solution to a friend? writings. The researcher also used the strategy of intersubjec-
6. Did you use a known procedure to solve the problem? tivity by having a colleague analyze the data from the inter-
7. For the second, third, fourth and fifth problems, the views using the coding technique developed by the researcher.
student was asked if he or she used a refinement/mod- The colleague coded and analyzed 30 random slices of journal
ification of an earlier strategy and whether he or she and interview data and came to the same conclusions as the
could detect any similarities in the problems or their researcher. For a given slice of data coded independently by
solutions. the colleague, there was an agreement of 89% for behaviors
These questions were formulated with the hope of that fell under orientation; 86% for behaviors under organiza-
enabling the students to verbalize their solution strategies. The tion; 93% for execution; 96% for generalization; and 91% for
researcher also wanted the students to justify their solutions reflection. This lends validity to the findings of this research.
and explain their reasoning. Questions 3 and 7 were specifically The researcher met reliability concerns by studying stu-
asked in order to trigger generalization. After each round of dents in the same ninth-grade algebra class. The researcher
interviews, the researcher recorded his impressions of the inter- documented his observations of the students over the school
views in his journal. The interviews were audiotaped, tran- year in his journal. In terms of putting sufficient time in the
scribed verbatim, and rechecked for errors. field, since the researcher was also the teacher of the ninth-
The journal writing and transcribed interview data were grade students, he was fully immersed in the culture of the
coded using techniques from grounded theory (Glaser & classroom. In addition, the personal interviews conducted with
Strauss, 1977). The constant comparative method was used to the students were tape-recorded, transcribed verbatim, and
look for patterns. The making of comparisons is an essential given to the students with a request for clarifications, omis-
feature of grounded theory methodology. Four categories—ori- sions, or additions.
entation, organization, execution, and verification—were oper-
ationalized from Lester’s (1985) problem-solving model. The Limitations
researcher compared action (behavior) to action (behavior) in The reader should be aware that the context of the study
order to classify data according to Lester’s conceptual model. contributed to the nature of the results. Hence, the researcher
Each behavior was then compared to other behaviors at the would like to point out the unique characteristics of this qual-
property level for similarities or differences and then placed itative study so that the reader can “judge” the applicability of
into a category. A category was characterized by properties or the results in other settings.
actions that defined or gave it meaning. When the data were The students in this study were freshman in an acceler-
coded and analyzed, similarities and differences were found in ated (honors) algebra class in a rural high school.
the problem-solving behaviors of the 9 students, as well as in Demographically speaking, they were all White, with middle-
behaviors that characterized the formation of generalizations class socioeconomic backgrounds. Eight out of the 9 students
for the five combinatorial problems. The categories of gener- aspired to finish high school with AP Calculus. The students in
alization and reflection emerged as a result of the study. the study were motivated to succeed in school and were will-
Generalization in this study was characterized as the ing participants in the research effort. Thus, student motiva-
process by which students derived or induced from particular tion and willingness to participate in the study may have
cases. It included identifying commonalties in the structure of influenced their effort levels and the results obtained.
the problems and their solutions, making analogies, and spe- The 9 students who participated in this study had a year of
cializing from a given set of objects to a smaller one. Reflection prealgebra in the eighth grade. In their mathematical back-
was characterized as the process by which the student ground prior to high school, these students had not been
abstracted knowledge from actions performed on the prob- exposed to constructing mathematical proofs, nor had they
lems. In other words, reflection consisted of thinking about been expected to construct general solutions to prealgebraic
similarities in the problems and solutions and abstracting these and algebraic problems. It is conceivable that, if students had
similarities over an extended time period. Finally, affect played been exposed to problems involving mathematical proof, they
an overarching role and influenced the success or failure of the would have been able to distinguish between problems that
students in forming the generality that characterized the class asked for existence solutions (problems 1 and 2) and those
of problems used in this study. The affective dimension include that asked for general solutions (problems 3, 4, and 5).
one’s attitudes, beliefs, feelings, opinions, and convictions The researcher had extremely high expectations for the stu-
(Burton, 1984; Mandler, 1984). dents in the accelerated algebra class. He expected them to

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 Sriraman

invest considerable time on the journal problems. The com- “numerous examples that don’t work.” This general scheme was
plexity of the five problems used in this study indicates that the used over and over again. These students had the attitude of
researcher expected extraordinary leaps of thought from high wanting to get the problem over with and were primarily con-
school freshmen. The researcher was also instrumental in cerned with executing and verifying the scenarios in the prob-
encouraging students to write out their strategies and reflec- lem situations.
tive summaries of journal problems over the course of the first The similarities and differences in the behaviors of the
semester. Thus, students already had this background when the students in the three subsets are presented in Tables 2, 3, and 4.
five combinatorial problems were given in the second semester. This is followed by a descriptive section, which focuses on
Thus, the articulateness found in journal writings is likely some of the solutions of Amy, John, Matt, and Hanna and
attributable to the researcher’s influence. includes interview vignettes that reveal their problem-solving
The students were told to work on the problems indepen- strategies and the mathematical experiences. Table 2 compares
dently without referring to books, friends, or other people in the problem-solving behaviors of the students in the three sub-
the class. The diversity in the journal solutions, the variety of sets in the orientation, organization, execution, and reflection
explanations during the interviews, and the inability of many phases of problem solving. Table 3 compares the generalization
students to find solutions to the last three problems indicate and reflective behaviors of the students in the three subsets.
that the students did not collaborate with one another. Finally, Table 4 compares the affective behaviors of the students
However, there is always the possibility that some students may in the three subsets. The purpose of these tables is to allow the
have talked to each other about the problems. reader to compare the problem-solving behaviors of the gifted
students (Subset A) with those of the nongifted students
(Subsets B and C).
Results

Qualitative analysis of the nine case studies resulted in The Mathematical Experiences
three subsets based on the problem-solving behaviors and gen- of the Gifted Students
eralizations developed by the students. Subset A included Amy,
John, Matt, and Hanna, who were successful in discovering the Problem#1: The Soda Problem
generality that characterized the solutions of the problems, Hanna began the problem by restating it in her own
namely the pigeonhole principle. They were able to isolate the words. She wrote:
similarities in the structure of the problems and their solutions.
They discerned that the solutions entailed two unequal quan- There are 6 listed sodas in Blaise’s Bistro. If one stu-
tities be “matched up” or compared and were then able to point dent placed an order for 1 soda, how many students
out how this played a role in the solution of the problem (i.e., would have to place an order so that 1 of the 6 sodas
any given slot was forced to have “more than one” of a quan- would be ordered by at least 2 students? In other
tity). These students showed great perseverance and curiosity words, how many students would it take to order a
and were motivated to pursue the problems and reflect on soda per student so that one soda was ordered at least
them over an extended period of time. As stated earlier, the twice?
researcher accessed the testing profiles of the 9 students after
data collection and analysis to find that the 4 students in Subset Hanna’s plan to solve the problem was “by making a list
A had been identified as mathematically gifted in their ele- of the 6 different sodas. She would then write “1st student, 2nd
mentary school. student, etc. to symbolize 1 student per order . . . and so on
Subset B included Bart, Jim, and Isabel, whose overall gen- for all 6 sodas.” Finally, Hanna’s solution consisted of a list with
eralization scheme was to employ algebraic operations on the the six sodas. She assigned one student per soda for the six
given numbers in the problems. These students focused on listed sodas, and then assigned the “7th student across from
the superficial similarities in the problems and tried to apply ginger ale to symbolize the 2nd student for one of the 6 sodas.”
procedures from algebra. Their comparisons of the problems From her work, Hanna “gathered that 7 students are required
often showed several inconsistencies. They were unable to pur- to order a soda, 1 soda per student, to ensure that at least 1 of
sue consistent strategy from one interview to the next over the the 6 sodas would get ordered by at least 2 students.” The
course of the five problems. researcher was impressed by Hanna’s journal. She had restated
Finally, Subset C included Jamie and Heidi, whose overall the problem in her own words, made a plan, executed the plan,
generalization scheme was “finding numerous examples that and then stopped once she had verified that her solution had
work,” and in some problems this was modified to include fulfilled all the problem conditions.

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 Mathematical Giftedness

Table 2

Comparison of Student Problem-Solving Behaviors

in the Orientation, Organization, Execution, and Verification Phases

Subset A Gifted Subset B Nongifted Subset C Nongifted

(Amy, John, Matt, Hanna) (Bart, Jim, Isabel) (Jamie, Heidi)

Orientation Consistently comprehending the Miscomprehending the problem Miscomprehending the problem
problem situation. situation. situation.

Assessing the adequacy of informa- Listing the “given” numbers in a

tion given in the problem situation. problem situation.

Identification of (and understand- Making up assumptions for the Poor understanding of assumptions
ing) the assumptions of the prob- given problem situation. for the given problem situation.
lem situation.

Distinguishing between interroga- Unclear distinction between inter- No distinction between interroga-
tive and declarative statements. rogative and declarative statements. tive and declarative statements.

Organization Global planning. Consistently Haphazard/vague global planning. Local planning.

planning to work their “way up” or
“starting out small.”

Controlling the variability of the Not controlling the variability of Not controlling the variability of
problem situation. the problem situation. the problem situation.

Execution Performance of correct local Performance of “unusual” local Performance of local actions.
actions. actions.

Continuously monitoring progress Not carefully monitoring progress Monitoring progress and consis-
and consistency of plans. and consistency of plans. tency of plans.

Verification Checking results of local actions. Inconsistencies in results of local Use of one particular case for veri-
actions. fication.

Verifying consistency of results with Inconsistency of results with imple- Use of examples/non-examples to
implemented plans. mented plans. reach conclusions.

Use of particular cases to better Use of particular cases to verify if a

understand why a phenomenon phenomenon occurred.
occurred

Problem#2: The Aspirin Problem make sure that all of the 45 aspirins were gone in 30 days. In
Matt began the problem by writing, “The problem is ask- order to solve the problem, Matt’s strategy was to make a list
ing to find a sequence of aspirin taken in 30 days and find out of the 30 days and then write the number of aspirin on the
if, in a number of consecutive days, he will take 14 aspirin.” He other side (see Figure 1).
understood this to mean “there must be a way for the person to Matt executed his plan by making a chart with 30 days
take 14 aspirin in any amount of consecutive days” and to on one side and the number of aspirin on the other. “I will

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Table 3

Comparison of Student Behaviors in Generalization and Reflection

Subset A Gifted Subset B Nongifted Subset C Nongifted

(Amy, John, Matt, Hanna) (Bart, Jim, Isabel) (Jamie, Heidi)

Generalization Identifying similarities in the struc- Identifying superficial similarities Identifying superficial similarities
ture of the problems. in the structure of the problems. in the structure of the problems.

Identifying similarities in the solu- Inconsistencies in verbalization of Contriving similarities in the solu-
tions of the problems. similarities in the solutions of the tions of the problems.
problems.
Using analogical reasoning.
Forcing connections with concepts
Refining methods where appropri- in algebra.
ate.

Extending the domain of validity.

Verbalizing common principles. Articulation barriers.

Reflection Conjecturing and examining plau- Conjecturing but not examining Little or no conjecturing.
sible examples and non-examples. the plausibility of a conjecture.

Relating to previous experience. Poor decision making during exe-

cution and verification.

Decision making during and after Putting aside problem after com- Putting aside problem after com-
execution and verification. pletion. pletion.

Thinking about similarities in the

problems and solutions.

Abstracting structural similarities in Abstracting superficial similarities No abstraction.

the problems and solutions over an from the wording of the problems
extended time period. and solutions.

try to write at least 1 pill in 1 day and put 2 pills on some I: Do you think this is the only way to do it?
days to make the 45 pills. I believe this should work.” Matt S: I think so.
assigned 1 pill for each of the 30 days and then assigned an I: So there is no other way?
additional pill for days 15–29. He then wrote, “It is possible, S: You mean, are these like the only kind of numbers you
the least amount of days it takes to get 14 aspirin is 7 days.” can use?
There was no further reflection on Matt’s part and no I: Yeah.
acknowledgement of the fact that other solutions were possi- S: No, ’cause you can use a couple of threes. Or you can put
ble. He also seemed to be convinced that the answer was 7 one day with a whole lot of them, and the rest of the days
days based on his solution. However, the following vignette you can just have ones.
reveals that Matt was aware of other solutions to the problem I: Okay, how long did you spend on the problem?
in addition to being able to identify the structural similarities S: I looked at it for a couple of days before, and I thought
in the first two problems (in all vignettes S= Student; I = about it for a while, and then in my study hall I wrote the
Researcher/Teacher). first two tasks. Then I went home, and the next day I spent

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 Mathematical Giftedness

Table 4

Comparison of Affective Behaviors

Subset A Gifted Subset B Nongifted Subset C Nongifted

(Amy, John, Matt, Hanna) (Bart, Jim, Isabel) (Jamie, Heidi)

Affect Perseverance. Lack of perseverance. Lack of perseverance

Confidence/Lack of confidence. Confidence/Lack of confidence. Lack of confidence.

Curiosity. Low degree of curiosity. Low degree of curiosity.

Excitement.

Frustration. Satisfaction. Satisfaction.

Valuing communication. Lack of eagerness to communicate. No eagerness to communicate.

Mathematics as a “way of thinking.” Mathematics as operations on Preconceived notion of “problem

numbers. solving” from middle school text-
book.

about half an hour trying to figure out how to do it. Days Pills Days Pills Days Pills
I: So, you think you spent more time on this compared to
the first one? 1 1 11 1 21 2
S: Yeah, the first one was a bit too easy. 2 1 12 1 22 2
I: Do you see any similarities in the two problems? 3 1 13 1 23 2
S: Mmm . . . What I noticed was going through and putting 4 1 14 1 24 2
one, then adding one for so many days, and you get the 5 1 15 2 25 2
answer. 6 1 16 2 26 2
I: Okay. 7 1 17 2 27 2
S: That’s what I did for all of them [pointing to the soda 8 1 18 2 28 2
problem], one had to have two, so I put two for one of 9 1 19 2 29 2
them. 10 1 20 2 30 1
I: And with the second problem?
S: I put in one in all of them, and then I changed half the Figure 1. Matt’s representation of the aspirin problem.
ones to twos.

Problem #3: The Number Sum Problem In order to start the problem, Amy’s plan was to make up a
Amy recorded her thoughts on what the problem was ask- set of 10 numbers between 1 and 100. She would then use
ing in her journal by writing, this set to find two ways of getting the same sum. She would
then make up another set and keep repeating this process. She
This problem is asking how something happens. I wrote, “Hopefully I will see a pattern. Then I can prove how
have to figure out how this happens and prove to you this happens.”
and myself that this will always happen. This may Amy’s first set was {3, 12, 23, 29, 53, 61, 70, 79, 81, 94},
seem impossible, but it will always work . . . it’s very and the sum that she very quickly found was 3 + 12 + 79 =
interesting . . . you would think that it wouldn’t work 94, a member in the set (i.e., the second sum is simply 94 =
in some sets of numbers, but it works in each and 94). Interestingly enough, her method for finding this sum was
every set. to first subtract 79 from 94 to get 15 and then to note the fact
that the sum of 3 and 12 was 15. Hence 3 + 12 + 79 = 94.

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 Sriraman

Her second set was {9, 12, 29, 41, 45, 71, 73, 88, 97, 98}, chosen set, such as 5, 17, 45, 50, 9 and 29, and she found sums
and this time her sum was 45 + 12 + 41 = 98. At this point, that equaled these numbers. Her conjecture was that seven num-
she decided to try something different. She would start with a bers were the maximum that one could have in a set without find-
set of 10 numbers, but now revise it somehow (i.e., to change ing a solution, namely two sums that were equal. She came to
some elements) with the hope of getting “different results.” She the conclusion that seven integers were the most one could choose
wrote that, at this stage, she was just experimenting with the without finding solutions, and, since the problem asked her to
hope of discovering something. She began with the set {5, 14, choose 10 integers, there would always be a solution.
16, 29, 44, 46, 53, 61, 80, 89} and looked for different ways She had abstracted the fact that going beyond a maximal
to get a solution, that is, sums that equaled an element in the number resulted in a certain phenomenon that fulfilled the
set or two sums that were equal. The different solutions that conditions of the problem. In the first problem, going beyond
she found were 5 + 14 + 61 = 80 and 46 + 5 + 29 = 80. She six students forced one soda to be ordered twice. In the sec-
then decided to “substitute” some numbers and “find new ond problem, going beyond 30 pills forced the person to take
results.” So, she changed the numbers 5, 14, 29, 46, and 80 to more than 1 pill a day and resulted in a string of consecutive
different numbers. She started with the set {5, 14, 16, 29, 44, days where exactly 14 pills were consumed. Now, in the case
46, 53, 61, 80, 89, which she labeled the “original” set, and of the 10-number set, Amy’s creation of a maximal set with 7
changed it to {6, 7, 16, 21, 44, 49, 53, 61, 82, 89}, which she carefully chosen elements resulted in two sums that were equal
labeled the “revised” set. She substituted 6, 7, 21, 49, and 82 when an eighth element was chosen. It was at this point that
for the numbers 5, 14, 29, 46, and 80 to see if this would result Amy reflected on her solution and wrote, “Since I tried to get
in a set that did not give any solutions. However, she quickly the most variation as possible in the set {1, 2, 4, 8, 16, 32, 64,
found more sums and expressed surprise that it still worked. . . .}, it reminds me of the first problem, where I tried to get the
“7 + 82 = 89; Wow, I found one right away!” She decided to most variety of orders possible, assigning one person to each
find more sums and found a number of sums, such as 7 + 21 soda. So, after all, these two problems are similar! Did you plan
+ 61 = 89, a member of the revised set; 16 + 49 = 21 + 44; 61 this?”
+ 21 = 82; and 7 + 16 + 21 = 44. Having reached another dead It was clear that Amy was beginning to develop an intuitive
end, she decided to try something new. She wrote, “I am going grasp of the hidden generality in the problems, namely the
to put some thought into choosing my 10 numbers.” pigeonhole principle. During the interview, she was able to
She finally created the set {1, 2, 4, 8, 16, 32, 64, . . .} as identify the structural similarities in the three problems and
follows: verbalize the pigeonhole principle.

I started out with 1, then I chose 2. Now, I didn’t want Problem#4: The Acquaintance Problem
to get a solution, so my next number obviously wasn’t John understood this problem to be asking “I could take
going to be 3, so I put 4 instead, because 1 + 2 = 3 20 people and prove whether or not the same person would
and I would have had a solution already. I continued have the same amount of friends as somebody else.” In order to
working this way. The next bigger number would be solve the problem, John wrote that he would label the 20 peo-
7, because 1 + 2 + 4 = 7, so I didn’t choose 7; instead, ple and then use “guess and check.” To solve the problem, John
I chose the next number 8. drew a figure (see Figure 2).
He explained his figure as follows. The top row stands for
She continued this pattern of carefully choosing the numbers, the people in the room (i.e., 1 stands for person 1 and so on).
until she came to 64. She also observed that she was doubling The second row stands for the number of friends. For example,
the previous number to choose the next number. according to the table, Person 1 had one friend, Person 2 had
two friends, and so forth. John wrote, “ I thought for a while
There are still no possible solutions, but if I doubled and decided if 1 knew 1 friend, then 2 knew 2 friends and so
64, I would get 128 . . . and the problem states that on. I tried not to have the same friend amount twice.”
the numbers have to be from 1–100, so I couldn’t use The researcher conjectured that John actually reflected on
128. I still have 3 numbers to go. Now, if I pick any the problem before deciding to proceed along the lines of not
random number, look what happens. assigning the same number of friends to the people in the
room. Proceeding along these lines when he came to the 20th
Amy picked numerous random numbers between 64 and 100, person in the room, he deduced that, since the 20th person
such as 87, 99, 68, 71, 84, and 92 and always found sums that couldn’t possibly know 20 people in the room, he or she would
equaled these numbers. For example: 87 = 64 + 16 + 2 + 1 + 4. have to know at least 1 person, or at most 19 people. Hence,
She then picked numbers between the numbers in her carefully two people would end up with the same number of friends.

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 Mathematical Giftedness

1 2 3 4 5 6 7 8 9 10
1 2 3 4 5 6 7 8 9 10

11 12 13 14 15 16 17 18 19 20
11 12 13 14 15 16 17 18 19 1–19

Figure 2. John’s representation of the acquaintance problem.

The researcher found this argument remarkable because it fourth problems. He verbalized this by saying that all the
applied the pigeonhole principle implicitly in order to conclude problems were either asking for a certain amount or if a cer-
that two people had to end up with the same number of friends. tain amount was possible. He realized that he had made a table
The researcher asked John to explain his solution strategy and for his solutions to 1, 2, and 4 and that there were two quan-
then asked him if there were any similarities in the problems or tities being compared with the characteristic that one quantity
the solutions to the problems, which eventually led to verbaliza- was “shorter” than the other. He also said that this allowed
tion of the pigeonhole principle, the generality that characterized the problems to be solved. Since John had implicitly stated
his solutions to problems 1, 2, and 4. The following vignette what enabled the problems to be solved, namely the pigeon-
illustrates John’s discovery of the pigeonhole principle. hole principle, the researcher made the pedagogical decision of
using John’s numbers to facilitate the verbalization of the
S: I made a table for all of them, and I listed all the people pigeonhole principle. It is important for the reader to note
and brands possible, and figured it out by placing numbers that the researcher made this decision only after the student
to where it works. had implicitly stated the pigeonhole principle in his or her
I: So, how is the solution the same? own words. In John’s case, this was evident after he had iden-
S: A table and list the people and figure out the numbers for tified the two quantities that were being compared, with one
each specific person. being shorter than the other, which enabled the problem to be
I: Is that the only thing in common? solved. This was his way of stating the pigeonhole principle.
S: Among the first, second, and fourth. Yeah. It’s all asking This will become evident to the reader in the following
for a specific amount . . . like for people or brand of soda. vignette.
They ask for a certain amount that can be possible. For this
one [pointing to the aspirin problem] the certain amount I: Since you threw all those numbers at me—6 and 7, 30 and
could be 20. 45, 19, and 20—why don’t I ask you this: What if you
I: You think there is some kind of an equation, a rule that used these numbers for holes and these for pigeons?
you talked about before, that is common to these? S: [Laughing] There will always be one more pigeon?
S: There are 6 soda pops and 7 people, 30 days and 45 I: Where?
aspirin. I am breaking down the numbers. S: There will be two in a hole?
I: So, what is going on? I: So, tell me what happens here?
S: There are more people than soda pops and more aspirin S: Then it is going to be . . . you could put 15 in one and
than days, and there are more friends known than people. put the rest in the others (pointing to the numbers 30 and
I: More friends known than people? 45). Some holes are going to have more than one pigeon?
S: Oh! Yeah . . . one is shorter. I: What about these numbers [pointing to the 19 and 20].
I: What is shorter? S: One hole will have two here.
S: More people than friends. They are never equal. There is I: You still haven’t told me that equation of yours.
always one thing that is more. S: [A long period of silence] Pigeons greater than holes. Let
I: So, how does all that help you solve the problem? x be the holes, then the pigeons will be greater than the
S: The problems say at least one, so I knew there would be number of holes.
more people than sodas. I: How could you say that using your x?
I: And what happened as a result? S: 1 + x.
S: [Silence. Writing down numbers] The problem could then I: So, you wrote down that you have x holes and 1 + x
be solved. pigeons and that would?
S: Then all the holes are filled, and one more than once . . .
At this point in the interview, John had correctly identi- I was surprised how it worked for the last problem (the
fied the similarities in the structure of the first, second, and acquaintance problem) . . . that worked out.

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 Sriraman

Thus, John arrived at the generality that characterized his but tried sets with less than 10 integers. During the execution
solutions of problems 1, 2, and 4. During this journey, the phase of problem solving, these students consistently per-
researcher noted that John had reflected on a possible “equa- formed correct local actions (manipulations) and monitored
tion” for 3 weeks. Unlike Amy, who just stumbled upon the their progress. Once the results of local actions were obtained,
generality during the course of solving the third problem, John they were checked for accuracy and consistency. In the verifi-
consciously looked for something that would capture the prob- cation phase of last three problems, these students consistently
lems in an equation. Note that John was very quick in his ver- made use of particular cases to gain insight into why a phe-
balization of the pigeonhole principle, which is attributable to nomenon occurred. In terms of forming generalizations for this
his implicit use of this principle in his solutions to three of the class of problems, the successful students correctly identified
four problems. similarities in the structure of three or more problems, as well
as similarities in their solutions. They were adept at using
Problem#5: The Band Problem analogies during their explanation of the similarities in the
None of the 9 students were able to solve the band prob- problems. They were also able to communicate effectively and
lem. Amy and Matt constructed plausible explanations about verbalize the common principle they thought characterized
how the shuffles occurred, but were unable to apply the three or more of the problems. In many instances, they spe-
pigeonhole principle to solve it. The solutions to the five prob- cialized from consideration of a given set of objects to a smaller
lems are located in Appendix B. subset contained in the given one.
This concludes the descriptive narratives of the problem- The generalization behaviors exhibited by Amy, John,
solving experiences of the 4 gifted students. Amy stumbled on Matt, and Hanna show several consistencies with the existing
the pigeonhole principle after solving the first three problems research literature. Krutetskii (1976) came to the conclusion
and was able to apply it to solve the fourth problem. John ver- that, in order for students to correctly formulate generaliza-
balized the pigeonhole principle after attempting the first four tions, they had to abstract from the specific content and single
problems by reflecting on and identifying the similarities in out similarities, structures, and relationships. Amy, John, Matt,
problems 1, 2, and 4. Finally, Matt and Hanna discovered the and Hanna were able to do this in varying degrees. Their reflec-
pigeonhole principle after attempting all five problems by iden- tive behaviors consisted of thinking about similarities in the
tifying the similarities in problems 1, 2, and 4. Among the problems and solutions and abstracting these similarities over
gifted subset, Amy was by far the most successful student. Her an extended time period. This finding also verifies the conjec-
creation of a maximal set in trying to solve the third problem tures of Piaget (1971) and Dubinsky (1991), who viewed gen-
was also deemed mathematically noteworthy by several math eralization as a process of “reflective abstraction.” Dubinsky
professors at a nearby university. claimed that generalization involved the combination of
objects and processes, which involved a high degree of aware-
ness of the part of the subject. In this study, the five problems
Conclusions and Implications were the objects, the solutions to the five problems were the
processes, and the generality that characterized the class of
The results indicate that the 4 gifted students, Amy, John, problems and solutions was the pigeonhole principle.
Matt, and Hanna, were successful in forming generalizations. There were other reflective behaviors that were shown by the
Broadly speaking, the major difference between the gifted stu- gifted students that are not explicitly mentioned in the research
dents and the others lay in the orientation, organization, and literature as aiding the process of generalization. In the context of
reflection phases of problem solving. The gifted students this research study, there was a considerable amount of deci-
invested a considerable amount of time in trying to understand sion-making behavior exhibited by the gifted students during
the problem situation, identifying the assumptions clearly, and and after the execution and verification phases of problem solv-
devising a plan that was global in nature. Although these stu- ing. Decision making could be viewed as “quick” reflective
dents never found general solutions to the five problems, they behavior during the problem-solving process that steered the stu-
did consistently work their way up by beginning with simpler dents toward a correct solution. Another reflective behavior was
cases that modeled the given problem situation. In doing so, conjecturing after attempting a given problem. After the students
they were controlling the variability of the problem situation. had attempted a given problem, they often formed a conjecture
In other words, they realized that the quantities in the given and then pursued their conjecture by examining plausible exam-
problems were not invariant (e.g., in problem 3, Amy con- ples. This was a striking feature of Amy’s journal writings and
trolled the variability of the problem by picking the integers was also observed in the others. For instance, John said that he
in their sets more and more carefully). In doing so, these stu- was looking for an equation that would solve the problems, and
dents did not restrict themselves to a set with just 10 integers, he finally came up with it after solving problem 4.

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The researcher mentioned earlier that there were subtle dif- orientation and organization in a problem-solving situation.
ferences in the quality of generalizations formed by the gifted This finding should be of great interest to teachers and coun-
students. Amy was the most successful student in this subset selors. The gifted students displayed conceptual understanding
because she discovered the generality after problem 3 and was because they were able to abstract similarities and form valid
also able to solve problem 4 by using the generality she had conceptual links. Affect played a major role in how they
constructed. She then tried to subsume problem 5 under the approached a problem situation. In particular, their beliefs
general method, but was unable to do so. In a way, she was about what constituted mathematics influenced how they tack-
operating at the level of a mathematician. A mathematician led a given problem. The gifted students may have found the
would view Amy’s generalization scheme as follows: First, a problems captivating enough that it created an affective need
general method was derived from problems 1, 2, and 3. The for them to get a sense of the underlying pattern that charac-
method was then formulated explicitly (as the pigeonhole prin- terized them (Burton, 1984; Mandler, 1984; Schoenfeld,
ciple) and considered as an entity by itself, and its structure was 1985).
analyzed. This structure was then used to include a different So, if teachers want students to become adept at forming
type of problem (problem 4) without making changes to the generalizations, the first and foremost challenge is to find vari-
original method (Skemp, 1986). John and Matt, on the other ous classes of problems with general solutions that are accessible
hand, were able to arrive at the generality after solving problem to the students and capture their interest. The findings here
4 and problem 5 respectively, but were unable to subsume indicate that gifted students are particularly capable of abstract-
problems 3 and 5 under the generality. In Hanna’s case, she ing similarities in the structure of problems and situations in a
arrived at an intuitive understanding of the generality by manner akin to mathematicians, as well as in formulating valid
abstracting similarities in problems 1, 2, and 4 and by con- mathematical generalizations. This makes it crucial for high
sciously excluding problems 3 and 5 from this process. Since school teachers to create learning opportunities that allow math-
abstraction is a premise of generalization (Davis & Hersh, ematically gifted students to develop and apply their talents.
1981; Dayvdov, 1990), the abstracting behaviors of these stu-
dents are similar to those exhibited by mathematicians and
enabled them to be successful in forming valid generalizations. References
The affective behaviors of the gifted students in forming
generalizations are consistent with the literature (Burton, 1984; Birkhoff, G. (1969). Mathematics and psychology. SIAM
Mandler, 1984). According to Burton, cognitive activity is Review, 11, 429–469.
charted by affective responses that can be observed as passing Brownell, W. A. (1942). The place and meaning in the teach-
through three phases: entry, attack, and review. The phase of ing of arithmetic. The Elementary School Journal, 4,
engaging a problem is called entry. Surprise, curiosity, or ten- 256–265.
sion creates an affective need that is resolved by exploration Burton, L. (1984). Mathematical thinking: The struggle for
(attack), which in turn satisfies the cognitive need to get a sense meaning. Journal for Research in Mathematics Education,
of the underlying pattern, which in this study was the pigeon- 15, 35–49.
hole principle. In the gifted students, the researcher noted the Davis, P. J., & Hersh, R. (1981). The mathematical experience.
powerful positive emotions that often went along with the con- New York: Houghton-Mifflin.
struction of new ideas (von Glasersfeld, 1987). This result is Davydov, V. V. (1990). Type of generalization in instruction:
consistent with research literature claims that most affective Soviet studies in mathematics education. Reston, VA:
factors arise out of emotional responses to the interruption of National Council of Teachers of Mathematics.
plans or planned behavior (Burton; Mandler; Schoenfeld, Dienes, Z. P. (1961). On abstraction and generalization.
1985). Besides showing surprise and curiosity, over the course Harvard Educational Review, 31, 281–301.
of the five problems the gifted students also showed remarkable Dreyfus, T. (1991). Advanced mathematical thinking
perseverance and experienced bouts of frustration. They valued processes. In D. Tall (Ed.), Advanced mathematical think-
communication and viewed mathematics as a “way of think- ing (pp. 25–40). The Netherlands: Kluwer Academic
ing.” Publishers.
The 4 gifted students had a natural predisposition Dubinsky, E. (1991). Constructive aspects of reflective abstrac-
(Shapiro, 1965) to engage in the manifested problem-solving tion in advanced mathematics. In L. P. Steffe (Ed.),
and generalization behaviors. Despite not being offered enrich- Epistemological foundations of mathematical experience (pp.
ment and acceleration opportunities during the middle school 160–187). New York: Springer-Verlag.
years, it was interesting to find that the gifted students showed English, L. D. (1992). Problem solving with combinations.
a high level of reflective behavior in addition to concern with Arithmetic Teacher, 40(2), 72–77.

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Ervynck, G. (1991). Mathematical creativity. In D. Tall (Ed.). Sternberg, R. J. (1979). Human intelligence: Perspectives on its
Advanced mathematical thinking (pp. 42–53). The theory and measurement. Norwood, NJ: Ablex.
Netherlands: Kluwer Academic Publishers. von Glasersfeld, E. (1987). Learning as a constructive activity.
Frensch, P., & Sternberg, R. (1992). Complex problem solving: In C. Janvier (Ed.), Problems of representation in the teach-
principles and mechanisms. Hillsdale, NJ: Erlbaum. ing and learning of mathematics (pp. 3–18), Hillsdale, NJ:
Gardner, M. (1997). The last recreations. New York: Springer- Erlbaum.
Verlag.
Glaser, B., & Strauss, A. (1977). The discovery of grounded the-
ory: Strategies for qualitative research. San Francisco: Appendix A :
University of California Press. The Problems
Greenes, C. (1981). Identifying the gifted student in mathe-
matics. Arithmetic Teacher, 28, 14–18. Problem 1
Kanevsky, L. S. (1990). Pursuing qualitative differences in the The soda menu of a bistro has 6 choices for sodas, namely
flexible use of a problem solving strategy by young chil- Cola, Diet Cola, Lemonade, Ginger Ale, Root Beer, and Diet
dren. Journal for the Education of the Gifted, 13, 115–140. Root Beer.
Kilpatrick, J. (1985). A retrospective account of the past How many students would be required to place soda
twenty-five years of research on teaching mathematical orders, 1 soda per student, in order to ensure that at least 1 of
problem solving. In E. A. Silver (Ed.), Teaching and learn- the 6 listed sodas would be ordered by at least 2 students?
ing mathematical problem solving: Multiple research per-
spectives (pp. 1–16). Hillsdale, NJ: Erlbaum. Problem 2
Krutetskii, V. A. (1976). The psychology of mathematical abili- A person takes at least 1 aspirin a day for 30 days. Suppose
ties in school children.(J. Teller, Trans.; J. Kilpatrick & I. he takes 45 aspirin altogether. Is it possible that, in some
Wirszup, Eds.). Chicago: University of Chicago Press. sequence of consecutive days, he takes exactly 14 aspirin?
Lester, F. K. (1985). Methodological considerations in research Justify your solution.
on mathematical problem solving. In E. A. Silver (Ed.),
Teaching and learning mathematical problem solving: Problem 3 (adapted from Gardner, 1997)
Multiple research perspectives (pp. 41–70). Hillsdale, NJ: Choose a set S of 10 positive integers smaller than 100. For
Erlbaum. example I choose the set S = {3, 9, 14, 21, 26, 35, 42, 59, 63,
Mandler, G. (1984). Mind and body: Psychology of emotion and 76}. There are two completely different selections from S that
stress. New York: Norton. have the same sum.
National Council of Teachers of Mathematics. (2000). For example, in my set S, I can first select 14, 63, and then
Principles and standards for school mathematics. Reston, VA: select 35, 42. Notice that they both add up to 77 (14 + 63 =
Author. 77; 35+ 42 = 77). I could also first select 3, 9, and 14 and then
Piaget, J. (1971). Biology and knowledge. Edinburgh University select 26. Notice that they both add up to 26 (3 + 9 + 14 =
Press. 26; and 26 = 26). No matter how you choose a set of 10 posi-
Piaget, J. (1975). The child’s conception of the world. Totowa, tive integers smaller than 100, there will always be two com-
NJ: Littlefield, Adams. pletely different selections that have the same sum. Why does
Polya, G. (1945). How to solve it. NJ: Princeton University this happen? Prove that this will always happen.
Press.
Schoenfeld, A. H. (1985). Mathematical problem solving. New Problem 4
York: Academic Press. There are 20 people in the room. Some of them are
Schoenfeld, A. H. (1992). Learning to think mathematically: acquainted with each other, some not. Prove that there are 2
Problem solving, metacognition, and sense making in persons in the room who have an equal number of acquain-
mathematics. In D. A. Grouws (Ed.), Handbook of research tances.
on mathematics teaching and learning (pp. 334–368). New
York: Simon and Schuster. Problem 5 (adapted from Gardner, 1997)
Shapiro, S. I. (1965). A study of pupil’s individual characteris- A rectangular array consists of rows and columns.
tics in processing mathematical information. Voprosy Consider a marching band whose members are lined up in a
Psikhologii, 2, 1–113. rectangular array of m rows and n columns (m and n can be any
Skemp, R. (1986). The psychology of learning mathematics. natural numbers). Viewing the band from the left side, the
Penguin Books. bandmaster notices that some of the shorter members are hid-

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 Mathematical Giftedness

den in the array. He rectifies this aesthetic flaw by arranging the Problem 3
musicians in each row in increasing order of height from left The Number Sum problem can be solved as follows: There
to right so that each one is of a height greater than or equal to are 210 = 1,024 subsets of the 10 integers, but there can be only
that of the person to his left (from the viewpoint of the band- 901 possible sums, the number of integers between the mini-
master). When the bandmaster goes around to the front, how- mum and maximum sums. With more subsets than possible
ever, he finds that once again some of the shorter members are sums, there must exist at least one sum that corresponds to at
concealed. He proceeds to shuffle the musicians within their least two subsets. Hence, there are always two completely dif-
columns so that they are arranged in increasing order of height ferent selections that yield the same sum.
from front to back.
At this point, he hesitates to go back to the left side to Problem 4
see what his adjustment has done to his carefully arranged The Acquaintance problem can be resolved as follows: If
rows. When he does go, however, he is pleasantly surprised there is a person in the room who has no acquaintances at all
to find that the rows are still arranged in increasing order of then each of the other persons in the room may have either 1, or
height from left to right! Shuffling an array within its 2, or 3, . . . or 18 acquaintances, or do not have acquaintances
columns in this manner does not undo the increasing order at all. Therefore, we have 19 “holes” numbered 0, 1, 2, 3, . . .
in its rows. Why does this happen? Prove that this will always 19, and have to distribute among them 20 people. Next, assume
be the case. that every person in the room has an acquaintance. Again, we
have 19 holes (1, 2, 3, . . . , 19) and 20 people. Thus, two peo-
ple will be forced to have the same number of acquaintances.
Appendix B:
The Solutions Problem 5
The Band problem can be proved via reductio ad absurdum
Problem 1 (proof by contradiction). Assume that all the columns have
The Soda problem has the obvious solution that 7 students been arranged, but there is a row in which a taller musician A
would be required to place soda orders, since the worst case sce- (column I) has been placed in front (or to the left) of a shorter
nario is each of the first 6 students orders a different drink, thus musician B (column J). Since the columns have been arranged,
forcing the seventh student to order a drink that has been pre- every musician in segment X from A back in column I is at
viously ordered. least as tall as A, and every musician in segment Y from B for-
ward in column J is no taller than B. Since A is taller than B,
Problem 2 this implies that members in segment X are taller than mem-
The Aspirin problem is commonly resolved by assuming bers in segment Y. Now consider the halfway point at which
that the person takes at least 1 aspirin pill a day. Therefore, in the rows have been arranged, but not the columns. To get to
30 days, the person would have consumed exactly 30 aspirin this point, one must move the musicians from segment X to
pills, thus leaving a surplus of 15 pills, which the person can their former positions throughout column I and return to seg-
randomly take in the 30-day cycle. Let ai be the total number ment Y to their positions through column J. The members of
of aspirin consumed up to and including the i-th day, for i = X and Y have to be distributed over the rows 1, 2, . . . m, as if
1, . . .,30. Combine these with the numbers a1 + 14, . . ., the m rows were the holes. The segments X and Y have total
a30 + 14, providing 60 numbers, all positive and less or equal length of m + 1. By the pigeonhole principle, two musicians
to 45 + 14 = 59. Hence, 2 of these 60 numbers are identical. must end up in the same row. They could not come from the
Since all ai’s and, hence, (ai + 14)’s are distinct (at least 1 same segment, so in some row there must be a member C from
aspirin a day consumed), then aj = ai + 14, for some i<j. X, ahead of a member D from Y. Since C is taller than D, this
Thus, on days i + 1 to j, the person consumes exactly 14 arrangement violates the established increasing order of rows.
aspirin. So, the conclusion follows by reductio ad absurdum

Spring 2003, Volume XIV, Number 3 165

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