A Model of High-Performing Students’ Mathematical Problem Solving

Carl Joshua T. Quines

11-C

Submitted to:

Mrs. Mary Ann Dayrit

2

Acknowledgements

Although this research is an individual undertaking, and the largest project I have

attempted to do myself, I have never felt individual throughout its conduct. There were

people who were utmost in their support of my efforts.

In particular, I would like to thank the finalists of the 19th Philippine Mathematical

Olympiad for being good company throughout the competition, for agreeing to be the

participants in the study, and for making multiple suggestions for writing the paper.

I would like to thank my colleagues in section C for their support for my individual

undertaking. Despite being the only individual project in the whole class, they were in good

humor with “my groupmates” not submitting their work on time, or “my groupmates” not

helping out in the project. They were also very helpful in reminding me when the deadline

was, and for sympathizing with my troubles in writing the paper.

Finally, I would like to thank my research adviser Mrs. Dayrit for providing her

assistance throughout the study and for providing corrections in the paper. This project

made me realize that qualitative research is not secondary to quantitative research, but

equally important to when it comes to the framework of scientific knowledge.

3

Abstract

Multiple models, which differentiate problem solving in terms of cognition, have

been proposed in various studies. This study differentiates problem solving in terms of

ability level, and creates a model by interviewing the participants of the 19th Philippine

Mathematical Olympiad. Sixteen participants were interviewed twenty-nine times,

narrating their solutions to the four problems in order to produce a step-by-step process of

how high-performing students solve mathematical problems. It was found that in general,

high-performing students mostly follow the same process in solving non-standard

problems, to a highly similar extent. Students performed exploration, followed by a

motivated trial-and-error cycle, followed by stopping, in the process of solving a problem.

The components of the model and two examples of its application are discussed, as well as

comparison to existing models. Further research is recommended to how students of

different skill levels solve problems, and to the extent of similarity.

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Table of Contents

Acknowledgements 2

Abstract 3

Table of Contents 4

List of Figures 5

Chapter 1: The Problem and Its Background 6

Chapter 2: Review of Related Literature 11

Chapter 3: Methodology 18

Chapter 4: Results and Discussion 22

Chapter 5: Summary, Conclusions and Recommendations 46

Bibliography 48

Appendix A: Letter of Consent 51

5

List of Figures

Figure 1. Barbacena and Sy’s metacognitive model 16

Figure 2. Two timelines of problem solving from Schoenfeld 17

Figure 3. The simplified version of the model. 31

Figure 4. The final problem solving model 32

Figure 5. Where verification proceeds to, in general. 39

Figure 6. The model applied to problem four 43

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Chapter 1

THE PROBLEM AND ITS BACKGROUND

Introduction

Many students find problems as a challenging part of the curriculum (Lugo, 2011).

But problems are a major part of mathematics, and indeed, thinking mathematically means

being able to think in terms of problems – how to solve them and what they mean. Problems

are part of the mathematical development of any student, and solving them is a major part

of what doing mathematics really is.

In the field of mathematics education research, much work has been done regarding

problem solving. Indeed, Romberg (1969), in his seminal review of the literature, has

classified studies of problem solving and creativity as one of the eight fields of mathematics

education research. The literature does not lack in modelling how students solve problems,

and various models have been proposed, as in the studies of Barbacena and Sy (2015), Lesh

and Doerr (2003), Shinn (2002) and Sweller (1988). All of these models deal with problem

solving in general aspects: a metacognitive model, problem solving in general, problem

solving for curriculum math, and a cognitive model, respectively.

These models differentiate problem solving in terms of cognition, the methods in

which students think. However, only initial work has been conducted in differentiating

problem solving in terms of the students’ relative skill. Such work has been conducted by

Schoenfeld (1984), who in his work compared two groups of students working on a non-

standard problem. The first group which stuck with the same approach did not solve it,

while the second group that tried different approaches managed to. This work is

particularly of interest as it shows that not all problem solving approaches are the same:

7

some approaches are different from others, and changing the approach can help in problem

solving, particularly in non-standard problems.

This idea is further extended by examining the problem solving of high-performing

students. This topic is of interest to the researcher by virtue of being in the same category.

By comparing how high-performing students solve non-standard problems, a general

model can be created to show how these students work on such problems. This model can

be helpful in the sense that it can provide deeper insight to how such students solve various

non-standard problems.

Such a model will be distinct from previous models in the sense that it examines

only students who performed well. The models existing in literature examine problem

solving in general, but Schoenfeld (1984) out that some problem solving models are

ineffective. By examining high-performing students we can produce a problem solving

model that is effective in the sense that high-performing students follow it to solve non-

standard problems with a degree of success.

In this paper the researcher compiles a model of mathematical problem solving for

high-performing students. Such a model would be helpful for students who regularly solve

non-standard problems in quizzes or exams to learn how to better do so, for students who

engage in mathematical competitions that involve such non-standard problems to improve

how their performance, for mathematics teachers to help guide their students in how to

better deal with non-standard problems, and for coaches in mathematical competitions to

train their students to adapt a better problem solving methodology.

Statement of the Problem

8

The main objective of this study is to develop a model that explains the specific

steps in the process that high-performing students take to solve mathematics problems.

Specifically, this study aims to answer the following questions:

1. How much do high-performing students follow the same process to solve non-

standard problems?

2. To what extent do high-performing students follow a similar pattern, and are there

multiple kinds of patterns that need to be modelled?

3. What are the general steps high-performing students take to solve non-standard

mathematical problems?

4. What process do students take to perform these general steps? In particular,

a. What order do these students take these steps?

b. Are these steps repeated, and when?

c. In what circumstances do students perform certain steps?

Scope and Delimitation

Of the population of high-performing students, the students to be examined are

those who have competed in the national stage of the 19th Philippine Mathematical

Olympiad. The problems of this competition are of the non-standard type and are excellent

for modelling non-standard problem solving; the rationale for this decision is further

expanded in the literature review. Of these 22 students, one student, the researcher, is

excluded.

This study is limited to creating a general model of how high-performing students

solve mathematical problems. In this respect, it is not concerned with the individual

differences that students have from one another, but rather the similarities. Furthermore,

9

this research is limited in the sense that it examines only how students solve a specific set

of problems. A final delimitation is that this study is conducted over a very short time scale,

which may unnecessarily bias results by providing an incomplete picture. Despite these

restrictions, it is anticipated that the results of this research can be generalized to non-

standard problem solving in general.

Significance of the Study

The main beneficiaries of the results of this research are students who engage in

mathematical competitions that involve such non-standard problem solving. The results are

of interest to these types of students as they can reveal insight to how high-performing

students, who often perform well in these types of competitions, solve non-standard

problems. The students can then adapt this method of problem solving as their own, and

consequently aid their problem solving ability. In particular, the researcher and his

colleagues are a part of this group and can potentially benefit from the results.

However, this model can also benefit students who have a difficult time solving

non-standard problems. Problems of this sort often appear in quizzes or exams, particularly

for harder courses or in schools with a higher standard of learning. In this sense, the model

can help students adapt a different perspective of problem solving, which has been shown

before to help students solve problems they could not have before. (Schoenfeld, 1984)

By extension, such a model will benefit teachers, coaches and tutors involved with

these kinds of students. Coaches who train students for problem solving competitions can

adapt their training to the results of this study. Teachers and tutors who attempt to teach

problem solving can encourage students to follow this model of problem solving, which

can help the students comprehend the material in a deeper level.

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It is also hoped that such a model will help future researchers in the field of

problem solving, for providing a framework for analyzing narratives of how a student

analyzes problems, and be a basis for future research in the field.

Definition of Terms

High-performing students are defined to be one that performs well in solving non-

standard problems.

Idiosyncrasies are differences between the method an individual uses to solve problems

in comparison to the model presented for problem solving.

Models are taken to be any description of a series of steps that a student uses to solve a

problem. In particular, a model consists of processes and an order of these processes.

Non-standard problems are taken to be problems that require deep conceptual

understanding and intuitive thought.

Problem solving is the act of solving non-standard problems, which requires creative

thought, critical thinking, or strong analytical reasoning skills.

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Chapter 2

REVIEW OF RELATED LITERATURE

Much research has been conducted regarding mathematical problem solving. This

field is commonly considered to be a part of mathematics education and education in

general, and it is of interest in particular to mathematics teachers and instructors who wish

to teach their students problem solving skills. In this review, the field of mathematics

education, studies on problem solving (in particular, mathematical problem solving), and

the currently available models on the subject, are discussed and compared. The review

concludes with how this study fits in the available literature.

Mathematics Education

Research on mathematics education is a well-established field. In his review of

current mathematics research, Romberg (1969) has characterized the field as “large in

quantity, …, improving in quality, and diverse.” Nearly five decades later this fact still

holds water: several journals in mathematics education are in circulation such as the Journal

for Research in Mathematics Education, the Mathematics Education Research Journal, the

IEJME–Mathematics Education, the Journal of Mathematics Teacher Education, among

others.

In the Philippines, the Mathematics Teachers Educators publishes Intersection, a

refereed journal that publishes research about mathematics education with a local setting

and context. The most recent issue, released in May 2015, includes articles such as “The

effectiveness of short-term in-service training in the context of teaching mathematics for

understanding” (Verzosa, 2015) and “Use of manipulatives to develop selected second-

12

year high school students’ understanding of equality and linear equations” (Rivera,

Sinfuego, Tudayan and Limjap, 2015).

Romberg (1969), in the same work, has categorized mathematical education

research into eight rough areas: (1) mathematical learning from an association learning

framework; (2) mathematical learning from an activity learning framework; (3)

mathematical problem solving and creative behavior; (4) mathematics teaching; (5) the

effectiveness of instructional programs; (6) the association of learner characteristics with

mathematical achievement; (7) attitudes toward mathematics; and (8) the evaluation and

measurement of mathematics achievement.

It is to be noted that the field has grown since then: one would add the recent studies

involving the use of technology in education. To this effect, the study of mathematical

problem solving forms a large component of mathematical education research.

Problem solving

The phenomenon of problem solving in general is well-understood. Research dates

back as early as Bloom, Engelhart, Furst, Hill and Krathwohl (1956), who categorizes his

taxonomy of learning objectives as knowledge, comprehension, application, analysis,

synthesis and evaluation. Problem solving is used to refer to the latter three, as a catch-all

term for analysis, synthesis and evaluation. This taxonomy was later revised in the seminal

book by Anderson, Krathwohl and Hill (2001), where the objectives were change to

remember, understand, apply, analyze, evaluate and create.

The term “problem solving” took on multiple meanings over time. Schoenfeld

(1983) gave five definitions of problem solving, including (1) “to train students to ‘think

creatively’ and/or ‘develop their problem solving ability’”, (2) “to prepare students for

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problem competitions”, (3) “to provide teachers with instruction in … heuristic strategies”,

(4) “to learn standard techniques in particular domains” and (5) “to provide a new approach

to mathematics or to try to induce ‘critical thinking’ or ‘analytical reasoning’ skills.”

Schoenfeld goes on to show how these five definitions are not at all distinct from

each other, and in fact related. In this case, distinguishing between these definitions does

not make a major difference in interpreting the results of this study. The primary definition

of “problem solving” used in this research is modeled after Schoenfeld’s first and fifth

definitions.

Non-Standard Problems

To fully understand the phenomenon of problem solving, a definition of problem

needs to be developed. In a different work on mathematics education, Schoenfeld (1992),

examines two kinds of problems: problems that are routine exercises, or problems that are

challenging and require a new method or approach.

The first kind of problems, routine exercises, are in this paper considered “standard

problems.” Tao (2006) states that these problems require students to “remember and carry

out methods that others have previously discovered.” These are the types of problems that

are considered routine exercises, or are examined in curricular mathematics.

It is the second kind of problems that are the concern of this research, which will

be considered “non-standard problems.” Tao (2006), in the same reference, said that these

problems require deep conceptual understanding and intuitive thought. A student must be

have skills in mathematical writing, argumentation and reasoning in order to solve these

kinds of problems, which are different from those previously encountered.

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These kinds of skills, in particular conceptual understanding and mathematical

reasoning, are analogous with Schoenfeld’s first and fifth definitions of problem solving

as “thinking creatively”, “critical thinking” and “analytical reasoning”. It is for this reason

that non-standard problems are the focus of this paper, as these measure problem solving

skills better than standard problems, with the chosen definition of problem solving.

Schoenfeld (1992), in the mentioned work, gives examples of non-standard

problems, all of them coming from mathematics olympiads. These olympiads are typically

composed of three to five problems to be answered in three to five hours, and are non-

standard in the sense that all of them require deep creative thinking. Tao (2006) also uses

olympiad problems as an example of non-standard problems. It is for this reason that the

sample of this study is taken from the national finalists of the Philippine Mathematical

Olympiad, as this contains non-standard problems, and thus requires strong problem

solving skill.

Aspects of Problem solving

Lester, Garofalo and Knoll (1989) list five aspects of mathematical problem

solving: (1) the knowledge base, (2) problem solving strategies, (3) monitoring and control,

(4) beliefs and affects, (5) practices. These five categories provide a framework for

evaluating mathematical problem solving.

The first of these is the knowledge base, which is defined as the “inventory of an

individual’s degree of knowledge of facts and procedures”. The second is problem solving

strategies, or heuristics, which Pólya (1945) describes as strategies like “analogy, auxiliary

elements, decomposing and recombining, induction, specialization, variation, and working

backwards”.

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The third, self-regulation, is described by Brown (1987) as one of three broad areas

under metacognition, listing self-regulation as the ability to control the amount of time

spent in each of the six activities: reading, analyzing, exploring, planning, implementing,

verifying.

Beliefs and affects refers to what Bloom (1956) described as “affective domains”.

These include student beliefs such as “mathematics problems have only one correct

answer”, or “there is only one way to solve the problem”; as well as teacher beliefs like

“mathematics is a subject of ideas than a subject of facts” and “mathematics can be best

understood by rediscovering ideas”.

The last is practices, which Resnick (1989) discusses as “the matter of acquiring

the habits and dispositions of interpretation and sense-making”. His work emphasizes the

importance of practice in mathematics, and gave evidence that taking time to implement

the four aspects outlined above helped in problem solving.

Problem solving Models

In the local field, Barbacena and Sy (2015) published a paper exhibiting a

metacognitive model in mathematical problem solving, where eighteen junior college

mathematics students from Bicol University were asked to solve a set of problems. The

students then listed what went in their mind while they solved the questions. With the

results, they were able to create the metacognitive model outlined in Figure 1.

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Figure 1. TS stands for thinking skills, MA stands for metacognitive

awareness, ME stands for metacognitive evaluation, MR stands for

metacognitive regulation. (Barbacena and Sy, 2015)

Other mathematical problem solving models that have been presented were of

Sweller (1988), Lesh and Doerr (2003) and Shinn (2002), all of which have focused on

presenting a general problem solving model, with similar results. These models are all

similar to Barbacena and Sy’s model in the sense that they centered on metacognition,

rather than the actual process taken by students in problem solving.

Schoenfeld (1989), in a different work, compares two groups of partnered students

solving a mathematics problem. The first group did not solve the problem; the second group

did. The difference can be clearly seen in both cases: the first group only explored the

problem, while the second group explored, planned, and implemented different plans,

leading to a solution which they verified.

This exhibits that there is a difference between typical students and high-

performing students with respect to the approaches they use to solve problems. However,

17

Figure 2. Left: time-line graph of two typical students attempting

to solve a non-standard problem. Right: time-line graph of two

different students solving a non-standard problem. (Schoenfeld, 1989)

Schoenfeld did not attempt to create a model to explain the step-by-step process involved,

the purpose of his study being only to describe the differences between the two groups of

students. The time-line graph of the two groups of students can be seen in Figure 2.

This study, instead, focuses on providing a detailed model that describes the

mathematical problem solving of high-performing students. It differs from previous studies

by tackling the problem solving methods of students who have performed well in problem

solving in the past, as evidence by their performance in a mathematics olympiad, and by

attempting to create a specified step-by-step process rather than merely describing or

comparing. The results are anticipated to contribute to the field by highlighting the problem

solving process of high-performing students, which has potential pedagogical applications.

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Chapter 3

METHODOLOGY

Research Design

This study is a combination of phenomenological design and grounded theory. As

the purpose of this study is to describe a step-by-step procedure of an individual’s thinking,

the phenomenological design is appropriate as it stresses the individual’s subjective

experience (Tesch, 1990). According to Mertens (1998), phenomenological design seeks

the individual perception and meaning of a phenomenon or experience. The descriptive

approach to phenomenological design is followed.

The study uses the grounded theory method to build a model of mathematical

problem solving, which is appropriate as involves the construction of a new theory, as there

is no existing theoretical framework. The data are tagged with codes, which have been

extracted from the data. Finally, the codes have been grouped into concepts, and then to

categories, becoming the basis of the new theory (Strauss & Corbin, 1998).

The study takes the constructivist approach to grounded theory, assuming that the

researcher constructs the data and theories as a result of interaction with the field and its

participants. For this reason, the suggestions of Charmaz (2006) in conducting interviews

are followed, letting the data emerge from the collection naturally, without following any

predetermined structure.

To follow the constructivist approach, several procedures were used to gather rich

data. Multiple interviews were conducted over a period of several days to fully document

the process of problem solving. Observations were coded as soon as they were made, to

identify points areas to explore in further follow-up interviews.

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Researcher’s Role

In this study, the researcher takes the role of gathering and analyzing the data. As

the researcher himself is a competitor in the setting of the research, he excludes himself

from the study. The researcher, being a student who often participates in competitions

which tests the non-standard problem solving skill that is the focus of this study,

acknowledges that the analysis of data may be biased toward his process of problem

solving. This is inherent in the grounded theory design, and the constructivist model

acknowledges that data is colored by the researcher’s perspectives (Charmaz, 2006).

Ethical considerations for the research included the concerns of informed consent,

anonymity and confidentiality. Participants for the study were fully informed of their role

and signed forms of consent. Respondents were fully informed that they will be recorded

and that the recordings will only be used for the purposes of transcription, and will not be

released publicly.

Data Collection

The respondents in this study were the national finalists of the 2017 Philippine

Mathematical Olympiad who consented to the study, with the researcher being excluded.

To reiterate the rationale for this sample, Schoenfeld (1992) and Tao (2006) use olympiad

problems as standard examples for non-standard problems, which are the main focus of

this study. The competitors of the mentioned competition have demonstrated such skill by

solving such non-standard problems by qualifying to the final round. Finally, due to the

proximity of the sample to the researcher, and the ideal problem solving environment of

the competition, it is expedient to choose these contestants as the respondents.

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The competition was held on a single day. Of the twenty-two national finalists,

sixteen consented to take part in the research. Four finalists declined to participate in the

study, one finalist was unable to attend the competition, and the remaining finalist is the

researcher. The written competition included four problems to be solved for four and a half

hours. Exit interviews were conducted after the competition.

Following Strauss and Corbin’s (1998) approach, data was collected until new data

does not change the emerging theory. To do this, follow-up interviews were conducted

with some of the participants for the week following the competition to elicit more

information about problem solving until the data did not change the emerging theory.

Data Recording

Data was collected through several means. A semi-structured face-to-face interview

was conducted on the day of the national finals of the competition. After the written round

of the contest, each contestant who consented to the study was interviewed about how they

approached each problem. Each interview lasted several minutes, with questions aimed at

understanding the process the participant followed. As the constructivist approach for

grounded theory is followed, there was only one fixed question for the interview, letting

the follow-up questions emerge from the response of the participant (Charmaz, 2006).

Multiple follow-up interviews were conducted with participants who consented to

do so. These interviews were done online, through a social networking site. Over several

days after the competition, more data was gathered in how the participants solved problems,

refining the questions asked based on the codes gathered in the initial interviews. Since the

grounded theory approach requires simultaneous data collection, recording and analysis,

this was repeated multiple times.

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Data Analysis

Strauss and Corbin (1998) state that data analysis should be done simultaneously

with data collection and recording. Data analysis was done in multiple stages, but the

general process was the same, similar to the method recommended by Charmaz (2006).

Any recordings or records were transcribed, identifying details removed, and a set

of observations was made for each part of the transcription, known as open coding. These

were then organized into codes and organized into concepts, a process known as axial

coding. These concepts were used as the basis of the data collection for the next stage.

After repeating the process several times, these codes and concepts were repeatedly

modified and synthesized. Finally, the data were selectively coded by organizing them into

categories, which were finally sorted to create a step-by-step model.

Data Validity and Reliability

The method of triangulation presented by Miles and Huberman (1994) was used.

This involves supporting a finding by showing independent measures agree with it, or do

not contradict it. This was done by comparing the results of the model with predictions

offered by other models, such as those by Barbacena and Sy (2015), Schoenfeld (1989)

and Pólya (1945), and verifying that no contradiction occurs.

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Chapter 4

RESULTS AND DISCUSSION

A total of twenty-nine interviews were conducted over the span of two weeks after

the competition proper. Eight of these interviews were conducted face-to-face, while

twenty-one were conducted online. Each of the sixteen participants was interviewed at least

once. The analysis of the resulting transcripts has led to the synthesis of a problem solving

model generalizing their descriptions of problem solving, which is presented in this chapter.

The competition proper consisted of four questions to be answered in four-and-a-

half hours. The first question is a number theory question, involving an inequality with the

number theoretic functions phi and sigma. The second question is an algebra question, an

equation involving the minimum of three terms. The third question is a combinatorics

problem related to coloring the integers. The final question is a geometry problem

involving the locus of a certain point.

All of the sixteen participants have remarked to the effect that the competition was

difficult; in particular, those who joined last year’s competition said that it was more

difficult this year. Some participants were at first reluctant to share how they solved a

certain problem, with the most common reason being that they were unable to solve it, and

thus had the belief that it was not relevant for the study.

The following discussion will proceed first by discussing the particulars of each

problem, presenting the most common solution paths and attempts, followed by discussing

the constructed model synthesized from the collected responses and the various

components of the model.

Problem One

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1. Given 𝑛 ∈ ℕ, let 𝜎(𝑛) denote the sum of the positive divisors of 𝑛 and

𝜑(𝑛) denote the number of positive integers 𝑚 ≤ 𝑛 for which gcd(𝑚, 𝑛) =

1. Show that, for all 𝑛 ∈ ℕ,

1

𝜑(𝑛)+

1

𝜎(𝑛)≥

2

𝑛

and determine when equality holds.

The first problem in the examination is an inequality involving the number theoretic

functions phi and sigma. A majority of contestants claimed to solve this problem.

The most common solution path, after trying small or particular cases, was to

consider the prime factorization of n, using the multiplicativity of both functions, and

applying the AM-GM inequality to prove the needed inequality. Alternative solutions were

to proceed with casework on n based on parity or prime factors.

The motivation for considering the prime factorization of n was often the fact that

both functions involved the prime factors of n. Such an observation either came from prior

knowledge or from observing small cases. For example, one participant said that “I tried

bashing with small numbers first. I tried analyzing patterns for adding more factors”. Upon

further questioning, it is revealed that some participants were trying to restate the functions

phi and sigma in a different manner. As one participant said, “it was to convert the problem

into something more familiar to me, which in this case is an inequality with multiple

variables instead of having two functions in n”.

Some participants directly substituted the formulas for phi and sigma in terms of

the prime factors directly, without making an initial observation from small cases. As one

participant shares, “it was number theory functions, I mean, naturally you would try to do

that,” and “there are very very few (if not just one) ways of expressing phi and sigma, and

24

the most obvious way is through their prime factors”, indicating that such a technique is

common for number theory problems of a certain kind.

Factoring n subsequently motivated considering the multiplicativity of both

functions, though few of the respondents explicitly stated so. Most of the participants only

involved a specific case of the multiplicativity of both functions, often involving the

products of primes; some participants used the general case of multiplicativity, reducing

the problem to prime powers; while others only implied multiplicativity through the

formulas for phi and sigma, which are well-known.

Participants who did not immediately factorize n attempted casework on the parity

of n. The motivation for this comes from the fact that phi for even n was at least half of n,

owing to the fact that half the integers less than n were odd. This observation was often

made from considering particular cases: “it clearly holds, because half of the equation is

already satisfied”. After such an observation, the only remaining cases were the odd ones.

One participant managed to proceed from this by considering the factorization of n,

returning to the first solution path.

After this, the problem has been restated in terms of the prime factorization of n.

The other step in the problem would be the application of the AM-GM inequality. There

were multiple motivations for this, the most common being cancelling terms: “I thought

that I needed to use an inequality to prove the problem, then I noticed the common terms

in numerator and denominator so AM-GM” as one participant describes. Other reasons for

doing so included “the 2ab sort of looked like the RHS of a well-known AM-GM result…

the a² reminded me of AM-GM as well”, or “naisip ko, if only this addition sign were a

multiplication sign”, or “as n² was computed, it was instinct to take the square root to reach

the goal of 2/n”.

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It is interesting to note that some participants attempted to use the AM-GM

inequality before observing or making use of multiplicativity, showing that these two main

insights were thus independent of each other.

Problem Two

2. Find all positive real numbers 𝑎, 𝑏, 𝑐 ≤ 1 such that

min {√𝑎𝑏 + 1

𝑎𝑏𝑐, √

𝑏𝑐 + 1

𝑎𝑏𝑐, √

𝑎𝑐 + 1

𝑎𝑏𝑐} = √

1 − 𝑎

𝑎+ √

1 − 𝑏

𝑏+ √

1 − 𝑐

𝑐.

Only three contestants interviewed claimed to fully solve this problem. Several

other participants tried the problem. Of these solutions, several steps were made in a non-

definite order: observing that an inequality needed to be used, attempting definite cases,

substituting the variables, ordering the variables, applying Cauchy-Schwarz, using calculus,

expanding and attempting to prove no solutions existed.

Although the second problem in the competition is an equality involving the

minimum of three terms, the question involved is actually an inequality. As one participant

describes, “when I see something like that (one equation, multiple variables), one of my

first tries it to simplify it to the equality case of an inequality… generally you need at least

n equations to solve for n variables”. Another participant shares that the motivation behind

this is the minimum on the left-hand side, which inspired him to turn it to an inequality.

Nearly all the contestants who tried this problem observed that it had to be converted to an

inequality in some manner or another.

Three participants described how they attempted to make everything equal, then

making two variables equal: “most cyclic inequalities have an equality case when all

26

variables are equal.” This is an example of trying definite cases. Several other contestants

also attempted to find solutions by substituting definite values and solving for the

remaining variable: “Halimbawa yung a, ginawa kong zero. ’Tas naghanap ako ng values

para sa b and c. Kaso lang mahirap, mahirap talaga siya.” Such a substitution also helped

in recognizing which side was larger: “I plugged in some values and then deduced that the

LHS > RHS.”

Most participants who attempted the problem said without loss of generality that

the variables are in a certain order. This was to “easily determine which one of the three is

the minimum”. One participant shares this is standard to do in non-typical symmetric

inequalities, and that the minimum in particular motivated him to do this.

All but two of the participants who attempted this problem performed the

substitution 𝑥 =1

𝑎, 𝑦 =

1

𝑏, 𝑧 =

1

𝑐. After making the substitution, the problem becomes

equivalent to a problem in literature, the fourth problem of the 2013 United States of

America Mathematical Olympiad. One participant recognized this during the competition,

but was unable to recall its solution, and proceeded with a different approach. The

inspiration for the substitution comes from noticing how the reciprocals of the variables

appear multiple times throughout the expression, and motivation included making the

problem simpler.

One participant managed to solve the problem by making a series of substitutions.

“The RHS makes it tempting to use the 𝑥 = 𝑝2 + 1, 𝑦 = 𝑞2 + 1, 𝑧 = 𝑟2 + 1 substitution,

because everything was square roots.” Indeed, after making the substitution, the right-hand

side of the inequality simplifies to the sum of the three variables.

27

Another participant solved the problem by using calculus. He claims that “it was

not a problem that needs powerful and common olympiad techniques. Therefore, I tried to

go back to classroom math.” Upon further prompting for the motivation, he replied that he

“felt I had to go back to the basics. Because I was running out of approaches and the partial

derivative looked nice.” This approach worked, and he managed to solve the problem

shortly.

Dead ends in particular included expanding without performing a substitution

beforehand, and proving there were no solutions. One participant shares that expanding the

expression but “it became dirty at some point.” Several participants tried to show there

were no solutions in some point during solving the problem, before finding a solution by

trying definite cases.

Problem Three

3. Each of the numbers in the set 𝐴 = {1, 2, … , 2017} is colored either red

or white. Prove that for each n ≥ 18, there exists a corresponding coloring

of the numbers in 𝐴 such that any of its arithmetic sequences having 𝑛 terms

contains numbers of both colors.

Of the participants interviewed, only one claimed to solve this problem fully. All

the participants who attempted the problem tried to construct a definite example of such a

coloring, though the ways in which they did so varied. The other approach tried in the

problem was proving the existence of such a coloring indirectly, without constructing one.

The motivation behind constructing such a coloring was that such a construction

would provide an affirmative answer for the problem. As one participant says, “prove there

exists” can be showing a construction, or showing that such a construction exists but

28

without knowing what the construction is. The first approach is more natural, and was the

first approach of most who attempted the problem.

A common approach was to consider a smaller case. “It was too hard to handle for

2017. So, I think I just tried to see why n less than eighteen couldn’t work.” As another

participant shares, “I started with two and three to see what kind of coloring worked. Two

was trivial, but four was too long to bash completely.” After attempting such a smaller case,

often through brute force, contestants attempted to generalize their solutions to the case of

eighteen through induction, but were unable to.

Another common approach was to use the number eighteen in the coloring. One

participant shared that he tried coloring according to the eighteenth color, but realized it

would not work when the common difference was divisible by eighteen. Other attempts

that involved the fact that eighteen terms were being avoided included placing the integers

in a table with eighteen columns and coloring per column, this being a “combinatorial way

to do it”.

Most participants who attempted the problem realized that such a coloring would

have to be near-random and non-recursive. Attempts to exploit this included coloring based

on prime numbers, coloring based on divisibility, coloring based on parity, alternating, the

Fibonacci sequence, among others. Participants who stopped coloring did so because the

rules for coloring became “ridiculous” and “complex”.

One participant initially tried coloring, but eventually settled doing a constructive

argument instead. After failing to color the sequence to fulfill the constraints, he read the

problem again and observed that it only asked to prove the existence, and recalled that the

29

use of pigeonhole was a common technique for proving existence. This lead him to

consider the idea of counting the number of sequences instead, which lead to a solution.

Problem Four

4. Circles 𝒞1 and 𝒞2 with centers at 𝐶1 and 𝐶2, respectively, intersect at two

distinct points 𝐴 and 𝐵. Points 𝑃 and 𝑄 are varying points on 𝒞1 and𝒞2 ,

respectively, such that 𝑃, 𝐵 and 𝑄 are collinear and 𝐵 is always between 𝑃

and 𝑄. Let lines 𝑃𝐶1 and 𝑄𝐶2 intersect at 𝑅, let 𝐼 be the incenter of ∆𝑃𝑄𝑅,

and let 𝑆 be the circumcenter of ∆𝑃𝐼𝑄. Show that as 𝑃 and 𝑄 vary, 𝑆 traces

out an arc of a circle whose center is concyclic with 𝐴, 𝐶1 and 𝐶2.

Of the participants interviewed, this is the problem that was attempted the least.

Three of the participants interviewed claimed to solve the problem. The various approaches

attempted were conjecturing based on the diagram, recognizing the Incenter-Excenter

Lemma, looking for cyclic quadrilaterals, angle chasing, coordinate geometry, trying

specific cases, looking for the center and attempting to reduce the number of points.

All participants who read the question first attempted to draw a diagram. This lead

to some natural conjectures, from most common: that 𝐴𝑅𝐶1𝐶2 is cyclic, that APSQR is

cyclic, that the midpoint of the arc 𝐶1𝐶2 in the circle 𝐴𝑅𝐶1𝐶2 is the center of the required

locus. Often participants tried to draw multiple diagrams, or multiple parts in the same

diagram; one participant describes “I tried to draw three sets, but it just made the drawing

super messy.”

Most contestants recognized the use of the Incenter-Excenter Lemma, alternatively

known as Fact Five. One participant thought that this was obvious: “I think almost

30

everyone noticed Fact Five.” It became clear after the interviews that this is not the case;

two participants were unfamiliar with the fact but managed to prove its conclusions without

knowledge that it was well-known.

Some participants commented that one of the things they tried was to look for cyclic

quadrilaterals and similar triangles. One participant describes “I tried to find the connection

of R, S, P and Q with the rest of the diagram”. Another participant explains the motivation:

“If there are properties you can find, that you can use, intuitively, you would, you should

use them.”

Two participants attempted using coordinates. “I think I first tried to bash it with

coordinates… it was too long,” said one participant.

One participant describes trying special cases. “I tried say setting PQ parallel to

𝐶1𝐶2… I tried the two endpoints, when PQ was off to the right or to the left.” He did not

claim this provided any further insight, however.

Most participants who made progress in the fourth problem adopted the strategy of

reducing the number of points. “…point R is ‘done’ when we found that A was on APQR.

R is a varying point – not very helpful in looking for something constant. If something

constant can replace it, then go.” Most participants did something similar, using different

terms to describe the process.

Problem Solving Model

In the following sections we discuss the model as applied to the last problem of the

competition, followed by discussing its various components, concluding with a comparison

to other models.

31

Figure 3. The simplified version of the model.

An initial, simplified model was first constructed. The core of students’ approaches

was a trial-and-error process, attempting to do something and discarding it if it does not

work. This is evidenced by the way the narratives of problems are structured: a contestant

would describe what they attempted to do, why they attempted to do so, and the results of

the attempt.

Before the trial-and-error process is exploration. This involves reading and

understanding the problem, drawing diagrams or tables and trying specific cases, in order

to get a “feel” for what the problem is asking. After the trial-and-error process is stopping,

determining whether or not to stop the repeated trial-and-error, which can be through either

solving the problem or failing to solve it.

The trial-and-error process can be described as a motivated trial-and-error process.

It can be seen that problem solving in general not only involves trial-and-error, but a key

part of the process are the motivations behind the attempts used. A normal trial-and-error

approach would haphazardly attempt to use various methods, as what Schoenfeld (1989)

describes what one group of student did. The main difference was that there were well-

defined reasons behind the approaches, which is named here as motivating factors.

Another component of the motivated trial-and-error involves verification of the

attempt. After trying to use a method, the student would then try to verify its results:

32

whether or not the results are correct, whether or not the results can be used in furthering

knowledge about the problem, whether or not the results are relevant in finding a solution,

and so on. This process is named as verification, and after verification a student would

either attempt the trial-and-error process once more or proceed to stopping.

It can be observed that a student would sometimes return to exploring the problem.

For example, one student describes reading the problem again after failed attempts because

“I was looking for shortcuts [because] time rush.” It can also be observed that after a

student would sometimes return the trial-and-error process after evaluating stopping, if

evaluation showed that attempting the problem would help. From these observations, the

final model was produced, as in Figure 4.

Figure 4. The final problem solving model.

It is to be noted that the students in the study consistently followed the process

described in the final model, an approach that consistent of motivated trial-and-error. There

were no deviations from the above pattern across problems or students, and most

idiosyncrasies were observed in the individual processes rather than the overall pattern.

This resolves the question of whether or not high-performing students follow the same

general model and to what extent they follow the model.

33

Exploration

The first component in the model is exploration. This was defined as the initial

steps a student would attempt to understand the problem. A student would explore until

finding motivation, which would move to the motivation step, and motivation would return

to exploration if no motivation is found.

Reading the problem. The first step of solving a problem is reading it, which all

contestants describe as their first step. Thus, the research often asked for the first step

attempted after reading the problem. Not much emphasis was given on reading the problem

in the narratives of each contestant, except when it played a crucial role, as in one

participant being able to solve a problem after rereading it. Students would often read the

problem multiple times to attempt to note the details of the problem.

Understanding the problem. Students attempt to understand the problem in various

ways. One participant describes that he was “trying to get a feel for the problem by

understanding what was being asked.” This can be done through other methods under

exploration such as drawing diagrams, trying specific cases, or filling up tables with known

results.

Drawing diagrams. After reading the problem, the first approach attempted after

the geometry problem was invariably drawing the associated diagram. Often participants

drew this diagram multiple times, with the first being an approximation and the second

being a refinement, or drawing the diagrams from multiple perspectives. Conditions for

drawing new diagrams involved, for example, the first diagram being a special case, or the

first diagram being inaccurate.

34

Small cases. Attempting specific, small, or extreme cases was a common first

approach for algebra or number theory problems. For problems asking to find all the

solutions for, small cases involved looking for trivial and solutions with a small magnitude.

For inequalities, it involved trying specific values to attempt to assess how the problem

operates. The code was also used to refer to trying specific constructions, or taking specific

points.

Making a table. Part of attempting small or specific cases was making a table to

record the data gathered. For the first problem, for example, one participant described

making a table of n and its phi and sigma, in order to find a pattern. Tables can also be used

to initially explore an attempted construction, as one participant described for the third

problem when he made a table for coloring.

Motivation

The second component in the model, and the first part of the motivated trial-and-

error cycle, is motivation. This referred to the various factors and reasons a student had in

attempting a certain approach. These are various, and the following discussion is focused

on intuition, recognizing past knowledge, simplifying the problem, aversion, and

observation, which accounted for most of the coded responses. Other codes that occurred

less frequently and will not be in the subsequent discussion included attempting to save

time, adding to knowledge of the problem, and trying to make something more elegant.

Intuition. Almost all of the coded motivations were intuition. This manifested itself

with different phrases such as “magic”, “lightbulb moment”, “sudden realization”. This

was also the code used when a participant claimed that the motivation behind an approach

was “hard to describe”, “unknown” or “with absolutely no motivation whatsoever”.

35

Participants who gave such a response sometimes followed it up with a question of its

validity as motivation, for example, one participant asked “is this region legit?” when they

described their motivation as a lightbulb moment.

As most participants were unable to describe intuition, probing questions were

asked to elaborate on motivation. Most of the explanations were coded with recognition of

past knowledge or recall of past experience, which leads to the conclusion that intuition is

recognition of something that is previously seen.

Participants who gave a response coded under intuition were asked to elaborate in

a follow-up interview what their view of intuition was. Multiple responses were collected,

with various perspectives on intuition. As intuition played a large role in motivating the

attempts of the students, these responses are listed below:

“It’s just gut feeling, maybe even luck when you put it into context. Intuition was

something like that gut feeling that ‘proving all these points are concyclic would

help’, ‘proving these points are collinear would help.’ It’s that weird feeling you

know isn’t coming from experience.”

“It’s pattern recognition from previous problems you’ve solved or looked at the

solutions to. An equivalence or similarity to a previous problem would trigger a

recognition response that would make you want to try something. The more similar

or equivalent something is, the more likely you are in trying it.”

“The source of intuition is past experiences, experiences can strengthen or weaken

pieces of intuition, and not all intuition is correct or beneficial. Intuition is more

accurately defined as a trigger-action pattern, which is basically a cognitive step

from x to y, where x to y isn’t obvious for most people.”

36

“Intuition is the invisible guiding force in a mathematicians’ attempts to solve

problems. Intuition comes from the mathematicians’ previous attempts, previous

endeavors, previous solutions. As comparison, I believe solving math problems is

like finding your way out of a dark room given a few hints. The intuition is the light

you have created from being in similar dark rooms before.”

It is to be noted that all of the responses were concordant with the conclusions

drawn from examining intuition through a probing question, indicating that while most

participants knew what intuition was, it was difficult for them to describe without probing

from the interviewer. When this was pointed out, one participant describes this occurs

because “all solution booklets and even books don’t include the motivation behind the

magic, and the work is therefore left to the reader.” In this sense, the conclusion can be

drawn that by default, the motivation behind any inventive step in a solution is not shown,

and thus is difficult to describe as such description is unfamiliar in writing solutions.

Recognizing past knowledge. As explained above, a large portion of responses

coded under intuition were further coded under recognition of past knowledge. This was

because of recognizing a problem seen before, or something familiar. One participant was

inspired to do a trigonometric substitution because “it looked like something familiar”.

This accounted for most of the coding, indicating that a large portion of problem solving

involved recognizing past knowledge, which was often built from “learning contest math”,

as one participant describes.

Some ways recognition manifested itself follows. In the first problem, recognizing

that phi and sigma were number theoretic functions with a formula based on prime

factorization motivated prime factorizing n. For the second problem, it was recognizing

that equalities were often turned to inequalities that largely motivated the first part of the

37

solution. For the fourth problem, it was recognition of the configuration of the Incenter-

Excenter lemma that subsequently motivated its application.

Simplifying the problem. Simplifying the problem, or attempting to make things

easier, accounted for the second largest number of codes under motivation. This code was

used when a participant prefixed an explanation with “to make this easier to work with”,

“easy to use”, “to reduce clutter”, among others. One participant described this as “fixing”

the terms to make subsequent manipulations easier.

Such an approach was often successful for the second problem, as most of the coded

motivations for that problem were under this category. Substitution was often motivated

by simplifying the problem to make things “easier to write”. This also played a large role

in the responses for the first problem, which substitution of prime factorization made things

“easier to work with”. This was also the motivation for reducing the number of points for

problem four.

Aversion. Aversion was the code used when a participant was actively avoiding to

make things difficult. This was the code often used when a participant said “gross” or

“ugly”, and came up most frequently in manipulating algebraic expressions. There was an

aversion to messy calculations, which motivated ways to simplify the calculation. As one

participant describes: “[I tried] to get rid of square roots, because they are hard to

manipulate. I also tried Holder’s, to get rid of the square roots.”

Observation. Observation was the code used for describing motivation that comes

while the student was examining something, as opposed to recognition or desire. Such an

observation often comes from examining a diagram and subsequently making a conjecture,

which motivated one participant to prove a certain point was the center in the last problem.

38

An observation can also come from making specific cases, which gave one participant the

idea to generalize from these specific cases and subsequently attempt to verify this

generalization.

Approach

The heart of the model is the approach, the central component of both the model

and the motivated trial-and-error process. An approach was anything specific that the

student tried in response to the motivation, which was subsequently subject to verification.

Approaches were subcategorized into three parts: concrete, abstract, and meta-approaches,

based on their focus.

Concrete approaches centered on the problem, abstract approaches centered on the

problem solver, while meta-approaches centered on the environment. Of the three kinds of

approaches, concrete approaches appeared the most, followed by abstract and meta-

approaches, however all participants demonstrated use of all of three kinds.

Concrete approaches. A concrete approach is differentiated by its attention toward

the problem and its solution. It is associated with approaches such as manipulation,

simplification, cancelling, factorization, coordinate geometry, angle chasing, length

chasing, applying a theorem, looking for collinearity, and trying specific cases. Concrete

approaches were not given much elaboration despite appearing most frequently, indicating

that these kinds of approaches were routine in solving a problem.

Abstract approaches. Abstract approaches are focused toward the solver. These

were approaches that involved thinking about the problem and what the solution involved,

and how the solver would approach toward that. These included introspection, looking

back, generalization, repairing a proof, hindsight, and conjecture. These were differentiated

39

from motivation from the fact that approaches were actions, while motivations were

reasons.

Meta-approaches. These were approaches that were centered on the environment

of the problem, and involved specifics about the competition or the solver himself. These

included past experience, looking toward the problem order, known strengths and

weakness, and questioning the validity of the conditions. Also coded under meta-

approaches were actions like stretching and taking breaks.

Verification

The fourth component of the model, and the last step of the motivated trial-and-

error cycle, was verification. After attempting an approach, the student then validated the

approach through various methods, such as checking the proof, trying specific cases to

confirm a conjecture, rereading a problem, or doubting the proof. Whether or not the

verification fails or succeeds determines the next action, as summarized in Figure 5.

Figure 5. Where verification proceeds to, in general.

40

If a verification fails, the student often falls back toward motivation. Failing

verification was also considered as motivation for the abstract approach of repairing a proof.

Failing motivation, the student would often go back toward exploration in an attempt to

find something to subsequently motivate an approach, as described by one participant who

“ran out of things to do” and “tried reading the problem again”.

A student can also fall back toward stopping if no motivation is found. This is

distinct from initial motivation, as there are no cases recorded of a student immediately

stopping if no motivation is found. This indicates that students in the study often tried

multiple approaches before consequently stopping after failed verification.

A successful verification can either lead to subsequent motivation for another

approach, through observation. Verifying an approach added it to the properties a student

had at disposal to solve the problem, and this was often desirable: “it’s all about taking

what’s given, trying to make more of what’s given. Sort of like, taking bread and trying to

make it fluff, fluffing bread.” A successful verification can also lead to a stopping factor,

which would subsequently end the problem solving process.

Checking proofs. The most common form of verification was checking the proof of

a result obtained through an approach. This was done through repeating the steps involved

in proving the result from the approach, or attempting to find a different or alternative

approach. A successful verification of a proof for a problem often lead to the stopping

condition of correct solution.

Specific cases. This was a verification process that involved attempting specific

cases in order to verify the current result. Consider the second problem, for example, after

the approach of making a conjecture that no solutions exist. One participant subsequently

41

tried to look for specific values that satisfied the problem, and found an example. This lead

to a failed verification, which lead to the motivation stage once again.

Reread problem. After an approach, a participant would sometimes reread the

problem to look for missed conditions and hypothesis. One participant describes that they

found their solution to be wrong after observing that they misunderstood the question, and

subsequently tried the problem again. This is distinct from reading the problem for

exploration, as this process is meant to verify the authenticity of a result from an attempted

approach.

Doubt at proof. One form of verification was coded as doubt at proof, or attempting

to disprove. This was often the method of verification used for the abstract approach of

conjecturing, and was often done by finding specific cases. An alternative method of doing

this approach was observing a flaw in the reasoning behind a proof, which subsequently

lead to discarding the entire approach.

Stopping

The final component in the model is stopping. This refers to the conditions and

factors that influenced a student to stop attempting a problem. Failing or succeeding

verification can lead to stopping conditions, and failed assessment of a stopping condition

lead back to verification, which could subsequently restart the trial-and-error cycle.

Successful assessments of stopping conditions would end the problem solving process.

Correct solution. The stopping condition that invariably resulted in a successful

assessment was after verification of a correct solution. Verification of a complete and

correct solution often only lead to rewriting the solution to make it clearer, a repeat of the

42

process of verification, or terminating the problem solving process by attempting a

different problem.

Judged as difficult. After a failed verification a participant may judge a problem as

too difficult to solve and will avoid the problem entirely. For example, a participant who

said that he “did not have much intuition in algebra”, a meta-approach of knowing strengths

and weaknesses, lead to a failed verification and a subsequent judgement of the problem

as too difficult, leading the participant to not attempt the problem for the rest of the

competition.

Not enough time. A participant may have a successful verification but not enough

time to finish the problem. This lead the participant to stop attempting the problem and

attempt a problem which they believed to have a higher chance of succeeding.

Attempt later. A participant who fails multiple times to have motivation for a

problem could decide to attempt the problem later, as in the case of one participant who

returns to problem three only after solving problem four and having time left.

Application

An application of the model in assessing solutions to problem four is presented in

Figure 6. Note that this is the model used in assessing the collection of all solutions to

problem four. The model can also be used to assess individual solutions to a single problem.

Comparison with Other Models

It is of interest to compare this model to ones presented by other researchers through

different perspectives. Barbacena and Sy (2015) presented a model based on metacognition,

43

Fig

ure

6. T

he

pro

ble

m s

olv

ing

model

appli

ed t

o p

roble

m f

our

in t

he

com

pet

itio

n.

44

Schoenfeld (1989) identified six activities in a solution, while Pólya (1945) described

problem solving through a list of heuristics.

The model presented by Barbacena and Sy (2015) was focused on metacognition,

and can be seen in Figure 1. In this case, thinking skills are parallel to concrete approaches,

metacognitive awareness is parallel to abstract approaches, metacognitive evaluation is

parallel to verification, while metacognitive regulation is parallel to meta-approaches.

Their model emphasizes the relationships between the approaches which is not given

emphasis in the model presented in this study.

Schoenfeld (1989) identified six activities in problem solving: read, analyze,

explore, plan, implement, and verify. Reading and exploring would correspond to

exploration in the presented model, analyze would correspond to abstract approaches,

planning would correspond to motivation, implement would correspond to concrete

approaches, and verification is retained in both models. Schoenfeld did not attempt to

create a model to explain problem solving, but rather only described the six activities that

were commonly attempted in the problem solving process. It is to be noted that Schoenfeld

did not include an activity corresponding to meta-approaches, as these did not arise in the

context of his study, which involved isolated problems rather than multiple problems in a

problem set.

Multiple heuristics in the work of Pólya (1945) correspond to motivations and

approaches identified in this study, such as generalization falling under abstract approaches,

drawing a figure falling under exploration and using all the data falling under motivation.

Pólya also identifies four phases in problem solving: understanding the problem, which

would fall under exploration, devising a plan, falling under motivation, carrying out the

plan, which is an approach, and looking back, which is verification and stopping. This

45

corresponds precisely with the model presented in the study, from a more general

perspective.

Generally, the predictions of the model are in concordance with predictions from

other models, and from the method of triangulation, presented in Miles and Huberman

(1994) this provides validity and reliability to the data collected from the study.

46

Chapter 5

SUMMARY, CONCLUSIONS AND RECOMMENDATIONS

Summary

After the conduct of the competition proper, twenty-nine interviews of the sixteen

participants were conducted over two weeks. An analysis of the resulting transcriptions led

to the synthesis of a problem solving model, consisting of exploration, followed by a

motivated trial-and-error cycle, followed by stopping. The components of the model were

discussed and the model was applied to analyzing problem narratives. The model was also

compared to other problem solving models and their implications.

Conclusions

The results of the study imply that high-performing students follow mostly the same

process in solving non-standard problems, to a highly similar extent. Idiosyncrasies were

found only in deviations in individual methods, rather than the global model, which held

for each participant and each problem. In general, students performed exploration,

followed by motivated trial-and-error, followed by stopping. Students would perform

exploration upon first seeing a problem or upon lack of motivation, and stopping upon

failed or successful verification. Steps in the motivated trial-and-error cycle are often

repeated in the course of solving a problem.

Recommendations

Further research is recommended on the topic of problem solving, particularly in

the field of modelling the problem solving process, as a noticeable gap in literature was

observed. This study is only a preliminary attempt at such a model, and it is more than

47

possible for other studies to build upon and expand this model or other existing models in

the field, as an attempt to reduce the possible bias contained in each model. It is also

recommended that the results of this study be adapted for pedagogical use, for example, in

curriculum development, lesson planning, or training for mathematics competitions.

Further research is necessary to the manner on which students of different skill levels solve

problems, and to the extent of similarity. Finally, a longer, longitudinal study on how

problem solving skills change over time would be helpful in understanding the different

mental models of the process.

48

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Appendix A

Letter of Consent

Consent to Participate in Research

A Model of High-Performing Students’ Mathematical Problem solving

I am a student at the Valenzuela City School of Mathematics and Science. For

Practical Research 1, I am planning to conduct a study, which I invite you to take a part in.

You are being invited to participate in this study because you are a national qualifier of the

19th Philippine Mathematical Olympiad.

The purpose of this study is to create a model for high-performing students’

mathematical problem solving. If you agree to be in this study, you will be asked to

participate in a short five-minute interview on the day of the finals itself.

After taking the written portion of the national stage, you will be asked how you

solved a particular problem in the exam. This will only take a few minutes at most, however,

you might not be the first one interviewed. In the event you are unable to do a face-to-face

interview, you can schedule for an online interview within the week.

There is no direct benefit to you anticipated from participating in the study.

However, it is hoped that the information gained from the study will help other students

understand how high-performing students like you solve mathematical problems, which

can help them solve problems as well.

If you consent so, we will record your responses using an audio recorder. Afterward,

we will transcribe the contents of the interview. If you do not consent so, we will instead

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write your responses. In either case, the audio recordings and the transcripts will not be

published and presented to anyone except the researcher.

Your study data will be handled as confidentially as possible. Individual names and

other personally identifiable information will not be used. To minimize the risks to

confidentiality, we will remove all recordings used in the study afterward.