mathematical problem solving- an evolving research and practice domain

8/3/2019 Mathematical Problem Solving- An Evolving Research and Practice Domain

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O R I G I N A L A R T I C L E

Mathematical problem solving: an evolving researchand practice domain

Manuel Santos-Trigo

Accepted: 20 July 2007 / Published online: 4 August 2007

FIZ Karlsruhe 2007

Abstract Research programs in mathematical problem

solving have evolved with the development and availabilityof computational tools. I review and discuss research pro-

grams that have influenced and shaped the development of

mathematical education in Mexico and elsewhere. An

overarching principle that distinguishes the problem solv-

ing approach to develop and learn mathematics is to

conceptualize the discipline as a set of dilemmas or prob-

lems that need to be explored and solved in terms of

mathematical resources and strategies. In this context,

relevant questions that help structure and organize this

paper include: What does it mean to learn mathematics in

terms of problem solving? To what extent do research

programs in problem solving orient curricular proposals?What types of instructional scenarios promote the students

development of mathematical thinking based on problem

solving? What type of reasoning do students develop as a

result of using distinct computational tools in mathematical

problem solving?

Mathematics instruction should help students develop

mathematical power, including the use of specific

mathematical modes of thought that are both versatile

and powerful, including modeling, abstraction, opti-

mization, logical analysis, inference from data, anduse of symbols (Schoenfeld, 1992, p. 345).

1 Introduction

Schoenfelds quotation summarizes fundamental aspects

associated with mathematical problem solving that will

help organize the content and structure of this paper.

What does it mean for students to develop mathematical

power? What are those particular modes of thought that

distinguish the processes of comprehending and devel-

oping mathematical knowledge? What types of problem

solving scenarios can help students develop habits (use

of different representations, identification of conjectures,

looking for arguments, use of particular notation, and

communication of results) that promote mathematical

thinking? What computational tools are important to helpstudents develop mathematical power? The discussion of

these questions involves addressing themes related to the

nature of mathematical thinking and problem solving,

mathematical instruction, the use of computational tools,

and the students development of mathematical problem

solving competences. In particular, as Schoenfeld (1992)

states, goals for mathematics instruction depend on

ones conceptualization of what mathematics is, and

what it means to understand mathematics (p. 334).

Thus, it becomes important to identify and discuss the

type of conceptualization of the discipline that is con-

sistent with principles associated with mathematicalproblem solving.

What aspects of problem solving research have influ-

enced and oriented curricular design and instructional

practices? Silver (1990) recognizes that research results in

mathematics education can be used to shape and support

instructional proposals. He elaborates on three research

components that have contributed to the design and

development of mathematical instruction:

M. Santos-Trigo (&)

Department of Mathematics Education,

Center for Research and Advanced Studies, IPN,

Av. IPN 2508, Sn Pedro Zacatenco,

07360 Mexico, D.F., Mexico

e-mail: [email protected]

123

ZDM Mathematics Education (2007) 39:523536

DOI 10.1007/s11858-007-0057-9


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1. Results or particular research findings that come from

research programs often can be used in educational

practices. [T]he results of systematic programs of

research can develop cumulative results that lend

themselves to substantive interpretation and important

implication for practice (Silver, 1990, p. 2);

2. Research methods that instructors or teachers can use

to explore and assess their students mathematicalknowledge. For instance, problem based interviews

have widely used to assess and foster students

development of mathematical thinking, and

3. Research frameworks that can provide information

about how students learning takes place. [M]uch

of the current interest in problem solving in mathe-

matics education is due in large part to the influence of

theoretical constructs, such as heuristic process, and

theoretical perspectives, such as an orientation toward

cognitive processes rather than cognitive products

(Silver, 1990, p. 6).

Thus, it becomes important to reflect on the extent to which

problem-solving programs have contributed to frame and

orient instructional practices that promote the construction

or development of students mathematical knowledge.

Thus, the aim of this paper is to identify and trace an

evolution of main aspects associated with research and

mathematics practices in problem solving. I argue that

mathematical problem solving as a research and practice

domain has evolved along the development and availability

of computational tools and, as a result, research questions

and instructional practices need to be examined deeply in

order to characterize principles and tenets that support thisdomain.

The paper starts with a background section that briefly

describes key issues and questions that have been central in

problem solving research programs. This information leads

to the description and contrast relevant features of some

research programs and their influence in instructional

practices.

Schoenfelds influential program (1985) inspired and

supported several problem-solving projects is revised to

trace the origin of current problem solving approaches.

Following, I sketch aspects of problem solving that have

influenced the mathematics curriculum and practice inMexico.

Finally, the students use of particular tools (dynamic

software, spreadsheets and hand-held calculators) in prob-

lem solving leads to examine principles and tenets

associated with problem solving approaches in terms of the

types of students mathematical behaviour that seems to be

enhanced with the use of these tools. It is argued that

typical problem solving frameworks that have emerged

from examining students problem solving approaches

based on the use of paper and pencil need to be adjusted in

accordance with what the problem solver shows with the

use of those tools (Santos-Trigo & Barrera-Mora, 2007).

2 Background: beyond multiple problem

solving interpretations

What is mathematical problem solving? How can problems

be characterized? Is there a single interpretation associated

with problem solving or are there many interpretations?

What is common to problem solving approaches? These

questions have been part of the research agenda in math-

ematical problem solving and the discussion of these

questions implies a reflection on the main tenets that dis-

tinguish the inquiry or approach based on problem solving.

The study of problem solving scenarios to learn math-

ematics has pervaded research agendas in mathematics

education and influenced mathematical practices during the

last three decades. Stanic & Kilpatrick (1988) pointed outthat problem solving has become a slogan encompassing

different views of what education is, of what schooling is,

of what mathematics is, and of why we should teach

mathematics in general and problem solving in particular

(p. 1). In this context, it is also recognized that there are

multiple interpretations of what mathematical problem

solving is or entails and ways in which research results in

this area have oriented mathematical instruction (Santos-

Trigo, 1998c). Indeed, Schoenfeld suggests that any

research program in this area should clarify the use of the

term problem solving:

The term [problem solving] has served as an umbrella

under which radically different types of research have

been conducted. At minimum there should be a de

facto requirement (now the exception rather than the

rule) that every study or discussion of problem

solving be accompanied by an operational definition

of the term and examples of what the author means.

Great confusion arises when the same term refers

to a multitude of sometimes contradictory and typi-

cally underspecified behaviors (Schoenfeld, 1992,

pp. 363364).

Frensch & Funke (1995) also recognize the great number ofdefinitions for the terms problem and problem solving. They

argue that the definition of problem solving may differ in

terms of scientific purposes in which there is interest to

investigate cognitive aspects and the sequences that an

individual follows to solve problems; engineering purposes

in which the research focus is on investigating problem-

solving scenarios and context or structure to optimize the

results; and humanistic purposes in which the research

focuses on documenting how personal interpretation of

524 M. Santos-Trigo

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events affects the problem solving process. They offer a

working definition of what they call complex problem

solving (CPS):

CPS occurs to overcome barriers between a given

state and desired goal state by means of behavioral

and/or cognitive, multistep activities. The given state,

goal state, and barriers between given state and goalstate are complex, change dynamically during prob-

lem solving, and are intransparent. The exact

properties of the given state, goal state, and barriers

are unknown to the solver at the outset. CPS implies

the efficient interaction between a solver and the

situational requirements of the task, and involves a

solvers cognitive, emotional, personal, and social

abilities and knowledge (p. 18).

How do the subjects cognitive barriers emerge? What does

it involve for the solvers to be aware of a given state? How

does the subject construct the desired goal? How does the

interaction between the solver and task become efficient?etc. These questions need to be addressed to develop ideas

and concepts associated with the working definition

provided by Frensch and Funke.

This paper does not pretend to review the many defini-

tions that researchers and practitioners have used for

problem solving. Instead, we will construct a problem

solving characterization around principles and ways of

thinking associated with problems solving and school

mathematics.

Lester & Kehle (2003, p. 510) characterize problem

solving as an activity that involves the students engage-

ment in a variety of cognitive actions including accessing

and using previous knowledge and experience:

Successful problem solving involves coordinating

previous experiences, knowledge, familiar represen-

tations and patterns of inference, and intuition in an

effort to generate new representations and related

patterns of inference that resolve the tension or

ambiguity (i.e., lack of meaningful representations

and supportive inferential moves) that prompted the

original problem-solving activity.

What does it mean for students to coordinate previous

knowledge and experiences to generate new knowledge? It

is evident that if students are to be engaged in problem

solving activities they need to develop a way of thinking

consistent with mathematical practices, in which problems

or tasks are seen as dilemmas that need to be examined in

terms of questions. Thus, students need to problematize

their own learning. In this process, they can use various

representations to identify and explore conjectures or

mathematical relations, look for distinct mathematical

arguments to support them, and develop efficient ways to

express and communicate their results (Santos-Trigo,

2006a).

Thus, solving even routine problems or comprehending

a particular situation (e.g. the definition of derivative) can

be approached in terms of dilemmas or questions that

students need to explore and resolve. This process can

eventually lead the students to look for connections and

extensions of the problem, or to examine and explore keyideas involved in a definition or in mathematical contents.

The problematizing idea goes beyond the discussion of

differences between routine and nonroutine problems since

students can problematize even routine problems to trans-

form them into nonroutine problem solving activities

(Santos-Trigo, 1998b). Arcavi (2000) utilizes a problem

solving approach to uncover, reflect on and communicate

his experiences and ways to frame research and work in

mathematics education research. How to select a research

problem? How to assess its relevance? How to formulate

research questions? What research designs or methods to

choose? These are some of the questions that Arcavi usesto identify and discuss crucial aspects in mathematics

education research. Thus, posing questions, looking for

various ways to represent and examine mathematical

relations, presenting arguments to support conjectures, and

communicating or presenting results are essential and

necessary activities to be engaged in problem solving

approaches.

3 Some research themes and results in mathematical

problem solving

I do not intend to present an extensive problem solving

literature review, instead I focus on identifying crucial

research questions and results that have contributed to the

development of this inquiry domain.

What are the most important themes or research ques-

tions that have been researched in mathematical problem

solving? What research methodologies have been used?

What are the main research results that have been pub-

lished? Is there any trend in research methodologies and

results? How have research results influenced mathematical

practices? A remarkable feature in the problem solving

research agenda is that the themes, questions, and research

methods have changed notably throughout the years, e.g.

Krutetskii (1976) and Schoenfeld (2000). The shift in

research themes is also related to shifts in research designs

and methodologies. Early problem solving research relied

on quantitative methods and hypothesis testing (statistical)

designs, but later approaches were and continue to be mainly

based on qualitative methodologies (Schoenfeld, 2000).

To summarize relevant results that have emerged from

research programs in mathematical problem solving during

Mathematical problem solving: an evolving research and practice domain 525

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the last 35 years, I have identified relevant research ques-

tions, research methods, findings, and their influence in

mathematical instruction. Tables 1, 2 and 3 show relevant

developments in the history of problem solving.

In terms of curricula proposals and instruction, the

document that best promotes the students development of

mathematical experiences based on problem solving

approaches is the Principles and Standards for SchoolMathematics (NCTM, 2000). The document is structured

around five content standards and five inherent processes of

mathematics practices. It posits that:

By learning problem solving in mathematics, students

should acquire ways of thinking, habits of persistence

and curiosity, and confidence in unfamiliar situations

that will serve well outside the mathematics class-

room. In everyday life and in workplace, being a good

problem solver can lead to great advantages Prob-

lem solving is an integrated part of all mathematics

learning, and so it should not be an isolated part of themathematics program (NCTM, 2000, p. 52).

Lester & Kehle (2003) pointed out that many researchers

often refer to this document to justify their problem solving

studies rather than identifying the themes that need to be

investigated. Apparently [the researchers] believe that

there no longer is a need to refer to the body of relevant

literature on problem solving to justify their work

instead, the Standards have become the authority

(p. 510). Lester and Kehles comment not only criticizes

the ways standard ideas are used to support research

projects, but also recognizes the need and importance to

investigate deeply problem solving instructional scenarios

that promote the values and principles associated with the

Principles and Standards.

4 Problem solving as a community of inquiry

In order to recognize and value a learning approach based

on problem solving, we must identify key or relevant

principles that need to be clear when following this

approach to learn and solve mathematical problems. I argue

that an overarching principle that characterizes any problem

solving approach to construct or learn mathematics is that

researchers, teachers and students conceptualize the disci-

pline as a set of problems or dilemmas that need to be

examined and solved through the use of mathematical

resources. Thus, problem solving is an inquiry domain in

which learners are encouraged to pose and pursue relevantquestions. To inquire means to formulate and pursue

questions, to identify and investigate dilemmas, to search

for evidence or information, and to present and communi-

cate results. It means willingness to wonder, to explore

questions and to develop mathematical understanding

within a community that values both collaboration and

constant reflection. A mode of inquiry involves necessarily

the challenges of the status quo and a continuous recon-

ceptualization of what is learned and how knowledge is

constructed.

[In a community of inquiry] participants grow intoand contribute to continual reconstitution of the

community through critical reflection; inquiry is

developed as one of the forms of practice within the

community and individual identity develops through

reflective inquiry (Jaworski, 2006, p. 202)

Thus, an integrating principle in all problem-solving

approaches is that students should have the opportunity

to pose questions around the problem or situation that lead

them to recognize relevant information needed to compre-

hend and explore meaning associated with concepts. These

questions are not just relevant during the entire solution

process; their importance relies in how they help to extend

the problem or think of other related problems.

Once you have learned how to ask questionsrele-

vant and appropriate and substantial questionsyou

have learned how to learn and no one can keep you

from learning whatever you want or need to know

(Postman & Weingartner, 1969, p. 23).

In this process, students constantly reflect on ways to

articulate and apply their ideas.

Table 1 Main developments of mathematical problem solving research during the 1970s and early 1980s

Type of research questions What is the role of the statement of problems in students problem solving approaches? What are

the characteristics of the tasks or set of tasks? What are the anticipated effects of changes in the

task characteristics? What are the problem-solving outcomes that one intends to measure?

(Goldin & McClintock, 1984). What are the aspects of problems that correlate with students

difficulties to solve them?

Research methods Quantitative approach and hypothesis testing (statistical) designs, the use of questionnaires.

Main findings Recognition that the content, the context, the structure, syntax and heuristics are variables

embedded in all problems and influence students problem solving approaches (Goldin &

Cladwell, 1984). The importance of heuristic methods in problem solving (Krulik, 1980).

Mathematical instruction Instruction centered on the teacher, introducing problems situated in diverse contexts.

526 M. Santos-Trigo

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Articulation requires reflection in that it involves

lifting out the critical ideas of an activity so that the

essence of the activity can be communicated. In the

process, the activity becomes an object of thought.

In other words, in order to articulate our ideas, we

must reflect on them in order to identify and

describe critical elements (Carpenter & Lehrer,1999, p.22).

The use of computational tools offers students appropriate

conditions to learn and solve problems within a community

of inquiry. Thus, for teachers to construct a learning

community, they need to provide instructional conditions

in which students engage in mathematical activities that

appreciate and value both individual and collaborative

work. The use of computational tools provides students the

opportunity to formulate and explore questions that may

lead them to identify mathematical results or relations

(Santos-Trigo, 2006b).

5 Mathematical problem solving developments

in the Mexican education system

To what extent problem solving approaches become rele-

vant in curricula and instructional proposals in Mexico? To

answer this question, I will first present information about

the makeup of Mexican education system. Following, I will

analyze current mathematical curricula for compulsory

education (PreK-9), and the main features of mathematics

textbooks. I should emphasize that textbooks are the

teachers main source to design and implement problem

solving activities in the classroom.

Table 2 Main problem solving developments during the 1980s and 1990s

Type of research questions What is mathematical thinking? How can students develop ways of reasoning that are consistent with the

development of mathematics? What features distinguish experts problem solving approaches from

students or neophytes approaches? What is the role of heuristic methods in students problem solving

processes? What is the role of metacognition in problem solving? (Schoenfeld, 1985). To what extent

the affective and students belief systems permeate their problem solving approaches? How can

problem-solving competences be developed in situated contexts? (Greeno et al., 1999).

Research methods Qualitative methods, problem based interviews, protocol analysis, and case studies.

Main findings Identification of fundamental categories to explain the individuals or students development of problem

solving competences: Knowledge base, cognitive and metacognitive strategies, belief systems and

affective relationships, and mathematical disposition (Schoenfeld, 1992, 1998; Santos-Trigo, 1996,

1998a).

Mathematical instruction Instruction centered on the students, small group discussions, the importance of students previous

knowledge, scaffolding instructional strategies, the construction of a learning community in the

classroom.

Table 3 Current problem solving directions

Type of research questions What type of mathematical reasoning do students develop as a result of using computational

technology in their problem solving approaches? To what extent does the use of technologyfavor the students reconstruction of mathematical relations or results? What curricular changes

are needed in order to promote the use of computational technology in students problem

solving approaches? What is the students process of transforming a device, the software, into a

mathematical problem-solving tool? What theoretical frameworks help explain the construction

or development of students new mathematical knowledge based on the use of computational

tools? How do these frameworks differ from those in which students work on problems using

paper and pencil?

Research methods Qualitative methods, problem based interviews, protocol analysis, case studies, design

experiments, didactical engineering, anthropological approaches, and ethnographic methods.

Main findings Distinction between material objects and instruments (psychological construct), the user

transforms the material object into an instrument through an appropriation process (Verillon &

Rabardel, 1995; Guin & Trouche, 2002, Guin, Ruthven & Trouche, 2005; Artigue, 2002;

Roschelle, Kaput & Stroup, 2000).

Mathematical instruction Instruction centered on the students, small group discussions, the importance of students previousknowledge, scaffolding instructional strategies, the construction of a learning community in the

classroom, the systematic use of different computational tool. Multiple representations of

mathematical objects.


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Mexicos population is about one hundred million and

during the recent school year 20052006 some 32 million

students attended public education at all levels. The average

schoolingfor the population is grade 8, and 7.7% of the adult

population is illiterate. During 20052006 the basic educa-

tion PreK-9 included about 25 million students distributed,

18% at Pre-K level, 57% in grades 16 (basic education),

and 24% in grades 79 (secondary school). It is important tomention that the basic education PreK-9 is compulsory for

all students. The federal government, through the Secretata

de Educacion (Ministrty of Education), establishes the

national curricula and designs or approves the correspond-

ing implementation material (textbooks, lesson guides,

assessment sheets, library material, etc).

After compulsory schooling is finished, grades 1012

involve diverse federal, state, and university regulated

schooling systems which do not share a curriculum. Each

system is based on particular needs and objectives. The

options available to students include technical (profes-

sional technicians) or general orientations (to continue onto university). For example, Mexicos National Autono-

mous University (UNAM) and the National Polytechnic

Institute (IPN) are two large institutions that have their own

1012 education options. During the 20052006 schooling

year the population at this level was about 4 million

students. At the university level, the population in the

20052006 year was about 2.5 million.

5.1 Problem solving and basic education

There is evidence that some of the ideas endorsed in the

NCTM (2000) document were used to justify and organize

the curriculum proposal for both PK-6 and 79. For

instance, for each grade the official curriculum that frames

the mathematics syllabus, identifies problem-solving

activities as key aspects in the pupils mathematics learn-

ing. It is recognized explicitly that pupils at the PK-6 level

should acquire basic mathematics knowledge to develop:

The ability to utilize mathematics as an instrument to

identify, formulate, and solve problems

The ability to predict and verify results

The ability to communicate and interpret mathematical

information

A spatial sense

The ability to estimate results and measurements

The ability to use measuring, drawing, and calculating

instruments

Abstract thinking that involves the systematization and

generalization of rules and strategies (http://www.sep.

gob.mx/wb2/sep/sep_121_matematicas, downloaded

March 24, 2007).

In addition, the curricula proposal at this level identifies six

major themes that are used to organize the content at each

grade:

Numbers, their relations and operations

Measurement

Geometry

Processes of Change

Data Analysis

Randomness and Prediction.

Comment: These themes resemble what the NCTM (2000)

proposes as the content standards, along with some

changes: The data analysis and probability standard is

divided into two themes, data analysis and prediction, and

the algebra strand is part of the processes of change theme.

In addition, the proposal does not include explicitly the five

process standards identified in the NCTM proposal. Thus,

the proposal fails to communicate the relevance and need

to relate and connect both the contents and mathematical

processes.For secondary school (grades 79) the 2006 official

curriculum document (SEP, 2006) recognizes that the study

of mathematics allows students to develop the way of

thinking needed to express mathematical situations

embedded in diverse contexts, and it encourages the use of

adequate resources to identify, formulate, and solve prob-

lems. The central focus is to invite and encourage the

students to work on activities in which they need to reflect

on and find diverse ways to solve problems, and to propose

arguments to validate their results. The content for these

three grades (79) is structured around three main themes

or axes:

Numerical sense and algebraic thinking that involve the

study of fundamentals of arithmetics and algebra. In

particular, the development and sense of a mathemat-

ical language is emphasized, as well as the study of

algebraic thinking in terms of connecting patterns

activities and pre-algebraic types of reasoning.

Form, space, and measurements that involve the

construction of geometric figures, measurements, and

the study of geometric properties.

Data analysis that involves the study of deterministic

and random phenomena using diagrammatic, graphical,or table representations.

Why are the six themes which structure the organization of

grades PK-6 curricula reduced to three in grades 79?

What are the advantages of organizing the contents only

around three themes? These questions are not addressed in

the document and, again, the process standards are not

explicit.

The contents for each of these three grades are orga-

nized into five content blocks. The idea behind using the

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block organization is to help teachers and students to

clearly identify and achieve partial goals. In addition, each

content block includes contents related to the three main

themes: numerical sense and algebra; form, space, and

measurement; and data analysis. However, the list of

contents and abilities included in each block seems to be

arbitrarily ordered and this makes it difficult to actually

identify main ideas from secondary contents.Comment: The official curricula proposals for PK-6 and

79 grades recognize the relevance of problem solving

activities in students mathematics learning but lack a

coherent presentation of key elements associated with

problem solving approaches. In particular, the proposals do

not show clearly the relevance for students to constantly

engage in inquiry approaches to identify conjectures and to

search for distinct arguments to support them. The exam-

ples used to illustrate the class development do not allow

the appreciation and value of the importance of problem-

solving strategies. For instance, a problem suggested to

help grade 9 students develop number sense and algebraicthinking is the square of this number minus 5 is equal to

220. What is the number? (SEP, 2006, p. 15).

6 Textbooks in compulsory education

Ideally, teachers at PK-9 grades should understand and

apply the fundamental principles associated with the cur-

ricula proposals. However, in practice what is more

relevant for teachers is the use of textbooks. I should

mention that official textbooks used in grades 16 are

written by a group of mathematics educators and teachersthat works for the Education Ministry.

At the secondary level (grades 79) private publishers

can send textbook proposals that are evaluated by an ad hoc

evaluation committee appointed by the Minister of Edu-

cation. Since textbooks are the main resources that orient

teachers practice, it is important to examine the extent to

which the books actually endorse a problem solving

approach in their contents presentation.

I describe the general features of two current textbooks,

one that is used in grade 4 and other in the first year of

secondary level (Matematicas, Cuarto Grado, revised edi-

tion, SEP 2000; Matematicas 1, primero de secundaria,2006, respectively). All textbooks are organized into five

blocks following the official curriculum proposal. Each

block includes around 18 lessons and usually each lesson is

presented in two pages. Blocks are defined in terms of

dividing the total list of mathematical contents to be cov-

ered in each grade into five parts.

All the lessons are identified with names that provide

their context. For example, some lessons names used in

the fourth grade textbook are: Trip to the Market, The

Market, The Raffle or Lottery, The Ferry Wheel, The

Circus, etc. Each lesson addresses particular contents of the

program. For instance, the lesson Trip to the Market is

associated with contents that involve reading and drawing

maps, point location, and trajectories identification. The

idea of approaching the contents in terms of discussing

activities within familiar contexts may be interesting for

students to engage in mathematical reflection. However,the linear organization of the content, and the uniformity of

the length dedicated to the development of each lesson

(two pages) may also limit the students development of

mathematical thinking. The main concerns identified with

the structure, organization, and presentation of the lessons

that may hinder or limit the students mathematical reflec-

tion include:

1. All the lessons have a similar format that describes the

context of each lesson and presents relevant data. The

students are asked to respond a series of questions that

require the use of the contents. In this context, studentshave no opportunities to pose and discuss their own

questions since their work is reduced to answer the

posed questions using only the information provided.

2. All the contents are introduced and presented in the

lesson at the same level of complexity. For each

content presented, the students have to respond one or

two related questions and as a consequence they have

little opportunity to reflect on the power of conceptual

ideas and the role of operations or mathematical

procedures. It may be difficult for students to identify

and differentiate main ideas or concepts from proce-

dures or rules in order to answer the questions.3. Each lesson seems to be set independently from the

others and students have little opportunity to construct

a line of thinking in which they examine deeply the

mathematical ideas embedded in each lesson. Instead,

they respond those questions already posed that often

require short answers without even exploring connec-

tions that may emerge while working on the activities

of the following lessons.

4. It seems that the design of each lesson is based on a list

of contents that students have to use to respond the

questions and fails to show relevant processes (exam-

ining particular cases, making conjectures, looking forarguments to support relations. and ways to commu-

nicate results) that students need to construct and use

in the problem solving process they should develop.

5. At the end of each block there is a review lesson in

which the students are asked to respond a set of

questions that are similar to those of previous lessons.

Thus, in this closing block lesson, students again have

to respond to a set of questions but they are nor asked

to reflect on how those questions are related, neither


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what problem solving strategies were used to solve

them.

In general, these observations can also be applied to the

textbooks used in grades 79. The lessons are developed

following a list of contents defined in the official programs

and there is little or no room for students to think of

connections, and mathematical processes involved inproblem solving approaches. I should mention that while

for grades 16 there is only one official textbook for each

grade that all students receive at no expense (it is paid by

the taxpayer money); in grades 79 there are some twenty

textbook options that teachers can choose. It is assumed

that all textbooks are evaluated and approved by an expert

panel that works for the Education Ministry. It is evident

that a lot of work needs to be done in the design and

structure of a curriculum proposal that clearly incorporates

problem solving approaches, and the development of

textbooks and auxiliary material that helps and guides

students in the development of a way of thinking consistentwith mathematical practice. For instance, a syllabus needs

to be organized around key fundamental ideas like

proportional reasoning, and the study of change in

connection with the development of habits of mathematical

thinking (Goldenberg, 1996).

Finally, it is also important to mention that the gov-

ernment has been implementing a program called

Enciclomedia to introduce the use of technological

media (electronic boards, computers, digital library, etc.) as

a way to enrich students learning experiences in the

classroom. So far, the program covers grades 5 and 6 and

includes a digital version of some textbooks and someactivities designed to complement the textbook lessons.

Teachers, in general, complain about the lack of training

programs to prepare them in the use of these tools, and the

impact of the program has been insignificant.

In terms of research programs that involve mathematical

problem solving, Hitt (2002) reports the importance for

students to articulate various types of representations to be

successful in problem solving approaches. His research is

based on the use of an enriched cooperative learning

environment methodology in which the participants not

only have opportunity to elicit their ideas and levels of

conceptual understanding; but also to deal with the cog-nitive conflicts that arise during their interaction with the

tasks. Estrada (2004) has investigated the type of strategies

and resources that high school and university students

develop in activities that promote problem formulation. His

research addresses ways to construct and improve the

students ideas that involve co-variation reasoning at col-

lege level.

Moreno & Santos (in press) document results from

research studies that involve the use of technology in

problem solving environments. They recognize that the use

of computational tools has begun to transform the educa-

tional system in terms of epistemological and cognitive

aspects since it has contributed to the development of a

new form of realism to deal with mathematical objects.

Moreno and Santos point out:

The presence and use of digital technologies hasintroduced new ways of looking at mathematical

cognition and as a consequence, the tools have

offered the potentiality to re-shape the goals of our

whole research field. Nevertheless the tension

between the local and the global mathematical

approaches emerges again and we have come to think

that presently, only local explanations are possible in

our field. Local theories might be the answer to the

plethora of explanations we encounter and need to

deal with. But even if local, a mathematics education

theory must be developed from a scaffolding

approach (perhaps a field of pragmatic evidences)that eventually crystallizes in the theory under

construction.

In this context, a case study is presented as a way to

illustrate activities and results of a research program that

aims to promote high school teachers and students use of

diverse computational tools in problem solving approaches.

Results are used to identify and discuss new research areas

that need to be explored. In particular, the importance of

revising and adjusting conceptual frameworks used to

explain the students construction or development of new

knowledge. This case study is part of an ongoing national

research project that began three yeas ago in which high

school teachers have participated in the process of design-

ing and using series of problems that foster the use of

computational tools.

7 A case study: the importance of computational tools

in students problem solving approaches

7.1 The context

The Case Study includes two related phases: (a) A research

team (formed by a math educator, two graduate students,and two high school teachers) selects or designs a series of

tasks and each task is discussed in depth to identify stu-

dents potential or hypothetical learning trajectories

(Simon & Tzur, 2004); and (b) the actual implementation

of the tasks in which 12 volunteer high school students

participated during one semester in a weekly 3 h problem-

solving sessions (Espinosa, 2006). The dynamics of the

sessions include students working on the task individually

or in pairs, presentations of their work to all participants,

530 M. Santos-Trigo

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and plenary discussions coordinated by the teacher. All the

paired students interactions were recorded, students pre-

sentations and plenary discussions were video-taped, and

students handed in their initial individual work and paired

work (including electronic files). We were interested in

identifying and discussing students problem solving

approaches that emerged during the development of the

sessions, rather than analysing in detail the studentsproblem solving behaviours. The problem posed comes

from Schoenfelds problem solving book and it is used to

illustrate and contrast students approaches based on paper

and pencil and those based on the use of dynamic software.

Problem 1. Your are given two intersecting straight

lines and a point P marked on one of them, Fig. 1, below.

Show how to construct, using straightedge and compass, a

circle that is tangent to both lines and that has the point P as

its point of tangency to one of the lines (Schoenfeld, 1985,

p.15).

What does it mean that a circle is tangent to two lines?

How can a tangent to a circle be drawn? How a tangent to acircle at one point and the line that passes by the center of

that circle and the tangent point are related? These are

some of the relevant questions that students discussed in

order to represent and explore the problem using dynamic

software. It is important to mention that this group of

students had experience in the use of the software. For

example, they had previously used the software to draw a

tangent circle with center P to a given line and P out of the

line.

Indeed, students initially drew a circle that is tangent to

line L1 at point P. What information is needed to draw a

circle? Where should the center of that circle be located?

These questions lead students to recognize that center must

lie on the perpendicular line to that passes by point P

(Fig. 2).

Comment: Two relevant aspects appear in the students

use of the tool, the accuracy of the representation and the

possibility of examining a family of cases rather than

the one just drawn. By moving some elements within

the representations students observed that the initial prop-

erties were maintained. When students moved the center of

the circle along the perpendicular, they visualized a solu-

tion to the problem (the circle tangent to both lines) and

reflected on what properties that this possible solution

would have. That is, an important heuristic that the students

used was to assume the existence of the solution and fromthere they identified relevant properties that helped them

approach and solve the problem.

Any object representation becomes an instance to

examine mathematical properties. For example, some stu-

dents could notice that drawing a circle tangent to line L1passing by P, with center at one point C on L3 (the per-

pendicular line to L1 that passes by P), then the line L4passing by the center of that circle and the tangency point

on L2 must be perpendicular to L2 (Fig. 3).

Within the representation, students observed that it was

possible to move some particular objects to identify in-

variants or relationships. For example, what is the locus ofpoint T when point C is moved along line L3? The software

again became a powerful tool to determine the locus of

point T when point C is moved along line L3. The inter-

section point T0 of the locus and line L2 determines the

tangent point of the circle on line L2. Thus, to draw the

circle tangent to both lines, it was sufficient to draw a

perpendicular line to L2, passing by T0 and the intersection

Fig. 1 Two intersecting lines and a point on one of them

Fig. 2 Any circle with center on the perpendicular to L1 that passes

by point P will be tangent to L1 at point P

Fig. 3 Line TC must be perpendicular to L2


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of this perpendicular with line L3 will be the center of the

tangent circle (Fig. 4).

Based on the above solution, some students asked: What

is the locus of point C (center of the tangent circle) when

point P is moved along line L1? (Fig. 5).

Students argued that the locus of point C when point P is

moved along line L1 is the bisector of angle PQT0. The

argument was based on showing that triangles PQC andT0QC are congruent. With this information, they concluded

that to draw a tangent circle to both lines, using straight-

edge and compass it was sufficient to draw the bisector of

the angle formed by the two lines, the perpendicular to L1passing by the tangency point, and the intersection point of

this perpendicular and the bisecting angle will be the center

of the circle.

Furthermore, students also proposed another non-

Euclidian solution by situating point Q on line L2 and

drawing perpendicular line L3 to L1 that passes by point P

and the perpendicular bisector L4 of segment PQ. The

perpendicular line L5 to L2 cuts L4 at point S. Which is the

locus of point S when point Q is moved along line L2?

(Fig. 6),

In this case, students argued that any point S on theparabola holds that the distance from point to S is the same

as the distance form S to line L2. This is because point S is

on the perpendicular bisector of segment QP. In addition,

the intersection point between the parabola and line L3 will

be the center of the circle that will be tangent to both lines

(Fig. 7).

Another important property that students found while

examining the construction was to recognize that the dis-

tance from the intersection point of lines L1 and L2 to both

tangent points needed to be the same. This led the students

to draw a circle with center Q and radius P and draw

perpendicular lines to lines L1 and L2 passing by points Pand P0 respectively. Here, they noticed that the interception

point (C) of those perpendicular lines was the center of the

required circle (Fig. 8).

Comment: There is evidence that the use of the soft-

ware allowed the students to explore the plausibility of

initial conjectures and to accept or reject them in terms of

empirical evidence. For instance, one student intended to

locate point Q on line L2 and identify PQ as the diameter of

the tangent circle (this construction appears also in

Schoenfeld, 1985, pp. 3637); but in this case, using the

graphic computational tool, the students construction

provided information to reject this conjecture. In addition,

the use of the software became important for students not

only to identify different ways to solve the problem, but

also to identify proper knowledge and resources to support

their results. Another important feature in using the tool is

that students used properties of mathematical objects like

the parabola to solve the problem (Fig. 7). That is, the

parabola became a resource to solve the problem. Later,

Fig. 4 Drawing the circle tangent to both lines at P and T 0

Fig. 5 Locus of point C when point P is moved along line L1

Fig. 6 The locus of point S when point Q is moved along line L2seems to be a parabola

Fig. 7 The center of the circle tangent to both lines is the intersection

of line L3 and the parabola

532 M. Santos-Trigo

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students explored other connections of this problem that

involve the study of other conic sections (Santos-Trigo &

Espinosa-Perez, 2002; Santos-Trigo et al., 2006)

8 Reflections

What types of problem solving approaches do students

exhibit when they systematically use some computational

tools? What types of problem solving strategies are favored

when students use those tools? To what extent the heuristic

methods used in problem solving approaches based on

paper and pencil are modified or extended when students

use computational tools? What types of arguments do

students show to support conjectures or relationships? To

what extent the dynamic representations of problems and

mathematical objects help students visualize and identifydifferent conjectures? These questions are used to discuss

main results that I have identified during the design and

implementation of problem solving approaches that

enhance the use of technology.

There is evidence that when students represent problems

using dynamic software, they engage in a line of thinking

in which they have the opportunity to identify conjectures

or mathematical relations. In this process, students often

find serendipitous results or ways to reconstruct mathe-

matical relations. For example, a simple construction that

involves a line L, a circle with centre at C (any point on L),

a point P on L, Q any point on circle with centre at C, and

point R the intersection point of lines PQ and Q0R (Q0 is the

reflected point of Q with respect to line L) can generate all

the conic sections studied in analytic geometry. What is the

locus of point R when point Q is moved along the circle?

With the use of the software, it is easy to find that the locus

corresponds to an ellipse (Fig. 9). When point P is moved

along line L, other conics sections appear (Fig. 10). Stu-

dents are encouraged to provide mathematical arguments to

justify the properties associated with those figures.

When students solved textbooks problems with the use

of different tools, it was interesting to observe that each

tool offered them distinct ways to represent and examine

the problem. For example, to approach the problem show

that among all rectangles with a given perimeter, the one

with the largest area is a square (Thomas & Finney, 1992,p. 215), students initially represented the problem dynam-

ically and observed that the maximum area of the family of

rectangle with a fixed semi-perimeter AB is reached when

there is a square with a side half of the length of AB

(Fig. 11).

The same problem was also approached with the use of a

graphic calculator. Here students graphed the function

A(x) = x2 + 4.8x and found directly its maximum value

(Fig. 12).

The same problem (as a general case) was approached

with the use of a calculator. Students determined the

derivative ofA(x) = x(p x) and found A0(x) = 0 (Fig. 13).It is interesting to observe that the dynamic representation

of the problem led students to identify the solution without

using algebraic resources. However, the use of the calcu-

lator demanded initially that students represent the problem

algebraically. Then, the graphic representation of the

problem allowed them to identify directly the side of the

square where the area is a maximum. Similarly, the use of

calculus resources (derivative concepts) led them to solve

the problem. At this stage, students had opportunity to

Fig. 8 What properties does a circle tangent to both lines hold?

Fig. 9 What is the locus of point R when point Q is moved along line

L?

Fig. 10 When point P is moved along line L other conics appear


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discuss relevant issues or concepts in terms of using dif-

ferent representations that appeared during the three

approaches used to solve this type of problems. There is

evidence that when students use various computational

tools to approach the problems, they develop distinct and

often complementary ways to represent and examine those

problems (Santos-Trigo, 2004b, 2006b).

9 Final remarks

In this paper mathematical problem solving is conceived as

a way of thinking in which problem solvers or, in this case,

students develop and exhibit habits, values, resources,

strategies, and a disposition consistent with mathematical

practice in order to comprehend mathematical ideas and

concepts, and to explore and solve mathematical tasks or

situations. It is also recognized that there are various paths

for students to develop mathematical thinking. However, a

common salient feature in those paths is the crucial role

that problematizing by the students themselves has on their

learning. By problematizing I mean the opportunity that

students have to think about problems, situations, contents,

and a knowledge base in terms of dilemmas or questions

that need to be pursued and solved. Thus, an overarching

principle in problem solving activities is the crucial role for

students to pose or formulate questions throughout their

interaction with mathematical tasks or contents (Santos-

Trigo, 2007).

In this context, even routine problems or exercises, that

initially may require students to access and apply only

formulae, rules, or well defined procedures, can be trans-

formed into nonroutine activities as a result of posing and

examining questions that lead students to identify andexplore connections or extensions of those problems (e.g.

general cases).

The students problem solving behaviours involve the

construction of a line of thinking or an inquiry process in

which they constantly reflect deeply on ways to represent

and explore mathematical ideas and solve problems (San-

tos-Trigo et al., 2007).

Research in mathematical problem solving has shed

light on the importance of relating the students mathe-

matical learning to the process of developing the discipline

(Schoenfeld, 1985, 1992, 1998). Thus, the students

development of problem solving experience is a slowprocess shaped by cognitive, metacognitive, and affective

variables.

Problem-solving performance seems to be a function

of several interdependent categories of factors

including: Knowledge acquisition and utilization,

Fig. 12 Graphic representation

Fig. 13 A calculus approach

Fig. 11 Diverse representations of the problem: object, graphic and

algebraic representations

534 M. Santos-Trigo

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control, beliefs, affects, socio cultural contexts,

implicit and explicit patterns of inference making,

and facility with various representational modes (i.e.,

symbolic, visual, oral, and kinesthetic) (Lester &

Kehle, 2003, p. 508).

Thus, students construct, develop, refine, or transform their

mathematical understanding and problem solving compe-tences as a result of formulating relevant questions and

pursuing them through the use of different media. It is also

recognized that students initial problem solving

approaches may be incoherent and limited; those

approaches get refined and improved when students openly

present and discuss their ideas within a learning commu-

nity that promotes and values mathematical inquiry.

How can we translate or transform problem solving

research results into curricula and instructional proposals?

There is evidence that some curricula frameworks and

instructional practices (NCTM, 2000; NRC, 1999; Kilpa-

trick, Swafford & Findell, 2001) have been framed andorganized around problem solving activities. However,

there is a pending need to discuss and reflect on the type of

changes and structure that distinguish both problem solving

based curricula and instructional practices. A case in point,

in Mexico, the presentation and justification of the current

curricula proposal for basic education recognizes implicitly

the relevance of problem solving approaches; however,

they fail to incorporate this problem solving view in the

organization and presentation of the mathematical contents

to be studied in elementary school. Thus, focusing on main

mathematical ideas (quantitative and proportional reason-

ing in elementary education) and the development ofmathematical habits or problem solving processes seems to

be a crucial component to structure and frame curricular

proposals for elementary school.

Finally, the systematic use of computational tools offers

the students the possibility of enhancing their problem

solving approaches. It is also recognized that different tools

may provide different ways to represent and explore

mathematical problems. For example, the students use of

dynamic software seems to favor the construction of

dynamic representations of mathematical objects or prob-

lems. As a consequence, students are likely to develop a set

of heuristics that involve measuring particular mathemati-cal attributes (segment lengths, angles, areas, perimeters,

etc.), dragging objects with a geometric configuration, and

the appropriate use of the Cartesian system to detect,

explore, and support mathematical relations or conjectures

(Santos-Trigo, 2004a, b, c). In this context, it becomes

important to document and analyze the type of reasoning

that students develop as a result of using more than one tool

in their problem solving experiences. In particular, current

problem solving frameworks to explain the students

construction or development of new mathematical knowl-

edge need to be re-examined in terms of the new types of

reasoning that may emerge as a result of using such tools.

Acknowledgments The author acknowledges the support received

from Conacyt (Consejo Nacional de Ciencia y Tecnologa, reference,

47850) during the preparation of this work.

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mathematical problem solving- an evolving research and practice domain

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