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Research in Mathematics Education
Series Editors: Jinfa Cai · James Middleton

Patricio Felmer
Erkki Pehkonen
Jeremy Kilpatrick Editors

Posing and
Solving
Mathematical
Problems
Advances and New Perspectives
Research in Mathematics Education

Series editors
Jinfa Cai
James A. Middleton

More information about this series at http://www.springer.com/series/13030

Patricio Felmer • Erkki Pehkonen
Jeremy Kilpatrick
Editors

Posing and Solving

Mathematical Problems
Advances and New Perspectives
Editors
Patricio Felmer Erkki Pehkonen
University of Chile University of Helskini
Santiago, Chile Helsinki, Finland

Jeremy Kilpatrick
University of Georgia
Athens, USA

Research in Mathematics Education

ISBN 978-3-319-28021-9 ISBN 978-3-319-28023-3 (eBook)
DOI 10.1007/978-3-319-28023-3

Library of Congress Control Number: 2016933779

© Springer International Publishing Switzerland 2016

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The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication
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The publisher, the authors and the editors are safe to assume that the advice and information in this book
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Introduction

Systematic research on problem solving in mathematics can be seen to have begun

over 70 years ago with the work of George Pólya, whose most famous publication
was likely the book How to Solve It (Pólya, 1945). Today there is a huge literature
on mathematical problem solving that includes research studies, descriptions, sur-
veys, and analyses. Among the most influential publications have been (and still
are) the book by Mason, Burton, and Stacey (1985); the book by Schoenfeld (1985);
and the paper by Kilpatrick (1987). The Mason et al. (1985) book emphasizes the
importance of creativity and highlights the many cul-de-sacs in problem solving as
well as the importance of a solver’s persistence. The book by Schoenfeld (1985) is
a well-known sourcebook. Younger researchers call it the “black book” of problem
solving. Kilpatrick’s (1987) paper underlines the connection between problem solv-
ing and problem posing, giving special emphasis to problem formulation. These
publications form part of the foundation on which this book rests.
The chapters in the book are based on presentations at the final workshop of a
comparative research project from 2010 to 2013 between the University of Chile
and the University of Helsinki. The project, whose title was On the Development of
Pupils’ and Teachers’ Mathematical Understanding and Performance when Dealing
with Open-Ended Problems, was initiated by Prof. Erkki Pehkonen (Helsinki) and
Prof. Leonor Varas (Santiago). In 2009, the Chilean CONICYT (Comisión Nacional
de Investigación Científica y Tecnológica) and the Finnish Academy opened a
cooperative program in educational research. Profs. Pehkonen and Varas worked
together on an application for a research grant whose leading idea was pupils’
development with open-ended problem solving. The project was funded and oper-
ated for 3 years. The final workshop, an integral part of the joint research project,
was originally designed as a forum to discuss the main results of the project.
However, with support from the Center for Advanced Research in Education
(CIAE) and the Center for Mathematical Modeling (CMM), both at the University
of Chile, a grant was obtained that enabled the workshop to be expanded well
beyond the project participants. The grant supported the invitation of more than 20
international specialists in the field of mathematical problem solving to join the
workshop. In the selection of additional participants, we tried to get a broad group

v
vi Introduction

of specialists from different parts of the world. After the workshop, all presenters
were offered an opportunity to contribute a chapter to the book, and almost all
accepted the invitation. Each paper was blind reviewed by two people—in most
cases an author of a different chapter, but in some cases an outside reviewer.
The program of the 4-day problem-solving workshop at the University of Chile
(Santiago) in December 2013 was as follows:

Tuesday 10 Wednesday 11 Thursday 12 Friday 13

9:00–9:45 Yan Ping Xin Leonor Varas
United States Chile
9:45–10:30 Peter Liljedahl Salomé Martinez Teachers’
Canada Chile workshop (CF)
11:00–11:45 Masami Isoda Hähkiöniemi Andras Ambrus Teachers’
Japan Finland Hungary workshop (CF)
11:45–13:00 Jeremy Kilpatrick Jinfa Cai John Mason Markku Hannula
United States United States England and Liisa Näveri
(CF)
Finland
15:00–15:45 Erkki Pehkonen Torsten Fritzlar Yew Hoong Valentina Giaconi
Finland Germany Leong and María
Singapore Victoria Martínez
(CF)
Chile
15:45–16:30 José Carrillo Susan Leung Wim van Dooren Alejandro López
Spain Taiwan Belgium and Paulina
Araya (CF)
Chile
17:00–17:45 Rosa Leikin Patricio Felmer Markku Hannula
Israel Chile Finland
17:45–18:30 Bernd Closing
Zimmermann ceremony with
Germany music from “Los
Bosquinos Band”

In the case of several authors, usually the first one gave the presentation.
The book is divided into three parts: (I) Problem Posing and Solving Today; (II)
Students, Problem Posing, and Problem Solving; and (III) Teachers, Problem
Posing, and Problem Solving.
Part I begins with the summary of the role of mathematical textbooks in problem
posing by Jinfa Cai et al. In the next paper José Carrillo and Jorge Cruz discuss the
role of problem posing and solving. Affect is also an important factor in problem
solving; this is dealt with by Valentina Giaconi et al. in the frame of Chilean ele-
mentary students. Nicolas Libedinsky and Jorge Soto Andrade examine the coop-
eration between affect and problem solving. Jeremy Kilpatrick opens a new aspect
in problem solving, discussing problem solving and inquiry. The section is closed
by Bernd Zimmermann who looks at the history of mathematics and reveals inter-
esting problems. The section review is given by John Mason.
Introduction vii

Part II begins with Jinfa Cai’s and Frank Lester’s overview on problem-solving
research results. Then András Ambrus and Krisztina Barczi-Veres consider the situ-
ation of problem solving in Hungary, especially from the viewpoint of average stu-
dents. Torsten Fritzlar explains the results of an exploratory problem implemented
by him. The next paper is from Erkki Pehkonen et al. who describe a new data
gathering method used in the Chile–Finland research project. Manuel Santos-Trigo
and Luis Moreno-Armella have used technology in order to foster students’ experi-
ences in problem solving. In the chapter of Tine Degrande et al., the modeling
aspects of problem solving are under focus. Yan Ping Xin deals with model-based
problem solving. Here Masami Isoda has written the section review.
Part III begins with John Mason’s considerations where he examines the concept
of problem from a new viewpoint. The paper of Patricio Felmer and Josefa Perdomo-
Díaz discusses Chilean novice teacher in problem solving. Leong Yew Hoong et al.
deal with problem solving in the Singaporean curriculum. Problem posing in the
elementary school program is examined by Shuk-kwan S. Leung. Edward A. Silver
discusses problem solving in teachers’ professional learning. Peter Liljedahl
explains on the conditions of teaching problem solving. The section review is given
by Kaye Stacey.
Finally we would like to thank a lot of peoples for their helping hands. Especially
we are grateful for those anonymous reviewers who helped us to improve the chap-
ters in the book. But above all we thank Gladys Cavallone for her huge job in practi-
cally organizing the workshop at the university and her efficient handling of the
papers of the book.

Santiago, Chile Patricio Felmer

Helsinki, Finland Erkki Pehkonen
Athens, USA Jeremy Kilpatrick

References

Kilpatrick, J. (1987). Problem formulating: Where do good problems come from? In A. H.

Schoenfeld (Ed.), Cognitive science and mathematics education (pp. 123–147). Hillsdale, NJ:
Erlbaum.
Mason, J., Burton, L., & Stacey, K. (1985). Thinking mathematically. Bristol: Addison-Wesley.
Pólya, G. (1945). How to solve it. Princeton, NJ: Princeton University Press.
Schoenfeld, A. H. (1985). Mathematical problem solving. Orlando, FL: Academic Press.
Contents

Part I Problem Posing and Solving Today

How Do Textbooks Incorporate Mathematical Problem Posing?
An International Comparative Study ........................................................... 3
Jinfa Cai, Chunlian Jiang, Stephen Hwang, Bikai Nie, and Dianshun Hu
Problem-Posing and Questioning: Two Tools to Help Solve Problems ...... 23
José Carrillo and Jorge Cruz
Affective Factors and Beliefs About Mathematics of Young Chilean
Children: Understanding Cultural Characteristics ..................................... 37
Valentina Giaconi, María Leonor Varas, Laura Tuohilampi,
and Markku Hannula
On the Role of Corporeality, Affect, and Metaphoring
in Problem-Solving.......................................................................................... 53
Nicolás Libedinsky and Jorge Soto-Andrade
Reformulating: Approaching Mathematical Problem
Solving as Inquiry ........................................................................................... 69
Jeremy Kilpatrick
Improving of Mathematical Problem-Solving: Some New IDEAS
from Old Resources ........................................................................................ 83
Bernd Zimmermann
Part 1 Reaction: Problem Posing and Solving Today .................................. 109
John Mason

Part II Students, Problem Posing, and Problem Solving

Can Mathematical Problem Solving Be Taught? Preliminary
Answers from 30 Years of Research .............................................................. 117
Frank K. Lester Jr. and Jinfa Cai

ix
x Contents

Teaching Mathematical Problem Solving in Hungary for Students

Who Have Average Ability in Mathematics ................................................. 137
András Ambrus and Krisztina Barczi-Veres
“Memorable Diagonals”: Exploratory Problems as Propositions
for Doing Mathematics ................................................................................... 157
Torsten Fritzlar
Pupils’ Drawings as a Research Tool in
Mathematical Problem-Solving Lessons ....................................................... 167
Erkki Pehkonen, Maija Ahtee, and Anu Laine
The Use of Digital Technology to Frame and Foster Learners’
Problem-Solving Experiences ........................................................................ 189
Manuel Santos-Trigo and Luis Moreno-Armella
Proportional Word Problem Solving Through a Modeling Lens:
A Half-Empty or Half-Full Glass? ............................................................... 209
Tine Degrande, Lieven Verschaffel, and Wim Van Dooren
Conceptual Model-Based Problem Solving .................................................. 231
Yan Ping Xin
Reaction: Students, Problem Posing, and Problem Solving........................ 255
Jeremy Kilpatrick

Part III Teachers, Problem Posing, and Problem Solving

When Is a Problem…? “When” Is Actually the Problem!.......................... 263
John Mason
Novice Chilean Secondary Mathematics Teachers as Problem Solvers ..... 287
Patricio Felmer and Josefa Perdomo-Díaz
Infusing Mathematical Problem Solving in the Mathematics
Curriculum: Replacement Units ................................................................... 309
Yew Hoong Leong, Eng Guan Tay, Tin Lam Toh, Khiok Seng Quek,
Pee Choon Toh, and Jaguthsing Dindyal
Mathematical Problem Posing: A Case of Elementary School
Teachers Developing Tasks and Designing Instructions in Taiwan ............ 327
Shuk-Kwan S. Leung
Mathematical Problem Solving and Teacher Professional Learning:
The Case of a Modified PISA Mathematics Task ........................................ 345
Edward A. Silver
Contents xi

Building Thinking Classrooms: Conditions for Problem-Solving .............. 361

Peter Liljedahl
Reaction: Teachers, Problem Posing and Problem-Solving ........................ 387
Kaye Stacey

Index ................................................................................................................. 393

 Part I
Problem Posing and Solving Today
How Do Textbooks Incorporate Mathematical
Problem Posing? An International
Comparative Study

Jinfa Cai, Chunlian Jiang, Stephen Hwang, Bikai Nie, and Dianshun Hu

Abstract This study examines how standards-based mathematics textbooks used in

China and the United States implement problem-posing tasks. We analyzed the prob-
lem-posing tasks in two US standards-based mathematics textbook series, Everyday
Mathematics and Investigations in Number, Data, and Space, and two Chinese stan-
dards-based mathematics textbook series, both titled Shuxue (Mathematics), published
by People’s Education Press and Beijing Normal University. All four textbook series
included a very small proportion of problem-posing tasks. Among the four series of
textbooks, the majority of the problem-posing tasks were in the content strand of num-
ber and operations, with a few in other content strands. Significant differences were
found between the Chinese and US textbook series as well as between the two text-
book series used in each country. Implications for the inclusion of mathematical
problem-posing tasks in elementary mathematics textbooks are discussed.

Keywords Problem-posing tasks • Curriculum • Textbooks • Mathematics education

reform • Comparative studies • China • United States

In recent years, interest in incorporating problem posing in school mathematics

instruction has grown steadily among mathematics education researchers and prac-
titioners (Australian Education Council, 1991; Cai, Hwang, Jiang, & Silber, 2015;
Singer, Ellerton, & Cai, 2013). Although historically, problem solving has been
more central than problem posing in school mathematics and mathematics education

J. Cai (*)
University of Delaware, 523 Ewing Hall, Newark, DE 19716, USA
e-mail: jcai@math.udel.edu
C. Jiang
University of Macau, Macau, China
S. Hwang • B. Nie
University of Delaware, Newark, DE, USA
D. Hu
Central China Normal University, Wuhan, China

© Springer International Publishing Switzerland 2016 3

P. Felmer et al. (eds.), Posing and Solving Mathematical Problems,
Research in Mathematics Education, DOI 10.1007/978-3-319-28023-3_1
4 J. Cai et al.

research, over the past several decades, curriculum reforms in many countries
around the world have begun to raise the profile of problem posing at different
educational levels (e.g., van den Brink, 1987; Chinese Ministry of Education, 1986,
2001a, 2011; English, 1997; Hashimoto, 1987; Healy, 1993; Keil, 1964/1967;
Kruteskii, 1976; National Council of Teachers of Mathematics [NCTM], 2000;
National Governors Association Center for Best Practices & Council of Chief State
School Officers [NGACBP & CCSSO], 2010). In part, this has been reflective of a
growing recognition that problem-posing activities can promote students’ concep-
tual understanding, foster their ability to reason and communicate mathematically,
and capture their interest and curiosity (Cai et al., 2015; NCTM, 1991). Because
problem posing and problem solving are often interwoven activities (Silver, 1994)
and success with one has been shown to be associated with success with the other
(Cai & Hwang, 2002; Silver & Cai, 1996), it makes sense to consider how problem
posing can be integrated as an effective part of mathematics instruction.
However, for problem posing to play a more central role in mathematics class-
rooms, teachers must have access to resources for problem-posing activities. In par-
ticular, mathematics curriculum materials should feature a good representation of
problem-posing activities. Although supplemental materials can partially address
the situation (e.g., Lu & Wang, 2006; Wang & Lu, 2000), it is important to have
problem-posing activities in the curriculum materials that teachers regularly use, as
curriculum can be a powerful agent for instructional change (Cai & Howson, 2013;
Howson, Keitel, & Kilpatrick, 1981). Thus, the significance of including productive
and robust problem-posing activities in curriculum materials should not be
overlooked.
Yet there is at present a lack of research that focuses on problem posing in the
textbooks that students and teachers actually use, as opposed to the curriculum
frameworks on which those textbooks are based. How has the inclusion of problem
posing in curriculum frameworks played out in real textbooks? Given the variety of
ways to engage students in one form or another of problem posing, how exactly do
textbooks include problem posing? What kinds of choices have textbook writers
and curriculum developers made in creating existing materials? In order to begin
addressing these questions, this study took an international perspective to examine
four mathematics textbook series, two of which are used in China and two of which
are used in the United States. All four series are based on reform curriculum stan-
dards from their respective countries (Chinese Ministry of Education, 2001a;
NCTM, 2000) which include problem posing as an important element.
Both China and the United States have engaged in similar reforms regarding
mathematics education, and problem posing has been explicitly included in the
reform documents that have guided the reforms in each country. Moreover, the
overall role of curriculum is quite similar in these two countries. It serves to deter-
mine what students are taught and, with respect to the design of textbooks, it con-
veys the ideas underlying the educational reforms. Thus, it seemed fruitful to
conduct a comparative study between the textbooks of the two countries in order to
provide an international perspective on the integration of problem posing into com-
monly used curriculum materials. Indeed, the field has long been interested in such
How Do Textbooks Incorporate Mathematical Problem Posing? An International… 5

comparative studies between China and the United States, whether they address
curriculum, classroom instruction, teacher education, or a myriad of other aspects
of the educational system (Cai, 1995). This research lies squarely in this compara-
tive tradition, taking a curricular perspective to analyze problem posing.

Background

Mathematical Problem Posing and Student Learning

A primary goal of research in mathematics education, including problem posing, is

to improve student learning. Researchers have noted the potential for problem pos-
ing to benefit student learning, both in mathematics (English, 1998; Lavy & Shriki,
2010; Silver, 1994; Toluk-Uçar, 2009) and in other areas such as reading
(Rosenshine, Meister, & Chapman, 1996). Problem-posing activities are often cog-
nitively demanding tasks (Cai & Hwang, 2002) that can require students to stretch
their thinking beyond problem-solving procedures to improve their understanding
by reflecting on the deeper structure and goal of the task. As tasks with different
cognitive demands are likely to induce different kinds of learning (Doyle, 1983),
the high cognitive demand of problem-posing activities can provide intellectual
contexts for students’ rich mathematical development.
In particular, because problem posing involves the generation of new problems
and questions aimed at exploring a given situation as well as the reformulation of a
problem during the process of solving it (Silver, 1994), encouraging students to
generate problems is likely to foster both student understanding of problem situa-
tions and the development of more advanced problem-solving strategies. Indeed,
using eight open-ended problem-solving tasks, Silver and Cai (1996) found a high
correlation between students’ mathematical problem-solving performance and their
problem-posing performance. More successful problem solvers were those who
generated more, and more complex, problems. Similarly, Cai and Hwang (2002,
2003) found links between students’ strategy use in problem solving and the types
of problems students posed. Clearly, the relationships between problem posing and
problem solving provide a rationale for recommendations to incorporate problem
posing into school mathematics at different educational levels (Chinese Ministry of
Education, 1986, 2001a, 2003, 2011; NCTM, 2000).
6 J. Cai et al.

Problem Posing, Mathematics Curricula, and Curriculum

Reform

Given the potential positive impact of including problem-posing activities in the

mathematics classroom, it is useful to consider how curriculum might support such
activities. Curriculum has historically been seen as a powerful agent for instruc-
tional change in the face of changing societal demands on the education system (Cai
& Howson, 2013; Howson et al., 1981). For example, a number of countries includ-
ing China and the United States have been undertaking similar mathematics educa-
tion reforms. The overarching goals of the reforms have been to improve students’
learning of mathematics and to nurture students’ innovation and creativity (Chinese
Ministry of Education, 2001b; NCTM, 2000). In the United States, NCTM (2000)
has placed a strong emphasis on students’ thinking, reasoning, and problem solving.
It calls for students to “formulate interesting problems based on a wide variety of
situations, both within and outside of mathematics” (NCTM, 2000, p. 258). In
China, students’ thinking and reasoning have also been emphasized in the mathe-
matics education reform. One of the six objectives of the new curriculum reform is
for students to be actively involved in inquiry-based activities in order to develop
their abilities to collect and process information, to attain new knowledge, to ana-
lyze and solve problems, and to communicate and cooperate (Chinese Ministry of
Education, 2001b). At the 9-year compulsory education stage, students are expected
to learn how to pose problems from mathematical perspectives, how to understand
problems, and how to apply their knowledge and skills to solve problems so as to
increase their awareness of mathematical applications (Chinese Ministry of
Education, 2001a). The high school mathematics curriculum is intended to enhance
students’ abilities to pose, analyze, and solve problems from mathematical perspec-
tives, to express and communicate mathematically, and to attain mathematical
knowledge independently (Chinese Ministry of Education, 2003). An additional
goal is for students to change their learning styles from passive to active through
being engaged in problem posing and problem solving (Chinese Ministry of
Education, 2001a, 2003).
Yet if, as these curriculum reform documents advocate, problem-posing activi-
ties are to become a more central part of mathematics classrooms, there must be
resources ready for problem-posing activities. Although teachers can take it upon
themselves to transform the problems and tasks in their existing curriculum materi-
als into problem-posing tasks, it is reasonable to posit that having ready-made
problem-posing resources available would facilitate teachers’ implementation of
problem-posing activities in their classrooms. One approach is to provide such
activities as supplementary materials. Lu and Wang (2006; Wang & Lu, 2000)
launched a project on mathematical situations and problem posing. They developed
supplementary teaching materials based on mathematical contexts and used them to
enhance students’ problem-posing abilities. These teaching materials were not
intended to replace textbooks; instead, they were used to supplement regular text-
book problems. Although helpful and potentially effective, it remains the case that
How Do Textbooks Incorporate Mathematical Problem Posing? An International… 7

teachers have easiest and most ready access to materials that are in their existing
curriculum materials. Moreover, particularly in countries like China in which teach-
ers carefully study their textbooks to guide and improve their teaching (Cai & Nie,
2007), the inclusion of problem-posing resources in those textbooks should be par-
ticularly powerful influences on classroom practice.
How, then, is problem posing represented in the mathematics textbooks that
teachers regularly use? Many current textbooks have been designed to implement
reform curriculum standards. For example, the NSF-supported projects that devel-
oped reform mathematics curricula in the United States based on the 1989 NCTM
Standards produced materials that were markedly different from the traditional text-
books that had preceded them. Among other features, the reform textbook series
included many more problems set in realistic contexts and more problems that could
be solved using multiple strategies (Senk & Thompson, 2003). Similarly, Chinese
textbook materials also evolved in response to reform guidelines in China. For
example, the 2004 edition of the Chinese elementary mathematics textbook series
published by the People’s Education Press (PEP) included a larger percentage of
problem-posing tasks than the 1994 edition (Hu, Cai, & Nie, 2014). However, more
generally it is not so clear where and how textbooks that have been designed to
implement reform curriculum standards include problem-posing tasks. Are problem-
posing tasks found broadly and systematically across the textbooks with respect to
both mathematical content and grade level, or are they distributed unevenly across
grade and content? To what extent do the textbooks embody the stances of the
reform standards toward problem posing? If reform standards portray problem pos-
ing as a theme that should run throughout mathematics education, it is useful to
examine the degree to which the actual textbooks exhibit this perspective.
Moreover, it is useful to consider whether the inclusion of problem-posing tasks
in reform-guided curriculum materials reflects a systematic approach to the devel-
opment of problem-posing abilities in students. For example, the inclusion of sam-
ple problems within problem-posing tasks may provide a window into the intent of
textbook designers. In earlier versions of Chinese mathematics textbooks, problem
posing was not included as a topic in its own right. Rather, problem posing was
treated as an intermediate step in problem solving. Newer, reform-oriented revi-
sions of the textbooks have included problem posing as a learning goal. To that end,
textbook designers have had to incorporate materials that can guide students through
the process of posing problems. One way to do this is to include sample problems
within problem-posing tasks for students to emulate. Thus, the degree to which
problem-posing tasks in textbooks include sample problems can be an indicator of
how intentional textbook designers were in building problem posing from the cur-
riculum standards.
Similarly, there are several types of problem-posing tasks that have been identified
in research on problem posing. Based on work by Stoyanova (1998) and Silver (1995),
Christou, Mousoulides, Pittalis, Pitta-Pantazi, and Sriraman (2005) describe five such
types defined by the nature of the problem students are asked to pose: a problem in
general (free situations), a problem with a given answer, a problem that contains
certain information, questions for a problem situation, and a problem that fits a
8 J. Cai et al.

given calculation. In addition, different problem-posing tasks may present given

information to students in several ways, including the use of visual and symbolic
modes of representation that may or may not be influenced by and consonant with
other design and pedagogical choices for a given textbook. Different types of tasks
thus reflect different qualities and priorities in problem-posing task design, such as
the degree to which the task is constrained for the student (e.g., Stoyanova, 1998)
or the role the task may play in relationship with problem solving (e.g., Silver, 1995).
Therefore, the manner in which different types of problem-posing tasks are incorpo-
rated into textbooks can provide further information about the degree to which these
materials systematically integrate problem posing from the curriculum standards and
to which they aim to develop particular aspects of problem posing for students.
On the whole, further work is needed to understand whether and how problem
posing is integrated into textbooks and the degree to which different ways of doing
so is effective in achieving the goals of curriculum reform. Of course, even when
problem posing is intentionally built into curriculum materials, it is still necessary
to study how problem-posing tasks are implemented by teachers in actual class-
rooms. The work that teachers do in transforming written curriculum materials into
live instruction depends on many other factors, including teachers’ knowledge and
beliefs. Nevertheless, as yet there has not been a substantial body of research exam-
ining whether and how the curricula themselves incorporate problem posing
(Cai et al., 2015). This study is intended to address the gap between the knowledge
about the incorporation of problem posing in curricula and textbooks. Specifically,
we address the following research question:
How are different problem-posing tasks included in recent US and Chinese
reform-oriented mathematics textbooks?
This study will provide researchers, curriculum developers, and textbook writers
with rich information about how to incorporate problem posing into school
mathematics.

Method

Materials

We examined two series of elementary mathematics textbooks used in China and

two series used in the United States. Of the two Chinese textbook series, one was
published by PEP, and the other was published by Beijing Normal University
(BNU). Both curricula were developed based on the new mathematics curriculum
standards (Chinese Ministry of Education, 2001a). We chose two popular series for
the textbooks used in the United States: Everyday Mathematics, developed by the
University of Chicago School Mathematics Project (UCSMP, 2012a, 2012b) and
Investigations in Number, Data, and Space (hereafter shortened as Investigations),
published by TERC, Cambridge, MA (TERC, 2008a, 2008b, 2008c, 2008d, 2008e,
2008f). These two series are generally taken to be examples of standards-based
How Do Textbooks Incorporate Mathematical Problem Posing? An International… 9

curricula (Riordan & Noyce, 2001; Senk & Thompson, 2003). In all four cases, the
textbooks represent the most widely adopted elementary mathematics curriculum
materials in their respective countries.

Task Analysis

We first checked every task in the four textbook series to identify those that were prob-
lem-posing tasks, including those cases where problem posing was included as a com-
ponent of a larger problem-solving task or activity. We then analyzed each problem-posing
task in terms of its (a) grade level, (b) content area, (c) presentation of given information
(e.g., with/without graphs, figures, tables, etc.) and whether there were sample questions
that students could imitate, and (d) types of problem-posing tasks.
With respect to the types of problem-posing tasks, we classified each problem-
posing task according to what it required students to do, relative to the information
provided in the task. These types were specified based on a holistic analysis of the
requirements in a problem-posing task. Special attention was paid to whether a
problem poser needed to provide information as givens and whether there was a
sample question that a problem poser could emulate to reproduce similar ones.
Five types of problem-posing tasks were identified. We describe these types below,
roughly ordered from the problem-posing task types that are the most mathemati-
cally constrained to those that are least mathematically constrained:
1. Posing a problem that matches the given arithmetic operation(s). Students are
asked to make up a story or a word problem that can be solved with a given arith-
metic operation. Tasks of this type provide the student with an explicit arithmetic
operation, and the student is expected to provide a context and pose a problem that
matches the operation. For example, write a story problem for 65 ´ 35 . Then solve
the problem and show how you solved it (TERC, 2008d, Unit 8, p. 29).
2. Posing variations on a question with the same mathematical relationship or
structure. Given a sample problem or problem situation (it is not necessary for
the sample to include a question), students are asked to pose a similar problem
complete with given information and question. The student can change the con-
text, the specific numbers, or even which quantity is the unknown quantity, but
the fundamental mathematical relationship or structure must mirror the sample.
For example, if six people share three apples, each person will get ½ of an apple.
Make up a problem about equal shares so that each person gets one fourth of
something (TERC, 2008c, Unit 7, p. 35).
3. Posing additional questions based on the given information and a sample ques-
tion. Students are asked to pose additional problems after solving a given problem
with sample question(s). The additional problems are expected to involve the
given information but are not required to mirror a particular mathematical
relationship. Although students may choose to provide additional information,
they may not change the given information. For example, on weekends, a father
and his son went climbing. The distance from the ground to the top of the mountain
10 J. Cai et al.

is 7.2 km. It took them 3 h to climb up and 2 h to walk down. What are the speeds
going up and going down? Can you pose additional mathematical questions
(People’s Education Press, 2001, 5a, p. 20)?
4. Posing questions based on given information. Students are provided with a problem
context and information but no sample problem. They are expected to generate
questions based on the given information. For example, four children (A, B, C,
and D) are practicing Chinese typing. The following table shows their practice
time every day and their records on a test where each of them could select an
article to type. Based on the data source, please pose two questions and try to
answer them (Beijing Normal University Press, 2001, 4a, p. 72).

A B C D
Practice time every day (in minutes) 20 30 35 60
Test records Time (minutes) 12 19 18 13
No. of words typed 384 931 846 728

5. Unconstrained problem-posing tasks. These tasks ask students to pose problems

to show the application of mathematics in real life but otherwise do not provide
given information or constraints on the structure of the problem. For example,
what mathematical problems could you find in your life? Please write them
down. Can you solve them? (Beijing Normal University Press, 2001, 1b, p. 98).
To establish interrater reliability for the coding of the problem-posing tasks,
30 problem-posing tasks from Chinese textbooks and 26 problem-posing tasks from
US textbooks were randomly selected and coded by two coders who are proficient in
both Chinese and English. For the Chinese textbooks, the two coders reached the fol-
lowing levels of agreement in each of the categories: (a) content area (100 %), (b) use
of various representations for the given information (e.g., with/without graphs, fig-
ures, tables, etc.) (92 %) and whether there were sample questions that students could
imitate (89 %), and (c) types of problem-posing tasks (82 %). Similarly, for the US
textbooks, the two coders reached the following levels of agreement in each of the
categories: (a) content area (89 %), (b) use of various representations for the given
information (e.g., with/without graphs, figures, tables, etc.) (88 %) and whether there
were sample questions that students could imitate (81 %), and (c) types of problem-
posing tasks (77 %). The discrepancies were resolved through discussion.

Results

Number of Problem-Posing Tasks at Different Grade Levels

The two Chinese textbook series and the US Everyday Mathematics series were
written for children in grades 1–6. However, the Everyday Mathematics textbooks
for children at grades 1 and 2 are combined. The US Investigations series was written
How Do Textbooks Incorporate Mathematical Problem Posing? An International… 11

Table 1 Total number of problems and percentage of problem-posing (pp) tasks in the four
mathematics textbooks series from grades 1–6
China United States
PEP BNU Investigations Everyday
Grade n % PP n % PP n % PP n % PP
1 527 3.61 570 5.96 490 0 –b –
2 565 6.73 549 5.65 741 1.62 1651 1.03
3 589 3.40 541 2.77 832 0.72 1322 1.06
4 621 4.83 561 2.85 760 1.97 1565 1.28
5 659 2.12 619 2.75 726 2.62 1896 1.16
6 627 1.75 545 2.94 –a – 1673 0.42
Total 3588 3.68 3385 3.81 3549 1.47 8107 0.99
Note: aInvestigations does not have grade 6 textbooks
b
For Everyday Mathematics of grades 1 and 2, we combined the data because there is only one
combined Student Reference Book for the two grades

for children in grades 1–5. For each textbook series, the total number of tasks (n)
and the percentage of those that were classified as problem-posing tasks are shown
in Table 1.
Overall, the percentages of problem-posing tasks were quite small for all four text-
book series. However, there were some differences across the series. The percentages
of problem-posing tasks in the two Chinese textbook series were more than double
those in the two US textbook series. The problem-posing tasks in the two Chinese
textbook series made up similar percentages of the total numbers of tasks in those
series, whereas the two US textbook series were significantly different from each
other in terms of percentage of problem-posing tasks. Specifically, a higher percent-
age of the tasks in the Investigations textbooks was problem-posing tasks compared
with that in the Everyday Mathematics textbook series (z = 2.25, p < 0.05).
The percentages of problem-posing tasks were also very different across differ-
ent grade levels. No grade had the largest percentage of problem-posing tasks across
the four series, and indeed the percentage rose and fell from grade to grade within
most of the series (although the grade-to-grade fluctuations within Everyday
Mathematics were comparatively small). Between the two textbook series in each
country, we compared the percentage of problem-posing tasks at each grade level.
There were no significant differences except between Investigations and Everyday
Mathematics at grade 5 (z = 2.69, p < 0.01).

Number of Problem-Posing Tasks in Different Content Areas

We classified the problem-posing tasks in the four textbook series by the content
area in which they were situated: number and operations, algebra, geometry, mea-
surement, and data analysis and probability, following the content areas used by
12 J. Cai et al.

Table 2 Percentage distribution of problem-posing tasks in different content areas in the four
mathematics textbook series
China United States
Content area PEP (n = 132) BNU (n = 105a) Investigations (n = 52) Everyday (n = 80)
Numbers and 73.48 76.19 90.38 91.25
operations
Algebra 0 1.90 5.77 1.25
Geometry 3.79 2.86 0 1.25
Measurement 0.76 2.86 0 0
Data analysis 21.97 16.19 3.85 6.25
and probability
Note: aIn several review sections in the BNU textbook series, there are problems like “What math-
ematical problems have you found in your life? Write them down and try to solve them.” Therefore,
the content areas they are related to cannot be determined. Twenty-four such problem-posing tasks
were excluded in this analysis

NCTM (2000) (Table 2). However, in several review sections in the BNU textbook
series, there were questions like “What mathematical problems have you found in
your life? Write them down and try to solve them,” for which the content area could
not be determined. The 24 free-structured problem-posing tasks of this type in the
BNU series were therefore omitted from the content area analysis. The percentage
distribution of problem-posing tasks in the five content areas was significantly
different across the four textbook series (chi-square = 31.22, df = 12, p < 0.01).
However, no significant difference was found between the two textbook series in
each country.
For all four textbook series, the majority of the problem-posing tasks were
related to number and operations. The percentages of number and operations
problem-posing tasks in the US textbook series were higher than those in the
Chinese textbook series (Investigation vs. PEP: z = 2.50, p < 0.05; Everyday
Mathematics vs. PEP: z = 3.15, p < 0.01; Investigation vs. BNU: z = 2.13, p < 0.05;
Everyday Mathematics vs. BNU: z = 2.68, p < 0.01). However, the difference in the
percentages of problem-posing tasks in number and operations in the two textbook
series in each country was not significant.
For the two Chinese textbook series, the second highest percentage of problem-
posing tasks was related to data analysis and probability. The difference in the
percentages of problem-posing tasks in data analysis and probability in the two
textbook series in each country was not significant. However, the percentages of
data analysis problem-posing tasks in the two Chinese textbook series were sig-
nificantly higher than those in the two US textbook series (PEP vs. Investigations,
z = 2.96, p < 0.01; PEP vs. Everyday Mathematics, z = 3.02, p < 0.01; BNU vs.
Investigations, z = 2.23, p < 0.05; BNU vs. Everyday Mathematics, z = 2.07,
p < 0.05).
For all four textbook series, very few problem-posing tasks were related to algebra,
geometry, or measurement, with the percentages all less than 6 %.
How Do Textbooks Incorporate Mathematical Problem Posing? An International… 13

Table 3 Percentages of types of problem-posing tasks in the four mathematics textbook series
China United States
PEP BNU Investigations Everyday
Types of problem-posing tasks (n = 132) (n = 129) (n = 52) (n = 80)
Posing a problem that matches the given 3.79 3.88 84.62 68.75
arithmetic operation(s)
Posing variations on a question with the 0 6.20 13.46 23.75
same mathematical relationship or structure
Posing additional questions based on the 65.91 56.59 1.92 5.00
given information and a sample question
Posing questions based on given 30.30 14.73 0 2.50
information
Unconstrained problem-posing tasks 0 18.60 0 0

Types of Problem-Posing Tasks

The problem-posing tasks in the four textbook series were classified into the following
five types based on what they required the student to do: (1) posing a problem that
matches the given arithmetic operation(s), (2) posing variations on a question with
the same mathematical relationship or structure, (3) posing additional questions
based on the given information and a sample question, (4) posing questions based
on given information, and (5) unconstrained problem-posing tasks. The percentages
of the problem-posing tasks of each type are shown in Table 3.
The data in Table 3 showed large discrepancies between the Chinese and US
textbook series and between the two textbook series in each country. Recall that the
types of problem-posing tasks were roughly ordered from most constrained to least
constrained. The percentages in Table 3 suggest that the Chinese textbooks had larger
percentages of problem-posing tasks that were comparatively less constrained,
whereas the US textbooks had larger percentages of tasks that were comparatively
more constrained.
For the two Chinese textbook series, the majority of the problem-posing tasks
required students to pose additional questions for given information after presenting
students with sample questions (e.g., On weekends, a father and his son went climb-
ing. The distance from the ground to the top of the mountain is 7.2 km. It took them
3 h to climb up and 2 h to walk down. What are the speeds going up and going down?
Can you pose additional mathematical questions?). Although the percentages of
problem-posing tasks of this type were not significantly different between the two
textbooks within either country, the percentages in the two Chinese textbook series
were significantly higher than those in the two US textbook series (BNU vs. Everyday
Mathematics: z = 7.52, p < 0.001). In contrast, for the two US textbook series, the
majority of problem-posing tasks required students to pose problems that matched
the given arithmetic operations (e.g., Write a story problem for 65 ´ 35 . Then solve
the problem and show how you solved it). The percentages of problem-posing tasks
14 J. Cai et al.

Table 4 Percentages of problem-posing tasks with/without sample questions and with/without

information presented in pictures, figures, or tables (PFT)
With sample questions Without sample questions
Textbook series With PFT Without PFT With PFT Without PFT
PEP (n = 132) 33.33 32.58 12.88 21.21
BNU (n = 131) 56.59 0.78 20.16 22.48
Investigations (n = 60) 3.85 5.77 3.85 86.54
Everyday Mathematics (n = 81) 17.50 10.00 6.25 66.25

of this type were not significantly different between the two Chinese textbook series,
but the percentage of problem-posing tasks of this type in Investigations was signifi-
cantly higher than that in Everyday Mathematics. The percentages in the two US
textbook series were significantly higher than those in the two Chinese textbook
series (BNU vs. Everyday Mathematics: z = 10.08, p < 0.001).
For the PEP textbook series, the second most common type of problem-posing task
was posing questions based on given information. The percentage of such tasks in PEP
was significantly higher than that in the BNU textbook series (z = 3.01, p < 0.01),
although this type of problem-posing task was the third most common type in BNU.
In turn, the percentage of such tasks in BNU was significantly higher than that in the
Everyday Mathematics textbook series (z = 2.86, p < 0.01). For the BNU textbook
series, the second most common type of problem-posing task was unconstrained prob-
lem-posing tasks. There were no such tasks in the other three textbook series.
For the Everyday Mathematics textbook series, the second most common
problem-posing task was posing variations on a question with the same mathemati-
cal relationship or structure. Although this percentage was not significantly higher
than that in the Investigations textbook series, it was significantly higher than those
in both Chinese textbook series (BNU: z = 3.68, p < 0.001). However, the percent-
ages of reformulation problem-posing tasks in BNU and Investigations were not
significantly different.

Presentation of Problem-Posing Tasks and Inclusion of Sample

Questions

Table 4 shows the degree to which the four textbooks included sample questions in
problem-posing tasks and to which they presented information in these tasks using
pictures, figures, or tables. Significant differences existed among the four textbook
series in both aspects (chi-square = 167.78, df = 9, p < 0.001). There were also sig-
nificant differences between the two Chinese textbook series (chi-square = 49.15,
df = 3, p < 0.001) but not between the two US textbook series.
Specifically, the two Chinese textbook series (PEP 66 %, BNU 57.37 %) had higher
percentages of problem-posing tasks with sample questions than the US textbook
How Do Textbooks Incorporate Mathematical Problem Posing? An International… 15

series (Investigations 9.62 %, Everyday Mathematics 27.50 %). The differences

between the two Chinese textbook series regarding inclusion of sample questions were
not significant. However, they are significant between the two US textbook series.
Of the problem-posing tasks included in the US mathematics textbooks, less than
half were presented with information in pictures, figures, or tables (Investigations
7.70 %, Everyday Mathematics 23.75 %). This was a lower percentage than in the
two Chinese textbook series (PEP 46.21 %, BNU 76.75 %). The two textbook series
within each country were significantly different in their percentages of problem-
posing tasks that included information presented in pictures, figures, and tables
(PEP vs. BNU, z = 5.06, p < 0.001; Investigations vs. Everyday, z = 2.38, p < 0.05).

Discussion

Problem Posing and Curriculum Reform

Curriculum reform has often been viewed as a powerful tool for educational
improvement because changes in curriculum have the potential to change classroom
practice and student learning (Cai & Howson, 2013). Reform-guided mathematics
curricula in both China and the United States have put great emphasis on problem
posing because of its potential to develop students’ creative thinking and ability to
innovate in the new century. Consequently, both Chinese and US textbook develop-
ers have made some effort to integrate problem-posing tasks into curriculum mate-
rials. Although our data show that the Chinese textbooks we examined do contain a
greater percentage of problem-posing tasks than the US textbooks, the percentage
of such tasks in each of the four textbooks we examined is still quite low.
The comparatively small representation of problem-posing tasks among a large sea
of problem-solving tasks may reflect, to some degree, the relative emphases and
placement of problem posing in the reform curriculum guidelines of the two coun-
tries. Problem posing was explicitly included as part of the problem-solving standard
for each grade band in NCTM’s (1989) Curriculum and Evaluation Standards that
guided the development of US reform mathematics curricula in the 1990s. In the
subsequent Principles and Standards for School Mathematics (NCTM, 2000),
problem posing was again part of the problem-solving standard in each grade band.
Given the strong focus on increasing the role of problem solving in reform mathemat-
ics curricula, it may be the case that problem posing was overshadowed. Indeed, the
recent Common Core State Standards for Mathematics (NGACBP & CCSSO, 2010)
only mentions problem posing once, whereas problem solving permeates the docu-
ment (Ellerton, 2013). The Chinese reform curriculum standards also include problem
posing as part of the overall objectives regarding problem solving (Chinese Ministry
of Education, 2011). In addition, they discuss the role of problem posing in assess-
ment and instruction. This broader inclusion of problem posing across the Chinese
reform curriculum guidelines may be connected to the somewhat greater inclusion of
problem posing in the two Chinese textbook series we examined.
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