AN ANALYSIS OF USING JAPANESE PROBLEM SOLVING ORIENTED LESSON STRUCTURE TO SWEDISH MATHEMATICS CLASSROOM DISCOURSE

Co-chairs:
Manuel Santos Trigo (Mexico)
Zahra Gooya (Iran)
Team Members:
Man Goo Park (Korea)
Chunlian Jiang (China)
Dindyal Jaguthsing (Singapore)
Liaison IPC Member:
Yuriko Baldin

12th International Congress on Mathematical Education
Topic Study Group 15: Problem solving in mathematics education
Aims
Mathematical problem solving (MPS) is a field of study with a long history and has
supported numerous research programs in mathematics education at all levels. Given
the importance of MPS, the orientations and structure of many curriculum proposals
and teaching models throughout the world have been either directly or indirectly
influenced by it. However, the variety of problem solving programs with different
agendas and practical implications makes it necessary to revise and reflect on their
common foundations, nature, and historical development, and to build up and
implement problem solving approaches to support and foster students' learning, and
the development of mathematical knowledge and competencies.
In addition, the availability and use of digital tools in both real world and school
environments require that researchers and practitioners review, analyze, and discuss
the ways in which the tools could help students enhance and build up their
mathematical knowledge. Thus, all the participants in the academic activities of the
Topic Study Group will have an opportunity to reflect on and discuss issues and
themes that address the relevance, research programs and results, current trends and
agendas, and developments in MPS. The initial list of themes to frame and structure
the sessions is presented here:
1. Origin, a historical overview, and characterization of mathematical
Problem Solving. The aim of this section is to document and reflect on the
roots and evolution processes of the MPS from philosophical, psychological,
social, and cultural perspectives. The systematic discussion of these aspects
will help us characterize and distinguish the rationale to think of and to relate
problem solving approaches to the processes of comprehending and
constructing the mathematics knowledge of students. As well, the discussion
provides us with an opportunity to talk about the nature and characteristics of
conflicting and controversial terms such as problem, assessment, and routine
and non-routine tasks.
2. Foundations and nature of mathematical problem solving. The purpose is
to identify and discuss principles or tenets that explain problem solvers'
cognitive behaviours to justify the development and construction of
mathematics knowledge in terms of problem solving activities.
3. Problem solving frameworks. The aim is to carefully review the extant
frameworks that are currently used to structure and support research and

curriculum reforms in MPS. In particular, the focus will be on discussing the
extent to which these frameworks either explain students' mathematical
problem solving behaviours, or serve as tools to explain why and how students
construct new mathematical knowledge. In addition, it will be fruitful for
furthering the field by making clearer distinctions among the existing
frameworks in terms of their particular characteristics.
4. Research programs in mathematical problem solving. The aim is to identify
and revise ways in which research programs have contributed to the
development of the field within many different contexts and research
traditions. In particular, a discussion on how research findings in MPS are
disseminated and used in different education systems.
5. Curriculum proposals. The purpose is to discuss the distinguishing features
of a mathematics curriculum that is structured around various problem-
solving approaches. In particular, to identify and examine feasible ways to
clearly relate problem-solving principles to the organization and structure of
mathematical contents, processes and habits of mathematics practices. In
addition, to address ways in which MPS could be integrated across curricula.
6. The influence of social and cultural perspectives on problem solving
approaches. The purpose is to discuss and document the extent to which
social and cultural perspectives shape the ways of conceiving and
implementing various problem-solving approaches.
7. Problem solving assessment. The purpose is to identify and discuss different
ways of assessing students' problem solving performances. In particular, to
discuss the extent to which international studies such as TIMSS and PISA
assess students' problem-solving processes and competencies. Furthermore, it
will be important to discuss the effect of promoting mathematical competitions
and in specific, mathematics Olympiads to enhance students' problem solving
approaches.
8. Problem solving and the use of digital tools (internet, computer software,
hand-held calculators, Ipads, etc.). The purpose of this section is to analyse the
different ways of reasoning that students might construct as a result of using
systematically such tools. For example, to analyze the extent to which the use
of the tools enhances heuristics and representations used in paper and pencil
environments.
9. Problem solving outside schools. It is recognized that a variety of activities
with which students are engaged outside the school environment could play an
important role in their mathematics learning. Thus, it is important to identify
and discuss different ways in which students can participate in out-of school
problem-solving activities that involve realistic and complex tasks.

10. The role of problem solving in teacher education (both pre-service and in-
service). The aim is to discuss the ways in which, pre-service and in-service
teachers could develop their mathematical and didactic knowledge for teaching
via problem solving approaches.
11. Problem solving and university / tertiary education. In this section, we will
address themes related to the use of problem-solving approaches to study the
content of university mathematics. In particular, the role of problem solving in
grasping big ideas such as infinity and proofs.
12. Future directions and advances. The purpose is to identify future trends and
directions in research, curriculum developments, and teaching of mathematical
problem solving as a field of study.
Guidelines for submission
We invite the mathematics education community to submit proposals addressing the
themes listed above and others related issues. The proposal should be around 8 pages
and should be sent by October 31, 2011 both via email to the group co-chairs and
through the on-line submission system at the Congress Website. The members of the
organizing group will review each proposal. The results and comments will be sent by
January 15, 2012. And, the final version of the contribution should be sent by April 10,
2012. Any question that you might have, please send an email to any member of the
group.
On-line submission
Go to<My Page> at the first page of the Congress Homepage http://icme12.org or
press <Submit your proposal> button on TSG 15 website in the Congress Homepage.
Organizers
Co-chairs: Zahra Gooya(Iran) zahra.gooya@yahoo.com
Manuel Santos Trigo(Mexico) msantos@cinvestav.mx
Team Members: Man Goo Park(Korea) mpark29@snue.ac.kr
Chunlian Jiang(China) cljiang@umac.mo
Dindyal Jaguthsing (Singapore) jaguthsing.dindyal@nie.edu.sg
Liaison IPC Member: Yuriko Baldin yuriko@dm.ufscar.br or
yuriko.baldin@uol.com.br

Paper List:
1428 An Analysis of Effects on Mathematical Problem Solving depending on
Analogical Conditions
Eunseob Ban, The Graduate school of Korea National
University of Education, Korea, hymnes@naver.com
Heechan Lew, Korea National University of Education,
Chung Buk, Korea, hclew@knue.ac.kr
Jaehong Shin, Korea National University of Education,
Chung Buk, Korea, jhshin@knue.ac.kr
1297 Identifying and Describing Cognitive and Social Factors Involved in
Problem Solving Processes. A Case Study
Matias Camacho-Machin, Department of Mathematical
Analysis- University of La Laguna, Spain, mcamacho@ull.es
Josefa Perdomo-Diaz, Department of Mathematical Analysis-
University of La Laguna, Spain, pepiperdomo@gmail.com
1260 The Role of Problem-based Learning and Problem Solving in the
Mathematical Preparation of Pre-service Elementary Teachers
L. Diane Miller, Middle Tennessee State University,
United States, diane.miller@mtsu.edu
Brandon Banes, Middle Tennessee State University, United
States, bcb3u@mtmail.mtsu.edu
1105 Elements to stimulate and develop the problem posing competence in pre
service and in service primary teachers
Uldarico Malaspina, Pontificia Universidad Católica del
Perú, Perú, umalasp@pucp.edu.pe
Cecilia Gaita, Pontificia Universidad Católica del Perú,
Perú, cgaita@pucp.edu.pe
Vicenc Font, Universidad de Barcelona, vfont@ub.edu
Jesús V. Flores, Pontificia Universidad Católica del
Perú, Perú, jvflores@pucp.pe
1102 Problem solving in and beyond the classroom: perspectives and products
from participants in a web-based mathematical competition
Helia Jacinto, University of Lisbon, Portugal,
helia_jacinto@hotmail.com
Susana Carreira, FCT, University of Lisbon, Portugal,
scarrei@ualg.pt

1050 Problem Solving in Remedial Mathematics: A Jumpstart to Reform.
Baker, William, Hostos Community College, CUNY,
wbaker@hostos.cuny.edu
Dias, Olen, Hostos Community College, CUNY,
odias@hostos.cuny.edu
Prabhu, Vrunda, Bronx Community College, CUNY,
Vrunda.prabhu@bcc.hostos.cuny
Czarnocha, Bronislaw, Hostos Community College, CUNY,
bczarnocha@hostos.cuny.edu
0925 Experimenting Mathematically - Fostering Processes of Inductive
Reasoning in Explorative Mathematical Situations
Timo Leuders, Institute for Mathematics Education,
University of Education Freiburg,Germany,
leuders@ph-freiburg.de
Kathleen Philipp, Institute for Mathematics Education,
University of Education Freiburg, Germany,
kathleen.philipp@ph-freiburg.de
0814 Problem Solving and its Relationship to Student Mathematical
Epistemology: A Case Study in South Korea
Christine Yang, Northwestern University, United States,
cky@u.northwestern.edu
Uri Wilensky, Northwestern University, United States,
uri@northwestern.edu
0714 Common and Flexible Use of Mathematical Nonroutine Problem Solving
Strategies
Yeliz Yazgan, Uludag University, Turkey,
yazgany@uludag.edu.tr
Cigdem Arslan, Istanbul University, Turkey,
arslanc@gmail.com
0633 Errors in Solving Word Problems on Speed: A Case in Singapore and
Mainland China
Chunlian JIANG, University of Macau, China,
cljiang@umac.mo
Yuanhua Fu, 19690629, China, 308260128@qq.com

0600 Problem-based Learning in a Grade 6 Mathematics Classroom: An
example from China
Yan Chen, Southwest University, China,
chenyan_54@yahoo.cn
Cynthia Nicol, University of British Columbia, Canada,
cynthia.nicol@ubc.ca
0591 What Problem Solving Ought To Mean and How Combinatorial
Geometry Answers This Question: Divertismento in Nine Movements
Alexander Soifer, University of Colorado, United States,
asoifer@uccs.edu
0554 Forces hindering development of mathematical Problem solving among
school children
Azita Manouchehri, The Ohio State University, United
States, manouchehri.1@osu.edu
Pingping Zhang, The Ohio State University, United States,
Zhang.726@buckeyemail.osu.edu
Yating Liu, The Ohio State University, United States,
liu.891@buckeyemail.osu.edu
0485 An Analysis of Using Japanese Problem Solving Oriented Lesson Structure
To Swedish Mathematics Classroom Discourse
Yukiko Asami-Johansson, Department of Electronics,
Sweden, yuoasn@hig.se
0481 The Effects of Drawings on Problem Solving
Amir Hossein Ashna, USM, Iran, ahashna@yahoo.com
Lim Chap Sam, USM, Malaysia, cslim@usm.my
0407 The Role of Socio didactic in Mathematics
Álvaro Poblete, Ciencias Exactas Universidad de los
Lagos, Chile, apoblete@ulagos.cl
Verónica Díaz, Ciencias Exactas Universidad de los Lagos,
Chile, mvdiaz@ulagos.cl
0291 Problem Solving Processes of Fifth Graders - an Analysis of Problem
Solving Types
Benjamin Rott, IDMP, Germany, rott@idmp.uni-hannover.de
0283 Taking a Closer Look at Mathematical Problem Solving
Lay Keow Ng, National Institute of Education (NIE),
Singapore, ng_lay_keow@moe.edu.sg

Paper List: [Poster15]
1750 Effect of Implementation of Lesson Study on Student’s Divergent
Thinking In Mathematics
Julaluk Jai-on, Khon Kaen University, Thailand,
jula_jai@hotmail.com
Auijit Pattanajak, Center for Research in Mathematics
Education, Thailand, auipat@hotmail.com
Utith Inprasit, Ubon Ratchathani University, Thailand,
scutitin@mail2.ubu.ac.th
1705 Students’ Number Sense: Computational Estimation in Mathematics Class
using Lesson Study and Open Approach
Siwarak Promraksa, Khon Kaen University, Thailand,
spromraksa@hotmail.com
Kiat Sang-Aroon, Centre of Excellence in Mathematics,
Thailand, skiat@kku.ac.th
Auijit Pattanajak, Khon Kaen University, Thailand,
auipat@hotmail.com
1671 The Variation of Emotions in Mathematical Problem Solving.
Ahn Yoon Kyeong, The Graduate School Ewha Womans
University, Korea, sange1004@naver.com
Kim Sun Hee, Silla University, Korea, mathsun@silla.ac.kr
1613 Problem Solving Tasks through Decomposition-Composition Operations:
a Conceptual Learning Approach for Proofs and Proving Processes and
the Concept of Infinity
Medhat Rahim, Lakehead University, Canada,
mhrahim@lakeheadu.ca
Haitham M AlKhateeb , Qatar University, Qatar,
halkhateeb@qu.du.qa
Hissa Albinali, Qatar University, Qatar,
h.albinali@qu.edu.qa
Tracy Shields, Lakehead University, Canada
tjshields@lakeheadu.ca
Radcliffe Siddo, Lakehead University, Canada,
rsiddo@lakeheadu.ca

1602 21st century youngsters: how do they solve mathematical problems
beyond the classroom?
Helia Jacinto, University of Lisbon, Portugal,
Susana Carreira, University of Lisbon, Portugal,
scarrei@ualg.pt
1553 Study on Activation of the Ability of Problem Solving: Elimination of the
Inhibitory Factor of Flexible Idea
Miyo Akita, Naruto University of Education, Japan,
akitam@naruto-u.ac.jp
Noboru Saito, Rissho University, Japan, nsaito@ris.ac.jp
1495 Teaching Problem Solving Strategies: Comparative Study Between
Sweden and Hungary
Eva Fulop, Department of Mathematical Sciences, Sweden,
evaf@chalmers.se
1481 Iranian Students’ Performance in Non-Routine Problem Solving Tasks
Abolfazl Rafiepour Gatabi, Shahid Bahonar University of
Kerman, Iran, Drafiepour@gmail.com
1323 Emotions to be aroused in teaching and to be embodied in learning. - On
the effect math teachers have on students’ learning
Zheng Linran, East China University of Science and
Technology, China, sh_zlr@163.com
0205 The Way to Cultivate Children's Spatial Senses
Ngai-Ying Wong, The Chinese University of Hong Kong,
China, nywong@cuhk.edu.hk
Shi-hong Xu, Teaching Research Office of Guangzhou
Bureau, China, xshhgz@126.com

! "#
$ % & ' ( % &) * ) + ) , ) - . ! /+ &
#
0 ) 1 / 2 ) % , 3
. 4 , ! - 5 6 & ! ) 0 7) -
& 8 6
- 5 6 & ! ) 0 7) -
287 7
3
- 5 6 & ! ) 0 7) -
" 87 7
! " #
. ! )
9(*# 2 ! ! & ! 6 ! 7 2
! 2 6 & ! 6
& 0 7) ** : ) 99$) **;: 1 & 7 <
- ) 9$=: 1 & 7 < . ) 99(: 6 7) 99(: 6 7 < 1 & 7) 99 : & ) 9(;)
9(=: > ) 9$=: > < ? < ) 9$(: > < 1 & 7 < , ) **;#
@ & & 9(;) 9(=#
& ! ! 7 2
! 6 6 & )
! 2 & ! 6
& & 2 ) !
6 2 6 & > ) 9$=: > ) 9$(#
) / **9#) 2 7

0 ) / 2) ,
) *
2 6 ! ) 2 !
6 2 &) 2 2 &
6 2 A 6
&
B ! ) ! & & & 6
! ! 2 ) 7 2 ! 6
6 & ) ! 7 & ? )
** #) & 6 &
) !!
7 & 6 & & 6 6
) & & !
! & &
!! ) 2 2 & "
! ? ) 2 2
7 2 6
, ! 6 2 6 2 ) 2 6
& 2 7 2 !
. & 6 !
6 ) & & !! !
2 & 7 2
. 6 2 ) 2 2 2 &
7 2) 2 2 & 6 @
6 7 2 6 @ 6
& ! 2 . )
7 2 & ) 2 6 & !
7 2 C 6 & ) 2 & D@
. ! E 6 7 < 1 & 7) 99 : C ) **F#
. ! 6 ! ! &
! ) 6 ) 4 ) 9$3: 6 7) 99(:
> ) 9$=# ! !
! ! 0 7) ** : 4 ) 9$3: 1 & 7 <
- ) 9$=# 4 9$3# G &G !
2 & 2 ! ! &
2 7 4 < 7 ) 99=: 4 7 < 1 & 7) 9$3: 1 & 7 < . ) 99(#
4 E
0& 2 &) 6 7 1 & 7 99 # ! ! 6
! 6 . & !

0 ) / 2) ,
) *
! ! !
! 6 2 & .
2 !! ! !
2 & 2
! !
+ ) 6
& & !
6 @ 6 ) **3# H
6 4 7 1 & 7 9$3#G A 2
@ 1 & 7 .
99(# 2 6 ) 6
6 !
7 2
C ) 1 ! ) 4 & ** # & !
. & !
6 ) ? ** # &
7 !
6 6 6 2 ! !
@ ) 6 ! 6
6 6 ! 1 )
9;(# / 7 2 )
! 6 6 & !
6 5 &) & 6 ! 6
6 & ! ! ! &) & 2 7 !
! @ ) . < 2 2 ) * *#) & < & @ 6 ) **3:
< 0 ) **(: , & ) ** #) & < < I )
**=: C ) **3: I < ) * *#
/ , * # !! ! " & + @ +
@ &# ! 7 2 !
2 ! ! 0&
) ! )
! 2 6 & 6
D 6 " E & / , * #
D E 2 ! & 6 7 1 & 7 99 # !
) 2 6 6
7 " ! 7 2 ) 6 )
! ! / 7 2 )
6 6 7 )
6 &) 2 & ! ! 6

0 ) / 2) ,
) *
6
! 6 ! 6 )
! & 2 ! & & 2 !
! 6 ! 6 !& !! 6 !
& 6
. ! & ! & ! 7 &
6 !& !! 6 ! 6 &
! ! 6 B 6
) ) 6
# 2 ! 6 B &
2 ! 2 J
B ) ! 6 2 6
) ) 6 #)
2 #
, ) ! 6 2 6
) 2 6
& ! &) 2 A &
!! K ! ! &
2 ! , % ) - " .
6 ! & 2 !
. $* 2 K & 6 !
6 K 6 & ! ! & !
!
6 ) " 6 )
6 ) ! K ! A
A , " 2
2 K ! A
#) 2 " 2 & & ! K !
6 #
C 2 2 2
K ! # . 2 A
A

0 ) / 2) ,
) *
. J A <
A
4 L *# 4 L *# 43 L *# 4; L *#
A 6
6
. A ! ) 6
& 2 A 6
A A A !
6 6 ! ) &
7 2 & E &
6 A
C 6 !& 6 2
6 & ! )
K ! K ! A
! . 2
" ! 6 ) 6 ) !
K ! A
. J !
.
K
K
!
K
! A
A
6 ! 6
! A
. ! A 2 ! ; 6 2 K
! A ) 6 ) 7 !
6 , " 6 2 A ) 6 !
A A ! K ! A C
2 ! 2 7 2 !
! ! . !
A 2 ( ( 6 2 K !
A ) 6 ) ( 7
3 6 #

0 ) / 2) ,
) *
6 !& ! & ) 2 6 !
6 )
) 6 # 2 #
6 !& & 2 & !! !
6 @ & ! !!
6 ) 6 !
2 & & K &
.2 ! ! & 2 ! 2 J
B ) ! 6 2 2 3(M#
6 ) )
6 #) 2 (M# # . 3
! 2 6 !
. 3J 6 !
A
L F*# L *#
! 39 9
M# 3( (
B 6 !
. & 6 A ) K
2 !
0
5
10
15
20
25
30
35
40
analogical condition non-analogical condition
Success
rate

0 ) / 2) ,
) *
6 2 6
! !! N *(#) 2 2 K F == L **9#
0& & ! A ) ! & 2
, ) ! 6 2 2 F*M# 6
) (M#
) *M# 6 . ; ! 2
! & !
. ;J 6 !
A
L *#
L *#
6
L *#
( ;
! $ ( F
M# F* ( *
B 6 !
. & 6 ) K 2
! & ! 6
& 6 6
! 6 6
2 6 !! N *(#) 2 2
K ( * 3 L * (# ) ! 6
6 2 6 6
0
10
20
30
40
50
60
70
visual representation
mapping
concept mapping retrieval
Success
rate

0 ) / 2) ,
) *
!! N *(#) 2 2 K F FF= L * *# 0 2 E &
!! 2 6
0& & ! A ) ! & 2
C ! !! ! 6 ! !
& @ ! A 4 ) 4 ) 43) 4; & 7
6 ) 6 ) ! K ! A 1 2 6 )
& 6 ) ) & (M# 2 2
6 2 & !
/ 2 ! 6 7 2 7 2 J
6 ) 6 ) ! K ) 7 2
. !
6 ! 6 ) A 6
6 ! ) 2
! A 7 2 . 7 ! 6 2 ! ! &
6 ! / **9#) / , * #) & 9(;) 9(=#) 2
! 7
! ! & 9(;) 9(=#
B ) 6 ! ! & 6 !! 6
! 7 2 7 2
D 6 " E & / , * # ! 2 J
@ ! ! K & )
! ! ! 2 ! ) 2
& 6 7 1 & 7 99 #
. ! & 6 ! 7 2 2
& & 6 !
6 2 & !
! @ ! & ! ) 7
& 2 6 &
! & 2
6 ! 6 & 2 ! 6
@ ) @ ) . ) > @ ) < 2 2 ) * *# H
2 6 ! ! ) $ #
% # & '& 3(F 3F
@ 6 ) @ **3# . ! 6 !
( # & )*) ( ;

0 ) / 2) ,
) *
0 7) ** # , 2 ? 4 ) - %
1 & 7) < 0 - 7 6 #) $
;* ;3;# ) @J .
) ) < 0 ) **(# . & ! 6 J @
6 ! 2 7 ( # & )')
;( =(
) % ) < I ) ? **=# 5 ! A 6 &
6 # ( + , & - #) (*
? ) ) < ) / ? ** # ! !
7 @ @ #) *../ 0 1
2 == $9# > ) H@J
! . !
? ) - ** # . AJ C & & & )
& !! & & &O ? 4 ) - % 1 & 7) < 0
- 7 6 #) $ 3 3 33;#
) @J .
) / ? 99$# > & & 6
1 & 3 #) ( ;F
) / ? # **;#
2 ) %J / 2 @
4 ) ? 9$3# , J @ ! 2 7 ! & 1
# & 4) (( =*
4 ) ? ) < 7 ) @ 0 99=# , & &
! $ & )* #) ;( (F
4 7) / ) < 1 & 7) - % 9$3# , ! 1
$ & () 3$
1 ) % 9;(# ) %J
5 6 &
1 & 7) - % ) < - ) - 9$=# , ! & !
5 1 & /) ;#) 33 3;*
1 & 7) - % ) < . ) 99(# ! )
@J .
/ ) - 1 **9# . ! 7 2 )
6 & /7 3#) 3(( 3F9
/ ) - 1 ) < , ) 0 * # " 6 6 &
( # & 4-) 3 ;*
6 7) / > 99(# , ! ! ! !
2 + & / #) ( 3*
6 7) / > ) < 1 & 7) - % 99 # 6 & & ,
( $ 8 & & 1 & /4 3#) 39$ ; (

0 ) / 2) ,
) *
) % 9(*# / J > -
& ) 4 9(;# 9" /& :
; J 5 6 &
& ) 4 9(=# < # J 5 6 &
> ) , - 9$=# @ ! 2 , (
$ 8 & & 1 & /= #) ; 39
> ) , - ) ? ) @ ) < ) 9$(# 5 ! ! !
6 2 , ( $ 8 & &
1 & // #) *F (
> ) / ) 1 & 7) - % ) < , ) % C **;# @ &
1 : & ** #) 3= F*
, & ) ? @ ** # + ! 6 & J . !
6 A 6 , % & */ 3#)
3*3 3 =
C ) > C **F# 1 > & &
& 1 7 ) %J % C & < , )
C ) C 1 ) 1 ! ) 4 , ) 4 &) 0 ) < ) , ** # . ,.@> !
& ? 4 ) - % 1 & 7) < 0 - 7 6
#) $ ( (=#
) @J .
C ) / **3# H J . 7 &
5 ) 5 6 & 2 7)
I ) ? ) < ) % * *# . ! & 6 J 5 !
A 6 8 & /* #) == *;

! "#
$ % & ' ( % &) * ) + ) , ) - . ! /+ & #
% ! 01 1 1
2 3 / /
4 4
! "
# $
%
$
%
& % ' ! "$
( ) * +
. ! & !
5 3 & 6 3
3 5 ! 3 5
3 ) ! ! 3
3 ) ) 6 5 & 3
) !
6 ) 5 6 3 ) 7 6 ) 889# . ! !
! 5 !! ! 5 3
. ! ! 3 ! !
3 ! ! !
+0 3 &
. 5 & &
3 5 6 ! 3 )
3 5 6 3 ) 5 &
3 6 5 3
3 5 3 5 ) &)
3 &) # . ! ) & 5 & ! ! )
3 ) ! 3

) , - # * , -
: ;< ) *
: ! ! 3 & #
5 #
= 7
= ) **$) (9#
. ! ) &
. 5 ! 5 !& ! 5
& 6 5 ! 3 ! #
3 5 6 ! # - 5 !
5 6 6 ) 3 !
3 ) 5 ! 5 3 3 ! &
! 5 6
. 3 ! ! )
! 3 # !
! & 5 &) #
5 ! . ! 3
& > & ! 5
! 5 ! +0 5 5 5
.
: 5 ! +0 ! & !
& 5 ? 3 & &
! 5 ! @ 5 ! 5 )
! > & !!
@ ? ! ) > !
& . ! @ !
! !! 3 3 !
! &? ! ) ) ) ) ) 3 ! )
- 6 ) **8# A 5 3 3 5 ! .
&
: ! ( ) 5 3 & 3
3 3 ) 6 . , 5 3
> # 5 6 3 ! *
5 ! ! 3 6
3 . 5 3 B & .
** !
@ 3 ! ! ) 5 5
& 0 ! ) 5
) & 5 @ . C 5

) , - # * , -
) * ;
& ! +0 ) )
) ! ! ! +0 ) 3 #
A ? 0 ! . ! ! +0
0 & !
#
&
( #
&
D #
>
> 0
!
> !
A
,
!
0 !!
!
)
5
>
!
, > !
!!
3
!
A 3 !
5 !!
3
0
0 !! !
3
&
. 5
! +0
)
. > !
+0
! !
3
.
!
!
& .
. > !
+0 !
! 3
. 3 !
!
+0 &
. & 3 ! ( E 5 6
! C 3 ) & ! 5 5 6
! & ! 5 6 B & .
** )
! 3 ! 3 : ! 5 5 6 ! )
& ) 5 & & &
! 3 5 ) 5 5
: ! 5 !& 3 5
3 ! . 1 ) 01 7 , . )
* # 3 ! > !
3 ! 5 5 !
5 > ! & ! ! ! ! !
? : > F
. ! : > F & 3 3
. ! 5 & ! !
! 5 6? 3 = = ) **$#
. & ! E 5 6 * 5 !& 3
& ! ! ) ! 3
# & ! ! & 5
&) ! " 3 #

) , - # * , -
: ;< ) *
!" #$
= !
. ! . 3 3
! !! ! ) 3 3 ) @ ) )
# : > F ) ) 5
! )
! 3 3 ! ! ) ! !
! ! @
. ! ! 6 !
& ! #) ! 3 3 ! ! G : >
F 5 ! ! !
3 &) & 5 @ &
3 3 ! ! 5 ! 3 )
3 3 3 3 !
5 3 3 3 ! 5 ! .
! 5 > 5 F 3 3 & ) 5 &
& 3 5 5 3 H" 3 3 H#
: > ) 5 ) ) 3 3
! 3 H 3 3 3 H#) >
!
, ! ! I 5 5 &
> J
F ? G ) 5 3 3 5
: > ? 5 3
F ? , " 3 3
: > ? 0 3 3 3
F ? C
: > ? . ! ! @
F ? +-
A
: & 3 3 &) !! ! 3 5
!! > . ! 5 > 5 5 & 3
& & ! ! 3 3 ! ! 6
! 3
G !! 5 5 ! 5 C # K
#
C −
= J
I ? . 6 5 &
F? I 5J / C
:? ! #
C −
= &
F? 0
:?

) , - # * , -
) * ;
F? , #
C −
= ! J
L: > M
G !! 5 5 ! 5 E #K
E # K J
:? #
C −
= ) ! 5 &) 5
−
=
#
C ) ! 3
F? +-
> ) 5 5 C # 3 J
:? . 5
F? G 3 5 ) =
#
C 5
) J N 5 3 ) 3
LF @ ? C # @ JM
A <
: : > F 3 5 3 !
5 ! 3 ! ! .
! 5 !& +0 ) ! @
#
**<
*
<
*
C −
= F @ 6 & ! +0
! 5 &
3 ! & # : > E !! & ! !
! ! ! ) 5 ! 5
! ! 3 5 : >
: > ? L <
*
#
*
<
#
C D
=
⋅
+ −
M 5 5 3 3 3
! ! G 3 & ! )
@
. ? 6 J
: > ?
. ? & OJ
: > ? J
. ? ) & E & ! ! J ! !
3 J L& % M ! ! 3 J
: > ? : ! ! 5 3
A &) < ! ) : > F 3 6 5
@ & 3 ! ! +0
8
*
C −
= ) 3 ! # ! 6
5 5
I
! < ! ) !
) 5 6 & ! 5 3 )
5 3 +0
8
*
C −
= . ! 5 5 ! ! : > F
5 @ ?
F ? ! @ ) 5 ! 3 3 ) 5 5 & 3 )

) , - # * , -
: ;< ) *
: > ? ) ) . 5
F ? ) @ 5 . @ : ! )
5 5 & 3 3 ) 5 5 & 3 / 6)
@ ) @ ) 5 ) 3 L M
F ? 6 ) 5 : 5 : 3 3 )
3 3 ) G 5 LF ! M
: > ? ) 3 5 : 3 3 ) 3 3
F ? ) 5 ! ) &
) ! 3 3 ! 3 3 L M
: > ? P
. ? G J
: > ? ! D8
*−
!!
. ? ) !! & ) 3
: > ? ) ! 3 ! 3 &
+ & ) 5
8
*
> 5 3 3 ! N ) &
! 5 : > > 5 3 & )
3 3 & ! ! 3 3
) & 5 @ 5 ! 5
!! 3 ! G ! ) & ! @
3 ) & 5 & 6 5 & !
5 > ? & ! ! +0 . > 5 5
! 3 , ! ) 88 #
Q 3
&) !! @ ! ) > ! !
! & ! +0 1 * # 5
5 5 &
! ) 5 ! !! 0
: > F 5 ! 3 ) 3
! ) & !
! . !! &
5 H 3 H ) 5 5 5
! 3 +0 **<
*
<
*
C −
=
A 9
. > ) ) 5 5 ! ) !
3 3 !
3 6
" !% !" #$
A 5 &

) , - # * , -
) * ;
. ! ! . !
5 & ! ) 5 ! !
A > ) ) 3 5 5
! &
3 !! > ! #
C A <#
! E 5 5 ! 3 . ! !
5 ! 3 3 ! 3 &
5 ! ! 5 & 5
! 6 ) > &) " !& 5 @ )
: > F 5 & 6 !
" ! . ! 5 3 !
5 A 5 6
0 ! & : > F )
! & 5 & 3 & 5 ! !
E 3 ! - 6 ) **8#) !
3 : > C ? H. /
$ $
H) 5 F
>
****
#
! ! & ! 6 5
: > 5 # ! ! & 6
. 3 5 E ) 3
! > ! +0
" 3
. ! : > F @ 6 & > ! !
# ) 5 & 3 " 3 ) )
! ) ! ) !
! ! ! +0
. ! 5 > 5 ! > ! #
! & 6 #) !& 5 !! 5
! & 3 6) 5 3
! &
/ E # ! & 6 & . !
3 I 5 & 6
J
F ? # @
: > ? ) ) # H # @ H . & ! &
F ? L. & M
: > ? G P 5 # < R ) < * < 5
* < & !! ! L M
F ? 5 < & 6 5 5 3 6 5 @
& 6 5 5 ! ! & : &

) , - # * , -
: ;< ) *
6 5 @ * S 0 & 6 5J 3 ! ! )
& 6 5 5 L M / C >
A S
: !! * ! . ) & !
& ! ! <)
5 : > F 3 5 5
! ! ) 5 !&
!! 5 & 6 ) ! > ) )
! ! 3
! !
, > ! 3
! 5 > ! & 5 !
: > ? L M G 6
5
F ? ) 5 5 6 L M G 6 !
) 3 , 5 6 @
6 ) J
&
: > ? * R ) ) F ? I & 3 &
: > ? < R ) * <
F ? / J
: > ? Q ) 5 * < R
F ? * < ) J
: > ? ) * G
0 1
F ? * < ) J
: > ? ) * G
F ? ) 5 *
I 5 & 6
: > ? *
F ? :
: > ? * <
A $
: > ! ! &
5 : > F 3 )
! <) 5 & > ! ! &
! ! #
) ) 6 5 D * & A 8#
F? G ) ! ! ! #
) &
:? ) 5 D * &
F? #
) = J ***
:? . ! ! J

) , - # * , -
) * ;
F? . ! ! ***
:? . ! ! 3 ,
F? 5 E 5 D * & #
) 3 & ***
:? : 5 & & #
) J
F? N #
) ! !
:? : #
) ! ! & L M . 5 D * & #
) 5
F? ) : >) ! & 3 D * & & ***) 3 & D *
& 3 *** & & & D *J 5 3 & ***
***
:? ) 5 ) 5 5 3 I 5 & J
F? . 3 L M
:? ! ! 3 & *** L M +6 &) 5 5 L M / C E
#
D *) 3 & ***P
F? 0 3 & ***J TD** 3 & ***J
:? G J
F? G ) ! & D * & ***
:? : 3 & ***
F? N& 5 ) 6 & 3 6 3 &
. 5 & ! & !
) 5 ! 5
) 6 ! 3 ! ! & - 6 )
**8#
. > ! !! : > F E !
. 5 ! 3 !
! 3 ) 5 E & !
! ) 5 5 !
! &? 3 - 6 )
**8#
. ! 5 5 5 ! .
3 ! +
5 5 & ! ! ) 5
Problem 1:
Decay of uranium
Problem2: Mercury Pollution Problem3: Population dynamics
Stage 1:
Understanding
Stage 2: Analysis Stage 3: Solution (a
particular case)
Stage 1:
Understanding
Stage 2: Analysis Stage 3: Solution (a
particular case)
Cognitive
Factors
Reorganization of
mathematicalideas
Reorganizationof
mathematicalideas
Reorganization of
mathematical ideas
Selectionof
heuristics
Giving meaning to
the mathematical
procedures
Social
Factors
Familiarity with the
terminology
Familiarity with theterminology
Familiarity withthe
terminology
Commitment to a
mathematical
objective
Control of response Control of response

) , - # * , -
: ;< ) *
A ) 5 3 !
3 ! ! !
) ! 5 & & # .
! 5 ! 3 ) 5 3 !
) 5 3 ! ! 3 !
) ) ! ! ! 3 5
! " & 3 5
3
# " $
) U 01 ) % U , . ) * # / V
V & V V ! 2 #
* $ .
1 ) U 01 ) % U , . ) * # : > !
, W - 5 N A + & 0 !! @
#) ( # *
?RR R R 5 X J K 7 K $
) 7 6 ) 889# 3 ) ) 3
> ! 3 ) ) < ) S( 8*
- 6) % ) ,5 !! ) % 7 A ) N # **8# . , !
! & & . 3 0 * 4 S # ( ((#
G ) 0 ? : &
= ) 7 = ) G **$# . ! 2 ? : ! ,
/ 2 2 @ . . 0 !! @ )
7 = ) #) * $ 5
3 $ . : ! :
YS< G ) 0
, ! ) : 88 # / 6 &? 3 ) )
6 Q 5 ) 0 # 0 5
4 $ & 6 *
) <<D <S*
: 6 5 ? . 5 6 & & = " !
& ! , 3 ! ;0; ! 3 ! , 5
! 02 **$ *( (D

12th
International Congress on Mathematical Education
Program Name XX-YY-zz (pp. abcde-fghij)
8 July – 15 July, 2012, COEX, Seoul, Korea (This part is for LOC use only. Please do not change this part.)
abcde
THE ROLE OF PROBLEM-BASED LEARNING AND PROBLEM
SOLVING IN THE MATHEMATICAL PREPARATION OF
PRE-SERVICE ELEMENTARY TEACHERS
L. Diane Miller
Middle Tennessee State University
Diane.Miller@mtsu.edu
Brandon Banes
Middle Tennessee State University
bcb3u@mtmail.mtsu.edu
One theme of TSG15 is to examine the role of problem solving in teacher education with the aim to
examine ways in which pre-service teachers can develop their mathematical didactic knowledge for
teaching via problem-solving approaches. This paper examines a classroom intervention aimed at
improving the construction of content knowledge and problem solving skills. The researchers used a
control-experimental group design to investigate the value of implementing a problem-based
learning curriculum in a university level mathematics class required of pre-service elementary
teachers. The data collected and results of this study are presented.
Problem-based Learning, Mathematical Problem Solving, Pre-Service Elementary Teachers
INTRODUCTION
The history of mathematics as an academic discipline is more than 300 years old. In contrast,
research in mathematics education is approximately 75 years old with research directed at
mathematical problem solving only about 50 years old. One factor for the relative youth of
research in mathematical problem solving in the United States of America is described in the
Handbook for Research on Mathematics Teaching & Learning (Schoenfeld, 1992) where the
emphasis between basic skills and problem solving is modeled by a 10-year cycle.
Historically, problem solving’s importance as a basic skill has been recognized by a variety of
groups. The National Council of Supervisors of Mathematics (1977) listed problem solving as
the principal reason for studying mathematics; An Agenda for Action (National Council of
Teachers of Mathematics, 1980) recommended that problem solving be the focus of school
mathematics in the 1980s; the Twelve Components of Essential Mathematics (National
Council of Supervisors of Mathematics, 1988) said, “Learning to solve problems is the
principal reason for studying mathematics;” the Curriculum and Evaluation Standards
(National Council of Teachers of Mathematics, 1989) has problem solving as a common
thread throughout the preK-12 curriculum; and, the Principles and Standards for School
Mathematics (National Council of Teachers of Mathematics, 2000) continues to reflect
problem solving as a topic of importance in the preK-12 curriculum.

Miller and Banes
Abcde+3 ICME-12, 2012
Clearly, mathematical problem solving remains a topic for inclusion in school and
post-secondary mathematics; however, the amount of research on problem solving appears to
be on the decline (Lesh & Zawojewski, 2007). Reasons for the decline in scholarship on
mathematical problem solving may vary from country-to-country; but, one worldwide trend
may be attributed to the emphasis on high-stakes testing of basic competencies.
Another fact supported by evidence in the documents listed above is that elementary teachers
are expected to teach their students how to solve problems. The classrooms in which they
have learned mathematics are often dominated by a teacher-centered learning environment in
which problem-solving abilities are developed by first teaching the concepts and procedures
of mathematics topics, then assigning one-step “story” problems that are designed to provide
practice on the content learned. Problem solving is then taught as a collection of strategies
applied to a set of problems that requires the mathematics learned in the first steps of this
continuous cycle. “When taught this way, problem solving (and its strategies) is portrayed as
concept- and context-independent processes, isolated from important mathematical ideas.”
(Lesh & Zawojewski, 2007, p. 765.)
A common belief held by many is that teachers teach as they were taught. Hoping to find an
intervention that would alter the current cycle of how problem solving and mathematics is
taught at the elementary school level, the researchers in this study took a different approach to
teaching mathematics and teaching problem solving. Their intervention focused on
connecting learning mathematics with problem solving. The impetus for this study includes a
personal belief that the study of problem solving needs to happen in the context of learning
mathematics and that learning mathematics needs to happen within a context of problem
solving. Other support for the design and implementation of this study follows.
The National Council of Teachers of Mathematics in the United States has promoted the
learning of mathematics through problem solving for many years (NCTM, 1989; NCTM,
2000). Elementary classrooms are ripe with opportunities for problem-solving activities but
many elementary teachers have not been exposed to teaching mathematics in this way and,
therefore, are more likely to teach mathematics using a traditional approach (Timmerman,
2004). Teachers need to have the experience of solving problems so that they know how to
create a problem-solving environment that challenges their students to think (Jacobbe &
Millman, 2009). Elementary students struggling to learn mathematics need help, but often
their teacher is not prepared to answer their questions because of their own limitations in
knowing mathematics (Gresham, 2010). Too often problem solving becomes the bane of
positive attitudes about mathematics because students are learning mathematics as a
collection of rules to apply without a context and, when a context is provided (e.g., a word
problem), students look for ways to implement problem-solving strategies recently taught
rather than interpreting the situation and thinking through a process for solution. In summary,
the process of learning mathematics and problem solving becomes meaningless and, with
time, many students develop a negative attitude towards mathematics, in general, and
problem solving (word problems) more specifically.
A teacher’s attitude towards mathematics has a direct impact on instructional effectiveness
(Malinsky, McJunkin, Pannells, & Ross, 2004). Situations like the ones described above,

Miller and Banes
ICME-12, 2012 abcde+2
create a cycle, “future teachers pass through elementary schools learning to detest
math…then return to teach a new generation to detest it” (Polya, 1945, p. ix). The researchers
conducting this study agree with the position stated in Schoenfeld’s (1992) summary, that
students “abstract their beliefs about formal mathematics – their sense of their discipline – in
large measure from their experiences in the classroom” and that “students’ beliefs shape their
behavior in ways that have extraordinarily powerful (and often negative) consequences” (p.
359).
Elementary education majors in the United States of America often have high scores on
mathematics anxiety rating instruments, indicating high levels of anxiety about anything
related to mathematics (Ashcraft & Krause, 2007; Lang & Lang, 2010; Malinsky, McJunkin,
Pannells, & Ross, 2004). Additionally, mathematics anxiety has been shown to be a key
ingredient in the development of negative attitudes toward mathematics and the perception
that teachers are ‘cold’ and detached during mathematics instruction (Ashcraft & Krause,
2007). Research has also found mathematics anxiety and mathematics performance to be
negatively correlated (Lang & Lang, 2010), meaning that high mathematics anxiety can result
in poor performance on mathematics tasks. While examining the relationships among
anxiety, confidence, and performance, Lang and Lang (2010) found that confidence is linked
to a student’s level of anxiety which, in turn, influences a student’s performance. Students
who are confident in their ability to do mathematics, in particular, to solve problems,
demonstrate a level of decreased anxiety and an increased ability to do mathematics (Lang &
Lang, 2010), a positive relationship mathematics educators need to nourish in all students.
Research questions in this study included analyses of change in students’ performance in
mathematics (concepts, processes, and problem solving), change in anxiety towards
mathematics, and change in attitude towards mathematics. While three research questions
were traditionally stated, the researchers practiced integrated approaches to studying the
co-development of mathematical concepts, problem-solving processes, metacognitive
functions, and beliefs or dispositions.
THEORETICAL FRAMEWORK
The researchers adopted a models-and-modeling perspective (Lesh & Doerr, 2003) for
conducting this study, a perspective aligned with constructivist and sociocultural
philosophies about the teaching and learning of mathematics. The characteristics of using a
models-and-modeling perspective guided the design and implementation of the study. First,
the conceptual models used to study and understand mathematical problem solving were
continually under design. Second, the researchers organized the intervention around products
(PBL reports) that were developed to instill change in how students learned mathematics
through problem solving. Third, the study engaged the researchers in theory development
including specific methodologies and tools designed to investigate how students can learn
mathematics through problem solving. Fourth, the researchers started the study embracing a
goal of producing results that are generalizable. This goal will not be achieved with one study;
instead, the researchers plan a series of studies that will work towards products and tools that
can be used in other studies striving for an intervention that can be implemented in similar
classrooms.

Miller and Banes
In addition to having a models-and-modeling perspective, the researchers couched the
research design within the paradigms of situated cognition and communities of practice.
Situated cognition refers to teaching and learning mathematics and problem solving in a
context (Greeno, 1998). Closely aligned with situated cognition is the recognition that the
construction of knowledge is contextually based and socially situated. The ability of
individuals to function well within a group is a highly prized attribute of employees.
Pre-service elementary teachers need to experience problem solving within the dynamics of a
group in order to design similar, meaningful experiences for their future students.
THE INTERVENTION
The focus of this study was to investigate the role of problem-based learning (PBL) and
problem solving in the mathematical preparation of pre-service elementary teachers. The
researchers developed a number concepts course for pre-service elementary teachers taught
through PBL experiences and problem solving. Topics in the course included, but were not
limited to, concepts and procedures engaging whole numbers, common fractions, decimals
and percents, proportional reasoning, number theory including least common denominator,
greatest common multiple, and tests of divisibility, and converting nonterminating, repeating
decimals to rational numbers. The instructional design of the course provided a
student-centered learning environment in which students could construct knowledge about
elementary mathematics concepts and procedures using PBL experiences and problem
solving.
The researchers use the term student-centered to imply that students’ learning experiences
occur by working problems individually or in small collaborative groups. A PBL experience
consists of posing a scenario (an event that occurs in real-life) that can generate multiple,
related problems to solve. The principal idea behind PBL is that “the starting point for
learning should be a problem, a query, or a puzzle that the learner wishes to solve” (Boud,
1985, p. 13). PBL has been used for years in the United States in medical and nursing schools.
More recently, PBL has been used in a variety of classes including biology, physics, nutrition,
and teacher education programs. There is a dearth of literature on the use of PBL in
mathematics instruction.
A meaningful definition for problem solving for this study comes from Lesh & Zawojewski
(2007), “a task, or goal-oriented activity, becomes a problem to solve (or problematic) when
the ‘problem solver’ (which may be a collaborating group of specialists) needs to develop a
more productive way of thinking about the given situation.” This definition takes problem
solving beyond memorized strategies that result in an answer. It embraces the position that
people can learn mathematics through problem solving and that they learn problem solving
through creating mathematical models. The design of the number concepts course in this
study placed students in a situation to begin their learning experience by interpreting, sorting
out, integrating, modifying, revising, or refining clusters of mathematical concepts from
various topics within and beyond mathematics. Through mathematical problem solving,
students were introduced to and reviewed basic mathematics concepts and procedures with
the intent to construct knowledge that will be long lasting.

Miller and Banes
An example PBL is given below to familiarize the reviewer with the type of life-based
context problems that were presented.
One hundred twenty-one dogs were rescued from a [Local] County puppy mill in March
2011 by the Animal Rescue Corps (ARC). They were temporarily sheltered at the [Local]
Fairgrounds until a court hearing was held to determine if the dogs should be returned to
the owner or become wards of the ARC and put up for adoption. The dogs were held at the
Fairgrounds facility for nine days. The ARC asked volunteers to donate food, water, crates,
beds, bowls, towels, money, and time to care for the dogs until the court date. The owner
was charged ten dollars per dog per day for the care and services received at the
Fairgrounds. (Note: This was a real event that students had knowledge about through local
media outlets.)
A sample problem presented with this statement follows.
According to the [Local] Animal Hospital, adult dogs need annual vaccinations for
distemper, hepatitis, Leptospirosis, Parvovirus, and Para-influenza (abbreviated
DHLPPC); rabies (required by law); giardia (an intestinal parasite); Lyme disease; and,
should be tested for heart worms. Assuming 75% of the 121 dogs rescued are adults ( > 1
year old), what is the cost of vaccinating the adult dogs?
Data provided:
Vaccination Cost
DHLPPC $22.75
Rabies $15.00
Giardia $16.75
Lyme disease $18.50
Heart worm test $29.90
As students completed the first PBL experience, they questioned how they were learning
mathematics by solving the problems. This surprised the researchers because the students
were not making the connection between solving problems within a relevant context with
learning mathematics in a classroom context. To further demonstrate to the students how
much mathematics is embedded in a life-based context, the researchers created a single
problem to solve based upon the ages of the researchers. This problem is presented below.
[Researcher A’s] age divided by [Researcher B’s] age creates a rational number, namely
2.185, that is a nonterminating, repeating decimal (185 is the repetend). If 2.185 is written
as a common fraction in the form a/b, [Researcher A’s] age will be the numerator and
[Researcher B’s] age will be the denominator. How old is [Researcher A]? How old is
[Researcher B]?
After students and Researcher A solved this problem, a discussion followed that resulted in a
list of concepts, skills, and generalizations reflected in this problem. The list included the
following topics.
Definitions for rational number, nonterminating decimal, repeating decimal, repetend,
common fraction, integer, numerator, denominator, plus others

Miller and Banes
Why can a denominator NOT be zero?
Writing a nonterminating, repeating decimal in common fraction form, a/b where a &
b are integers and b ≠ 0.
Reducing fractions to lowest or simplest form
Finding a GCF (greatest common factor)
Tests of divisibility
Using tests of divisibility
Primes & composites
Arithmetic operations
METHODOLOGY
The researchers began this study with a focus on the effect the intervention strategy (learning
mathematics through PBL experiences and problem solving in a student-centered
environment) has on pre-service elementary teacher’s construction of content knowledge,
change in mathematics anxiety, and change in attitude towards mathematics. The initial study
looks at the “whole picture” to see if learning mathematics through problem solving and
learning problem solving through mathematics has an effect on these three domains before
drilling down to specific facets of the intervention that suggest cause-and-effect relationships.
The need to decrease the mathematics anxiety of pre-service elementary teachers and to
improve their attitudes toward mathematics has similar merit to the need to improve the
construction of content knowledge, the relationship among the three domains having
previously been documented.
The researchers began with the premise that pre-service elementary teachers in the United
States have negative attitudes toward mathematics, high anxieties about studying
mathematics, low confidence in their mathematics ability, and have a record of poor
performance in mathematics. Additionally, the researchers accepted the premise that these
attributes may largely be the result of the K-12 mathematics instruction they have
experienced, instruction that reflects mathematics as a collection of facts and processes to be
memorized, rather than understood, and practiced only in the classroom and not in life-based
contexts. The study investigated an intervention intended to reverse the negative cycle that
seems to pervade the mathematical preparation of elementary school teachers; one that
improves future teachers’ attitudes and beliefs about mathematics, reduces their anxieties
toward mathematics, and increases their content knowledge before they enter the classroom.
The intervention strategy implemented was teaching mathematics content through PBL
experiences and problem solving within a student-centered learning environment.
The study was conducted at a regional university in the southeastern United States of America
during fall 2011. To determine potential benefits of teaching through PBL experiences and
problem solving, the researchers completed comparative analyses between students from
both a traditional, lecture taught number concepts course (control group) and students from a
number concepts course taught using PBL experiences and problem solving (experimental
group) in a student-centered learning environment. The research questions follow:

Miller and Banes
Will university students preparing to become elementary school teachers learn basic
mathematics concepts, skills, and generalizations better by participating in a
student-centered learning environment utilizing PBL experiences versus a traditional
lecture methodology?
Will university students preparing to become elementary school teachers decrease
their mathematics anxiety by participating in a student-centered learning environment
utilizing PBL experiences versus a traditional lecture methodology?
Will university students preparing to become elementary school teachers improve their
attitudes toward mathematics by participating in a student-centered learning
environment utilizing PBL experiences versus a traditional lecture methodology?
The Mathematics Anxiety Rating Scale (MARS) (Suinn & Winston, 2003) was used as a pre-
and post-test instrument for both groups to document the change in mathematics anxiety from
the beginning of the semester to the end. The Attitudes Toward Mathematics Inventory
(ATMI) (Tapia & Marsh, 2004) was given as a pre- and post-test instrument to measure
changes in attitudes toward mathematics for both groups. A researcher designed test was used
to measure growth in the construction of content knowledge. The same test was used as a pre-
and post-test instrument to measure changes in content knowledge.
The experimental group started with thirty-eight students majoring in elementary or special
education. The number concepts course is one of two required mathematics courses in these
majors. Thirty-three students were female and five students were male. During the semester,
two females and one male dropped the course for personal reasons (8% loss) leaving
thirty-five subjects for final comparison to the control group. The control group (consisting of
two classes of the number concepts course) started with fifty-eight students majoring in
elementary or special education. Fifty-one students were female and seven students were
male. During the semester, eight students dropped the course for personal reasons (13% loss)
leaving fifty subjects for final comparison to the experimental group.
The unit of analysis for this study is a group, control group versus experimental group. The
researchers compared a traditional, teacher-centered lecture approach to teaching
mathematics to pre-service elementary teachers to a non-traditional, student-centered
approach that utilized problem-based learning and problem solving as the mode of
instruction. The research questions focused on whether learning and reviewing mathematics
in a meaningful context, that is, mathematics couched within life-based problems, and
working with other students to solve problems would decrease students’ anxieties towards
mathematics, improve their attitudes toward mathematics, at the same time they were
constructing knowledge about mathematics concepts and processes.
RESULTS
In the beginning of the fall 2011 semester both the control (labelled with a subscript of one)
and experimental group (labelled with a subscript of two) took pre-tests to assess their
beginning content knowledge (M1 = 15.68, M2 = 14.6, SD1 = 6.44, SD2 = 5.45), mathematics
anxiety (M1 = 68.94, M2 = 67.91, SD1 = 19.50, SD2 = 16.23), and attitudes (M1 = 126.92, M2 =
134.01, SD1 = 34.01, SD2 = 27.57). In order to compare the progress of the two groups, t-tests

Miller and Banes
on their pre-tests for content knowledge, mathematics anxiety, and attitudes were performed
to test the null hypothesis that there is no difference between the means of the groups’ scores
on the pre-tests respectively. No differences were found on content knowledge (t(83) = .809,
p = .421), anxiety (t(83) = .255, p = .799), and attitudes (t(83) = -1.021, p = .310).
To determine if the groups were different at the end of the semester t-tests were performed on
the groups’ post-test scores on content knowledge (M1 = 29.1, M2 = 26.71, SD1 = 5.74, SD2 =
5.03), mathematics anxiety (M1 = 67.68, M2 = 71.69, SD1 = 19.23, SD2 = 19.99), and attitudes
(M1 = 125.16, M2 = 121.23, SD1 = 34.73, SD2 = 25.89). The null hypothesis was that there was
no difference between the means of the groups’ scores on the post-tests respectively. No
differences were found on content knowledge (t(83) = 1.982, p = .051), anxiety (t(83) = -.930,
p = .355), and attitudes (t(83) = .568, p = .572).
To determine if one group improved significantly more than the other, a new variable was
created for each of the instruments used. The new variable was the change from pre-test to
post-test. These variables were named content knowledge change (M1 = 13.42, M2 = 12.11,
SD1 = 4.75, SD2 = 6.07), anxiety change (M1 = -1.26, M2 = 3.77, SD1 = 15.87, SD2 = 15.43),
and attitude change (M1 = -1.76, M2 = -12.79, SD1 = 24.76, SD2 = 22.88). A t-test was
performed to test the null hypothesis that there was no difference between the means of the
groups’ change variables respectively. No differences were found on content knowledge
change (t(83) = 1.112, p = .270) and anxiety change (t(83) = -1.455, p = .149). The
experimental group’s attitude towards mathematics decreased during the semester and their
attitude change was statistically different from the control group (t(83) = 2.084 , p = .04)
Problem solving and the ability to analyze information and communicate solutions were not
included in the research questions, but the researchers thought that a comparison of problem-
solving ability and communication of one’s solution should be variables of interest. At the
end of the semester a short problem-solving assessment, consisting of three “non-routine” and
context based problems, was given to both groups (M1 = .94, M2 = 1.3714, SD1 = .62, SD2 =
.73). The results were recorded and a t-test was performed to test the null hypothesis that there
is no difference between the means of the groups’ problem-solving scores. The experimental
group was significantly better on these three problem-solving tasks (t(83) = -2.932, p = .004).
CONCLUSIONS
Problem-based learning experiences were found to be equally effective as traditional
instruction for students’ construction of content knowledge in a number concepts course for
pre-service elementary teachers. Considering the mathematical struggles of elementary
teachers in the United States, the researchers feel that “equally effective” should not be the
goal of new instructional methods; however, we have shown that PBL and problem solving
can be used without harm to students’ acquisition of content knowledge.
Both groups showed little change in their mathematics anxiety. The number of highly
mathematically anxious students was low in both groups and this could partly explain the
overall lack of a decrease in mathematics anxiety. That is, students who are not
mathematically anxious will show less or no response to an intervention. However, the aim of

Miller and Banes
mathematics preparation programs for pre-service elementary teachers should be to lower
mathematics anxiety.
Students working in a collaborative, problem-based learning environment showed a
significant decrease in their attitudes toward mathematics. The researchers speculate that this
result may be due to students’ negative attitudes toward the additional effort that they had to
exert in the PBL experiences. Future research is needed to determine if a cause-and-effect
relationship exists.
The most positive result of this pilot study was that students learning through PBL
experiences were better problem solvers than those students learning via a traditional lecture
approach.
LIMITATIONS
The researchers acknowledge certain limitations of this study. Specifically, the number of
participants is low and the selection of participants was a sample of convenience rather than
random assignment. Thus, any conclusions made from this study may not be generalizable to
the entire population of pre-service elementary teachers in the United States.
Collecting and analyzing only numerical data are additional limitations of this study. Data
sets including students’ responses to directed writing prompts as a means to solicit their
thoughts while solving problems and samples of students’ work as problems are attempted
and eventually solved will create a rich source for qualitative analyses. Future studies
including a qualitative component for data collection and analyses will further the
researchers’ ability to develop theories around how students learn mathematics through
problem solving.
FUTURE RESEARCH
The researchers are continuing to study the effectiveness of teaching pre-service elementary
teachers through problem-based learning experiences and problem solving. In spring 2012,
the researchers conducted a second pilot study in another section of the same number
concepts course. The objectives of the second study were: to implement PBL experiences
used in fall 2011, with revisions; to create and pilot new PBL experiences; and, to collect
student work and responses to directed writing prompts to begin discussions about how
analyses of qualitative data sets can contribute to the development of a theoretical framework
for how pre-service elementary teachers learn mathematics through problem solving. The
same pre- and post-test measures for attitudes, anxieties, and content knowledge were
collected. The spring 2012 study did not include a control group; thus, the researchers are
only looking at students’ change in attitudes, anxieties, and content knowledge. At the time
this paper was submitted, the results of the spring 2012 study were unavailable.
References
Ashcraft, M., & Krause, J.A. (2007). Working memory, math performance, and math anxiety.
Psychonomic Bulletin & Review, 14(2), 243-248.
Boud, D. (1985). Problem based learning in perspective. PBL in Education for the
Professions, 13.

Miller and Banes
Greeno, J. (1988). The situativity of knowing, learning, and research. American Psychologist,
53(1), 5-26.
Gresham, G. (2010). A study exploring exceptional education pre-service teachers’
mathematical anxiety. IUMPST: The Journal, 4, 1-14.
Jacobbe, T., & Millman, R.S. (2009). Mathematical habits of the mind for preservice
teachers. School Science and Mathematics, 109(5), 298-302.
Lang, J.W.B., & Lang, J. (2010). Priming competence diminishes the link between cognitive
test anxiety and test performance: Implications for the interpretation of test scores.
Psychological Science, 21(6), 811-819.
Lesh, R., & Doerr, H. (2003). Foundations of a models and modeling perspective on
mathematics teaching, learning, and problem solving. In R. Lesh & H. Doerr (Eds.),
Beyond constructivism: Models and modeling perspectives on mathematics problem
solving, learning and teaching (pp. 3-34). Mahwah, NJ: Erlbaum.
Lesh, R., & Zawojewski, J. (2007). Problem solving and modeling. In F.K. Lester, Jr. (Ed.),
Second handbook of research on mathematics teaching and learning (pp. 763-804).
Charlotte, NC: Information Age Publishing. (A project of the National Council of
Teachers of Mathematics)
Malinsky, M., McJunkin, M., Pannells, T., & Ross, A. (2004). Math anxiety in pre-service
elementary school teachers. Education, 127(2), 274-279.
National Council of Teachers of Mathematics. (1989). Curriculum and evaluation standards
for school mathematics. Reston, VA: Author.
National Council of Teachers of Mathematics. (2000). Principles and standards for school
mathematics. Reston, VA: Author.
Polya, G. (1945). How to solve it: A new aspect of mathematical method. New York:
Princeton University Press.
Schoenfeld, A. H. (1992). Learning to think mathematically: Problem solving,
metacognition, and sense making in mathematics. In D. Grouws (Ed.), Handbook of
research on mathematics teaching and learning (pp. 334-370). New York: McMillan.
Suinn, R., & Winston, E.H. (2003). Mathematics Anxiety Rating Scale (short version).
Tapia, M., & Marsh, G.E. II. (2004). An instrument to measure mathematics attitudes.
[Electronic Version]. Academic Exchange Quarterly, 8(4).
Timmerman, M.A. (2004). The influences of three interventions on prospective elementary
teachers’ beliefs about the knowledge base needed for teaching mathematics. School
Science and Mathematics, 104(8), 369-380.

12th
8 July – 15 July, 2012, COEX, Seoul, Korea (This part is for LOC use only. Please do not
change this part.)
abcde
ELEMENTS TO STIMULATE AND DEVELOP THE PROBLEM
POSING COMPETENCE OF PRE SERVICE AND IN SERVICE
PRIMARY TEACHERS
Uldarico Malaspina
Pontificia Universidad Católica del Perú
umalasp@pucp.edu.pe
Cecilia Gaita
cgaita@pucp.edu.pe
Vicenç Font
vfont@ub.edu
Universidad de Barcelona
Jesús V. Flores
jvflores@pucp.pe
This paper presents exploratory studies on the competence of creating problems within the
framework of a project to be developed extensively in 2012, using the onto-semiotic approach to
cognition and mathematics instruction (OSA) as the theoretical framework. It describes
the importance of competence to create problems in mathematics teachers at all educational
levels and particularly in primary education. Studies with students of the Faculty of Education
and students of the Masters in the Mathematics Teaching Program at the Pontificia
Universidad Catolica del Peru, provide elements to make proposals to stimulate and develop the
skills that require pre service and in service primary teachers to create and propose math problems.
From the results obtained in this first stage and the contributions of other researchers in this line of
thought, we formulate some proposals.
Key words: posing problems, creating problems, teacher competences, didactical suitability.

Malaspina, Gaita, Font & Flores
INTRODUCTION
We consider that is very important to study and investigate the creation and the proposal of
problems, in parallel with problems solving. This issue is fundamental because of the
following points: 1) The contributions of research about problem solving in
learning mathematics can not be realized if problems are not good and not suitable; 2) Not
only there is lack of problems in textbooks – usually there are just exercises – but the
few problems that there are hardly correspond to the specific needs of the teachers
who develop their teaching in particular contexts and seek to encourage the learning
of students with particular experiences and motivations; and 3) Discovery learning leads the
child to imagine situations that the teacher could use creatively, making “new problems” that
may encourage both the understanding of the concept being treated and the development
of child’s self-esteem.
We agree with Malaspina (2011b) that the activity of creating mathematical problems
complements problem solving well because it encourages further creativity and
contributes to the clarification of the problem situation, language, concepts,
propositions, procedures and arguments, expected to be handled by students within
an appropriate epistemic configuration.
In addition, the task of creating problems should not be an exclusive activity of teachers, but
should also be an activity of their students. Teachers should stimulate their students as part of
the activities in problem solving: seeking variations to a given problem, special cases,
generalizations, connections and contextualization. This generates an interesting dynamic in
classes because it usually leads to new difficulties, created by the students, that require the
introduction of new concepts and techniques to overcome the difficulties, or an awareness of
the limitations of mathematical resources available and the importance of learning new
fields of mathematics.
In this paper we show the first results obtained in relation to the meanings that teachers give
in regards to the role that problems play in the mathematics teaching, the characteristics that
the problems must possess to be considered "good problems" and the role that the teachers
assign to the task of creating problems as a part of their professional practice.
FRAMEWORK
There are various theoretical approaches to investigate the creation of
problems (Silver 1994; Charalambos, C., Kyriakides, L. & Philippou, G., 2003, Chua
and Yeap, 2008). Stoyanova (1998) proposes three types of situations related to the
creation of problems: a) free situations, b) semi-structured situations c) structured
situations. The first relates to situations without any restrictions, for example, when asked to
propose problems for mathematical Olympiads; the second refers to situations where
students must propose a problem by having some information; and the third refers to
situations where students create problems by reformulating other problems that have already
been solved, or by varying the conditions or questions of given problems.
On the other hand Christou, Mousoulides, Pittalis, Pantazi and Sriram (2005)

ICME-12, 2012
abcde+2
propose a model in which the creation of problems activates several processes: editing
quantitative information, selecting quantitative information, comprehending quantitative
information and translating quantitative information. This framework provides elements for
the analysis of cognitive processes that come into play when creating problems.
In order to make a holistic analysis of the teaching of mathematics, we have considered
the Onto-Semiotic Approach to knowledge and mathematics education (OSA)
(Godino, Fuller and Font, 2007; Font, Planas and Godino, 2010). Here, mathematical activity
plays a central role and is modeled in terms of systems of operative and discursive practices.
From these practices different types of related primary objects (language, arguments,
concepts, propositions, procedures and problems) emerge, building epistemic or cognitive
(depending on whether the adopted point of view is institutional or personal) configurations
among one another (see hexagon in Figure 1).
Figure 1. An onto-semiotic representation of mathematical knowledge
Problem-situations promote and contextualize the activity while languages (symbols,
notations, graphs) represent the other entities and serve as tools for action, and arguments
justify the procedures and propositions that relate the concepts. Finally, the objects that
appear in mathematical practices and those which emerge from these practices depend on the
“language game” (Wittgenstein, 1953) in which they participate, and might be considered
from the five facets of dual dimensions (decagon in Figure 1): personal/institutional,
unitary/systemic, expression/content, ostensive/non-ostensive and extensive/intensive. Other
theoretical elements of OSA are the suitability criteria of mathematical instruction process:
epistemic, cognitive, interactional, mediational, emotional and ecological suitability. We
believe that these suitability criteria may be used to assess the didactic quality of a problem as
it provides explicit assessment elements that go beyond purely subjective considerations.
Thus, based on experiences in different educational levels, Malaspina (2011a) has identified

some characteristics that “good” problems should have from the didactical point of view,
considering the suitability criteria of the OSA.
a) It is not very difficult and the solution is seen to be attainable by the students
(cognitive suitability).
b) It favors discovering an intuitive way to get the solution or conjecturing a solution
(interactional, emotional and cognitive suitability)
c) In order to accept or reject the conjectures, it favors making some verifications
with the eventual help of calculators or computers (interactional and meditational
suitability)
d) Solving the problem is perceived as interesting and useful (emotional and
ecological suitability).
e) It enables mathematical connections among various mathematical topics and real
situations or other fields of knowledge (epistemic and ecological suitability).
f) What the problem is about is clearly perceived (to determine something, to prove,
to show, etc) (interactional and cognitive suitability).
g) It favors the use of logical relationships rather than the mechanical use of
algorithms (epistemic suitability).
h) It encourages the creation of new problems by making some changes that lead to
significant situations, both didactically and mathematically (epistemic suitability)
Malaspina states that the epistemic suitability has to do with "doing mathematics". In this
sense, he associates the epistemic suitability with e, g and h since the establishment of math
connections, the use of logical relationships and the creation of new problems are essential in
mathematical activity.
On the other hand, interactional competence has to do with the "path" that will overcome the
difficulties in finding the solution. In this sense, there is an association with b, c and f. While
f allows seeing the beginning of the path, in b we have the path, and in c this path is built.
Beside the link that characteristic b has with interactional suitability, there is a link of this
characteristic with the emotional suitability too. Since there is a path established to solve the
problem, there would not be frustration. It is also, to some extent, cognitive, because it opens
the possibility to solve the problem.
Finally, as ecological suitability has to do with the educational project and with the context, it
is linked to d and e.
OBJECTIVES
To identify the meanings that pre service and in service teachers give in regards to the role
that problems play in the teaching of mathematics and to the role of creating problems in the
teaching practice.
To relate the didactic suitability criteria with the characteristics that teachers think good
problems should have.

ICME-12, 2012
abcde+2
METHODOLOGY
In a first stage, a problem situation was designed to identify how important the activity of
creating problems from their own experience is for a mathematics teacher.
This problem situation was first applied to preservice teachers. A second time it was applied
to in service teachers. We performed the contrast between the answers given by these two
groups. In a second stage, a questionnaire was designed and administered to in service
teachers. The aim was to obtain elements to characterize the problems to be created for the
primary level, and to collect items about the presence of the activity of creating problems in
teaching; in particular, about spontaneity to create problems from specific situations (an intra
mathematical situation and a contextualized situation)
DESIGN, IMPLEMENTATION AND RESULTS
The problem situation
We designed a problem situation in which the context was an everyday experience. A didactic
experience of a primary teacher with a proactive attitude to create problems was presented.
The problem involves simple arithmetic and is open to various interpretations because of lack
of clarity in its statement.
The problem situation presented is as follows:
Teacher Ramirez went to buy some bread one day. From his experience, he invented the
following problem and proposed it to his fifth grade students of primary school.
In the neighborhood bakery, six loaves of bread are sold for S/. 1.00, but each loaf costs S/.
0.20. Mary bought 9 loaves of bread and paid S/. 2.00. How much change should she receive
back?
Then, some students’ answers are given:
Peter: S/. 0.20; Daniel: S/. 0.50; Carmen: S/. 0.70; Juan: S/. 0.40; and Zoila: it cannot be
known.
In relation to this problem, it was asked to solve the following questions:
a) Enter a comment that you would make to each of the students for his or her answer.
b) Make a comment freely on
o What it was done by teacher Ramirez.
o The problem
c) Write some suggestion you would make for teacher Ramirez
This problem situation was proposed to students of primary school teaching courses and
mastery students in mathematics education, at different times. Their answers allowed us to
determine, in a direct or indirect way, different aspects related to the creation of problems

such as the assessment of creating problems to be brought to the classroom, the assessment of
the connection between problems and the everyday context, the opening to admit several
correct solutions to a problem as a consequence of the different interpretations given to a
statement; and the willingness to spontaneously propose a better statement as a way to
express willingness to create problems.
In table I, we present some quantitative results about the answers of undergraduate students
(pre service teachers) and graduate students (in service teachers).
Table 1: Answers of pre service and in service teachers
Pre service
teachers
(pre grade
students) n = 34
In service
teachers
(Mastery
students) n = 14
a) Value the fact that the teacher creates a
problem 26% 33%
b) Value the fact that the problem considers an
extra mathematical context.
56% 50%
c) Consider the problem to have an unclear
statement
65% 33%
d) Make suggestions to improve the problem 24% 8%
e) Consider only one of the given answers to be
the correct one.
70% 42%
Some comments:
It is noteworthy that the percentage of in service teachers who states a lack of clarity of the
problem is almost half the percentage of pre service teachers. In this sense, the pre service
teachers appear to be more critical than in service teachers regarding the clarity of the text.
This suggests that in the in service teachers a strong predominance of a pre established
institutional meaning is manifested over personal meanings.
On the other hand, since the 67% of in service teachers do not state that there is lack of clarity
on the statement of the problem, one would expect a similar percentage to solve the problem
by obtaining only one answer as correct. However, 42% of this group consider that there is
only one correct answer. Indeed, 25% believe that none of the answers given by students is
correct (this information is not listed in the table). Moreover, 70% of pre service teachers
consider that only one of the answers to be correct and consider Juan´s response as the most
accurate (96% of them said that the change that Mary should receive is S/. 0.40) .
In both groups, less than a half of each group explicitly express their appreciation for the
creation of the problem by the classroom teacher (26% in the pre service teachers and 33% in
the in service teachers). In the free comments they are asked about what was done by the
teacher. The majority in both groups, refer particularly to the problem itself or

ICME-12, 2012
abcde+2
contextualization without giving more importance to the fact that a teacher gives his or her
students a problem created by him or her that is inspired by an everyday experience.
The questionnaire
In the second phase, a questionnaire with more specific questions related to the creation of
problems was developed. In addition, two specific suggestive situations to create problems
were presented without being explicitly asked to do so. In this way, the questionnaire sought
to investigate whether the creation of problems could arise spontaneously.
One situation corresponds to an intra mathematical or formal context and another corresponds
to an extra mathematical or real context. However both of them, according to the
classification made by Stoyanova (1998), are semi-structured situations, because they
provide a context from which, problems would be created.
In table II the questionnaire is shown. Items 7 and 8 correspond to the suggestive situations.
Table II. The questionnaire
1. What is the source of most of the problems you present to your students in the development of
the course you teach? (books, handouts, internet, etc)
2. What is the main use that should be given to the problems of mathematics when developing a
course of mathematics in basic education?
3. Write three words or a short phrase that express what you think a mathematics problem should
have in order to facilitate learning of this subject.
4. List three conditions that favor the creation of math problems for elementary school teachers,
to be used in their classes.
5. If you think that the creation of problems by primary teachers for their classes would have
some advantages, list some of these advantages.
6. List three fundamental characteristics that you think a problem created for a math primary
class should have.
7. It is said that a natural number is a capicua if it remains the same when it is read from left to
right and from right to left. Thus, the number 363 is a capicua. It is also observed that the sum
of its digits is 12. How would you use this with primary students? Give a specific example.
8. There are two boxes of the same shape and size. In one of them 8 oranges can fit while in the
other box 12 oranges can fit. How would you use this with primary students? Give a specific
example.
This questionnaire was proposed to in service teachers. Their answers allowed identifying
certain characteristics that they consider that good problems must have.
Some of the results
Thirteen of fourteen teachers said that the problems they used in their classes are transcribed
from some source. Considering the classification made by Stoyanova, only one of them

declares to have a creative attitude towards semi-structured situations, and sometimes, in free
situations. However, that same teacher proposes an exercise with a direct solution from a semi
structured situation (intra mathematical) given in item 7 and does not propose an exercise or
problem from a semi structured situation (contextualized), given in item 8.
Regarding the characteristics that good problems should have for the learning, the answers (to
item 3) are dispersed. A total of sixteen characteristics are mentioned and from all of them, to
be contextualized is the most frequent (6). Three of fourteen teachers mentioned to be
motivating as one characteristic.
About the assessment given to the creation of problems, seven of fourteen teachers said that
one advantage of creating problems for their classes would be that the problems would be
adapted to the level of their students. Two of them said it would improve learning.
In relation to the characteristics that good problems created for a mathematics class in
elementary school should have, answers (to item 6) are very disperse. They mentioned a total
of eighteen characteristics. Out of these eighteen, the characteristics that the teachers
mentioned most frequently (six) were those characteristics of having a clear text and being
motivating to students. Five of fourteen teachers mentioned the characteristic of being
contextualized.
In the situation described in item 7 (semi structured and intra mathematical), three teachers
proposed a problem: (One of the problems was “Given the number such that the addition
of its digits is 15, what is the maximum value of ab?) Three teachers proposed an exercise that
has an immediate solution (for example, “Write a number of three digits in which the digit on
the hundreds place is equal to the digit on the units place”). Two other teachers suggested
activities related to the situation and one other teacher wrote a trivial text which seems to be a
problem but with some writing deficiencies (If is a capicua number with , what
condition should a and b have in order to be a capicua number?). Another teacher gave
evidence of having tried to create a problem and other four teachers did not answer.
We noticed that providing quantitative information in a mathematical context invites teachers
to create at least one routine exercise which reflects some competence to understand and
organize the mathematical information received. However, only three of fourteen teachers
proposed problems. This is indicative of deficiencies in creating problems.
In the situation described in item 8 (semi-structured, contextualized), two teachers tried to
propose a problem (for example: “Find the ratio between the sum of the areas of both boxes if
the radius of the oranges that are in the first box is twice the radius of the oranges in the
second box”), three teachers proposed immediate answer questions (Are all oranges of the
same size?); one teacher proposed an activity related to the situation and eight teachers did
not answer.
We noticed that this problem situation presents a major challenge compared to the previous
one. This could be because the context requires the creator of the problem to develop in
translating quantitative information process, according to the model of Christou et al (2005)

ICME-12, 2012
abcde+2
which requires creativity, as well as an adequate mathematic background, to edit the
information and propose a relevant question.
FINAL CONSIDERATIONS
In relation to the criteria of suitability for the analysis, we have found that for both groups,
(pre service and in service teachers), the problem presented (related to loaves of bread) has
cognitive and affective suitability. Both groups considered the problem to be solvable by
students and to be a problem that students would have an interest in solving.
One aspect we found positive for the development of competence to create problems is that a
significant percentage of both groups welcomed the fact that the problem proposed to the
students is contextualized (in the problem situation as well as in the questionnaire). This fact
is considered in the epistemic suitability of the problem, since it allows identifying a
mathematical conception about the mathematical task based on experience-“doing
mathematics”- and this is going to be associated to the importance of creating problems from
everyday situations.
The fact that in both groups there was recognition that the problem situation presented by the
teacher is not clear enough is also favorable to the competence of creating problems. We
believe that as a part of the competence to create problems is to recognize that the text of the
problem should express the conditions established and the asked questions in a clear way.
In relation to actions that are aimed to develop the competence to create problems, a warning
about the predominance of institutional meaning over personal meaning must be considered.
It will be essential to keep in mind the role that language plays as well as the analysis
according to the dual aspect of OSA personal / institutional. One hypothesis to investigate
further is that having been trained under a didactic contract with a high predominance of
institutional meanings over personal ones and without appropriate interactions, makes that a
critical spirit is not strong and then, despite having noticed deficiencies in the clarity of the
text of a problem, it is considered that there is an interpretation that the student should use to
solve it.
The low value given to the creation of problems is consistent with the lack of initiative of the
teachers in suggesting improvements to the proposed problem, despite having noticed the
problem’s lack of clarity.
In general, there is a passive attitude of teachers towards the creation of problems. Because
of that, it is vital to find ways to positively modify that attitude. In this respect, it is important
to encourage the use of ecological suitability criteria in all teaching activity, so that by
creating these problems, these ones pick up specific aspects of the classrooms.
Finally, we consider that an essential condition to create problems is to have sufficient
knowledge of the corresponding mathematical topic and its relationship with other contexts

(epistemic and cognitive competence). Also, the competence of creating problems has to be
encouraged from courses in pre service training and it must be cultivated continuously by
developing workshops that integrate the mathematical, creative and didactical aspects in a
participative way.
Acknowledgements
This study was conducted under the research project EDU 2009-08 120 and the research
project 53133-IREM-PUCP 2012
References
Charalambos, C., Kyriakides, L., & Philippou, G. (2003). Testing a comprehensive model for
measuring problem solving and problem posing skills of primary pupils. In N.A. Pateman,
B.J. Dougherty & J. Zilliox (Eds.), Proceedings of the27th PME, vol 2 (pp. 205-212).
Honolulu : PME.
Chua, P. and Yeap, B. (2008) Problem posing performance of grade 9 students in Singapore
on an open-ended stimulus. TSG 19. Mexico: ICME 2008.
Christou, C., Mousoulides, N., Pittalis, M., Pantazi, D.P., y Sriraman, B. (2005). An empirical
taxonomy of problem posing processes. Zentralblatt für Didaktik der Mathematik, 37(3),
Font, V., Planas, N., and Godino, J.D. (2010). Modelo para el análisis didáctico en educación
matemática [A model for didactic analysis in mathematics education]. Infancia y
Aprendizaje, 33(1), 89-105.
Font, V. (2011). Competencias profesionales en la formación inicial de profesores de
matemáticas de secundaria. Revista Iberoamericana Unión, 26, 9-25.
Godino, J. D., Batanero, C. and Font, V. (2007). The Onto Semiotic Approach to Research in
Mathematics Education. ZDM-The International Journal on Mathematics Education,
39(1-2), 127-135.
Malaspina, U. (2011a) Resolución de problemas y estímulo del pensamiento optimizador en
la educación básica. Conferencia. Brasil: Actas de la XIII Conferencia Interamericana de
Educación Matemática.
Malaspina, U. (2011b). Intuición y resolución de problemas de optimización. Un análisis
ontosemiótico y propuestas para la educación básica. Alemania: Lap Lambert Academic
Publishing GMBH & Co.KG -Editorial Académica Española.
Silver, E. A. (1994). On mathematical problem posing. For the Learning of Mathematics,
14(1), 19-28.
Stoyanova, E. (1998). Problem posing in mathematics classrooms. In A. McIntosh & N.
Ellerton (Eds.), Research in Mathematics Education: a contemporary perspective (pp.
164185). Edith Cowan University: MASTEC.
Wittgenstein, L. (1953). Philosophical investigations. New York: The MacMillan Company.

12th
8 July – 15 July, 2012, COEX, Seoul, Korea (This part is for LOC use only. Please do not change this part.)
abcde
PROBLEM SOLVING IN AND BEYOND THE CLASSROOM:
PERSPECTIVES AND PRODUCTS FROM PARTICIPANTS IN A
WEB-BASED MATHEMATICAL COMPETITION1
Hélia Jacinto
Research Unit of the Institute of Education
of the University of Lisbon
Susana Carreira
FCT, University of Algarve & Research
Unit, Institute of Education, Univ. of Lisbon
scarrei@ualg.pt
Being aware that school is just one of many places where youngsters learn, the research community
has been stressing the need to deepen knowledge about the role and relevance of beyond school
mathematical learning contexts. In this paper we are focusing on the mathematical problem solving
activity that occurs at Sub14 - a beyond school web-based competition. Our purpose is to describe
and understand how do participants engage in solving mathematical problems within the
competition, and what are their perspectives regarding the differences and the similarities of the
problem solving activity in the mathematics classroom and in the competition Sub14. Participants’
perspectives show an overlap between mathematical activity developed in those two contexts. The
analysis of participants’ productions illustrate how they merge knowledge about problem solving
acquired in this beyond school context with mathematics content knowledge, learnt at school.
Keywords: mathematical problem solving; beyond school learning; web-based competition.
INTRODUCTION
Portuguese schools provide a large number of mathematical competitions for students to
enroll in, according to their personal preferences and adding up to all class oriented activities.
In this paper we discuss how a particular mathematical competition is influencing the
participants’ problem solving s s in the south of Portugal. Sub14 is a web-based
mathematical problem solving competition, addressed to 13-14 years old students, and it is
organized by the Mathematics Department of the Faculty of Sciences and Technology of the
University of Algarve (http://fctec.ualg.pt/matematica/5estrelas/).
Sub14 comprises two stages. The Qualifying consists of ten problems, each one posted online
every two weeks. The Organizing Committee posts every new problem and provides updated
information on the website, which also allows participants to send their answers using a
simplified text editor and attach a file of any format, containing their productions on solving
the problems. The participants may solve the problems working alone or in small teams, using
their preferred methods and tools, but they must send their solution and a complete
explanation of their reasoning. The Committee assesses every answer and replies to each
participant with some constructive feedback about the given answer. Participants are
welcome to complete or improve their answers, according to the feedback received.
Participants must have answered correctly to, at least, eight problems, to attend to the Final.

Jacinto, Carreira
In this paper we will focus on part of a broad research study, anchored in the Sub14’s
mathematical problem solving activity, and underline some implications of the results
obtained which are triggering new paths for research. Particularly, we will present the
perspectives of the participants regarding the differences and the similarities of the problem
solving activity, which they engage in at the mathematics classroom and at Sub14. There are
four focuses in the theoretical approach that we used: (a) looking at mathematics as a human
activity, (b) taking problem solving as an environment to develop mathematical thinking and
reasoning, (c) exploring the concept of being mathematically and technologically competent
and finally (d) considering the role of home technologies in beyond school mathematics
learning. In this paper, we will focus on mathematical knowledge and problem solving skills,
fostered both in the classroom and in the online mathematics competition – Sub14.
THEORETICAL BACKGROUND
Mathematical knowledge and problem solving
As Stanic and Kilpatric (1989) stated, “problems” have been a constant in the mathematics
curricula, but only in recent decades the debate has arisen regarding “problem solving” and its
importance for the learning of mathematics (NCSM, 1978). The work that has most inspired
the development of problem solving theories is undoubtedly that of George Polya. Ever since,
problem solving has inspired a multitude of studies, with just as many purposes, and a
common thought: the notorious importance of improving the ability to solve problems to the
development of mathematical skills (APM, 1988/2009; APM, 2007; ME, 2007).
Polya (1978) described problem solving as the art of discovery. He considered that a student
is doing mathematics, when solving a problem that “challenges his curiosity and evo his
creative abilities” (p. V). Polya’s widely own problem solving model emphasizes the
importance of spotting and exploring a brilliant idea, through a questioning stand that allows
looking at the problem from different perspectives. Such model comprises four stages: (i)
understanding the problem – the solver should read and analyse the problem carefully,
interpret and identify some relevant aspects such as the data or the conditions; (ii) devising a
plan – the solver must decide which is the best strategy for that specific problem; (iii) carrying
out the plan – the solver must be persistent in executing the devised plan; (iv) retrospective
analysis – the solver should reconsider and review the path taken and check the solution.
In addition to Polya’s model, we considered three other problem solving models, due to its
relevance: the three-step strategy proposed by Schoenfeld (1985); the model proposed by
Lester, Garofalo and Kroll (1989, quoted by De Corte, 2000), which establishes the cognitive
processes involved in problem solving; and the five stages model of self-regulated learning
created by Verschaffel (1999). A brief analysis, focusing on the processes included in these
four models, reveals that the different phases almost match each other. The main differences
between these models lies in the theoretical assumptions and motivations that support them
but, at the level of the structure and organization of processes, the distinction lies in the extent
and refinement of each phase of each model. The model presented by Polya seems to be the
one that best captures the consistency of the ideas proposed by the other authors.

Jacinto, Carreira
Problem solving as a major activity for the learning of mathematics
Learning to solve problems is the main reason to study mathematics, as claimed by the NCSM
in 1978. In fact, “problems” have taken a prominent place in school mathematics (Stanic &
Kilpatrick, 1989) for such a long time that, at the heart of the western society, endures the idea
that a good student is necessarily a good problem solver. In Portugal, in line with international
trends, problem solving is seen in curricular documents as a of “major activity in
mathematics” (APM, 1988/2009, p. 42), for “students to strengthen, broaden and deepen their
mathematical edge” (ME, 2007, p. 6).
This view on mathematical problem solving entails a conception of mathematical knowledge
that is not reducible to proficiency on facts, rules, techniques, and computational skills,
theorems or structures. It moves towards broader constructs that entails the notion of
mathematical competence (Perrenoud, 1999; ME, 2001) and problem solving as a source of
mathematical knowledge. In solving a problem there are several cognitive processes that have
to be triggered, either separately or jointly, in pursuing a particular goal: to understand, to
analyse, to represent, to solve, to reflect or to communicate. One of the purposes of
mathematical problems should be to introduce and foster mathematical thinking (Schoenfeld,
1992), adopting a mathematical point of view, which impels the solver to mathematize: to
model, to symbolize, to abstract, to represent and to use mathematical language and tools.
Many authors suggested several approaches to teaching problem solving, thus, it can be seen
in various forms and its teaching can assume several functions in the mathematics classroom.
Hatfield (1978) presented three perspectives: teaching for problem solving – which focuses
on the acquisition of mathematical tools (algorithms, techniques and procedures) that are
obviously useful to solving problems; teaching about problem solving – focused on the
teacher and how to guide students in processes that lead to the solution, often using Polya’s
model; and teaching through problem solving – that focuses on the discovery of new content
or mathematical techniques, resulting from the problem solving activity, i. e., mathematical
concepts are introduced with a problem (Schroeder & Lester, 1989). This latter perspective
reflects the kind of teaching that Polya and other researchers emphasize in their work.
Sub14’s mathematical problem solving activity
Since Sub14 is a competitive activity, the Organizing Committee recognizes its importance in
facilitating mathematical problem solving and in complementing the curricular tasks.
Therefore, the Committee encourages creativity at several levels namely in the reasoning,
finding a strategy, or communicating those ideas. However, every participant must present a
complete and detailed explanation of the reasoning developed during the solving process, and
such mathematical thinking should be as clear as possible.
Another Sub14’s feature is the nature of the feedback sent to the participants by the
Committee aimed at appreciating their ideas and stimulating self-correction (Jacinto, Amado
& Carreira, 2009). Such feedback may complement the work that teachers provide in their
classrooms regarding problem solving. Some teachers neglect the creativity involved in
finding a path and, sometimes, student’s self-confidence is shattered to the point that he is not
fully able to appreciate mathematics and its nature.

Jacinto, Carreira
The problems posed by the Committee are “word problems”, in the sense that Borasi (1986)
states: the context is fully expressed in the statement, the formulation of the problem is unique
and explicit; the desired solution is almost exclusively unique and precise, and the strategic
approach involves combining several algorithms or techniques known (p. 134). Although
there is not a strong concern in following the syllabi and its contents, the Committee considers
the mathematical knowledge needed to solve each problem, since Sub14 is addressed at
students from two different school levels.
METHODOLOGY
This paper reports part of a larger study focused on the mathematical competition described
previously, and that had as major objective the understanding of the participants’ perceptions
regarding (i) the mathematical activity, (ii) the competences involved in taking part in the
competition and (iii) the role of the technological tools they have used. In particular, this
study aimed at understanding how the students engaged in solving mathematical problems
within the competition, and what were their views on problem solving when comparing
school to the competition environment.
The methods we used in that larger study were essentially qualitative, since we were looking
for the participants’ perspectives regarding their activity while involved in Sub14 and to their
particular approaches to the problems proposed. The data collecting involved the gathering of
documental data, such as the electronic files containing solutions from participants of several
editions of the competition and the feedback sent to the students, by e-mail. We then
administered a questionnaire to 86 participants who attended the Final, aiming at a broader
identification of their personal views regarding their habits of using ICT, on their attitudes
towards mathematics, problem solving and their participation in the Sub14.
We also applied semi-structured interviews to eleven of the Finalists. We selected those
interviewees intentionally in order to obtain diversity of perspectives, since they showed
different problem solving and technology skills along the qualifying stage of the competition.
The interview aimed at bringing out the competitors’ perceptions regarding mathematics,
problem solving and the use of ICT in two different settings: the classroom and the Sub14.
This paper presents data from these three sources since we seek to combine the quantitative
results from the questionnaire, with short excerpts from interviews to elucidate a particular
point of view and the analysis of the solution sent by a team of three participants to a
numerical/algebraic problem. Combining such information, we hope to describe the problem
solving activity that participants experience at the Sub14 and at their classrooms.
DATA ANALYSIS AND MAIN RESULTS
Participants’ perceptions on problem solving in and beyond the classroom
According to the participants, the problems that teachers occasionally propose in their
mathematics classes differ from those that they solve for the Sub14. Besides being scarce,
problem solving activity in the classroom is often associated with a specific mathematical
content or procedure that the teacher aims to develop. Therefore, the challenging aspect of the
problem diminishes considerably since the students already know what procedure or rule they

AN ANALYSIS OF USING JAPANESE PROBLEM SOLVING ORIENTED LESSON STRUCTURE TO SWEDISH MATHEMATICS CLASSROOM DISCOURSE

AN ANALYSIS OF USING JAPANESE PROBLEM SOLVING ORIENTED LESSON STRUCTURE TO SWEDISH MATHEMATICS CLASSROOM DISCOURSE

Recommended

Recommended

More Related Content

Similar to AN ANALYSIS OF USING JAPANESE PROBLEM SOLVING ORIENTED LESSON STRUCTURE TO SWEDISH MATHEMATICS CLASSROOM DISCOURSE

Similar to AN ANALYSIS OF USING JAPANESE PROBLEM SOLVING ORIENTED LESSON STRUCTURE TO SWEDISH MATHEMATICS CLASSROOM DISCOURSE (20)

More from Joe Andelija

More from Joe Andelija (20)

Recently uploaded

Recently uploaded (20)

AN ANALYSIS OF USING JAPANESE PROBLEM SOLVING ORIENTED LESSON STRUCTURE TO SWEDISH MATHEMATICS CLASSROOM DISCOURSE