Creative Education
2013. Vol.4, No.11, 700-704
Published Online November 2013 in SciRes (http://www.scirp.org/journal/ce) http://dx.doi.org/10.4236/ce.2013.411099
Open Access
700
Key Universal Activities of Mathematical Learning in Problem
Solving Mathematics Classroom
Saastra Laah-On1, Pimpaka Intaros2, Kiat Sangaroon3
1Master Program in Mathematics Education, Faculty of Educ at i o n , Khon Kaen University, Khon Kaen, Thailand
2Doctoral Program in Mathematics Education, Faculty of Education, Khon Kaen University, Khon Kaen, Thailand
3Faculty of Science, Khon Kaen University, Khon Kaen, Thai l a nd
Received September 25th, 2013; revised October 25th, 2013; accepted November 2nd, 2013
Copyright © 2013 Saastra Laah-On et al. This is an open access article distributed under the Creative Commons
Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the
original work is properly cited.
To enhance students’ problem solving potential, an important skill for 21st century, teachers should con-
cern what kinds of authentic-mathematics experiences that students can get through problem solving (Cai,
Mamona-Downs, & Weber, 2005). In addition, mathematics classroom approach has changed radically
from a drill-and-practice approach to more insight-based problem oriented approach (Van Oers, 2002).
According to a problem solving mathematics classroom, in which an open approach is used as a teaching
approach, students are required to adapt what they have learned to solve problem situations (Inprasitha,
2010). These problem situations are designed based on students’ experiences. Regarding these points, a
purpose of this study was to investigate key universal activities, which are based on Bishop (1988) who
presented the key universal activities as foundations for students’ mathematical learning. Case study was
employed in this study. Video and audio tape recording, and field note taking were used as methods for
collecting data of a targeted group including six of grade 1 students in 2010 academic year of a school
participating the Project for Professional Development of Mathematics Teachers through Lesson Study
and Open Approach. Data were analyzed by using descriptive statistic and analytic description. The re-
sults showed that there were various key universal activities in each problem situation occurring in prob-
lem solving mathematics classroom. These key universal activities have been enhancing the students to
solve the problems efficiently.
Keywords: Key Universal Activities; Problem Solving Classroom; Open Approach; Lesson Study
Introduction
In the 21st century, there are more increasing of a complexity
of problems that one could encounter, both in a country and in
the world. This has been requiring more dedication of preparing
students to be problem solvers as providing the students to
learn some strategies that could help them to cope with such
problems (Barell, 2010). The purpose of learning has not been
knowledge itself, but rather that the students should have abili-
ties to learn in every time and develop themselves constantly
(Bellanca & Brandt, 2011). Therefore, classrooms must be
changed from “traditional classroom” that devotes most of in-
structional time to teacher’s recitation and practices, to “reform
classroom” that provides wider task demand to inquire stu-
dents’ effective problem-solving strategies and communication
practices (Forman, 1996). According to this point, what is
critical is not whether the teachers should use problem solving
as an approach to teach mathematics, but rather what kinds of
authentic experiences concerning mathematics that students can
get through problem solving (Cai, Mamona-Downs, & Weber,
2005). Recently, the classroom approach to mathematics has
changed radically from a drill-and-practice approach to more
insight-based problem oriented approach. In other words, the
problem solving approach to mathematics is introducing stu-
dents into the culture of mathematics practice (Van Oers, 2002).
In socio-cultural points of view, as Bishop (1988) has real-
ized that mathematics is a cultural phenomena, a mathematical
enculturation process then is a process in which concepts,
meanings, processes and values are shaped according to certain
manner, and also emphasizing on social context where there is
an interaction between those who participate in this process.
The goal of this process is developing a way of knowing in
each individual by promoting the transformation from tech-
nique or a way of doing to meaning or a way of knowing.
Therefore, it is an intentional process of shaping ideas. More-
over, this process is based on experiences formed in key uni-
versal activities that are counting, measuring, locating, design-
ing, building, playing, and describing. These activities are the
foundations for the development of students’ mathematical
ideas. In other words, these activities promote the transforma-
tion from technique, or a way of doing, to meaning, or a way of
doing.
A mathematics classroom using an open approach (Figure 1)
as a teaching approach which encourages the students to learn
mathematics by themselves along 4 phases are composed of 1)
Posing open-ended problem, 2) Students’ self learning, 3)
Whole class discussion and comparison, and 4) Summarization
through connecting students’ mathematical ideas emerging in

S. LAAH-ON ET AL.
Summariza tion through
connec ti ng st udents’
mathematical ideas emerged
in the classr oom
Whole c lass discussion
and comparison
Students’ self learning
Posing open-ended problem
Figure 1.
Open approach as a teaching approach (Inprasitha, 2010).
the classroom (Inprasitha, 2010).
Illustration of classroom activities using the open approach
(Table 1) could be described as 1) Posing open-ended problem
—the open-ended problems or problem situations are posed in
the classroom and the students are often asked about a meaning
of the problems and challenged to solve the problems; 2) Stu-
dents’ self learning—this phase consists of a combination of
two parts: individual work and discussion by the whole class; 3)
Whole class discussion and comparison—the students’ activi-
ties are crucial to further development of a lesson in which the
teacher should try to identify those students who do not under-
stand the problems and provide more suggestions to stimulate
the students in a whole class to think according to the problems;
and 4) Summarization through connecting students’ mathe-
matical ideas emerging in the classroom—the teacher should
include all students’ prepositions and concentrate on one point
view, lead to a conclusion by integrating and arranging them
according to particular point of view, and also facilitate a
smooth transition to the next lesson (Inprasitha, 2010).
Moreover, the teachers mostly start the classes with a prob-
lem situation which is designed by using open-ended problems
and is closed to the students experience or what the students
have learned, and the learning organization in this classroom is
considered as a interaction process between a teacher and stu-
dents, and among students themselves where the teacher or-
chestrate the students’ mathematical ideas resulted from pro-
moting the students to think and solve the problems in their
own way. Therefore this process can be described by social and
cultural aspects (Inprasitha, Pattanajak, & Prakaikam, 2007).
These mean that the approach in this classroom necessarily
nurtures the students to learn mathematics in meaningful ways
according with the students own experiences.
As a result, what is needed for the teachers or mathematical
cultivator is a broad understanding of mathematics as a cultural
phenomena (Bishop, 1988). Therefore, the teachers should be
conscious about what kind of experiences the students could
learn best in mathematical culture. According to these points,
deep insight of the problem solving mathematics classroom is
very important for the teachers. Lesson study, consequently, is
necessary for the teachers to do their practices along with a
cycle of the lesson study (Figure 2). Collaboratively observing
the research lesson (Do) and collaboratively reflecting on
teaching practice (See) would support the teachers to compre-
hend their classroom where they could analyze the activities
occurring in the classroom and promote the students’ learning
according with the activities.
Research Methodology
Theoretical Frameworks
The theoretical frameworks used to conduct this research
Collaboratively de sign
research l esson
(
Plan
)
Collaboratively observe
the research lesson
(Do)
Collaborati vely reflect on
teaching p ractice
(See)
Figure 2.
Lesson study in Thailand (Inprasitha, 2010).
were composed of 2 theoretical frameworks; 1) Open Approach
as a Teaching Approach (Inprasitha, 2010) used to analyze
phases of the classroom that emphasize on problem solving,
and 2) Key Universal Activities (Bishop, 1988) used to ana-
lyzed activities occurred in each phase of the classroom
whether these activities composed of the key universal activi-
ties which are the foundations for students’ mathematical learn-
ing.
Objective of the Study
This research was aimed to investigate key universal activi-
ties, in which are the foundations for students’ mathematical
learning based on Bishop (1988), in the problem solving
mathematics classroom, in which the open approach is used as
the teaching approach based on Inprasitha (2011), that would
yield more realization for the teachers of how to promote the
students’ learning in meaningful ways.
Target Group of the Study
The target group in this research included six of grade 1 stu-
dents, who were studying at Ban Bueng-neum-bueng-krai-noon
School, Khonkaen province, and attending 5 learning activities
of the Length Comparison learning unit in the second semester
of the 2010 school year. The school has been participating in
the Project for Professional Development of Mathematics
Teachers through Lesson Study and Open Approach since 2007
school year, the teachers have been organizing learning activi-
ties by using the open approach as a teaching approach which is
supervised by the Center for Research in Mathematics Educa-
tion (CRME), Khon Kaen University.
Data Collection and Analysis
In this research, the lesson study team, including the teacher,
observing teachers, the author as a researcher, and a researcher
assistant, cooperatively designed the learning activities by using
a Japanese textbook “Study with Your Friends Mathematics for
Elementary School 1st grade” which emphasizing on “students’
how to learn” that supports students’ self learning (Inprasitha,
2010). Several methods were used to collect and analyze the
data in the classroom; video and audio recording, and field note
taking were used as methods for collecting data, the collected
data were then analyzed by using descriptive statistic and pre-
sented by using analytic description.
Results and Discussions
In the problem solving mathematics classroom of the learn-
ing unit of comparing length, there were all of 6 key universal
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S. LAAH-ON ET AL.
Open Access 702
Table 1.
Classroom activities using the open approach.
Teacher’s Actions Students’ R esponses
Posing open - ended problem Ask about a meaning of the problems and
challenge to s olve the pr oblems Interpret pr oblem situatio ns, addre ss thei r proble m and speak out lo ud
Students’ self learning Collect em erging students’ ideas Find their own problem resolutions and prepare to
discuss with other students
Whole class discussion and comparison Conduct discussion and sometimes share ideas
with students Present and discuss ide a s
Summarization through connecting students’
mathematical ideas emerged in the classroom Highlight and summary students’ ideas Take and share ideas with teacher and other students
activities occurred in the classroom and there are variously key
universal activities occurred in each phase of the classroom as
shown in the table as follows.
Information (Table 2) revealed that all kinds of key universal
activities occurred in the problem solving mathematics class-
room especially in the phase of Students’ self learning in which
the students were encouraged to think about the problem situa-
tions and solve the problems by themselves, although there was
no the key universal activity occurred in some phases of the
classroom. Examples of those scenarios which the students
participate each key universal activity as follows.
Counting A ctivity
In the activities of constructing a paper chain that the stu-
dents in each group cooperated to make the paper chain and
comparing them among all groups, the students discussed about
how to compare paper chains’ length. The students tried to
solve the problem related to a certain number of the papers used
to make paper chain whereby they started to count one by one,
and then changed to count by two pieces of papers while they
were struggling in re-counting the papers, as a following pro-
tocol.
Student A: 39 and then 40.
Student B: 48, 49, 50.
Teacher: Oh, why all of you didn’t get the same number of
paper?
Student C: Let’s count them again.
Student B: 2, 4, 6, 8, 12, 14, 16, 18. (Student B stopped to
count.)
Measuring Activity
In the activities of comparing a length of two strings which
were straight and tied, respectively, one of the students com-
pared two strings’ length. The student tried to solve the prob-
lem related to length comparison whereby she untied the string
first and compared them directly as putting the strings’ end to
be on the same level. This was resulted from her experience in
an activity of comparing two straight things (pencils) and iden-
tifying which one was longer, as a following protocol.
Teacher: Okay, there are two strings here. Which one is
longer? (Figure 3)
All students: Red./Blue.
Teacher: How could we do? (Student F brought the strings
from the teacher and then put a couple of their ends on the same
level.)
Student D: Untie that string. (Figure 4)
Teacher: Like this? (Student F untied the string, and then
stretched them from the bottom to the up side.)
Locating Activity
In the activities of measuring and comparing the students’
part of body and the students have to put a ribbon represented
their arms’ length on a presenting paper, one group of the all
the students’ groups tried to solve the problem related to refer-
ring point whereby they used an upper edge of the presenting
paper to be the referring point for making these two lines were
parallel and easy for comparison. This was resulted from their
experience in an activity of comparing two strings that have to
put their ends on the same level, as a following protocol.
Student B: (Student B was putting the ribbon representing a
length of her arm on the presenting paper)
Student A: It’s enough? (Student A asked student B who was
putting the ribbon representing a length of her arm on the pre-
senting paper).
Student B: Hey, it’s not straight. (Student B tried to stretch
the ribbon after she used the upper edge of the presenting paper
to be the starting point of comparing the ribbons.)
Designing and Building Activity
In the activities of comparing the ribbon represented the
length of the students’ arms on a presenting paper, the students
tried to solve about geometric attribute whereby they put one of
the ends of the ribbon on a point beyond a starting point of the
presenting paper by using tessellation, since the length of her
arm was longer than the presenting area. This was resulted from
their experience in an activity of comparing two strings while
they were adapting to change the starting points for comparing
them, as a following protocol.
Student D: Why is a paper not enough? (Student D asked the
teacher after she put the ribbon representing a length of her arm
on the presenting paper by put one of the ribbon’s ends on the
same line of perpendicular line in the presenting paper.)
Teacher: Does it mean that your arms are longer than the
presenting paper?
Student D: (Student D moved out the ribbon from the pre-
senting paper and tried to put it back on the presenting paper.)
Teacher: Where is it? (Figure 5) (Student D put the ribbon
on the presenting paper again whereby putting one of the ends
beyond the perpendicular line in the presenting paper.)
Playing Activity
In the activities of constructing a paper chain, one group of
the all the students’ groups tried to solve the problem related to

S. LAAH-ON ET AL.
Table 2.
Percentile of occurrence of the key universal activities in each phase of the classrooms.
Posing Open-Ended
Problem Students’ Self Learning Whole Class Discussion and Compa rison Summarization through Conne ction Percentile of Occurr ence
Counting 0 3.96 1.98 0.99 6.93
Measuring 10.9 21.78 17.82 1.98 52.48
Locating 0.99 9.9 2.97 0 13.86
Designing a nd Building 0.99 2.97 0 0 3.96
Playing 0 1.98 0 0 1.98
Describing 7.92 4.95 3.96 3.96 20.79
Total 20.8 45.54 26.73 6.93 100
Figure 3.
Teacher used ribbon strings as instruc-
tional material.
Figure 4.
Student compared length between 2
ribbon strings.
Figure 5.
Student put down a ribbon sting on
representing paper.
planning and rule whereby one of the students in the group
planned for taking less time of making the paper chain by ask-
ing another students to put some glue on the ends of papers
before she will compound them together, as a following proto-
col.
Student E: I can’t compound them.
Student D: Yes, we can. (Student D glued one side of the
paper’ ends.) Glue it like this, I will compound them. (Student
D asked student E to glue the papers.)
Describing Activity
In the activities of comparing a length of two strings and the
students have to show their reasons to support their answer, one
of the students tried to solve the problem related to reasoning
whereby she put one couple of the strings’ ends on the same
level and then the difference of the couple strings was shown.
This could enhance her describing reasonably of which one was
longer, as a following protocol.
Teacher: What should be in the same level? (Student D
stretched the strings whereby one couple of the ends was in the
same level.) Oh, is here be in the same level? What’s about the
longer one?
Student D: This part is over then… (Figure 6) (Students D
folded the remainder part down at the shorter one’s end.)
Conclusion
This information could be the empirical evidences which
support an important characteristic of a new approach used in
the problem solving-mathematics classroom that provides the
opportunity for the students to learn mathematics in the mean-
ingful ways for their own experiences. These would be pro-
moted by the key universal activities which are the foundation
for the students’ mathematical learning. In other words, the
students could recall what they have learned based on the key-
universal activities to use as the way to solve the problems or
do mathematics, and then use as resources to construct the ways
of knowing or making sense for themselves. The students can
regulate to do problem solving activities by themselves. There-
fore, interestingly, the open approach can shift the mathematics
classroom from the traditional classroom in which the teachers
are the center and students only practice and drill, to a place
where the students are able to do mathematics by themselves
via the key universal activities.
Acknowledgements
This research is supported by the Higher Education Research
Promotion and National Research University Project of Thai-
land, Office of the Higher Education Commission, through the
Open Access 703

S. LAAH-ON ET AL.
Figure 6.
Student tried to use instructional mate-
rial to explain way of think i ng.
Cluster of Research to Enhance the Quality of Basic Education
and this research is granted by the Center for Research in
Mathematics Education, Khon Kaen University, Thailand.
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