An Investigation Of Secondary Teachers Understanding And Belief On Mathematical Problem Solving

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An Investigation of Secondary Teachers’
Understanding and Belief on Mathematical
Problem Solving
To cite this article: Tatag Yuli Eko Siswono et al 2016 J. Phys.: Conf. Ser. 693 012015
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An Investigation of Secondary Teachers’ Understanding and
Belief on Mathematical Problem Solving
Tatag Yuli Eko Siswono1
, Ahmad Wachidul Kohar2
, Ika Kurniasari3
,
Yuliani Puji Astuti4
Faculty of Mathematics and Natural Science, Surabaya State University, Indonesia
E-mail: 1
tatagsiswono@unesa.ac.id; 2
bangwachid@gmail.com
3
ika.kurniasari@gmail.com; 4
yuliani.matunesa@gmail.com
Abstract. Weaknesses on problem solving of Indonesian students as reported by recent
international surveys give rise to questions on how Indonesian teachers bring out idea of
problem solving in mathematics lesson. An explorative study was undertaken to investigate
how secondary teachers who teach mathematics at junior high school level understand and
show belief toward mathematical problem solving. Participants were teachers from four cities
in East Java province comprising 45 state teachers and 25 private teachers. Data was obtained
through questionnaires and written test. The results of this study point out that the teachers
understand pedagogical problem solving knowledge well as indicated by high score of
observed teachers‘ responses showing understanding on problem solving as instruction as well
as implementation of problem solving in teaching practice. However, they less understand on
problem solving content knowledge such as problem solving strategies and meaning of
problem itself. Regarding teacher’s difficulties, teachers admitted to most frequently fail in (1)
determining a precise mathematical model or strategies when carrying out problem solving
steps which is supported by data of test result that revealed transformation error as the most
frequently observed errors in teachers’ work and (2) choosing suitable real situation when
designing context-based problem solving task. Meanwhile, analysis of teacher’s beliefs on
problem solving shows that teachers tend to view both mathematics and how students should
learn mathematics as body static perspective, while they tend to believe to apply idea of
problem solving as dynamic approach when teaching mathematics.
1. Introduction
It has already been agreed that problem solving is an essential issue deeply discussed in mathematics
education in recent decades since its practical role to the individual and society. Problem-solving as
one of five standard competences in mathematics mentioned by NCTM (National Council of Teachers
of Mathematics) [1] not only develops individuals’ conception about aspects of mathematics, but also
it helps to adapt to various problems in many aspects of their lives. NCTM [1] also recommended that
problem solving be the focus of mathematics teaching because it encompasses skills and functions
which are an important part of everyday life.
This issue brings a variety of responds which regards to both story of success and difficulties in
applying it within practical mathematics teaching and learning in many countries. It is generally
known that mathematics curriculum in some countries can be ascertained put problem solving as a
ICMAME 2015 IOP Publishing
Journal of Physics: Conference Series 693 (2016) 012015 doi:10.1088/1742-6596/693/1/012015
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main component, such as in Singapore [2,3], United States [2], England [2], Australia [2), Finland [4],
and Indonesia [5]. The success story, for example, was reported by [6] which showed that problem-
solving, based on open tasks in specific topic is feasible and effective in the classroom. Students are
challenged and actively involved in cognitive and physical activities as well as in interesting
discoveries. In Finland, the country which often puts their students in top rank in international survey
like PISA (Programme for International Student Assessment) [7], problem solving has been a part of
the Finnish curriculum for a couple of decades but still it has not found its place in the classroom
reality [9]. Singapore curriculum, on the other hand, encourages teachers to reduce the content taught
through direct teaching but instead engage students in meaningful activities so that they use
knowledge to solve problems and whilst solving problems extend their knowledge through inquiry [3].
However, the issue of mathematical problem solving in Indonesia, especially, has not yielded a
satisfactory result although it has been becoming one of curriculum contents of school mathematics
since year 1968 [5]. Fact shows that the students' ability to solve mathematical problems is still weak
as pointed out by the results of international studies on education, such as TIMSS (Trends in
International Mathematics and Science Study) and PISA. Result of TIMSS 2011 [8] participated by
secondary students grade 8 reported that Indonesia ranked 38th
out of 42 countries with score 386 of
center point of TIMSS of 500, while the latest PISA report in 2012, participated by 15-year old
secondary students, Indonesia was reported to get rank of 64 out of 65 countries. In the latter study,
even almost all Indonesian students (98.5%) were only able to reach level 3 of 6 levels of the task
examined [7,10]. Research on Indonesian students’ performances on problem solving, particularly,
also supported this issue. The study of Siswono, Abadi, and Rosyidi [11] to the fifth grade students as
many as 202 students from five elementary schools in Sidoarjo showed that students’ ability in solving
problems (especially the open-ended problem) is still low as indicated by the data that the students'
ability to solve problems that show fluency is only 17.8%, novelty 5.0%, and flexibility 5.4%. This
result is also supported by studies of Kohar & Zulkardi [12] and Wijaya, van den Heuvel-Panhuizen,
Doorman, & Robitzschc [13] each of which qualitatively investigate the extend to which secondary
students perform their mathematical problem solving stages on context-based task which point out that
the student participants mostly failed in early stages of contextual problem solving steps, namely
devising strategies to transform contextual situation into precise mathematical model.
This condition is supposed to be influenced by various factors such as students’ background
regarding mathematics performance, the available of learning resources, teacher beliefs, the ability of
teachers, facilities, learning process, as well as national educational policies. Teacher as a crucial
factor in the development of students’ learning has a role to build knowledge of mathematical problem
solving for themselves as problem solvers and to help students to become better problem solvers.
Thus, some previous studies revealed the relationship between teacher’s knowledge and practice about
problem solving with students’ performance on problem solving. Grouws and Cebulla [14] in their
study found that students’ performance can also be influenced by teachers’ teaching practices.
Teaching practice, on the other hand, is influenced by teaching knowledge and beliefs [15,16].
The knowledge is important to identify students’ mathematical problem solving proficiency within
practical teaching. For instance, being able to identify the possible strategies used by students in
problem solving allows teachers to interpret why a particular problem could be difficult. Moreover,
being able to choose a suitable problem and understand the nature of it is also an important part of a
problem solving lesson [17]. This knowledge, as Franke & Kazemi [18] stated, could help teachers to
understand which characteristics make problems difficult for students and why. Other scholars, Ball,
Thames, and Phelps [19], suggested that general mathematical ability does not fully account for the
knowledge and skills needed for effective mathematics teaching. Teachers, they said, need a special
type of knowledge to effectively teach problem solving which should be more than general problem-
solving ability. These studies suggest that teachers need to have knowledge of a variety of problems
that are relevant for teaching problem solving. Hence, it is important to investigate what knowledge
does teacher have regarding this matter. Meanwhile, teachers’ beliefs may also play a role in students’
problem solving because teaching practice is often affected by what teachers think about the teaching
2

and learning of mathematics [20]. Teachers’ beliefs about students’ ability and learning greatly
influence their teaching practices [21]. The study of Stipek, Givvin, Salmon, MacGyvers [16] show
that there is a substantial coherence among teachers' beliefs and consistent associations between their
beliefs and their practice. Thus, it is important to know what beliefs that teachers in Indonesia
typically have, particularly, in order to understand their influence toward teaching practice.
Hence, this issue gives rise to the questions of how Indonesian teachers bring out idea of
problem solving in their understanding related to both content and pedagogical knowledge as well as
how they experience difficulties, and how are their beliefs on this issue. As an initial step to address
these questions, this present study aims to investigate secondary teachers’ understanding and belief on
mathematical problem solving.Therefore, the following research questions were addressed:
1. How did secondary teachers understand mathematical problem solving regarding problem,
problem solving as instruction, problem solving steps, problem solving strategies, and
instructional practice of problem solving as well as level of their performance on problem
solving task?
2. What are secondary teachers’ difficulties which are regarded to understand mathematical
problem solving?
3. How are secondary teachers’ beliefs on mathematical problem solving?
2. Theoretical Background
2.1 Teachers’ understanding on the nature of problem solving
Teaching problem solving needs some understandings which are related to some points of knowledge.
Chapman [22] mentioned three types of knowledge for teaching problem solving: problem solving
content knowledge, pedagogical problem solving knowledge, and affective factors and beliefs. (see
table 1). Structuring some knowledge within this table was then confirmed as follows. First, figuring
out what it means by problem. Chapman [22] argued that teachers should understand problems based
on their structure and purpose in order to make sense of how to guide students’ solutions including
understanding on types of tasks, such as cognitively demanding tasks; multiple-solution tasks; tasks
with potential to occasion/promote mathematical creativity in problem solving; demanding problems
that generally allowed for a variety of problem-solving strategies; rich mathematical tasks, and
particularly open-ended problems. Second, understanding problem solving in instruction which means
teachers are encouraged to foster their students in completing problem solving steps precisely.
Third, coming up with idea of instructional practices for problem solving, teachers need to
consider a series of activities which give students opportunity to solve problems which needs
challenging complex thinking and logical reasoning. The activity depends on how the teacher's ability
to prepare a problem. Crespo & Sinclair [in 23] explained that teachers who are able to create
questions in the initial situation will be more successful learning rather than the teacher who asked the
problems spontaneously.Thus, teaching for problem solving, teachers should be proficient in as well as
understand the nature of it in order to teach it effectively. In general, this categorization appears to
give insight on emerging teachers’ conception of problem solving theoretically to teachers’ actual
teaching practically.
3

Table 1. Knowledge needed in understanding problem solving
In addition to the specific issue related to problem solving content knowledge, teachers should
also be proficient to deal with a variety of problem solving task, such as completing problem solving
steps as well as applying problem solving strategies on various types of mathematical task. Level of
understanding regarding this issue was discussed by some frameworks, such as Polya’s problem
solving step [24], and some error analysis guidelines on performing mathematical tasks developed by,
for instance, Wijaya et al [13] who adapted from three main frameworks, namely Newman’s error
[25], Blum and Less’ modelling stages [as cited in 26], and PISA’s mathematization stages [27], and
Kohar & Zulkardi [12] who adapted from Valley, Murray, & Brown [28] and PISA’s mathematical
literacy [29]. To figure out the possibility of those frameworks are applied to investigate level of
teachers’ proficiency in solving a variety of problem solving task, the following table shows
characteristics of some of those.
Table 2. Comparing frameworks on performance level in solving mathematical tasks
Polya’s Problem Solving Step
[24]
Newman Analysis [25]
PISA’s Mathematical
Literacy [29]
Understanding the problem:
identify the unknown, data, and
condition related to the information
given in the problem
Reading: recognize of words
and symbols
Formulate: recognise and
identify opportunities to use
mathematics and then
provide mathematical
structure to a problem
presented in some
contextualised form
Comprehension: understand the
meaning of a problem
Devising a plan: find the
connection between data and the
unknown, consider auxiliary
Transformation: transform a
word problem into an
appropriate mathematical
Type of
knowledge
Knowledge Description
Problem
solving
content
knowledge
Mathematical
problem solving
proficiency
Understanding what is needed for successful mathematical problem
solving
Mathematical
problems
Understanding of the nature of meaningful problems; structure and
purpose of different types of problems; impact of problem
characteristics on learners
Mathematical
problem solving
Being proficient in problem solving
Understanding of mathematical problem solving as a way of thinking;
problem solving models and the meaning and use of heuristics; how to
interpreting students’ unusual solutions; and implications of students'
different approaches
Problem posing Understanding of problem posing before, during and after problem
solving
Pedagogical
problem
solving
knowledge
Students as
mathematical
problem solvers
Understanding what a student knows, can do, and is disposed to do
(e.g., students’ difficulties with problem solving; characteristics of good
problem solvers; students’ problem solving thinking)
Instructional
practices for
problem solving
Understanding how and what it means to help students to become better
problem solvers (e.g., instructional techniques for heuristics/strategies,
metacognition, use of technology, and assessment of students’ problem
solving progress; when and how to intervene during students’ problem
solving).
Affective factors and beliefs Understanding nature and impact of productive and unproductive
affective factors and beliefs on learning and teaching problem solving
and teaching
4

Polya’s Problem Solving Step
[24]
Newman Analysis [25]
PISA’s Mathematical
Literacy [29]
problem if an immediate
connection cannot be found, and
finally obtain a plan of the solution
problem
Carrying out the plan Process skills: perform
mathematical procedures
Employ: apply mathematical
concepts, facts, procedures,
and reasoning to solve
mathematically-formulated
problems to obtain
mathematical conclusions
Looking back: examine the solution
obtained
Encoding: represent the
mathematical solution into
acceptable written form
Interpret/evaluate: reflect
upon mathematical
solutions, results, or
conclusions and interpret
them in the context of real-
life problems.
In general, from table 2 we can notice that element of steps among those three frameworks
corresponds each other. Specifically, steps of understanding problem and devising strategies,
simultaneously, has likely similar idea with steps of reading, comprehension, and transformation in
Newman analysis, while this idea also appear on mathematical literacy, i.e. formulate. As early stages
in solving mathematical task, they end up with determining precise mathematical model or strategies
before performing further steps of solving problem. Likewise, each idea of carrying out step in Polya’s
process, process skill in Newman, and employ in PISA’s mathematical literacy deals with undertaking
mathematical procedure to find mathematical results, such as performing arithmetic computations,
solving equations, making logical deductions from mathematical assumptions, performing symbolic
manipulations, or extracting mathematical information from tables and graph. Furthermore, the last
step of Polya’s, i.e. looking back, corresponds to final stage of Newman analysis, i.e. encoding and
PISA’s mathematical literacy, i.e. interpretation. The idea of this stage is interpreting the mathematical
result to the initial problem such as checking the reasonableness of the answer or considering other
strategies and solution of the problem. The difference, obviously, only appear on the type of the tasks
examined where PISA’s mathematical literacy specifies on contextual task [29], while Polya and
Newman respectively deals with general mathematical problem [24] and written mathematical task
[25]. Comparing those three frameworks, it is known that Polya’s problem solving steps, which was
introduced before the other two frameworks, have an agreement with both Newman analysis and
PISA’s mathematical literacy. Thus, Newman’s error categories can be used to analyze teachers’ level
of performance in solving context-based mathematical problem solving tasks, which were used in this
study.
The typical teaching of problem solving should also need to be considered as important parts of
understanding instructional practices for problem solving pedagogical knowledge. As examples,
typical learning in Japan [30] consists of (1) discuss the previous lesson, (2) presents a problem, (3)
ask students work individually or in groups, (4) discussion of problem-solving method, and (5)
provides a summary and discussion of an important point. Shimizu [30] explains that underline and
summarizes the activities or "Matome" has the function (1) underline and summarize the main points,
(2) to encourage reflection of what we have done, (3) determine the context to recognize new concepts
or terms based on previous experience, and (4) make the connection (connection) between the new and
the previous topic. Similarly, teachers in Finland as reported by Koponen [31] in his study carry out
problem solving lesson by firstly introducing the problem, then working on the problem in pairs or in
small groups, instructing the individual students on their solutions, and the final whole-classroom
discussion.
5

Based on the above discussion it is necessary to know actually how teachers understand the
nature of problem solving that includes problem itself, problem-solving, problem-solving strategies,
problem solving approach brought into classroom practical teaching, and posing problem solving task.
Besides, it is also necessary to look into teachers’ performance in solving problem solving task.
2.2 Teachers’ beliefs on mathematical problem solving
Mathematical beliefs, as Raymond said, are regarded as “personal judgments about mathematics
formulated from experiences in mathematics [32]. They play role as prerequisite development of
problem solving itself [33]. Regarding their categorization related to other interrelated fields,
Weldeana & Abraham [34] summarized frameworks of teachers’ mathematical belief systems into
smaller subsystems, including beliefs about the following: (a) the nature of mathematics, (b) the actual
context of mathematics teaching and learning, and (c) the ideal context of mathematics teaching and
learning. These subsystems are wide-ranging, including, for examples, teachers’ views on
mathematical knowledge; the role of learners and learning; the role of teachers and teaching; and
nature of mathematics activities. This categorization was also conceptualized by Ernest [35] who
described three views of nature of mathematics: the instrumentalist view, the platonist view, and the
problem solving view. The instrumentalist believes mathematics is useful and collects unrelated facts,
rules and skills. The platonist views mathematics as a consistent, connected and objective structure,
which means mathematics is a unified body of knowledge that is discovered, not created. The problem
solving view sees mathematics as a dynamically organized structure located in a social and cultural
context.
In attempt to simplify these views, Beswick [30] has tried to make connections among the
nature of mathematics, mathematics learning, and mathematics teaching as follows.
Table 3. Connections among belief about mathematics, mathematics teaching, and mathematics
learning
Beliefs about the nature of
mathematics
Beliefs about mathematics
teaching
Beliefs about mathematics
learning
Instrumentalist Content focussed with an
emphasis on performance
Skill mastery, passive reception of
knowledge
Platonist Content focussed with an
emphasis on understanding
Active construction of understanding
Problem solving Learner focussed Autonomous exploration of own
interests
The relationship between teachers’ beliefs about mathematics and its teaching and learning
practice, has been investigated in many studies. Ernest [in 31] claimed that a teacher’s personal view
of mathematics underpins beliefs on the teaching and learning of mathematics. In line with this claim,
Schoenfeld [37] stated that the teacher’s sense of the mathematical enterprise determines the nature of
the classroom environment that the teacher creates. Thus, beliefs influence teaching and learning
practice. Other study, Ruthven’s [38] study, for instance, argued that teachers need to broaden their
perspective about ability and quality in mathematical learning which is probably more easily changed
by changing practical teaching in classroom first, as the teachers’ understanding of mathematics
teaching and learning. Thus, teaching and learning practice influence teacher’s belief.
Studies on how deep mathematics teacher develop beliefs on mathematics and its practice in
teaching and learning were then investigated by some researchers. A study of Zhang and Sze [39]
comparing preservice teachers’ belief on mathematics in China and Thailand, for instance, revealed
that generally teachers participating in this study believe that mathematics is about thinking, logic, and
usefulness, rather than a subject of calculableness and preciseness. Regarding their belief on
mathematics teaching and learning, the results showed that Chinese preservice teacher’s beliefs are
6

more like constructivist. Beswik [40] in his study on two mathematics teacher’s view on the nature of
mathematics and mathematics as school subject regarding on the three views: platonist,
instrumentalist, and problem solving, suggested that more attention needs to be paid to the beliefs
about the nature of mathematics that the teachers have constructed as a result of the cumulative
experience of learning mathematics. Thus, we get insight on how important these views offer
opportunities for teachers to rethink their own beliefs and get to know more about teaching practices.
These beliefs, as Schoenfeld [37] argued, give impact on students’ belief in learning
mathematics which then obviously influence their mathematics performances. The cause, for instance,
is that teachers rely on established beliefs to choose pedagogical content and curriculum guidelines
[e.g. 17]. If teachers tend to believe mathematics as a set of tools that contain facts, rules, and skills,
the lesson will likely to be centered on teachers instead of students [41]. Furthermore, The importance
of students 'and teachers' beliefs about the role of problem solving in mathematics is a prerequisite
development of problem solving itself [41]. Romberg in [42] shows the relationship of elements in the
teaching of mathematics as follows.
Figure 1. Relationship among elements of teaching mathematics
Figure 1 point out that not only teacher’s mathematical content, but also teachers’ beliefs influence
students’ performance. This view illustrates the importance of types of teacher beliefs which are
needed to be investigated as attempt to improve teachers’ proficiency dealing with problem solving.
3. Methods
This is a descriptive explorative research which aims to explore teachers' understanding on
mathematical problem solving and their belief.
3.1 Participants
Participants were secondary teachers who has minimum a bachelor degree, have taught more than 5
years, from four cities: Surabaya, Sidoarjo, Gresik, and Mojokerto. There were 25 private teachers and
45 state involved in this study.
3.2 Data collection
Data were collected from questionnaire and problem solving task. The questionnaire consisted of 21
multiple choices questions. Each item provided 4 to 17 choices. Some of those questions had large
number of choices because of a need to cover as many as possibilities of teacher’s responses, both
correct and incorrect. For instance, the question item: “In my opinion, the best way to teach
mathematics are...” had 17 choices consisting 9 correct answers, 4 partially correct answers, and 4
incorrect answers. Thus, teacher could choose more than one choices. To explore the understanding of
teachers in problem solving, there were 15 items categorized into 7 groups of questions. The
categorization of these groups was based on Chapman’s type of problem solving knowledge described
in table 1. The groups are (a) problem solving content knowledge: meaning of problem (1 item), open
ended problem (1 item), problem solving as instruction (1 item), problem solving steps (3 items),
problem solving strategies (2 items), and (b) pedagogical problem solving knowledge: instructional
Mathematical
Content
Teachers
Belief
Planning Teaching at
Classroom
Students’
Performance
7

practice of problem solving (3 items), and designing problem solving task (3 items). Other three items
are categorized to identify the difficulties of teachers in problem solving while the other three items
are categorized to explore teacher beliefs, i.e regarding mathematics, how to teach mathematics, and
how students should learn mathematics.
Meanwhile, the problem solving tasks were designed to explore teachers’ understanding
regarding contextual tasks which do not likely need any higher prerequisite formal mathematical
knowledge. See the tasks at appendix. The following is the description on the tasks’ demands.
Table 4. Description of problem solving tasks
No Unit Description Source
1 Futsal score Interpret information about a score achieved by a futsal
team in a tournament which is implicitly stated from the
information given in a table
Modified version
from Kohar and
Zulkardi [12]
2 World online
mathematics
contest
Formulate a mathematical model to determine a perfect time
to hold an online mathematics contest participated by
students from different countries in different continents
Developed by
authors
The difference between those two task, particularly, appears on the most dominant stages which are
likely more needed to be performed when solving the task. Here, task 1 needs more performance on
final stage of problem solving, i.e. interpreting mathematical result to initial problem, while task 2, in
opposite, demands more likely performance on early stages, i.e. from understanding the task to
devising strategies or mathematical model of the task.
3.3 Data analysis
Descriptive analysis on investigating teachers’ understanding and belief was carried out by using score
given on each group of questions. Each option on a question has score either 1 (not understanding), 2
(partial understanding), or 3(full understanding). As an example, we give one question contained in
group of question: problem solving content knowledge including its options and its score as follows.
An open-ended mathematics task is the task which...
A. contains open sentences (score 1)
B. produces a variety of strategies to find out the solutions (score 3)
C. gives an open opportunity to persons who want to solve (score 1)
D. contains higher level of difficulty so that needs higher mathematical skill as well (score 2)
E. has more than one solutions (score 3)
F. has opportunity to be developed into other type of tasks by changing information or
requirements from the solved task (score 2)
The score varies to show level of understanding from 1.00 (do not understand), to 3.00 (fully
understand), while the other scores varies to show level of beliefs on mathematical problem solving
from 1.00 (platonist view/as tool) to 3.00 (problem solving view). The score is given to each
participant on each question based on the following formula.
options
chosen
of
number
score
total
obtained
(S)
Score =
In detail, we have developed a guideline to categorize these levels as shown by the following table.
Table 5. Scoring category level of teachers’ understanding and beliefs
Score (S) Level of understanding on
mathematical problem solving
Level of beliefs toward mathematical
problem solving
67
.
1
00
.
1 <
≤ S not understand (NU) as tool/instrumentalist view
33
.
2
67
.
1 ≤
≤ S partially understand (PU) as body static/platonist view
00
.
3
33
.
2 ≤
< S fully understand (FU) as dynamic/problem solving view
8

Regarding teachers’ performance on problem solving task, we used framework on investigating
individual’s performance based on stages of mathematical modelling adapted from Wijaya [13], and
Kohar [12] since the task is in the category of context-based problem solving task. This is shown as
follows.
Table 6. Level of teachers’ performance on problem solving task
Type of responses Sub-type Codes Explanation
Comprehension error Misunderstanding the
instruction
C-1 Teacher incorrectly interpreted what they were
asked to do.
Error in selecting
information
C-2 Teacher was unable to differentiate between
relevant and irrelevant information (e.g. using all
information provided in a task, neglecting
relevant information, or adding other unrelated
information not given in the task ) or was unable
to gather required information which was not
provided in the task.
Transformation/
devising strategies
error
Procedural tendency
T-1 Teacher tended to use directly a mathematical
procedure (such as formula, algorithm) without
analyzing whether or not it was needed
Taking too much
account of the context
T-2 Teacher’s answer only referred to the
context/real world situation without taking the
perspective of the mathematics
Wrong mathematical
operation/concept
T-3 Teacher used mathematical procedure/concepts
which are not relevant to the tasks
Mathematical
Processing error
Algebraic error P-1 Error in solving algebraic expression or function.
Arithmetical error P-2 Error in calculation.
Measurement error
P-3 Teacher could not convert between units (e.g.
from hour to minute)
Unfinished answer
P-4 Teacher used a correct formula or procedure, but
she/he did not finish it.
Interpretation error I Teacher was unable to correctly interpret and
validate the mathematical solution in terms of
the real world problem. This error was reflected
by an impossible or not realistic answer.
Full understanding
Arithmetical approach
F-1 Teacher performed complete and correct steps of
solving the task by applying arithmetical
approach dominantly (e.g. applying standard
arithmetics operation such as adding,
subtracting, multiplying, or dividing number) to
get solution
Algebraic approach
F-2 Teacher performed complete and correct steps of
solving the task by applying algebraic approach
dominantly (e.g. building algebraic equation,
manipulating algebraic form) to get solution
Unknown U Type of response could not be coded since its
limited information from teacher's work
9

4. Results
4.1 Teachers’ understanding on problem solving
4.1.1 Teachers’ understanding in questionnaire results
There were seven categories of questions which were tested to measure teachers’ understanding on
problem solving. Each category could contain more than one questions. For instance, category of
problem solving strategies contained two questions, while category of experience in designing
problem solving tasks contained three questions. Table 7 shows teachers’ average score on the
questionnaire.
Table 7. Teachers’ understanding on mathematical problem solving
Category of teachers’ understanding on
problem solving
Score Interpretation
Private State Total
Meaning of problem 2.17 2.46 2.36 FU
Open-ended problem 2.65 2.80 2.75 FU
Problem solving as instruction 2.84 2.96 2.91 FU
Problem solving steps 2.41 2.50 2.47 FU
Problem solving strategies 1.89 1.80 1.83 PU
Implementation of steps and strategies of
problem solving in teaching
2.43 2.45 2.44 FU
Experience in designing problem solving task 2.66 2.63 2.64 FU
Table 7 shows that the lowest score of understanding problem solving appears on category problem
solving strategy (1.83) which means teachers did not really.understand toward this group of questions.
In giving response on questions related to problem solving strategy, data show that most teachers
chose wrong options related to type of problem solving strategy should be applied on a given
information on the question. Moreover, teacher did not really show good understanding on the
meaning of problem as indicated by its score, i.e., 2.36, which could be interpreted as low score of full
understanding based on scoring category on table 5. However, higher score appears on category
problem solving as instruction (2.91) and designing problem solving task (2.64), both of which are
related to practical knowledge of problem solving. It shows even though teachers are aware of the
importance of problem solving as the focus of learning but there are still weaknesses in selecting a
task question as a problem and solution strategies. Thus, regarding Chapman’s category of knowledge
needed to understand problem solving, we then could note that the teachers had relatively better
understanding on pedagogical problem solving knowledge rather than problem solving content
knowledge.
4.1.2 Teachers’ understanding in performing problem solving task
In total, we had 140 possible responses (number of tasks done by all teachers in total) which included
35 correct responses (25%), 80 incorrect responses (57.15%), i.e., no credit or partial credit, and 25
missing responses (17.85%). Each incorrect response had an opportunity to be coded by more than one
sub-type code since its different errors found from this response. For instance, a response could be
coded as mathematical processing error subtype algebraic error (P-1) and interpretation
simultaneously (I). Thus, the total number of responses was no longer 140 items, instead we found
176 coded responses. Then, the percentage of each category of responses is given as follows.
10

Table 8. Frequency of teachers’ performance on problem solving task
Type of
responses
Sub-type Codes Task 1 Task 2 N %
Total
per
type
%
Comprehension
error
Misunderstanding the
instruction
C-1 6 9 15 8.52% 37 21.02%
Error in selecting
information
C-2 11 11 22 12.50%
Transformation/d
evising strategies
error
Procedural tendency T-1 7 4 11 6.25% 41 23.30%
Taking too much
account of the
context
T-2 1 3 4 2.27%
Wrong mathematical
operation/concept
T-3 9 17 26 14.77%
Mathematical
Processing error
Algebraic error P-1 1 1 2 1.14% 8 4.55%
Arithmetical error P-2 0 3 3 1.70%
Measurement error P-3 1 0 1 0.57%
Unfinished answer P-4 1 1 2 1.14%
Interpretation
error
I 18 12 30 17.05% 30 17.05%
Full
understanding
Arithmetical
approach
F-1 22 10 32 18.18% 35 19.89%
Algebraic approach F-2 3 0 3 1.70%
Unknown U 11 14 25 14.20% 25 14.20%
Table 8 shows that transformation/devising strategies error constitute to the most frequently found
from teachers’ work (23.30%), while mathematical process error, conversely, become the least
frequently observed (4.55%). Morover, they also performed comprehension error highly, i.e., 21.03%.
This point out that teachers found some difficulties in early stages of problem solving steps.
As examples on how teachers perform those errors, the following figures show comprehension error
and transformation-interpretation error, respectively on task 1.
Figure 1. Examples of errors on task 1
On comprehension error, teacher at figure 1a was unable to distinguish between relevant and irrelevant
information given from the table. He only considered information in column won, lost, and drawn
without giving more attention to the column goals for and goals against to convey a calculation. Thus,
we coded it as C-2. Meanwhile, transformation error was performed by teacher at figure 1b who
tended to use directly a mathematical procedure, i.e. addition and subtraction without analyzing
a. C-2 b. T-1 and I
11

whether or not it was needed and did not provide precise mathematical argumentation supporting the
procedure he used. Consequently, he got the score in negative number, which is not possible to happen
in real world setting. Thus, we also coded it as I error.
Regarding transformation error and comprehension error on task 2, we give these following
examples.
Figure 2. Examples of errors on task 2
Figure 2a was coded as transformation error because the teacher provided wrong mathematical model
to find the best time chosen for holding the competition. She seemed trying to find LCM of the
differences of hours from the three cities which is not suitable to find the solution. Hence, this error
was coded as T-3. Meanwhile, figure 2b actually shows a unique strategy, i.e. using algebraic
approach, which was not found from other teachers’ work. However, the teacher did not perform
carefully in selecting information related to model of inequality for Ankara time, i.e. writing
12 x+14 23, instead of 12 x+2 23. Thus, we coded it as P-1 since it contains algebraic error of
finding solution of inequality.
Teachers’ complete performance in solving the tasks is also interested to be discussed. Here, we
found two approaches, i.e. arithmetical and algebraic approach. Here are examples of these approaches
from task 1.
a. T-3
b. P-1
Translation:
Finding least common multiples from differences
of hours among those three countries, i.e. LCM
from 4,15, and 5 = 60 hours
So, it should start from 12.00 a.m. to 12.00 p.m.
Translation:
let x is time for Greenwich, then allowed time for other cities to
hold the competition is given as follows.
Jakarta: 12 ≤ x+7 23 5 x 16
Sydney:12 x+11 23 1 x 12
Ankara:12 x+14 23 -2 x 9
Since the duration of competition is 1 hour, then range of time
that can be used is from 5 to 8 a.m (Greenwich time)
12

Figure 3. Examples of teachers’ complete performance on the tasks
Figure 3a shows that teacher used symbols representing number of goals successfully shot to another
team, then carrying out a well-operated algebraic form by seeing about information of goals for and
goals against from the table in task 1 to find out each of goals produced by each of teams.
Interestingly, the teacher started to symbolize score of Mentari vs Surya by a : a, which means he
know that number of goals produced between Mentari and Surya is same based on information of
number of drawn match given in the table. Similarly, teacher at figure 3b also applied this information
to find each of score between two teams of each match, but he applied arithmetical approach by listing
some possibilities of score and then carrying out simple operation (addition) of number to find out the
score of Mentari vs Surya FC.
4.2 Teachers’ difficulties on understanding problem solving
We categorized teachers’ difficulties into three type of questions, i.e. problem solving steps, designing
problem solving task, and causes of difficulties in designing problem solving task. Table 7 show
teachers’ responses on this issue.
Table 9. Frequency of teachers’ difficulties on problem solving
Questions Options
The number of teacher choosing
the options
Total Percentage
Private % State %
Most difficult
steps of
read the task 4 16.00% 7 15.56% 11 15.71%
understand problem 6 24.00% 18 40.00% 24 34.29%
Translation:
Let score of Mentari vs Surya FC = a : a
Mentari vs Rajawali = b : c
Surya FC vs Rajawali = d : e
Then
...............................
(algebraic operation same with shown figure)
...............................
So, Mentari vs Rajawali is 4 : 2
b. F-2
a. F-1
Translation:
Analysis
Mentari
vs Surya
Mentari
vs
Rajawali
Rajawali
vs Surya
possibility 1:1 6:1 4:7
goal againts: true
goal for: true
goal against: false
goal for: true
goal against: false
goal for: true
13

Questions Options
The number of teacher choosing
the options
Total Percentage
problem solving determine precise
mathematical model and
strategies
15 60.00% 20 44.44% 35 50.00%
employ mathematical
procedures and facts
2 8.00% 2 4.44% 4 5.71%
interpret result to look back to
initial problem
4 16.00% 11 24.44% 15 21.43%
Difficulties in
designing
problem solving
task
determine real world context
on mathematics topic
21 84.00% 29 64.44% 50 71.43%
make correct written sentence 5 20.00% 7 15.56% 12 17.14%
design task with more than
one solution
2 8.00% 2 4.44% 4 5.71%
one alternative strategies
5 20.00% 8 17.78% 13 18.57%
one alternative solution and
strategies
3 12.00% 9 20.00% 12 17.14%
design simple task but
challenging
16 64.00% 23 51.11% 39 55.71%
Causes of
difficulties in
designing
problem solving
task
not frequently solve higher
order thinking-based tasks
9 36.00% 10 22.22% 19 27.14%
never learn how to design
problem solving task
10 40.00% 12 26.67% 22 31.43%
frequently make set of
evaluation only testing on
calculation and procedure
6 24.00% 6 13.33% 12 17.14%
not frequently read references
about problem solving task
10 40.00% 18 40.00% 28 40.00%
frequently use mathematical
task provided in text book
5 20.00% 10 22.22% 15 21.43%
too focus on guiding students
in solving task using one step
solution
10 40.00% 17 37.78% 27 38.57%
difficulties regarding a variety
of students' ability in
classrrom
2 8.00% 2 4.44% 4 5.71%
From the table, regarding steps of problem solving, teachers especially find difficulties mostly in step
of determining strategies/mathematical model (50.00%), followed by understanding
problem/comprehension (34.29%), and looking back at initial problem (21.43%). The difficulty in
designing a problem-solving task, especially are determining the context of real problems of
mathematical topics (71.43%) and designing simple task but challenging (55.71%). The top four of
cause of these difficulties are almost evenly not frequently read references about problem solving task
(40.00%), too focus on guiding students in solving task using one step solution (38.57%), never learn
how to design problem solving task (31.43%), and not frequently solve higher order thinking-based
tasks (27.14%).
14

4.3 Teachers’ beliefs on mathematical problem solving
Belief in mathematics can be classified into three parts, namely mathematics as a tool, mathematics as
body static and dynamic mathematics as human creations. Table 7 shows teachers’ average score of
beliefs on mathematics, teaching mathematics, and learning mathematics.
Table 10. Score of teachers’ beliefs on mathematical problem solving
Type of beliefs Private State Total Interpretation
mathematics 2.35 2.02 2.14 body static/platonist
teaching mathematics 2.29 2.41 2.36 problem solving
learning mathematics 1.93 1.89 1.90 body static/platonist
In general, the table above shows that teachers still tend to view mathematics, teaching mathematics,
and learning mathematics as body static to problem solving which agrees to platonist-problem solving
view as indicated score range varying from 1.90 - 2.36. However, belief in learning mathematics has
the lowest score compared to the other two beliefs. Moreover, its score, i.e. 1.90, is rather closer to
scoring range for tool view. Hence, teachers’ belief on learning mathématics is rather instrumentalist.
Supporting this data, by calculating percentage of frequency of teacher choosing ‘instrumentalist’
option, for instance, we noted that most teacher (38 out of 70 agree that students should learn
mathematics by starting to work on simple questions to harder questions. On the other hand, in
viewing mathematics, for instance, 38 out of 70 view mathematics as only tool to learn and develop
other disciplines, while in viewing teaching mathematics, we noted that there were many teachers (22
out of 70 teachers) still believe that teaching mathematics should explain topic systematically followed
by giving some examples and asking students to work on some questions. However, in viewing
teaching mathematics, teachers relatively agree to apply problem solving/dynamic approach, as
indicated by the highest score among three types of beliefs shown in table 7, i.e. 2.36. As an example,
we noted that most teacher (47 out of 70) agree that teaching mathematics should start from posing
contextual mathematics problem.
5. Discussion and Conclusion
Results of studying teachers’ understanding on problem solving show that most teachers were likely
found difficulties on problem solving content knowledge, especially in identifying problem solving
strategy. Analysis of their performance in solving problem solving task also reveal that errors mostly
happened when they started to devise strategies or precise mathematical model. These findings were
then supported by their opinion on questionnaire that the most difficult step of carrying out problem
solving step is devising strategies. Moreover, understanding this knowledge needs likely more
attention to deeply comprehend interrelated content knowledge regarding certain topic. For instance,
when investigating prospective elementary teachers’ common thinking about open-ended,
Chapman [17] found that they actually know that open ended problem meant more than one answer
but they were doubt about what this meant mathematically such as inability to provide example of
other answers on a given open-ended task.
Regarding teachers’ belief, the fact that teachers tend to view both mathematics and how students
should learn mathematics as body static perspective, while they tend to believe to apply idea of
problem solving as dynamic approach when teaching mathematics shows that belief in mathematics is
not the only factor affecting the practice of teaching and the views of how students should learn.
Raymond [in 4] describes other factors besides belief in mathematics such as teacher education
programs, teaching social norms, teachers’ life outside the classroom, characteristics of teacher's
personality, the situation in the classroom, and student life outside the classroom. Teachers also may
15

have a tendency toward a view on teaching and learning mathematics which includes encouraging
students to be actively involved in solving problems in various contexts. However, our findings about
teachers’ view on teaching problem solving seemed contrast with the study of Wijaya, et al [20] and
Maulana [43] which point out that teachers’ view of teaching mathematics task in their study admit to
mostly gave context-based tasks which explicitly provide the needed procedures and contain only the
information that is relevant for solving the tasks. This teachers’ view does not support the idea of
teaching mathematics as problem solving as our finding on teachers which agreed to teach
mathematics as problem solving. Furthermore, teachers in their study prefer to teach using directive
teaching approach by mostly explaining a topic while students write, listen, and answer closed
questions. Thus, there is an inconsistency between teachers’ actual practical teaching and their view on
teaching problem solving. A conjecture of this issue could be related with teacher’s understanding on
problem solving knowledge, particularly on the lack of teachers’ problem solving content knowledge
as we found in this study since there is a significant association among teachers’ understanding,
beliefs, and teaching practice on problem solving [16, 44, 45].
To sum up, we argue that: (1) teachers’ understandings on problem solving content knowledge
was less than those on pedagogical problem solving knowledge, (2) teachers more believed on
mathematics and mathematics learning as body static, while in practice, they tend to believe to the
views that they should teach mathematics as a dynamic/problem solving activity. The implication of
this study recommends the need to develop a program of teacher professional development in
understanding and applying problem solving in teaching practices taking into consideration the
difficulties experienced by teachers primarily on problem solving content knowledge. Thus there will
be a balance between teachers’ view and teachers’ actual knowledge and practice toward mathematical
problem solving.
Acknowledgements
We would like to thank Ministry of Higher Education and Research on fundamental research grant in
2015, the Rector of Surabaya State University and the Dean of the Faculty Mathematics and Natural
Science, Surabaya State University.
References
[1] NCTM 2000 Principles and Standards for School Mathematics (NCTM:Reston, VA)
[2] Stacey, Kaye 2005 The Place of Problem Solving in Contemporary Mathematics Curriculum
Documents. Journal of Mathematical Behavior 24 341–350
[3] Kaur, Berinderjeet. & Yeap Ban Har 2009 Mathematical Problem Solving in Singapore. In
Mathematical Problem Solving: Year book 2009, Association of Mathematics Educator edited
by Berinderjeet Kaur, Yeap Ban Har & Manu Kapur. Singapore: World Scientific, page 3-13
[4] Goos, Merrilyn, Stilman, Gloria., Vale, Colleen 2007 Teaching Secondary School Mathematics:
research and practice for 21st
century (Crows Nest, NSW: Allen & Unwin)
[5] Depdiknas 2009 Perkembangan Kurikulum Indonesia (Jakarta: Departemen Pendidikan
Nasional, Direktorat SMP)
[6] Sulivan, Peter., Mousley, Judith, & Jorgensen, Robyn 2009 Task and Pedagogies that Facilitate
Mathematical Problem Solving. In Mathematical Problem Solving: Year book 2009,
Association of Mathematics Educator edited by Berinderjeet Kaur, Yeap Ban Har & Manu
Kapur (Singapore: World Scientific) page 17-42
[7] OECD 2013 PISA 2012 Results in Focus (Paris: OECD Publishing)
[8] Mullis, I.V.S., Martin, M.O., Foy, P., & Arora, A 2012 TIMSS 2011 International Result in
Mathematics (Chestnut Hill, USA: TIMSS & PIRS International Study Center)
16

[9] Pehkonen, E., Hannula, M., & Björkqvist, O 2007 Problem solving as a teaching method in
mathematics education. In Pehkonen, E., Ahtee, M. & Lavonen, J. (Ed.), How Finns Learn
Mathematics and Science page 121–129 (Rotterdam: Sinse)
[10] National Center for Education Statistics USA. 2013 PISA 2012 Data Tables, Figures, and
Exhibits. Retrieved March 28, 2014, from http://nces.ed.gov/pubs2014/2014024_tables.pdf
[11] Siswono, Tatag Y.E., Abadi, Rosyidi, A.H 2008 Pengembangan Model Pembelajaran
Matematika Berbasis Pengajuan dan Pemecahan Masalah untuk Meningkatkan Kemampuan
Berpikir Kreatif Siswa Sekolah Dasar (Surabaya: Penelitian Universitas Negeri Surabaya)
[12] Kohar A.W & Zulkardi 2015 Developing PISA-like mathematics tasks for investigating
Indonesian students’ profile of mathematical literacy, in pursuit of quality mathematics
education for all, Cebu City May 11-15, in Chaterine Vistro Yu (Ed) Proceedings of the 7th
ICMI-East Asia Regional Conference on Mathematics Education (Quezon City: Philippine
Council of Mathematics Teacher Educators (MATHTED), Int), page 159-168
[13] Wijaya, A., van den Heuvel-Panhuizen, M., Doorman, M., & Robitzsch, A 2014 Difficulties in
solving context-based PISA mathematics tasks: An analysis of students’ errors The Mathematics
Enthusiast 11 555-584.
[14] Grouws, D. A., & Cebulla, K. J. 2000. Improving student achievement in mathematics.
Brussels: International Academy of Education.
[15] Polly, D., McGee, J. R., Wang, C., Lambert, R. G., Pugalee, D. K., & Johnson, S. (2013). The
Association between Teachers' Beliefs, Enacted Practices, and Student Learning in
Mathematics. Mathematics Educator, 22(2), 11-30.
[16] Stipek, D. J., Givvin, K. B., Salmon, J. M., & MacGyvers, V. L. 2001. Teachers’ beliefs and
practices related to mathematics instruction. Teaching and teacher education, 17(2), 213-226.
[17] Chapman, O. 2013. Mathematical-task knowledge for teaching. Journal of Mathematics Teacher
Education, 16, 1-6.
[18] Franke, M.L., & Kazemi, E. 2001. Learning to teach mathematics: Focus on student thinking.
Theory into Practice, 40(2), 102–109.
[19] Ball, D. L., Thames, M. H., & Phelps, G. 2008. Content knowledge for teaching what makes it
special?. Journal of teacher education, 59(5), 389-407.
[20] Wijaya, A., van den Heuvel-Panhuizen, M., & Doorman, M. 2015. Teachers’ teaching practices
and beliefs regarding context-based tasks and their relation with students’ difficulties in solving
these tasks. Mathematics Education Research Journal, 27(4), 637-662.
[21] Borko, H., & Shavelson, R. 1990. Teacher decision making. In B. F. Jones & L. Idol (Eds.),
Dimensions of thinking and cognitive instruction (pp. 311–346). Elmhurst, IL: North Central
Regional Educational Laboratory and Hillsdale, NJ: Erlbaum.
[22] Chapman, O 2015 Mathematics teachers’ knowledge for teaching problem solving. LUMAT 3
19-36
[23] Vistro-Yu, Catherine P. 2009 Using Inovvation Techniques to Generate ‘New’ Problems. In
by Berinderjeet Kaur, Yeap Ban Har & Manu Kapur (Singapore: World Scientific) page 185-
204
[24] Polya, G 1973 How To Solve It (2nd
edition) (New Jersey: Princeton University Press)
[25] Clements, M. A 1980. Analyzing children’s errors on written mathematical task. Educational
Studies in Mathematics 11 1-21
[26] Maass, K 2010 Classification scheme for modelling tasks. Journal für Mathematik-Didaktik,
31(2), 285-311
[27] OECD 2010 PISA 2009 results: What students know and can do. Student performance in
reading, mathematics, and science (Vol. I) (Paris: OECD)
[28] Vale, P., Murray, S., & Brown, B 2012 Mathematical literacy examination items and student
errors: an analysis of English Second Language students' responses. Per Linguam: a Journal of
Language Learning= Per Linguam: Tydskrif vir Taalaanleer 28 65-83.
17

[29] OECD 2013 PISA 2012 Assessment and Analytical Framework: Mathematics, Reading,
Science, Problem Solving and Financial Literacy (Paris: OECD Publishing)
[30] Shimizu, Y 2009 Japanese Approach to Teaching Mathematics via Problem Solving. In
by Berinderjeet Kaur, Yeap Ban Har & Manu Kapur (Singapore: World Scientific) page 89-101
[31] Koponen, M 2015 Teacher’s instruction in the reflection phase of the problem solving
process. LUMAT 3 55-68
[32] Raymond, A. M. 1997 Inconsistency between a beginning elementary school teacher's
mathematics beliefs and teaching practice Journal for research in mathematics education 28
550-576.
[33] Wilson, James W., Fernandez, Maria L., & Hadaway, Nelda 1993 Mathematical Problem
Solving. In Research Ideas for the Classroom: High School Mathematics edited by Patricia S.
Wilson (New York: Macmillan) page 57-78
[34] Weldeana, H. N., & Abraham, S. T 2014 The effect of an historical perspective on prospective
teachers’ beliefs in learning mathematics Journal of Mathematics Teacher Education 17 303-
330
[35] Ernest, P 1991 The philosophy of mathematics education (London: The Falmer Press)
[36] Beswick, K 2012 Teachers' beliefs about school mathematics and mathematicians' mathematics
and their relationship to practice Educational Studies in Mathematics 79 127-147
[37] Schoenfeld, A. H 1992 Learning to think mathematically: Problem solving, metacognition, and
sense making in mathematics Handbook of research on mathematics teaching and learning,
334-370.
[38] Ruthven, K 1987 Ability stereotyping in mathematics Educational Studies in Mathematics 18
243–253
[39] Zhang, Q.P., Sze, I.N.L., & Y 2015 Pre-service teachers’ beliefs about mathematics in China
and Thailand, in pursuit of quality mathematics education for all, Cebu City May 11-15, in
Chaterine Vistro Yu (Ed) Proceedings of the 7th ICMI-East Asia Regional Conference on
Mathematics Education (Quezon City: Philippine Council of Mathematics Teacher Educators
(MATHTED), Int), page 517-524
[40] Beswick, K. 2012 Teachers' beliefs about school mathematics and mathematicians' mathematics
and their relationship to practice Educational Studies in Mathematics 79 127-147.
[41] Carpenter, T. P., Fennema, E., Peterson, P. L., Chiang, C. P., & Loef, M. 1989. Using
knowledge of children’s mathematics thinking in classroom teaching: An experimental
study. American educational research journal, 26(4), 499-531.
[42] Anderson, Judy., White, Paul., Sulivan, Peter 2005 Using a Schematic Model to Represent
Influences on, and Relationships Between,Teachers' Problem-Solving Beliefs and Practices
Mathematics Education Research Journal 17 9-38
[43] Maulana, R., Opdenakker, M.-C., Den Brok, P., & Bosker. 2012. Teacher-student interpersonal
behavior in secondary mathematics classes in Indonesia. International Journal of Science and
Mathematics Education, 10(1), 21–47.
[44] Da Ponte, J. P., & Chapman, O. 2006. Mathematics teachers’ knowledge and
practices. Handbook of research on the psychology of mathematics education, 461-494.
[45] Ball, D. L., Lubienski, S. T., & Mewborn, D. S. (2001). Research on teaching mathematics: The
unsolved problem of teachers’ mathematical knowledge.Handbook of research on teaching, 4,
433-456.
18

Appendix
Task 1
Futsal Tournament
In the end of this year, a subdistrict government held a futsal tournament involving some futsal teams from
schools within the subdistrict. To prepare the tournaments, three futsal teams from Eksakta school held a futsal
training in their schoolyard once a week. Each of them face each other in a match exactly once. The following
table presents the result of the training this week.
Team Won Lost Drawn Goals
Goals For Goals Againts
Mentari 1 0 1 7 5
Surya FC 1 0 1 8 6
Rajawali 0 2 0 5 9
Note:
Won :The number of matches won this week
Lost :The number of matches lost this week
Drawn :The number of matches drawn this week
Goals for: The number of goals scored this week
Goals against: The number of goals conceded this week
What is the score of the match Mentari vs Rajawali? Explain your reason.
Task 2
World Online Math Literacy Contest
An online Maths Literacy Competition was attended by participants from several cities, namely, Jakarta
(Indonesia), Sydney (Australia) and Ankara (Turkey). The competition lasts for 1 hour at the same time on each
of the local time. Here is the time difference in each country
Since all participants are students, so they cannot follow the contest at 07:00 a.m. to 12:00 a.m on each local
time for having school and at 11 p.m. to 6:00 a.m. for bedtime. Find the appropriate range of time that the
committee should select for the contest. Explain your strategy.
Ankara (early morning)
Greenwich (midnight) Jakarta (morning) Sydney (noon)
19

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