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Copyright ? 2006-2013 Scientific Research Publishing Inc. All rights reserved.
2012. Vol.3, No.8, 1380-1383
Published Online December 2012 in SciRes (http://www.SciRP.org/journal/ce) http://dx.doi.org/10.4236/ce.2012.38201
Copyright © 2012 SciRe s .
Teachers’ Creativity in Posing Statistical Problems from
Effandi Zakaria1,2, Faridah Salleh1
1Department of M ethodology and Educ ational Practice, Faculty of Education, Universiti Kebangsaan Malaysia,
2Institute of Space Science, Universiti Kebangsaan Malaysia, Bangi, Malaysia
Received October 27th, 2012; revised November 29th, 2012; accepted December 10th, 2012
Choosing a quality problem in mathematics is a challenge for many teachers. Teachers cannot rely on
textbooks for good problems. They have to be able to pose their own problems in order to promote
mathematical thinking among students. This study was conducted to explore the creativity of 175 teachers
in terms of fluency, flexibility, and originality in posing statistical problems. Participants consisted of
secondary school teachers from twenty schools in Peninsular Malaysia. Teaching experience was ranged
from 1 to 33 years. The features of the problems posed by these teachers were also studied. The partici-
pants were provided a stimulus, which was a set of ungrouped discrete data, and they were asked to pose
as many problems as they could. The posed statistical problems were supposed to promote mathematical
thinking and to increase students’ understanding. Findings showed that participants were able to pose a
total of 270 (74%) statistical problems within the time given. The mean of the creativity score was 11.08
(s.d. = 6.76). Analysis showed no significant difference in creativity between gender and the value of t =
–.346, p = .73, where p > .05. Analysis showed significant differences in the teachers’ creativity scores
for three groups of teachers: F(2172) = 6.83, p = .001, p < .05.The results also showed that 115 (31.5%)
posed problems focuses on the statistical content measure of central tendency. The study provided expo-
sure to the teachers to pose problems that can trigger students’ thinking in solving statistical problems.
Keywords: Statistical Problems; Creativity; Fluency; Flexibility; Originality
NCTM (2000) has strongly recommended both problem pos-
ing and problem solving activities are implemented in the
teaching and learning of mathematics. The quality of the prob-
lems can serve as an index of how well a person can solve
problems (Kilpatrick, 1987). Gonzales (1996) found that there
was a correlation between mathematical competence and prob-
lem posing. Therefore, teachers should have skills in both pos-
ing problems and solving problems in order to help students to
learn mathematics (Brown & Walter, 1983; Kilpatrick, 1987;
Problem posing is a creation of a new problem or the refor-
mulation of problem through a given situation (Leung, 1993).
Problem posing also involve the generation of new problems or
questions in order to explore a given situation as specially as a
complex problem (Silver, 1994) In posing new problems, the
emphasis is not to produce solutions but the structure of the
problems itself (Lowrie, 2002). A study by Silver et al. (1996)
shows that the activity of problem posing can be done either
during the process of problem-solving or after the process of
problem solving. According to Chua (2004), Jensen (1973), and
Noraini (2001), Guilford and Torrance have identified four
types of creative thinking: 1) originality, which is the process of
creating new ideas and original; 2) fluency, which is the proc-
ess of creating a lot of ideas at one time, where the generated
ideas do not necessarily to be too different; 3) flexibility, which
is the process of producing a range of ideas and different cate-
gory altogether; and 4) elaboration, which is the process of
adding one idea to another idea or process to see something in
detail. Balka (1974) used to measure creativity in mathematics
in terms of their flue ncy , fle xibi lity, and t he ori ginali ty of posed
Several studies found that mathematics instruction is still us-
ing the traditional methods (Tengku Zawawi, 2005; TIMSS,
2007; Yusminah, 2009; Zakaria & Iksan, 2007). In fact, ac-
cording to Tengku Zawawi (2005) and Yusminah (2009), teach-
ers are not able convey knowledge and conceptual skills to
students effectively. Students were less active in teaching and
learning process due to the teacher-centered approach. Student
interaction with and among teachers is quite limited. Students
prefer to listen for information without making any contribution.
They are not willing to ask or to give an opinion (Ministry of
Malaysian Education, 2008). Furthermore, Cunningham (2004)
states that student serve mostly as listeners and have little re-
sponsibility for constructing their own knowledge during les-
sons. Promoting posing problems in class would give students
the experience of having a series of related problems per each
topic. As teachers become proficient in problem posing, they
will be more willing to have their students engage in such ac-
tivities (Silver et al., 1996). Students would be able to form and
to develop their skills to identify strategies that provide solu-
tions to the problems. According to Perez (1985), research
shows that, if students are able to pose a good problem, then the
probability that they can solve problems is high. Understanding
of mathematical concepts learned by the students can be shown
through the works from problem-generating activities (English,
1997; Stickles, 2006).
E. ZAKARIA, F. SALLEH
NCTM (2000) suggested that all students should have knowl-
edge of statistics so that it was not limited to a group who were
interested to conduct research only. Initiatives to improve teach-
ing and learning of statistics and probability have been high-
lighted by the document of NCTM since 1989. More emphasis
should be given to these issues so that students can apply the
skills in data handling (Brumbaugh & Rock, 2006). However,
this topic is often not taught widely. Studies conducted in the
US shows that high school students said they get little exposure
or indirectly exposed to the topics of statistics or probability
(Brumbaugh & Rock, 2006).
According to the study by Leung and Silver (1997), pre-ser-
vice teachers were able to pose mathematical problems, but
mostly the ir problems we re lack ing in math ematic al compl exity ,
while Crespo (1998) shows that teachers need experience to
pose problems so that they can create a good and challenge
problems. Giving an original pro blem to students to solve w oul d
be the most effective way to teach mathematics (Perrin, 2007).
Skill in posing statistical problems should be seen as an ap-
proach to identify how teachers are able to pose good problems
and use the information from environment that leads to creative
thinking. The ability to pose a variety of good problems can
improve self-confidence in promoting the posing activities in
class (Stickles, 2006). To pose a problem requires active think-
ing processes. A stimulus is needed as a trigger for the ideas
(Slavin, 2000). In this study, a stimulus, which was the un-
grouped discrete data, was given to teachers. There were five
steps involved in the process of posing statistical problems.
First, a problem poser would try to identify the problem by
looking at the information available to suit his/her goal. Second,
the poser would identify the source of the problem. In the third
step, the problem poser determined what was required for the
cognitive domain of thinking, such as the level of knowledge,
understanding, or even evaluating (Winograd, 1991). The fourth
step carried out posing an interesting problem (Grundmeier,
2003). Finally, the fifth step was checking or looking back at
what had been done (Polya, 1973).
In this study, statistical creativity was measured in terms of
fluency, flexibility, and originality of the problems (Balka, 1974).
The fluency of the problem refers to the number of problems
that can be posed in a certain time, not the quality of ideas that
are emphasized. Flexibility refers to the number of posed prob-
lems different from the quality within the posed problem. Origi-
nality refers to uncommon posed problems between participants.
Problems features referred to the content of the statistical syl-
labus of Mathematics Lower Secondary and Upper Secondary,
including Additional Mathematics. The features also referred to
the cognitive level of Bloom’s taxonomy (Bloom, 1976) which
refers to the six levels of cognitive skills needed in teaching and
learning mathematics. The hierarchy is arranged from the lower
level of thinking to the highest level of thinking, as follows:
remembering, understanding, applying, analysing, synthesis, and
creating (Anderson & Krathwohl, 2001).
The purpose of the study was to identify the creativity of sta-
tistical problems in terms of fluency, flexibility, and originality.
In addition, this study also tried to identify the features of the
posed problem. Specifically, the objectives of this study were
1) To identify the features of the posed problems among
teachers in terms of statistical content.
2) To determine whether there were any statistically signifi-
cant differences in creativity to pose problem among teachers
according to their teaching experience.
This study involved a total of 175 mathematics teachers from
twenty schools in Peninsular Malaysia. Samples of this study
were carried out randomly. Teaching experience of these teach-
ers was ranged from 1 to 33 years. All of them had experience
in teaching mathematics and Additional Mathematics. There
were several new teachers who had taught less than a year, but
they had received in-house training from time to time. Data was
collected by the researcher from each school at the selected
time. A written instrument was used to collect posing problems.
This instrument included a stimulus consisting of 20 ungrouped
discrete data. Before the study was conducted, the teachers in-
volved were informed. Through the given stimulus, teachers
were asked to pose as many statistical problems that can trigger
students to think creatively. The problems were not evaluated
for their solution, but for activities that increase the under-
standing in learning a particular topic. Teachers were reminded
that they did not need to solve the posed problems. Teachers
were given 20 minutes to carry out their tasks. The posed prob-
lems were analyzed in three steps. The first step was to identify
whether the problem can be solved. This was due to the logical
problem and covered the content in the syllabus. The second
step was to identify the statistical content, and the third step was
given a score of fluency, flexibility, and the originality of the
problem in accordance with a rubric.
Findings and Discussion
The first finding concerns the features of posed problems
among teachers in terms of statistical content to answer the
research question, “What are the features of the posed problem
in terms of the statistical content?” The answers are provided in
Table 1, which shows the distribution of statistical problems by
Statistical content is divided into five sub-categories: content;
basic statistics such as frequency, score/value, percentage, ratio,
interval classes, class size, frequency tables; measure of central
tendency which consists of the mean, mode, median; dispersion
measurement which consists of range, quartile range, standard
deviation, variance; cumulative frequency tables, bar graphs,
line graphs, pie charts, histograms, frequency polygon, ogive;
an opinion, conclusion/summary.
Teachers had posed a total of 365 problems, but after the
analysis only 270 (74%) was categorized as a statistical prob-
lem. The problem posed by these teachers, mostly centered on
the one sub-main statistical content with 173 (47.4%) problems.
The percentage was highest in the central measurement, 115
(31.5%) problems. A total of 71 (19.5%) posed problems were
the combination of two sub-content. For example, there were
40 (11.0%) posed problems in the category of “The combina-
tion of BS and one of the CT, DM, CG, and OC”. While only
26 (7.1%) of the posed problems were a combination of three
sub-contents. The posed problems that involved student opin-
ions, making conclusions, and discussion were very limited.
For example, in the main categories of sub-content, only 5
(1.4%) problems posed in the “OC” category. A total of 30
(8.2%) of the problems for “The combination of CT and one of
the DM, CG, OC” were in two sub-content category, and only 1
(.3%) problem was in the category of “The combination of CT
Copyright © 2012 SciRe s . 1381
E. ZAKARIA, F. SALLEH
Copyright © 2012 SciRe s .
Number of problems according to categories of statistical content.
Statistical content category Number of problems
5sub-mai n content
Basic statisti cs (BS) 36 (9.9%)
Central of tende ncy (CT) 115 (31.5%)
Dispersion measurement (DM) 14 (3.8%)
Charts and gra phs (CG) 3 (.8%)
Opinions and conclusi ons (OC) 5 (1.4%)
The combination of two sub-content
BS and one of the CT, DM, CG, OC 40 (11.0%)
CT and one of DM, CG, OC 30 (8.2%)
DM & OC 1 (.3%)
The combination of 3 sub-content
BS with any 2 from CT, DM, CG, OC 19 (5.2%)
CT with any 2 of the DM, CG, OC 7 (1.9%)
Unaccepted problems 95 (26.0%)
& OC”. While the combinations of three sub-contents were
from category “The combination of KMS with any 2 of the
KMP, C & G, P & K,” only seven (1.9%) problems were posed.
The following are examples of posed problems:
“The data above represent number of goals by a football
team for 1 season. What is the average goal for the football
team for that season?”
“Data below represent the number of books read by 20 stu-
dents in January 2011.
1) Construct a frequency table for the given data.
2) By using data from 1), draw a pie chart to represents the
3) State the mode for the number of books read.”
“The above data shows the number of children of a group of
staff in an office. Find:
1) The mean number of children of the staff.
2) The standard deviation of children of the staff.”
This study shows that even if teachers were asked to pose
problems that lead to higher-order thinking, some teachers
posed problems in the lower category (Table 2). Percentages of
18.1% (66 problems) on understanding level, 48.5% (177
problems) on applying level and only 7.4% (27 problems) on
analysing level. A study by Senk et al. (1997) found that 68%
of posed problems were in low-level category, While, Harpster
(1999) found that 60% of the teachers posed low-level prob-
lems. According to Thompson (2008), even if the teacher can
identify the problem in the form of low-level thinking or
higher-order thinking (HOT), but, when they were asked to
pose problems of higher-order thinking, only 45% of teachers
will posed higher-level problems. Most of the teacher’s experi-
enced this problem posing activity for the first time. The teach-
ers were used to creating test questions by reformulating prob-
lems that were commonly found in the textbook or in references.
This is in line with the findings of Lowrie (2002), in which the
inclination problems generated for the first time will usually be
similar to exercise or routine problems. Therefore, the consis-
tency of posing problems was essential. A study by Cespo
(2003) found that teachers can change the patterns of posed
problems and become better quality problem posers if they do
this activity often.
Category of bloom taxonomy.
Level of tax onomy Bloom Task
Understanding 66 (18.1%)
Applying 177 (48.5%)
Analysing 27 (7.4%)
Unaccepted problems 95 (26.0%)
Total 365 (100%)
The second research question: “Are there any statistically
significant differences in creativity to generate problem among
teachers according to teaching experience?” Table 3 showed
the summary for one way between-groups ANOVA.
Testing for homogeneity of variance for the three groups
showed the values of the Levene’s = 1.258 (Sig. = .287), which
is >.05. ANOVA analysis was used to explore the impact of
teachers’ experience in creativity scores through the ungrouped
discrete data. Teachers were grouped into three groups accord-
ing to years of teaching (group 1: less than 6 years; group 2: 6
to 10 years; group 3: more than 10 years). Analysis showed sig-
nificant differences at p < .05 in creativity scores for three
groups of teachers: F (2172) = 6.83, p = .001. Post-hoc compa-
risons using the Turkey HSD test showed a group 1 mean score
(M = 8.53, SD = 5.85) were significantly different from group 3
(M = 13.15, SD = 6.94). Group 2 (M = 10.59, SD = 6.53) did
not differ significantly from group 1 or group 3. Experience
teachers (>10 yrs) are more creative in generating problems
than other teachers. This may be due to the exposure to more
statistical knowledge during their teaching career.
Teachers who participated in this study were asked to pose a
statistical problem. However, from 365 posed problems, only
270 (74%) problems were accepted for this study. The excluded
problems were not logical, did not have a solution, or were
incomplete. The features posed problems were identified in
terms of statistical content. The findings showed that not all
teachers tend to posed higher-order thinking problems. There
were differences in terms of experience; the differences be-
tween the group of teachers whose experience ranged fewer
than six years and other groups were clearly shown.
E. ZAKARIA, F. SALLEH
One way between-groups ANOVA in creativity scores based on teachers’ experience.
Variables Teachi ng experience N Mean F p
Teachers’ creativity K1: <6 yrs 43 8.53 6.831 .001
K2: 6 - 10 yrs 64 10.59
K3: >10 yrs 68 13.15
Teachers should be exposed to problem-generating activities
before they can perform this activity with students. Teachers
should engage students in class to participate in problem-posing
(Zakaria & Ngah, 2011). Creating a new problem or reformu-
lating a problem is not an easy task, and it requires planning,
skill, and personal strength to do this activity effectively. Pre-
vious studies have shown that problem posing can increase stu-
dents’ comprehension, encourage communication of mathema-
tics, improve students’ self-confidence, and establish positive at-
titudes towards mathematics. Problem posing should be given
serious consideration by teachers in preparing lesson activities.
Anderson, L. W., & Krathwohl, D. R. (Eds.) (2001). A taxonomy for
Learning, teaching, and assessing: A revision of Bloom’s taxonomy
of educational objectives. New York: Addison Wesley Longman.
Balka, D. S. (1974). The development of an instrument to measure
creative ability in mathematics. Ph.D. Thesis, Main St . Durha m, NH:
University of New Hampshire.
Bloom, B. S. (1976). Human characteristics and school learning. New
Brown, S. I., & Walter. M. I. (1983). The art of problem posing. Phila-
delphia: The Franklin Institute Press.
Brumbaugh, D. K., & Rock, D. (2006). Teaching secondary mathemat-
ics (3th ed.). Hillsdale, NJ: Lawrence Erlbaum Associates.
Chua, Y. P. (2004). Creative and critical thinking styles. New York:
University Putra Malaysia Press.
Crespo, S. (2003). Learning to pose mathematical problems: Exploring
changes in preservice teachers’ practices. Educational Studies in
Mathematics, 52, 243-270. doi:10.1023/A:1024364304664
Cunningham, R. F. (2004). Problem posing: An opportunity for in-
creasing students responsibility. Mathematics and Computer Educa-
tion, 38, 83-89
English, L. D. (1997). Promoting a problem posing classroom. Teach-
ing Children mathematics, 4, 172-180
Gonzales, N. A. (1996). Problem formulation: Insights from student ge-
nerated questions. School Science and Mathematics, 96, 152-157.
Grundmeier, T. A. (2002). University students’ problem posing and
attitudes towards mathematics. Primus: Problem, Resources, and Is-
sues in Mathematics Undergraduate Studi es , 12, 122-134.
Grundmeier, T. A. (2003). The effects of mathematical problem posing
providing experiences for k-8 pre-service teachers: Investigating
teachers’ beliefs and characteristics of posed problems. Ph.D. Thesis,
Main St. Durham, NH: Universty of New Hampshire.
Harpster, D. L. (1999). A study of possible factors that influence the
construction of teacher-made problem that assess higher-order think-
ing skill. Ph.D. Thesis, Bozeman: Montana State University.
Jensen, L. R. (1973). The relationships among mathematical creativity,
numerical aptitute and mathematical achievement. Ph.D. Thesis. Au -
stin: University of Texas.
Kilpatrick, J. (1987). Formulating the problem: Where do good prob-
lems come from? In A. H. Schoenfeld (Ed.), Cognitive Science and
Mathematics Education (pp. 123-147). Hillsdale, NJ: Lawrence Erl-
Leung, S. S. (1993). Mathematical problem posing: The influence of
task formats, mathematics knowledge, and creative thinking. In I.
Hirabayashi, N. Nohda, k. Shigematsu, & F. Lin (Eds.), Proceedings
of 17th International conference of International Group for the Psy-
chology of Mathematics Education (pp. 33-40). Tsukuba, J apan.
Leung, S. K., & Silver, E. A. (1997). The role of task format, mathe-
matics knowledge, and creative thinking on the arithmetic problem
posing of prospective elementary school teachers. Mathematics Edu-
cation Research Journal, 9, 5-24. doi:10.1007/BF03217299
Lowrie, T. (2002). Designing a framework for problem posing young
children generating open-ended task. Contemporary Issues in Early
Childhood, 3, 354-364. doi:10.2304/ciec.2002.3.3.4
Ministry of Malaysian Education (2008). Inspectorate and quality as-
surance. Annual Report, Putra Jaya.
NCTM (2000). Principles and Standards for School Mathematics. Res-
ton, VA: NCTM.
Noraini, I. (2001). Pedagogy in mathematics education. Kuala Lumpur:
Utusan Publications & Distributors Sdn Bhd.
Perez, J. A. (1985). Effects of students generated problems on problem
solving performance (writing word problems). Ph.D. Thesis, New
York: Columbia University College.
Perrin, J. R. (2007). Problem posing at all levels in the calculus class-
room. School Science and Mathemati c s , 107, 182-192.
Polya, G. (1973). How to solve it: A new aspect of mathematical me-
thod (2nd ed.). Princeton, New Jersey: Princeton Univeversity Pres s.
Senk, S. L., Beckmann, C. E., & Thompson, D. R. (1997). Assessment
and grading in high school mathematics classroom. Journal for Re-
search in Mathematics Education, 28, 187-215. doi:10.2307/749761
Silver, E. A. (1994). On mathematical problem posing. For the Learn-
ing of Mathematics, 14, 19-28.
Silver. E. A., Mamona-Downs, J., Leung, S. S., & Kenney, P. A. (1996).
Posing mathematical problems: An exploratory study. Journal for
Research in Mathematics Education, 27, 293-309.
Slavin, R. E. (2000). Educational psychology: Theory and practice (6th
ed.). Boston: Allyn & Bacon.
Stickles, P. R. (2006). An analysis of secondary and middle school
teachers’ mathematical problem posing. Ph.D. Thesis, Bloomington:
University of Indiana.
Tengku Zawawi, T. Z. (2005). Pedagogical content knowledge of frac-
tion among primary school mathematics teacher. Ph.D. Thesis, Bangi:
Faculty of Education, Universiti Kebangsaan Malaysia.
TIMSS (2007). International mathematics report. Boston: International
Thompson, T. (2008). Mathematics teachers’ interpretation of higher-
order thinking in Bloom’s taxomomy. Electronic International Jour-
nal of Mathematics Education, 3, 96-109.
Winograd, K. (1991). Writing, solving and sharing original math story
problem. Case studies of fifth grade children’s cognitive behaviour.
The Annual Meeting of the American Educational Research Associa-
tion, Chicago, 3-7 April 1991.
Yusminah, M. Y. (2009). A case study of teachers’ pedagogical content
knowledge of functions. Proceedings of the 3th International Con-
ference on Science and Mathematics Education. Penang, 10-12 No-
Zakaria, E., & Iksan, Z. (2007). Promoting cooperative learning. Eurasia
Journal of Mathematics, Science & Technology Education, 3, 35- 39.
Zakaria, E., & Ngah, N. (2011). A preliminary analysis of students’
problem-posing ability and its relationship to attitudes towards prob-
lem solving. Research Journal of Applied Sciences, Engineering and
Technology, 3, 866-870.
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