Teachers’ Creativity in Posing Statistical Problems from Discrete Data

doi:10.4236/ce.2012.38201

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Creative Education

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

1380

Teachers’ Creativity in Posing Statistical Problems from

Discrete Data

Effandi Zakaria1,2, Faridah Salleh1

1Department of M ethodology and Educ ational Practice, Faculty of Education, Universiti Kebangsaan Malaysia,

Bangi, Malaysia

2Institute of Space Science, Universiti Kebangsaan Malaysia, Bangi, Malaysia

Email: [email protected]

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

Introduction

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;

Silver, 1994).

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

problems.

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

as follows:

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.

Methodology

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

category.

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

E. ZAKARIA, F. SALLEH

1382

Table 1.

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%)

Total 365

& 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:

 Example 1:

“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?”

 Example 2:

“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

data.

3) State the mode for the number of books read.”

 Example 3:

“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.

Table 2.

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.

Conclusion

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

Table 3.

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.

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