Teachers’ pedagogies and their impact on students’ mathematical problem solving

Journal of Mathematical Behavior 24 (2005) 238–252

Teachers’ pedagogies and their impact on students’mathematical problem solving�

Kai Fai Hoa,∗, John G. Hedbergb

a Centre for Research in Pedagogy and Practice, National Institute of Education, Nanyang Technological University,7-03-118G, 1 Nanyang Walk, Singapore 637616, Singapore

b Australian Centre for Educational Studies, Macquarie University, Australia

Available online 30 September 2005

Abstract

This paper examines the classroom practices of three teachers teaching mathematics at the 5th grade level in three Singaporeschools. Using a video-coding scheme, a series of lessons was coded into relevant phases comprising problem solving, teachingconcepts/skills, going over assigned work, and student activities. It explores the teachers’ pedagogical experimentation in theirteaching of mathematical problem solving after an analysis of their current practices. It concludes with a review of the effects suchchanges have on students’ problem solving successes as reflected in pre- and post-problem-solving tests.© 2005 Elsevier Inc. All rights reserved.

Keywords: Mathematical problem solving; Teachers’ pedagogies; Video-coding; Classroom practices

1. Introduction

This study stems from the curriculum framework outlined in the Primary Mathematics Syllabus for all Singaporeschools (Ministry of Education, 2000). It places mathematical problem solving (MPS) as the central focus of thecurriculum (Fig. 1), where “mathematical problem solving includes using and applying mathematics in practical tasks,in real life problems and within mathematics itself” (MOE, 2000, p. 5). This entails a wide range of problem types— “from routine mathematical problems to problems in unfamiliar contexts and open-ended investigations that makeuse of the relevant mathematics and thinking processes.” There is purportedly less emphasis on the tell-show-doparadigm, and more emphasis on instructional practices that encourage problem solving, practical and investigativework, and communicative aspects of mathematics learning. Central to this focus is the teacher. AsHowson, Keitel,and Kirkpatrick (1981)have pointed out “one cannot truly talk, then, of a ‘national curriculum’, for it depends uponindividual teachers, their methods and understanding, and their interpretation of aims, guidelines, texts, etc. The partplayed by the individual teacher must, therefore, be recognized” (p. 2).

The mathematics curriculum may be viewed from three different perspectives, i.e., theintended curriculum, theimplemented curriculum, and theattained curriculum (Howson & Malone, 1984; Robitaille & Dirks, 1982). Thus,while it is intended that problem solvingshould be the central focus of the curriculum, the role of the teacher is

� This paper is drawn from a funded project CRP01/04 TSK, “Developing the Repertoire of Heuristics for Mathematical Problem Solving,” Centrefor Research in Pedagogy and Practice, National Institute of Education, Nanyang Technological University, Singapore.

∗ Corresponding author. Tel.: +65 6790 3370; fax: +65 6316 4787.E-mail address: [email protected] (K.F. Ho).

0732-3123/$ – see front matter © 2005 Elsevier Inc. All rights reserved.doi:10.1016/j.jmathb.2005.09.006

K.F. Ho, J.G. Hedberg / Journal of Mathematical Behavior 24 (2005) 238–252 239

Fig. 1. Framework of the Singapore Mathematics Program.

pertinent to its effective implementation and the question remains: to what extent is MPS emphasized in the classroom,and with what degree of success?

1.1. Teachers — the implementers

In a 2-year study in two Singapore schools to investigate pedagogical practices in the elementary mathematicsclassroom,Chang, Kaur, Koay, and Lee (2001)found that traditional teaching approaches predominated amongstthe teachers. The typical teaching approach was expository, followed by students practicing routine exercises toconsolidate the concepts, knowledge and skills. Chang et al.’s study involved video taping five 1-h mathematicslessons for four teachers, two from an elite school and two from a local school. In another study,Foong, Yap, and Koay(1996)described how a number of teachers expressed their concern over their perceived lack of skills for the teachingof mathematics using a problem solving approach. They found that teachers felt inadequately prepared to teachMPS when the examples were nonroutine problems that had several possible solutions. They doubted their abilityto communicate the multiple concepts required by students to understand without being confused by the number ofmethods and heuristics suggested in the newly released syllabus, and the teachers expressed unease with the emphasison open-ended problem solving. Such lack of confidence led to the general belief that there was an over reliance ontextbooks and a narrow range of problem types used in classroom examples, both contrary to the official syllabus (MOE,2000).

Teachers’ development in giving problem solving instructions has been insufficiently explored by researchers(Chapman, 1999; Lester, 1994). In one study,Norton, McRobbie, and Cooper (2002)investigated how nine teachersresponded to a reform curriculum (Board of Senior Secondary School Studies, 1992) in line with reforms initiated bythe National Council for Teachers of Mathematics (NCTM, 1989, 2000) and the Australian Education Council (AEC,1990). They sought to find if teachers who were using aninvestigative approach that involved students actively engagingin: “making sense of new information and ideas” (Curriculum Council, 1998, p. 1), “investigating mathematicalprocesses situated within meaningful contexts” (Australian Association of Mathematics Teachers Inc., 1996, p. 4), and“construction of meaning” (Anderson, 1994, p. 1). The teachers varied their pedagogical approaches differentially forstudents with different abilities, notwithstanding the stated goals of conceptual understanding for more able studentsand predominately calculational goals for less able students. Three teachers still favored the “show and tell” approachfor both groups of students, while another three employed a mix of “explain” and “show and tell” approaches. Ofthe remaining three, two used the investigative approach for the more able students and “show and tell” for the lessable. Only one teacher out of the nine used an investigative approach asintended in the curriculum for both groupsof students. The researchers observed that while the teachers expressed support for the investigative approach and theobjective of teaching for conceptual understanding, other factors (particularly preparation for high-stake examinations)appeared to influence the goals and approaches adopted in classrooms.

240 K.F. Ho, J.G. Hedberg / Journal of Mathematical Behavior 24 (2005) 238–252

Teachers’ understanding of problem solving, their interpretations of how to teach it and how much time to spendon it vary (Grouws, 1996). Possible conceptions of teaching problem solving include: teachingabout, teachingfor,and teaching via problem solving (Schroeder & Lester, 1989); problematizing mathematics as a way to think aboutproblem solving (Hiebert et al., 1996). The teacher’s role in implementing a curriculum that emphasizes problemsolving involves more than just expressions of support on the part of the teachers (Senger, 1999). Possible supportiveingredients in the process might include interventions that explorewhat approaches teachers could adopt, other suitablelesson formats or problem tasks.

In studies of approaches to teaching problem solving, teachers were often “assigned” particular approaches byresearchers who then proceeded to investigate the implementation and subsequent effects each approach has on stu-dents’ learning. InSigurdson, Olson, and Mason’s (1994)study, the effects of classroom teaching that incorporateda problem-solving dimension on student learning of mathematics were investigated. Three approaches were imple-mented: algorithmic practice, teaching with meaning, and a problem-solving approach. The problem-solving approachinvolved teaching with meaning (Sigurdson & Olson, 1992) plus a daily insertion of 10 min of problem-process work.The three approaches were assigned to the 41 teachers in the study. Preparation of teachers involved 10 h of work-shops for the algorithmic practice approach and 25 h of workshops for the other two approaches, all spread over theimplementation period of 5 months. The outcomes of their study were somewhat complex, with the analysis donealong the three approaches and the students in each approach divided into low-, medium-, and high-achievers. Theyclaimed, among other things, that the meaning and problem-process approaches in teaching were important, result-ing in more students’ learning with improved achievement and positive attitudes. They also noted that the higherachieving students benefited more. However, in another study about two classes, one high-ability and the other low,Holton, Anderson, Thomas, and Fletcher (1999)found that lower ability students seemed to benefit more from theintroduction of problem solving lessons. While such research on problem solving has significant implications, theextent and the way in which problem solving is implemented in the classroom remains largely unexplored. Empiricaldata about the way that teachers taught before their involvement in the project, observations of pedagogical practiceswithin in the classrooms during implementation, and the salient features of different approaches that impacted studentswas not collected. These aspects are important to a better understanding of the process of curriculumimplementa-tion.

The current study aims to address these issues. In particular, this study posits links between describing what ishappening in the classrooms and subsequent changes in teachers’ classroom practices without imposing or assigningany particular approach for teachers to adopt and follow. We also address the following issue: how students’ learningof problem solving skills is impacted by the changes in the teachers’ classroom practices.

1.2. Students — the “attainers”

Foong and Koay (1997)found interesting consequences of teachers’ lack of preparation in using the new approachesrecommended in a revised syllabus. Using eight pairs of items, each pair consisting of a standard word problem typicallyfound in textbooks and a realistic word problem where the student needed to consider the realities of the context of theproblem statements, the researchers found that students tended “to disregard the actual situation described” in wordproblems and “instead, go straight into exploring the possible combinations of numbers to infer directly the neededmathematical operations” (p. 73). Earlier,Koay and Foong (1996)many of the nearly 300 lower secondary studentsthat they examined failed to make connections between school mathematics and everyday life. These studies suggestthe teaching of MPS did not apply mathematics in practical tasks and real world problems, as mandated in theintendedcurriculum. Students’ attainments are falling short of the intentions.

Cai’s (2003)exploratory study suggests that most students were “able to select appropriate solution strategies tosolve” the tasks, and chose “appropriate solution representations to clearly communicate their solution processes”(p. 733). He explored fourth, fifth and sixth grade students MPS skills, using four tasks which were mathematicallyrich, and were “embedded in different content areas and contexts, and allowed Singaporean students’ thinking fromvarious perspectives.” Further, Singaporean students’ repeated top ranking performance in mathematics on the Trendsin International Mathematics and Science Study (TIMSS-2003) (Mullis, Martin, Gonzalez, & Chrostowski, 2004)suggests that the current syllabus (MOE, 2000) is working well.

Such seemingly contrasting findings about students’ attainment warrant a need for a closer look at teachers’ class-room practices and their possible impact on students’ learning.


2. Method

This study began by identifying the elements MPS pedagogical practice that exists in typical elementary mathe-matics classrooms. In particular, it addressed the main research question: What teachers’ classroom practices supportmathematical problem solving development in their students?

Following the collection of a systematic, evidence base describing current mathematics instruction practices, anintervention was designed to raise the teachers’ awareness of MPS ideas and processes and to support an increasedemphasis on the centrality of problem solving in the Singapore Mathematics Program. The intervention had threecomponents and followed a design research approach: First, teachers were interviewed to review salient features of theirclassroom practices and prompted to give their own descriptions and interpretations of events. Second, we conducteda workshop which discussedPolya’s (1957)four phases of problem solving — understanding, planning, executing,and looking back. Third, after the workshop and some lapse of time, informal post-lesson interviews were conductedwhen we returned to observe the teachers again. These were short discussions immediately after observations. Thepurpose was to talk about the lesson, its degree ofsuccess (in terms of problem solving instruction) and the teachers’own assessment of the outcomes. The intervention lead to a second research question: Given a reflective interventionemphasizing MPS instructional emphasis, do teachers change teaching strategies and does it result in increased students’problem solving successes?

From the analysis of the initial observations, we saw that the teachers, when introducing problems, tended to read theproblems quickly and proceed to immediately execute the solutions, with little or no strategic planning. They also didnot reflect on the solution or its success. As the goal was to explore the explicit development of students’ metacognitiveaspects of problem solving, when conducting the workshop, the importance of explaining during the reading of theproblem and the possibility of being more explicit about planning and reflecting were highlighted. The need to employmore rich and authentic problem tasks was also highlighted. Several examples were shown and their solutions wereworked through thoroughly following Polya’s four phases.

While it is not possible to trace the individual trajectory of each and every student’s learning, an adaptation ofthe Cobb, Stephan, McClain, and Gravemeijer (2001)approach was used to document the collective mathematicaldevelopment of a classroom community over the extended periods of time covered by instructional sequences. Cobbet al. resolved the issue about “the trajectory of. . . student’s learning” and the “significant qualitative differencesin their mathematical thinking at any point in time,” by proposing “a hypothetical learning trajectory as consistingof conjectures about the collective mathematical development of the classroom community” (p. 117). Likewise inthis study, it is difficult to ascertain the casual relations or direct impact between theintervention, possible teacherchange and students’ learning outcomes. Hence instructional impact on students was viewed in terms of a collectivemathematical development through their responses in a pre- and post-set of problem solving tasks.

2.1. Data sources

The chief source of data was from classroom observations which were recorded on video and audio, and thentranscribed. In addition, short discussions with teachers immediately after an observation contingent upon teachers’availability and the researchers’ field notes were secondary sources. For the question of whether there had been anincrease in students’ problem solving successes, a quantitative approach was used. Repeated paired-samplet tests ofstudents’ responses in the pre- and post-tests are used to test for significant performance differences.

The period of video recording spanned over 5 weeks for the pre-invention stage. The pre-test was administered to thestudents at the end of this stage. The reflective practice intervention, teachers’ workshop, interviews and discussions,occurred over the next month. Several weeks later, the researchers returned to video record some lessons in the post-intervention stage. Typically this spanned 3 weeks, at the end of which the post-test was administered to the students.The total time for the study spanned approximately 4 months.

2.2. The sample

The focus is on three teachers from three different schools who were involved in a larger study. Their involvementcame about on an opportunistic basis through a meeting with the schools’ superintendent. The classes they taught weredeemed academically weak for the Grade 5 level at their respective schools. Teacher A was from an upper band school


(i.e., above national average in academic terms). She has been teaching for 10 years mainly at the lower primary levels.At the time of the study, it was her first time teaching Grade 5 mathematics in a class of 37 students. Teacher B wasfrom a middle band school. He has been teaching for 3 years at the Grade 5 level. He worked previously in the financialindustry for about 10 years before obtaining a teaching diploma. His class had 36 students. Teacher C was from a lowerband school and has been teaching for 10 years. She had 33 students in her class — some of them had opted to be inher class even though they only qualified for a lower stream. All three teachers are form teachers for their classes andalso teach English and Science to their class. The analysis began with these teachers mainly because the researcherswanted to know the impact of the intervention on the weaker students first.

2.3. The coding scheme

A coding scheme was developed using the Grounded Theory approach (Glaser & Strauss, 1967) where ideas ofpedagogical phasesemerged as the video recordings were reviewed. After several iterations, a coding scheme thatdivided each lesson into five phases was developed. These phases include problem solving, teaching of concepts andskills, going over assigned work, student activities, and other classroom events.

The problem solving category refers to a phase where time is spent solving a problem. This usually happens whenthe teacher presents a problem as part of a problem-solving episode. This can be teacher-led, teacher-student-question-and-answer, or student-led. This phase was further analyzed usingPolya (1957)’sfour stages of problem solving,namelyUnderstanding, Planning, Executing, andReflecting. (This strategy was chosen as it seemed easiest for theteachers to begin their exploration compared to more recent adaptations, e.g.,Schoenfeld, 1985.)

The teaching of skills and concepts phase relates directly with the MPS curriculum framework (seeFig. 1). Teachingof skills occurs when the teacher uses class time to teach skills such as arithmetic or algebraic manipulation, estima-tion/approximation, mental calculation, communication, use of mathematical tools, handling data, etc. For example,the teacher might recall some common mistakes the class made in assigned work, and proceed to show and explain thecorrect steps. This teaching of skills is distinguished from the problem solvingExecution phase where skills areusedto solve a problem after some precedingUnderstanding/Planning has taken place. For the teaching of concepts, classtime is spent showing, demonstrating, defining, and explaining the numerical, geometrical, algebraic, or statisticalconcepts.

The “going over assigned work” phase occurs when the teacher uses class time to go over work that has been assignedpreviously. Students would have had spent some time on the assigned work. Common examples include given exercisesor worksheets, tasks/homework, test paper/assessment, etc. Three types are distinguished: reworking, procedural andquick check. Reworking happens when the teacher goes over the assigned work thoroughly, i.e., understanding theproblem, planning, executing and reflecting the problem. It has a “corrective” or a “remedial” tone which is differentfrom the problem solving instruction in the problem solving phase. The instruction also tends to be “algorithmic”versus the more “heuristic” type of instruction in the problem-solving phase. In the procedural type, the teacher goesover the assigned work to focus on the correctness of the procedures needed to solve the problem. The intention is veryoften for students to know the steps to solve the problem. The third type refers to the teacher going over the answersquickly for students to check against their own answers.

The student activities phase refers to spending class time with students doing assigned tasks. There were threecommon types of student activities,presentation, group work, andseat work. Student(s) present their solutions ofassigned problems to their classmates/teacher either using the white board or overhead projector, or by standing upto present orally. Group work occurs when groups of students are given tasks to do. Seat work refers to studentscompleting assigned tasks individually. If a few students are called up to show their answers on board while the rest arecontinuing with their seat work, then it is considered seat work but if the teacher stops the students’ working and tellsthe class to pay attention to what their classmates are writing/doing in front, then it is students’ presentation. When theteacher goes over the students’ work on the board, then a change of phase to “going over assigned work” is deemed tohave occurred.

The last phase of “other classroom events” is a catch-all category. Generally there are two types: those that arerelated to the on-going lesson, for example, rearranging the seating arrangement as part of the transition to group work,and those unrelated to the lesson such as the teacher making a general announcement. The inclusion for this phase isfor a more accurate accounting of class time.

Thus, the video-coding scheme comprises five main phases and a total of 14 sub-categories in the five phases.


Table 1Total number of lessons, time and the average time per lesson in the pre-intervention stage

Teacher Band of school Number of lessons Total time observed(to the nearest min)

Average time per lesson(nearest min)

A Upper 9 7:38:00 51B Middle 8 6:49:00 51C Low 6 5:41:00 57

Overall 23 20:08:00 53

2.4. Student assessment tasks

Two other aspects were important to the development of the evidence base: the type of mathematical word problemsused as teaching examples and the range of heuristics that could be used to solve each problem. When identifyingproblem types, it was noted that the teachers generally usedstandard word problems typically found in textbooks (Foong& Koay, 1997). Other word problem types were deemednon-standard, and were not typically found in textbooks. Therange of heuristics to be coded in the scheme was taken directly from the syllabus (MOE, 2000, p. 6). Examples include“act it out,” “use a diagram or a model,” “guess and check,” “make a systematic list,” and so onTable 1shows thecomplete list.

This study used ten problems for the pre-test. Half of these problems were standard word problems while the otherhalf were non-standard. In the post-test, the problems were similar, except the tasks were amended slightly in termsof wording and numbers. To illustrate, the following standard problem — “In 2003, Dan was 16 and his mother was44. In which year was Dan’s mother eight times as old as Dan?” was recast as “Mary received 14 vitamin pills andJohn 29 vitamin pills on 2nd August. They both take only one pill daily from that day. What is the date when John hasfour times more pills than Mary?” in the post-test. For the following non-standard problem in the pre-test — “Thereare 100 buns to be shared by 100 monks. The senior monks get three buns each and three junior monks share one bun.How many senior monks are there?,” the post-test’s equivalent problem appeared as “The zoo keeper gave 80 bananasto 50 monkeys. The big monkeys got two bananas each, and three small monkeys shared 2 bananas. How many bigmonkeys are there?” For these two pairs of problems as well as the others, the processes required to solve each of themin both the pre- and post-tests were kept as similar as possible.

3. Results and discussion

The most commonly adopted approach by the teachers focused on the textbook/workbook. The teachers followeda scheme of work endorsed by the respective schools’ departmental heads: the teacher introduced the topic, explainedthe concepts, and gave students time to practice the related skills. Word problems were conceived as separate eventsand only introduced near the end of the topic. The topics observed were whole numbers and fractions.

In the pre-intervention phase, 23 lessons of the three teachers were recorded and coded. Each lesson averaged about53 min. The followingTable 1summarizes the time of observations.

3.1. Intercoder agreement

To analyze the classroom observations, the researchers used the coding scheme as the main analytical tool. Theissue of intercoder reliability or more specifically intercoder agreement for such an analysis is fundamental (Bakeman& Gottman, 1997; Neuendorf, 2002). Out of the 23 lessons, six were coded by two different coders from a pool offive including one of the researchers. The coders were trained over a period of at least 3 weeks. Percentage agreement,defined as the proportion of the number of agreements to the number of agreements and disagreements, is used tocalculate intercoder agreement. To account for agreement expected by chance, the Cohen’s� (Cohen, 1960) is alsocalculated. Three variables were examined: the agreement for the five main phases, the overall agreement for all 14sub-categories, and the agreement within the problem solving phase. For the five main phases, the agreement was97% and the�, 0.96. The agreement across the 14 sub-categories was 87% and the�, 0.85. The agreement withinthe problem solving phase is 78% and the�, 0.72. On the whole, the coders were able to identify reliably the five


Table 2Amount of time each teacher spent on each phase

Teacher Problem solving (aspercent of total)

Teaching conceptsand skills (aspercent of total)

Going over assignedwork (as percent oftotal)

Student activities(as percent of total)

Others (aspercent oftotal)

Total timeobserved

A 03:04:15 (40%) 00:37:35 (8%) 0:53:20 (12%) 2:45:20 (36%) 0:17:25 (4%) 7:38:00 (100%)B 00:30:15 (7%) 01:02:50 (15%) 1:14:10 (18%) 2:57:40 (44%) 1:04:25 (16%) 6:49:00 (100%)C 00:07:05 (2%) 00:29:45 (9%) 1:45:00 (31%) 2:55:40 (52%) 0:23:10 (7%) 5:41:00 (100%)

main phases and the 14 sub-categories. For the four sub-categories within the problem solving phase, the� of 0.72is deemed acceptable by some researchers likeLombard, Snyder-Duch, and Bracken (2002)for exploratory studies,while Fleiss (1981)would characterize it as “good.” Notwithstanding, this “borderline reliability” when seen withinthe overall frame of the study allows for some conditional analysis for looking into the teaching of problem solving inclassrooms.

3.2. The results

It was clear that all three teachers shared some common ways of teaching but differed in the amount of time spenton the various phases (Table 2).

To illustrate the proportion of time as a percentage of the overall, each column of the graph is stacked as one unit,i.e., 100% (Fig. 2).

Generally Teacher A spent proportionately more class time (40%) on problem solving than the other two teachers.In the middle band school, B spent 7%. C spent the least at 2%. A and C used solely whole-class teaching. B spentsome of his instruction time doing problems using a group work configuration.

3.3. The problem solving phase

Each problem-solving phase was further deconstructed usingPolya’s (1957)four stages ofUnderstanding,Planning,Execution, andReflection. The followingFig. 3shows the expansion on each of the teachers, bearing in mind each ofthem spent 40, 7, and 2% of their overall observed class time on problem solving respectively.

Notably the teachers spent much of their time during the problem-solving phase on understanding the problem andexecuting the steps required to solve the problem. Teacher A spent 17% of her problem solving time on planning whileC spent proportionately less time at 7%. Teacher B did no solutionplanning at all.

Fig. 2. Distribution of lesson time of the three Grade 5 teachers.


Fig. 3. Proportion of time teachers spent on each stage of problem solving.

3.4. Types of word problems used

Of the 78 questions used by the three teachers during the period of observation, 74 were the standardword problem typically found in textbooks (Foong & Koay, 1997). A distinction was made between prob-lems useddirectly for problem solving instruction and those problems that were assigned as homework or classwork, and later used by the teacher for problem solving instruction. The latter type of problems was classi-fied as “Going over assigned work — Reworking.”Table 3 shows the number of problems used by the teach-ers.

Teachers B and C gave instruction on problem solving more through the mode of “going over assigned work,” whichhad a more remediation tone than tackling a fresh problem. For A, her MPS instruction occurred through presentingproblemsdirectly. She spent more time on problem solving than the other two teachers and she used more problemsand spent more time per problem.

3.5. Range of heuristics

The problem solving heuristic most commonly observed in the three classrooms was the Diagram/Model (used inabout 74% of the problems). Of the remaining heuristics listed in the syllabus, they were seen once or twice or notat all (seeTable 5). TheThink of a related problem heuristic is not listed in the syllabus but Teacher A used it as aheuristic for four problems.

3.6. Teaching of concepts and skills phase

The time spent in this phase ranged from 8 to 15%. As the topics covered during the period of observation weremainly whole numbers and fractions, the concepts taught were numerical concepts. The teachers mainly coveredarithmetic manipulation, taught the skills involved in adding and multiplying fractions.

Table 3Number of problems used in the observed lessons

Teacher Number of problems used Total numberof questions

Problem solving Going over assignedwork — Reworking

A 37 6 43B 5 13 18C 1 16 17


Table 4Distribution of time in the “Going over assigned work” phase

Reworking Procedural Quick Phase total As percent of overall total

A 00:45:10 (85%) 00:07:05 (13%) 00:01:05 (2%) 00:53:20 (100%) 12B 00:32:39 (44%) 00:27:18 (37%) 00:14:15 (19%) 01:14:12 (100%) 18C 01:05:55 (63%) 00:31:35 (30%) 00:07:30 (7%) 01:45:00 (100%) 31

3.7. Going over assigned work phase

Going over assigned work enables students to check their work against the teacher’s. As discussed in the section onthe coding scheme (see p. 5), the going over assigned work phase has three sub-categories: reworking, procedural andquick check.Table 4shows the time spent in this phase are distributed over the three categories.

The percentage of time teachers spent in this phase ranged from 12 to 31%, with Teacher C spending the most time.According to discussions with C, she needed togo over assigned work many times to show her studentshow to do theassigned work. All three teachers mainly reworked assigned problems though B and C spent relatively more time thanA going over the procedures of solving word problems and checking answers quickly.

3.8. Types of student activities

The various types of student activities observed were grouped in the following three categories: students’ presentationmainly in the form of standing in front of the class and talking about their work; group work including pair work; andindividual seat work.Fig. 4shows the distribution of the types of student activities in each class.

All teachers assigned individual seat work during class time. In A’s case it was the only form of student activitycoded. For group work, only B used the small group configuration in class. B’s students did presentation during ourobservations. The common mode was students writing their solutions on board; and occasionally being asked to explaintheir answers. This was typically followed by the teacher going over the written answers. Teacher C’s 1% of studentpresentation amounted to less than 2 min, suggesting that it was not part of her regular practice.

3.9. Post-intervention observations

After a reflective practice intervention comprising interviews and a workshop (see p. 4), the teachers were observedagain about once a week over 3–4 weeks. These were lessons that teachers deemed typical andinvolved some problemsolving. The general approaches used by the teachers remained similar to the earlier observations, with teacherscontinuing with their scheme of work as planned at the beginning of the year. In terms of proportion of time both

Fig. 4. Percentage of time spent on the three types of student activities.


Fig. 5. Comparing patterns in problem solving phases before–after intervention.

Teachers A and B did not change significantly their distribution of time over the five phases. However, C increasedher amount of time spent on the problem-solving phase significantly from 2.1% (or about 7 min of 5.75 h) to 7.5% (oralmost 11 min of 2.5 h). Other notable qualitative changes in the post-intervention observations are discussed in thefollowing sections.

3.9.1. Changes in patterns in the problem solving phaseTeacher A spent proportionately much more time on reflection than before. Notably there were explicit attempts to

look back for alternative ways of solving the same problems. B factored in some time for planning whereas before hedid not and he also spent relatively more time working through the execution. The notable change for both A and Cis their increase in the proportion of time for reflection. B spent less time on reflection. The changes are illustrated inFig. 5.

3.9.2. More non-standard word problems usedBefore, about 95% word problems discussed in class were standard problems. After intervention, teachers use of

non-standard problems increased from 5 to about 45%.

3.9.3. Range of Heuristics used more evenly distributedAlthough the Diagram/Model heuristic remained the most commonly used at approximately 50% of the time, more

of theother heuristics were observed after the intervention in the work of Teachers A and C, but not Teacher B (SeeTable 5).

For Teachers A and C, there is generally a wider “spread” of heuristics after the intervention. Teachers A and Cusedother heuristics more than before. Teacher B did not make any notable changes.

3.9.4. Other qualitative changesIn post-intervention observations, Teacher A made attempts to incorporate some pair and small group work in her

class, and her students were given opportunities to present and talk about their solutions. In her students’ presentationof solutions there was increased evidence of the use of heuristics whereasbefore there was none.

Teacher B made several changes. His classroom was rearranged from cluster seating (more suitable for group work)to columns and rows (more suited to whole class teaching). He usedonly whole class teaching followed by individualseat work in theafter stage. He stopped doing group work. According to him, he had thought his students were notbenefiting from group work and reverted back to whole class teaching for more effective learning. His students hadnot done well in the school’s continual assessment. He also constrained his ventures into non-standard problems byfocusing more on textbook based problems.

Teacher C made her first foray into group work for her class after the intervention. She also experimented with non-standard problems. Her students were not familiar with working in groups. After the initial difficulties, they managedsome degree of success after a few weeks. C now spent more time teaching problem solving. As before, C’s studentsdid not get to present their solutions in front of the class. Instead she provided morescaffolding to facilitate learningin groups of students.


Table 5Range of heuristics teachers used in class before and after intervention

Range of heuristics observed(before and after intervention)

Teachers (before and after intervention)

A B C Total

Before After Before After Before After Before After

1. Act it out 2 22. Diagram/model 30 12 3 1 7 3 40 163. Guess and check 1 2 1 1 1 2 44. Systematic listing 3 1 4 1 75. Look for patterns 2 26. Work backwards 1 1 1 1 27. Before–after concept 1 1 1 18. Make suppositions9. Restate problem 1 110. Simplify problem 1 1 211. Solve part problem 1 112. Think of related problems 4 1 4 1

Total 38 18 8 5 8 11 54 34

3.10. Students pre- and post-tests

The number of students who took the pre- and post-test is summarized inTable 6.The sets of tasks were scored into three categories: correct, attempted (but incorrect or incomplete) and blank. The

overall scores for the three classes are summarized inTable 7.Fig. 6 illustrates the changes in the students’ performances from pre-test to post-test.The number of correct answers went up about 5.6% while the number of blank responses went down about 3%.

There was also a slight drop of 2.5% in the “Attempted” category, i.e., there were fewer incomplete or wrong answers.

3.11. Details of results in individual classes

There was a notable increase in the number of correct answers in Teacher A’s class from 73 to 98 even with oneabsentee who did not take the post-test. The number of blank responses also went down significantly (Table 8).

Table 9includes thet test results for Teacher A’s class. The mean correct answers increased by 0.72, or by about36%. Thet value of 3.224 andP at .003 indicates a significant increase with a moderate effect size of 0.54.

The changes in Teacher B’s students were moderate — an increase in the number correct from 59 to 69, and adecrease in the number of blank responses from 138 to 93 (Table 8). The mean for Teacher B’s class increased by 0.44or about 28% but was not statistically significant.

Table 6The number of students in each class and the number who took the tests

A B C Total

Number of students 37 36 33 106Pre-test 37 36 33 106Post-test 36 34 31 101

Table 7Overall results

Correct Attempted Blank Total (number)

Number Mean SD Number Mean SD Number Mean SD

Pre-test 167 1.58 1.14 558 5.26 2.24 335 3.16 2.26 1060Post-test 216 2.14 1.29 506 5.01 2.20 288 2.85 2.01 1010


Fig. 6. Overall pre-test and post-test results.

For Teacher C’s students the number of correct answers increased from 35 to 49. It was the only class that had anincrease in blank responses, but as the post-test was administered 15 min late, the students had less time to complete it.The mean increased 0.52, or about 48% which was statistically significant. This occurred despite the circumstances inthe post-test administration mentioned above. Further, these students were from a lower band school, and the weakestclass of the grade level. The effect size of 0.42 was moderate.

In summary, students in A and C’s classes showed statistically significant increase in the number of correct answersin their post-tests. Overall, there was a significant increase in the number of correct answers. There were less incom-plete/wrong answers and blanks — an indication that the students did better and were more willing to try to solve theproblems than before.

Table 8Individual teacher’s class results

Correct Attempted Blank Total (number)

Number Mean SD Number Mean SD Number Mean SD

A’s classPre-test 73 1.97 1.07 176 4.76 2.31 121 3.27 2.10 370Post-test 98 2.72 1.32 165 4.58 2.36 97 2.69 1.95 360

B’s classPre-test 59 1.64 1.33 163 4.53 1.63 138 3.83 2.37 360Post-test 69 2.03 1.31 178 5.24 1.71 93 2.74 1.88 340

C’s classPre-test 35 1.06 0.75 219 6.64 2.15 76 2.30 2.10 330Post-test 49 1.58 0.92 163 5.26 2.46 98 3.16 2.24 310

Table 9Class paired-samplest test for the number of correct answers

Mean SD t value df P d

Teacher A (n = 36)Pre-test 2.00 1.07 3.224 35 0.003 0.54Post-test 2.72 1.32

Teacher B (n = 34)Pre-test 1.59 1.31 1.69 33 0.100 0.29Post-test 2.03 1.31

Teacher C (n = 31)Pre-test 1.06 0.73 2.327 30 0.027 0.42Post-test 1.58 0.92


4. Conclusion

This study has gone some way in recording and analyzing particular interactions of three teachers’ classroompractices in the teaching and learning of mathematics with a particular focus on problem solving. The video-codingscheme enabled the identification of five phases: problem solving, the teaching of concepts/skills, going over assignedwork, student activities and other class events. The amount of time each teacher spent on each of the phases describestypical classroom practice. However, traditional patterns of teaching a topic, explaining concepts and giving studentsexercises to practice related skills prevailed. Similar patterns were observed both before and after the reflective practiceintervention. Such patterns match whatSchroeder and Lester (1989)describe as teachingfor problem solving. To someextent we observed a narrow interpretation of this approach with teachers doing problem solving with students onlyafter the introduction of concepts or following work on computational or procedural skills.

According toSchroeder and Lester (1989, p. 34), teaching via problem solving brings out best the curricula focus onproblem solving. Ideally, the intervention should be to shift the teachers’ approach towards it. But such a major reformchange would require something more involved and complex (Senger, 1999). Given the short time the current studyspanned, it was more realistic to take the first steps to get teachers teachabout problem solving highlightingPolya’s(1957)model of problem solving, with the longer term goal of moving towards teaching mathematics via problemsolving.

The results suggest that the teachers had at least started to question their own views about problem solving and how itcan be emphasized in their mathematics teaching. We noted that Teachers A and Cloosen some of their predominatingteacher-centered approach to include some pair and small group work, resulting in greater student interaction and moreactive participation. There was also abetter spread of time through the four (Polya) phases of problem solving afterthe intervention, breaking theunderstand–execute pattern to include more time onplanning andreflection. Notably, Bwas more explicit aboutplanning while Teachers A and C spent more time onreflection.

One unexpected change was Teacher B stopped using group work in his class and reverted back to whole classteaching. His concerns were that his class’ continuous assessment was slipping and they were not benefiting from thegroup work he had done before. B’s concerns reflected the belief many teachers have — that it was far more importantto prepare students for formal assessment than to implement problem solving lessons (Norton et al., 2002).

The range of heuristics introduced to students and their ability to try more than a limited range of heuristics to solvemathematical problems has been shown to be possible and the implementations vary greatly between classrooms andproduce different effects. There are lessons for both the development of mathematical thinking in real classrooms aswell as those who seek to change pedagogical practice. The analysis has shown great individual variation in the wayteachers implement a simple problem solving strategy.

The range of heuristics employed and the type of problem were also important aspects of changing MPS classroompractice. All teachers used more non-standard problems in class after the intervention as they realized that they wereproviding a limited experience for their students. This change seems to be compatible with the findings ofCai (2003)about the kinds of problem solving tasks that should be used. For this effect, the degree of support for teachers interms of resources and examples is critical. In fairness to the teachers, the testing and assessment systems and thefact that one heuristic strategy (diagram/model) can be used by students to achieve solutions for at least 75% oftheir assessment works actively against increasing other heuristic use. Teachers A and C did widened their range ofheuristics, employing other heuristics like “guess and check,” “look for patterns,” and “systematic listing.” Resourcessuch as suitable non-standard problems did help support teachers extend the range of heuristics in the teaching andlearning of MPS. To that end providing a set of problems that can be solved by more than one heuristic strategy is animportant starting point for teachers to consider changing their basic approach.

The overall results seem to suggest that the intervention strategy was effective, although the benefits of the interven-tion were small. The Teachers A and C made qualitative changes to their teaching. Their students also demonstratedmoderate but statistically significant improvements in students’ learning. We suggest that the changes to Teacher Aand C’s teaching styles may have accounted for this improvement. Other possible factors that might have accountedfor the improvement include students’natural mathematical development over the 4-month span of study or simplyfamiliarity with the post-test items after having seen the pre-test. These factors are acknowledged here and we can-not claim students’ improvement wassolely due to the qualitative changes in the teachers’ practices. Nonetheless, theintervention appears to provide a first-step in developing instruction to help students better engage in authentic problemsolving.


In addition, better student performances in Teachers A and C classes may have something to do with the fact thatthey spent more time onreflection in their problem solving for both the teachers and the students. This lends supportto the few studies on the important role reflection plays in problem solving (cf.Cifarelli, 1998; Simon, Tzur, Heinz, &Kinzel, 2004). To some extent the results of the current study demonstrated in actual classrooms the effectiveness ofan intervention that is designed to promote increased reflection in the students. There are important implications forclassroom application that need to be explicated through further analysis and research.

Clearly, observations of classroom practices can be viewed through many lenses (e.g.,Bourke, 1985; Cobb etal., 2001; Spillane & Zeuli, 1999; Stigler & Hiebert, 1999). Perhaps one important area is to look at classroomtasks and discourse patterns in the Singapore mathematics classroom context. Would the patterns so identified becomparable with the three patterns identified bySpillane and Zueli (1999), namely, “Conceptually grounded tasks andconceptually centered discourse,” “Conceptually oriented tasks and procedure-bounded discourse,” and “Peripheralchanges, continuity at the substantive core”? This avenue would enrich further insights into classroom practices.Another area could be studying the observations for a “teaching script” (Stigler & Hiebert, 1999) culturally specific toSingapore. Given that the present sample size is small, it remains to be seen if by extending the investigation such a“script” can indeed be differentiated.

Finally, the focus of the larger study is to systematically enlarge the evidence base and specifically work with thechanged explicit class discussion on metacognitive strategies that will assist students in planning and reviewing theirapproaches. Most students had not been explicitly pushed to discuss their problem solving solutions in terms other thanprocedural steps. Working with the teachers in this study covered a realistic period with authentic school assessmenttasks and has resulted in distinct behavioral changes in the students’ approaches. From cries of “this is too difficult”to greater attempts and less blank responses we have seen changed attitudes to the process of solving mathematicalproblems. Our results suggest that with an emphasis on metacognitive strategies and working with an explicit planningapproach, students can change their approach and experience greater success.

Acknowledgments

We acknowledge the invaluable contributions of the late Teong Su Kwang to the conceptualization of the project,Mrs. Chang Swee Tong for her support, the teachers who so willingly gave their time and cooperation, and to LuisLioe and John Tiong for assisting in the data collection.

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Teachers’ pedagogies and their impact on students’ mathematical problem solving