The Integration of Problem Posing in Teaching and Learning of Mathematics

A Dissertation Proposal by Roslinda Rosli Texas A&M University

Overview Chapter IV: IIII: II : Proposed journal article 2 13 Chapter I -Problem statement -Purpose of study -Theoretical framework Method -Journal selection

Chapter IV: Proposed journal article 3

Chapter II : Proposed journal article 1

Chapter III: Proposed journal article 2

Chapter I

What is problem posing? Problem posing

Problem posing -can take place before, during or after the problem solving process

Problem generation (Silver, 1994)

Problem reformulation

â€˘ National Council of Teachers of Mathematics (2000) highlighted the importance of problem posing as means of classroom instruction along with problem solving. â€˘ Help students to develop their mathematical knowledge and abilities by exploring their curiosity about specific concepts presented (English, 1997; Joyce, Weil, & Colhoun, 2000).

Problem Statement â€˘ Students of all ages, including those who subsequently become teachers, have limited experience in problem posing (Crespo & Sinclair, 2008). â€˘ Preservice and inservice teachers posed many low quality problems (Silver, Mamona-Downs, Leung, & Kenney, 1996).

Purpose of the Study â€˘ Utilizing three-article format of dissertation to develop a better understanding about the potential of the problem posing approach in improving teaching and learning of mathematics.

Proposed Articles Article 1

• The effects of teaching and learning mathematics with problem posing : A meta-analysis

Article 2

• A mixed analysis of middle grade preservice teachers’ mathematical problem solving and problem posing

Article 3

• Examining elementary preservice teachers’ knowledge of teaching fraction: A mixed research study

Theoretical Framework â€˘ Constructivist learning theory

(von

Glasersfeld, 1989).

â€˘ The openness feature of the problem posing tasks can reveal how an individual learns mathematics (Kulm, 1994).

Method â€˘ Use of multiple approaches and paradigms â€˘ Quantitative methods are the dominant component of the proposed studies

Journal Selection • Two potential journals for each article are blind peer reviewed indexed in Scopus (impact factor/ranking) and Cabell educational databases • Table 1 (brief information of the proposed journals)

Chapter II The Effects of Teaching and Learning Mathematics with Problem Posing : A Meta-Analysis

Rationales â€˘ A number of studies has examined many research emphases and methodological aspects on problem posing activities (e.g., Cai, 1998; 2003; Cai & Hwang, 2002; Moses et al., 1990; Silver et al., 1996; Stoyanova, 1999; Yuan & Sriraman, 2011).

â€˘ Many of these studies present positive outcome of problem posing on studentsâ€™ mathematical knowledge, problem solving and problem posing skills, creativity, and disposition toward mathematics.

Purpose â€˘ To date, there is not any meta-analytic study on this topic. â€˘ To systematically synthesize empirical findings (since 1989) on problem posing as a classroom intervention for improving teaching and learning of mathematics.

Research Questions â€˘ How extensive is the empirical evidence on the effectiveness of problem posing in mathematics classrooms? â€˘ What are factors (e.g., instructional features, methodological, student characteristics) affect the effectiveness of problem posing in mathematics classrooms?

Search Criteria

Keywords: Databases-ERIC, Manual Google Careful Scholars examination search-references, problem to ProQuest cross-examine posing, of eachleading Dissertation problem empirical the journals formulating, available study-will and in Theses, education/mathematics empirical include problem JSTOR, only studies. generation. quantitative and Educational education studies Text. that pr

Coding Procedure

â€˘ Will develop a coding sheet iteratively to illustrate the features of the studies. â€˘ Calculate the effect sizes in order to combine and compare various measurement procedures used.

Statistical Analysis • Effect size calculations

(Lipsey & Wilson, 2001):

Cohen’s d for contrasting intervention and nonintervention group means. Odds ratio for frequencies and proportions of dichotomous variables (e.g., kind of problems posed before and after problem posing intervention).

Statistical Analysis • Comparing effect sizes among studies by converting the standardized mean difference effect sizes to odds ratios or vice versa (Hasselblad & Hedges, 1995) .

Eor = e

π Ed ( ) 3

• Compute the standard errors and confidence interval of effect sizes.

Statistical Analysis • The effect sizes will be grouped into similar research outcomes (e.g., belief, problem solving abilities). • Then, compare the effect of problem posing intervention – positive effect size reflect a positive effect of treatment (Lipsey & Wilson, 2001).

Chapter III A Mixed Analysis of Middle Grade Preservice Teachersâ€™ Mathematical Problem Solving and Problem Posing

Rationales • According to NCTM (2000), “Good problem solvers tend naturally to pose problems based on situations they see” (p. 53). • The results from previous studies were mixed suggesting a complex relationship between problem solving and problem posing success (Cai & Hwang, 2002; Chen, Can Dooren, Qi, & Verschaffel, 2010).

Rationales â€˘ With respect to teacher education, there is scant research in the literature emphasizing the link between mathematical problem solving and problem posing (Chen et al., 2010; Silver et al., 1996).

â€˘ As problem solving and problem posing becomes central to the learning of mathematics (NCTM, 1991, 2000), teachers have a crucial role in helping students develop a repertoire of associations for proficient problem posing and effective problem solving (Moses et al., 1990).

Purpose (a) explore select middle grade preservice teachersâ€™ abilities in solving a particular mathematical problem; (b) examine the select middle grade preservice teachersâ€™ capabilities in generating new problems based on a given situation or after they have solved the given problem; and (c) find the link between their problem solving and problem posing abilities.

Conceptual Framework • Silver’s (1994) problem posing framework Problem generation-occur before the problem solving process Problem reformulation-extension of problem solving activity

• Throughout problem solving and problem posing activities, students can make connections among mathematical ideas, then construct and restructure their knowledge based on prior ones (Moses et al., 1990).

Participants and Setting • Participants : approximately 50 middle school preservice teachers in a problem solving course. • The course professor will emphasize Polya’s four-step problem solving process and will integrate problem posing activities regularly during class instruction. • This study is part of regular class activities- permission to conduct the study are granted through the Texas A & M University Institutional Review Board (IRB).

Instrument and Procedures • A pair of problem solving and problem posing task is adapted from Cai and Lester’s (2005) The Block Pattern Problem . • Pilot study was conducted in Spring 2011 and several changes were made to the task and administration of the task. • Participants will be given 15 minutes for each task. • Multilink cubes will be provided to assist them.

Instrument and Procedures â—?

Solve thesolving Generate Problem posing three pattern new block problems problems with a variety of difficulty levels (easy, mod

Data Analysis • A sequential mixed analysis will be utilized to analyze the data (Onwuegbuzie & Teddlie, 2003) wherein the qualitative data will be transformed into a numerical form (Tashakkori & Teddlie, 1998). Phase 1 • Problem solving rubric with 1-6 points (Oregon Department of Education, 2011).

• Problem solving strategies: find a pattern, make a table, draw a picture/diagram Phase 2 • Problem posing rubric with 1-4 points

Data Analysis • Phase 3 Descriptive statistics of preservice teachers problem solving and problem posing performance will be generated using SPSS version 17.0. Spearman’s correlation coefficient, r will be computed to illustrate whether the relationship is direct, high, indirect, inverse, low, moderate, negative, perfect, positive, strong, and weak (Huck, 2007).

Legitimation • Threats to mixed methods validity, for example: Conversion – interpreting and quantitizing qualitative data (written responses). Minimize the researcher personal bias, 10% of the responses will be randomly selected and coded independently by another coder for inter-rater agreement.

Sequential-reversing the sequence of the quantitative and qualitative phases

Chapter IV Examining Elementary Preservice Teachersâ€™ Knowledge of Teaching Fraction: A Mixed Research Study

Rationales • There has been continued debate over fraction instruction in classrooms - difficult concept to teach and learn. • Teachers have insufficient knowledge of subject they are teaching that should have been acquired during teacher preparation program (e.g., Ball, 1993; Newton, 2008). • Very little is known on problem solving and problem posing tasks that can be used to measure knowledge for teaching.

Purpose of the Study • To measure the effect of a specific fraction instruction (Mathematics TEKS Connections [MTC] Module) on elementary preservice teachers’ content knowledge and pedagogical content knowledge.

• To explore elementary preservice teachers’ problem solving and problem posing strategies.

Conceptual Framework

• The complex nature of knowledge for teaching mathematics (Shulman, 1986). content knowledge (CK) – the amount of mathematics knowledge pedagogical content knowledge (PCK) - the ways to make the learning of specific concepts comprehensible

Research Questions (1) What is the effect of an instructional unit developing conceptual understanding of fractions using concrete models in a mathematics methods course on the levels of elementary preservice teachersâ€™ content knowledge for teaching fractions?

(2) What is the effect of an instructional unit developing conceptual understanding of fractions using concrete models in a mathematics methods course on the levels of elementary preservice teachersâ€™ pedagogical content knowledge for teaching fractions?

(3) How do the elementary preservice teachers describe the strategies that they use for solving and posing fractions tasks?

Research Design • A fully mixed concurrent dominant status design-more weight on quantitative method (Leech & Onwuegbuzie, 2009). • Pragmatist research paradigm will drive this quasi-experimental study: Post positivism (assess cause-and-effect relationship) Constructivist (seek individual and collective reconstructions)

Research Design Quasi-experimental design

(Shadish et al., 2002)

â€˘ one-group pretest-posttest design with a nonequivalent dependent variable (Measure B) as a way of decreasing the threats to internal and external validity. {O1A, O1B } X {O2A, O2B } â€˘ the outcome of Measure A (i.e., fraction knowledge, DV) is expected to change because of treatment, X (i.e., the fraction instruction, IV) but not for measure B (i.e., geometry knowledge).

Participants and Setting Population •

•

•

Approximately 140 elementary PTs in a public university in Texas Mathematics methods course (Fall 2011), have completed most of the coursework, will be student teaching in Spring 2012 Three instructors are teaching four class sections

Sample • 72 PTs from the same professor (convenience sampling/identical sample) • Information sheet (IRB) – ethical consideration •Quantitative and qualitative data will be gathered concurrently to ensure complementarity and triangulation •An adequate sample size (G*Power 3)

Instruments

Procedures

Data Analysis

•

score items based on the degree of correctness (0-1 point)

•

descriptive statistics (means, standard deviation, percentages)

•

analysis of differences (paired t tests) - check the underlying assumptions

•

effect sizes (Cohen’s d)

Stage I • written responses will be transformed into numerical valuesbased on the degree of correctness (1-4 points) • code 10% for the inter-rater agreement • Statistical analyses similar to QUAN and compare results with quantitative phase Stage II • use constant comparison analysis for coding the written responses and organize into similar strategies/concepts used

Parallel Mixed Analysis for Triangulation and Complementarity

Limitations â—?

Internal

Limitations

References Brown, S. I., & Walter, M. I. (2005). The art of problem posing. (3rd ed.). Mahwah, NJ: Erlbaum. Cai, J. (1998). An investigation of U.S. and Chinese students’ mathematical problem posing and problem solving. Mathematics Education Research Journal , 10(7), 37–50. doi:10.1007/BF03217121 Chen, L., Van Dooren, W., Chen, Q., & Verschaffel, L. (2010). An investigation on Chinese teachers’ realistic problem posing and problem solving abilities and beliefs. International Journal of Science and Mathematics Education, 9, 130. doi:10.1007/s10763-010-9259-7 Shulman, L. S. (1986). Those who understand: Knowledge growth in teaching. Educational Researcher, 15(2), 4-14. Retrieved from http://www.jstor.org/journals/0013189X.html Silver, E. A. (1994). On mathematical problem posing. For the Learning of Mathematics, 14(1), 19-28. Retrieved from http://www.jstor.org/action/showPublication? journalCode=forlearningmath