The role of optimism in collaborative problem solving in mathematics: building problem solving capacity

Personnel: 

Chief Investigator:
Dr Gaye Williams 

Research Assistants:

Katie Fox, Michelle Hines, Daniel Fox

Funding: 

Australian Research Council Discovery Project (with APD Fellowship)

Project Outline:

Australian primary students’ mathematical performances on international benchmark tests have slipped to average, with students’ lack of mathematical confidence a major contributing factor. In her PhD research (awarded the University of Melbourne Chancellor’s Prize for Social Sciences, and the AARE Doctoral Award), Gaye found optimism (resilience) to be a characteristic of successful individual problem solvers that related to confidence. The effects of optimism on creative interactions needs to be examined. The whole class teaching experiment (Engaged to Learn Approach), piloted in 2006, elicited creative group thinking that provided opportunities tostudy optimism-enactment. This video stimulated student interview study arising from this pilot study will produce findings to inform the development of pedagogies to strengthen mathematical performances and build mathematically literacy. This study crosses research boundaries by examining optimism for its potential to promote learning rather than its role in building mental well-being. It also contributes to research on building mental well being by identifying optimism-building situations. This study examines how the relative optimism of group members influences their potential to think creatively to develop new mathematical knowledge, and it illustrates how students of varying abilities can solve mathematical problems creatively. The project has been designed to find whether optimism can be built over time as students progressively gain successes through their intellectual effort during mathematical problem solving. In addition, it examines whether increased optimism does (as expected) lead to increased problem solving capacity. Publications reporting preliminary findings from this study for researchers and for teachers are listed along with the papers reporting research findings that raised the questions that provided the impetus for this study. 

Project Design:

This longitudinal study follows students from Grade 4 to Grade 5 to Grade 6 (2009-2011). Three classes are studied each year:

• 2009: 3 classes containing Grade 4 students
• 2010: 3 classes containing as many as possible 2009 Grade 4 students in Grade 5
• 2011: 3 classes containing as many as possible 2010 Grade 5 students in Grade 6

Nine complex problem-solving tasks that are accessible to students with varying mathematical backgrounds are undertaken over the three-year period. Each year three tasks are undertaken.

• Task 1: 3 eighty-minute sessions (approx April each year)
• Task 2: 2 eighty-minute sessions (approx June each year)
• Task 3: 1 eighty-minute session (approx October each year)

Four cameras in the classroom (each with the potential for audio feeds from two groups) capture the activity of the 6-7 groups of students in the class, and group reports to the class as a whole. The classroom videos assist in the analysis of what each student knew at various points in the lesson, and what influenced changes in their understandings. These videos also enable analysis of the enactment of optimism or lack of optimism and the effects this can have on group interactions and opportunities for creative thinking during the collaborative development of new knowledge. 

Individual video-stimulated interviews are undertaken with four students after each lesson. The student controls the ‘two up’ video containing images of their group, and group reports to the class. They find parts of the lesson that were important to them, and talk about what they thought was happening, what they were thinking, and what they were feeling. These interviews provide opportunities for students to reconstruct the thinking they did in class, and identify what assisted / inhibited new learning. The interviews also provide indicators of optimism or present lack of optimism (discourse analysis). Interviews with students in the second and third year probe the similarities and differences students perceive between the tasks they have undertaken, whether they perceive their ways of approaching these tasks have changed over time, and what they think has contributed any such changes.

The data collected enables ‘backtracking’ to examine the activity undertaken by any student over the period of the study. Thus, if a student’s indicators of optimism begin to change, there is opportunity to re-examine the activity of this students and the groups to which they belonged over time. Similarly, if a student is found to inhibit creative interactions in a group, their activity in previous groups can be examined to see whether there are aspects of group composition that influence the ways such students interact. In other words, the rich data set provides opportunities to examine the past activity of students and groups when such activity becomes important later in the study.

Engaged to Learn Approach 

Students work in groups of three or four on problem solving tasks for an extended period of time. Groups give brief reports to the class at intervals, and each group decides for themselves whether anything they have heard could be useful to their own exploratory activity. Groups have opportunity to think about mathematical ideas developed by their peers and adapt them for their own purposes. Such classroom interactions provide many opportunities for students to experience successes overtime that should theoretically be optimism (resilience) building. Key aspects of this approach that contribute to student perceptions of successes include: a) tasks that are accessible through a variety of representations and levels of mathematical sophistication; b) the rotation of group roles between reporting sessions; c) whole group contributing to priming reporter; d) sequencing reports in an order that gives every group the opportunity to report something new to the class; e) valuing rather than praising each report by identifying something within which contributes to class knowledge; f) making explicit that communicating the process of thinking about a problem is an important aspect of problem solving; and g) valuing a groups ability to identify what they do know and what they still need to find out more about. 

Pilot Study (2006)

Australian primary students' mathematical performances on international benchmark tests have slipped to average, with students' lack of mathematical confidence a major contributing factor (TIMSS, 2002). To overcome this problem, mathematics education research needs to focus on psychological as well as mathematical domains. Problem solving has been found to be an effective way for students to develop a deeper understanding of mathematical ideas and concepts (Cobb, Wood, Yackel, & Mc Neal, 1992) but students are not always inclined to try to solve unfamiliar problems. Individual students' inclination to solve unfamiliar problems (Williams, 2003) has been associated with optimism or resilience and, optimism can be built through achieving success in overcoming challenges (Seligman, 1995). How does optimism influence group rather than individual problem solving though? To enhance mathematics learning through group problem solving we need to know how the relative optimism (or lack thereof) of group members could influence group problem solving performance. This could inform the composition of groups likely to enhance student learning. Such information could assist teachers, teacher educators, task designers, and policy makers interested in finding ways to raise the mathematical performance of students. It should also stimulate further research in this important area.

Collaborative problem solving was studied in classrooms in which a high frequency of optimism building activity was expected to occur and it was found that it did. Students in Grade 5/6 classes in a government school were studied as they undertook three collaborative problem-solving tasks that extended over one or more lesson during 2006. Three cameras capture the activity of the six groups of four students. Microphones capture the talk of three of these groups (with opportunity to change the microphone focus). A fourth camera is available to capture reporting activity not clearly visible elsewhere. After each lesson, four students took part in video-stimulated interviews. They discussed what they had learnt, and what had influenced their learning. Situations that should theoretically build optimism were identified.

Gaye Williams

Gaye Williams taught/coordinated secondary mathematics for more than 25 years in rural and metropolitan schools (girls, boys, and co-ed: Government, Independent, and Catholic). She has provided Professional Learning for primary and secondary teachers in Australia, the USA, the Philippines, and Sweden on task design, and task implementation focused on collaborative group work. The perspectives she developed as a teacher have influenced the way she analyses group interactions and student learning in her research. Her contributions to research on engaging students in the learning of mathematics have been recognised through the 2007 University of Melbourne Chancellor’s Prize (Social Sciences), and the 2006 Australian Associations of Educational Researchers Doctoral Award for her PhD thesis.

Gaye is a fulltime staff member at Deakin University: Email This e-mail address is being protected from spambots. You need JavaScript enabled to view it She is the recipient of an Australian Postdoctoral Fellowship from the Australian Research Council, which supports her present research hosted by the International Centre for Classroom Research (University of Melbourne). Gaye is an honorary Senior Fellow of the Melbourne Graduate School at the University of Melbourne. 

Research Publications Arising From Project and Pilot Study

Williams, G. (in press). Symbiosis between creative mathematical thinking accompanied by high positive affect, and optimism. Paper accepted for publications by the International Research Group for the Psychology of Mathematics Education, Belo Horizonte, Brazil, July 2010.  

Williams, G. (in press) Abstracting by constructing then revising a ‘Partially Correct Construct’: A case study. Paper accepted for publication Mathematics Research Group of Australasia Conference Book. Fremantle, July, 2010. 

Williams, G. (2009). Engaged to learn pedagogy: Theoretically identified optimism-building situations. In R. Hunter, B. Bicknell, & T. Burgess. Crossing Divides: Mathematics Education Research Group of Australasia 32 Conference Proceedings, (Vol. 2. 595-602). Wellington, NZ: MERGA.

Williams, G. (2009). Spontaneous student questions: informing pedagogy to promote creative mathematical thinking. In Tzekaki, M., Kaldrimidou, M., & Sakonidis, H. (Eds.). Proceedings of 33rd conference of the International Group for the Psychology of Mathematics, (Vol. 5, pp. 345-352). Thessaloniki, Greece: PME.

Williams, G. (2008). Group Composition: Influences of optimism and lack thereof. In O. Figueras, J. Cortina, S. Alatorre, T. Rojano & A. Sepulveda (Eds.), Proceedings of the 2008 Joint Meeting of the International Group for the Psychology of Mathematics Education and the Group for the Psychology of Mathematics Education Meeting of North America, (Vol. 4, pp. 425-432). Mexico: Cinvestat-UMSNH.

Williams, G. (2008). Links Between Optimism-Building and Problem Solving Capacity. Paper presented at the eleventh conference of the International Congress for Mathematics Education in Topic Study Group 26, Mexico, July 2008. Accessed at http://tsg.icme11.org/tsg/show/27

Williams, G. (2008). How Group Composition Can Influence Opportunities for Spontaneous Learning.  In M. Goos, R. Brown, & K. Maker (Eds.). Mathematics Education Research Group of Australasia 31 Conference Proceedings, (Vol. 2, 581-588). Brisbane, Australia: University of Queensland.

Williams, G. (2007). Classroom teaching experiment: Eliciting creative mathematical thinking. In J. Woo, H. Lew, K. Park, & D. Seo (Eds.). Proceedings of the 31st conference of the International Group for the Psychology of Mathematics Education (Vol. 4, pp. 257-364). Seoul, Korea: PME.

Research Communicated as Teacher Papers 

Williams, G. (in press). Building optimism in prospective mathematics teachers: Psychological characteristics enabling flexible pedagogy. In Orit Zavlasky and Peter Sullivan (Eds.), Constructing knowledge for teaching secondary mathematics: Tasks to enhance prospective and practicing teacher learning. Springer Publications.

Williams, G. (2009). Why some groups work and some do not. Mathematics of Prime Importance, Proceedings of the 2009 MAV Conference, pp. 271-278. Mathematical Association of Victoria, Melbourne.

Williams, G., Menzel, B., & Sheridan, B. (2009). Suddenly we had engaged Middle Years students! How did that happen? Mathematics of Prime Importance, Proceedings of the 2009 MAV Conference, pp. 279-287. Mathematical Association of Victoria, Melbourne.

Williams, G. (2009). A deep learning approach. Shine, 9, 52-53.Williams, G. (2007). Deep understanding of ‘big ideas’ in mathematics: what does this ‘look like’? In J. Vincent, J. Dowsey, & R. Pierce (Eds.). Mathematics – Making sense of our world, Proceedings of the 2005 MAV conference, pp. 326-339, Mathematical Association of Victoria, Melbourne.

Williams, G. (2003). Student inclination to work with unfamiliar challenging problems: the role of resilience. In B. Clarke, A. Bishop, R. Cameron, H. Forgasz & W. Seah (Eds.), Making Mathematicians (pp. 374-385). Melbourne, Victoria: Mathematical Association of Victoria.

Research Papers that Raised Questions: Impetus for Study 

Williams, G. (2006). Autonomous Looking-In to support creative mathematical thinking: Capitalising on activity in Australian LPS classrooms. In D. Clarke, C. Kietel, Y. Shimizu (Eds). Mathematics classrooms in twelve countries: the insider's perspective (pp. 221-236), Sense Publications.

Williams, G. (2006) Impetus to explore: Approaching operational deficiency optimistically. In J. Novotna, H. Moraova, M. Kratka, N. Stehlikova. Proceedings of the 30th conference of the International Group for the Psychology of Mathematics Education (Vol. 5, pp. 393-400). Prague, Czech Republic: PME.

Williams, G. (2006). Student-engineered ‘Space to Think’. In P. Grootenboer, R. Zevenbergen, & M. Chinnappan (Eds.), Identities cultures and learning spaces (Vol. 2, pp. 567-576). Canberra, ACT: Mathematical Education Research Group of Australasia.

Williams, G. (2003). Associations between student pursuit of novel mathematical ideas and resilience. In L. Bragg, C. Campbell G. Herbert & J. Mousley (Eds.), Mathematical Education Research: Innovation, Networking, Opportunity,  (Vol. 2, pp. 752-759). Geelong, Victoria: Deakin University.