The teaching/learning need

Thinking mathematically

A traditional approach to teaching mathematics is to get students to spend most of their time practicing paper-and-pencil computations.  Students come to believe that the goal of mathematics it to be able to make calculations as quickly as possible.  This does not help them to apply these skills or to be able to "recognize problems that call for these skills" (Connected Mathematics). The purpose of mathematics is to be able to solve problems.  To be able to do this the students needs to spend time "solving problems that require thinking, planning, reasoning, computing, and evaluating" (Connected Mathematics).

The traditional approach to problem solving is to show the student how to solve a problem and then get them to practice that method on similar problems.  However, the Cognitive Learning Theory tells us that students will be able to make sense of the mathematics if the skills and concepts are embedded within a context or a problem.  A PBL approach requires the student to explore, examine and reflect on methods so that they will develop a deeper understanding of mathematical concepts and related procedures.  

A "spiraling" mathematics curriculum often has the problem that students do not remember the methods or skills they have previously learnt.  This breeds a view of mathematics as a collection of unconnected techniques and algorithms to be memorised.   Not only does PBL help students to make sense of mathematics but it also helps them to process it in a more retrievable way because it is connected to prior knowledge.  This connectedness means that they are more likely to retain what they have learnt and apply it to future learning.  This is how PBL encourages deeper understanding of concepts and how they are connected.

Meaningful Learning in Mathematics

The recommendations made by (Everybody Counts, cited in Schoenfeld, 1992. p. 4) are: 

• Seeking solutions, not just memorizing procedures;

• Exploring patterns, not just memorizing formulas;

• Formulating conjectures, not just doing exercises.

To encourage meaningful learning the teacher or instructional designer needs to identify obstacles and challenges to teaching and devise strategies to overcome them (Project TALENT, P. 293).  The aim  is to help the student think mathematically about the world they live in.

For another view of Meaningful Learning see Jonassen's model

Control and Motivation

Contemporary students need to have a sense of control over their learning process.  

“[A] degree of control over their own learning can provide challenge, motivation and engagement for a wide range of student groups.” (Hennessy et al., 2007. p.140) 

They expect to be an initiator and as well as a respondent in this process.  They want learning to be "conceptually and intellectually engaging" (Hennessy et al., p.12).  Satisfying these needs will motivate students and promote meaningful learning. 

An ICT Mediated Solution

Computer assisted Collaborative Learning (CSCL) arose in the 1990s from the potential for social interaction made available by computer networks and the Internet (Stahl et al., 2006).  Rochelle & Teasley (1995) argue that collaborative problem solving consists of two concurrent activities: solving the problem together and building a shared conception of the problem in relation to a joint problem space.  They describe computers as a cognitive tool for group discussion and negotiation directed towards the construction of shared meanings and solutions to problems.

“Mathematics instruction should provide students the opportunity to explore a broad range of problems and problem situations, ranging from exercises to open-ended problems and exploratory situations.” (Schoenfeld, 1992. p. 32)

PBL offers a way to address the recommendations made by Everybody Counts.  While  prescription of how a problem is solved in a worked example still has it's place in learning mathematics, much more can be learnt when students engage in "solving complex and ill-structured problems as well as simple, well-structured problems" (Jonassen, 1997, cited in Jonassen et al 2003, p. 8).  Adhering to the constructivist learning principles, a student who is allowed to apply the content of their studies to their own life and situation, will be able to construct their learning based on their own experiences" (Barb et al, 2003, P. 123).  It gives the student the "opportunity to contextualise the "content" to be learned" (Barb et al, 2003, P. 127).  A flexible PBL environment will support learners in their "investigation of issues and the building upon experiences particular to their own lives" (Barb et al, 2003, P. 114)

In PBL the students become the designer and active creator of a collaborative knowledge building process.  PBL encourages student-student and student- teacher dialogue about the problem. By collaborating in small groups on problem-solving tasks, they are learning mathematicaly and socially. (766 words)

Next- The Learning theories that underpin Problem Based Learning.

PBL and the Knowledge Creation Metaphor

Back to Problem Based Learning

To Instructional Design Australia

References

Barab, S.A., Thomas, M.K. and Merrill, H. (2001) Online Learning: From Information dissemination to fostering Collaboration.  Journal of Interactive Learning Research (2001) 12(1), 105-143

Connected Mathematics Project retrieved from http://showmecenter.missouri.edu/showme/cmp.shtml on 8 May

Johassen, D, H, (1996) Computers in the Classroom, Mindtools for Critical Thinking. Prentice-Hall

Johassen, D, H, Char, C. and Yueh, Hl (1998) Computers as mindtools for engaging learners in critical thinking. TechTrends; Mar 1998; 43, 2; Career and Technical Education

Jonassen, D. H. (2000) Computers as Mindtools for Schools, Engaging Critical thinking Second Edition.Prentice hill

Jonassen, D.H. Howland, J., Moore, J., Marra, RM. (2003) Learning to solve Problems with Technology. A Constructivist Perspective

Pea, R, D.   (1985) Learning to Think Mathematically retrieved from http://www.stanford.edu/~roypea/RoyPDF%20folder/A24_Pea_85c.pdf 15 April 2007

Project TALENT: Infusing Technolgoy in K-12,

Rochelle, J. and Teasley, S.D. (1995)  The construction of shared knowledge in Collaborative Problem sovling.  In Computer supported Collaborative Learning Edited by O'Malley 1995

Schoenfeld, A. H. (1992). Learning to think mathematically: Problem solving, metacognition, and sense-making in mathematics. In D. Grouws (Ed.), Handbook for Research on Mathematics Teaching and Learning (pp. 334-370). New York: MacMillan.

Stahl, G., Koschmann, T., & Suthers, D. (2006) Computer-supported collaborative learning: an historical perspective. In R.K. sawyer (ED,) Cambridge handbook of the learning sciences (pp.409-426).  Retrieved from http://www.cis.drexel.edu/faculty/gerry/cscl/CSCL_English.pdf