by Ng Kit Ee Dawn
What types of tasks can teachers use to enhance collaborative learning in the Math classroom? And how does working in groups enable students develop mathematical decision-making skills? Find out how contextualized tasks can help.
- What is the value of contextualized tasks for mathematical learning?
- How can such tasks help to foster collaborative decision making?
- What difficulties do teachers need to look out for?
Contextualized tasks are mathematical tasks embedded in the experiences of students. They are open-ended and non-routine, and may be interdisciplinary in nature.
Project work, application tasks and mathematical modelling tasks are examples of contextualized tasks.
Such tasks help students to relate school-based learning with real-life use of mathematics. They provide opportunities for flexible, adaptive applications of what students know.
The knowledge and skills that students bring to the task, whether taught or self-discovered, determines their personal interpretations of the context.
Mathematical Learning in Context
Contextualized tasks help students learn important and interlinked mathematical skills, such as:
- making sense of context (e.g., drawing assumptions and specifying conditions or constraints);
- making connections between mathematical concepts and skills and real-world needs; and
- making appropriate and reasonable mathematical decisions based on the contextual demands.
Students also learn collaborative skills as they often have to work in groups when dealing with contextualized tasks. This encourages them to learn with and from one another.
A research study was carried out to investigate how students apply mathematical concepts and skills to a contextualized task.
Two aspects of the study are described: the difficulties students faced during mathematical decision making; and effective strategies for interpersonal monitoring of mathematical thinking during group collaboration.
The students had to work in groups of four to complete an interdisciplinary project over a 14-week period. They were tasked to design an environmentally friendly building in a location of their choice in Singapore.
They met with their facilitating teachers on a weekly basis and were given mini-tasks to complete along the way. Three of the mini-tasks required them to apply mathematical knowledge and skills:
1. deciding on the design, size, location, features and facilities of the building;
2. calculating the cost of fitting out an area in the building; and
3. making scale drawings of the building.
Each group also had to construct a physical model of their building according to their scale drawings.
Surveys were administered to consenting participants. Additional data was gathered through interviews, lesson observations and video-taped lessons of 10 voluntary groups (5 groups from each stream) across the 3 schools.
Understanding Students’ Mathematical Difficulties
The students were found to have difficulties in at least three areas: spatial visualization; awareness of the purpose and use of scale drawings; and integration of real-world knowledge with mathematical decision making.
Difficulties with spatial visualization
Some students had difficulty making realistic and reasoned estimations of lengths and areas. One student suggested that the dimensions of the school hall floor should be 5m by 5m. A group member questioned how the entire school population would fit in an area of 25m2 during weekly assemblies.
Some students could not visualize how the various views of the building (e.g., top, side, front), when put together, would form a coherent image of the whole building.
Limited awareness of the use of scale drawings
Several students had difficulties distinguishing between scale drawings and measured drawings (i.e., drawings with only dimensions indicated). There was also evidence that they did not understand the purpose of scale drawings. Some groups made unrealistic scale drawings from which they could not make physical scale models.
Lack of integration of real-world knowledge
A few students did not apply their understanding of furnishings and budgeting to the decisions they made. For example, one student insisted on having a bathtub in a school toilet, to the intense objection of group members who felt this was unrealistic and out of the budget.
Interpersonal Monitoring of Mathematical Thinking
As the students worked together, they also engaged in interpersonal monitoring of mathematical thinking.
These strategies were identified based on an analysis framework adopted for this study, which drew upon other established frameworks (Artzt & Armour-Thomas, 1992; Goos, 2002; Schoenfeld, 1985).
Students helped to monitor each other’s mathematical thinking by questioning their group members on:
- the repertoire of mathematical knowledge and skills they could bring to the project;
- whether there was important information missing in the given project description;
- the suitability of their mathematical approaches;
- the progress in their chosen approaches; and
- The appropriateness, logic and accuracy of their decisions and calculations.
Providing Effective Scaffolding
These findings can help teachers facilitate mathematical learning more effectively when using contextualized tasks.
For example, teachers can tap the potential of contextualized tasks through careful and purposeful scaffolding. Scaffolding can focus on the interlinked aspects of mathematical learning highlighted above.
Teachers can also encourage group members to question each other critically in order to activate effective interpersonal monitoring of mathematical thinking.
References
Artzt, A. F., & Armour-Thomas, E. (1992). Development of a cognitive-metacognitive framework for protocol analysis of mathematical problem solving in small groups. Cognition and Instruction, 9(2), 137-175.
Goos, M. (2002). Understanding metacognitive failure. Journal of Mathematical Behavior, 21(3), 283-302.
Ng, K. E. D. (2009). Thinking, small group interactions, and interdisciplinary project work. Unpublished doctoral dissertation, University of Melbourne, Australia.
Schoenfeld, A. H. (1985). Mathematical problem solving. Orlando, FL: Academic Press.
