Mathematics Tasks that Promote Conceptual Understanding

 

Mathematical tasks or exercises are common in many mathematics classes. Our classrooms are loaded with exercises and drills. Students are usually given a lot of exercises thinking that students will learn when they repeatedly do the same exercise. However, teachers need to look at mathematical exercises in terms of what is available for the learner to notice (Marton, Runesson & Tsui, 2003). This is done  by asking progressively and systematically “what  changes and what stays the same” (Watson & Mason, 2006). According to Simon and Tzur (2004)  a well-designed sequence of tasks invites learners to reflect on the effect of their actions so that they recognize key relationships. It also pointed out that mathematics is learned by becoming familiar with tasks that manifest mathematical ideas and by constructing generalizations from tasks. However, Watson and Mason (2006) caution that learning does not necessarily take place solely through learners observing some patterns in their work, even if they have generalized them explicitly. They argue this is so because “learners can do this by focusing on surface syntactic structures rather than deeper mathematical meaning – just following a process with different numbers rather than understanding how the sequence of actions produces an answer” (Watson & Mason, 2006, p.93). Also, Doerr (2000) recognizes the limitation of a sequence of tasks if students‟ focus is on the calculations because the attention may be shifted from making generalizations to “rote drilling”. Unless specific steps  are taken to promote students‟ engagement and higher order thinking, advanced conceptual understanding will not be constructed. To minimize the possibility of random collections of items treated as individual tasks by students and to maximize the possibility of non-arbitrary relationship building, Watson and Mason (2006) suggest that teachers aim to constrain the number and nature of the differences they present to students and as a result, increase the likelihood that students‟ attention will be focused on the intended mathematical concepts.

References:

Doerr, H. (2000). How can I find a pattern in this random data? The convergence of multiplicative and probabilistic reasoning. Journal of Mathematical Behaviour, 18, 431-454.

Gu, L., Huang, R., & Marton, F. (2004). Teaching with variation: A Chinese way of promoting effective Mathematics learning. In L. Fan, N. Y. Wong, J. Cai & S. Li (Eds.), How Chinese learn Mathematics: Perspectives from insiders. Singapore: World Scientific Publishing.

Marton, F., Runesson, U., & Tsui, A. (2003). The space for learning. In F. Marton & A. Tsui (Eds.), Classroom discourse and the space for learning (pp3-40). Mahwah, NJ: Lawrence Erlbaum Associates, Inc.

Simon, M., & Tzur, R. (2004). Explicating the role of mathematical tasks in conceptual learning: An elaborating of the hypothetical learning trajectory. Mathematical Thinking and Learning, 6, 91-104.

Watson, A., & Mason, J. (2006). Seeing an exercise as a single mathematical object: Using variation to structure sense-making. Mathematical Thinking and Learning, 8(2), 91-111.