From "Easy" to "Difficult" or Vice Versa: The Case of Infinite Sets
Tsamir, Pessia, Focus on Learning Problems in Mathematics
It is commonly agreed that students' ways of thinking should be taken into account when planning instruction and that teachers should choose or construct sequences of instruction for use and discussion in class. The NCTM document, "Principles and Standards for School Mathematics" (2000, p. 17), for instance, specifies that teachers "need to understand the different representations of an idea, the relative strengths and weaknesses of each, and how they are related to one another ... They need to know the ideas with which students often have difficulty and ways to help bridge common misunderstandings". However, the shift from such theoretical knowledge to the design of teaching sequences and to practice is not a trivial one in the least.
For example, one important facet of mathematical knowledge is the ability to move flexibly among different representations of a given mathematical notion, process or problem. Still, it has been widely documented that different representations of the same mathematical problem often trigger different and sometimes even conflicting solutions (e.g., Even, 1998; Janvier, Girardon, & Morand, 1993). Research findings clearly indicate how students tend to respond to the different representations of the same mathematical task and their justifications for these responses (Duval, 1983; Hart, Johnson, Brown, Dickson, & Clarkson, 1989; Tirosh & Tsamir, 1996). A question that arises is how the accumulated data on students' reactions to different representations of given mathematical tasks can be applied to instruction.
1. Using Different Representations in Instruction
Two methods that use the accumulated findings indicating that different representations lead to different results, are teaching by analogy and teaching by cognitive conflict. In both methods, students are presented with different representations of the same task: one is the target task, known to commonly trigger an incorrect response (a difficult task), and the other is the anchoring task--a task known to intuitively trigger a correct response (an easy task). Teachers then attempt to promote students' awareness of the fact that they are being presented with the same task, albeit in different representations.
When going "From Difficult to Easy" in the cognitive conflict teaching approach, students are first asked to solve the target task and then the anchoring task. The students frequently reach two different solutions. Thus, in a subsequent discussion, teachers try to lead them to identify the conflicting elements in their different solutions, and to resolve the conflict according to the relevant mathematical theory. The challenges are to promote students' awareness that the two tasks are essentially the same, and that their responses are incompatible with each other, then to lead them to resolve the conflict in accordance with the formal, mathematical framework (see, for instance, Swan, 1983; Tirosh & Graeber, 1990). It should be noted that, in the literature the terms "anchoring task" and "target task" were defined and used with reference to the From Easy to Difficult teaching (by analogy) approach.
When going "From Easy to Difficult" in the teaching by analogy approach, the students are first asked to solve the anchoring task and then the target task. The teacher preferably chooses or constructs a sequence of bridging tasks that gradually lead from the anchoring task to the target task. Such a sequence of steps may assist the students in reaching the same correct solution to the same task presented in different ways. Indeed, it has been reported that by following such a "From Easy to Difficult" sequence of instruction, students frequently reach correct solutions to all tasks presented in the sequence, including the target task. The challenge is to find suitable anchoring and bridging tasks, and to verify that at each stage students make the mathematical connections that allow them to grasp the "sameness" of the tasks (see, for instance, Clement, 1993; Stavy, 1991). …