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GORDON ELDRIDGE: LESSONS IN LEARNING
In Mathematics, Maximize Learning for All Students
By Gordon Eldridge, TIE Columnist 06-May-14
As part of a three-year investigation into barriers that may inhibit the mathematical learning of some students, researchers from Australia’s La Trobe University identified a set of lesson elements they believe can be used to maximize the learning of all students in mathematics. These elements are: • A goal task (open-ended) with a series of less complex, problem-like tasks that scaffold development towards the goal task. The intermediate tasks anticipate a set of hypothetical cognitive processes leading to the goal task. In this case, the goal task was as follows: given isometric paper and a simple, two-dimensional representation of two side views of a building, students were asked to draw what the building might look like. The initial intermediate scaffolding task was for students to draw on isometric paper two different cakes that could be made by joining three Lamingtons (Australian cakes typically shaped like a rectangular prism). • A set of enabling prompts, which anticipate the potential difficulties students may have in achieving the tasks and can therefore be used to support students unable to solve the task without help. The prompts typically remove one of the factors contributing to the difficulty and avoid the need for the teacher to direct the student towards a particular solution strategy. For example, some students had trouble with the initial task because they did not appreciate the way isometric paper could be used. These students were given an additional isometric sheet with two cubes already drawn. Another prompt available was a set of cubes, which students could use to model what the three Lamington cakes might look like when joined. Students are only given the enabling prompts if they encounter difficulty with the task. • A set of tasks that could be posed as extension tasks to students who complete the original task. These always extend thinking around the concepts central to the goal task. For example, students who finished the initial task were asked to draw some cakes that could be made with four Lamingtons. • A list of specific pedagogies that are made explicit to students. There is some evidence from research that the use of open-ended tasks can disadvantage students who are less familiar with the goals of schooling. To help alleviate this, students were told the purpose behind each of the steps which built toward the goal task. For example, when introducing the initial task with the Lamingtons, the teacher said, “This is for you to see if you can use isometric paper to draw cubic shapes. You will learn to draw different shapes in different ways using isometric paper. There is more than one possible answer.” • Opportunities to share strategies and thought-processes and discuss them at the end of the process. The study was conducted in stages, and the stage of the study reported here involved the trialing of a lesson incorporating these elements. The lesson is based on the premise that all students in a class, regardless of prior experience and ability, should have the same goal focus and should share a common set of learning experiences. The researchers believe that this should enable all students to develop a sense of communal experience that will help them (a) participate in substantive conversations about mathematical concepts, and (b) feel successful in their mathematical learning. What were the results of the study? • The enabling prompts allowed students who experienced difficulties to re-engage with the tasks and come up with a possible solution. • The nature of the enabling prompts meant that other students were often not aware that a particular student was having difficulty with a task. In fact, the lesson observer who was in the classroom as part of the research often did not notice that these interventions had taken place. • All students were engaged with the tasks and were able to come up with a possible solution to the goal task (though some solutions of course were simpler than others), and therefore had the possibility of contributing to the final discussions. • Interviews with the teacher and some students indicated that students had felt successful in the lesson. • Analysis of observations and student products of 10 similar lessons yielded similar results, suggesting that the elements of the lesson may have broad applicability. What does this mean for our classrooms? The fact that all students were able to produce a suitable solution to the problem is a significant result, considering the lesson was conducted with a class of 55 students across a broad spectrum of ability levels. The elements outlined here seem to present not only a viable alternative to the traditional didactic math lesson, but one that also seems to motivate students to become involved in mathematical thinking and problem solving. Obviously, the idea of pre-planned enabling prompts requires a great deal of preparation on the part of the teacher in analyzing the task at hand and the potential difficulties it presents, then devising suitable prompts to enable students to think their way through those difficulties independently. Each of the sub-tasks in the lesson analyzed in this study had a variety of different prompts, available for the teacher to use with students having varying difficulties. Some of these were never used, but those that were used enabled the teacher to avoid giving the kind of lengthy explanations that, even when effective, can rob the student of the opportunity to think things through for themselves. Making pedagogies explicit to students also requires forethought on the part of the teacher, but this study suggests there may be significant rewards for using these strategies in terms of creating a communal experience for students; allowing all students to experience success; and guiding students towards taking more responsibility for their learning. If we really believe all children in our classroom can learn to think independently, then techniques such as this are a useful addition to the mathematics teacher’s toolkit. Reference Sullivan, P., Mousley, J., and Zevenbergen, R. (2006) “Teacher actions to maximize mathematics learning opportunities in heterogeneous classrooms.” International Journal of Science and Mathematics Education 4, pp. 117-143.
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