Reflecting on CCRA - Pedagogy and Mathematics
Discussion of the practice of creating, critiquing and revising arguments:
The segments of Ms. Hauser’s class presented offer a rich example of students engaged in creating, critiquing, and revising arguments.
It is important to think about CCRA both as a mathematical practice and as a way to develop students’ mathematical knowledge.
As mathematicians generate new knowledge, they create, critique and revise arguments. A mathematical argument includes assumptions, logical deductions (inferences), and, often, calculations or other mechanics involved in its execution. Some arguments might contain a flaw in a step in the reasoning or calculations, but rest on solid assumptions. Other arguments might contain a flaw in the assumptions.
For her argument, Amanda assumed that she could scale up a hexagon train of a given length and that the perimeter would scale up proportionally. Given this assumption, her reasoning and calculations were correct. However, there was an important flaw in her assumption.
A teacher (and mathematician) has a choice when faced with an argument that is based on a false premise or assumption. Perhaps the assumption is “fatally flawed,” meaning that it is flawed in such a way that it is impossible to adjust the argument to make it work. Perhaps, however, some revision process – e.g., extending the argument, restricting the domain – would shore up the argument. As a teacher, it is critical to elicit students’ arguments to hear their assumptions and reasoning in order to make some kind of determination about where the argument is on solid ground and what aspect may need revision or complete rethinking. This work then informs the teacher’s pedagogical decision making.
In working individually with Amanda, Ms. Hauser tried to prompt her to see that her method was flawed, since the assumption of proportionality was flawed. This pedagogical approach, however, did not produce any revision to the argument at that point. In the whole-class setting, Ms. Hauser asked the students to take up analyzing the scaling argument. The student critiques of Amanda’s (and other students’) method identified that it systematically overlooked the “connections” between the groups of hexagons and consequently overcounted the perimeter. This classroom forum of critique and review ultimately led to a revision that resulted in a complete, logical argument for the perimeter of the 25th figure.
Ms. Hauser reflects on teaching with CCRA
Ms. Hauser describes how her students use their classmates’ reasoning after it has been publicly presented.
Ms. Hauser gives her perspective on the value of the mathematical habit of creating, critiquing, and revising arguments.
Ms. Hauser describes pedagogical strategies for choosing which student arguments to make public.
Deepening and extending the mathematics
This set of slides (PDF) offers additional information about the mathematics of the Hexagon Train Task and provides some extension tasks.