Students’ identification and expression of relations between variables in linear functions tasks in three curriculum contexts

ABSTRACT

Researching students’ responses to tasks at different year levels and in varied curriculum contexts can provide insights that relate their understandings to prior learning experiences and teaching approaches. In this article, we discuss evidence of students from three curriculum contexts (English, Australian, and Israeli) (n = 350) ways of attending to the functional relationship between two variables at different levels of secondary school in their responses to two linear functions tasks. We found that the students from an English national curriculum context were more likely than the other cohorts to focus only on the dependent variable when presented with a table of values. With a figural pattern generalization task, no examples of invalid proportional reasoning were found among student responses from the Israeli curriculum context, unlike those in the English and Australian curriculum contexts. A high percentage of the Year seven-eighths responses from the Israeli curriculum context evidenced symbolic generalization whereas several responses from the Australian curriculum indicated explicit descriptive, but not symbolic, generalization. Twenty teachers also participated in the study. Analysis of national curriculum content and task examples from each context, along with the teachers’ expectations of their students’ responses to the tasks and described teaching approaches for linear functions, led to our exploration of possible reasons for differences and similarities found among the responses. We suggest possible implications for teaching and learning linear functions and avenues for further research.

KEYWORDS:

• Linear functions
• relations between variables
• rate of change
• generalization
• secondary mathematics education

Acknowledgments

The authors would like to acknowledge with appreciation the contributions of Anne Watson and Steve Lerman to the early phases of the research project on which this paper is based.