This thesis comprises three original papers about learning rate of change in school mathematics. The overall aim of the thesis is to contribute with knowledge in this particular area of research. Within this aim, each of the original papers contribute with its own perspective. The theoretical framework used in the thesis is variation theory of learning (Marton & Booth, 1997; Marton, 2015), by which learning is seen as experiencing a phenomenon in a new way. This theoretical point of departure is reflected in the research question of the thesis: What is critical to discern to use and express rate as a measure of the relation of changes in two quantities? The empirical study was conducted as a learning study (Pang & Marton, 2003). A learning study is an iterative, interventional research arrangement in which teachers and researchers collaboratively explore a specific ability, the object of learning, worthwhile for the students to learn. The object of learning related to this thesis, ‘to express the quantitative rate of change of a linear relation’, was explored in a series of three research lessons at a secondary school. Data consists of students’ responses to written pre- and post-tests, and lesson videos. Some data have been analysed during the on-going empirical study and some data have been analysed after it was concluded. Principles from variation theory have been used as tools for analysis throughout the study. Main results of Paper I demonstrate how two critical aspects are identified and revised through the process of learning study. In Paper II the results indicate that qualitatively different questions in a task may affect students’ ways to relate changes in two quantities. The results of Paper III suggest how different perspectives of slope may promote homogeneity as an aspect of rate. Results also comprise six critical aspects of the object of learning, four of which was identified by revisiting the results of Paper II. In summary, the critical aspects also specify the meaning of a covariational perspective of rate. Results are discussed in relation to previous educational research about rate of change, covariation of quantities and students’ conceptions of rate and slope. Further research directions are suggested.

The aim of this study is to compare the impact of different phrasings of questions as of how students relate and express relations between different quantities in student tasks concerning rate of change. Through a qualitative analysis we compare how students in ninth grade (age 15) respond to two different framings of questions, concerning how fast the volume of fluid in two medicine bags change. The analysis indicates that a comparing question (”which changes fastest?”) can open up a wide outcome space, in which we could observe five qualitatively distinct ways of solving the task. Furthermore, a question that requests a value (“how fast does it change?”) seem to encourage students to make multiplicative comparisons, which is close to the mathematical meaning of rate of change. Finally, we discuss the potential of each question for pointing to different aspects of rate of change, and how they could be used by teachers for different purposes in teaching situations.