A Developing Approach to Studying Students’ Learning through Their Mathematical Activity

Learning trajectories in mathematics: A foundation for standards, curriculum, assessment, and instruction

“…Pieces of the necessary knowledge are nevertheless available, and the standards-based reform movement of the last few decades is shifting the norms of teaching away from just delivering the content and towards taking more responsibility for helping all students at least to achieve adequate levels of performance in core subjects The state content standards, as they have been tied to grade levels, can be seen as a first approximation of the order in which students should learn the required content and skills However, the current state standards are more prescriptive than they are descriptive They define the order in which, and the time or grade by which, students should learn specific content and skills as evidenced by satisfactory performance levels But typically state standards have not been deeply rooted in empirical studies of the ways children's thinking and understanding of mathematics actually develop in interaction with instruction 2 Rather they usually have been compromises derived from the disciplinary logic of mathematics itself, experience with the ways mathematics has usually been taught, as reflected in textbooks and teachers' practical wisdom, and lobbying and special pleading on behalf of influential individuals and groups arguing for inclusion of particular topics, or particular ideas about "reform" i. iNTrOduCTiON 3 We favor the view that students are active participants in their learning, bringing to it their own theories or cognitive structures (sometimes called "schemes" or "schemata" in the cognitive science literature) on what they are learning and how it works, and assimilating new experience into those theories if they can, or modifying them to accommodate experiences that do not fit Their theories also may evolve and generalize based on their recognition of and reflection on similarities and connections in their experiences, but just how these learning processes work is an issue that requires further research (Simon et al , 2010) We would not, however, carry this view so far as to say that students cannot be told things by teachers or learn things from books that will modify their learning (or their theories)-that they have to discover everything for themselves A central function of telling and showing in instruction is presumably to help to direct attention to aspects of experience that students' theories can assimilate or accommodate to in constructive ways or "the basics " Absent a strong grounding in research on student learning, and the efficacy of associated instructional responses, state standards tend at best to be lists of mathematics topics and some indication of when they should be taught grade by grade without explicit attention being paid to how those topics relate to each other and whether they offer students opportunities over time to develop a coherent understanding of core mathematical concepts and the nature of mathematical argument The end result has been a structure of standards and loosely associated curricula that has been famously described as being "a mile wide and an inch deep" (Schmidt et al , 1997) Of course some of the problems with current standards could be remedied by being even more mathematical-that is, by considering the structure of the discipline and being much clearer about which concepts are more central or "bigger," and about how they connect to each other in terms of disciplinary priority A focus on what can be derived from what might yield a more coherent ordering of what should be taught And recognizing the logic of that ordering might lead teachers to encourage learning of the central ideas more thoroughly when they are first encountered, so that those ideas don't spread so broadly and ineffectively through large swaths of the curriculum But even with improved logical coherence, it is not necessarily the case that all or even most students will ...…”

Measuring Messy Mathematics: Assessing learning in a mathematical inquiry context

“…If educators assumed that what students learned was the result of the instructional practices of teachers, then it would be unnecessary to assess learning (Wiliam, 2010a); teaching does not cause learning (Simon et al, 2010). In order to assess student learning through inquiry, I will firstly clarify views of learning that align with this pedagogy.…”

“…In inquiry, the complex interactions that reflect students participating involve a wide range of characteristics. Simon and his colleagues (Simon et al, 2010) stated that learning occurred in the context of mathematical communication with peers, such as negotiating meanings, or sharing and comparing solutions. In these situations, students are required to justify their thinking when challenged by others, establishing shared mathematical ideas that provide the basis for subsequent work.…”

“…For constructivists, a problem is a roadblock or problematic, in relation to the solver (Confrey, 1991;Fielding-Wells & Makar, 2008;Koellner, Jacobs & Borko, 2011;Stigler et al, 1996). These roadblocks or perturbations are useful in activating assimilatory schemes that the students have available, becoming an important part of a teaching process (Simon et al, 2010). Often inquiry situations can be engineered by the teacher to evoke misconceptions and to capitalise on any unanticipated problems that arise (Makar, 2008;.…”

“…For the mathematics learner in inquiry, classroom peers become the audience as well as problem posers and fellow learners (FieldingWells, 2010). Educational research which strives to understand mathematics conceptual learning does so with the hope of informing teaching practice and pedagogy to support student learning (Simon, 1995;Simon et al, 2010;Steffe, 2003). Although various aspects about learning mathematics in inquiry classrooms has been explored, this has not been considered in relation to classroom elements of assessment and teaching in inquiry settings, even though changes in one element could potentially influence all others.…”