We take mathematical structure to mean the identification of general properties which are instantiated in particular situations as relationships between elements or subsets of elements of a set. Because we take the view that appreciating structure is powerfully productive, attention to structure should be an essential part of mathematical teaching and learning. This is not to be confused with teaching children mathematical structure. We observe that children from quite early ages are able to appreciate structure to a greater extent than some authors have imagined. Initiating students to appreciate structure implies, of course, that their appreciation of it needs to be cultivated in order to deepen and to become more mature. We first consider some recent research that supports fllis view and then go on to argue that unless students are encouraged to attend to structure and to engage in structural fllinking they will be blocked from fllinking productively and deeply about mathematics. We provide several illustrative cases in which structural fllinking helps to bridge the myfllical chasm between conceptual and procedural approaches to teaclfing and learning mathematics. Finally we place our proposals in the context of how several writers in the past have attempted to explore the importance of structure in mathematics teaching and learning. Theoretical FrameWe take mathematical structure to mean the identification of general properties which are instantiated in particular situations as relationships between elements.These elements can be mathematical objects like numbers and triangles, sets with functions between them, relations on sets, even relations between relations in an ongoing hierarchy. Usually it is helpful to think of structure in terms of an agreed list of properties which are taken as axioms and from which other properties can be deduced. Mathematically, the definition of a relation derives from set theory as a subset of a Cartesian product of sets. Psychologically, a relationship is some connection or association between elements or subsets which have been themselves been discerned. When the relationship is seen as instantiation of a property, the relation becomes (part of) a structure. For example:The relation between a whole number and its double can be denoted by {[n, 2n]: n = 1, 2, 3,...} or by n > 2n among other ways; being a pair consisting of a number and its double is a property instantiated here among the whole numbers, but other numbers are possible. The set {[1, 2], [2, 5], [3, 2], [1, 4], [4, 4]} is also a relation, though not one that deserves a special name, and unlikely to be an instantiation of a specific property other than that it specifies this relation! Recognising a relationship amongst two or more objects is not in itself structural or relational thinking, which, for us, involves making use of relationships as instantiations of properties.Awareness of the use of properties lies at the core of structural thinking. We define structural
We sought to examine empirically the prevailing assumption that changing assessment can leverage curricular reform. This assumption has been signi® cantly con-® rmed by our research for the case of mandated high-stakes assessment. Two studies were conducted in the two most populous Australian states, New South Wales and Victoria. In the ® nal two years of secondary school in both states, courses of study and assessment arrangements are mandated for all schools, including the private sector, by the state' s Board of Studies. Congruence between mandated assessment and schoolwide instructional practice was found in two states whose high-stakes assessment embodied quite contrasting values. The assessment agendaWhatever metaphor is employed to characterize the function of assessment within or upon the curriculum, the centrality of assessment is universally acknowledged. This paper makes use of the ® ndings from two related studies to establish and elaborate two fundamental observations regarding the function of assessment:. attempts at curriculum reform are likely to be futile unless accompanied by matching assessment reform; and . assessment can be the engine of curriculum reform, or the principal impediment to its implementation.
ABSTRACT.This paper examines one mode of mathematical communication: that of student journal writing in mathematics. The focus of the discussion is a study of four years' use of journal writing in mathematics involving approximately 500 students in Grades 7 through 11 in a particular Victorian secondary school. The evaluation of the experimental use in one school of journal writing in mathematics provides a powerful demonstration of the link between language and mathematics and suggests a relationship between students' mathematical writings and their perceptions of mathematics and mathematical activity.
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