Max Stephens

Japanese lesson study in mathematics : its impact, diversity and potential for educational improvement

Japanese Lesson Study in Mathematics Methods and Types of Study Lessons Trends of Research Topics in Japan Society of Mathematical Education Diversity and Variety of Lesson Study International Cooperative Projects.

Assessment: the engine of systemic curricular reform?

We sought to examine empirically the prevailing assumption that changing assessment can leverage curricular reform. This assumption has been significantly confirmed 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 final 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 states Board of Studies. Congruence between mandated assessment and schoolwide instructional practice was found in two states whose high-stakes assessment embodied quite contrasting values.

Appreciating Mathematical Structure for All.

We takemathematical 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 this view and then go on to argue that unless students are encouraged to attend to structure and to engage in structural thinking they will be blocked from thinking productively and deeply about mathematics. We provide several illustrative cases in which structural thinking helps to bridge the mythical chasm between conceptual and procedural approaches to teaching 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.

The Ripple Effect: The Instructional Impact of the Systemic Introduction of Performance Assessment in Mathematics

Mathematics assessment recently has been primarily directed towards increasing the degree of correspondence between the intended curriculum, the taught curriculum, the learned curriculum, and the assessed curriculum. In the past, countries such as Australia, in attempting to assess the taught curriculum, have limited their assessment to timed and written tests, where the major focus has been on facts or skills. It is becoming increasingly evident that exclusive reliance on this form of testing is inappropriate: misrepresenting mathematics, at odds with contemporary curricula, misleading in the information it provides teachers, and potentially destructive in its effects on some learners (Clarke, 1992).

Graph Interpretation Aspects of Statistical Literacy: A Japanese Perspective

Many educators and researchers are trying to define statistical literacy for the 21st century. Kimura, a Japanese science educator, has suggested that a key task of statistical literacy is the ability to extract qualitative information from quantitative information, and/or to create new information from qualitative and quantitative information. This article presents research that offers a theoretical basis using the SOLO Taxonomy to capture students’ ability to create new information from qualitative and quantitative information. This research shows that the “creation of dimensionally new information” is a complex construct requiring further research and a deeper analysis than Kimura appears to have used.

Regulating the Entry of Teachers of Mathematics into the Profession: Challenges, New Models, and Glimpses into the Future

This chapter draws attention to different methods being used in the preservice training of teachers of mathematics in the United States of America, Australia, Japan, and the Netherlands. The analysis presented reveals that there are qualitative and quantitative differences in the approaches the nations are taking to the preparation of future teachers of mathematics, especially in relation to practicum experience during teacher education courses. In some nations, great emphasis is placed on giving student teachers as much time as possible in front of real classes of students, but in other nations (e.g., Japan), more emphasis is given to the quality of lesson planning and less to the amount of time actually spent in classrooms. There are also between-nations differences in the extent to which preservice teachers collaborate with their peers in their practicum experiences.

The Australian Curriculum: Mathematics—How Did it Come About? What Challenges Does it Present for Teachers and for the Teaching of Mathematics?

The Australian Curriculum: Mathematics which incorporates the content descriptions and proficiencies from Foundation Year to Year 10 came into being in December 2010 when all Australian governments—the national government and the governments of the eight States and Territories—gave their approval to the draft which had been in circulation for nearly two years. Prior to that, each State and Territory had responsibility for developing and implementing its own curriculum. In 2008, an Australian Curriculum and Reporting Authority (ACARA) was also established to coordinate and oversee the development of national curricula in all areas of compulsory schooling, and to move towards an agreed upon national curriculum for Years 11 and 12. The formation of ACARA and the adoption of an Australian Curriculum: Mathematics (2010) are interpreted as a result of major transformations of an Australian federalist model over the past twenty years, shaped in large degree by the demands of national assessment and school reporting. This chapter examines how this came about, what has been achieved within Australia’s ongoing federalist framework, and also points to some future challenges for teachers in implementing the national curriculum in mathematics.

Making Connections Between Modelling and Constructing Mathematics Knowledge: An Historical Perspective

This study will look at a surprising resolution of the tension that arises in trying to strike a balance between modelling and pure mathematics by examining Japanese textbooks for the junior high school nearly 70 years ago. Three characteristics are found: (1) two distinct roles –first as objects to mathematize in order to solve real world problems and second as evidence by which to test the validity of mathematical concepts; (2) repeated instances of the same contexts through which new phases of mathematization could be developed; and (3) a series of real world questions focussed on the reason for solving a real world problem.

Reconsidering the Roles and Characteristics of Models in Mathematics Education

A model is generally assumed to be built by translating a real world problem into a mathematical representation. We attempt to re-construct this interpretation and point to at least two distinct meanings: (Role 1) models as hypothetical working spaces, and (Role 2) models as physical/mental entities for comparing and contrasting. This leads us to draw attention to four different perspectives of modelling: (a) where modelling is interpreted as interactive translations among plural worlds not between two fixed worlds; (b) where models have the potential to incorporate scenarios beyond the initial problem situation; (c) where the mathematical world is used as a source of mental entities for comparing and contrasting; (d) where modelling competency means knowing how to balance between these different roles. Perspectives (a) and (d) are concerned with Role 1 and (b), (c) and (d) are concerned with Role 2.

Comparing an Analytical Approach and a Constructive Approach to Modelling

This study investigates the effects when an analytical approach is used in modelling as compared to a constructive approach. In the first approach, students were given a simple mathematical representation of the situation being modeled, while in the second, students were given some key questions to guide them in creating a suitable representation. Although the results were not totally in favor of a constructive approach, it does appear that students using an analytical approach tended to focus too much on the given mathematical representation without paying sufficient attention to the assumptions and limiting conditions implicit in the situation.