I7 Using Workplace Practice to Inform Curriculum Change Geoff Wake Julian Williams University of ... more I7 Using Workplace Practice to Inform Curriculum Change Geoff Wake Julian Williams University of Manchester University of Manchester geoff. wake@ man. ac. uk Julian. williams (o; man. ac. uk We report here implications for curriculum and pedagogy drawn from the research ...
We report a study of repairs in communication between workers and visiting outsiders (students, r... more We report a study of repairs in communication between workers and visiting outsiders (students, researchers or teachers). We show how cultural models such as metaphors and mathematical models facilitated explanations and repair work in inquiry and pedagogical dialogues. We extend previous theorisations of metaphor by Black; Lakoff and Johnson; Lakoff & Nunes; and Schon, to formulate a perspective on mathematical models and modelling and show how dialogues can manifest (i) application of 'dead' models to new contexts, and (ii) generative modelling. In particular, we draw in some depth on one case study of the use of a double number line model of the 'gas day' and its mediation of communication within two dialogues, characterised by inquiry and pedagogical discourse genres respectively. In addition to spatial and gestural affordances due to its blend of grounding metaphors, the model translates between workplace objects on the one side and spreadsheet-mathematical symbols on the other. The model is found to afford generative constructions that mediate the emergence of new understandings in the dialogues. Finally we discuss the significance of this metaphorical perspective on modelling for mathematics education.
The paper will report on the current debate on the educational interfaces between mathematics and... more The paper will report on the current debate on the educational interfaces between mathematics and industry at school level. Especially the approaches, which plead for the inclusion of mathematical modelling into mathematical education at school are described. Based on this description the educational consequences for a stronger connection between mathematics at workplace and school mathematics are discussed.
ABSTRACT This paper reports on a case study in which we detail how a college mathematics and chem... more ABSTRACT This paper reports on a case study in which we detail how a college mathematics and chemistry student struggles to make sense of the graphical output of an experiment in an industrial chemistry laboratory. The student's attempts to interpret unfamiliar graphical conventions are described and contrasted with those of college mathematics. Our analysis of this draws on activity theory to assist in understanding the position of the student in both the college and the workplace. This highlights the limitations of the experience of the student at college and we question how the mathematics curriculum might be adapted to assist students in making sense of workplace graphical output.
There is increasing recognition that current mathematics curricula do not adequately equip people... more There is increasing recognition that current mathematics curricula do not adequately equip people to use and apply mathematics effectively in different spheres of their lives, for example, as learners, citizens and workers. For a flavour of the national
Lesson study is increasingly used throughout the world as a model of collaborative professional l... more Lesson study is increasingly used throughout the world as a model of collaborative professional learning. In this article we describe the theoretical foundations of an ongoing study which seeks to explore how we might inform collaborative learning with a focus on the development of curriculum coherence. The study involves collaboration between teachers and researchers in both the United Kingdom and Japan. We draw on the theory of expansive learning as a development of Cultural Historical Activity Theory and give insight into the potential this has to inform our work with a range of didactical devices or artefacts that provide connections across topics of conceptual understanding of mathematics and development over time. Expansive learning as a theoretical lens helps us understand our exploratory work in this area and provides insights into how this may be taken forward over the course of the study
In mathematics education in England, as in other countries, there is often tension in the on-goin... more In mathematics education in England, as in other countries, there is often tension in the on-going debate about the nature of the curriculum between proponents of “back-to-basics” who argue for a focus on the learning of fundamental knowledge, concepts, skills and techniques and those who take the view that the development of problem solving skills through a more inquiry-oriented approach are essential and will motivate learning of the basics. This might be characterised as a debate between those prioritising ‘content’ on the one hand and on the other those who favour ‘process skills’. In general over recent years the former have held sway in England. Consequently we find that English mathematics teachers in many instances have themselves been learners and have become teachers in a culture in which lessons might be characterised as focussed on content and in practice lessons often start with teacher exposition of “how to” use rules and procedures followed by students mimicking what ...
Social Interactions in Multicultural Settings, 2009
We acknowledge the support of the ESRC-TLRP programme of research into widening participation, fu... more We acknowledge the support of the ESRC-TLRP programme of research into widening participation, funded by ESRC.RES-139-25-0241.
The starting point of this article is the question, “how might we inform an epistemology of numer... more The starting point of this article is the question, “how might we inform an epistemology of numeracy from the point of view of better preparing young people for workplace competence?” To inform thinking illustrative data from two projects that researched into mathematics in workplace activity and the teaching and learning of modelling in the classroom is used albeit, by necessity, briefly. Analysis draws attention to the crucial role that understanding the structure of the contextual situation plays in developing a mathematical model of this. It is at this coupling of reality and mathematics that insight into, and understanding of, both mathematics and reality can be developed—or not. This important issue is illustrated with reference to two specific workplace situations that draw on understanding of fraction as gradient and explore how we might use a model of this to scaffold understanding of both reality and mathematics and how each might support the other. With a focus on model formulation during classroom activity attention is then drawn to how students tend to work towards reaching a solution to a particular problem with the consequence that their mathematical representation of the reality does not easily allow for consideration of variation of key factors. In conclusion a research agenda is proposed that seeks to inform an epistemology of numeracy by focussing on numerate activity at the nexus of reality and mathematics by (1) structuring mathematical knowledge using mathematical models that might be used to provide insight into a range of models of workplace realities, and (2) requiring student engagement with repeated use of models in ways that emphasise exploration of variability in key factors of the realities that the models represent.
This article analyses aspects of the process of developing 'functional' assessments of mathematic... more This article analyses aspects of the process of developing 'functional' assessments of mathematics at the end of compulsory schooling in England. A protocol that was developed for scrutinising assessment items is presented. This protocol includes an indicator of the 'authenticity' of each assessment item. The data are drawn from scrutiny of 589 assessment items from thirty-nine formal unseen examinations taken by students aged sixteen, and the article illustrates ways that mathematics is presented in different contexts in examinations. We suggest that currently the 'human face' of the questions may serve to disguise routine calculations, and we argue that in formal examinations connections between mathematics assessments situated in context and functional mathematics have yet to be established.
Previous work on transition between institutions suggests transitional problems can be dangerous ... more Previous work on transition between institutions suggests transitional problems can be dangerous for individual learners‟ progress. Drawing on the literature, and on our CHAT perspective, we anticipated that transition points would be significant to students narrative construction of ...
I7 Using Workplace Practice to Inform Curriculum Change Geoff Wake Julian Williams University of ... more I7 Using Workplace Practice to Inform Curriculum Change Geoff Wake Julian Williams University of Manchester University of Manchester geoff. wake@ man. ac. uk Julian. williams (o; man. ac. uk We report here implications for curriculum and pedagogy drawn from the research ...
We report a study of repairs in communication between workers and visiting outsiders (students, r... more We report a study of repairs in communication between workers and visiting outsiders (students, researchers or teachers). We show how cultural models such as metaphors and mathematical models facilitated explanations and repair work in inquiry and pedagogical dialogues. We extend previous theorisations of metaphor by Black; Lakoff and Johnson; Lakoff & Nunes; and Schon, to formulate a perspective on mathematical models and modelling and show how dialogues can manifest (i) application of 'dead' models to new contexts, and (ii) generative modelling. In particular, we draw in some depth on one case study of the use of a double number line model of the 'gas day' and its mediation of communication within two dialogues, characterised by inquiry and pedagogical discourse genres respectively. In addition to spatial and gestural affordances due to its blend of grounding metaphors, the model translates between workplace objects on the one side and spreadsheet-mathematical symbols on the other. The model is found to afford generative constructions that mediate the emergence of new understandings in the dialogues. Finally we discuss the significance of this metaphorical perspective on modelling for mathematics education.
The paper will report on the current debate on the educational interfaces between mathematics and... more The paper will report on the current debate on the educational interfaces between mathematics and industry at school level. Especially the approaches, which plead for the inclusion of mathematical modelling into mathematical education at school are described. Based on this description the educational consequences for a stronger connection between mathematics at workplace and school mathematics are discussed.
ABSTRACT This paper reports on a case study in which we detail how a college mathematics and chem... more ABSTRACT This paper reports on a case study in which we detail how a college mathematics and chemistry student struggles to make sense of the graphical output of an experiment in an industrial chemistry laboratory. The student's attempts to interpret unfamiliar graphical conventions are described and contrasted with those of college mathematics. Our analysis of this draws on activity theory to assist in understanding the position of the student in both the college and the workplace. This highlights the limitations of the experience of the student at college and we question how the mathematics curriculum might be adapted to assist students in making sense of workplace graphical output.
There is increasing recognition that current mathematics curricula do not adequately equip people... more There is increasing recognition that current mathematics curricula do not adequately equip people to use and apply mathematics effectively in different spheres of their lives, for example, as learners, citizens and workers. For a flavour of the national
Lesson study is increasingly used throughout the world as a model of collaborative professional l... more Lesson study is increasingly used throughout the world as a model of collaborative professional learning. In this article we describe the theoretical foundations of an ongoing study which seeks to explore how we might inform collaborative learning with a focus on the development of curriculum coherence. The study involves collaboration between teachers and researchers in both the United Kingdom and Japan. We draw on the theory of expansive learning as a development of Cultural Historical Activity Theory and give insight into the potential this has to inform our work with a range of didactical devices or artefacts that provide connections across topics of conceptual understanding of mathematics and development over time. Expansive learning as a theoretical lens helps us understand our exploratory work in this area and provides insights into how this may be taken forward over the course of the study
In mathematics education in England, as in other countries, there is often tension in the on-goin... more In mathematics education in England, as in other countries, there is often tension in the on-going debate about the nature of the curriculum between proponents of “back-to-basics” who argue for a focus on the learning of fundamental knowledge, concepts, skills and techniques and those who take the view that the development of problem solving skills through a more inquiry-oriented approach are essential and will motivate learning of the basics. This might be characterised as a debate between those prioritising ‘content’ on the one hand and on the other those who favour ‘process skills’. In general over recent years the former have held sway in England. Consequently we find that English mathematics teachers in many instances have themselves been learners and have become teachers in a culture in which lessons might be characterised as focussed on content and in practice lessons often start with teacher exposition of “how to” use rules and procedures followed by students mimicking what ...
Social Interactions in Multicultural Settings, 2009
We acknowledge the support of the ESRC-TLRP programme of research into widening participation, fu... more We acknowledge the support of the ESRC-TLRP programme of research into widening participation, funded by ESRC.RES-139-25-0241.
The starting point of this article is the question, “how might we inform an epistemology of numer... more The starting point of this article is the question, “how might we inform an epistemology of numeracy from the point of view of better preparing young people for workplace competence?” To inform thinking illustrative data from two projects that researched into mathematics in workplace activity and the teaching and learning of modelling in the classroom is used albeit, by necessity, briefly. Analysis draws attention to the crucial role that understanding the structure of the contextual situation plays in developing a mathematical model of this. It is at this coupling of reality and mathematics that insight into, and understanding of, both mathematics and reality can be developed—or not. This important issue is illustrated with reference to two specific workplace situations that draw on understanding of fraction as gradient and explore how we might use a model of this to scaffold understanding of both reality and mathematics and how each might support the other. With a focus on model formulation during classroom activity attention is then drawn to how students tend to work towards reaching a solution to a particular problem with the consequence that their mathematical representation of the reality does not easily allow for consideration of variation of key factors. In conclusion a research agenda is proposed that seeks to inform an epistemology of numeracy by focussing on numerate activity at the nexus of reality and mathematics by (1) structuring mathematical knowledge using mathematical models that might be used to provide insight into a range of models of workplace realities, and (2) requiring student engagement with repeated use of models in ways that emphasise exploration of variability in key factors of the realities that the models represent.
This article analyses aspects of the process of developing 'functional' assessments of mathematic... more This article analyses aspects of the process of developing 'functional' assessments of mathematics at the end of compulsory schooling in England. A protocol that was developed for scrutinising assessment items is presented. This protocol includes an indicator of the 'authenticity' of each assessment item. The data are drawn from scrutiny of 589 assessment items from thirty-nine formal unseen examinations taken by students aged sixteen, and the article illustrates ways that mathematics is presented in different contexts in examinations. We suggest that currently the 'human face' of the questions may serve to disguise routine calculations, and we argue that in formal examinations connections between mathematics assessments situated in context and functional mathematics have yet to be established.
Previous work on transition between institutions suggests transitional problems can be dangerous ... more Previous work on transition between institutions suggests transitional problems can be dangerous for individual learners‟ progress. Drawing on the literature, and on our CHAT perspective, we anticipated that transition points would be significant to students narrative construction of ...
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