Research design and methods Introduction to research paradigms

Overview What is research (in mathematics education)? Research questions and how they emerge. What makes a researchable question? Generate examples for use later on. Philosophy in (mathematics education) research Ontology and Epistemology Assumptions and values Research paradigms How does this all relate to mathematics education? Next time: introduction to methods in research Barbara Jaworski MEC 2008 1

What is research? Barbara Jaworski MEC 2008 1

Two main goals of research in mathematics education To enhance knowledge in the field knowledge about mathematics , learning mathematics , teaching mathematics , doing research in learning mathematics … To enhance practice in the field to enable better learning and teaching of mathematics , better researching … Barbara Jaworski MEC 2008 1

What is education? Mathematics Learning Teaching

Mathematics Education How mathematical topics can be made accessible for learners; How we learn mathematics successfully; How we teach mathematics in order that learning can be successful; How we develop mathematics teaching and educate mathematics teachers; How research in the field informs and promotes development. Barbara Jaworski MEC 2008 1

Research area What is the (proposed) area of your research? Here I will either go round the class getting each person to talk about their thoughts/plans OR divide the class into groups of 3 or 4 to talk together about their research. Barbara Jaworski MEC 2008 1

Research focus Within your research area, where or on what are you focusing (or do you want to focus) particularly? Here I will get people to write down a short statement -- 2 or three lines, and then we will go round, reading them out. Barbara Jaworski MEC 2008 1

Research questions What questions are you addressing, or would you like to ask and find out the answers to? Each person writes one or more questions independently – 5 minutes. In small groups, they have to look at the 3/4 questions and discuss what they could do to find out answers to those questions. Barbara Jaworski MEC 2008 1

Research questions What do I want to find out? What is my field of research? Can I express this more succinctly? Are there specific questions I want answers to Refining …. I use this after the discussion to emphasise that research questions are never set in stone. Barbara Jaworski MEC 2008 1

Questions I will take one or more questions from each group Barbara Jaworski MEC 2008 1

Barbara Jaworski MEC 2008 1

Researchable questions What are the problems that students face in learning calculus? What, as teachers, do/can we do to enable students to know mathematics? Researchable questions What do we want for our students and how does this relate to how we ourselves know and see mathematics? Is it possible to find answers to your questions? What might such answers look like? How might you go about finding answers? The nature of questions at different times in the research process The development of research questions during the research. Having very clear research questions for quantitative research, whereas in an interpretative paradigm questions evolve. This will be whole group discussion considering what it means to be able to answer the research questions. Barbara Jaworski MEC 2008 1

Philosophy underpinning research

What is? Ontology

Activity Write down three things that you know, related to your area of research. Try to make each one different in some fundamental way (that you can justify) from the others. Lehrer, 1990, p. 2-5 Barbara Jaworski MEC 2008 1

What knowledge? Mathematics Pedagogy … Didactics … … Barbara Jaworski MEC 2008 1

Epistemology Status of knowledge What is true? Barbara Jaworski MEC 2008 1

What are your epistemological and ontological positions? What kinds of knowledge? What IS? What kinds of interpretations are likely to be involved and how do you expect to deal with them? Do you seek objectivity? Do you expect to address human issues and values, and if so what is your stance on these? What questions of bias might arise? What kinds of ethical issues are likely to be involved? Barbara Jaworski MEC 2008 1

Assumptions Barbara Jaworski MEC 2008 1

Values Pring, 2000, Chapter 4 Bishop, 2001

Paradigms Kuhn, 1970 (p. 103-5 & 109-10) Carr & Kemmis, 1986, p. 71-75 Pring, 2000, pp. 108 - 115

Paradigms Positivist Interpretivist Critical theorist Constructivist Post modernist Pring Carr and Kemmis Lincoln and Guba Barbara Jaworski MEC 2008 1

Positivism Seeking objectivity and truth through defining social situations in scientific terms usually involving quantification, measure and logic: defining measurable variables; designing comparable situations; giving absolute values; not leaving open to interpretation. Justification most often through statistical analysis or study of carefully controlled experimental conditions . Barbara Jaworski MEC 2008 1

Interpretivism Recognising social situations as complex and seeking to describe and characterise them through interpretation: seeking meaning in observed actions and interactions; gaining insight to people’s perspectives on who they are and what they do. Justification through detailed description and multiple sources of explanation and evidence to support interpretation and throw light on what is studied being critical about the perspectives one brings to interpretation Barbara Jaworski MEC 2008 1

Critical theory Going beyond descriptive interpretation to recognise that social situations embody deeply political human issues and power relationships that research should seek to uncover and address such issues: revealing relationships which limit or oppress; bringing critical analysis to accepted traditions to offer opportunities for change. Justification through action and interaction that examine deeply and overtly ways of thinking, reveal factors and conditions that suppress individuals or groups and provide emancipatory/empowering opportunity through giving voice, enabling and enfranchising. Barbara Jaworski MEC 2008 1

Constructivism Making sense of social situations through constructing accounts, stories or narratives to describe and explain; going beyond interpretation to offer explanatory constructions of situations and events to understand meanings and relationships. Justification through acknowledging alternative ways of seeing, and looking critically at the authentic nature of what is presented. Barbara Jaworski MEC 2008 1

Postmodernism Going beyond modernism which rationalises, structures and seeks to explain by categorising and compartmentalising: bringing and valuing multiple perspectives and methods; questioning the dominance of any one view of the world, deconstructing to reveal the limiting nature of imposed structures; revolt against control. Justification in revellation; coversation and negotiation, opening up; not pretending to compartmentalise; revealing complexity and chaos. Barbara Jaworski MEC 2008 1

Paradigm taxonomies Carr and Kemmis (Chapters 2, 3 &5) Pring (chapter 5) Guba and Lincoln Lather Barbara Jaworski MEC 2008 1

Mathematics How do we see mathematics? Absolute and ‘out there’ Abstract and impenetrable Formal and rational Humanly constructable Fallible What else? Davis & Hersh Ernest Barbara Jaworski MEC 2008 1

Mathematics for learning Preconstructed and predigested versus Constructable and Fallible Do students have to reconstruct all that has been constructed before? How is, or could this be, possible? Can we give pre-constructed mathematics to students? How? What are the alternatives? Barbara Jaworski MEC 2008 1

Mathematics for Teaching How can a teacher make mathematics available for students? What does/can it mean to ‘make available’? What, as teachers, do/can we do to enable students to know mathematics? What does it mean to know mathematics? What do we want for our students and how does this relate to how we ourselves know and see mathematics? Barbara Jaworski MEC 2008 1

What are we trying to do in our research? For example: Know more about the kinds of difficulties students experience with particular mathematical topics such as integration or vector spaces Know more about the kinds of settings that both provide access to mathematics and engage students actively and enjoyably and foster understanding and skill What paradigmatic issues arise when we start to construct research that will lead to such knowing? Barbara Jaworski MEC 2008 1