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+ MERGA 33: Shaping the future of mathematics education Technology, research and practice in mathematics education Barry Kissane The Mathematics Education Research Group of Australasia

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+ Outline Technology in mathematics education What technology? Policy statements Technology and research in mathematics education Trends over twenty years Big pictures and big ideas Technology, research and practice in mathematics education (How) is practice informed by research? (How) might we do better?

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+ Clicker 1: Who are we today? 1. Classroom teacher (in a school) 2. Head of department (in a school) 3. Teacher educator (in a university) 4. Researcher (in a university) 5. Maths teacher (in a university) 6. Other

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+ Technology in mathematics education

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+ Three roles for technology Computational To provide answers to mathematical questions Experiential To provide a means for students to interact with and explore mathematical ideas not otherwise available, to provoke and support mathematical thinking Influential To be considered as a significant factor when decisions are made about the nature of the curriculum

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+ Policy positions on technology ACARA Shape Paper on Australian Curriculum An important consideration in the structuring of the curriculum is to embed digital technologies so that they are not optional extras. (p.9) National Council of Teachers of Mathematics Position Paper Technology is an essential tool for learning mathematics in the 21st century, and all schools must ensure that all their students have access to technology. Effective teachers maximize the potential of technology to develop students understanding, stimulate their interest, and increase their proficiency in mathematics. When technology is used strategically, it can provide access to mathematics for all students. (2008)

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+ AAMT statements AAMT Statement on the Use of Calculators and Computers for Mathematics in Australian Schools It is recommended that: 1. All students have ready access to appropriate technology as a means both to support and extend their mathematics learning experiences … (1996) AAMT Communiqué on graphics calculators and school mathematics There is a compelling case for the advantages offered to students who use graphics calculators when learning mathematics. They are empowering learning tools, and their effective use in Australias classrooms is to be highly recommended. (2000)

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+ Digital Education Revolution Australian government initiative to provide laptops for students Increased access to high speed broadband anticipated Mathematics Framing paper: Digital technologies allow new approaches to explaining and presenting mathematics, as well as assisting in connecting representations and thus deepening understanding. The continuing evolution of digital technologies has progressively changed the work of mathematicians and school mathematics (consider the use of logarithm tables and the slide rule), and the curriculum must continue to adapt. Digital technologies are now more powerful, accessible and pervasive. (p.9)

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+ What technology for students? Hand-held devices Four-function calculators Scientific calculators Graphics calculators CAS-enabled graphics calculators Interactive devices Casio ClassPad, TI-Nspire PDA devices iPod Touch, iPhone, iPad Computer software Spreadsheets Dynamic geometry Cabri Geometry, Geometers SketchPad, GeoGebra, etc. Statistics Fathom, TinkerPlots, etc … iPod Touch, iPad The Internet Worldwide web Learning online (HOTMaths)(HOTMaths Maths by Email The Le@rning Federation Social networking, Web 2.0, etc.

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+ What technology for teachers? Hand-held devices As for students With demonstration versions Networked versions Computer software As for students Demonstration software E.g., Autograph The Internet The Le@rning Federation Online learning E.g., HOTMaths As for students Teaching technology Interactive white boards Graphics tablets Audience response devices (clickers)

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+ Clicker 2: Mathematics, technology and me Which one best describes you? 1. I teach maths with technology and do some research related to technology 2. I teach maths with technology but dont do research related to technology 3. I dont teach maths with technology but some of my research is related to technology 4. I neither teach maths with technology nor do research related to technology

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+ Computers, calculators, Internet, … It is clear that there are large differences between what is available to students and teachers Schools are differentially resourced Some excellent software is expensive Staff have preferences as well External constraints can be dominant (especially in senior secondary school) Graphics calculators portability, cost and exam acceptability Home Internet access is very high, and rising for many communities, but still SES differences

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+ A big picture 1990-2010 Seymour Papert in the early 1980s observed that the computer laboratory was Schools defence against technology. Graphics calculators were designed solely for mathematics education and broke down this defence (for many) Software available on all computers (i.e. spreadsheets) began to be used too Purpose-built software for mathematics education was developed The Internet Laptop computers and home access to technology

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+ The big picture 2010-2030

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+ A personal opinion about graphics calculators My engagement with graphics calculators began in 1986, when it was clear that there was no more efficient way of ensuring access to technology in many, if not most, US schools. It continues to be the case in 2010 that a technology that is individually affordable (to many), flexible, powerful, portable and acceptable to high-stakes exam authorities offers the best prospect of taking technology seriously and thinking of universal access. Despite its many limitations This will not always be the case

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+ Technology examples? Not really time Many are familiar Graphics calculators CAS Interactive devices Geometry Statistics Internet

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+ The Internet There is a large and increasing number of opportunities for students to engage with mathematics on the web

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+ Some iPod examples

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+ Some more examples

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+ Technology and research in mathematics education

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+ Technology and research: A naïve question Teachers (and others) would like an answer to the naïve question: Does it work? That is, if we use this technology with students, will they learn mathematics (better)? Yes? No? Of course, it is never that simple …

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+ Does it work?

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+ Why does it work?

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+ Why doesnt it work?

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+ Why does it work only sometimes?

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+ Why does it work only sometimes with my Year 10 class?

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+ Why does it work only sometimes with Jane Smiths Year 10 class?

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+ Would it work with Jane Smiths Year 10 class?

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+ Would it work with Jane Smiths Year 10 class in NSW?

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+ Technology and research: Does it work? It depends … on many things The classroom The teacher The curriculum The student The technology itself There is no panacea

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+ Changing research perspectives on technology in Australasia MERGAs RIMEA series 1988-1991: Calculators and computers in teaching and learning of mathematics 1992-1995: ?? 1996-1999: Technology-assisted instruction in mathematics education 2000-2003: Computers, multimedia and the Internet in mathematics education; Calculators and computer algebra systems 2004-2007: Teaching and learning with technology: Realising the potential 2008-2011: ??

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+ Stages in research on technology Developmental work, drawing on research in various disciplines Early empirical studies concerned with proof of concept Case studies Comparative studies involving quasi-experimental designs Larger studies with randomised, controlled trials

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+ A balance of approaches While research in a wide range of areas could directly or indirectly facilitate the effective utilization of educational technology within our nations K-12 schools, much of the research that the panel believes to be most important falls into one of the the following three categories: 1. Basic research in various learning-related disciplines and fundamental work on various educationally related technologies; 2. Early-stage research aimed at developing new forms of educational software, content and technology-enabled pedagogy; and 3. Empirical studies designed to determine which approaches to the use of technology are in fact most effective. (PCAST, 1997, Executive Summary) (p. 443) Ferrini-Mundy, J. & Breaux, G.A. (2008) Research, policy and technology use. In Blume, Glendon W. & Heid, M. Kathleen (2008) Research on technology and the teaching and learning of Mathematics: Volume 2: Cases and perspectives. (pp 427-448) USA: Information Age, NCTM.

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+ Should technology have a role in school mathematics? In the Panels judgement, the principal goal of such empirical work should not be to answer the question of whether computers can be effectively used within the school. The probability that elementary and secondary education will prove to be the one information-based industry in which computer technology does not have a natural role would at this point appear to be so low as to render unconscionably wasteful any research that might be designed to answer this question alone. (PCAST, 1997, Section 8.3: Priorities for Future Research) (p. 444) Ferrini-Mundy, J. & Breaux, G.A. (2008) Research, policy and technology use. In Blume, Glendon W. & Heid, M. Kathleen (2008) Research on technology and the teaching and learning of Mathematics: Volume 2: Cases and perspectives. (pp 427-448) USA: Information Age, NCTM.

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+ What might research offer us? An opportunity to understand things better But rarely an unambiguous answer to important questions of teaching and learning An opportunity to explore the boundaries of relevance of a theoretical framework to understand practice An opportunity to put (competing) theories to a test New phenomena to explore Most research projects generate as many fresh questions as answers Further research is needed to …

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+ Problems with research on technology in particular A moving target, as the technology is changing (very rapidly), as Jim Kaput remarked in 1992: Anyone who presumes to describe the roles of technology in mathematics education faces challenges akin to describing a newly active volcano the mathematical mountain is changing before our eyes, with myriad forces operating on it and within it simultaneously. (p. 515) Unavoidable novelty effects Teacher effects Curriculum (including external examination) effects especially in senior secondary school and undergraduate mathematics? Time span (longitudinal research?) Up-scaling and generalisability problems

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+ The place of reviews of research For some of the foregoing reasons, research results rarely (if ever) lead to uncomplicated, unequivocal solutions to problems The gold standard of empirical scientific research, the randomised experiment, is clearly unattainable in this field (yet) … if in any branches of mathematics education … So, systematic reviews of research are important, and meta- analyses even more important, to try to reconcile differences in findings These are major undertakings (eg RIMEA)

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+ What does research tell us? Some sources RIMEA series Every four years, focusing on Australasia NCTM Handook of Research Key constructs NCTM Research Syntheses volumes Systematic, structured compilations MERGA conferences and journals Some recent highlights

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+ RIMEIA 2004-2007: Some big pictures Thomas, M. & Chinnappan, M. (2008) Teaching and learning with technology: Realising the potential. In H. Forgasz, A. Barkatsas, A. Bishop, B. Clarke, S. Keast, T.S. Wee, T. S. & P. Sullivan (Eds.) Research in Mathematics Education in Australasia 2004-2007. (pp 165-193). Rotterdam: Sense Publishers. … a high level of enthusiasm from both students and teachers to embrace a variety of technologies … A focus on … the crucial role of the teacher when employing technological tools…

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+ Organising constructs Affordances E.g., Presence of technology Constraints Student or teacher instrumentation Time available Curriculum content Pedagogical technology knowledge (PTK) principles, conditions and techniques required to teach mathematics through the technology (p.167)

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+ Teacher variables Metaphors for technology (Goos, Galbraith, Geiger, et al) Master Servant Partner Extension of self Professional development variables Teacher confidence Technical expertise PTK Use of CAS Teacher privileging CAS as a conceptual tool, not just a crutch

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+ Some big pictures? One factor that consistently needs attention is whether the success reported in studies can translate to teachers in general, or whether the research participants are exceptional in some ways. (p. 170) Research and teaching community are enthused … but teachers need support and guidance in classroom implementation Both pre-service and in-service. (p. 183) Conflicting results regarding CAS

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+ A perspective of constructs This recent major review of the field suggested a number of constructs as organisers of the research, evolved from collections of studies. Rose Mary Zbiek, M. Kathleen Heid, Glendon W. Blume & Thomas P. Dick (2007) Research on technology in mathematics education: A perspective of constructs. In F. K. Lester Jr. (ed.) Second handbook of research on mathematics teaching and learning. (pp 1169-1207). USA: Information Age, NCTM.

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+ Which constructs? Technical and conceptual activities Cognitive tools Tools and mathematical activity Externalised representation Mathematical fidelity Cognitive fidelity Student-Tool relationships Instrumental genesis

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+ More constructs Students and mathematical activity Exploratory activity Expressive activity Methods of working Technology and practice Pedagogical fidelity (Teacher) privileging Technology and curriculum: Constructs that capture the opportunities for change in curriculum facilitated by technology Representational fluency Mathematical concordance Amplifiers and reorganisers Sequencing and emphasis: Microprocedures and macroprocedures

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+ Research syntheses Heid, M. Kathleen & Blume, Glendon W. (2008) Research on technology and the teaching and learning of Mathematics: Volume 1: Research syntheses. USA: Information Age, NCTM. Rational number Algebraic understanding Geometry Calculus Mathematical modelling Practice Equity

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+ Algebra Technology in conjunction with technology-based curricular approaches can effectively change the content and processes of school algebra. (p. 97) Technology in conjunction with technology-based curricular approaches can affect the processes of mathematical activity in an algebraic setting. Many of these effects are related to the representational capacity of technology. (p. 97) Technology in conjunction with technology-based curricular approaches can affect the acquisition of algebraic concepts and procedures (p. 98) Heid, M. Kathleen & Blume, Glendon W. (2008) Algebra and function development. In Heid, M. Kathleen & Blume, Glendon W. (Eds.) Research on technology and the teaching and learning of Mathematics: Volume 1: Research syntheses. (pp 55-108) USA: Information Age, NCTM.

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+ Geometry There is evidence that computer environments can support learning and teaching in geometry in new and dynamic ways, as well as complementing and enriching traditional strategies. (p. 141) There is not yet a critical amount of research devoted to long-term teaching with regular use of DGS. Moreover there is currently a lack of computer- supported geometry teaching. (p. 191) The computer provides a window on students [geometric] understandings. (p.189) In a DGS, construction tasks induce the need to use geometrical knowledge. (p. 190) DGS offers a new perspective in addressing the issue of the teaching and learning of proof. (p. 190) Hollebrands, K., Laborde, C. & Straser, R. (2008) Technology and the learning of geometry at the secondary level. In Heid, M. Kathleen & Blume, Glendon W. (Eds.) Research on technology and the teaching and learning of Mathematics: Volume 1: Research syntheses. (pp 155-205) USA: Information Age, NCTM.

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+ Probability and statistics Statistics was not mentioned in the Research Syntheses publication, and Friels chapter emphasises the relative recency of attention to research on statistics education RIMEA 2004-2007 review also noted relative dearth of research about statistics with technology in Australasia (at that time) Research with educational software (such as Fathom and TinkerPlots) is relatively new, with results (case studies, design studies) informing conceptions of an appropriate curriculum. Technology is an assumed part of the developing EDA conception of statistics, with a focus on understanding data. Friel, S.(2008) The research frontier. In Blume, Glendon W. & Heid, M. Kathleen (Eds.) Research on technology and the teaching and learning of Mathematics: Volume 2: Cases and perspectives. (pp 279-331) USA: Information Age, NCTM.

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+ Teachers and technology Survey research has provided some helpful information about secondary mathematics teacher use of technology and professional development needs The best recent example is: Goos & Bennison (2008) Surveying the technology landscape: Teachers use of technology in secondary mathematics classrooms. Mathematics Education Research Journal, 20(3), 102-130. Computers, graphics calculators and the Internet Clear effects of mandatory use of technology (graphics calculators) More use of technology in senior school than below Marginal use of computers and the Internet Professional development is important and can be influential Bennison & Goos (MERJ, 2010) note that effective integration remains patchy, with a number of teacher issues identified Thomas surveys (1995 & 2005) in NZ highlight access issues for computers

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+ The Internet (and beyond) There seems to be relatively little empirical research yet on the use of the Internet by students and teachers Internet as a source of information about mathematics seems to have no place in the curriculum? (yet seems likely to be of interest to many students?) There are very rapid changes in technology outside mathematics classrooms Web 2.0 and the ubiquitous Internet Mobile phones with computer capabilities in an interconnected world Podcasts and video A curriculum that seems oblivious or impervious to these must seem increasingly quaint to students How does research keep up?

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+ Undergraduate teaching In many places, it seems that the use of technology in early undergraduate mathematics differs sharply from the use of technology in schools Wood, L. (2008) University learners of mathematics. In H. Forgasz, A. Barkatsas, A. Bishop, B. Clarke, S. Keast, T.S. Wee, T. S. & P. Sullivan (Eds.) Research in Mathematics Education in Australasia 2004-2007. (pp 73-97). Rotterdam: Sense Publishers. On computing tools, the majority of authors espouse the use of professional software and hardware tools. Such as Excel, CAS and computers rather than teaching-only tools such as graphics calculators. (p. 91) There is a distinct split between universities that favour computing tools for mathematics learning and those who work only with pen and paper. (p.91)

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+ Proficiencies and technology The draft Australian Curriculum – Mathematics identifies four proficiencies: Understanding Fluency Problem solving Reasoning Teachers might reasonably expect to see clear guidance, advice and descriptions about the (different) role of technology in these

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+ The nature of the curriculum There seems to be limited evidence of technology influencing the nature of the curriculum (at least in the Australian Curriculum drafts, in my personal opinion) Technology is mostly interpreted as pedagogy and thus the prerogative of the teacher? Computation is recognised, and there is encouragement to use available technology to change the teaching and learning experience Coherence of teaching, learning and assessment is worthy of closer research, as it seems highly likely that what is used in assessment is likely to determine what is generally used for teaching and learning.

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+ K-10 draft, Australian Curriculum Information and communication technologies (ICT) allow students to solve problems and perform tasks that previously have been onerous. Calculators of all types from the simple four operations versions to the more complex graphical and CAS calculators allow students to make calculations, draw graphs and interpret data in ways that previously have not been possible. There are spreadsheets, dynamic geometry programs and other software that can engage students and promote understanding of key concepts. It is expected that mathematics classrooms will make use of all available ICT in teaching and learning situations. [ACARA, 2010; emphasis added]

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+ 11-12 draft, Australian curriculum The Shape of the Australian Curriculum – Mathematics states that available technology should be used for teaching and learning situations. Technology can include computer algebra systems, graphing packages, financial and statistical packages and dynamic geometry. These can be implemented through either a computer or calculator. Technology can aid in developing skills and allay the tedium of repeated calculations. For example a technology can be used to complete recursive calculations. The decision about using technology in assessment programs is not within the province of the curriculum, jurisdictional assessment agencies will make that decision.

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+ Adding-on technology? Fey, J.T., Hollenbeck, R.M. & Wray, J.A. (2010) Technology and the mathematics curriculum. In Reys, B.J., Reys, R.E. & Rubenstein, R. NCTM 72 nd Yearbook: Mathematics curriculum. (pp 41-49). Reston, VA: NCTM offered an opinion on this question: Curriculum specialists and other interested parties should examine objectives to determine whether technology can enhance students learning of mathematics. However, technology should not be an add- on to curricula. Using technology to cover topics that are just as accessible through other approaches may actually interfere with learning and undermine the benefits of technology. Given the urgency of providing strong mathematical preparation for students who will enter and live in a technologically sophisticated society and workplace, such study and experimentation by all involved in the enterprise of mathematics teaching should be a high priority for our field. (2010, p.48) (Emphasis added.)

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+ Some examples of curriculum influence? E.g., changing the emphasis in statistics from mathematical statistics to data analysis, using real data and real problems, using suitable technology tools E.g., approaches to probability beyond the formal classicist approach (in terms of sample spaces and equally likely outcomes, sets and combinatorics); study of risk E.g., numerical approaches to calculus problems such as finding relative extrema or numerical solutions to differential equations E.g., Focus on construction and interpretation of integrals, rather than methods of integration, in an age of CAS

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+ Some more examples E.g., explorations with geometric software to encourage and motivate conjecturing, reasoning and proof E.g., some focus on numerical solution of equations rather than only on exact solutions of equations E.g., use of reducible interest (which is what occurs in practice) rather than simple and compound interest (which usually dont occur in practice) Emphasis in the draft Australian Curriculum seems to focus on using technology to teach the same curriculum to which we have become accustomed … a form of retrofitting … rather than reconsider the scope and sequence of the curriculum in the light of available technologies This is of course an opinion, not an empirical finding

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+ Technology, research and practice in mathematics education

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+ Research and practice How does research influence practice? In general, not only for the particular case of technology What are the problems? How might we strengthen the links?

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+ A litmus test? Julie is teaching her Maths 2D class next semester, starting a unit on calculus with a group of students not in the strongest stream. She has been teaching for six years now and is a competent user of technologies. Should students use the CAS calculator? Why? How? For what? Will some computer software be useful? Which? How should she use it? Could the Internet be useful here? How? For what? Could her Interactive White Board be used? How? Why? What will research tell her about such things? Where should she look?

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+ Clicker 4: Consulting research Think of some maths you have taught to students recently with technology. Which of the following best describes you? 1. I consulted a research source for advice before I started. 2. I had previously consulted research, so didnt need to do so again. 3. I did not consult any research. 4. I havent taught maths to students recently with technology.

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+ Possibility 1: Practitioners accessing researchers Attend the MERGA conference (in their home city) Attend the joint AAMT-MERGA conference(s) In Alice Springs next July Interrogate the MERGA website for conference or journal publications Obtain published research advice Research journals are usually not written for the audience of teachers Very expensive and inaccessible in most schools Interpretations of impact within ERA focus on research colleagues not colleagues in schools Voices from the field in MERJ is a welcome initiative

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+ Possibility 2: Researchers advising practitioners Researchers can advise practitioners directly What research says monographs? Earlier research syntheses published by NCTM 67 th NCTM Yearbook: technology-supported mathematics learning environments (2005) is a good example Association of Mathematics Educators (Singapore) Handbooks Write advice papers based on research in journals for teachers It is hard to write these; partly because research findings often do not readily translate to practice Not many people try to do this, as the rewards are few Conduct targeted conferences (eg ACER conference 2010) for the purposeACER conference 2010 Impact unavoidably limited to those who can attend

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+ Clicker 5: Advising practitioners In the last two years, have you submitted a paper based on your research to a publication meant for maths teachers? 1. Yes, and it was accepted 2. Yes, but it was rejected 3. No

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+ Possibility 3: Materials development Classroom materials and curricula can be developed following classroom-based research CAS-CAT project TinkerPlots, Geometers SketchPad materials Hillary Shuard project Calculator Aware Numeracy materials (Some) calculator manufacturer materials are based on work in schools MATHS300 software Materials themselves can be researched Even trialing seems rare for Australian school textbooks? UCSMP experience

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+ Possibility 4: Professional development Pre-service teacher education Informed by research (eg Goos, Stillman & Vales Teaching Secondary School Mathematics: Research and Practice for the 21 st Century) Limited short-term impact on the field, as most teachers are already teaching! Teacher conferences Seems rare for research to be the basis of presentations? Rare for researchers to see these as important? Even rarer for their institutions to do so in the world of ERA? Teacher courses (Eg 2008 Summer School) Happen rarely and impact on only a few?

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+ Clicker 6: Teacher meetings In the past year, have you attended a conference or meeting of teachers in order to discuss your research? 1. Yes 2. No 3. Ive not been involved in research in the past year.

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+ Possibility 5: Researchers and practitioners working together Action research projects Eg, AGQTP, ASSISTM Classroom-based research generally Teaching experiments, case studies, field trials Funding? Time-span? Queensland team (Goos, Geiger, Renshaw, Galbraith, …) is a very good example We need more good examples

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+ Clicker 7: Working together In the past year, have you worked in a school with a team of colleagues on a research project? 1. Yes; I am a school teacher member of the research team 2. Yes; I am a member of the research team, but not a school teacher 3. No; although my students were involved in a research project 4. No

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+ Possibility 6: Web-based support ACARA intentions are to provide significant online advice and support to teachers How can advice informed by research on technology be best included in that? Who will do it? Especially in light of the limited inclusion of technology into the curriculum itself to date MERGA website is outstanding, although the materials are not written with practice in mind NCETM example in the UK seems to have much to commend it. Here is an example about Interactive White Boards. NCETM Here Is there scope for an Australian version? Very significant funding is needed Not only about technology, of course

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+ Concluding remarks

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+ If … The technology is designed to capture important mathematical ideas faithfully; and It is improved with the aid of suitable research with students; and The curriculum is written and assessed on the assumption that technology is available; and Curriculum materials and tasks have been developed accordingly; and The teacher is adequately supported to use the technology confidently and well in the classroom; then It will work

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+ Bringing it all together: some final observations There is a rich resource of research on technology already available … with many gaps It is already clear that technology has much to offer While the world of technology itself keeps changing rapidly Much of the research is not written directly for teachers Focus of some research is on teacher practices, recognising that what happens in classrooms is of great importance, not only the technology itself Professional development is a direct object of study Building partnerships between research and practice is a critical part of making joint progress … so, finally, what is the relationship between research and practice…?

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+ Research and practice Practice Research

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+ Clicker 8: Did you like that picture? 1. Yes 2. No 3. I didnt understand it, so I cant tell.

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+ Thank you B.Kissane@Murdoch.edu.au http://wwwstaff.murdoch.edu.au/~kissane

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