Technology, research and practice in mathematics education

1. The Mathematics Education Research Group of Australasia Technology, research and practice in mathematics education Barry Kissane

2. 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?

3. 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

4. Technology in mathematics education

5. 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

6. 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)

7. 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 Australia’s classrooms is to be highly recommended”. (2000)

8. 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)

9. 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, Geometer’s SketchPad, GeoGebra, etc. • Statistics • Fathom, TinkerPlots, etc … • iPod Touch, iPad • The Internet • Worldwide web • Learning online(HOTMaths) • Maths by Email • The Le@rning Federation • Social networking, Web 2.0, etc.

10. 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’)

11. 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 don’t do research related to technology 3. I don’t 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

12. 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

13. A big picture 1990-2010 • Seymour Papert in the early 1980’s observed that the computer laboratory was School’s 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

14. The big picture 2010-2030

15. 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

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

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

18. Some iPod examples

19. Some more examples

20. Technology and research in mathematics education

21. Technology and research: A naïvequestion • 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 …

22. Does it work?

23. Why does it work?

24. Why doesn’t it work?

25. Why does it work only sometimes?

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

27. Why does it work only sometimes with Jane Smith’s Year 10 class?

28. Would it work with Jane Smith’s Year 10 class?

29. Would it work with Jane Smith’s Year 10 class in NSW?

30. 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

31. Changing research perspectives on technology in Australasia • MERGA’sRIMEA 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: ??

32. 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

33. 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.

34. Should technology have a role in school mathematics? “In the Panel’s 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.

35. 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 …”

36. 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

37. 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)

38. 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

39. 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…”

40. 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)

41. 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

42. 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

43. 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.

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

45. 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

46. 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

47. 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.

48. 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 student’s [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.

49. Probability and statistics • Statistics was not mentioned in the Research Syntheses publication, and Friel’s 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.

50. 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: Teacher’s 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