Presentation to Umeå Workshop 2010

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Presentation to Workshop on Design Research held at Umeå Mathemtics Education Research Centre (UMERC), 16 - 17 December 2010.

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  • On the contested (and distorted) nature of mathematics If we look at the history of the development of the discipline of mathematics it can be seen that the nature of the subject itself has long been contested. This has profound implications for school mathematics - for example is it an abstract subject for an elite or should mathematics be for all? In his analysis Lakatos distinguishes between the deductivist approach and the heuristic approach which he describes as "the logic of proofs and refutations". With regard to the former it is argued that Euclidean methodology has developed a certain obligatory style of presentation which is described as deductivist style: This style starts with a painstakingly stated list of axioms, lemmas and/or definitions. The axioms and definitions often look artificial and mystifyingly complicated. One is never told how these complications arose. The list of axioms and definitions is followed by carefully worded theorems. These are loaded with heavy-going conditions; it seems impossible that anyone should ever have guessed them. The theorem is followed by proof.  Lakatos (1976, p 142) Mathematics is compared with a conjuring act according to this "Euclidian ritual" and the student is obliged to accept this without asking questions about the underlying assumptions. In this deductivist style, under which all propositions are true and all inferences valid, mathematics is presented as an ever-increasing set of eternal, immutable truths. I wish to argue that it is not simply the deductivist approach which is a problem but the way in which this has become distorted into a form of fundamentalism that is akin to religious fundamentalism. Such a fundamentalism promotes an authoritarian view of mathematics, which hides the struggle and adventure involved. Also such authoritarianism is the very antithesis of the conditions needed to foster independent and critical thinking. An alternative perspective, which I would describe as mathematical fallibilism, argues for a view of mathematics as human activity and it is this human mathematical activity that produces mathematics. However when this is presented in textbooks this product of human activity "alienates itself" (ibid, p146) from the very human activity, which produced it. The mathematics educator, Geoff Giles used the expression which is very vivid for me of “dead geometry entombed in text books” If we look to our second activity this is an example in mathematics of the opportunities afforded by the use of dynamic geometry software to enable students to independently study the invariant (unchanging) relationships between points, lines and circles, forming their own conjectures and testing them out visually, which is the very essence of geometry.
  • Presentation to Umeå Workshop 2010

    1. 1. What is design research, what is the point of it and what do we want to do? Brian Hudson and Sheila Henderson Umeå Workshop on Design Research 16 th December 2010
    2. 2. Structure of the presentation <ul><li>On the nature of Design Research and its relation with Didactical Design and teachers’ professional work </li></ul><ul><li>On the context for the project Developing Mathematical Thinking in the Primary Classroom </li></ul><ul><li>On the Dundee study into student primary teachers’ levels of mathematics competence and confidence </li></ul><ul><li>On the contested (and distorted) nature of mathematics and student teachers’ attitudes and beliefs </li></ul><ul><li>Developing Mathematical Thinking in the Primary Classroom – the project aims, overview of the CPD module design and of the design research project plan </li></ul>
    3. 3. The Integrative Didactical Design (IDD) framework <ul><li>The main goal of the Integrative Didactical Design (IDD) framework (Hudson, 2008) is to support the construction of propositions for actions in relation to both teaching and learning and to design and construct teaching situations, pedagogical activities and learning environments that enable both teachers and learners to put these propositions into practice. </li></ul><ul><li>Specifically, the framework aims to address the ways in which we might systematically create, test, evaluate and disseminate teaching and learning interventions that will have a maximum impact on practice and contribute significantly to the development of theory about both teaching and learning. </li></ul><ul><ul><li>Hudson, B. (2008) Didactical Design Research for Teaching as a Design Profession. In Hudson and Zgaga (Eds.) Teacher Education Policy in Europe: A Voice of Higher Education Institutions . University of Umeå/University of Ljubljana, 345-365. </li></ul></ul><ul><ul><ul><li>http://www.use.umu.se/digitalAssets/44/44502_a-voice-of-higher-education-institutions.pdf </li></ul></ul></ul>
    4. 4. On the nature of Didactical Design <ul><li>The focus of the Didactical Design cycle is on the design of teaching-studying-learning processes which is a central role of the teacher in the promotion of student-centred learning processes </li></ul><ul><li>We can consider the process of Didactical Design in the form of a cyclical process of Analysis, Design, Development, Interaction and Evaluation leading through to a subsequent process of re-design … </li></ul><ul><ul><li>Reference </li></ul></ul><ul><ul><li>Hudson, B. (forthcoming) Didactical Design for Technology Enhanced Learning. In Hudson, B. and </li></ul></ul><ul><ul><li>Meyer, M. (Eds.) Beyond Fragmentation: Didactics, Learning and Teaching in Europe , </li></ul></ul><ul><ul><li>Verlag Barbara Budrich, Opladen and Farmington Hills. </li></ul></ul>
    5. 5. On higher order thinking, creativity and design as central to teachers’ professional work http:// edorigami.wikispaces.com /Bloom%27s+and+ICT+tools
    6. 6. On the context for Developing Mathematical Thinking in the Primary Classroom <ul><li>Despite past initiatives to improve the teaching and learning of mathematics, such as the Cockcroft Report which was published nearly 30 years ago and the 5-14 Mathematics Guidelines 10 years later, most mathematics lessons in Scotland still tend to feature some form of teacher-led demonstration followed by children practising skills and procedures from a commercially produced scheme (SEED 2005). </li></ul><ul><li>These findings were confirmed by TIMSS (IEA 2008) which found that 72% of both P5 and S2 pupils were taught using a textbook as the primary resource compared to the international average of 65% and 60% respectively. </li></ul><ul><li>The most recent SSA (Scottish Government 2009) also reported that pupils using textbooks and working quietly on their own was the most common form of activity in mathematics classes in Scotland. </li></ul><ul><li>This background context would seem to suggest that attempts to move to more constructivist models of teaching and more active approaches to learning mathematics have not been as successful as might have been hoped. </li></ul>
    7. 7. On the Dundee study into student primary teachers’ levels of mathematics competence and confidence <ul><li>The findings highlighted that students’ subject knowledge was often lacking when assessed using the online assessment. </li></ul><ul><li>It was also found that those students possessing more advanced mathematics qualifications at SCQF level 6 (SCQF 2003) were less likely to display competence in primary mathematics and that their confidence levels in the subject were lower than their less well qualified peers. </li></ul><ul><li>References </li></ul><ul><ul><li>Henderson, S. and Rodrigues, S. (2008) Scottish student primary teachers’ levels of mathematics competence and confidence for teaching mathematics: Some implications for national qualifications and initial teacher education, Journal of Education for Teaching, 34 (2), 93-107. </li></ul></ul><ul><ul><li>Henderson, S. (2010) Mathematics Education: The Intertwining of Affect and Cognition . Unpublished doctoral thesis. D.Ed. University of Dundee. </li></ul></ul>
    8. 8. On the contested nature of mathematics <ul><li>Is it an abstract subject for an elite or should mathematics be for all? </li></ul><ul><li>This (deductivist) style starts with a painstakingly stated list of axioms, lemmas and/or definitions. The axioms and definitions often look artificial and mystifyingly complicated. One is never told how these complications arose. The list of axioms and definitions is followed by carefully worded theorems. These are loaded with heavy-going conditions; it seems impossible that anyone should ever have guessed them. The theorem is followed by proof.  </li></ul><ul><ul><li>Lakatos (1976, p 142) </li></ul></ul><ul><li>When this is presented in textbooks this product of human activity &quot;alienates itself&quot; (ibid, p146) from the very human activity, which produced it. </li></ul><ul><li>“ dead geometry entombed in text books” (Geoff Giles, 1981) </li></ul>
    9. 9. On the contested (and distorted) nature of mathematics and student teachers’ attitudes & beliefs <ul><li>Mathematical fundamentalism </li></ul><ul><li>Infallible and authoritarian </li></ul><ul><li>Dogmatic and absolutist </li></ul><ul><li>Irrefutable and certain </li></ul><ul><li>Strict procedures </li></ul><ul><li>Rule following </li></ul><ul><li>Right and wrong answers </li></ul><ul><li>High stakes testing </li></ul><ul><li>Boring </li></ul><ul><li>De-motivating </li></ul><ul><li>Fear and anxiety </li></ul><ul><li>Alienation from the subject itself </li></ul><ul><li>Mathematical fallibilism </li></ul><ul><li>Fallible and liberating </li></ul><ul><li>Critical thinking, growth & change </li></ul><ul><li>Refutable and uncertain </li></ul><ul><li>Multiple solutions </li></ul><ul><li>Creative reasoning </li></ul><ul><li>Errors and mistakes </li></ul><ul><li>Evaluation & self assessment </li></ul><ul><li>Engaging </li></ul><ul><li>Motivating </li></ul><ul><li>Enjoyment and fulfilment </li></ul><ul><li>A creative human activity </li></ul><ul><ul><li>Reference </li></ul></ul><ul><ul><li>Lakatos, I. (1976) Proofs and Refutations . Cambridge: Cambridge University Press . </li></ul></ul>
    10. 10. More recent findings in relation to attitudes and beliefs <ul><li>Online survey of students enrolled on Years 1 to 4 of a BEd Honours degree programme and a Post-Graduate Diploma in Education (Primary) (PGDEP) course during the week beginning 12th September 2010. </li></ul><ul><li>It was completed by 148 students from a population of 388, representing a response rate of 38%. </li></ul><ul><li>Significant minority (25-30%) of students in September 2010 who continue to hold more fundamentalist beliefs </li></ul><ul><li>Many students “sit on the fence” in stating their beliefs </li></ul><ul><ul><li>An issue related to confidence? </li></ul></ul><ul><ul><li>Hudson, B. and Henderson, S. (2010) What is Subject Content Knowledge in Mathematics? On the Implications for Student Teachers’ Competence and Confidence in Teaching Mathematics, Teacher Education Policy in Europe (TEPE) Network Conference , University of Tallinn, 30th September – 1st October 2010. </li></ul></ul>
    11. 11. More recent findings in relation to attitudes and beliefs – some responses from students <ul><li>… my low confidence in maths was down to teachers in my standard grade year who weren't helpful, encouraging or positive in their teaching. </li></ul><ul><li>I have always been scared of doing mental maths and did not enjoy maths at school … </li></ul><ul><li>I recall the attitude of my class teacher in primary school … if you answered a mathematical question incorrectly your were belittled in front of your class mates and made to feel very inadequate. </li></ul><ul><li>Fear of maths is a learned state which is the result of poor teaching. </li></ul><ul><li>I have always believed that the wrong mode of teaching maths can have an extremely negative impact on the perception of maths for the learner. </li></ul><ul><li>I have always enjoyed maths. I had a great maths teacher throughout my time in secondary and this motivated me to learn more and achieve in maths. </li></ul>
    12. 12. Developing Mathematical Thinking in the Primary Classroom – the project aims <ul><li>The project will focus on the collaborative development, with teachers, of a Postgraduate course of Continuing Professional Development (CPD) </li></ul><ul><li>It will form the first stage in a Masters level programme in Mathematics Education for Primary Specialists in Mathematics and be accredited as a module worth 30 credits leading towards a Postgraduate Certificate in Mathematics Education. </li></ul><ul><li>Funded by Scottish Government (2010-11), £46,331 </li></ul><ul><li>Curriculum (for Excellence) Development Partnership with a Core Group of 3 teachers from 3 different local authorities and 20 teachers to take part in the CPD </li></ul><ul><li>A maths specialist in every primary school as an ultimate goal? </li></ul>
    13. 13. Overview of the CPD module design <ul><li>Work process defined within a project and work-based approach to learning combined with classroom-based Action Research </li></ul><ul><li>Milestones established based on phases of Orientation & Planning, Working & Reflecting and Evaluation in a blended learning environment </li></ul><ul><li>Based around 3 key questions: </li></ul><ul><ul><ul><ul><li>What is mathematics? </li></ul></ul></ul></ul><ul><ul><ul><ul><li>What is mathematical thinking? </li></ul></ul></ul></ul><ul><ul><ul><ul><li>What is good mathematics teaching? </li></ul></ul></ul></ul>
    14. 14. Overview of the design research project plan <ul><li>A design research approach in the context of a development in mathematics teacher education involving an intervention in the continuing professional mathematical learning of teachers </li></ul>
    15. 15. Thank you for your attention

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