How we Teach : Inquiry in teaching and learning mathematics

1. How we Teach:Inquiry in teaching and learning mathematics Barbara Jaworski

2. 1 2 3 4 5 6 7 1 2 ? , 2 , … 7 A Starting Problem

3. Filling a rectangle with squares Starting with any rectangle, fill with squares of side equal to the length of the shorter side of the rectangle until no more will fit. Repeat with any rectangle remaining. How many times do we need to do this before the rectangle is filled up? How many squares do we insert at each stage?

4. 1 2 3 4 5 6 7 1 2 7, 2, …

5. Questions and Actions Try it with a rectangle of 5x19. What sequence? Can you find a rectangle that will give a longer sequence of squares? Are such sequences always finite? Can you find a rectangle that will generate any given sequence? e.g., 3, 1, 5? http://home.hia.no/~byrgeb/bb/classes/English/Euclid.html 3, 1, 4

6. What such a task offers Ready engagement Easy access – everybody can engage Capturing interest, engaging curiosity, stimulation Questions to tackle Basic questions that guide initial engagement Potential to stimulate further questions as a result of initial engagement Possibility to engage with serious mathematical concepts Geometric and numeric relationships Algebraic formulation Standard results (such as Euclid’s algorithm) Convincing and proving

7. Investigational work in learning and teaching mathematics 1980s – International research interest in problem solving in mathematics learning and teaching (e.g., Polya, Mason, Schoenfeld) In the UK – use of investigative tasks in informal ways in classrooms 1982: Cockcroft Report encouraged investigational work in classrooms. 1984: BJ: Transition from school teaching to Open University: How can we use investigational tasks to teach mathematics more generally? PhD study: Characterising investigational approaches to teaching mathematics. (Thesis 1991, Book 1994)

8. Working with teachers Focus on how teachers create opportunity for pupils to engage with mathematics seriously and enjoyably In CPD programmes In initial teacher education programmes In research programmes Use of inquiry approaches For pupils doing mathematics in the classroom For teachers developing teaching For academics working with teachers to develop teaching

9. Many years of thinking about … What it means to teach mathematics to enable students’ conceptual learning Processes and strategies in teaching mathematics Design of tasks for mathematical work Use of resources, including ICT Dealing with the real world of schools and classrooms How as teachers we develop mathematics teaching How research can be integral in promoting development and charting its progress.

10. Most of this … … has been in a school context. Except: Part time tutorial teaching for 10 years for the Open University [M101 – the mathematics foundation course] ESRC study of tutorial teaching of mathematics at Oxford

11. A central concern What does it mean for students to engage with mathematics with fluency and (conceptual) understanding? How as teachers do we create opportunity for this? How do we know what sense students are making of what we offer? How can we develop teaching practice in informed ways?

12. For example … We can set and mark homework, set and mark tests. We can sit with students (watch their work, listen to them, ask probing questions) for enough time to glean what they are doing and how they are thinking. We can set up group activity in which students work on and discuss mathematics together, fostering attention to key mathematical ideas and mathematical argument.

13. Research has involved … Clinical interviews with one or more students, recording and probing their understanding through getting them to work on tasks and articulate their thinking Classroom studies in which the reality of the classroom is preserved (as far as research will allow) and students’ classroom participation is recorded and analysed.

14. This brings me to LU … I am teaching mathematics I am teaching students The environment is very different from a school environment The resource is handled differently The environment is different Expectations are different The culture is different Same as in schools

15. Feeling de-skilled!! Despite the knowledge and long experience of thinking about issues in learning and teaching mathematics New kinds of practical situations for which I do not have finger-tip practical knowledge Different kinds of issues to consider A new culture of practice

16. Some factors Large number of students in a lecture I can’t easily know what individual students are doing and where they stand with the material Teachers’ interactions with the learner are of a different kind from a school environment Students who don’t come to lectures Students who don’t do the requisite study to support lecture material There feels to be a much greater distance between teaching and learning

17. Dealing with these factors

18. 1. Becoming acculturated How do those who have been used to working within this environment think about what they do? Talking with individuals How we Teach seminars Observation & reflection Research projects Developing awareness of the new culture – its norms, characteristics, expectations, ways of being. Being drawn into a culture of practice

19. 2. Thinking critically Which of my ways of thinking about mathematics learning and teaching need to change? For example, formative processes in the M-L-T relationship Engagement involves interaction between learners and teacher from which the teacher creates (with students) norms for ways of working, thinking and interaction can listen closely to learners’ use of M language and articulation of M. concepts designs tasks to challenge problematic conceptualisation and create opportunity for students to engage with M. ML SS MC

20. The teaching triad Management of Learning Sensitivity to Students Mathematical Challenge ML SS Harmony MC ML

21. 3. Action Lectures come and lectures go – can’t sit around thinking forever! (But that’s the same in classrooms) So, what do I DO? Overtly inquiring into doing – and learning from the process DOING • A teaching cycle • Plan • Act • Reflect • Feedback • An inquiry cycle • Plan/replan • Act and Observe • Reflect and Analyse • Feedback to planning

22. Specific details Mathematics for materials students 73 students Wide range of experience – GCSE to A at A level 2 lectures and 1 tutorial per week – whole cohort Difficulties of access Getting to know 70+ students (very different from 16 last year) Difficulties of directing input What are the key concepts and how much depth of focus? (e.g., trig. of rt. angled triangles) Where to prove and where to focus just on how to use and apply (e.g. sine and cosine rules)

23. Approaches Use skillful presentation to highlight concepts and emphasis key questions (PP placed on LEARN) e.g. circular functions : powerpoint display, dynamic GeoGebra demo, animation from the web (takes a lot of time to prepare). Get students inquiring at an appropriate level -- offer tasks that can be tackled at several levels, providing easy access and support, and more depth and challenge for the more experienced e.g., tutorial sheet Use GeoGebra to create an environment for exploration CAA tests provide formative and summative assessment Use HELM materials as back-up for all, especially those who need more basic emphasis and practice. The TT?? ML MC SS MC SS MC SS ML SS ML

24. Issues Students need to take appropriate levels of responsibility in working with the materials and resources offered. How to judge? Feedback is minimal, so it is hard to evaluate how successful the approach is being – how students are responding to it. (Questionnaire?) I get sucked into giving good presentations/explanations at the expense of engaging and challenging students Student engagement becomes “following”, “keeping with” rather than grappling with key ideas Gut reactions change from week to week as I try to discern what students are experiencing and doing.

25. Linear Algebra Research 200+ students – mainstream mathematics – good A level results. 2 lectures & 1 tutorial per wk for whole cohort Pre-prepared notes on LEARN with gaps for examples from lectures 1 small group tutorial per wk for all students CAA tests Marked coursework

26. In conclusion Considering “how we teach” is central to a process of being good teachers and developing teaching Developing teaching is not straightforward or comfortable It requires addressing hard questions, making compromises, being critical It’s helpful to know more about its impact on students. How can we achieve this?