1. http://www.migen.org The MiGen system and its impact on students and teachers: a review of the research dilemmas and positive outcomes through the eyes of a teacher and a researcher Eirini Geraniou (Institute of Education) Helen Humble (Amery Hill School)

2. http://www.migen.org OUTLINE 1.Rationale of the MiGen project (5 mins) 2.What is the MiGen system? (5 mins) 3.Research Methodology (5 mins) 4.Two student case studies(10-15 mins) 5.Our Dilemmas and Positive Outcomes (5 mins) 6.Discussion (15-20 mins) 7.Final Remarks (5 mins)

3. http://www.migen.org Generalisation as… the process of identifying and expressing relationships that underpin pictorial patterns 2(w + 2) + 2h For n red squares, there are 3n + 1 blue and 2n + 2 green squares

4. Students Difficulties Pattern-Spotting Specific/General case Linear Scaling S tudents are often taught methods of finding the general rule using the term- to-term method and fail to conceptualise relationships between variables, justify them and use them in a meaningful way Tasks presented on paper encourage students to focus on the specific case and fail to see the general case through the particular examples Students struggle to understand how to use letters to represent any value and how to use mathematical notations to express generality. Also, they can articulate the recursive rule, but not the algebraic one. Mathematical Language How many tiles for n=6? How many tiles for n=12? How many tiles for n=10? How many tiles for n=100?

5. http://www.migen.org AIM - to co-design, build and evaluate with teachers and teacher educators a pedagogical and technical environment for improving 11-14-year- old students learning of algebraic generalisation

6. http://www.migen.org The MiGen system comprises: - a microworld (eXpresser) - Intelligent support for students using eXpresser - Teacher Assistance tools

7. http://www.migen.org A MiGen task Task Question Construct the Train-Track model. Use more than one pattern to make the model. Use different colours for each pattern to show other people how you made your model. Find a rule for the number of tiles for any model. Share your rule and match it with your model. Collaborative Task Convince each other that your model and rules are correct Can you explain to each other why the rules look different but are equivalent? Discuss and write your explanations.

8. Addressing students difficulties Pattern-Spotting –Encourage students to look for the structure of the pattern to help them derive a general rule Specific/General Case –(Randomised) Animated Task Presentation Mathematical Language –Students with the help of the eXpresser language develop an understanding of the notions of variables and constants and make the transition to symbols easier Linear Scaling –The MiGen activities provide opportunities for engaging with counterexamples http://www.migen.org

9. Research Methodology In general –Iterative Design –Feedback from students –Close collaboration with teachers (& teacher educators) In Particular –Amery Hill School –Sample (year 7 – July 2009 – 16 students) (year 8 – July 2010 – 16 students) (year 9 – Nov 2010 – 30 students) –Activity Sequence Introductory Tasks – Individual Tasks – Collaborative Tasks –Classroom Discussion –Individual PCs http://www.migen.org

10. Research Methodology Data Collection Audio record (teachers and researchers interventions, students collaboration and task-based interviews) Video record classroom sessions Dialogue between teacher-researcher Before, during and after each session Teachers role Organise sessions, find students, introduce the system, explain the tasks, run classroom discussions, intervene when needed to support students, inform research team of students learning profiles Researchers role Support the teacher in the organisation of the studies, design activities in collaboration with the teacher, write a suggested lesson plan, familiarise teacher with the system, technical support during lessons, inform teacher of previous research studies and results http://www.migen.org

11. Student case studies: Steve and Laura Construct models Show intelligent support Grouping Tool http://www.migen.org

12. eXpresser – Train Track task http://www.migen.org

13. Steves model

31. List of prompts (Steve) Draw Attention/ Nudge –There are no unlocked numbers (Comment) –The pattern is not coloured because you need the exact number of tiles in the pattern (Hint) Create Cognitive Conflict/Challenge –Your pattern might not be coloured for a different number of building blocks (Comment) Reflection –How many tiles does the building block have? How many building blocks does your pattern have? (Question) –Thats correct but what is the link between these numbers? (Question) http://www.migen.org

32. Lauras model Add messing-up in the General Model...

33. Lauras model Add messing-up in the General Model...

34. Lauras model Add messing-up in the General Model...

35. Lauras model Add messing-up in the General Model...

36. Lauras model Add messing-up in the General Model...

37. Lauras model Add messing-up in the General Model...

38. Lauras model Add messing-up in the General Model...

39. Lauras model Add messing-up in the General Model...

40. Lauras model Add messing-up in the General Model...

41. Lauras model Add messing-up in the General Model...

42. Lauras model Add messing-up in the General Model...

43. Lauras model Add messing-up in the General Model...

44. Lauras model Add messing-up in the General Model...

45. Lauras model Add messing-up in the General Model...

46. Lauras model Add messing-up in the General Model...

47. Lauras model Add messing-up in the General Model...

48. Lauras model Add messing-up in the General Model...

49. Lauras model Add messing-up in the General Model...

50. Lauras model Add messing-up in the General Model...

51. Lauras model Add messing-up in the General Model...

52. Lauras model Add messing-up in the General Model...

53. Lauras model Add messing-up in the General Model...

54. List of prompts (Laura) Draw Attention/ Nudge –The General Model is messed-up (Comment) Create Cognitive Conflict/Challenge –Would the model be messed-up if you change the unlocked number? (Question) –Check whether your model will be messed-up by changing the unlocked number (Suggestion) –The unlocked number changed. Think: How did you repair the pattern? (Question) –The unlocked number changed again. How can you link the two patterns? (Question) Explicit Advice –Link the two patterns together (Suggestion) http://www.migen.org

55. Grouping Tool http://www.migen.org S L 0.9

56. This workshops focus 1. What were the dilemmas of the teacher during the studies? 2. What were the dilemmas of the researcher? 3. What were the positive outcomes of the collaboration between the teacher and the researcher for them and for the students? http://www.migen.org

57. Our dilemmas and positive outcomes... Research Focus http://www.migen.org Practical IssuesLearning Outcomes Dilemma e.g. Interventions (when, what, how, consistency) Positive Outcome e.g. Pre-empting linear addition Dilemma e.g. time-spent Positive Outcome e.g. reflective questionnaires or system development Dilemma e.g. balance between verbal and written responses Positive Outcome e.g. greater insight into the way students are thinking

58. Our dilemmas and positive outcomes... Research Focus http://www.migen.org Dilemmas - consistency of interactions and interventions - conscious of what you say as it affects students learning - withhold help (verbal and written explanations) and avoid teaching similar ideas just prior to research period - as a teacher you try a wide range of ways to help students understand without being concerned about which particular stimuli had the desired effect, whereas as a researcher you analyse each item separately to determine effectiveness - what really happened instead of what the teacher or the researcher thought happened

59. Our dilemmas and positive outcomes... Research Focus http://www.migen.org Positive outcomes - pre-empted the linear addition by not giving them in a sequence (as students natural inclination is to scale up) - individual attention - clearer view of what was causing the problem - teacher tools to review/reflect - almost running a mini research project every lesson [You ask yourself: Was that effective? What went wrong? How can you improve it next time?

60. Our dilemmas and positive outcomes... http://www.migen.org Practical Issues Dilemmas - time spent - half a class (how could you make it manageable for 30 students?) - 4 researchers and 1 teacher - curriculum requirements

61. Our dilemmas and positive outcomes... http://www.migen.org Practical Issues Positive outcomes - reflective questionnaires reveal students understanding and misunderstandings and how far along they are in the process of generalisation - teacher benefitted from using the technology - students benefit was larger time spent/think in depth when tackling these questions - system developed as we observed its use in the classroom

62. Our dilemmas and positive outcomes... http://www.migen.org Learning Outcomes Dilemmas - balance between students verbal and written responses - emphasis on learning outcomes (understand what students have learned) - not being explicit in learning outcomes - generalisations - what they are and how to work them out - questionnaire design for teachers benefit - help students clarify what they have learned but not moving them on to next step

63. Our dilemmas and positive outcomes... http://www.migen.org Learning Outcomes Positive outcomes - teacher: greater insight into how their students are thinking - teacher: reflect and change teaching approach of number sequences - in depth understanding of students difficulties - unearthed issues, e.g. Doubling - students feel special, increased their confidence - students benefitted from (guided) collaboration - immediate feedback from the microworld

64. Our Research Questions We were looking at two very different aspects of the lesson: how do students interact with the software? how is students understanding developing? Two sets of outcomes : Is the software working in the way it was intended? Do students gain insight which they can then apply to future problems? http://www.migen.org

65. Reflective Questions Constant Dialogue between teacher and researcher How do you capture all the data? How do you capture all the issues relating to the focus of the research that influence the lesson? What is being observed? What is the teachers role? What is the researchers role? How do these overlap? How do you organise the session? What is the plan? What activities do you use? How do you design these activities (having in mind the school curriculum and the projects research goals)? What are the aims in terms of data collection? What were we investigating? What did we want to find out? What did we miss? What were the teaching strategies? When should the teacher intervene? Should the teacher intervene when testing whether the intelligent support provides useful feedback to the students? http://www.migen.org

66. Final Remarks Benefits of doing classroom research Teacher –detailed observation of misconceptions and barriers to understanding of individual students Researcher –gathering of wide range of complementary data using a variety of means for later detailed analysis Collaboration between teacher and researcher gave the teacher greater understanding of what had been observed and the researcher greater knowledge of the context within which the research was conducted. THANK YOU

67. Appendix... http://www.migen.org

68. Teacher Assistance Tools 1. Student Tracking Tool 1. Student Tracking Tool (summary) 2. Student Tracking Tool2. Student Tracking Tool (timeline) 3. Grouping Tool

69. http://www.migen.org 1. Student Tracking tool (summary)

70. http://www.migen.org 2. Student Tracking tool (timeline)

71. 3. Grouping Tool http://www.migen.org