This document summarizes a research project that designed and tested an online environment called eXpresser to help students learn algebraic thinking and mathematical generalization. The project involved: 1) Developing eXpresser as an interactive microworld for building patterns and expressing rules that govern the patterns. 2) Conducting design experiments with 11-12 year old students to test eXpresser and gather feedback. 3) Interviewing students and finding that eXpresser helped them articulate relationships between quantities in patterns and validate rules through animation. 4) Designing group tasks for students to share and compare patterns/rules, finding most could determine if rules were equivalent.

1. An intelligent microworld as an alternative way to learn algebraic thinking Professor Richard Noss London Knowledge Lab Institute of Education, University of London.

2. Longitudinal Proof Project 1999-2003(Hoyles & Küchemann: http://www.ioe.ac.uk/proof/) analysing students’ learning trajectories in mathematical reasoning over time (age 12/13-15) annual surveying of high-attaining students from randomly selected schools 3000 students (age 13) from 63 schools tested June 2000 in number/ algebra & geometry same students tested again in 2001 & in 2002 some questions from the previous test some new or slightly modified questions

3. seeing & generalising structure Question A1 Yrs 8, 9 & 10 18 6 Pattern spotting approach 60

4. High achieving 11-12 years (N= 3000) Modest improvement then plateau

5. Some more research background While generating figural patterns, students display recursive strategies difficulty with geometric representations lack of rigour and commitment to justify simply “arithmetic exercises” with little understanding of relationships implied in underlying structure often from the way task is set, there is implicit indication that the way of change from one item to the next has to be kept constant and is determined by the first (mostly three) items

7. all ideas are treated as Improvable

9. Level 4 Level 4/5

10. The MiGEN team Eirini Geraniou (IOE) Sergio Gutierrez (BBK) Ken Kahn (IOE) Manolis Mavrikis (IOE) Darren Pearce (BBK) Niall Winters (IOE) PhD students Mihaela Cocea (BBK) Boon Chua Liang (IOE) Richard Noss (IOE) Alex Poulovassilis (BBK) Celia Hoyles (IOE) George Magoulas (BBK) School A (Hackney) School B (Leamington Spa) School C (Islington) School D (Alton) Teachers Teacher Educators Stakeholders Advisors John Mason (Open Univ.) Lulu Healy (Univ. UNIBAN, Sau Paulo, Brazil)

12. Intelligent Support to encourage generalisation through

13. feedback (generic & adaptive) to students

14. support for teachers to monitor students’ progress

15. tools to support collaboration

16. group tasks for sharing & discussing constructions and rules

23. small group sessions

25. do this at least 3 times using small numbers when they can calculate the product in their head

27. how did you get the answer?

28. make sure students are aware that there is more than one "correct" way to see the patterns

29. seek to establish what students are trying to do & remind them of relevant technical aspects

30. ‘mess up’ their constructions or provide them with counter-examples as a way to challenge them

31. draw students’ attention to General World (to show impact of their actions)

33. Overview of the Study 1. A generalisation pre-task (Train Track) on paper 2. Familiarisation Session through video tutorials 3. An introductory taskin eXpresser 4. A generalisation task (Train Track) in eXpresser 5. 1 to 1 interviews with 16 students 6. A generalisation post-task set on eXpresser with answers on paper BRIDGES ARCS

35. Did you make any links between numbers? Why?

36. When building patterns in ‘My World’, what happens in the ‘General World’? What is the same and what is different between the two worlds?

37. What is the play button for? When and why did you click it?

38. What is your rule (for Train Track)? Are there any numbers for which your rule doesn’t work? How do you know that your rule would work for any number?

40. Summary of findings to date 3. Reactions to student-controlled validation of constructions and rules by animation Kathy: “.. My World is like a plan and the General World is what comes to life and actually moves. It [General World] makes it live and animates it. The only difference is that on My World you put the figure number and it doesn’t change, whereas in the General World, the figure number changes”. Nancy pointing at the General World]: “It is not joined up… because you only have 1,2,3,4 of them and it’s got to be, one more extra. If there is 1,2,3,4,5 of these, that means that you always have to add one more. You might have 5 then you need to add an extra line, so that’s 6”

41. 4. Last Interview Question… How many green tiles are there for Fig Nos 6, 12, 1 and 600? 14 students used their rule to work out the answer on paper: they substituted their unlocked number with 6, 12, 1 and 600 and each time were able to give the correct answer 1 student didn’t seem to understand the question and was in need of a lot of assistance to answer. 1 student used the rule for Fig No 6, but then doubled her answer to find the answer for Fig No 12. After the researcher’s intervention, she used her rule to find the correct answer and reflected on her misconception

42. Nancy’s work

43. The Study with eXpresser 1. A generalisation pre-task (train-track) on paper 2. Familiarisation Session through video tutorials 3. An introductory taskin eXpresser 4. A generalisation task (train-track) in eXpresser 5. 1-to-1 interviews with 16 students 6. A generalisation post-task(arcs or bridges) on paper BRIDGES 7. ‘Group tasks ARCS

45. tools to support sharing & discussing models & rules for a patternSame or Different?

46. 16 students grouped in pairs based on their constructions and the corresponding rules they derived Students with apparently different, yet equivalent, rules were grouped asked 1. convince each other that your models & rules are correct (or not) 2. are rules equivalent or not? 3. decide which rule(s) you want to share with the rest of the class. Write down the reasons for your choice. A look at the eCollaborator

47. Nancy and Kathy Nancy’s rule and model Kathy’s rule and model

48. ‘Same or Different’ Kathy showed her rule: Nancy found this rule easy and straight-forward: She immediately saw the ‘structure’ of Kathy’s model by reading her derived rule. She saw the C-shape being repeated 4 times and that the 3 represented the number of tiles in the column Nancy’s rule was harder for Kathy: Nancy needed more tools to help her to explain

49. Themes for further exploration student responses at different phases: modelling, reflection on rules, reading others’ rules, exploring equlivalence more investigation of exploiting the dynamic medium as a model of generalisation building more support within and external to eXpresser for students and teachers supporting more general pattern making

50. If you want to join in our research ,you could have a look at eXpresser (the MiGen microworld). migenproject.wordpress.com links to latest version and tutorials