1. Jo Boaler Professor Mathematics Education Stanford University
The Many Colors of Algebra – Engaging Disaffected Students Through Collaboration and Agency.
Jo Boaler
Professor Mathematics Education
Stanford University
while I was a prof at Stanford, where I was for the last 9 years, I studied an unusual approach, not known in the UK, that resulted in very high achievement in mathematics as well as exceptionally good behaviour and respect among students, in an urban school.
When I took visitors to this school who were used to inner city schools, we would walk through the deprivation but then they would be open mouthed when they saw the way the students were working together, with impeccable behaviour, respect among students, for each other and ideas.
So today I am hoping to sketch the approach for you, briefly, and to show you some video clips from the school, as they give you a very clear sense of how it worked, but perhaps also to spend some time considering what this could mean for British schools.

2. When students are engaged in ..
Mixed ability, heterogeneous, rather than tracked groups
Problem solving, rather than rehearsing methods
Discussing ideas and reasoning

3. A case of teaching
Jack Dieckmann, Stanford University Tesha Sengupta-Irving, UCLA Nick Fiori – Yale University

4. Exploratory Algebra Class

5. Exploratory Algebra Class
Algebra as a problem solving tool
Integrating mathematical practices with algebraic content

6. The Students - ethnicity:
39% Latino
34% White
11% African-American
10% Asian
5% Filipino
1% Native American

7. The students - achievement (prior math class)
40% A or B
20% C
40% D or F
Disaffected?

9. Reasons for attending summer school?
10% involved in choice
90% ‘made’ to come by parents / teachers

10. 4 Teaching principles Engage students as active and capable learners
Teach mathematical practices – reasoning, organizing, representing, generalizing
Develop a collaborative, mathematical community
Give opportunities for student voice

11. Active and Capable Learners
Heterogeneous groups
Agency
Andrew Pickering
The ‘dance of agency’

12. Mathematical Practices
exploring, orienting, representing, generalizing, questioning, organizing mathematical thinking

13. Develop a collaborative, mathematical community
Groups
Pairs
Student presentations and discussions eg four 4’s

14. Student Voice

16. Research Data Summer school applications Lesson observations
Student surveys & reflections
Student interviews – 35 during the summer, 15 in the fall
Class materials – posters, work
MARS assessments
Grades in fall and winter

17. Results
Achievement
Engagement & Enjoyment
Future Success

18. A 24% increase.

19. Engagement: How much have you enjoyed this math class?

20. Has this class been more / less useful than regular math class

21. Kit

24. Rochelle

27. An example of the teaching.
Week 4. Menu Activities.

28. An example of the teaching, Week 4, Menu Activities
An example of the teaching, Week 4, Menu Activities. Things to watch out for….
Reluctant students
Encouragement of collaborative community
Teacher attempts to involve all students – even quiet ones, Charles
Alonzo (army jacket)

29. Alonzo

30. What do you see students learning
in this 5 minute clip of teaching?

31. Collaboration & Agency

32. The silent math class
“For the past year, math year was the hardest because you’re not supposed to talk, you’re not supposed to communicate.”
“In other classes it used to like be hard doing my work cause it used to be so boring…and I used to get frustrated and stuff and like right here we get to do group work and we get to talk and stuff and that like helps it not be so boring.”

33. Increased access to understanding
“in normal school you don’t get to do this, but it helped me understand things more”
“it helps me see how they see it and to see if I could understand it”
“I kind of build on other people’s ideas, I really do respect what other people say.”

34. Multiple Methods
“I used to use only one way the teacher taught me and not really understand it. Now I use different ways until I get it.” “When I don’t know how to solve a problem the way the teacher does it, I have other ways to solve it.”

35. Mathematical Seeing
“When we would see the problem in different ways we would understand it better.”
“ It’s like the way – the way our schools did it is like very black and white, and the way people do it here, it’s like very colorful, very bright. You have very different varieties you’re looking at. You can look at it one way, turn your head, and all of a sudden you see a whole different picture.

36. Mathematical tinkering
“I have learned that after finding a pattern you can stretch it in many ways instead of just staring at it. I have learned to think beyond the answer to the problem ”
“Generalizing helped me to look beyond the problems and make challenges for myself”
‘When I’m done, I think of something harder to do”

37. Common Core Standards: Mathematical Practices.
Make sense of problems and persevere in solving them
Reason abstractly and quantitatively
Construct viable arguments and critique the reasoning of others
Model with mathematics
Use appropriate tools strategically
Attend to precision
Look for and make use of structure
Look for and express regularity in repeated reasoning
Long, complex, challenging
Open-ness
Need to elicit different ideas and perspectives – in support of these? …Organisation, small cases, representations

38. Supporting Practices
Organisation
Taking a smaller case
Representation

39. How many squares are on a chessboard?

40. Organizing “I have learned to organize my work – write it all down”
“I learned to organize my work by making T-tables, making charts, also I learned that I should label important information in directions etc”
Organizing

41. Trying a smaller case
“Patterns were very helpful because sometimes the question was asking about a huge number, so then I would just start with some smaller numbers, find a pattern and predict the answer without just taking a lot of time and effort to do the one big problem”

42. Beans and Bowls.
How many ways are there to arrange 3 beans into two bowls?

43. Representing Answers to ‘What have you learned’:
“I learned to say what I’m thinking (in words).”
“taking notes, to remember info and drawing pictures to see what’s going on”
“I learned to see patterns a lot better and how to understand how it gets bigger (or smaller).”

46. Common Core Standards: Mathematical Practices.
Make sense of problems and persevere in solving them
Reason abstractly and quantitatively
Construct viable arguments and critique the reasoning of others
Model with mathematics
Use appropriate tools strategically
Attend to precision
Look for and make use of structure
Look for and express regularity in repeated reasoning

47. A case: mathematical practices & heterogeneity
Gauss

48. How many blocks are in case 100?

49. Common Core Standards: Mathematical Practices.
Make sense of problems and persevere in solving them
Reason abstractly and quantitatively
Construct viable arguments and critique the reasoning of others
Model with mathematics
Use appropriate tools strategically
Attend to precision
Look for and make use of structure
Look for and express regularity in repeated reasoning

50. Heterogeneity
This class has been more useful because we take the time to make sure everybody understands everything and we use different methods of learning.

51. Observations of Fall Classes:
Students sitting in rows, teacher presents, students work through worksheets. In silence.

52. The good news… significant improvement in math grades

53. The bad news… it didn’t last.

54. The students wanted: To be given hard challenges
To gain understanding through discussions
To be able to ‘stretch’ problems and determine mathematical pathways
To add some color to their mathematical landscapes

55. Back in their math classes:
I would say…the only way to describe summer school is very colorful and then this class is just still, ugghhh, black and white. And you just wanna ask ‘Can I have a little bit of yellow?’

56. Common Core Standards: Mathematical Practices.
Make sense of problems and persevere in solving them
Reason abstractly and quantitatively
Construct viable arguments and critique the reasoning of others
Model with mathematics
Use appropriate tools strategically
Attend to precision
Look for and make use of structure
Look for and express regularity in repeated reasoning

57. 2 student cases
Alonzo
Jorge

58. A book for teachers and parents..

59. Panel Discussion
After listening to the speakers, what additional or clarifying questions do you have regarding:
The Common Core State Standards for Mathematics (CCSS-M)
Implementing the 8 Standards for Mathematical Practice in the classroom
The changes in formative and summative assessment in your district/classroom
Plant a question about how to start implementing the CCSS-M in their districts or classrooms (focus on classroom practice and the 8 Standards for Mathematical Practice and formative assessment such as MARS or other performance assessments.