The Many Colors of Algebra – Engaging Disaffected StudentsThrough Collaboration and Agency.

Jo BoalerProfessor Mathematics EducationStanford University

When students are engaged in .. Mixed ability, heterogeneous, rather

than tracked groups

Problem solving, rather than rehearsing methods

Discussing ideas and reasoning

A case of teaching

Jack Dieckmann, Stanford University

Tesha Sengupta-Irving, UCLA

Nick Fiori – Yale University

Exploratory Algebra Class

Exploratory Algebra Class

Algebra as a problem solving tool

Integrating mathematical practices with algebraic content

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

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

Disaffected?

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

teachers

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

Active and Capable Learners

Heterogeneous groups Agency Andrew Pickering The ‘dance of agency’

Mathematical Practices exploring, orienting,

representing, generalizing, questioning, organizing mathematical thinking

Develop a collaborative, mathematical community Groups Pairs Student presentations and

discussions eg four 4’s

Student Voice

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

Results

Achievement Engagement & Enjoyment Future Success

A 24% increase.

Engagement: How much have you enjoyed this math class?

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

Kit

Rochelle

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)

Alonzo

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

Collaboration & Agency

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.”

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.”

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.”

Mathematical Seeing

“ 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.

“When we would see the problem in different ways we would understand it better.”

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 ”

‘When I’m done, I think of something harder to do”

“Generalizing helped me to look beyond the problems and make challenges for myself”

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

Supporting Practices

OrganisationTaking a smaller caseRepresentation

How many squares are on a chessboard?

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”

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”

Beans and Bowls.

How many ways are there to arrange 3 beans into two bowls?

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).”

 

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

A case: mathematical practices & heterogeneity Gauss

How many blocks are in case 100?

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

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

Heterogeneity

Observations of Fall Classes: Students sitting in rows,

teacher presents, students work through worksheets. In silence.

The good news… significant improvement in math grades

The bad news… it didn’t last.

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

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?’

Back in their math classes:

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

2 student cases

Alonzo

Jorge

A book for teachers and parents..

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