The Many Colors of Algebra –
Engaging Disaffected Students
Through Collaboration and Agency.
Jo Boaler
Professor Mathematics Education
Stanford University
When students are engaged in
..
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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
“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.
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”
Common Core Standards:
Mathematical Practices.
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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
 Organisation
 Taking
a smaller case
 Representation
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
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.
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
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?’
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
