The Many Colors of Algebra

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

• Jo Boaler
• Professor Mathematics Education
• Stanford University

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When students are engaged in ..

• Mixed ability, heterogeneous, rather than tracked
  groups

• Problem solving, rather than rehearsing methods

• Discussing ideas and reasoning

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A case of teaching

• Jack Dieckmann, Stanford University
• Tesha Sengupta-Irving, UCLA
• Nick Fiori Yale University

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Exploratory Algebra Class
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Exploratory Algebra Class

• Algebra as a problem solving tool
• Integrating mathematical practices with algebraic
  content

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The Students - ethnicity

• 39 Latino
• 34 White
• 11 African-American
• 10 Asian
• 5 Filipino
• 1 Native American

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The students - achievement (prior math class)

• 40 A or B
• 20 C
• 40 D or F
• Disaffected?

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Reasons for attending summer school?

• 10 involved in choice
• 90 made to come by parents / teachers

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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

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Active and Capable Learners

• Heterogeneous groups
• Agency
• Andrew Pickering
• The dance of agency

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Mathematical Practices

• exploring, orienting, representing, generalizing,
  questioning, organizing mathematical thinking

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Develop a collaborative, mathematical community

• Groups
• Pairs
• Student presentations and discussions eg four 4s

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Student Voice
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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

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Results

• Achievement
• Engagement Enjoyment
• Future Success

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A 24 increase.
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Engagement How much have you enjoyed this math
class?
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Has this class been more / less useful than
regular math class
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Kit
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Rochelle
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An example of the teaching. Week 4. Menu
Activities.
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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)

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Alonzo
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What do you see students learning in this 5
minute clip of teaching?
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Collaboration Agency
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The silent math class

• For the past year, math year was the hardest
  because youre not supposed to talk, youre not
  supposed to communicate.
• In other classes it used to like be hard doing
  my work cause it used to be so boringand 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.

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Increased access to understanding

• in normal school you dont 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 peoples ideas, I
  really do respect what other people say.

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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 dont know how to solve a problem the
  way the teacher does it, I have other ways to
  solve it.

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Mathematical Seeing
When we would see the problem in different
ways we would understand it better.

• Its like the way the way our schools did
  it is like very black and white, and the way
  people do it here, its like very colorful, very
  bright. You have very different varieties youre
  looking at. You can look at it one way, turn
  your head, and all of a sudden you see a whole
  different picture.

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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 Im done, I think of something harder to
do
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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

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Supporting Practices

• Organisation
• Taking a smaller case
• Representation

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How many squares are on a chessboard?
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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

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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

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

• Answers to What have you learned
• I learned to say what Im thinking (in words).
• taking notes, to remember info and drawing
  pictures to see whats going on
• I learned to see patterns a lot better and how
  to understand how it gets bigger (or smaller).
•  

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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

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A case mathematical practices heterogeneity

• Gauss

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How many blocks are in case 100?
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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

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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.
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Observations of Fall Classes

• Students sitting in rows, teacher presents,
  students work through worksheets. In silence.

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The good news significant improvement in math
grades
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The bad news it didnt last.
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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

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Back in their math classes
I would saythe 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?
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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

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2 student cases

• Alonzo
• Jorge

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A book for teachers and parents..
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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