The Many Colors of A l g e b r a – Engaging Disaffected Students Through Collaboration and Agency.

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 ’ teachers to come by parents /

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 It ’ ’ s like the way – the way our schools did it is like very black and white, and the way people do it here, 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.

        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?

Video 1 – teaching organizing, representing, generalizing … through girl and ways and beans

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..