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The Colors of Algebra
Rigor and Resilience at a "Last Chance" School


Joanne da Luz, Life Learning Academy, San Francisco USD

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Teaching Key Algebraic Concepts through Interdisciplinary Mathematics Teaching Students Who Have Failed in – and Been Failed By – Traditional Schooling

Context
Where do I teach?

Content
What are my students learning?

Teaching Practice
What's my approach?


Student Work

Reflections

Resources

Standards


Archive

Archive of Practice

Documentation of Teaching/ Classroom Videos

Document Commentary
Preparing for Presentations
Helping boys to prepare for presentations
I emphasize presentations as not only a form of assessment but as an opportunity for students to demonstrate their learning.
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Preparing for Presentations

Helping boys to prepare for presentations
I emphasize presentations as not only a form of assessment but as an opportunity for students to demonstrate their learning. Over the course of the first semester, these three boys have prepared and given several presentations and are well practiced in using presentation rubrics to prepare their content and delivery.

They have been painting, writing, and talking about multiplying expressions that result in quadratic equations for several days. I assumed that making the transition from their experiential practice to delivery would be easy given last semester's success with presentations. What I do not realize is that although they have the experience, language, and concepts, they are lacking organization and order.

You will notice that I pass the yellow sheet to Kenny. It was his effforts during preparation time that brought this to my attention. He could vocalize the importance of dimensions, color choices, and their results but struggled to decide on what to talk about first and why.

While the boys have a clear outline, you can hear Steve referring to his painting while voicing his equation. A close up on his painting reveals the terms that he is combining like terms and when a certain color is left out of his equation, you hear him saying with concern, "I forgot those!" Similarly, Donny is actually touching the terms on his painting that he is combining and then recording. Kenny begins to concentrate on using his toolkits to address content requirements. He knows that the audience will be looking for him to talk about multiplying, area, dimensions, combining like terms, and factoring.

As the boys are interviewed about their preparation, it is clear that "audience" is not only a motivating factor but also provides a foundation for accountability. Steve is keenly aware that his success depends on whether or not he can accurately describe this process to classmates who did not do this project.

When Donny asks "What does she mean by dimension choices?" Kenny's confidence begins to emerge as he incorporates "symmetry" into his justifications. He applies his understanding of symmetry versus balance to the other boy's paintings. He is becoming an excellent resource for Donny and Steve.

During an interview, Kenny lists all of the math concepts he will be presenting and describes how he is going to use what the interviewer summarizes as "artistic terms and mathematical terms" to explain his painting. She is modeling for him how to use language to describe his efforts and he reciprocates by saying "I'm trying to explain to the new people about the art of it in mathematical terms. Hopefully I can get both of them down."

Building negative with Lab Gear
While Kenny, Steve, and Donny are feverishly preparing for their presentation, the newer algebra students are modeling "minus" for the first time.
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Meanwhile, I'm working with the other students, talking about how to show "negative" and how to "make zeros." The new algebra students have practiced multiplying using the dimension tool and lab gear but without the painting project. I am taking this opportunity to introduce negatives into their multiplication models while Kenny, Steve, and Donny are preparing their presentations.

Not on this clip, Ricky elsewhere described his apathy with my math class as feeling like "it's just gonna happen again" (meaning that he expects failure) Coupled with some encouraging words from other staff members and students, it seems like the opportunity to use Lab Gear has lightened his mood. And it is important to note that Ricky's sense of doom is not different from many of the other boys. He is just willing to vocalize it.The mood generated by Lab Gear and Minus--"Can We Build It? Yes We Can!"

Scaffolding their Preparations
Kenny searches for the right word. Here is where he relates the technique of combining like terms with his version of "making it shorter."
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Although I feel confident that Kenny has a solid grasp of the topics he will address and the poise to do it, I have to remain patient while I help him to verify his wording. Simplify means making simple. Kenny and I solidify the use of the mathematical term "simplify" as a substitute for "making shorter."

You will notice Donny's posture and body language showing his lack of patience with not only this process but my constant attention toward Kenny. Reflecting on this video clip makes me realize now that Kenny is using all the right resources to help him build the rightquestions that will engage me as a teacher. Helping to build Donny's confidence will be a challenge for me.

Final Presentations Introduction
What does the audience get out of this?
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The boys you see in the audience are new students to the school and were enrolled at the beginning of the second semester. Although they have been in my Algebra class for only a month, they have received the message that "toolkits" are important. You will hear one student say "so we'll need toolkit papers?" because he knows that recording and maintaining them on the special gold colored paper is critical to his success in the class. In fact, I use gold colored paper purposely and often introduce toolkits as resources that are "like gold."

The recording of toolkits is one way that I impart a sense of value to the students about mathematical language. I often give participation points during discussion based on how well students refer to and use their toolkits in their explanations and conjectures. By introducing the presentations as an opportunity to obtain toolkit information, I am sending a message that "the math that is actually happening" in the painting project has vocabularly that is important.
Final Presentations: Steve
In this clip, you will hear Steve's unique understanding of a variable in relation to his color choices.
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Final Student Presentations: Steve

"Not Right"
He decided that it was especially important for the audience to know about a mistake he made in his first sketch. He emphasizes that assigning one color to both the constants and the variable "is not right."

Throughout the painting project, Steve always appeared to be working with ease and catching his own mistakes. I assumed that his presentation would run smoothly. But you will hear me coaching him through the content of his presentation.Although Steve is not using the words "variable" and "constant" in this clip, I remember making a conscious effort not to interrupt with the words "variable " and "constant."

At the time of the presentation, I felt confident that his understanding would carry over to the work I anticipated for setting up and solving equations. I felt that an interruption would undermine his emphasis on conceptual understanding.
Final Presentations: Kenny
Kenny had developed a good sense of many of the toolkits.
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Final Student Presentations : Kenny

 

 

Final Presentations: Donny
Donny expressed his mathematical decisions articulately.
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Final Student Presentations : Donny

 

Discussion: New Toolkits
Learning about new toolkits, and discussion of Kenny's painting.
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Final Discussion of Student Presentations

 

Reflections on Teaching

Document Description Commentary
Reflective Interview
About the unit, the students, and the school.
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Reflective interview about the unit and the students  
interview: Donny
Donny's very insightful commentary regarding his educational history provides any teacher with perspective.
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Interview: Donny

He points specifically to what many educators refer to as "gaps in education" and how he realized that one "nice" teacher in particular wanted to address them.
His responses to the interview clearly describe how a fifteen year old boy has internalized failure and is struggling to overcome it. However challenging Donny is for me, I constantly have to remind myself (if Donny hasn't already done so in his own way) that his improvement and ultimate success will happen over time and not right when I want it to happen or expect that it should happen.

interview: Kenny
Reflective interview with Kenny about his prior experiences with math.
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Interview: Kenny

 

interview: Steve
Reflective interview with Steve about his prior experiences with math.
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Interview: Steve

 

interview: Justin Warren
In this clip, my colleage Justin, the art and photography teacher at Life Learning Academy, reflects on his own experience with algebra and explains how "space and dimension" help students to conceptualize the process of multiplication whether it be with numbers and/or variables.
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Interview: Colleague Justin Warren
Justin is an amazing photographer and photography teacher. He is able to provide students with opportunities and insights that I could not and it is evident in the conversations he has with the boys while they are painting and trying to incorporate both aesthetics and algebra into their decisions.

 

 

 

 

 

 

Site last updated January 18, 2006