The Algebra Project community celebrates Marcus Hung’s recent completion of the National Board for Professional Teaching Standards (NBPTS) Certification. A NBPTS certificate is “achieved upon successful completion of a voluntary assessment program designed to recognize effective and accomplished teachers who meet high standards based on what teachers should know and be able to do” (from the NBPTS website). National Board candidates in mathematics are required to complete a rigorous assessment (covering everything from Algebra to Calculus) and submit four portfolio entries that demonstrate clear and convincing evidence of highly accomplished teaching. We applaud Marcus’ use of mentoring, teaching, and leadership in the Algebra Project, which provided a framework for his work in this process.

Marcus also leads the AP’s Teacher Curriculum Team (TCT) which has gathered various AP teachers to work on the Trip Line module, and is a member of the project’s Teacher Resource Materials (TRM) working group, which has recently been working on aligning Algebra Project curriculum modules with the new Common Core State Standards for Mathematics.

Also, this Fall, Marcus began a 5-year Master Teacher Fellowship with Math for America at U.C. Berkeley, which he will use to further explore teacher leadership and work to incorporate Algebra Project pedagogy in professional development support of other teachers.

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The partnership marks the first school-based collaboration with ETS. Dr. Michael Nettles is Senior Vice President and Edmund W Gordon Chair at ETS’ Policy Evaluation and Research Center. Dr. Nettles met Moses during the latter’s fellowship at Princeton last year. Nettles and his team are interested in AP/YPP strategies for raising the floor of math literacy for previously low-performing students, and would like to see these initiatives grow in ways measurable by researchers.

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The AP/YPP group used the Park City sessions to deconstruct some of the module lessons to see how an after school enrichment program could be organized and sequenced for classroom instruction. Finally, the group worked on aligning the Flagway module lessons with the new Common Core State Standards in Mathematics standards and practices. Says Tienda, “It was a very intensive, very useful three weeks of mathematical enrichment and professional growth.”

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The AP is proposing to quarterback the creation of a national coalition of whole schools dedicated to raising the floor for students who previously have been performing in the lowest quartile on standardized exams. Algebra Project curricula has been used in classrooms but not in whole schools. The AP is developing ties to programs, like the National Writing Project so that students in the lowest quartile can have access to quality education in additional academic subjects. When more schools and more teachers experience success educating studentsnot currently thriving, one ingredient for a national dialogue on the right to quality education will be in place. A second meeting to advance this coalition is being planned for the spring of 2013 at ETS.

]]>On May 9-10^{th}, over 100 people from 4 universities and 27 national and Mississippi organizations attended a “We the People” forum with the theme of “constitutional personhood” – the free citizenship guaranteed to all when slavery was abolished, and the most precious fruit of abolitionist struggle and Union victory in a devastating Civil War.

Every citizen should be able to think critically, to participate meaningfully in the political process, and enjoy the liberty to be harmoniously diverse. This is why education for critical thought and opportunity for meaningful political participation are inalienable and positive human rights. Every child has an affirmative right to be educated in ways that facilitate critical thought, active citizenship, and responsible independence in life choices. Every person has an affirmative right to a meaningful voice in the political process.

After the Civil War, the struggle against slavery became a movement to expand the meaning of “We the People” in a reconstructed nation. The antislavery movement became a civil rights movement. The “We the People” forum studied and honored accomplishments of the civil rights movement since the 1960s and discussed “where do we go from here?” in four sessions:

(1) documenting and honoring the stories of civil rights workers in Mississippi in the 1960s and the courageous people and families who housed them and built local movements; (2) addressing the federal government’s role in protecting civil rights and the right to vote, with John Doar, former First Assistant Attorney General for Civil Rights; (3) examining the intersection of the labor movement and civil rights. Derrick Johnson of the Mississippi State Conference NAACP; Douglas Blackmon, author of “Slavery by Another Name”; Bob King, President of the United Auto-Workers; and Frank Smith of the African American Civil War Museum connected the convict labor system and contemporary union struggles. (4) charting a path forward in the quest for constitutional personhood, we established key principles to ground work in the future:

Education — for critical thought, civic participation, basic information and skills — is a fundamental human right to be recognized as implicit, or made explicit, in the U.S. Constitution and those of every state. This is a matter of life or death for the nation’s youth and a prerequisite for a healthy and economically just society.

College readiness should be a goal of K-12 schooling, and college should be publically provided or funded

Civics training and voter registration should be explicit components of K-12 schooling and of community-based education programs.

Voting restrictions based on histories of criminal conviction should be eased or removed, and “Zero Tolerance” practices that feed the school to prison pipeline should be curtailed or ended. At the same time, educational opportunities in prisons and detention facilities should be enhanced.

Public school curricula should be culturally relevant to the populations they serve, engaging students in solving problems in their own lives and that of their communities.

Civic engagement is required at all levels to promote and refine these principles and to hold public officials accountable for counterproductive education and voter qualification practices.

The forum was co-sponsored by Princeton University, Tougaloo College, the Mississippi State Conference of the NAACP, the Veterans of the Mississippi Civil Rights Movement, the SNCC Legacy Project, the Young People’s Project (YPP), and the Algebra Project. It’s work will continue in upcoming 50^{th} Anniversary Commemorations of civil rights accomplishments in the 1960s.

Bob Moses recently noted, “the Algebra Project has shown that there is a way for students at the bottom to learn math, graduate high school in four years, and be ready for college math for college credit. And once there are enough examples of success coming from a national coalition, we can dispel the myth that ‘these kids don’t want to or can’t learn.’ In so doing, I believe we will be in position to launch a movement, in collaboration with the Young Peoples’ Project, toward quality education as a constitutional right.”

In this issue, you’ll read about our efforts to grow this education movement, with new partnerships, and highlights from our national network.

]]>will be published in March 2013 issue of the Journal of Mathematical Behavior 32 (2013) 83 101. For a pre-publication draft PDF, click here

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**Introduction**

In the United States and throughout the world, there is a cacophonous debate about the quality of mathematics education from pre-school through collegiate levels. We have local, national and international exams, many curriculum projects and a plethora of rhetoric about many issues. One of those issues concerns the mathematics education of students from under-represented and under-served populations. This means, generally, students at the lowest level of academic achievement and socio-economic status. It has been well documented that, while it is not the only factor, there is a strong correlation between socio-economic status and achievement in mathematics for elementary and secondary school students (Jordan, 2007 and Jordan and Levine, 2009). Anyon (1981) points out that even with a standardized curriculum, social stratification of knowledge along socio-economic lines takes place. There are also often references in the literature that algebra serves as a “gatekeeper” subject and that mathematics literacy is rapidly becoming a pre-requisite for economic access (Kamii, 1990, Moses et. al, 1989). In addition, there is evidence of a strong overlap between socio-economic status and students in the bottom quartile of mathematics achievement (National Mathematics Advisory Panel, 2008). At the same time there has been little research on the needs of students in these groups in relation to specific mathematics concepts (Lubienski and Bowen, 2000). The authors of this paper chose to focus on the students in the bottom quartile of mathematics achievement because of the general lack of attention given to this group, even in the conversation around providing mathematics for all. For a while, the debate was about whether the education of students in this target population should focus on skills and procedural facility or on conceptual understanding. Then it seemed that people began to realize that both were essential, but still there are arguments about which should come first, which of the two potential kinds of learning depends on and is based on the other. There is considerable “data” that buttresses these arguments, but almost all of it comes from scores on mandated tests of various kinds. Unfortunately, although these tests may be effective in measuring, in some sense, skills and procedural knowledge, they tell us very little about conceptual understanding which is much harder to evaluate. Indeed, there is very little data about the conceptual understanding of students from the aforementioned target populations.

This paper reports the results of one example of an overall approach, the Algebra Project, designed to begin to fill that lacuna. We are not claiming that the results of this study are anything like a complete solution to the problems described in the previous paragraph. We are not even addressing the full range of difficulties. Rather, we are focusing on one very narrow part of the overall problem. The goals of the Algebra Project are: to help high school students from the target population in one country (the United States) to pass all state and federally mandated examinations; graduate from high school “on time”; do well enough on college admissions examinations to be accepted into college; and to have knowledge and understanding of high school mathematics sufficient to place into, and succeed, in credit-bearing college mathematics courses. We give a single example, an existence proof if you will, of one class that used a curriculum and pedagogy developed by Bob Moses’ Algebra Project. As a result of this approach, most of the students developed sufficient procedural knowledge to achieve most of these goals. Moreover, using qualitative research techniques from Mathematics Education research, it was possible to see that they made a strong start on developing an understanding of one of the most important topics in high school mathematics — the function concept. In particular, we will consider several difficulties that, according to the literature, high school and college students have in developing their understanding of the function concept and see what effect our curriculum and pedagogical strategy for functions had in helping our participants overcome these difficulties.

In Section 1 we review some of the literature on learning and teaching the concept of functions, focusing in particular on some specific conceptual difficulties that are reported. We will also refer to APOS Theory, which gives a description of a possible process by which the function concept can be learned. APOS Theory can be used, and in many studies has been used, successfully, as a strictly developmental perspective (e.g., Breidenbach et al. 1992), or as a strictly analytical evaluative tool (e.g., Dubinsky et al., to appear), or as both (e.g., Weller et al., 2011). In this study, it is used as a strictly analytical evaluation tool. APOS Theory focuses on models of what might be going on in the mind of an individual when he or she is trying to learn a mathematical concept and uses these models to evaluate student successes and failures in dealing with mathematical problem situations. We chose APOS Theory for this study because of its effectiveness in previous studies over the past 28 years (see Weller et al., 2003). In Section 2 we describe the Algebra Project, which is the national program under which this research was conducted. In Section 3 we give a detailed description of the study including the formal research question, the participants, the instructional treatment, and the research methodology. Our results will be presented in Section 4. Here we will consider five of the aspects of the function concept studied in the literature as reported in Section 1. Finally, in Section 5 we compare results on our participants’ apparent understanding of aspects of the function concept as reported in Section 4 with that of other, often more mathematically sophisticated, participants as reported in the literature review in Section 1. In this last section, we will also draw some conclusions, discuss the limitations of the study, consider what might be some topics for future research in this area, and mention some possible Implications for teaching practice.

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***...continued in pre-publication draft PDF: click here
also available at ScienceDirect.com:
http://www.sciencedirect.com/science/article/pii/S0732312312000582
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***and, posted earlier this Fall:

Ed Dubinsky, author of Algebra Project curriculum and retired professor of mathematics and mathematics education, is engaged in two exciting international partnerships.

An international team of seven researchers in mathematics education residing in four different countries are coming together to write a book on APOS Theory. Dubinsky and his colleagues have written that APOS “begins with the hypothesis that mathematical knowledge consists in an individual’s tendency to deal with perceived mathematical problem situations by constructing mental *actions*, *processes*, and *objects *and organizing them in *schemas *to make sense of the situations and solve the problems.” Springer will publish the book in the fall of 2013. Meanwhile, this fall Dubinsky is consulting with the Center for Successful Teaching and Learning of Hochschule Braunschweig/Wolfenbutel in Germany. He will advise the institution on the revision of their teaching of mathematics.

Dubinsky is co-author, with Bob Moses, of “Philosophy, Math Research, Math Ed Research, K–16 Education, and the Civil Rights Movement: A Synthesis”, which appeared in the March 2011 issue of the AMS Notices, available for PDF download at www.**ams**.org/**notices**/**2011**03/rtx110300401p.pdf