Iowa State University Capstones, Theses and Graduate Theses and Dissertations Dissertations
2020
Mathematics reborn: Empowerment with youth participatory action research EntreMundos in reconstructing our relationship with mathematics
Ricardo Martinez Iowa State University
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Recommended Citation Martinez, Ricardo, "Mathematics reborn: Empowerment with youth participatory action research EntreMundos in reconstructing our relationship with mathematics" (2020). Graduate Theses and Dissertations. 18181. https://lib.dr.iastate.edu/etd/18181
This Dissertation is brought to you for free and open access by the Iowa State University Capstones, Theses and Dissertations at Iowa State University Digital Repository. It has been accepted for inclusion in Graduate Theses and Dissertations by an authorized administrator of Iowa State University Digital Repository. For more information, please contact [email protected]. Mathematics reborn: Empowerment with youth participatory action research
EntreMundos in reconstructing our relationship with mathematics
by
Ricardo Martinez
A dissertation submitted to the graduate faculty
in partial fulfillment of the requirements for the degree of
DOCTOR OF PHILOSOPHY
Major: Education
Program of Study Committee: Ji Yeong I, Major Professor Christa Jackson Mollie Appelgate EunJin Bahng Julio Cammarota
The student author, whose presentation of the scholarship herein was approved by the program of study committee, is solely responsible for the content of this dissertation. The Graduate College will ensure this dissertation is globally accessible and will not permit alterations after a degree is conferred.
Iowa State University
Ames, Iowa
2020
Copyright © Ricardo Martinez, 2020. All rights reserved. ii
DEDICATION
This is dedicated to everyone that existed before me and everyone that will live after me.
I want to start by saying, if not for the life work of Gloria E. Anzaldúa, I would not be the person
I am today, and this dissertation would have never been. In many ways, I would not be able to dedicate this dissertation to all the people that existed with me without the awareness of myself and the world I gained from Anzaldúa’s words and stories.
Thus I dedicate this to everyone I grew up with from Delano – this is for everyone who’s parent(s) worked in the fields and for anyone associated, affiliated or adjacent to gang life because gangs (you) are not the problem and that is why I work to change and challenge systems of oppression through education. We must all first understand the root cause of societal pain and trauma, where gangs are only a symptom. I say that because too many people I grew up with either don’t exist, are in jail, or spent and are spending too much time behind bars. For that reason, this work is dedicated to the multiple future revolutions that we must support together.
iii
TABLE OF CONTENTS
Page
ACKNOWLEDGMENTS ...... v
ABSTRACT ...... vii
CHAPTER 1. INTRODUCTION ...... 1 An Introduction To REALM ...... 1 Mathematics Researcher Positionality ...... 3 Introduction ...... 5
CHAPTER 2. LITERATURE REVIEW ...... 14 Critical Pedagogy and Mathematics Education...... 14 Feminism Pedagogy and Critical Pedagogy ...... 18 Spiritual Activism and Conocimientos...... 20 Critical Consciousness ...... 23 Ethnomathematics ...... 24 Identity, Mathematics, and Dehumanization in Mathematics Education ...... 25 Equity, Mathematics and “Social Justice” ...... 26 Critical Literacy, Mathematics, and Critical Consciousness ...... 28 Youth Participatory Action Research and Mathematics ...... 29
CHAPTER 3. FRAMEWORK ...... 31 (Y)PAR EntreMundos Epistemologies ...... 31 Conocimientos when reading and writing the word ...... 37 Autohistoria-teoria-mathematica ...... 42 Conocimientos: The Story ...... 43 Chapter Summary: Conocimientos and Mathematical Shift ...... 46
CHAPTER 4. METHODS ...... 48 History ...... 48 Researchers Role ...... 49 Recruitment ...... 50 Context ...... 51 REALM Structure ...... 51 Data Collection ...... 55 Data Analysis ...... 56
CHAPTER 5. RESULTS PART I - NARRATIVE ...... 59 Narrative of Ten-Day Program ...... 59 Day 1: The Formation of a Collective ...... 60 Day 2: Generative Themes Emerge ...... 65 Day 3: Identifying Key Stakeholders ...... 69 Day 4(5): Critical Social Science Research Training ...... 74 Day (5): Reviewing Generative Themes ...... 78 iv
Day 6: Data Collecting ...... 80 Day 7: Revisiting theoretical framing of conocimientos ...... 82 Day 8: Identifying Patterns and Themes ...... 85 Day 9: Bringing Everything Together ...... 88 Day 10: Presentations and Next Steps ...... 92 Mathematical Turns: Towards an Epistemology of YPAR EntreMundos ...... 93 Turn 1: Our Roots, Ancient Mathematical Wisdom ...... 94 Turn 2: Our Voice, A Mathematical Creative Praxes ...... 95 Turn 3: Our World, A Lens of Mathematical Critical Literacy ...... 97 A Balance of Action and Reflection ...... 98
CHAPTER 6. RESULTS PART II ...... 100 Results of Individual Youth ...... 100 Results of Group Analysis ...... 110 Results of Collective Analysis ...... 112 Chapter Summary ...... 114
CHAPTER 7. DISCUSSION & CONCLUSION ...... 116 Discussions of the Seven Stages of Conocimientos ...... 116 el arrebato: a catalyst for mathematics ...... 117 Nepantla: Embracing Infinity Nested in Infinity ...... 119 El Compromison: Mathematics can change the world ...... 122 Putting Coyolxauhqui Together: Reading Mathematics to Know Self and World ...... 125 The Blow Up: Meeting other to write Mathematics ...... 128 Shifting Realities: Counting our Acts of Resistance ...... 130 Coatlicue: The Power and Fear of Knowing Mathematics ...... 132 Conscious Raising Mathematics ...... 135 Conocimientos with|in Mathematics ...... 138 Conclusion ...... 140
CHAPTER 8. FINAL REMARKS: A POETIC RETURN ...... 143 Mathematics Me – Therefore You ...... 144
REFERENCES ...... 145
APPENDIX A. I AM MATH POEM TEMPLATE ...... 154
APPENDIX B. DAILY JOURNAL PROMPTS ...... 155
APPENDIX C. IRB APPROVAL MEMO ...... 157 v
ACKNOWLEDGMENTS
I want to acknowledge the youth that I have worked during my time in Iowa because
The youth taught me how to do the work that I do. I was lucky enough to have the opportunity to work with three different groups of young people form three different cities in Iowa. Many thanks to Al Exito that gave me multiple opportunities to also interact with youth across the state of Iowa. I want to acknowledge all the APEX students I was able to teach during my time at
Iowa State. I am thankful to have had the opportunity to also work with Science Bound scholars and staff. The experience I gained from Al Exito, APEX, and Science Bound cannot be measured. I want to acknowledge all the theorizing done in the mountains with Greg
Wickenkamp and DJ Dr. J Won – everything we learned from listening to music, to learning how to survive helped me throughout my time at Iowa. I want to thank Ms. Carla for showing me what it means for your job to be life work that impacts everyone you come into contact with. I cannot thank Dawn enough for allowing me to work with Al Exito, and I want to acknowledge
Alexis Campbell for making the commitment to work with Dawn, Cindy Delgado, and myself in creating the playground of this dissertation. I hope Cindy does know that she was vital in our program being a success. I want to acknowledge my two mentees that taught me so much and I cannot wait to see how, Trinity Dearborn and Araceli Lopez-Valdivia change this world.
A special shout out goes to all my fellow graduate students, Hilda Makori, Kari
Jurgenson, Tyanez Jones, Craig Vanpay, Rudi Motshubi, Carmen Jones, Claudia Young, Alade
McKen, Jeanne Connelly, Dariana Glasco-Boyd, Betsy Araujo Grando, Brandon Clark, Cueponi
Cihuatl Espinoza, Maria Espinoza, and Kassandra Diaz all of you supported me in one way or another – so thank you. I especially want to thank Yi Jin for showing me what it takes to get a
PhD in leading by example when I started my program, and Wesley Harris who helped me and vi my thinking through the many nights writing at Café Diem. I then want to acknowledge Heather
Lindfros-Navarro, Melissa Adams-Corral, Gabriela Vargas, Vijay Williams, Karla Torres, Saja
Ibram, Malika Davis, Zoe Thornton, Teddy Chao, Beckie Sue Abraham, Ben Gleason, Gabriel
Rodriguez, Nick Tanchuk, Özlem Karakaya, Laryssa Nadolny, Jihee Yoon, Lorena San Elias,
Kait Ogden, Breanne Ulloa, Nehemias Ulloa, Israel Almodóvar, Jocelyn Del Pilar, Diana
Figueroa, the Farren family and Carla Mcnelly Valladares for being part of my story.
I would like to thank my committee, Ji Yeong I, Christa Jackson, Mollie Appelgate,
EunJin Bahng and Julio Cammarota because each of you have imprinted at least one positive memory that I know will help me grow as a person and all of you showed me how to mentor future graduate students. A big special thank you to Ji Yeong - You are what every major professor should be in that I always felt as if I mattered as a person first, then as a scholar - you helped me by being truthful and transparent in honoring my knowledge, identity and abilities while motivating me to be a better scholar.
This dissertation is for all my friends and family. I want to thank my brother, Jesus for being my biggest support when getting my undergraduate degree because without those years, I would have never been an educator. I want to thank my brother, Vinny for teaching me about resistance where I gained insight that no book can teach. Above all this dissertation is dedicated to my sister Camelia, who I say is the reason I am a good person, and without her, I do not even want to think where I would be on this world.
I want to end by acknowledging the history of mathematics, which is the history of this plant. It is the earth that bounds all of us together and we must being to see that mathematics is one way for all of us to be connected. Thus when I acknowledge you, I acknowledge the world and simultaneously acknowledge myself. vii
ABSTRACT
In this dissertation mathematics is positioned as guiding epistemology with participatory action research (PAR) EntreMundos epistemology that simultaneous explores mathematical youth empowerment and the nature of mathematics within youth participatory action research
(YPAR). Anzaldúa’s theory of conocimientos allows for the exploration of consciousness while youth engage in a mathematics (Y)PAR EntreMundos summer program. The first main research explorations focus on understand how mathematics can be lived within YPAR and the second question gives insight on how the seven stages of conocimientos manifest in the learning of mathematics. (Y)PAR EntreMundos as such acts as the epistemology and methodology of this study. Narrative inquiry and context analysis are two methods used with thematic analysis to make meaning of the data.
YPAR creates a rich education space for young people to explore societal issues, in that mathematics is introduced to enhance the YPAR experience. Mathematics is epistemologically defined through indigenous cosmologies as a living entity, to further view mathematics as an ontological relationship, foregrounds in an axiology of mathematics being an experience.
(Y)PAR EntreMundos and conocimientos then allows for an expansion and development of individual-collective consciousness. The relationship between self, mathematics and others creates a full embodiment of mathematical learning.
The dissertation, chapter 5 is a narrative of the ten day summer mathematics (Y)PAR
EntreMundos program that was a partnership between myself, a Latinx community based organization and a University college pathway program for underrepresented students majoring in AgSTEM. The narrative an overview of each day of the program. Chapter 6 contains results for coding, where context analysis was utilized. Chapter 7 discusses the mathematical viii manifestation of the seven stages of conocimientos. The youth who participated in this project, had a clear shift in their consciousness in living with mathematics as a collective with (Y)PAR
EntreMundos. Mathematics with (Y)AR EntreMundos showed to transformative because it provides a space for both mathematical reflection and mathematical action.
1
CHAPTER 1. INTRODUCTION
An Introduction To REALM
Poem by Ricardo Martinez
To begin with zero –
is the origin of our emptiness.
So, we count – in order,
to number our memories;
to find the right frequency;
to sum the distance between us;
to make nil our differences -
all
to tell a story with mathematics.
So, we look at the readers and writer of this poem
(of this dissertation) in listening for a call of transformation,
in transposing our collective history to
the limelight of a-ha moments.
We begin with zero the day we are born.
Counting is akin to breathing and
Emptiness comes from being forced to believe a lie.
The forced motion that you, me and we
are non-mathematical -
creates the pain placed upon learning math. 2
Call healing and hope an experience of -
You as Me and Me as You – the reflexive property -
Reflecting on the truth that mathematics is alive
and dancing to the fractions of strings.
The music tells us
to know math is to know self - and
to know math is to know others – the transitive property.
Where the proof is already inside of you:
A fulcrum seen inside a kaleidoscope,
is the balancing act between what was and what will be -
An infinite future, created by
A finite life – living forever with math
meaning we are never alone ...
The above poem was written as a reflection on the program and experience created from my dissertation study. The poem titled "An Introduction to REALM" was written after the first draft of this dissertation was written with the intent of it being the start of the dissertation.
REALM is the pseudo name for the program and research site that mean Reflection Equals
Action in Liberatory Mathematics. Poetry is an intimate process that allows the writer to construct and reconstruct themselves as they reflect then became the ideal way of starting this dissertation. This poem will be discussed at the end of this dissertation in tandem with the poem
"Mathematics Me – Therefore You."
3
Mathematics Researcher Positionality
I begin with my positionality as a researcher to address my frustration with the current situation in education as a whole but especially mathematics education with respect to equity and/or “social justice”1 and the lack of praxis in the field. Praxis is the authentic union of action and reflection towards social change, where action without reflection results in nothing more than verbalism, and action without reflection can only lead to a revolution that does not liberate the people (Freire, 1976). In mathematics education, there are too many accounts of how teachers and researchers feel about teaching for “social justice” with no focus on action.
Conversations about “social justice” and mathematics have turned into a state of sharing stories that either make the writer feel better or make it easier for the readers to “understand” the
“difficultly” of teaching for “social justice.” Feel-good stories of “social justice,” in my opinion, are rarely addressing substance. More recently, multiple resources have been published, providing teachers with lessons focusing on social justice, and more researchers are talking about the need to humanize mathematics, both great steps moving forward. Yet, as I read mathematics education literature and engage in conversations with scholars at conferences over the four years, that few lessons for social justice focus on root causes of societal injustices, and even though researchers are using the language of humanization, there appears to be a lack of epistemic synergy. It is as if everyone has their own definition of humanization, and given a vast majority of mathematics education researchers have multiple forms of privilege, are they even capable of seeing the barrier towards humanization.
1 "social justice" will be in quotes to acknowledge a potential flaw in the aims of "social justice." Jose Media (2013) mentions when focusing on an ideal, it is easy to ignore imperfections of current societal issues, which in turn fails to recognize the root causes making proposed solutions epistemically flawed. Social injustice is more of an appropriate term for what this paper seeks to investigate. Therefore, the quotes around "social justice" serve as a reminder to address the root causes of social injustices in society. 4
What I have seen over the last four years at conferences is a lack of talking about changing structures that perpetuate injustices within mathematics education. We do have scholars in mathematics education doing research on "social justice," but few are engaging in social justice, creating a disconnect between academia and reality. I ask, "where are the accounts of people using mathematics to dissent and transform society?” or “why has/is ethnomathematics still located in the margins of mathematics education research?” Now, I understand that I am early in my career, and I may simply not be exposed to individuals taking a praxis approach to mathematics education, but once again, that highlights a problem. As an emerging scholar in mathematics education that wants to focus on "social justice," why is it so difficult to see critical work in mathematics education?
I acknowledge that I may be wrong. But this sentiment is expressed to emphasize the frustration and urgency of what I wish to do with my career as a researcher. Take the example of using Census data to use mathematics to understand the world. On the surface, the data looks like a great playground for mathematical exploration, but do we ever ask what Census data is and whether it impacts some if not all of us? For example, look at the words of poet Ariana Brown
(2017),
I take a good look at the census document that mocks me holder of indisputable truth a white man and his measuring tape blood counter, fraction distributor, half this, quarter that. My body is not an equation, mathematics is for colonizers
Do we ever take into consideration the lived injustices that people of color live with every day, or do we question nothing and continue to do what has always been done? I wonder how difficult it would be to explore mathematics from the Census if it is a constant reminder of trauma. We 5 need to learn that we cannot understand an individual’s identity without understanding the historical context that created the present or the individual's identity in relation to others. So, I turn to the words of Gutierrez (2002) as the inspiration behind this attempt of healing mathematics,
For a long time, we have blamed teachers, the curriculum, and even the system of inequalities in mathematics education. In our analyses, however, rarely do we consider that we (the mathematics education research community) could be part of the reason why very little in the way of significant changes ever arise in the kinds of achievement, participation, and relationships with mathematics that so many oppressed students experience (p. 179).
Coupled with the words of Gloria E. Anzaldúa,
Soy un amasamiento, I am an act of kneading, of uniting, and joining that not only has produced both a creature of darkness and a creature of light, but also a creature that questions the definitions of light and dark and gives them new meanings. (Anzaldúa, 2012, p. 232) as I begin my dissent in questioning the philosophical definition(s) of mathematics, and in questioning “social justice” both in and outside of mathematics education. Furthermore, I question myself and my role in perpetuating injustice. I look forward to questions posed by the reader, I look forward to being wrong, and I look forward to being neither right nor wrong, only to continue to question and critique myself. It is my positionality as a critical scholar that looks to make no claims of absolute truth as I dissent; I only seek to find roads that lead to authentic forms of social justice within mathematics education. I end in quoting Kamaal Ibn John Fareed
(2008) in that “seeing ain’t believing it’s the feeling that we needing. Believe in each other put the question to the system.”
Introduction
Undergraduate students studying mathematics as a major in college learn mathematics is built on logical proofs. Mathematics, in its purest form, is logic, void of any culture. Yet, how individual cultures arrive to this purest form of mathematics is a question with multiple answers. 6
More so, how mathematics is taught depends on the culture(s) of both the teacher and the student
(Darder, 1991; 1993). For example, if I am teaching how to find the area of a pyramid, do I show images of Egyptian pyramids, or do I show one of the many pyramids from Mexico, Central, or
South America? Is this a question of mathematics, culture, or of both? If mathematics is taken as culture-free, then it would be taught using only equations to understand any arbitrary pyramid.
Only using equations is not the current direction of mathematics education, which can be seen in the fourth Standard of Mathematical Practice of the Common Core State Standards for
Mathematics (National Governors Association & CCSSI, 2010), model with mathematics, where students solve problems from everyday life (Leinwand, 2014). Back to the questions like, do I show students pyramids from Africa or the Americas? Or do I show students how a military engineer named Archimedes approximated the value of pi, or do I mention how a tribe in Nova
Scotia used pi to construct fishing boats? Both military and fishing arrive at the same mathematical conclusion that pi is an irrational number; the only difference is how and why pi was developed. d'Ambrosio (Bishop, 1999) explained how pivotal Greek geometry was to Greek philosophy (and vice versa), which influenced Greek culture to emphasizes that we cannot separate mathematics from culture.
The question of what example and what culture to highlight when teaching mathematics through a lens of multiculturalism arrives at issues of equity in mathematics education, and the example of what teachers do use illustrates how teaching mathematics is a political act (Aguirre et al., 2017; Gutierrez, 2002). Teaching as a political act is derived from the nature of education informing (or not informing) the learner about the real world (Giroux, 2000); what the teacher chooses not to teach gives learners an incomplete understanding of the world, thus making them unaware and unprepared to make informed decisions about the world (Shor & Freire, 2002). 7
Mathematics cannot be separated from culture because mathematics cannot be separated from the real world. Mathematical learning in K-12 is rooted in logical systems, but we cannot forget that people, humans, first had to see mathematics in nature (and themselves) before "creating2” such a system.
Researchers should not only ask questions of why but should pose questions of how
(Foucault, 2013). One such question would be how mathematics has been developed into what
Bishop (1998) calls a weapon of Western imperialism. In the United States, mathematics is taught almost exclusively from a Eurocentric perspective. Wherein deciding to teach mathematics only from a Eurocentric perspective, unintendedly or naught, maps Eurocentric thinking on to every learner, a form of Western or Eurocentric imperialism. So, exploring the earlier questions of either sharing how the Greeks discovered pi or how the Mi’kmaq tribe of
Nova Scotia utilized pi becomes a question of teaching ideology and enters the realm of "social justice." Multicultural mathematics should be celebrated because of its logical construction and its ability to connect cultures. By utilizing, “social justice in mathematics education,” also known as “critical mathematics” defined by Wagner and Stinson (2012) as
a means for student and teacher self-empowerment to organize and reorganize interpretations of social institutions and traditions, and to develop proposals for more just and equitable social and political reform (p. 11)
Mathematics can then be used to connect cultures. Revisiting the question of which development of pi to share, the answer should be both with a commitment to explore how multiple cultures (more than two) developed and continue to develop mathematics.
Understanding how multiple cultures arrive at the same logical-mathematical conclusions
2 The word create is in quotes to emphasize that logic and order is already present in the universe, therefor mathematician can neither create nor discover mathematics. What is created is a way of describing math mathematics and the natural world. 8 reveal how cultural mathematics are connected to each other, thus showing one way that all people are interconnected. Mathematics is unique because regardless of how an individual culture constructs its system of mathematics, there are common mathematical axioms or similarities in all cultures’ mathematics (Bishop, 1999; 2000). The use of non-Eurocentric mathematics is reflective of the development of Ethnomathematics (d'Ambrosio, 1985; 2001); and more recently in the work of Marta Civil in regard to the use of funds of knowledge in the mathematics education community (Civil, 1998; González, Andrade, Civil & Moll, 2001).
Where funds of knowledge are the historically developed aspects of an individual’s identity, developed at home that makes themselves manifest in the classroom (Gonzalez, Moll, Tenery,
Rivera, Rendon, Gonzales, & Amanti, 1995). Funds of knowledge are further developed in The
TEACH Math project (Aguirre, 2018), which “focuses on integrating children's mathematics thinking and children's home and community-based funds of knowledge in mathematics methods courses” (p. 1). What we begin to see is that by attending to funds of knowledge in mathematics classrooms, multicultural learners are viewed as assets while utilizing their funds of knowledge to enrich the mathematical learning of the classroom. A focus on funds of knowledge allows for the empowering of mathematics learners by allowing them to see themselves as constructors and owners of mathematical knowledge.
In this dissertation, I identify a gap in mathematics education in terms of critical mathematics with respect to youth participatory action research (YPAR) and Conocimientos3 showing how minimal research has been done focusing on how consciousness is formed while learning mathematics. During the 2016 Psychology of Mathematics Education in North
3 Conocimiento is a Spanish word that means knowledge or awareness. Conocimientos the plural of conocimiento, refers to a theory and will always appear in its plural form throughout.
9
American Chapter (PMENA-38) Sin Fronteras: Questioning Borders with(in) Mathematics
Education, plenary speaker Sandra Crespo (2016) spoke on the need of critical consciousness in mathematics while suggesting to the audience that everyone should read the work of Paulo
Freire. Freire’s work has been influential in much of the “social justice” literature in mathematics education (Stinson & Wagner, 2016). Further exploring the work of Paulo Freire, we see a gap in mathematics education in understanding how mathematics contributes to an individual’s development of critical consciousness (Freire, 1973). Critical consciousness requires a socio- political-historical understanding of the world, with one’s understanding of their own human agency in contributing to the transformation of the world. Being critically conscious is not a mindset that once achieved is permeant; it is a constant process and understanding of an elastic world (Freire, 2005). Consciousness in mathematics is not well researched, but multiple researchers have explored identity development in mathematics education (Aguirre, 2013; Berry,
2011, Nasir, 2002; Woods, 2013). Where, if we take identity as the manifestation of consciousness, we can then begin to gain a deeper understanding of identity by exploring how consciousness let alone a critical consciousness is developed in the learning of mathematics.
In investigating how students’ consciousness shifts in the process of mathematical learning, I will utilize the theory of conocimientos (Anzaldúa, 2012), as it represents a form of conscious raising knowledge (Reza-Lopez, 2016). This dissertation will explore how consciousness is developed in the learning of mathematics for social change. Consciousness ranges from the rational modes of thinking related, to the natural world, and to imaginative modes of consciousness related to fantasy and the intangibles of the natural world (Anzaldúa,
2009) where consciousness is complex with multiple modes of consciousness shifting and overlapping between rational and imaginative modes (Anzaldúa, 2009). In order to understand 10 consciousness, the seven stages of conocimientos (Anzaldúa, 2004) will be used to represent one way of understanding how consciousness develops and shifts towards action upon the world
(Keating, 2016). In understanding how learners engage with mathematics with YPAR, this dissertation will explore the epistemological, ontological, and axiological nature of mathematics towards creating spaces where mathematics can be part of self, social and mathematical transformation in the development of critical consciousness. The epistemological view of mathematics takes mathematics as a living construct similar to Mathematx (Gutierrez, 2017).
Specifically, my focus will be on the ontological nature of mathematics, which will be developed by focusing on relationships made while learning mathematics and the axiological nature of mathematics, which will explore the ability to create entire mathematical experiences of self- exploration.
This dissertation will place emphasis on the individual aspects of a students’ identity and how those identities change while learning mathematics. Identity will be defined as the manifestation of consciousness, where consciousness is the awareness of self, others (not-self), and the world (self plus not-self). In other words, identity is the observable action, and consciousness is the history and lived experience that leads to identity being developed, along with any hopes, thinking, or projections for the future. The overall goal of this dissertation is to understand how consciousness is developed while learning mathematics towards understanding social injustices related to mathematics and the world. The priority of the dissertation was to create a space for young people to explore mathematics while engaging in YPAR. The conceptual framing for this dissertation will pair critical literacy with spiritual activism
(conocimientos), where the seven stages of conocimientos will act as the analytical framework for exploring mathematics as a relationship derived from viewing mathematics as a living entity. 11
Furthermore, the lifework behind this undertaking is intertwined with the (Y)PAR EntreMundos
Epistemologies. The two main research questions for my dissertation are:
1. How can mathematics be an experience within youth participatory action research?
2. How do high school students experience the stages of conocimientos while engaging in a mathematics-based youth participatory action research EntreMundos project during a summer program?
Thus, a (Y)PAR EntreMundos project (Ayala, Cammarota, Berta-Avila, Rivera, Rodriguez &
Torre, 2018) will be lived focusing on mathematics as a critical literacy (reflection) toward social
(mathematical) action. In education and educational research, YPAR is inherently a praxis approach as it empowers youth in developing a critical consciousness taking research to action/practice (Cammarota, 2018; Izzarry, 2009). A mathematical YPAR EntreMundos project provides a rich experience in understanding relationships between individuals and mathematics.
As mentioned earlier, research on critical consciousness development in mathematics education has not been thoroughly researched, but we cannot gain an understanding of consciousness in the learning of mathematics without first understanding what mathematics is epistemologically, ontologically, and on the axiological level of knowledge construction. Conocimientos provides an interesting intersection in that critical consciousness cannot be formed without multiple conocimientos. Conocimientos allow for the hybridity needed to understand both consciousness and the complexities of mathematical learning. Where without fully understanding what mathematics is, any attempts may further perpetuate social injustices (Medina, 2013) within mathematics education. Thus, by bridging YPAR with mathematics, an exploration of mathematics as praxis or what I call mathematical praxis allows for a more robust understanding of mathematics. A mathematical praxis requires mathematical reflection and mathematical action towards social transformation with the mathematical agency of knowing anyone can do math. A 12 mathematics (Y)PAR EntreMundos project simultaneously allows and welcomes a new look at the nature of mathematics and its role in youth developing a critical consciousness
In relation to mathematics education research, this dissertation simultaneously expands
Gutstein's (2004) reading and writing the world with mathematics (RWWM) and Gutierrez
Mathematx (2017) by diving deeper into the epistemological underpinnings of both. RWWM pulls directly for reading and writing the word (Freire & Macedo, 2004), and Mathematx pulls form paralleling indigenous cosmologies. Notably, nepantla is one of the three theories behind
Mathematx and is also one of the stages of conocimientos. This dissertation, which will work on constructing what I call conocimientos (with|in)4 mathematics, an ontological notion of mathematics as a relationship and axiological departure towards mathematics as an experience, noted as (C|M). More so, I will use the stages of conocimientos as an analytic framework in observing how conocimientos and mathematics manifests both physically and metaphysically within a YPAR collective to further explore (C|M). Specifically, the underpinnings of PAR-
EntreMundos, described later, will guide the dissertation investigating the following sub-research questions of the dissertation:
“How can mathematics as a critical literacy and critical action lead towards a revolutionary praxis?”
“How can mathematics be a conscious raising experience?”
To answer how can mathematics as a critical literacy and critical action lead towards a
4 Within, will be written as "(with|in)" for two reasons. The first reason pulls form conditional notation in mathematics where "A" given B" is written as (A|B). Therefore, when I write conocimientos (with|in) mathematics or (C|M), I am saying conocimientos first given mathematics and (M|C) represents mathematics first given conocimientos. The second reason like Anzaldúa's (2015) theory of nos/otra where the "/" represents the bridge that divides "us" from "others" is to highlight that conocimientos and mathematical learning are separated, and the "|" in (C|M) represents the multiple barriers need to have a true vision of conocimientos within mathematics. 13 revolutionary praxis, I documented the mathematical knowledge construction done throughout the summer YPAR program called REALM. The documentation of this project is significant because a study of its kind has not been done and will add richness to current youth studies literature as well to mathematics education research. Chapter 5 consists of a narrative that will show it is possible mathematics to be infused in all parts of YPAR. The two sub-research questions will be used to answer the second main research question. The first sub-question will explore mathematical praxis, where mathematics will be used to better understand society and critical theory in leading to action that addresses systemic inequities in society. Such a balance of action and reflection within mathematical knowledge construction is a revolutionary praxis that represents the epistemology underpinning of YPAR. Chapter 6 contains the results of using the stages of conocimientos as an analytic framework to show how consciousness shits throughout
REALM. In Chapter 7, the discussion, the power of viewing mathematics as alive will show how
(C|M) withing (Y)PAR EntreMundos is possible and transformative. This dissertation ends with a discussion guided by the two poems "An Introduction to REALM" and "Mathematics Me –
Therefore You" that represents my reflection on engaging in this work.
14
CHAPTER 2. Literature Review
It is important to begin with a brief overview of critical pedagogy and how it is connected to mathematics education. Further exploring critical pedagogy places emphasis on how/what knowledge is co-constructed when interacting with youth and arrives at conversations on critical consciousness in how it has (or has not) manifested in mathematics education. A focus on multicultural mathematics and Ethnomathematics will be examined prior to a review of "social justice" in mathematics education. Insofar to say, looking at the nature of mathematics requires looking at the nature of the self, or in other words, identity development in the learning of mathematics is important and will be reviewed.
Critical Pedagogy and Mathematics Education
The brief overview begins with critical pedagogy in the United States and various definitions of critical pedagogy, followed by the philosophical principles of critical pedagogy and their brief connection to mathematics education. Duncan-Andrade and Morrell (2008) provide a vision of critical pedagogy in relation to its Freirean roots by stating, "critical pedagogy … begins with the problem of dehumanization and seeks, through dialogue and praxis, to develop an individual who is able to participate in the transformation of society (p.134)" this emphasis of dehumanization aligns with current trends in mathematics as seen in the 2018
National Council of Teachers of Mathematics Annual research perspective, Rehumanizing mathematics: Rehumanizing mathematics for Black, Indigenous, and Latinx students (Gutiérrez,
Goffney, & Gutiérrez, 2018). Critical pedagogy is a broad term, therefore exploring the philosophical principles gives us a deeper understanding of the epistemological and pedagogical roots of critical pedagogy. Focusing on philosophical principles, described later, makes it easier to identify pockets of critical pedagogy within mathematics education. 15
In the United States educational context, critical pedagogy began with the publications of
Paulo Freire’s work, which was translated into English in 1970 (Darder, 2017). Particularly, mathematics and critical pedagogy have a long history even though mathematics education research rarely uses the term critical pedagogy. We see this in Marilyn Frankenstein’s (1983) article Critical mathematics education: an application of Paulo Freire’s epistemology, which calls for a focus on dialogue and a problem posing education towards developing a critical consciousness. Where critical pedagogy in mathematics education took the form of critical mathematics literacy (Aslan Tutak, Bondy & Adams, 2011). The influence of Freire did not stop there, as Gutstein (2006) used Freire and Macedo's (2005) literacy framework to construct
Reading and Writing the World with Mathematics (RWWM), a critical literacy lens of/for mathematics. RWWM centers on the role of mathematics in understanding the current socio- political construction of the world by understanding how the world came to be (Gutstein, 2006;
2017). Critical mathematic literacy is important as it allows for an examination of the power and privilege, inherently, in mathematics education (Frankenstein, 1989).
Returning to critical pedagogy, it was a term first used by Henry Giroux, a contemporary and friend of Freire (Kincheloe, 2007). Giroux (2011) describes critical pedagogy as a way to analyze and question the social agency of learning, asserting that students can actively participate in how and what they learn. This is done by critiquing and analyzing official levels of power.
Through critical pedagogy, students and teachers question the school curriculum along with national educational policies that act to perpetuate domination through a pedagogy of control. A central role of critical pedagogy is the creation of knowledge from both the teacher and the learner. Ira Shor (2012) emphasizes this relationship between teacher and learner in what he calls democratic critical pedagogy, where the teacher no longer acts as the authoritative holder of 16 knowledge but works to create “contact zones” or what Vygotsky (1980) calls zones of proximal development, where teacher and students learn from each other, as places to construct their identity. As such, teachers transform into a democratic authority that questions the status quo and work with students in negotiating the rules of how and what everyone learns (Shor, 2014).
Examples of critical pedagogy are exemplified in YPAR and will be discussed later.
The philosophical principles of critical pedagogy according to Darder (2017) are (a) cultural politics or cultural representation of schools; (b) political economy; (c) historical knowledge; (d) dialectical theory; (e) ideology and critique; (f) hegemony; (g) resistance and counter-hegemony; (h) praxis: (i) the alliance of theory and practice; and (j) dialogue and conscientization Cultural politics is a commitment to supporting the empowerment of those who have been culturally marginalized and economically disenfranchised. In mathematics education this principle can be seen in the work of multiple scholars (e.g., Averill, 2009; Bonner, 2009) who focus on culturally responsive teaching in mathematics education to incorporate and recognize the importance of including a student’s culture in the learning process (Ladson-
Billings, 2014). Political economy contends that schools work to continue to perpetuate the status quo, working against the interests of the politically and economically oppressed. In the book Rethinking Mathematics (Gutstein & Peterson, 2005), Andrew Brantlinger created an activity that compares the number of community centers to the number of liquor stores in various cities, hinting at economic resources of the community. The historicity of knowledge is when knowledge is being created within a historical context that gives meaning to the human experience. Related to the human experience is a dialectical theory which "embraces a dialectical view of knowledge that functions to unmask the connections between objective knowledge and the cultural norms, values, and standards of the society at large" (Darder, 2017, 17 p.11). The historicity of knowledge and dialectical theory are two principles that are not heavily researched in mathematics education outside of Ethnomathematics, which continues to be at the margins of mathematics education research in the United States. Ideology and critique center the role of whose ideology is being shared within the current educational curriculum. Hegemony is when a dominant group uses its power to assert its leadership (control) over subordinate groups to maintain superiority (Gramsci, 1971). Counter-Hegemony and resistance look to examine hegemony through resistance to transform classroom power relationships making central the voices of the learner. An example of counter-hegemony can be seen in the work done regarding teaching mathematics to emergent bilinguals in support of cultural and linguistic differences as intellectual resources (Celedon & Ramirez, 2008). Praxis is the balance between action and reflection for social transformation or liberation (Freire, 1973). Regarding praxis: the alliance of theory to practice, it is the idea that both theory and practice are linked in understanding the world. Dialogue and conscientization are both ideas pulled from the work of Paulo Freire and are significant aspects of critical pedagogy (Darder, 2017). For Shor and Freire (1987), it is only through dialogue that questions can be posed, discussed, developed, and further questioned to facilitate authentic moments of learning. Dialogue itself is a form of praxis, a balance of reflection (thinking what will be communicated), and action (the act of communicating). The development of questions and the act of questioning the world is a requirement of conscientization. Conscientization is the process where an individual is socio-culturally- politically aware of societal issues (requires a historical understanding), along with the awareness of individual human agency and the ability to act upon and change societal issues.
Conscientization is synonymous with critical consciousness. As mentioned earlier, little to no research has been done in understanding how consciousness is developed within mathematics 18
education. Vital to developing critical consciousness is dialogue and communication; it is dialogue and the role of communication in mathematics education that has not been fully researched. Yes, substantial research has been done regarding mathematical discussion (Smith & Stein, 2011), but discussion is not always dialogue and rarely reaches a point of praxis. For example, in dialogue, learners question the task before taking part in a mathematical discussion and continue to question not only the final answer but the context of what is learned. Back to mathematical discussion, it is a form of communication that results in an answer, not the communication of mathematics to those outside of the classroom. In so far, a mathematical dialogue is communication that critiques the final answer (Vithal, 2012). Additionally, substantial work has been done around discourse in mathematics education (see: Herbel-Eisenmann, Choppin,
Wagner, D., & Pimm, D., 2011), but little work has focused on directly on any of the principles of critical pedagogy.
Feminism Pedagogy and Critical Pedagogy
Feminist pedagogy must be mentioned and acknowledged due to its interconnectedness with the life/work of Gloria Anzaldúa (Moraga, 2015) and its inherent similarities to critical pedagogy. Insofar, the work of Anzaldúa is the main inspiration and theoretical underpinnings of this research. Seminal to women of color feminisms is the book The bridge Called my back by
Cherríe Moraga and Gloria E. Anzaldúa, in that is centered voices in the margins. "The bridge called my back furthermore countered white "mainstream" feminisms, also known as whitestream feminisms that placed a focus on the white middle-class experience. Whitestream feminism has failed "to acknowledge both the intersections of race, class, gender, and sexuality and the historic dispensations of whiteness" (Grande, 2003, p.330). This contrasts with a Black
Queer feminist pedagogy (Lewis, 2011), a re-centering pedagogy that focuses on the teaching of intersectional realities through interdisciplinary practices. Mel (2011) states, "[m]y identity 19 informs and constructs the classroom both in its difference from expected teaching identity, and its creative pedogeological power (p. 2011)." Understanding difference allows for the understanding of similarities (Anzaldúa, 2009). In the book The bridge we call home, a follow up to The bridge called my back, Anzaldúa and Keating (2013) explore the transformative power of difference towards transformation and interconnectedness. Another contrast to traditional feminisms comes in the form of a Chicana feminist pedagogy that looks at multiple identities around issues of domination and resistance around race, class, gender, and sexuality (Elenes,
Gonzales, Bernal, & Villenas, 2001) while incorporating indigenous perspectives and/or a mestiza consciousness focusing on the resiliency of self and the collective power of being part of a larger group (Elenes, 2002). The word mestiza is a term that evolved to be a form of Chicana consciousness, quoting Bernal, it is a consciousness "that straddles cultures, races, languages, nations, sexualities, and spiritualties – that is living with ambivalence while balancing opposing powers" (Elene, 2002). In discussing how to use a Chicana feminist epistemology, towards a pedagogy, Bernal (1998) mentions the importance of history, of whose history and how history is erased to address the importance of identifying and addressing colonial forces. Chicana epistemologies represent a lineage of how Anzaldúa impacted education (Pérez, 1999). Attaining to learners Marta Civil (2014) states “pretending that we can “improve” marginalized students’ mathematical learning opportunities without taking their lived experiences is educationally naïve at best” (p.11) which requires an intersectional lens of seeing students to fully understand students and their agency to change the colonial nature of mathematics (Bishop, 1990).
Feminism pedagogy and critical pedagogy both share the goal of humanization. Critical pedagogy does this by examining society structurally and historically through resistance. A feminism pedagogy focuses on understanding the multiple identities of the individual and works 20 to explicitly recenter those marginalized by society. An intersectional view of multiple identities allows us to see gender, sex, and sexuality as different constructs, which is central in feminism pedagogy, and more recently, it is being incorporated within critical pedagogy (Darder, 2017). In short, a feminism pedagogy can be viewed as a further development of critical pedagogy, but critical pedagogy cannot always be a feminist pedagogy. Critical pedagogy and feminism pedagogy both seek to liberate and humanize where a feminism pedagogy explicitly begins from the perspective and lived experience of women (those at the margins of society). Both feminism and critical pedagogy are concerned with power, its history, and resistance. The difference is critical pedagogy's early development did not consider multiple forms of identity, nor did it focus on the role of sexuality in society or the recentering of individuals in the margins.
Spiritual Activism and Conocimientos
Spiritual activism,
at the epistemological level, posits a metaphysics of interconnectedness and employs non-binary modes of thinking. …At the ethical level, spiritual activism requires concrete actions designed to intervene in and transform existing social conditions. Spiritual activism is spiritually for social change, a spirituality that recognizes the many differences among us yet insists on our commonalities and uses these commonalities as a catalyst for transformation. (Anzaldúa, 2015, p.323)
Spiritual activism, like critical pedagogy, advocates for action to transform and change current societal issues (Keating, 2008), and like feminism pedagogy, spiritual activism focuses on the multiple identities of the individual. Specifically, non-binary modes of thinking position people to think past labels and respect future unknown labels. It is non-binary modes of thinking that can connect critical pedagogy, feminism pedagogy (and future pedagogies) by refusing to separate them and realize both are being engaged along with other forms of resistance. Spiritual activism focuses on changing both the individual and society while learning how they are interconnected, rooted in indigenous cosmologies. 21
The theories of Autohistoria-teoria, nos/otras, nepantla, conocimientos, and new tribalism/El Mundo Zurdo are five culminating theories of Anzaldúa's work that come together as the epistemological underpinnings of spiritual activism with conocimientos being her final work before her death, that best encompasses the complete vision of spiritual activism (Keating,
2002). An Autohistoria-teoria is a technique of spoken-word-art-performance; it is a personal story that theorizes that can be part fiction, part theory and part lived experience (Keating, 2005).
Theories of nos/otras signify the constant dividing of an individual's consciousness with respect to others. The separation between self and others serves to emphasize the multiple forms of consciousness inside each individual along with the need of others, to make meaning of one's self (Keating, 2006). With the "/" in between nos (us) and otras (others) representing the bridge that still needs to be crossed to be nosotras (we) in order to begin to heal the fragmented self and unrealized world (Anzaldúa, 2013).
Nepantla, a Nahuatl word meaning "in-between space" for Anzaldúa (2006), represents a transitional stage made from the clash of worlds both physical and metaphysical. In general, life and learning are not a static process; therefore, we are always in nepantla; people are always learning and traveling between worlds or spaces. Nepantla signifies all forms of interaction and exchange/development of ideas. New tribalism represents an affinity-based approach to identity formulation by making alliances, which is different from assimilation and separatism. Alliances are central in creating shared meaning and knowledge. The manifestation and convergence of consciousness embody new tribalism and the collective actions taken to create a shared world. El
Mundo Zurdo translates to the left-handed world, and represents the interconnected difference needed to make revolutionary change (Keating, 2006) and can be viewed as the other half new tribalism. 22
Conocimientos consists of seven stages that result in a shift/change of consciousness
(Reya-Lopez, 2014). An individual must reach or experience all stages to develop conocimientos, a change in consciousness. The seven stages act as a path, with each stage providing guidance or the insight needed to internally and externally transform. In developing a particular stage of conocimientos, it is possible to develop/discover a different or multiple conocimientos (Anzaldúa, 2004). Conocimientos are based on social constructions of the world, and for that reason, conocimientos are a continual process. The seven stages rested in and outside of nepantla, and provide one way of understanding how all of Anzaldúa's theories lead to a consciousness of spiritual activism. The five theories guiding spiritual activism are non-linear, overlapping, and can, at times, be independent of each other. This can be seen in nepantla being a standalone theory as it can describe a conversation between two people that does not change the consciousness of either person. If the same conversation leads to a change or shift of consciousness, then nepantla as a theory overlaps and extends into conocimientos. More so, if the conversation is about one's lived experience, it enters the realm of Autohistoria-teoria. If collective understanding is made, then new tribalism and El Mundo Zurdo are engaged.
Anzaldúa's theory of spiritual activism and each of the five theories mentioned earlier
(used either individually or collectively) have influence countless feminist scholars laying the groundwork for Chicana feminism (Bernal, 1998; Elenes, 2001; Sandoval 2000) such that we can connect spiritual activism to both feminism pedagogy and critical pedagogy. Furthermore, in order to reach spiritual activism, multiple conocimientos are required. Multiple conocimientos are required to attain a critical consciousness that is inherently implied in the epistemological roots of spiritual activism and further links to critical pedagogy. The seven stages of conocimientos are a non-linear, non-sequential, and cyclical (both a cycle and/or repeating) 23 process of consciousness that are described in Table 1. Sub notation: C1, … , C7 will be used to abbreviate and emphasize the seven stages in the table. Conocimientos, the continual transformation, and shift in consciousness allow for the bridge between nos (us) and otras (them) to be repaired with others, from alliances made in El Mundo Zurdo, arriving at a vision of spiritual activism.
Critical Consciousness
Critical consciousness is defined as a socio-historical-political-cultural understanding of society with the self-awareness of being able to contribute to change in society (Freire, 1970).
Critical consciousness is an elastic concept given society is constantly in flux (Freire, 1970;
2000; 2018). Prior to the development of critical consciousness, is what Freire (1973) calls a transitional consciousness. Given a critical consciousness is not permanent, those with a sense of critical consciousness are always in-between transitional and critical consciousness. Individuals not seeking critical consciousness are said to have a naïve consciousness (Freire, 1973) because they have been rendered objects due to dehumanization (Freire, 1970). Relative to developing critical consciousness are ideas of praxis, critical literacy, and conocimientos. Furthermore,
Freire and Macedo (2005) provide a critical literacy approach in reading and writing the word
(described later), which inherently allows for a socio-historical-political-cultural understanding of the world. The praxis behind reading and writing the word is a revolutionary praxis (Freire,
1970) that requires a critical consciousness. Related to a transitional consciousness, Anzaldúa
(2004) offers conocimientos, seven stages of how consciousness changes (described later) that provide a more refined way of mapping transitional to critical consciousness development.
Understanding more than one form of mathematics provides an expansion to develop a critical consciousness around mathematics, with one such example being ethnomathematics. 24
Ethnomathematics
Ubriatan D'Ambrosio began his program ethnomathematical in the 1970s, and the term ethnomathematics was born as a mathematics of the third world that would transform into non-
Eurocentric forms of mathematics (Gerdes, 2001) connecting mathematics to culture
(D'Ambrosio, 1998). The initial rise of Ethnomathematics can be seen in the formation of the
International Study Group on Ethnomathematics in 1985 (D’Ambrosio, 2006), where we see varying definitions of/for Ethnomathematics. The various interpretations of Ethnomathematics are welcomed according to D’Ambrosio because culture is always in never static, requiring it to constantly be changing (2001). Pulling from multiple ideas, Powell and Frankenstein (1997) define Ethnomathematics,
to include the mathematical ideas of peoples manifested in written or non-written, oral or non-oral forms, many of which have been either ignored or otherwise distorted by conventional histories of mathematics (p. 9).
Furthermore, Borda (1990) reminds us that "ethno" in Ethnomathematics refers to cultural groups, not concepts of race, and mathematics represents mathematical activities like measuring, ordering, and modeling. The mathematics within Ethnomathematics would be the common axioms found in all forms of mathematics (Bishop, 2002). In simple terms, "ethnomathematics is used to express the relationship between culture and mathematics" (D’Ambrosio, 2001, p. 308).
By then focusing on the relationship and how it connects culture to mathematics, it shows the complexity and richness of Ethnomathematics (Moschkovich, 2012). Furthermore,
taking an ethnomathematics stance means seeing student mathematical activity in the classroom not as a deviant or novice version of academic or school mathematics but instead as a case of students' everyday activity, where participants use social and cognitive resources to make sense of the situation (Moschkovich, 2012, p. 9).
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Identity, Mathematics, and Dehumanization in Mathematics Education
Learning how to socialize is inherent in learning mathematics because teaching mathematics is a political act that impacts students understanding of the world (Gutierrez, 2002;
2018). Knowing that mathematics is part of the socialization and identity development of learners helps educators identify factors that contribute to students’ futures (Jackson, 2010).
Insofar to say, the connection between identity and learning is not an unknown relationship
(Martin, 2010). Wherein utilizing the identities of students in the learning of mathematics has a growing body of research (Aguirre, 2018; Celedon & Ramirez, 2012; Civil, 1998; 2014; Martin,
2010; Wood, 2013) but additional research is still needed as Martin (2011) states, “a focus on learning mathematics while Black is itself incomplete without a meaningful integration of identity-related issues and knowledge-focused concerns (p.58).” Understanding race in mathematics is only one aspect of identity that needs further exploration. Specifically, Aguirre,
Mayfield-Ingram and Martin (2013) “define a mathematics identity as the dispositions and deeply rooted held beliefs that students develop about their ability to participate and perform effectively in mathematical contexts and to use mathematics in powerful ways across the context of their lives” (p. 14). Although students' identities need our attention, we must also consider the role of the teacher and how their identities impact students. Specifically, if we do not pay attention to students' identities, then learning becomes a dehumanizing experience (Gutierrez,
2018, p. 3) where students are never fully seen or heard and where teachers cannot fully meet the needs of students (Civil, 2014). Teachers (and researchers) need to focus on the development of young learners by paying attention to identity, which should be part of the daily life work of teachers (Aguirre, 2013).
The idea of humanization and dehumanization stems from the work of Paulo Freire regarding dialogue, and the learning process is known as a problem posing education, a 26 humanizing approach, and banking education, a dehumanizing approach (Freire, 1996; 1973;
2000). A dehumanizing approach to education silences students, ignoring their identities and views students as mere objects. Shifting to mathematics education, Gutierrez (2018) offers eight dimensions of humanizing practices for mathematics educators with ideas of participation/positioning, cultures/histories, creation and ownership, all having a direct connection to identity development (Nieto, 2018). Furthermore, a full discovery of identity and humanization allows for individuals to discover or (re)discover their own agency and allows for the development of a critical consciousness (Shor & Freire, 1987). As previously mentioned, viewing the identities and even roles of teachers and students as interchangeable not only negotiates the power of the classroom (Shor, 2004) but also echoes Fasheh (1982) in that,
[t]here is a price for teaching math in a way that relates it to other aspects in society and culture which may result in raising the "critical consciousness" of the learners. And that price that the teacher usually pays varies directly with the power of authority and with the effectiveness of the teacher. The fear of paying the price is one main factor, in my opinion, that diverted education from its "natural" course and forced it to take detached and meaningless forms (p. 288).
Equity, Mathematics and “Social Justice”
Many students feel like they need to leave aspects of their identities outside of the mathematics classroom, but it is critical to create spaces where students’ identities are positioned as valued intellectual strengths (Jilk, 2012). A student’s identity is influenced by their culture, and this connection to culture reflects the need for equity in mathematics. An early step towards equity in mathematics curriculum was seen in Ethnomathematics challenging one-dimensional
Eurocentric views of mathematics education curricula across the world (Bishop: 1998; 2002).
The curriculum has control over learning in classrooms, as Valenzuela (2016) states,
27
every day in America, children are still required to “leave” home, albeit symbolically, because of the culturally chauvinist curriculum to which they are routinely subjected, and most typically by teachers and school systems that systematically fail to construct a meaningful educational practice out of students’ languages, cultures, community-based identities, or real-world experiences (p.5).
Another step for equity work in mathematics education can be found in the book Mathematics for Equity, a framework for successful practice (Nasir, Cabana, Shreve, Woodbury, & Louie,
2014). The book represents a case study of a school is successful in creating an equitable, high school mathematics learning space on the following core principles; all teachers are learners, a focus on student’s strengths and making spaces to be vulnerable, redefining what it means to be smart, redefining what it means to do math in school, and the importance of relationships. Equity in mathematics is more than just including multiple cultures into the curriculum; equity in mathematics is about creating spaces that contribute to the collective knowledge of the community (Barajas-Lopez & Bang, 2018). The act of creating spaces moves the conversation of equity in mathematics towards efforts of “social justice” in mathematics. Mathematics can then be used to connect to cultures by showing how multiple cultures arrive at the same axiomatic mathematical principles (Bishop, 1999). Attempts to define "social justice" in mathematics education can recently be seen in a joint position statement by the National Council of
Supervisors of Mathematics and TODOS: Mathematics for ALL in connecting "social justice" to eliminating deficit view on learning mathematics, to mathematics as a gatekeeper, the need of professional developed on mathematics and social justice, and engaging in a sociopolitical turn in acknowledging the socio-culture-political world (NCAM & TODOS, 2018). Attempts to enact
"social justice" can be seen in the Association of Mathematics Teacher Educator's (AMTE’s)
Collective Actions for AMTE to Develop Awareness of Equity and Social Justice in
Mathematics Education (AMTE, 2016). Mathematics and "social justice" is a relatively new area 28 while identifying injustice and calling for change has a rich neglected history, take the example of Ethnomathematics at its core being decolonial mathematics.
Critical Literacy, Mathematics, and Critical Consciousness
Mathematics as a form of critical literacy can best be summarized by Gutstein’s adaptation of Freire and Macedo’s (2005) Literacy: Reading the word and the world framework that Gutstein (2008) calls reading and writing the world with mathematics (RWWM). Freire and
Macedo’s framework is a way to understand the world by understanding the previous world through critical literacy (Freire, 1998). Critical literacy is not just the ability to read and write. It is the ability of communication and reflection inherent in being able to read and write (Freire,
2000). The ability to reflect and communicate is a necessity for revolutionary praxis. Gutstein
(2008) developed RWWM and incorporated the role of mathematics as a way to read the current and previous world.
Figure 2.1
Reading and Writing the World with Mathematics (Gutstein, 2016) 29
Figure 2.1 shows the role of mathematics in reading the previous reading of the world, also known as reading the mathematical word and how it leads to writing the mathematical world
(Gutstein, 2016). A “[c]ritical mathematics literacy enables the oppressed to use mathematics to accomplish their own ends and purpose” (Lenorard, 2010, p. 326).” Figure 1 provides such an example where reading the world is the process that engaged mathematics as a critical literacy in knowing how to use mathematics to read the word (understanding history) in reading the world with mathematics. More so, reading and writing is a cyclic process because understanding the word (reading) leads to action (writing) upon the world. Being able to read and write the world within a critical literacy paradigm leads to empowerment and understanding of one's own agency
(Freire, 1973; 1998; 2000). Knowing how to read the world and understanding one’s agency relates to critical consciousness and (re)humanizing mathematics. Where a critical mathematics agency requires individuals to see themselves as capable and powerful mathematical learners that have the option of participating in mathematics in both personal and social ways (Aguirre, 2013).
Without critical mathematics literacy and the understanding of identity, an understanding of a critical mathematics agency is impossible.
Youth Participatory Action Research and Mathematics
YPAR is a form of participatory action research (PAR) where the participants are youth.
PAR has a rich research history (Fals-Borda, 1988; Whyte, 1991; Brown, 2009; Cammarota &
Fine, 2008; 2018: Ayala, et al., 2018), where “[a]n immediate objective of PAR is to return the people the legitimacy of the knowledge they are capable of producing through their own verification systems, as fully scientific, and the right to use this knowledge” (Rahman, 1991, p.15). A goal of YPAR is youth empowerment, where “YPAR teaches young people that conditions of injustice are produced, not natural; are designed to privilege and oppress; but are 30 ultimately challengeable and thus changeable” (Cammarota & Fine, 2008, p. 2). YPAR seeks to empower youth by viewing the research process as more than a method and a methodology but as primarily a critical epistemological commitment (Cammarota, 2017) or, in general, by being a philosophy of life (Rahman & Fals-Borda, 1991). As for YPAR and mathematics, currently, only two papers directly connecting mathematics and YPAR exists. The first paper titled
Mathematics, critical literacy, and youth participatory action research (Yang, 2009) shows how youth can use statistics within YPAR where “mathematics was more than just a tool to critique social inequalities. It was also a sociocultural activity through which to examine complex codes of power, identity, and culture in human society” (p.144). The second paper titled, Striving toward transformational resistance: youth participatory action research in the Mathematics
Classroom (Raygoza, 2016) highlights the struggles of doing YPAR during traditional class time because of the pressures of meeting standards and other time constraints. Classroom limitation may rob students of the agency of developing their own research questions about their community, which is a crucial aspect of YPAR.
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CHAPTER 3. FRAMEWORK
The theoretical framing of this dissertation begins with (Y)PAR EntreMundos epistemologies to bridge conocimientos (Anzaldúa, 2009) with critical literacy (Freire &
Macedo, 2005). (Y)PAR EntreMundos, conocimientos, and critical literacy come together to revisit the philosophical nature of mathematical learning; to explore how consciousness is developed when learning mathematics as high school-aged youth experience YPAR during a summer program. After discussing YPAR, PAR EntreMundos, mathematical learning, and the conceptual framework of conocimientos within mathematics in reading (reflection) and writing
(action), I will share an autohistoria-teoria (Anzaldúa, 2009) as an epistemological extension to the theoretical groundings of this dissertation study.
(Y)PAR EntreMundos Epistemologies
YPAR is, first and foremost, an epistemology, a life-choice and commitment to revolutionary action upon systemic inequities of society (Cammarota & Fine, 2008: Fals-Borda,
1991). The action component of YPAR comes after youth generate and research their own research question(s) that directly affects their lives. The process of conducting and presenting research is an empowering experience for young people as they engage in transformational resistance and develop a critical consciousness (Cammarota & Fine, 2018). Transformational resistance is a form of resistance that requires a commitment to social justice and a critique of social oppression (Solozano & Bernal, 2001).
The epistemology of YPAR is central to empowering youth and creating an educational opportunity to liberate themselves by being owners of knowledge construction. An understating of YPAR as a guiding ideology must be acknowledged for a YPAR study to be a YPAR study. A focus on mathematics throughout the YPAR process still needs to be explored in general because 32
“[m]athematics in youth PAR has been underexplored and is often reduced to the practice of making statistically based statements about injustice. But mathematics can be more than a tool of advocacy. It can lead to increased skills across literate domains” (Yang, 2009, p.116). This study is the first of its kind in situating mathematics as a guiding epistemology alongside (Y)PAR
EntreMundos epistemologies and methods.
As YPAR is an expansion of PAR, PAR EntreMundos is an expansion of PAR by bringing Latinx theorizing to the forefront of what and how PAR is and has been used/lived and theorized (Ayala & Torre, 2009). Insofar,
a PAR EntreMundos seeks personal and collective transformation, engages the richness of our multiplicities, and works from a position of integration, amidst, or perhaps because of choques or conflict… The work of a PAR EntreMundos then, can in part be a way to heal communities and ourselves by “wholing” the fragmentations imposed in us… PAR EntreMundos as a methodology and epistemology transgresses, edge walks, and bridge builds (Cammarota, et al., 2018, p. 30). PAR EntreMundos pulls the work of Fals-Borsa, Solorzano, Bernal, Anzaldúa, and many others, in what Ayala et al. (2018) refer to as the southern tradition, which centers on Latinx scholars that have historically contributed to the development of PAR that is specific to the Latinx lived experiences. The southern tradition fully encompasses the epistemological reach of the theoretical lineages of PAR-EntreMundos. The theoretical foundation of PAR-EntreMundos consists of the southern traditions, critical race praxis, feminist (Latina) theorizing, indigenous ways of being, and critical consciousness of the collective (Ayala et al., 2018). Precisely, PAR-
EntreMundos consists of eight guiding principles, see Table 3.1, that guided the (Y)PAR process in this dissertation (see Chapter 4).
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Table 3.1
PAR-EntreMundos Guiding Principles
Guiding Principle Description
Participation. Practitioners and stakeholders should be involved in all steps of research (design, data collection, analysis, dissemination). Critical inquiry. The work needs to be grounded in critical-race and decolonizing theories that examine the socio-historical, socio-political and material contexts and conditions of our lives.
Knowledge co- Knowledge that informs action is produced in collaboration with construction. communities, where researchers and researched become a collective of knowledge-producers/actors. Power with(in) The collective critically reflects on its own process, fosters trusting relationships of mutuality between members, examines power within the group, and engages in deep self-inquiry.
Indigenous In the spirit of an approach to PAR that is EntreMundos and that grows cosmologies from the southern tradition, we see it as a way to reclaim and reimagine indigenous ways of knowing and engaging in this work as a healing process for the individual and community.
Creative praxes. The methods for collecting and presenting data are embedded in the cultural and creative productions of the local community, which may include poetry, music, dance, theater, and other forms of cultural and artistic expressions.
Transformational There is a commitment to conscious action and social change using action. creative praxes and engaged policy. Concientizacion This work is part of a movement, not simply separate sets of isolated para la colectiva. actions, whose goals include critical consciousness, social justice, and mutual liberation/emancipation from oppression.
Note: Information for this table directly from (Ayala, Cammarota, Berta-Avila, Rivera, Rodriguez & Torre, 2018)
Conocimientos and living spiritual activism represent one way of bringing together the PAR
EntreMundos guiding theoretical lineages. Table 3.2 shows the seven stages of conocimientos.
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Table 3.2
Stages of Conocimientos
Stage Description el arrebato, C1 This is the stage that kicks you out of your comfort zone. A rupture in the timeline that leads to a new beginning, ending, or both. Fragmentation nepantla, C2 Where meaning is made, it is the clash of two worlds/ideas. The difference and distance between physical and metaphysical worlds. Torn between ways.
Coatlicue, C3 This is desconocimiento, reflecting the pain of knowing and need to change. Acknowledging and reflecting on one’s own history and trauma from knowing the truth.
Compromison (el This is the realization that nothing is fixed, and that change. The steps that lead compromise), C4 to internal transformation or the external attempts to learn more about yourself. Acknowledging multiplicity in history, self, others and the future putting This is where you reinvent yourself as a new person. Requires a complete Coyolxauhqui understanding of the old self and the current self in dismantling the difference together, C5 between the two in creating a new self. Shifting from old ways to new ways. the blow up, C6 This is the realization that you are no longer the person you were based on interactions with others. More change is sustained by connections/bridges made with others. Learning from defeat and continuing to grow. shifting realities, C7 A spiritual transformation, being conscious of yourself and others, and the action taken in committing to collective change.
Note the information in this table comes from Gloria E. Anzaldúa’s (2002).
The seven stage’s of conocimientos assist in the philosophical foundation of this study in view of mathematics as an ontological experience. Spiritual activism contribute to the framing of mathematics as a living entity and nepantla, a single stage of conocimientos contribute to the focus of mathematics defined as an experience, see Table 3.3. 35
Table 3.3
Philosophical roots (C|M)
Philosophy Mathematics as… Theoretical Underpinnings Epistemologically Living Mathematx (Gutierrez, 2017) (Knowledge of) Spiritual Activism (Anzaldúa, 2009) Action and Knowledge (Fals-Borda & Rahman, 1991)
Ontologically A Relationship On Culture and Mathematics (Gerdes, 1998) (The being of) Conocimientos (Anzaldúa, 2004) PAR EntreMundos (Ayala et al., 2018)
Axiologically An Experience Ethnomathematics (d’ Ambrosio, 1992) (The value of) Nepantla (Anzaldúa, 1987) YPAR (Cammartora & Fine, 2008)
YPAR creates a rich educational playground where mathematics can be an experience and where youth can empower themselves by challenging both self and world. PAR
EntreMundos provides mathematical learning both the physical and metaphysical space to better understand how relationships can be formed with mathematics. As such, mathematics is transformed into a collective process that solidifies the formation of relationships between self, others, and mathematics. As mathematics YPAR and PAR EntreMundos work together, the epistemological being of mathematics becomes alive as we nurture and respect it only to be treated in return the same way. As previously mentioned, spiritual activism represents non-binary modes of thinking and being towards social transformation that hinges on the interconnectedness of us all (Anzaldúa, 2009). Mathematics can be a bridge in connecting people across the globe.
The interconnected nature of mathematics is revealed when we look at key axioms or structures in which mathematics is built upon (Bishop, 1991).
36
For example, the concept of addition and counting are two instances which are axiomatically always true in mathematics regardless of cultural contexts, how they are communicated is the only difference.
Furthermore, mathematics begins to reveal its collected spirit when understanding how mathematics is developed and built form logical proofs, constructed from previous mathematical knowledge. Therefore, when a new proof or form of mathematics is discovered (based on the work of prior individuals), it must be reviewed and verified by others before it is accepted by all, showing that mathematics in isolation, as a product of only individuals, is not possible. Yet, interject neo-liberalism in education, and the learning of mathematics is transformed into a mechanical clog of the workforce perpetuating individualistic dreams that if YOU work hard enough, YOU can make because of your talent alone.
Individualistic ways of thinking are further imposed on mathematics by K-12 curriculum, which only represents Eurocentric ideals of mathematics, referred to as Western Mathematics
(Bishop, 1990). More so, seeing only one perspective of how mathematics is learned and developed further fragments learners by keeping them away from additional forms of mathematics. The epistemology of (Y)PAR EntreMundos and mathematics represents the readiness and yearning to engage with non-Eurocentric mathematics, more specifically ethnomathematics where ethnomathematics represents the connection between a given culture and that culture’s mathematics (d’Ambrosio, 2001). The collectiveness of PAR that we strive to see in mathematics looks to value the mathematics of all cultures but also aims to connect and grow as a collective sharing mathematical and self-knowledge amongst each other to develop mathematics further, creating an embrace among self, others, and mathematics. Thus, by bridging conocimientos and mathematics, we move away from individualism in mathematics and 37 begin to envision a form of mathematical learning that honors the collective histories of mathematics across the plant. Conocimientos with|in mathematics, one of the main subjects of this dissertation will focus on the experiences and relationships that must be created while learning mathematics to create mathematical experiences of interconnectedness. A mathematical
(Y)PAR EntreMundos looks to further develop, heal, and strengthen the mathematician in each individual. Conocimientos with|in mathematics is for the interconnected social transformation of all of us, and YPAR and PAR EntreMundos provides the theoretical backbone to transform mathematics.
Conocimientos when reading and writing the word
In understanding social and structural problems, Freire and Macedo (2005), developed reading and writing the word, a critical literacy where the reading or knowledge of the world requires the reading of the word where to read the word one must first read the previous reading of the world (historical awareness), see Figure 3.1.
Figure 3.1
Reading and writing the word
Figure 3.1 starts in the center with attempting to read or understanding the current world then moves to left to read the word in gaining historical knowledge. Historical knowledge, along with current understanding (reading the world), then proceeds to the right of in writing new words and worlds (the future). Reading and writing the word leads to a cycle of action and reflection that 38 allows for the development of a critical consciousness because, in order the read the world a socio-cultural-historical-political understanding of the world is need to then, write new words
Freire, 2005). The conceptual framework for this dissertation does not separate the mathematical world and our current world; instead, this dissertation will situate mathematics as the word itself in the reading and writing of the word to better align with the philosophical framing of mathematics as a living entity. Mathematics, as the word itself, where the learning of mathematics is simultaneously an act of learning more about mathematics, the world, and self, allows for the exploration of how consciousness develops while learning mathematics. Insofar to say, the world and words cannot be understood without understanding how the individual fits in said world. The understanding of self is what requires the addition of conocimientos(s) in the process of reading and writing. Where reading represents reflection upon self, others, and the world, and where writing represents action upon self, others, and the world. To better understand how consciousness shifts in relation to self, others, and the world, the seven stages of conocimientos are utilized within reading and writing the word. Figure 3.2 revisits the non- linear, non-sequential, nested nature of conocimientos.
Figure 3.2
Cyclical nor-linear and nested properties of conocimientos 39
In Figure 2.2 the blue arrows represent one possible sequence or cycle, and the purple arrows emphasize the non-linear progression of the seven stage of conocimientos. The red (cycle) highlights the nested properties of conocimientos where, when at any stage, the possibility of a different conocimientos forming is possible. For example, when gaining conocimientos is learning about racism, you can come across an event or case that leads to an understanding of sexism before an understanding of racism. Furthermore, the initial entry stage of gaining conocimientos is arbitrary in that anyone stage can lead to each of the other stages.
The bridge between the conceptual and analytic framework can be seen in Figure 3.3.
Figure 3.3
Conocimientos within Mathematics in reading and writing the word 40
Figure 2.3, the conceptualization of conocimientos within mathematics in reading and writing the word was developed explicitly for this dissertation. The reading and writing of the world cannot be done without reading and writing mathematics (horizontal axis), which leads to the development of a critical consciousness (vertical axis). As mentioned earlier, critical consciousness cannot be developed without conocimientos; hence the stages of conocimientos are embedded within the paradigms of reading and writing the word (mathematics) and levels of critical consciousness. Furthermore, for this study, learning will be viewed through the seven stages of conocimientos in the reading and writing of the word (mathematics), reflecting how consciousness shifts towards the development of critical consciousness.
Returning to the conceptual framework (Figure 3.3), Figure 2.3 shows four arbitrary cases of individual cycles of conocimientos.
Figure 3.4
Arbitrary cases of how conocimientos manifest within the four quadrants. 41
Within all four quadrants (and within every single quadrant), there is an infinite number of possible cases—the arbitrary four cases will be used to explain the framework. Case A, located in quadrant II is situated solely in reading mathematics or mathematical reflection and is representative of mathematical learning emphasizing refection in understanding the socio- cultural-historical world. If the level of reflection in Case A did not represent a socio-cultural- historical understanding the it would be in Quadrant III with Case B. Furthermore, in Case A one of the stages of conocimientos is in Quadrant III a reminder that a single conocimientos does not imply critical consciousness. Case A like Case D are both above and below the horizontal divider between naive and critical consciousness as they represent a transitional consciousness. Case B also represents reflection upon the world, but this refection is not indicative of full critical consciousness. Case B is entirely in Quadrant III, which represents reflection that is still stuck in a naïve consciousness. For example, Case B could represent a change in consciousness and understanding of patriarchy but is still not enough to develop individual agency to be able to dismantle the patriarchy. Case D also lacks the development of a full critical consciousness but begins to reveal more of a commitment to challenging injustices because Quadrant IV like
Quadrant I represents action associated with conocimientos. Quadrant I represents action associated with a transitional consciousness that is closer to a critical consciousness, where quadrant II represents action associated with a transitional consciousness. Case E is indicative of a critical consciousness because it is in both Quadrants I and II, meaning both action and reflection are presents. Naïve, transitional, and critical consciousness across the four quadrants can be seen in Table 3.4. Column two of table 3.3 indicates if a single stage of conocimientos is present in each Quadrant of the framework for this dissertation.
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Table 3.4
Naïve, transitional, and critical consciousness within mathematical praxis
Level of Consciousness Presence of the Stages of Conocimientos
Naïve QIII or QIV or QIII and QIV
Transitional QI or QII or Q1 and QIV or QII and QIII or QI and QII and QIII
and QIV
Critical QI and QII
Autohistoria-teoria-mathematica
This chapter ends with an autohistoria-teoria-mathematica to bridge theoretical knowledge with practical wisdom. This story represents mathematics, axiologically as an experience, in flushing out the seven stages of conocimientos by centering the epistemology guiding the life work and theorizing of this dissertation. The story brings together legend, my lived experience, and the theorizing of mathematics as an experience. The theorizing of my (and any) autohistoria-teoria comes from both the writer and the reader. The story creates pockets of questioning where vagueness allows the reader to envision more of what mathematics should be.
In general, autohistoria-teoria was heavily used by Anzaldúa in establishing her multiple theories, as can be seen in the incorporation of the Aztec moon goddess, Coyolxauhqui, in theorizing how fragmentation leads to identity construction (Anzaldúa, 1999). For Anzaldúa, an autohistoria-teoria is an exploration of identity and consciousness (Keating, 2000). Thus, I share an autohistoria-teoria-matemática, an autohistoria-teoria, about my experience learning mathematics in a third-grade classroom.
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Conocimientos: The Story
Long ago, I came across the story of Quetzalcoatl, a feathered serpent and Aztec god of both the wind and learning, started on a journey living as a man. After walking for days, he became tired due to hunger. Fatigued with water now gone, he knew his death would soon come.
A rabbit grazing in the field saw the man and offered herself as nourishment, sacrificing her life for his. The man (Quetzalcoatl), taken by the rabbit’s choice, honored her by raising her to the moon, leaving her imprint on the surface for everyone to see and be acknowledge. To continue honoring the rabbits’ sacrifice, we shift to her descendant’s, bunnies, baby rabbits, that created a mathematical experience that allowed me to view how conocimientos can live with mathematics and how mathematics can come alive with conocimientos.
My Story
My third-grade teacher, who was also a farmer, created my first mathematical experience by bringing baby bunnies to class. I recall waiting outside of his class, wondering why a normally open door was locked. He then opened the door and told us all to go directly to our seats and not to go to the back of the classroom. In the back of the classroom, we saw multiple buckets and could smell an aroma of nature in the room. The teacher then reached into a bucket and lifted a baby bunny so all could see. So now, in groups, we lifted the bunnies from their place to measure their length and width. We examined baby bunnies’ ears, paws, and overall size. We also took the rabbits weight using a triple beam scale. We did this over what felt like the entire year, but it was probably only at most a couple of months. The teacher placed butcher paper on the walls because every time we collected data, we placed the information on charts we drew on the paper for all to see as we plotted mathematical and animal growth. I remember being excited to take measurements in comparing it with other groups to see who had the fattest bunny and whose bunny was growing the fastest. As a third-grade student, I was excited about 44 collecting and comparing (analyzing) data. Mathematically I was comparing values, making estimates, learning about units and measurement, creating graphs all while modeling growth with mathematics in a real-world context. Twenty years later, I still vividly recall this mathematical learning experience because the numbers were more than just numbers, the bunnies imprinted mathematics in my memory.
This mathematical learning experience mentioned above is a form of mathematics within conocimientos. Now let us look at this mathematical experience from two different perspectives.
The first perspective is represented by an imaginary individual who knows the numerical attributes but is afraid of the context of the activity. Her story is a fear of bunnies. The second story is from the perspective of mathematics being the barrier keeping the individual from engaging in the event; his story is a fear of mathematics.
Her Story
The moment the bunnies enter the room, the class was thrown into nepantla, this is where
Camelia would make meaning of the bunnies’ growth. She, like others in the class, was not comfortable with the small creature because a life that fits in one hand can easily be crushed.
More so, this real-world excursion was not a regular classroom activity; it was el arrebato, a shock to the system. Camelia typically does well in math class, but she is not comfortable holding small creatures because of the fear of causing harm to them. On the first day of the experience, she refuses to lift or even touch the bunny because she thinks it has nothing to do with math. Working in a group and being a third-grader, it is only natural that everyone tells her she should hold the bunny. Luckily, she is not alone, and other students are also afraid of holding the small, fragile creature. By working in a group with others, it allows her to let another student touch the bunny while she does the math. At this moment, the real-world context and Camelia’s consciousness are keeping her from engaging deeply in mathematics. 45
The student acts as a reviewer of data, not a researcher in the process of modeling animal growth. The Coatlicue stage for this individual is her fear and the anxiety of being scared to do something that everyone else does. Next, this individual begins to see other students that were initially afraid to pick up and hold the baby bunnies, and she knows she must change.
Simultaneously she picks up the bunny and passes through the compromision stage and beings to put Coyolxauhqui together, recreating herself to see that nothing is fixed just because the fact that you could not do something yesterday does not mean you cannot do it today. What follows is a new person that can lift not just small bunnies but other creatures. Passing the blow up,
Camelia realizes this experience has shifted her realities because picking up a little, fragile creature has made her gentler, nurturing, and more durable, it has given her the comfort to live and explore the rest of the world with new mathematical knowledge. In this story, mathematics was not the barrier; the context of the problem was. Consistently working with mathematics acts as a catalyst for change allowing for conocimientos in allowing for the learner to engage with the real world.
His Story
Mathematically speaking, let us assume that Vicente struggles with mathematics; for him, el arrebato is not the small creature but the mathematics classroom and what he believes mathematics to be. As he holds the tiny creature, nested in nepantla each week, he avoids seeing the math, yet he notices the bunny is growing waiting to leave an imprint. Vicente learns of the bunnies change and sees the world is not fixed, as he enters the compromision, and wants to know how much/fast the bunny is growing. He knows he must change and learn math because he wants to be able to talk about the small creature when not holding it by explaining how the creature has grown. Additional el arrebato’s appear in class when not measuring the little creature because others are mathematically describing the bunny and making predictions of how 46 much it will grow in the next measurement. Vicente hears students say, “I want to measure its ears in a year.” In silence, he knows he must reinvent himself, but in order to do so, he first must commit to trying to understand mathematics. This is the compromison, and for him to move forward, he must know that it is okay not to know the mathematics he needs to know because mathematical knowledge is not a fixed phenomenon. This reflects the real pain and struggles many students who fall behind in math class encounter as they continue to fall deeper and deeper. Vicente changes as he asks one of his group mates for help because he can no longer be the silent student that sits in the back of the classroom, thinking the teacher has forgotten him and that the teacher is the only one that can teach him. Growing from this valuable lesson of asking for help, Vicente shifts his reality and asks for help in other mathematical topics that he is uncomfortable with. He asks for help because he has learned that mathematics can help him gain a deeper understanding of others and self. Conocimientos began with the context and guided the mathematical learning that started by lifting the bunny allowing for awareness and hunger to learn even more mathematics.
Chapter Summary: Conocimientos and Mathematical Shift
Gaining conocimientos and mathematical knowledge is committing to spiritual activism by focusing simultaneously on the individual self and the world. In mathematics education, the seven stages of conocimientos have not researched with respect to mathematical learning. Ways of utilizing the stages of conocimientos is a current movement with the full theorizing of conocimientos recently appearing in 2013 (Anzaldúa, 2013). An example of conocimientos (and explicitly the stages of conocimientos) in education can be seen in Nepantlera pedagogy (Reza-
Lopez, 2014), which pairs the seven stages of conocimientos with Freire’s ideas of critical consciousness. As described earlier, Conocimientos represent a spiritual interconnected form of knowledge. Pairing mathematics and conocimientos allows for mathematics to become a 47 conscious-raising experience that represents both spiritual and scientific knowledge. This work positions itself in discourse with Gutstein’s RWWM, and Gutierrez’s Mathematx. Mathematx, is an epistemological imaging of mathematics as a living entity, which heavily pulls from
Anzaldúa’s theory of nepantla (Gutierrez, 2017). Mathematx is the epistemological definition of mathematics as a living entity. When conocimientos become part of mathematics, conocimientos within mathematics are the interconnected nature of individual identity and mathematical learning paired with the human agency to transform how mathematics is learned, developed, and used. The bridge between identity development, mathematics, and changing the world is towards what I call a mathematical consciousness, defined as being aware of inequities of the world and engaging in transformational action in relationship with mathematics, to dismantle structural oppression simultaneously changing self, mathematics and the world.
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CHAPTER 4. METHODS
The teaching of mathematics should reflect the social and physical environment when learning, with such an approach augmenting a deeper understanding and imagination (Joseph,
1987). The physical environment is continuously engaged when learning math directly or indirectly, as is the political nature of learning mathematics (Gutierrez, 2018). Furthermore, as
Martin (1997) states, “exposing the links between mathematics and social interests should not be a threat to mathematics” (p.175). In welcoming the connection between mathematics and the social world, the conceptual framework of reading and writing the word (Freire & Macedo,
2005) will be utilized and paired with the conocimientos (Anzaldúa, 2003; 2015) putting an emphasis on mathematics as a way to discover self, others and the world. The methodology of this study lead by the epistemology described in Chapter 3. (Y)PAR EntreMundos is both the epistemology and methodological structure of data collection. Before describing data collection and methods of analysis, it is essential to give a brief history of the program and context because the connections and relationships made to do this dissertation impact the entire study.
History
For the three years prior to this study, I had been involved with El Sol (pseudonym), a
Latinx community-based organization that provides programing for youth on Latinx history and culture in preparing students to become leaders within their community. El Sol provides educational services in eight different Midwestern cities and runs over 32 programs for middle to high school-aged youth. I was a program facilitator for two years in a town that is 50 miles away from the REALM site, where I conducted this dissertation research. Previously, I had started a math-YPAR project in the city 50 miles away that I abruptly ended due to issues not related to the YPAR. The first attempt of a math-YPAR program did not conclude, but it gained the 49 attention of the Director of El Sol. I had continued to discuss the prospect of doing a YPAR project with the Director of El Sol because she saw the value of the work. After much discussion,
I was going to lead a math-YPAR project with El Sol during their standard programming time of once a week after school. While planning and waiting to begin the weekly YPAR project, Sophia
(pseudonym), an El Sol youth, was trying to create a three to four-day writing and science camp for El Sol and La Luna (pseudonym) scholars. La Luna is a AgSTEM college pathways program that provides underrepresented students with a more learning opportunity in STEM starting in the eighth grade while high school La Luna students must meet yearly academic and service requirements to earn a 4-year tuition scholarship at the local state university. Sophia was creating this program as a requirement for La Luna. The Director of El Sol then informed me of Sophia’s ideas, and recommended doing the math-YPAR project over the summer. I then met with Sophia and the Director of La Luna and we decided to do a 10-day summer program where I would have four hours each day to engage students in REALM. REALM represents the first partnership between El Sol and La Luna.
Researchers Role
I was the sole creator and curator of the REALM curriculum, descried later in this chapter. Additionally, the history of REALM, previously described is essential in understanding my role as a participant-observer (Creswell & Poth, 2016) fully engaged with, as I was the sole facilitator of REALM activities. PAR is as much a life choice as it is a methodological approach to qualitative inquiry (Fals-Borda, 1987; Fine, 2008), and as a form of investigation, YPAR brings together and forms collectives between researchers, students, community activists and/or various other community members. More so, YPAR reflects a research commitment towards reflexive science (Burawoy, 1998). Reflexive science “elevates dialogue as its defining principle and intersubjectivity between participant and observer as its premise” (Burawoy, 1998, p. 14) 50 and has four principles; intervention, process, structuration and reconstruction. The first principle, intervention, situates all participants as being biased by being engaged and invested in how the research findings impact the lives of the participants. In YPAR, intervention is given because the researcher is part of the research collective with the youth. The second principle of reflexive science is process and reflects multiple ways of understanding any event where each person can arrive to understanding the event in a different way. The next principle, structuration, holds that we cannot observe everything, and we cannot control all factors. For this reason, it is vital to use a theory to explain unobservable social forces. The final principle, reconstruction, places the priority of the social situation over the individual (Burawoy, 1998) in that this dissertation is guided by conocimientos, a collective process of consciousness development.
Recruitment
To participate in REALM, students had to either be part of El Sol or part of La Luna. The number of participants was set to a maximum of 30 youth, based on my experience with YPAR and my experience as a classroom teacher. El Sol recruited internally via a word of mouth approach where the Director of the program individually informed students about the program.
La Luna had an internal recruitment process where students had to apply to part take in the program. With my assistance, Sophia sent out a flyer giving an overview of the program to La
Luna scholars. All youth that applied were allowed to participate in REALM. Sophia who was part of El Sol for four years and who is part of the community recruited for both La Luna and El
Sol, where all students present the first day of the program, had some form of communication with Sophia. If a student was not part of El Sol or La Luna they could still participate by joining
El Sol on the day they arrived to REALM. Requiring students to join El Sol if they were not part of La Luna or El Sol was strictly done for liability reasons. In total 28 youth signed up for the
REALM, but only 17 participated in REALM with 13 being part of this study. Three youth were 51 part of El Sol, and the rest were part of La Luna, with one individual being active in both programs. Before REALM, none of the youth participated in a YPAR program. It must be noted;
La Luna has a summer requirement of 40 hours of participation in a service or education program, where this program could count for those forty hours.
Context
The study takes place in a predominantly white mid-large midwestern city. Youth were not asked to self identify any demographic information because that is not the focus of the research. All youth were either of Latinx or African descent and represent an ethnically minoritized group in society and STEM fields. All youth are from the same school district, which has 63 schools, including 38 elementary schools, 11 middle schools, five comprehensive high schools, and ten schools that provide a range of specialized and alternative educational programs.
Within those participating in REALM represents five different high schools. The daily meeting place for REALM was a middle school within the school district. This site was selected by El
Sol, through their partnership with the school district. We had two classrooms that we used for
REALM activities. The classes had individual desks, a whiteboard, and a projector. Additionally, the middle school was used by the local Boys and Girls club. El Sol and the Boy and Girls Club were the only two youth groups in the building, but we had no daily interactions. The other individuals we would see were those doing construction throughout the school.
REALM Structure
An overview of REALM can be seen in Figure 4.1, followed by detailed bullet point.
09
Figure 4.1
REALM Daily overview with guiding epistemologies
53
I. Day 1: Establishing group expectations, defining YPAR, and being the formation of a
collection.
a. Mathematicas de las Americas is a presentation that focuses on the mathematics
and science history of the Aztec, Maya, and Inca peoples. The presentation was
created to connect both the guiding principles of indigenous cosmologies and
allow youth the opportunity to reflect on the power with(in).
b. Patolli, the oldest known board game from the Americas, was played to allow
youth to get to know each other (formation of a collective) by playing a game that
allowed them to explore probability, specifically expected values of outcomes.
Additionally, this connects mathematics to indigenous cosmologies.
II. Day 2: Developing a critical lens and identifying generative themes
a. Infographics centered mathematical literacy (knowledge) as a way to develop a
critical understanding of the world while learning about societal injustices.
b. Music, as a creative praxes, was used to help youth reflect on societal injustices
and issues in their math class. Youth were asked what they would do if they were
the president of the country and then if they had complete control of their math
class.
III. Day 3: Developing a critical lens and reflecting on generative themes
a. “Mathematics in the news” is an activity modified after Skovsmose (2011)
activity of have learners identify or find the mathematics in the newspaper. This
activity allowed youth to do critical inquiry of the youth’s communities.
b. Poems were created using an I am Math poem template (see Appendix A) as a
creative praxes to identify generative themes. 54
IV. Day 4 and 5: Developing a research question and researcher training
a. Youth were trained to be Researcher.
i. How to take field notes
ii. How to write observation memos
iii. How to conduct interviews and focus groups
iv. Survey design
1. Open-ended (qualitative)
2. Yes/No, check all that apply, ranking and Likert (quantitative)
v. Survey pilot testing
V. Day 6: Developing a critical lens, collect data and beginning analyzing data
a. Ratios of power activity connects the gender wage gap to the disproportionate
number of men to women in positions of power. The activity shows how ratios
can help in developing a critical lens but also the importance of knowing how to
read and communicate with mathematics.
VI. Day 7: Developing a critical lens and analyzing data
a. Mathematics problems contextually grounded around the Delano grape strike and
the Black Panther Party’s free breakfast program connect history to math.
VII. Day 8: Developing a critical lens, analyzing data and creating reports
a. Comparing slopes of students and teachers of color over the last seventeen years
allowed for youth to see at what rate their school is changing. The activity allows
for developing a critical lens on how to use data to compare variables over time.
In this case, the rate of change of students of color versus teachers of color in their
school district. 55
VIII. Day 9: Analyzing data and creating reports
a. Origins of mathematics, a decision of the oldest known mathematical artifact the
Lebombo Bone, circa 35,000 BC along with the game Gebet’a (also known as
mancala), circa 700 BC, showed the African origins of mathematics.
IX. Day 10: Present reports and reflect on the next steps.
a. Youth presented the graphs, charts, art, and infographics they created.
Data Collection
Table 4.1. shows what data was collected and used to answer the following research
questions:
1. How can mathematics be an experience within youth participatory action research? 2. How do high school students experience the stages of conocimientos while engaging in a mathematics-based youth participatory action research EntreMundos project during a summer program? 2.1. How can mathematics as a critical literacy and critical action lead towards a revolutionary praxis?” 2.2. “How can mathematics be a conscious raising experience?” The data in Table 4.1, excluding teaching memos and teaching slides, was collected as youth
worked in groups during REALM activities. Youth assigned themselves to groups based on their
own interests. All groups were in close enough proximity to hear and speak with other groups.
Each group had an audio recorded placed in or around their working space, and an audio
recorder was set up in front of the classroom to capture whole group discussions. Each youth was
given a notebook, that acted as their journal. After every activity or at the end of the day youth
were given a suggested prompt to address, see Appendix B for each prompt. Youth were told to
write as much as they feel comfortable sharing and that they did not have to address each
question. Youth having the right not to write aligns with the guiding epistemology of the
dissertation. 56
Table 4.1
Data used for each research question
Data Description Research Question Youth Journals At the end of each activity or day (except the first 22.12.2. journal), youth were asked to reflect on what they learned and how the learning made them feel. Before any activity, youth were asked to journal based on prompt “what is math?” and “what do you expect over the next two weeks The journal is where youth wrote and worked on for all but three activities. Group Audio Each group was audio recorded during the entire 12.2 duration of REALM Whole Audio An audio recorder was placed in between me and 12.2 the youth, capturing the audio of the entire room. Teaching After each day, I wrote a memo reflecting on the 122.12.2 Memos lesson. Additional Additional artifacts consisted of activities done 12 2.1 Artifacts outside of the youth journals and consisted of problem trees, I am Math Poems, Newspaper from the math in the news activity and worksheets used for the Gender Ratio activity Teaching Slides The PowerPoints used to teach each day of the 1 program were saved. Final The final products or presentations that each group 1 2.1 presentations of youth created.
Data Analysis
Thematic coding, a rigorous approach in analyzing themes (Fereday & Muir-Cochrane,
2006) was used to analyze youth journal entries. Each sentence was coding to honor every word written by each participant. Braun and Clarke (2006) provided six phases for thematic analysis:
(1) Familiarization with the data, (2) Coding, (3) Searching for themes, (4) Reviewing themes,
(5) Defining and naming themes and (6) Writing up. This analysis method is flexible enough to use with pre-established codes i.e., the seven stages of conocimientos, the analytic framework of this study. (1) Familiarization with the data was done by reviewing each journal, all teaching 57 presentations, audio of whole group sessions, teaching memos, and all additional artifacts. (2)
Coding using the seven stages of conocimientos was completed (see Table 4.2) to further explore how consciousness shifts.
Table 4.2
Codebook with example
Stage Simplified Description Excerpt Example el arrebato, C1 Any things that causes Although I love a-ha moments, one word [to discomfort describe math in school]! Confused nepantla, C2 Comparisons and math can be used in many ways to show definitions of worlds records from sports of all sorts and to also the diversity of places like schools. Coatlicue, C3 The pain of knowing | Why don’t our teachers tell us the “truth” tied to (negative) history about our Ancestors and where math actually comes from. compromison, C4 The process of change or Math can be used to describe the world we knowing change can live in and change it for the better. happen putting Comparing the past to I wasn’t only learning slope but I was learning Coyolxauhqui the present self - the the rate of admin and teachers of color together, C5 connection between decreasing from schools. worlds/ideas. the blow up, C6 Being aware of others It surprised me how it really connected to who and learning from them. I am and who we are Growing from push back. shifting realties, Enacting Spiritual Because of this program I can tell my math C7 Activism - collective teacher to teach us of our deep math’s history action to encourage us to use this skill to better our community and better the lives of those around us.
Moreover, content analyses (Neuendorf, 2019) was done to provide a summative view of the observed codes in both (3) searching for themes and (4) reviewing themes while searching for any patterns across codes. While all data assisted in the writing of a to write the 58 narrative, content analysis was done only on the youth journals because they contained reflection on learning, which is the best data to explore the formation of consciousness. Journals were coded line by line to honor the time youth spent writing. The narrative, along with the multiple graphs and tables from the content analysis, assisted in (5) defining and naming themes. (6) the wrap of coding represents the implications and discussion of the data in this study.
Validity and reliability
Merriam (1998) suggested six basic strategies to strengthen internal validity: (1) triangulation, (2) member check, (3) long-term observation, (4) peer examination, (5) participatory or collaborative modes of research, and (6) researcher’s biases. This study used triangulation along with the rich-think description (Creswell & Poth, 2016) in writing the narrative in order to share accurate results. Additionally, participatory modes of inquiry were used along with prolonged-time with participants where I spent 80 hours with the youth. Member checking of the pre-established codes was done with two emerging mathematics education researchers, both with prior k-12 mathematics teaching experience. Additionally, thematic analysis (Braun & Clarke, 2006) helped structure the coding process along with content analysis
(Neuendorf, 2019) in order to ensure reliability.
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CHAPTER 5. RESULTS PART I - NARRATIVE
The research question “how can mathematics be an experience within YPAR” is ethnographic in nature and as such a narrative is vital in addressing the question. This chapter represents part 1 of the results for this dissertation, and additionally, provides context to understand the results of the content analysis in Chapter 6. The challenge of connecting mathematical learning and (Y)PAR EntreMundos is multifaceted because of the limitations of mathematical learning in K-12 classrooms and YPAR being under-utilized and under-researched within mathematics education. Most mathematical moments with YPAR have been confined to statistical usage of mathematics to show findings after analyzing data. For example, it is common for youth to create surveys and represent findings in percentages, graphs, tables, etc. To understand how mathematics can be a crucial component to YPAR, mathematics became a key epistemology of a (Y)PAR EntreMundos study conducted without the limitation of the K-12 classroom and the normal school year. A project of this scope had never been documented until this point in educational research. Thus, this chapter will address the first main research question by giving a brief overview of each day and a further description of key mathematical activities or turns that show mathematics can be a guiding YPAR epistemology. This
Narrative of Ten-Day Program
The mathematics-(Y)PAR EntreMundos program will be referred to as Reflection Equals
Action with Liberatory Mathematics throughout this chapter. All activities were facilitated by me, as I was an active participant-observer. The youth worked in groups or pairs for all activities.
In this section, I will briefly describe the purpose of each day by giving an overview of the activities youth engaged with as it related to the mathematics explored within the YPAR structure. After a daily overview, special attention will be given to crucial turns throughout 60
REALM to show mathematical learning at all stages of a YPAR experience. Each activity is connected to the theoretical framing of PAP EntreMundos, described in Chapter 3.
Day 1: The Formation of a Collective
Day 1 consisted of four main activities, as shown in Figure 5.1 to form a collective and giving youth an overview of what to expect over the next ten days.
Matematicas The What & Patolli and Name Tents de las Day 1 Why of YPAR Probability Americas
Figure 5.1
Sequence of actives for day 1
I began the day with a modified name tent activity inspired by Ira Shor’s (2005) teaching with the intent of starting to cultivate students’ funds of knowledge by intentionally asking students questions related to their lives. Figure 5.2 was shown on the screen as youth arrived along with instructions on the board on what to do once completed the name tent.
Figure 5.2
Name tent image I created for day 1 introductions activity
The activity asks four questions; (1) favorite place in town; (2) a phobia you have; (3) a favorite
TV show or movie; (4) a random fact about them; and then youth had to find four other people in the classroom such that what they wrote on their name tents connected to everyone. The name tent activity helps create learning space grounded in democratic critical pedagogy because it 61 provides youth with the opportunity talk amongst each other versus just listening to me, the facilitator. Next, I asked the youth how we were going to introduce each other and how many of the four questions we would share. The first student elected to go to the front of the room and introduce themselves, and the rest followed. Similar to Shor’s life work, this activity resulted in the youth-students doing most of the talking for the first fifteen minutes of REALM and the youth were able to make an early, initial decision for the learning space that demonstrates a co- ownership of REALM. The purpose of the activity was shared with the youth to highlight the co- ownership and show how I teach/facilitate. I, as the holder of power, could not fool the learner; thus, I must reveal the intent of the activity to show that I would be transparent with everyone. It was after the name tent activity that I had youth do their first journal entry. This journal entry,
Journal 0, acted as a pseudo-pre-test focusing on how youth saw mathematics and what they expected for the next two weeks.
Following Journal 0, I described what is YPAR and the goal of REALM by stating that mathematics will be experienced alongside YPAR. In defining YPAR, I gave examples of non- mathematics YPAR from the literature along with a YPAR study that I am a part of in a different city, 30 miles away. I then did an expectations activity where youth determined expectations they have for themselves, their peers, and me. I left the room, while the youth came up with their expectation on their own, and they then presented what they expected for the next two weeks, which was to be challenged, to be respected, to learn new things, and to have fun. I told them that I expect the same out of them.
After expectations were agreed upon, I knew it was important for the youth to learn more about me, the person that they would be working with. It is important to note that I did not know any of the students prior to day 1. I began by saying where I grew up using math to tell a story. I 62 used a simple surface area example to show the land my former elementary, middle, and high school occupies and compared it to the land occupied by two prisons 1.5 miles away from my old middle school. I compared the area by constructing ratios of Education Land to Prison Land to Total Land to talk about the land used for schools versus the land used for prisons. The choice of relating schools to prisons was intentional because I knew I would share an infographic that focused on funding used for school and prisons on day 2. Additionally, ratios are a powerful way to communicate information, and they a complex mathematical concept.
After sharing about myself, I did a formal presentation titled Matematicas de las
Americas to focus on mathematical history and knowledge created by Latinx ancestors. Even though the title focuses on the Americas, I was intentional in stating that African mathematics is the oldest type of recorded mathematics and that Latinx people cannot deny the African legacy of Latin American people post-colonization. At the same time, I told the youth they cannot deny and choose to forget the indigenous blood that resided in them. The presentation compared base
10 number systems to talk about base twenty number systems of the Aztec, Mayan, and Inca peoples to foreground indigenous cosmologies and spoke about the history of zero in mathematics. I then talked about pyramids to show the interconnectedness of us all. I explicitly stated pyramids were created by Black Egyptians, not White Egyptians, as we see on television.
The reference to TV and media was to lay a foundation of critiquing systems versus individuals.
The presentation continued to show interconnectedness by showcasing pyramids across the
Americas, see Figure 5.3., and pyramids across the world.
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Figure 5.3
Slide used on day 1 of pyramids across Latin America
I then shifted to the Incan Quipi and ropes used in Africa that showed a connection between mathematics and language. The Incan Quipi was a series of ropes and knots that was used to organize values similar to, how we use matrices today. Similar ropes with knots have been found in Africa. The Quipi was used to show deeper hints at the interconnected nature of mathematics and the origin of mathematics from the Americas.
The first main math activity was Patolli & Probability, which was more successful then I could have predicted. The youth enjoyed playing the game so much that they played during lunch. While playing the game, we collectively looked at sample space and expected value. As a group, everyone recorded the first ten tosses, and together, we estimated the theoretically expected value of tossing beans compared to using a die. Figure 5.4. shows the total sum or number of tosses recorded along with the average of what each group had for the expected value. 64
Figure 5.4
Patolli expected values of each group
We then combined all tosses to get a more accurate approximation of the theoretically expected value for Patolli, E(x) = 2.5. As you can see one group was close to 2.5 with 2.55, a difference of 0.05, and all other groups was not as close, but all groups together had an expected value of 2.475, which had a smaller gap (2.5-2.475 = 0.025) than the best individual group
(0.025 < 0.05). I do wish I had more time to do the extension that focuses on the probability of each possible outcome. Still, the difficulty of finding such a probability would take too much time, and this would be a great example/activity for a statistics course. I did explain the possible extension to the youth and if more students had taken statistics we would have explored further.
Patolli and probability represent the first significant turn for unifying mathematics as a YPAR epistemology in understanding mathematics and YPAR as a unified learning experience. Patolli and probability will be further discussed later in this chapter. 65
Day 2: Generative Themes Emerge
Day 2 served as an entry to developing a critical lens, an essential aspect of YPAR that is the knowledge/language learned to address the root causes of societal injustices. The four central moments of day 2 consisted of learning the power of words, such as minoritized over minority5, followed by an art-based activity to introduce the seven stages of conocimientos to the youth, a
Venn diagram activity comparing schools with prisons and a music guided writing prompt of what they (the youth) would do if they were president of my math class (see Figure 5.5).
Conocimientos If I was We are not a Schools and Arte de las President of Day 2 Minority Prisons Mathematica my Math Class
Figure 5.5
Sequence of actives for day 2
The first image youth saw when they walked in the classroom was the mural, We are not a minority by El congreso de Artistas Cosmicos de las Americas de San Diego (see Figure 5.6).
Figure 5.6
We are not a minority slide used on day 2
5 Minority implies a comparison or ranking when compared to majority, where minority implies be less than the majority. No person should be ranked with another because all people should be equal. Furthermore, minoritized is a more accurate framing as it represents an action that was done to a person by a system. Thus, I do not say a person of color is a minority because they are no less than anyone else, what should be said is that people of color have been minoritized because it is the system that is problematics not the individual. 66
I decided to use the mural because I wanted to talk about language moving forward to talk about dehumanization. In that, the term minority creates a comparison and ranking with the majority. I introduced the framing of minoritized, which acknowledges that people (are equal), but some have been made into minorities by a system. Where we do not say minorities, we say minoritized people. I continued defining other relevant words and terms.
Next, we did a collective art activity based on the seven stages of conocimientos. The exercise consisted of youth drawing an image base on one of the stages of conocimientos, and then they would pass their paper to the right and draw another image based on a different stage of conocimientos contributing to what a different young person had already drawn. Youth were encouraged to try and think of how math and science related to the stage of conocimientos, if at all possible but were not required to do so. Once all seven stages had a drawing it was returned to the person who drew the initial image. Next, each young person came up with a title and narrative for their art. Overall the activity was received positive feedback form the youth, an example is in Figure 5.7.
Figure 5.7
Collective art based on conocimientos created by youth 67
Figure 5.7 shows the collective art created by seven different youth. Each image was based on a different stage of conocimientos and this piece was titled False Reality: The end of lies by one of the youth that contributed the initial drawings that started this piece of art. Additional we can see the description of this piece on the right of Figure 5.7 reads
Glass breaking represents the rupture in society. Graphs: shows the shift of one persons perspective. The clash: clash between pro-choice and pro-life represents an issue and the clash of both sides
The description for each piece of art was useful in showing the youth that they can collectively create art, and the narratives and title of each piece helped to pull generative themes. For example, we can see one of the youth that contributed to Figure 5.7 wrote that the images represent a class between pro-choice and pro-life, with the theme of reproduction rights being a potential topic to explore further.
I then utilized multiple infographics to talk about crucial concepts focusing on the theory of intersectionality. The use of infographics is an example of mathematical critical media literacy. The first image showed 56 times the United States has been part of a conflict in Latin
America and was used to talk about colonization and Imperialism. I wanted the youth to see that sometimes a significant number is all that is needed to prove a point. Next, the infographic on why trans-people need more visibility was used to show how percentages, multiplication, pie charts, and ratios can be used to highlight social issues. The infographic “incarceration vs education” and “education incarcerate” infographics were paired to show further how mathematics used to highlight the social problems but also to talk about the history of slavery.
The final infographic about school to prison was initially going to be used to talk about the difference between using math and not using math, but this led to an activity where the youth listed ten similarities and ten differences between schools and prisons. One of the students ended 68 up reading and using the infographic to talk about the differences in funding between prisons and schools. The impromptu activity forced me to push back the “math in the news activity” to a later day. The activity of finding differences and similarities between schools and prisons ended being great in utilizing the infographics. At one point, a young person referenced the actual amount of money difference – $10,000 for students and $100,000 for prisoners. What emerged was an unplanned activity of having youth create Venn diagrams to compare schools with prisons. This activity was done solely because of the interest students had on the topic. After creating Venn diagrams in groups, each group presented to the whole group. I usually do this activity with undergraduate college students that I teach, and this was the first time doing the activity with high school-aged youth. I will say what the youth created was similar to the quality of work done by college students. This activity also helped to pull additional generative themes.
I wanted to use music to help identify generative themes, a creative praxes approach. I have done a variation of this activity with youth multiple times, so I then modified it to reflect how students learn mathematics. I played the song “if I was president” by Las Cafeteras because it creates a nice transition to ask students what they would do if they were president. This time I then asked what you would do if you were president of your math class, and one student wanted me to clarify and asked if I meant what they would do if they were there math teacher, and I said yes. While students were working, I played the song “if I was president” by Wyclef Jean. I did this to talk about the historical nature of oppression and how the idea of a Black president being assassinated was something many people thought would happen when Obama was elected president. As students finish writing their journal, I then told them they have until the end of the song “a change is gonna come” by Sam Cooke. I used music to end the activity with a sense of hope because the world is not fixed, and I wanted the youth to see change as possible. The youth 69 enjoyed the music, and what they wrote in their journals was used to pull additional themes for future activities.
Day 3: Identifying Key Stakeholders
The purpose of day three was to continue to identify emerging themes and focus on mathematics as a critical literacy through critical praxes.
Newspaper Math & "I am from" "find the Reading Day 3 Math Poem math" Infographics
Figure 5.8
Sequence of actives for day 3
Day 3 began with the Newspaper “find the math” activity that was intended for day 2. For this physical activity the copies of a local newspaper and the USA Today were purchased across two days. It was important for youth to have physical copies versus viewing the news on their phone or computer because it allowed them to write in the newspaper collaboratively. I did not want the youth to be distracted because of online ads, and students confirmed later that they, in fact, would have been distracted doing the activity digitally. I selected a local and national newspaper so youth could see a difference between local and national issues.
I had not looked over the newspapers before passing them out to each group. The instructions were as follows,
In your group find the math in the newspaper. What numbers stood out and how were they represented? Are there any stories/articles that you think could benefit from having numbers/math in it? What types of math are used to talk about the real world (in general)? Could any of the article be turned into an infographic?
The intent was to do this activity directly after sharing some infographics but I ended up to like that there was an entire day difference between sharing infographics and looking for math in the 70 newspaper. What I did plan was to share additional infographics after the activity. During the newspaper math activity, students were able to write on the newspapers with pens, crayons and highlighters. Each group then shared some math they found in the paper. In comparing two pages of the newspaper one group noticed that one page had nothing but text, and the other had an image. As a class, we concluded that we tend to skip over things if they are just words that could lead to us missing out important information.
Next, I shared the “student life in America” infographic to show further how data could be shared with bar graphs and percentages (see, left of Figure 5.9).
Figure 5.9
Slide used to show student life in America (left) and queer youth of color (right) infographics
The next infographic used, Queer Youth of Color was shared to show how percentages, bar graphs, and pie-charts can be used to highlight social issues (see, right of Figure 5.9). Youth were then told that the Queer Youth of Color infographic was created by an organization, Trans 71
Student Educational Resources, that was founded by a 14 and a 16-year-old trans person. This was done to highlight the agency of youth in transforming the world. The previous two infographics were also shared to emphasize REALM space would be safe for LGBTQIA+ people. The infographics gender injustice (see Figure 5.10) was then paired with the women in congress infographics to discuss patriarchy but also to show a connection between math and taking action.
Figure 5.10
Slide used to show gender injustice (left) and women in congress (right) infographics
To further illustrate how ratios can be used in an infographic, the infographic mental health and incarceration (see Figure 5.11) was used to discuss the complexities of why people are forced into prisons. The intent in using prisoners with mental health issues was to connect it to the conversation on the parallels between education and incarceration from the previous day.
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The use of infographics, coupled with the newspaper activity, allowed youth to see various ways mathematics can be used to communicate information.
Figure 5.11
Prison and mental illness infographic
I had noticed some of the youth were interested in the environment so continued to show how data can be represented a shiny6 (an interactive plot of points). For example, the shiny
Climate change in major cities (1900-2014) was used to show the youth another way of using mathematics to highlight social issues (see Figure 5.12). Further this was used to show the political nature of how graphs can be deceiving or forgiving. I also made sure to show scatter plots with the line plots to relate everything to lines of best fit because they should have learned about lines of best fit in algebra. Additionally, I also showed a mosaic graph of students at a given high school and compared it with the teachers at the same high school. The students were
6 A shiny is interactive graph made using the programming language R 73 aware of this high school and the demographics of the city but were still surprised to learn the actual comparison between students and teachers.
Figure 5.12
Screenshot of slide utilizing shiney with Berkeley Earth data
Youth were told that Figure 5.13 was made using public data to show data accessibility to compare the demographics of students and staff from the same high school. I showed this mosaic’s because I wanted to see if students would start to question how this looks for their school and because I wanted to share as many ways to share out information with mathematics.
The final activity of the day was an “I am Math” poetry activity. This activity was vital in pulling generative themes from each young person while positioning mathematics as a creative praxis.
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The mathematically rooted poem represents the second major turn to understanding how mathematics can be infused within YPAR, which will be discussed at the end of this chapter.
Figure 5.13
Mosaic comparison of students and staff from the same high school
Day 4(5): Critical Social Science Research Training
Sponge Bob Genereative Image Dehumanzing and Focus Day 4(5) Themes Analysis Language Groups
The Counter Types of Revolution Narratives Problem Tree Day 5(4) Surveys (Dancing) with Poetry
Figure 5.14
Sequence of actives for day 4 separated as Day 4(5) and Day 5(4) 75
The notation of Day 4(5) represents the first half of day 4 and notation Day 5(4) represents the second half of Day 4, which was initially the plan for day 5. The change of schedule was due to a day-long trip for the youth to the local university. Thus, everything before lunch will constitute the theoretical day 4, and everything after lunch will constitute the theoretical day 5.
Day 4 did not focus on mathematics but instead it ensured youth developed research/problem statements and learned how to conduct social science research. Due to the time limitation of ten days, this was the make or break point for the YPAR structure of the study because I needed students to write a research question or at least determine what they wanted to research and who needs to see what they unravel (identify stakeholder). Before day 4 I looked over their journals to identify emerging themes. I then showed the youth the following themes,
Immigration, School Safety, Gender Equality, Women’s Rights, Impact on Environment, Quality of math instruction (real history of math, connecting math to real-world, assessment),
Surveillance/school as a prison, Teacher/Student relationships, and School Funding that had previously emerged through their activities. I wanted to front-load students with the themes as I went over research training and to prepare them for the problem tree activity we would be doing later in the day.
The research training began with an overview of document analysis. I share an image of
Native American boarding schools in asking students what they noticed and how they would describe the image. I used the image to discuss differences between writing a description of an image, attempting to make meaning of an image, ascribing meaning to an image and writing or expressing how the image make you feel. Not only was this to state that images can be data but also to foreground taking observation field notes. The second image (Figure 5.16) shared was 76 that of my high school to show pictures can support arguments and act as evidence in making claims. On the previous day many of the youth said that schools do not have metal fences with one student saying “yes, the majority of schools have fences,” and a majority said “no”.
Figure 5.15
Slide of my high school and image of the metal fence surrounding my high school
So, I shared an image from my high school to show how fences look elsewhere and ask them to describe what they saw in the image and how it compared to the previous image. This was done to indirectly reference patterns emerging for the data, in this case, a connection to the prison.
Next was an observation and mock focus group activity. I asked six of the youth, which we will call Group A to leave the room and go our second room down the hall. I then asked the remaining youth if there was a TV show that they all knew. All but one person had watched
Sponge Bob (I knew of the show and character Sponge Bob, but I had never watched an episode). I then told the group still in the room that we would be doing a focus group and asked 77 which two people would like to lead it (i.e. be the people asking questions). I then told the two young people to go into the hallway. With the remaining youth, which will call Group B, I said to them that they would each have to play a role. The roles were as follow: (1) blindly loves the show, (2) clearly hates the show, (3) has never seen the show but pretends to know it, (4) tries not to talk, (5) tries to always talk about their dog, and (6) tries to cut people off and talk about random topics. I then went into the hallway and gave the two youth that would be interviewing
Group B in the mock focus group leading suggestion on what type of questions to ask and the overall format. I then went into the room down the hall with Group A and told them they would be observing a focus group. I said to them that that half of them would focus on the interviewees
(Group B) and the rest would focus on the interviewers. Their goal was just to observe the event.
Once everyone was ready, we did the mock focus group. We then debriefed how the interviewers went, and the observers (Group A) tried to guess what roles of Group B. The mock focus group resulted in the youth asking me about my research and why I do it.
It was meaningful that they asked me why I am committed to doing YPAR project because it impacts them, and they have the right to know given REALM is a shared space for all of us. Which by the youth asking me means they wanted to learn more about me and were beginning if not already accepting me as part of the YPAR collective. I told the youth why I love and hate mathematics; in that mathematics made me choose to either think of myself as smart or think of myself as a person. Where I hate what math has been turned into because math is beautiful and found everywhere in this world. I then used this as an opportunity to also address an issue that happened previously and talk more about dehumanization, specifically objectification. The issue that had occurred was that a young man called a young woman the b- word, so I made sure to tell everyone where the word comes from and how if you are a b-word, 78 you are an animal, less than human and thus you are treated unequally. One student brought up the use of illegal in referring to undocumented people as an example, which tells me what this student is thinking. I then told the youth about spiritual activism and conocimientos in that both bodies of work guide my own life-work. I believe this inspired the youth so much so that they asked if they could learn more about the seven stages of conocimientos.
Day (5): Reviewing Generative Themes
Previously during a break on day 3 a student started dancing cumbia and the idea of him showing all of us some necessary dance steps emerged. A different young person then asked if the person dancing could teach all how to dance. So, I asked the young person if they felt comfortable doing so, and they ended up taking the first 30 minutes after lunch on day 4, showing use cumbia, bachata, and nortenas dance moves. After the revolution, which is what we called dancing, we talked about basic graph theory and how it can relate to the movement of dancing and also talked about mapping the dance moves to find the optimal number of people that could dance in a given dance hall or room, see Figure 15.16.
Figure 5.16
Mapped movement of dance steps
We talked about how to set up the problem of finding the optimal numbers of people in a dance hall and the value of knowing the optimal space both for fire-code purposes and for the love of 79 dancing because an over-crowed area does not let you dance.
Back to learning about research, two poems were used to talk about the difference between a narrative and a testimonio. First, I had students watch the first poem “can you see it.”
While students watched it, I also used it as practice for taking field notes by asking them to observe everything. Poetry competitions are ideal for making observation be poets incorporate facial expression, hand/body gestures, and tone of voice. I then told them why it was a narrative and asked them to observe the next poem to see if they could deduce what is a testimonio. We then talked about counter-narratives and counter storytelling. The youth picked up on these concepts quickly. Next, we went over different types of surveys. I showed examples of Likert scale, open-ended, multiple choice, and true-false surveys. We talked about the pros/cons and that it is okay to use a combination of different types of questions.
We then ended the day with a problem tree activity. I attempted to add a component to the activity where students would also identify numeric variables associated with root causes and symptoms of the problem they were addressing. The youth then developed research questions related to the following topics (1) Gender Equality and LGBTQIA+, (2) Immigration, (3) School to prison nexus, and (4) mental health. The problem tree activity took longer than anticipated due to the creative prowess of the students. Three groups placed themselves in competition on who could make the best problem tree, and with each group having a different definition of “best”, as seen in Figure 5.17. In the below figure on the left, you see labels on the roots and branches detailing symptoms and root cause of the issue, and on the right we have an artistic tree that only represents the trunk or main issues students will explore; where the tree on the left is the best in terms of detail and the tree on the right is the best in terms of art.
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The day ended with youth identifying key stakeholders where I told them that over the weekend, they need to think about who needs to be informed about their topic or who they want to upset
(in a good way).
Figure 5.17
Two different approaches to problem trees
I told the youth when I say upset, I mean educate because for many people the truth upsets them at first, but then they can only learn from it.
Day 6: Data Collecting
After day 5 one of the groups contacted me that they had already created a survey and wanted feedback. I gave them feedback; they made changes and then asked if they could start distributing their survey. The youth attended a community event the next day (Sunday) and wanted to take advantage of being around multiple people, specifically. I told the youth that I would meet them at the event to discuss a few things before they asked people to complete their 81 survey. I had already planned to attend the event to be present on behalf of El Sol. The group that collected data at the community event came to REALM on day 6, with over 30 people responding to their survey where I was not expecting this, but I was happy to see them taking the initiative. It was because this group was focusing on gender equality; I decided to use my gender wage people in power ratio activity. Where day 6 consisted of four activities, see below figure.
Ratios of Native What is a Gender & Language or Pilot Survey Day 6 Ratio? Power Slang
Figure 5.18
Sequence of actives for day 6
This activity was created initially for preservice teachers (PSTs) as part of a study that explored how an activity can be used to show PSTs can experience the seven stages of conocimientos while learning mathematics (Martinez & I, 2019)
The original activity was created using data from 2017, which was updated to reflect data from the same year REALM took place. Previously I had only selected 44 careers, but this time I used every job that had completed data. Each student was given a different list of careers. This was done to encourage discussion due to no one person having all the information. Before starting the activity, I did do a light review on how to find the value of a ratio (unit rate) and what a ratio is. All but one group were actively having a conversation about the activity. One of youth wrote in their journal that the activity helped them better understand ratios. I then debriefed the activity by showing each “answer” for the total average of medians.
After the gender wage activity, I played a poem title “slang” to emphasize that young people have their own language and ways of constructing knowledge. I told the youth moving forward that the outcome of their research can be displayed in multiple formats. The poem was 82 also used as a mild break to transition into practicing how to collect data. Each group spent time and created survey items, both quantitative and qualitative. Once each group had survey questions ready, their peers answered the questions. This was done so each group could pilot their surveys. The rest of the time was work time, so groups could finalize their research questions, identify who/how they would collect data from, and complete their surveys. With the group that already had data, I gave them a brief overview lesson on how to try and identify themes and patterns. I was not planning to talk about analyzing data until the next day, but since they were ahead and every other group was working, it did not create any issues.
Day 7: Revisiting theoretical framing of conocimientos
The priority for this day was to give students ample work time so they could send out their survey’s (see Figure. 5.19 for a day 7 overview)
Black Panthers Grape Strike Stages of Collect Data & Mathematical Mathematical Day 7 Conocimientos Work Time Resistance Resistance
Figure 5.19
Sequence of actives for day 7
The idea of a counter-narrative had come up a couple of times, and the idea of being an activist had been overheard by me multiple times. It is here that I decided to create and use mathematics problems situated in historical acts of resistance while simultaneously talking about counter-narratives and history they do not learn in schools. Additionally, I wanted to talk about various youth-led social movements, so I created the first set of math problems rooted in world- historical context. The first set of problems focus on the Black Panther Party. The first question cannot be solved without additional information, which was the intent in it allowed me ask the young people what other information would be needed to answer the posed math question. The 83 only way that a solution could be created would require additional variables based on assumptions. The need for more information was intentional to tell the students that mathematics is not just about a final answer. The next question also cannot be solved but is slightly more direct on what information is missing. The last problem (see figure 5.20) in this series could be solved only because I imposed the assumption of equal growth across each day.
Figure 5.20
Slide used to show math problem with Black Panther Party context
The problem highlights the free breakfast program of the Black Panther Party, the first of its kind across the United States. The focus on a social program like the free breakfast before school and free ambulance for the community showed counter-narratives because most people only see militant images of the Black Panther Party positioned as overly aggressive.
The next series of questions focus on the Delano grape strike. The first question talks about how Filipinos were the first to strike, which led to the formation of United Farm Workers and a march across California. To connect the distance marched from Delano to Sacramento to a distance, the youth would be more familiar with I showed a google maps image that showed the same distance across two well-known cities in the area. The next questioned focused on the number of people that supported the strike, while showing marching as a form of resistance 84 aside. The question, as shown below, references Cesar Chavez and his hunger strike for 25 days in losing 35 pounds. Where a hunger strike represents another form of protest, after students journaled and reflected on these questions, we talked about various youth-led movements throughout history.
Figure 5.21
Slide used to show math problem with Delano field strike as context
Talking about youth led movements was done to remind the youth of their agency, and I think it worked. They all look super motivated in being frustrated that they did not learn about any of this in their history classes. The choice of referencing Gandhi and King in the problem was to show commonality between the thinking of three different people. More so, the image shows solidarity among people with different racial and ethnic origins in that Kennedy, a privileged white man, supported Chavez.
With solidarity on everyone’s mind, we transitioned to talk about the stages of conocimientos because I had noticed that multiple students wanted to learn more about them after the collective art activity from day 2. So, I took the time to go over each stage. I started this by sharing a poem to ground the conversation. While talking about each stage, I would ask students how they saw the stage in their math class. The discussion led many insights on how 85 students view their mathematics class and school in general. After the discussion, one student requested a copy of Anzaldúa’s book Borderlands because they wanted to learn more.
The rest of the time was given for the youth to work on survey design, and I was surprised how far along the groups got in one day. I thought we were going to be falling behind, but they ended on pace to finish strong.
Day 8: Identifying Patterns and Themes
Slope of Identifying Students and How to Code Work Time Day 8 Patterns Teachers
Figure 5.22
Sequence of actives for day 8
On day 3, I showed the youth two mosaics that show the demographics of a school comparing teachers to students, which resulted in many students ponder how it looks in their school/district. On day 3, I told the students that I have the raw data and that we could do something math-related at a later time. Day 8 is the day I decided to use the data is using rates of change or slopes to compare data. After this activity, the youth were shown how to analyze qualitative data and had work time. The slope of students and teachers activity represents a key turn for mathematics within YPAR and will be discussed later, what follows is an overview of the activity. I wanted to have students look at the number of students of color and teachers, administrators and superintendents of color. I wanted to do this to focus on the real change in demographics in their city and state. As mentioned earlier, I used public data and I had youth plot points without telling them what they were plotting. I also had them find the slope and interpret what the slope means, contextually for the following question,
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Plot and label the following points, Point A: (1, 1.8), Point B: (16, 2.4) and Point C (17, 2.5) Make a line and find the slope between Point A and Point C Make a line and find the slope between Point B and point C What does the slope represent in the real world?
Additionally I asked them to repeat this process with the sets, {D:(1, 3.5), E:(16, 3.1), F:(17,
3.5)}, {G:(1, 0.9), H:(16, 1.1), I:(17, 0.7)} and {J:(1, 10), K:(16, 23), L:(17, 24)}. While working
I noticed the youth were very disconnected, and this was the moment that youth were least engaged in REALM. The learning in this activity is done in Algebra I and most young people in the class had already taken Algebra II. Yet, after fifteen minutes no one had completed the task and one group even started playing patolli while working. I approached two of the youth, and I asked how everything was going, and one of the youths said, “this feels like a normal math class.” I then revealed to students what the points represented by showing them, Figure 5.23.
Figure 5.23
Slide sued for characteristics of Iowa teachers
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After seeing the context of their learning the overall interest of the group changed. It was revealed to the youth that the x-coordinate represents the school year, with 1 representing the
2000-2001 school year, and the y-coordinate representing the percentage of minoritized people within each set. Where set {A:(1, 1.8), B:(16, 2.4), C:(17, 2.5)} represents teachers, set {D:(1,
3.5), E:(16, 3.1), F:(17, 3.5)} represents school principals, set {G:(1, 0.9), H:(16, 1.1), I:(17,
0.7)} represents superintendents and set {J:(1, 10), K:(16, 23), L:(17, 24)} represents students. A visual recreation of each set can be seen in Figure 5.24.
Figure 5.24
Recreation of minoritized teacher, principles, superintendents and students’ graphs
We then talked about what it means for the overall diversity of schools and the disproportionate and lack of change over the last 18 years, as being a problem. After students journaled about the activity and I told them all of this data is publicly available, they requested to see the data for their schools. I showed them where to find the raw data, how it looks, and how it can be cleaned for analysis.
Next, I wanted to go over how to do open coding with students. I created six hypothetical 88 replies to the question, “do you feel safe at school?” The hypothetical responses using pseudonyms are as follow:
• Littlefoot: No, I don’t feel safe because people call me names. • Cera: No I don’t feel safe because teachers are stupid • Petrie: I sometimes get followed by other students • Hopper: I know we have police coming to the school, but it just makes me think that the worst can happen and I don’t know how that makes me feel. • Ducky: My teachers is always yelling at me and I think they are out to get me because I get to school late. • Spike: Yes, but that is because no one is going to pick on me. If a student says something I will make them feel unsafe and if a teacher says something well I will just them to shut the F up what can they do to me.
As a group, we then coded each one and talked about how to organize and arrange themes. I then shared an example of using predetermined codes. I once again showed the seven stages of conocimientos and told students that they are an example of predetermined codes. Then
I showed them that they could organize the codes in a table to help see patterns. The rest of the time of Day 8 was work time as I walked around helping each group.
Day 9: Bringing Everything Together
Day 9, see the figure below, was mainly reserved to give youth work time with their group.
The Bone & If we must Patolli, Senet History of Work Time be Day 9 and Mancala Math Americans
Figure 5.25
Sequence of actives for day 9
I knew students needed this as a workday, so I only planned a brief math history presentation. I wanted to make sure to honor the youth of African descent because I knew a majority of the discussion focused on Latinx history. Thus, I created two slides to inform about 89 the Lebombo and Ishango Bones that are the first evidence of mathematics in the human history.
Both were found in Africa and are dated 35,000 BC and 20,000 BC. Next, to continue the theme of using games and counter-history, I wanted to make sure the youth were aware that Mancala originated in Africa. Most were familiar with the game but did not know it is dated to 700BC, and none had seen how some of the original game “boards” looked, see Figure 5.27. I thus showed the images of mancala “boards” made out of and carved in to stoned.
Figure 5.26
Slides from a history of mathematical bones discussion
Figure 5.27
Slide used to show Mancala origin and history 90
The rest of the time was student work time. One group spent all two and a half hours coding. I was unaware at that time, but each person in the group looked at about 11 survey responses overnight. Each person then analyzed the responses, and they shared out what they found as they then agreed on how to code the responses. The group conducted open coding in arriving at their themes, as seen in Figure 5.28. The group spent over two hours coming to an agreement on all codes and themes.
Figure 5.28
Group coding
In figure 5.28 you can see the youth used sticky note to write the themes that emerged and then they wrote which responses form their survey corresponded to each them. A second group produce a piece of art, a poem, and made various charts/graphs. Two of the youth in the group 91 worked in a quiet room (cafeteria), while two created graphs in outlining the infographic they wanted to make. A third group was frustrated in waiting for responses. They wanted to compare youth to older people and were having difficulty finding adults to take the survey. It may be because it is related to LGBTQIA+ rights. At the end of the day when students were about to go to lunch, the director of El Sol entered the room with four administrators the school district. The director asked if the youth would share what they have been doing. The youth said yes, and this represents a key turn for mathematics within YPAR. I then asked all adults to step into the hallway to ask the youth if they were okay in talking to them because the youth had to have the option of not speaking with the adults. I then told the youth that they could say whatever they want because I would put the administrators in their place if needed. It was important for the youth to know that they were free to say whatever they wanted and that they were supported. The youth seemed excited and energized after I said I would support them, and they agreed to share their work with the administrators. I then invited the adults back in, and two groups shared. The group that spent 2.5 hours coding all of their data (see figure 5.28) spoke about racial discrimination in the school district. They had just looked at how over 50 high school youth responded to their survey and were ready to talk. The administrators were surprised that in two days they had over 50 respondents. The second person to speak, represented a different group, had been working on charts. They shared some of the graphs, and one administrator in particular loved the graphs and said that this is the type of data that all administrators in the district should be looking at during their yearly training. Next the second group shared, and all administrators had wide-open eyes when the young person said that they had 99 students take the survey and at least ten students from the five high schools in the district. A similar example of what the youth showed the administrators can be seen in an updated version of what they presented at a regional 92 conference, see Figure 5.29. The youth presenting to administrators was a powerful moment of the youth taking action in presenting their work to key stakeholders. I then decided to debrief with these students directly after lunch. They were excited and proud of what they did, and they even talked about the next steps. The one person explicitly said they noticed how powerful charts and graphs could be, and was glad they did share out in acknowledging if they were alone, they might not have said anything.
Figure 5.29
Slide titled Do you feel safe at school? Created by youth
Day 10: Presentations and Next Steps
Individual Presentation Collective Group Next Celebration Day 10 to Peers Next Steps Steps
Figure 5.30
Sequence of actives for day 10 93
The overall schedule for Day 10 was to have individual meetings with groups to talk about the next steps and then have each group present for 10-15 minutes to each other. Every group agreed and were excited about presenting at the Latinx Excellence conference in October.
All decided to try and meet two times with me and multiple times amongst themselves. I was happy to hear them say, “can we meet without you,” and I said yes and explained that I could also give them feedback via email. All groups created a group text so we can communicate. Each group then shared their work with their peers before our celebration. To end REALM a potluck was organized by Sophia, the person who helped recruit youth to the program. In the next section key mathematical turns7 will be used to connection mathematics to YPAR.
Mathematical Turns: Towards an Epistemology of YPAR EntreMundos
To address the first research question, “how can mathematics be incorporated throughout a YPAR experience?” YPAR as a research methodology, YPAR as a method towards pedagogical practice, and YPAR as an epistemology must be addressed. Mathematics as a YPAR methodology, functioning as a method for research, is reflected in Chapter 4. Mathematics as a method or specifically mathematics as pedagogy through YPAR can be seen in connecting
Chapter 5 with Chapter 4. Mathematics as a YPAR epistemology begins theoretically in Chapter
3 and will be further addressed utilized key mathematical turns in the next section. Additionally,
YPAR, as a mathematical action, will be emphasized in the discussion, Chapter 7, to highlight the overall revolutionary Praxis of mathematical reflection by discussing the seven stages of conocimientos with|in mathematical learning.
7 Mathematical turn uses the word turn to emphasize a change in course where in to say a change in how mathematics has been learned needs to occur to understand how mathematics and YPAR can be congruent. 94
Turn 1: Our Roots, Ancient Mathematical Wisdom
As mentioned earlier, youth played Patolli to learn about probability and expected values.
Patolli is the oldest board game of the Americas (200AD), and like all games is imbedded with mathematics in learning how to count and being comfortable with numbers. Patolli is played by taking five beans, marking one side of the bean, and tossing them on the board. The number of marks that are facing up represents how many paces you can move. What is interesting is the possibility of zero because if no marks were visible, then you could not move any spaces, which represents early, if not fundamental, interactions with the concept of zero. Patolli was/is the mathematics of native people of the Americas and represents third world mathematics known as ethnomathematics (Gerdes, 2001). If we look at a game board for patolli, see Figure 5.31, we can see that Patolli was also deeply intertwined with spiritual/cosmological understandings of the universe in that the board incorporated the Gods.
Figure 5.31
Patolli game board
Patolli was played as a fun way to learn about the expected value of the outcome of tossing five beans, each with a probability of 0.5 that resulted in youth getting to know each 95 other on the first day of REALM. By engaging in ancient mathematical wisdom, defined “as the cultural, historical, spiritual and logical forms of mathematics that are collectively created to interconnect us to other forms of mathematics” (Martinez et al., in press/under review). Patolli allowed Latinx youth to have fun together playing a game rooted in the ethnomathematics of their ancestors, and it allowed them to learn more about Eurocentric mathematics (probability) in learning and reflecting on the mathematics that is already a part of them. Paired with the presentation on “the mathematics and science of the Americas” youth begin to see that mathematics is not independent of themselves and their culture. Mathematics within YPAR must be epistemologically rooted in ethnomathematics from the beginning of the program.
Turn 2: Our Voice, A Mathematical Creative Praxes
The use of the “I am math” poem activity represents mathematics as creative praxes in challenging stereotypical views of mathematicians being unable to be non-logical and poetic.
Poetry is an intimate process that allows the poet to reflect upon their own lives and feelings.
Each young person composed a poem that they then shared with their group. Youth shared with the whole class if they wanted. I was delighted when one student wrote, “I am from patolli,” showing how meaningful and guiding ethnomathematics as an ontology can be. To say the experience for the previous day created a relationship between the young person and patolli. As I was walking around the room, I was delighted/excited when I would hear what they were saying furthermore, many students did say they were surprised how much the poem made them think about their lives. What follows is one of the youths’ poems:
I am from __Prism__’s (Your favorite shape) from where my ____brother____ taught me, (Family member) to add ____Courage the cowardly dog____ (An object form growing up) Celebrating the difference between me, my friend ______LT______96
(The name of one of your friends) and _____Frida Khaldo______is like math, (A cultural figure of your people) When trying to learn ______AP Stats______(A High School Math or college Class) or enjoying ___Matrix Systems____ (A math concept that you enjoyed doing) and fighting to solve____finding congruence______(A math concept you did not like doing) We are from math and _____Any Garden____ (A game you like playing) Hidden numbers, I spy ______flowers______(A real-world object that matches your favorite shape) seeing patterns everywhere, especially at ______my dog______(Your favorite place growing up) and when I smell ______flowers______(something that smells nice) even when smelling ____my dog______(Something that smells bad) Math is me, ______resilient______(A positive characteristic that describes you) And math is you ___understanding__ and ______short______(a characteristic that you like in others) (another characteristic that you like in others)
From negative to positive infinity (a characteristic that you like in others) math is us, as we become ____world changer___’s (Your future career) Fore, I am math
And
together we will change _____climate and prejudice______(Something in this world you want to change)
Connecting mathematics to a person’s own life along and the future is key to living mathematics within YPAR. In general, if the future cannot be seen with mathematics, then there is no future where mathematics is alive. Mathematics as a YPAR EntreMundos epistemology breaks the monopoly of who is a mathematician by being a reflective critical praxis.
Additionally, the poem shows youth mathematics is part of their history and daily lives, the mathematics with(in). 97
Turn 3: Our World, A Lens of Mathematical Critical Literacy
For mathematics to be an epistemological force within YPAR, the bridge between the history of mathematics and an individual’s own lived mathematics must be traveled. An essential aspect of YPAR is the development of critical ways of seeing the world and then being able to use those lenses to identify and address the root cause of societal injustices. In the first four days, various infographics lead to learning more about societal issues. The infographics utilized numbers (mathematics) for communicating information and allowed us to have a conversation that dove deeper into the matter; for example, on Day 4 the infographics on incarceration versus education led to an unscheduled activity in comparing schools to prisons. Mathematical critical literacy is the reflection need to push mathematical YPAR into a mathematical critical action.
The knowledge construction with mathematics within YPAR must comprise of mathematics as both a reflection and action.
On day 8 when youth found the rate of change of minoritized people within their school district, they used mathematics to see their own community and its theoretical growth in seeing the future. The youth saw that the change over time of teachers of color was near zero when the growth of students of color was a consistent growth over time. When youth look at initial values, they begin to show inequity in that students made up 10% of the student population seventeen years ago. Now they make up 24% indicating that there has been a growth of students of color yet 17 years ago teachers of color made up 1.8% and now they make up 2.5%, a significantly smaller increase. This problem then allows students to ask questions that lead to understanding the root cause of such inequity. This activity used the youth’s schools showed the history of change and the possibility of the future if trends continue. However, by using the youth's school over their entire life time, it allows them to situate themselves in the problem further. The deeper learners dive into a problem, the deeper they can reflect on both the content of the problem and 98 their own lived experiences. Vital in understanding mathematics within YPAR is how the mathematics represents the history, present, and futures of the youth engaged in the work. A critical mathematics literacy helped them focus on both the world and self.
A Balance of Action and Reflection
Reflecting on mathematical learning is something that should be done (and, for the most part, is done) in schools. The struggle of mathematics within YPAR is not the reflection but the action.
Without action upon the world, YPAR lacks the opportunity to learn, develop agency and empowerment and cannot be considered YPAR. Thus, mathematical action is a necessity for mathematics within YPAR. One example of mathematical action is when a young person asked for a copy of the patolli game board to share with their siblings. The action of asking for a game board with the action of showing someone else how to play patolli is mathematical action upon the world because the learning done in REALM reaches further out into the world. A second example captures the essence of YPAR can be seen on Day 9 when youth presented their findings to district administrators. Youth experience the key turns mentioned above and then shared what they discovered, with key stakeholders at the end of Day 9. A final example that exemplifies mathematical critical reflection and mathematical critical action is the use of infographics throughout REALM. The reflection is the use of infographics as a mathematical critical literacy and as a way to learn/develop a crucial lens in viewing the world. The action of infographics is when youth construct their own infographics, as seen in an infographic made by one of the groups (see Figure 5.32). Figure 5.32 figure shows the use of percentages and ratios as part of how this group chose to report their findings. Mathematics, as part of YPAR reports, alone does not constitute how mathematics can be a YPAR experience. The epistemological underpinnings of mathematics to/with PAR EntreMundos (Chapter 3 and Chapter 4) are equally as vital in ensuring mathematics allows young people to reflect on their mathematical history and 99 future. Allowing for ethnomathematics to anchor mathematical learning is one way that allows space for youth to explore the mathematics that is already a part of them. More so, mathematics within YPAR should create learning experiences that welcome collaborative learning in connecting mathematics to the world, to self, to other and to other forms of mathematics.
Figure 5.32
Mathematical action infographic
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CHAPTER 6. RESULTS PART II
This chapter will show how the seven stages of conocimientos emerged across the ten days of REALM. As stated previously, the seven stages of conocimientos represent a shift or change in an individual’s consciousness and align with the epistemological and theoretical framings around PAR EntreMundos. In observing how the stages of conocimientos emerged, I can answer the second research question in understanding how consciousness shifts and developed when engaging in mathematical learning through the use of the seven stages of conocimientos as a conceptual framework. By taking a look at how pre-established codes based on the seven stages of conocimientos were observed for each individual (collectively) across the ten days of REALM, the second research question becomes addressed. The chapter will begin by giving a summative overview of how the seven stages of conocimientos (codes) emerged throughout the youth’s journals access the ten days of the programs. This allows us to see all the codes to shows when all the codes were observed and if all the seven stages of conocimientos were engaged at least once by all youth. Since conocimientos is a collective process, the codes will are viewed by each group rather than individuals in comparing patterns across all three groups. The chapter will end by then combining the collective codes of all youth across the ten days of REALM to examine the collective shared codes for all participants.
Results of Individual Youth
Through the journal entries, more stages of conocimientos were observed in the second week of the program with the highest number of codes occurring on the last day, see table 6.1.
Table 1 represents the observed frequency of each stage of conocimientos across each journal entry (where J0 represents the first journal entry before) of all participants during REALM programming. In the table, Jn with {n=1,2,3,4,5,6,7,8,9,10,} represents a journal entry written by 101 all youth at the end of day “n,” of the ten days of REALM. The numbers in Table 6.1 are the frequency of each code that appeared in each day’s journal of all participants.
Table 6.1
Frequency of Codes Across Journals
Code J0 J1 J2 J3 J4 J5 J6 J7 J8 J9 J10 Total C1, el arrebato 10 7 7 11 13 0 0 2 3 3 4 60 C2, nepantla 17 14 12 23 10 8 9 8 7 7 29 144 C3, Coatlicue 3 5 4 1 4 1 2 4 3 5 15 47 C4, compromison 4 8 5 7 1 5 8 10 4 2 13 67 C5, putting Coyolxauhqui 3 1 11 8 14 4 3 6 5 12 16 83 together C6, the blow up 1 2 2 7 6 1 9 7 4 18 17 74 C7, shifting realities 1 1 3 6 1 1 5 4 0 7 12 41 Total 39 38 44 63 49 20 36 41 26 54 106 516
In total, 516 codes emerged with nepantla having a frequency of 144, which is 61 more occurrences than the second most frequent code of putting Coyolxauhqui together. A high frequency for nepantla is expected due to nepantla being a metaphysical space that is constantly engaged at all moments of life and the space between two physical places or objects. Nepantla appeared the most on the last day as youth reflected on how mathematics related to themselves, their standard school classroom, and their community (see Figure 6.1).
14 12 10 8 6 4 2 0 Journal 0 Jouranl 1 Journal 2 Journal 3 Journal 4 Journal 5 Journal 6 Journal 7 Journal 8 Journal 9 Journal 10
Figure 6.1
Counts of nepantla code across journals 102
On Day 10 (J10), the highest number of shifting realities, 12, representing 29% of all shifting realities codes, was observed. Shifting realities represents the code with the lowest frequency, yet we see a pattern of a higher number of frequencies the second week (n = 28) over the first week (n = 13) with the two highest occurrences on Day 9 (J9) and Day 10 (J10) (see Figure 6.2).
8
7
6
5
4
3
2
1
0 Journal 0 Jouranl 1 Journal 2 Journal 3 Journal 4 Journal 5 Journal 6 Journal 7 Journal 8 Journal 9 Journal 10
Figure 6.2
Counts of shifting realities code across journals
The blow up stage had a similar pattern to shifting realities where we see the blow up stage having 35 occurrences, about 47% of all occurrences the last two days (see Figure 6.3). The only other pattern to emerge is an inverse pattern from the blow up and shifting realities, with the el arrebato stage (see Figure 6.4), where the occurrence of the first week (J0~J5) is 48 versus 12 in the second week (J6 ~ J10). The El arrebato, code emerged more in week, reflects the YPAR structure of REALM in pulling generative themes for the youth. In week one multiple activities were done to pull generative themes from youth that reflect injustices which cause el arrebato.
The average of the daily frequency of all codes is about 39.7%.
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14
12
10
8
6
4
2
0 Journal 0 Jouranl 1 Journal 2 Journal 3 Journal 4 Journal 5 Journal 6 Journal 7 Journal 8 Journal 9 Journal 10
Figure 6.3
Counts of blow up 6.3 code across journals
12
10
8
6
4
2
0 Journal 0 Jouranl 1 Journal 2 Journal 3 Journal 4 Journal 5 Journal 6 Journal 7 Journal 8 Journal 9 Journal 10
Figure 6.4
Counts of nepantla code across journals
However, it should be noted that J0 and J1 occurred on the same day with J0 happening at the start of the day and J1 at the end of Day 1. J4 and J5 also happened on the same day as well with J4 representing the first four hours of day 4 and J5 serving as the second fours of the program. By omitting J0 and combining J4 with J5, we receive an average of 53 codes collectively per day from all seven stages; if we do not omit J0 we have an average of 51.6 codes per daily journal entries; if we do not combine J4 and J5 but still omit J0 we have an average
47.7 for every 4 hours of REALM programing; and if we just take the average of each journal 104 entry we have an average of 47 codes per collective journal entry. The appearance of multiple codes each day shows that throughout REALM consciousness was shifting collectively because of the multiple stages of conocimientos youth engaged.
Table 1.2
Averages of codes across each journal entry
Stage of Conocimientos Totals Averages el arrebato 60 4.615385 nepantla 144 11.07692 Coatlicue 47 3.615385 compromison 67 5.153846 putting Coyolxauhqui together 83 6.384615 the blow up 74 5.692308 shifting realities 41 3.153846 Total codes 516 39.69231
Table 1.2 shows the average number of each code appeared for both total and each youth.
The averages shown are consistent with the previous mention of nepantla being engaged the most compared to all other stages. The fact that the average is above one for each stage and on average each youth experience about 39 instances of any of the seven stages of conocimientos is essential because even one case signifies a potential change in consciousness. Minimum and maximum values and standard deviation for the table 1.2 were omitted because the nature of the stages of conocimientos being a collective process. Averages with overall frequencies in table
1.1 can be used to compare individuals to the collective to understand who contributed the totals but become confounded with additional complex statistical analysis.
Results of Individual Youth
Moving from individual days/journals, Table 6.3 shows the observed frequency of the stages of conocimientos for each young person (where Y1=Youth 1, Y2=Youth 2, and so forth in 105 the table). Table 6.3 shows that all but two individuals experienced all of the seven stages of conocimientos.
Table 6.3
Frequency summative codes per youth
Code Y1 Y2 Y3 Y4 Y5 Y6 Y7 Y8 Y9 Y10 Y11 Y12 Y13 C1 5 9 6 1 2 6 2 5 2 5 5 6 6 C2 13 10 10 8 7 19 9 8 12 8 17 9 14 C3 5 7 6 0 1 8 3 3 2 3 4 3 6 C4 3 7 4 4 1 11 5 5 3 6 4 6 8 C5 7 9 8 4 2 10 6 6 6 6 4 6 9 C6 8 7 6 3 1 7 6 6 7 6 4 5 8 C7 3 7 1 0 0 8 2 4 4 3 5 1 3 Total 44 56 41 20 14 69 33 37 36 37 43 36 54 Note. C1 =el Arrebato, C2 = nepantla, C3 = Coatlicue, C4= compromison, C5 = putting Coyolxauhqui together, C6 = the blow up C7 = shifting realities
Y4 had zero occurrences of the Coatlicue stage and the shifting realities, while Y5 had zero shifts in realities. Both Y4 and Y5 were the youngest members of all youth; additionally, they both had the least total number of codes due to writing the least (see Table 6.4).
Table 6.4
Characters written by each youth
Y1 Y2 Y3 Y4 Y5 Y6 Y7 Y8 Y8 Y10 Y11 Y12 Y13 Number of 7647 9426 8007 5366 3529 14134 7491 8646 8823 7968 11218 7468 10998 Characters
The numbers in Table 6.4 show the number of characters that each youth wrote in all of
11 journal entries. On average, all youth excluding Y4 and Y5 wrote 9,256 (least largest integer) characters with a minimum of 7491, which shows Y4 and Y5 wrote significantly less. Based on the results in Table 6.3. and Table 6.4. the number of characters written by each person did 106 impact the number of observed codes where those that wrote the most did have a higher frequency of codes. More youth participants would be needed to determine a direct correlation statistically.
The frequencies from Table 6.1 are to percentages in Table 6.5 along with a density mapping of each column.
Table 6.5
Percentages and densities
Y1 Y2 Y3 Y4 Y5 Y6 Y7 Y8 Y8 Y10 Y11 Y12 Y13 C1 11.36 16.07 14.63 5.00 14.29 8.70 6.06 13.51 5.56 13.51 11.63 16.67 11.11 C2 29.55 17.86 24.39 40.00 50.00 27.54 27.27 21.62 33.33 21.62 39.53 25.00 25.93 C3 11.36 12.50 14.63 0.00 7.14 11.59 9.09 8.11 5.56 8.11 9.30 8.33 11.11 C4 6.82 12.50 9.76 20.00 7.14 15.94 15.15 13.51 8.33 16.22 9.30 16.67 14.81 C5 15.91 16.07 19.51 20.00 14.29 14.49 18.18 16.22 16.67 16.22 9.30 16.67 16.67 C6 18.18 12.50 14.63 15.00 7.14 10.14 18.18 16.22 19.44 16.22 9.30 13.89 14.81 C7 6.82 12.50 2.44 0.00 0.00 11.59 6.06 10.81 11.11 8.11 11.63 2.78 5.56 Note. C1 =el Arrebato, C2 = nepantla, C3 = Coatlicue, C4= compromison, C5 = putting Coyolxauhqui together, C6 = the blow up C7 = shifting realities
The density mapping, sometimes referred to as a hot map ranking, for each value in each column used the darkest color for the highest value and the lightest for the lowest value. For example, in
Y6 column, the cell in row C2 has the darkest background because 27.54 is the greatest percentage among all cells in the column, and row C6 in the same column has the lightest background because 10.14 is the least greatest percentage. Table 5 provides a visual that shows nepantla as the densest with shifting realities and Coatlicue having the least density. If we remove nepantla (C2), Coatlicue (C3), and shifting realities (C7) from the table, we see no major pattern across all youth, as Table 6.6 shows. Furthermore, the lack of a clear pattern among all youth can be seen in Figure 1. REALM can be seen in Figure 6.5, followed by detailed bullet points.
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Table 6.6
Middle densities
Y1 Y2 Y3 Y4 Y5 Y6 Y7 Y8 Y8 Y10 Y11 Y12 Y13 C1 11.36 16.07 14.63 5.00 14.29 8.70 6.06 13.51 5.56 13.51 11.63 16.67 11.11 C4 6.82 12.50 9.76 20.00 7.14 15.94 15.15 13.51 8.33 16.22 9.30 16.67 14.81 C5 15.91 16.07 19.51 20.00 14.29 14.49 18.18 16.22 16.67 16.22 9.30 16.67 16.67 C6 18.18 12.50 14.63 15.00 7.14 10.14 18.18 16.22 19.44 16.22 9.30 13.89 14.81 Note. C1 =el Arrebato, C2 = nepantla, C3 = Coatlicue, C4= compromison, C5 = putting Coyolxauhqui together, C6 = the blow up C7 = shifting realities
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Figure 6.5
Conocimientos across journals and you 109
Figure 6.5 shows the percentage of each stage of conocimientos per journal entry across each journal entry for every young person. Each bar is an area representation of the percentages with each color representing a specific stage of conocimientos. On the top left of Figure 6.5, all youth codes are combined, labeled collective follow by each youth that participated in REALM. If no bar is present, then the youth was absent on that day with the exception of Youth 5 Group 1, who did not write anything for J7 and J9 although Youth 5 attended on that day. Any young person had the option not to write anything if they chose to do, and their choice not to write was not and cannot be questioned, due to the epistemological groundings of the study. Figure 6.5 shows the individual multiplicity of how the seven stages of conocimientos emerged in that no individual had the same experience on any day. For example, Figure 6.6 shows two youth from the same groups, and no two days are the same.
Figure 6.6
Codes across journal of two youth from the same group
The uniqueness of how each of the youth experienced the seven stages of conocimientos was expected due to the nature of the stages of conocimientos focusing on how consciousness shifts.
Recall, conocimientos are a collective process; hence it is vital to see how each group 110 collectively experienced conocimientos. For that reason, to better find patterns, a group analysis was done where youth codes were combined based on the group they work in, in order to capture the collective presence of the codes.
Results of Group Analysis
I analyzed the data to compare the codes in Week 1 and Week 2 (see table 6.7 because the first week heavily focused on identifying an issue that the youth were going to address and the second week had the youth investigating the problem by doing research.
Table 6.7
Group comparison of weekly sums
Week 1 Total Week 2 Total G1 G2 G3 G1 G2 G3 C1 15 14 19 8 1 3 C2 27 26 31 21 22 17 C3 4 7 7 11 9 9 C4 7 9 14 12 15 10 C5 13 17 11 17 11 14 C6 6 6 7 19 20 16 C7 3 6 4 8 12 8 Total 75 85 93 96 90 77 Note. C1 =el Arrebato, C2 = nepantla, C3 = Coatlicue, C4= compromison, C5 = putting Coyolxauhqui together, C6 = the blow up C7 = shifting realities
Table 9 shows that 75 to 96 stages of conocimientos emerged each week. Each group did experience more el arrebatos (C1) the first week and the blow up (C6) stage occurred more the second week. Group 3 did experience about half as many occurrences of nepantla (C2) in the second week, which may reflect Group 3 have a decrease in overall code occurrences (93 in
Week 1 and 77 in Week 2, a difference of 16). The reduction of codes for group 3 reflects one of 111 the youth only have four journal entries the second week. Both Group 2 and Group 3 had an increase of 21 and 15, respectively of codes from Week 1 to Week 2. Understandably, codes increased the second week as youth became more comfortable with sharing their thoughts.
Overall, codes are condensed by each group in Figure 6.7. and show the distribution across each journal entry in a side by side comparison.
Figure 6.7
Group Comparisons of codes
The collective nature of conocimientos allows us to treat each group as an individual, given they have shared experiences. We see that for all groups except for Group 1on J0, putting
Coyolxauhqui together did not start to emerge for all groups until the end of day 1 for two in J2.
For all groups, the blow up occurred more the second week with J0 having the least occurrence.
Group 1 experience all seven stages of conocimientos on J2, J7 and J10, while group 3 112 experienced all stages on J3 and J10. Group 2 did not experience all seven stages on any individual journal, but given J4 and J5 occurred on the same day, Group 2 did experience all seven stages on Day 5, along with Group 3. A comparison of code frequencies of each group shows a similar number of all codes, 171, 175, and 170 for each group.
Unlike Figure 6.5, Conocimientos across journals and youth, Figure 6.7, Group
Comparisons of codes does show a pattern among groups for example, group 2 and group 3 had similar distribution for J4 with the exception of the occurrences of shifting realities on J4 for groups 4. Additionally, all groups had the least number of different codes on J5. J4 and J5 occurred on the same day that consisted of researcher training where all the activities where not done in their regular groups. The similarities of each group both by the number of unique codes for each journal entry and by the proportional comparisons seen in figure 6.7 allows us to further expand the collective by combining all youth codes across all journal entries.
Results of Collective Analysis
Given all youth were in a shared learning space, it is essential to understand how all stages of conocimientos shifted across the duration of REALM. Figure 6.8 represents all code occurrence for the youth collective across each journal entry, where, how much area each color covers represents how frequently each stage of conocimientos emerged from the journal entries.
In Figure 6.8, the similar trends of nepantla being the most engaged stage and the decrease of the number of observed instances between Week 1 and Week 2. All seven stages of conocimientos emerged in all journal entries excerpt Journal 5 and Journal 6. Collectively the sifting realities stage and the Coatlicue stage appear balanced and consistent throughout REALM except for
Journal 8 where no shifting realities stages were present. Furthermore, the multiplicity of the seven stages of conocimientos did result in excerpts being double or even triple coded. In Figure
6.8, putting Coyolxauhqui together decreases as the blow up increases because putting 113
Coyolxauhqui together represents individual transformation, and the blow is the end of internal transformation due to collective understanding.
100% 90% 80% 70% 60% 50% 40% 30% 20% 10% 0% J0 J1 J2 J3 J4 J5 J6 J7 J8 J9 J10
el arrebato nepantla Coatlicue compromison putting Coyolxuhqui together the blow up shifting realities
Figure 6.8
Collective codes of all youth across journal entries
By the end of the program, youth were more comfortable with each other and had formed a collective, making engagement in the blow up easier to manifest.
To see how the multiplicity occurred during the coding process, I created a matrix of co- occurrences of codes (see Table 6.8). Table 6.8 shows the co-occurrences of the seven stages of conocimientos, and as expected, about 30% of all co-occurrences are related to nepantla. What this means is 30% of the time, nepantla emerged simultaneously with another code. In Table 6.8, nepantla and compromison had the highest frequency of co-occurrence, which represents any mention of mathematics changing the future. By saying the future can change youth are in the compromison stage and because youth are talking about the world of mathematics and the future 114 world, it also represents nepantla, the clash, comparison, and distance between worlds. In total
260 of the 516 codes represents more than two codes, which will be further discussed in the next chapter along.
Table 6.8
Co-occurrences of codes
C1 C2 C3 C4 C5 C6 C7 SUM el Arrebato, C1 0 11 4 2 7 1 2 27 Nepantla, C2 11 0 10 23 16 10 7 77 Coatlicue, C3 4 10 0 3 4 2 4 27 Compromison, C4 2 23 3 0 5 3 6 42 putting Coyolxauhqui C5 38 together, 7 16 4 5 0 3 3 the blow up, C6 1 10 2 3 3 0 4 23 shifting realties, C7 2 7 4 6 3 4 0 26 SUM 27 77 27 42 38 23 26 260
The multiplicity of each stage of conocimientos creates an infinite number of combinations because of the endless possibilities of nepantla. The next chapter will continue to explore the co- occurrence of the stages of conocimientos in further discussing how consciousness shifts in the learning of mathematics.
Chapter Summary
This chapter is quintessential in seeing the possibility of the seven stages of conocimientos manifest within a mathematics infused (Y)PAR EntreMundos program. Observed instances of codes per youth shows the uniqueness of how individuals experience the stages of conocimientos and begin to answer the second research question.
How do high school students experience the stages of conocimientos while engaging in a mathematics-based youth participatory action research EntreMundos project during a summer program, behind this dissertation. While chapter 5 showed how mathematics and YPAR can function 115 together, chapter 6 builds off of the context of the experience described in chapter 5 by showing how the stages of conocimientos represent consciousness shifting throughout REALM.
Furthermore, by viewing groups as individuals and then viewing all youth as one collective allowed for patterns to emerge in showing how the stages of conocimientos shift across the duration of REALM in answering the second research question. The next chapter discusses the seven stages of conocimientos within mathematical reflection and action related to mathematical learning.
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CHAPTER 7. DISCUSSION & CONCLUSION
In this chapter, the stages of conocimientos are discussed related to mathematical learning. Understanding how each stage shapes and is shaped by mathematics allows for a greater understanding of how consciousness shifts in answering the second research question,
“How do high school-age youth experience the stages of conocimientos while engaging in mathematics-based youth participatory action research EntreMundos project during a summer program?” Each stage of conocimientos represents a particular junction for viewing mathematics as an embedded motion towards an ontological understanding of mathematical learning. This chapter will explore an embodiment of mathematics ranging from feelings and emotions connected to mathematical knowledge construction along with mathematics as a form of critical literacy in understanding the rich histories of mathematics. This research shows how the stages of conocimientos come together to allow for an exploration of mathematical knowledge construction that results in a change of consciousness. This chapter will discuss each stage of conocimientos in relation to mathematics education. After each stages is given attention, an example of one individual's journal entry will highlight how all seven stages come together in discussing the implications of conocimientos within mathematical learning.
Discussions of the Seven Stages of Conocimientos
In this section, the youths voice will be central in discussing the impact the stages of conocimientos have on mathematics. Each section begins with a quote form Anzaldúa to provide a departure point in honoring the philosophical lineage of framing mathematics epistemologically as a living entity, ontologically as a relationship and axiologically as an experience within a (Y)PAR EntreMundos paradigm. The chapter ends by conceptualizing the seven stages of conocimientos with|in mathematics. 117 el arrebato: a catalyst for mathematics
Your relationship to the world is irrevocably changed: you’re aware of your vulnerability, wary of men, and no longer trust the universe. This event pulled the linchpin that held your reality/story together and you cast your mind to find a symbol to represent this dislocation. (Anzaldúa, 2013, p. 546)
The start of learning is at the heart of el arrebato where dislocation leads to questions and where the process of finding answers comes from relationships made with others and the world.
El arrebato can be found at the end of learning as it can lead to new questions asked as people reflect upon what they just learned, or they can be shaken up by what they do not still know in the middle of mathematical learning. El arrebato is as much as a starting point, as it is a continuation of education. More so, at any aspect of the learning process, an el arrebato can manifest due to the context of the learning and/or how it impacts the learner's world. For example, on Day 2, when one of the youth looked for mathematics in the newspaper, they stated,
“looking at the newspaper it surprised me how easily things can be overlooked.” I coded the act of being surprised as an el arrebato, and it is occurred when the information within the content of the activity is being overlooked. When foreign and abstract context is used, learners feel distant and disengage, causing an additional shakeup. The result of this research revealed that El arrebatos are signifiers of opportunities to learn mathematics connected to either positive or negative emotions when learners are engaged in learning with mathematics and live with the other stages of conocimientos. A positive example of an el arrebato is a statement of one youth,
“it surprised me how much math you can find in a newspaper about everyday things.” Both positive and negative reactions to learning led individual youth to asking more questions and engaging with the other stages of conocimientos. In a journal entry, one young person wrote about learning math in one word, “Although I love a-ha moments, so one word! Confused.” This shows the joy of being surprised by mathematics with a-ha moments but also shows the 118 confusion of learning mathematics to emphasize that el arrebatos are actions and emotions that lead the youth to learning.
When mathematics connects to strong emotions, el arrebato is engaged. One of the strongest emotions is joy, as was seen when one young person described mathematics by saying
“yayyyy!!!:)” and when a different young person acknowledged mathematics is “educational, fun, and useful.” Both young people articulate excitement by saying yay and fun but what is equally powerful is noting that mathematics is something that can be fun and useful. Not being aware that mathematics can be helpful represents an el arrebato that will never be experienced.
El arrebato is important because it represents the shake-up needed to have learners realize that mathematics is useful, fun, cool, confusing, and more than emotionless logic. The multiple ways people see mathematics is inherent because of the complexities of how mathematics manifests in each individual, and el arrebato along with the rest of the stages of conocimientos are one way to understand mathematics, self, and the world better. Take the example of a youth stating “so [I] don’t see myself as a mathematician just because I’ve never been good at it, I’ve always struggled or end up frustrated in the end.” We can see how struggling and being frustrated leads to a shake-up on how they see themself as a mathematician and if this el arrebato is left alone then it will only continue to affect this young person in a not positive way. The same young person wrote the following journal entry on the last day of the program
The difference between the math we’ve learned in the past 2 weeks and what we learn at school is that, the math we learned here was connected to some part of history or poetry and I’ve learned something new while doing math but at school they don’t tie math to outside stories, it’s just all about math. … One thing I would tell my math teacher to change [is] the way they teach would be that connecting math to the real world have so many kids to get involved, it would have their attention more and maybe more students would pass.
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The el arrebato of not being a mathematician because of its difficulty can be challenged by shifting realties stage in connecting mathematics to the real world (nepantla), so it can be more engaging. Connecting mathematics to history and current “outside” stories can lead to the compromison in changing the classroom (and world) because both are not fixed conditions.
Without el arrebato people cannot act to change the world (compromison, the blow up and shifting realities) or to change themselves (Coatlicue, putting Coyolxauhqui together, the blow up) and nepantla, an infinite zone of possibilities cannot be experienced. As you continue reading this chapter, note these words from one of the REALM youth that represents el arrebato,
“I promised myself to come out of my comfort zone and I did.”
Nepantla: Embracing Infinity Nested in Infinity
You can’t stand living according to the old terms-yesterday’s mode of consciousness pinches like an outgrown shoe. Craving change, you yearn o open yourself and honor the space/time between transitions. (Anzaldúa, 2013, p.549)
Nepantla is multilayered that it represents all forms of epistemic friction between any two people and ideas that are different. Every time you read something, you are engaged in nepantla in that your ideas (worlds) meet the ideas (worlds) of the writer. The differences between the reader and writer can be absolute, minimum, or near zero, where the distance between the two is nepantla. Learning mathematics within nepantla acknowledges the differences between teacher and leaners, along with the differences between learners. Furthermore, nepantla welcomes the differences between individuals and ideas as it relates to the multiplicity of mathematics and the multilayered facets of identity. An example of how nepantla is nested and represents the comparison of multiple worlds can been seen in one youth writing “math is everything, from the structure of our bodies to the layout of a building, math can be applied to everything” where see the idea of mathematics being everything is comparing the world of mathematics (theoretically positioned as everything and as applied form of mathematics i.e. theoretical versus applied) with 120 the structure of body and the structure of buildings. Although nepantla emerged throughout the project, the interaction between mathematics and self is seen when youth wrote their “I am
Math” poems, as one youth wrote how they were “compared with a [geometrical] shape in a poem.” The interaction between mathematics and poetry was one-way that the youth could reflect upon mathematics related to themselves due to the intimacy involved when constructing a poem.
Nepantla is an exploration of worlds through connections between worlds, where nepantla can be used to explain all facets of life. In life, it is common for numbers to signify mathematics; thus nepantla codes that look at the worlds (concept) of mathematics and numbers show how mathematics connections a wide range of ideas. Table 7.1 shows how youth bridged mathematics to other worlds through their journal entries about what they did and learned from this project, while Table 7.2 shows how numbers connect to a wide range of worlds.
Table 7.1
Mathematics and other worlds
Excerpt from Journal Entries Engaged Worlds want to learn more about circles in mathematics and how math connects to art Math, circles and art different ways people used to do math. Math and different ways Finding the similarity between math and everyday life really opened my eyes Math and everyday life what games correspond to how they learned math math and games Along with learning math I learned more about history while reading questions. Math, history and reading To me math is a way of understanding life. It's like language that can be used to decode, Math, life and language math can be used in many ways to show records from sports of all Math, diversity and sorts and to also the diversity of places like schools. sports records Math, science, art and Math + science are my favorite subject besides Art favorite subject what we have been learning is more important than normal math Math and math
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As we can see in Table 7.1 connects nepantla and mathematics to art, history, reading, language, life, science, diversity, sports games and different ways of solving various types of math. It is essential to view mathematics through multiple lenses to understand how each form of mathematics connects to all people. During REALM, a youth wrote “what we have been learning is more important than normal math,” where they are comparing and adding value to two different ways of experiencing mathematics. The idea of normal math (classroom mathematic) is problematic as it creates a divide between the mathematics learned at school and the mathematics experience/lived outside of the classroom. Nepantla is a focus on the paths that connect these multiple worlds.
Table 7.2
Numbers and other worlds
Excerpt from Journal Entries Engaged Worlds calculate numbers for our living and learning. "Math is a concept" Number and calculations number system of ancient civilizations Number and history math is a way of using numbers to solve problems. numbers and problems number systems from different tribes and places. Numbers, culture and places The use of numbers and formulas. Numbers and formulas variable in a problem solving situation. Variables can vary from Numbers, letters and letters to numbers problem solving Math is whatever and wherever you can put 2 or more # together Numbers and Numbers Mathematics is a series of patterns and numbers. Numbers and patterns It can be found in plants, art and everywhere. Rather than giving us simple math problems w/ no context, Numbers and reading numbers and solving equations. numbers and equations Numbers create a ratio or percentage that back up a person's Numbers, ratios, statement. percentages and validity
In Table 7.2 we can see that numbers, a universal signifier of mathematics, also produce the following multiple worlds: calculations, history, problems, culture, places, formulas, letters, different types of numbers, patterns, equations, ratios, percentages, reading and claims of 122 validity. We can see an overlap between mathematics with numbers, which shows how multiple worlds can be connected.
Nepantla and mathematics is more than numbers in that it represents the culture of mathematics as well. The excerpt of one youth, “I don’t feel like I’m as smart as other kids,” represents what one person believes to be their ability compared to what they believe is the ability of others. The comparison of ability within nepantla and mathematical learning, along with the other stages of conocimientos should not lead to students feeling bad about comparing abilities with each other. Nepantla is the bridge between you and your ability where the bridge between you and a fellow learner also becomes a bridge the connection between them and their ability. Nepantla is space that allows knowledge to be shared, explored, and constructed with others. Nepantla and mathematics acknowledge multiple worlds and the possibility to connect to those worlds. Teachers are continually engaging in nepantla when teaching students and when teaching mathematics, but are they aware of or they creating more distance between themselves, mathematics, and their students? Nepantla is crucial for teachers in knowing the multiple worlds
(concepts) of mathematics can be connected to learners abilities. As best said by one of the youth when asked if they were president of their mathematic class, “I would try to instill the in- depthness of each topic because many kids always wonder what each thing is for and why it is used.”
El Compromison: Mathematics can change the world
You re-member your experiences in a new arrangement. Your responses to the challenges of daily life also adjust. As you continually reinterpret your past, you reshape your present. Instead of walking your habitual routes you forge new ones. The changes affect your biology. The cells in your brain shift and, in time, create new pathways, rewiring your brain. (Anzaldúa, 2.13 p. 556)
Mathematics and el compromison focuses on changing society as stated by one of the REALM youth, wherein “trying to discover better ways to improve problems in the world” with the 123 acceptance that changing the future is possible. Two different young people wrote, “[m]ath can be used to describe the world we live in and change it for the better” and “math can change the world because we use it every day to do challenging problems” best articulate the significance of el compromison. Mathematics should not be limited to finding a better or more efficient approach to a problem, and a better approach needs to lead to changes. el compromison has inherently been part of mathematics since its origin, in that new types of mathematics have always been possible. The evidence from REALM with respect to el compromison is related to learning mathematics, diversity, and the history of multicultural mathematics.
Mathematical learning must connect to the real world for learners to connect and know they can change the world. Youth wanted to learn new ways of connecting mathematics because they were already aware that new skills can be useful for their future. Mathematical learning allows for multiple futures to relate to the multiplicity of each individual and their experiences.
When asked what is one thing you would tell you, teacher, on the last day of REALM one young person wrote “[o]ne thing I would tell my math teacher about changing the way they teach is to understand how their students learn and to fit that into their experiences.8” El compromison thrives in embracing the multiplicity of a person’s identity and lived experience because multiple histories represent multiple potential futures. Mathematics can welcome the histories of students and incorporate them into the classroom, or we can choose to do nothing and create additional barriers for students to learn about the real world when learning mathematics.
Part of the Anzaldúa’s quote used to open this section stated: “as you continually reinterpret your past, you reshape your present,” whereby in reshaping your present, you are creating a new future. The desire of youth to want to learn about multiple histories of
8 This quote and others throughout this chapter were initially used more than once to emphasize a single passage having multiple codes representing the multiplicity of the seven stages of conocimientos. 124 mathematics has a deep impact on how youth see mathematics which can be seen as one young person wrote “math is a huge part of our world and no one acknowledges the importance of history it has on it which is fun to know,” where not only is history important but it is fun. For marginalized people, el compromison and mathematics allows for the reclaiming of ancestral knowledge. For the Latinx REALM youth learning about their ancestors and mathematics came from Patolli on the first day. Patolli being one of the youth’s favorites activities, along with multiple students writing that they learned the true origins of mathematics while posing why they do not learn about their cultures’ mathematical history in school, shows the impact el compromison can have. One youth wrote, “I learned about history in various cultures by solving equations of what they go through. I loved what they did [by] incorporating history with math because it’s a double learning experience” showing the potential of learning about mathematics and history being a rich learning experience. El compromison is an acknowledgment of ethnomathematics as it empowers marginalized youth by allowing them to know they come from a mathematically advanced people. Due to systemic racism, Black and Brown K-12 students are still taught to believe they are lesser when compared to their non-student of color peers
(Valenzuela, 2016). Ethnomathematics is then vital as it shows multiple historical and cultural forms of mathematics. The multiplicity of mathematical origins is the root of knowing multiple futures are possible, i.e., the world is not fixed.
Ethnomathematics and its history have always been a third world mathematics (Gerdes,
2004), and due to the overwhelmingly Eurocentric mathematics curriculum in the United States, ethnomathematics represents a challenge to the status quo. Ethnomathematics seeks to celebrate diversity in a system that still views diversity and change as a problem. El compromison is needed in order to accept diversity and to challenge injustices. This was seen during REALM 125 when youth worked with ratios and made generalizations using data taken for the National
Federal Bureau of Labor Statistics in looking at the median weekly income of men and women in the United States of the first quarter of the current year. As one youth mentioned, “mathematics is a series of patterns and numbers. It can be found in plants, art and everywhere” and “math is a creative outlet it can be used to bring out problems within our society the wage gap, gender inequality, racism.” The act of “bring out problems” requires a historical awareness of the societal issues and numbers and patterns allow for a greater understanding of the problem in extrapolating a solution. As one young person stated, “This is a real world situation that women get treated less than men which is an issue waiting to be solved.” When this young person wrote waiting to be solved, it shows that mathematical problem-solving views the world as dynamic
(and not fixed) because a solution is possible for any problem and that possibly made in to a reality will change society. Mathematics that lives with el compromison can best be summarized by these two youth voices:
Youth Voice 1: Seeing where math comes from makes me feel like a mathematician because math in everywhere. Youth Voice A: I think that inequality should be taught in math class because people need to be informed about the truth in society.
Being informed about the mathematical truth requires an examination of multiple mathematical histories along in uncovering inequalities and dismantling barriers that keep us from the potential of an equitable future.
Putting Coyolxauhqui Together: Reading Mathematics to Know Self and World
To treat the wounds and mend the rifts we must sometime reject the injunctions of culture, group, family, and ego. Activism is the courage to act consciously on our ideas, to exert power in resistance to ideological pressure – to risk leaving home. Empowerment comes from ideas … [b]y focusing on what we want to happen we change the present. (Anzaldúa, 2009, p.246).
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Coyolxauhqui represents transformation in deconstructing and reconstructing one’s own identity. Where el arrebato and mathematics represent how emotions are drawn out in the process of learning, putting Coyolxauhqui together focuses on how reflecting upon emotions, feelings and desires constructs identity. During REALM, after the Ratio’s of power activity, which looked at the gender wage, a young person reflected by writing “This activity taught me and replenished my math skill that I have learned the statistics for the Latinx community.” The idea of learning and replenishing a person is a reconstruction of their ability. Additionally, for young people to learn more about their community it is vital for them to understand problems in their community at deeper levels. The el compromison and mathematical learning is for understanding the past and the future, where putting Coyolxauhqui together allows mathematics to aid learners in understanding how the present self fits between the past and future. During
REALM a young person wrote “as people learn math now a days, they don’t learn how useful it can be to predict or determine statistics to paint a picture of someone’s struggle” when reflecting on the finding the rate of change of students, teachers and principals of color over time in their school district. It captures the importance of mathematical critically literacy in understanding issues within society. Putting Coyolxauhqui together is mathematical critical Literacy and the transformation of self, others and mathematics.
Critical Literacy in mathematics directly connects mathematics to the real world, as stated by a REALM youth, “the math we did lead me to think more about how math can be used in the world around us.” Furthermore, when young people learn about the world in the classroom, they begin to see that mathematics exists outside of the classroom diminishing the divide between mathematics in the classroom and outside of the classroom as two different entities. As one youth wrote, “math can be used outside the math classroom. Using statistics, we 127 can understand populations on the larger scale” where such an understanding is at the core of critical literacy. Critical literacy is central to developing a critical consciousness and REALM youth had multiple opportunities as they learned history in learning about the mathematics of the
Aztec people; they learned the history behind social programs created by the Blank Panthers
Party; they learned about the Delano Grape strike; they learned about the wages of men and women in the United states; they explored mathematics found in local and national newspapers; and they learn about the demographics of the students, teachers and administrators in their schools. Critical Literacy was an essential aspect of REALM and putting Coyolxauhqui together that shifted mathematical critical Literacy from an understanding of the world to an understanding of how an individual transforms and constructs their consciousness due to better understanding the world, a critical self literacy.
REALM allowed youth to see how mathematics can be applied to every aspect of our lives — even our feelings by creating a space described by two different young people, where
“I’ve grown way closer to my peers because I allowed myself to gradually open up on my own time and because of that I had a great time connecting with others” and where “I learned about people’s insights and different perspectives. I also learned more about myself as a person.”
Putting Coyolxauhqui together and mathematics is the simultaneous literacy of the world and self. When one young person wrote their “I am from Math” poem they connected with mathematics from their own history, and it allowed them to explore how they saw themselves in the world, as one young person said, “One thing I enjoyed from writing my poem was the feeling of being able to express who I am and what make me.” Discovering who you are and the agency each individual has is an empowering process that shows the transformative power of putting
Coyolxauhqui together. REALM was transformative in that it allowed youth to see mathematics 128 different, as one young person wrote on the last day, “[the] difference between the math we learned [in REALM] vs school math is [that] the way it was taught here really changed my perspective.” A change of perspective when learning mathematics should be occurring in every classroom, putting Coyolxauhqui together works to transform mathematics so there is no difference between home mathematics, classroom mathematics and the mathematics learned during REALM. It is a change in perspective (inherent with critical literacy) that can empower learners, as mention by a REALM youth, “learning math these last two weeks made me feel empowered to have such a skill.” Putting Coyolxauhqui together as a way of learning mathematics is central to the formation of constructing a relationship with mathematics in shaping an individual’s mathematical identity.
The Blow Up: Meeting other to write Mathematics
[N]o longer feeling ourselves “sick,” we snap out of the paralyzing states of confusions, depression, anxiety, and powerlessness and we are catapulted into enabling states of confidence and inner strength (Anzaldúa, 2009, p.122).
Identity and consciousness do not form in a vacuum, where putting Coyolxauhqui together represents the process of individual transformation, the blow up is the internal change at the end of the process of transformation because of others. The blow up is when the individual acknowledges collective learning and how learning has implications for everyone's future, occurring in mathematical learning when an individual sees beyond themselves. The understanding of mathematics being a collective learning process is exemplified in a young person reflecting on what is math on the last day of REALM, "we understand the world with math because we try solutions to make the world livable for society with advance technology and more" which led another youth to say "It made me feel so empowered to have such a revolutionary skill." The first quote highlights an understanding of the world (critical literacy) leads to solutions for all of us, as we understand the world with math, and the second quote 129 returns to the individual where they are now empowered. Putting Coyolxauhqui together and the blow up are deeply intertwined yet differ because the blow up focuses on others, and it is a collective understanding that signifies the results of internal transformation.
The impact of the blow up in mathematics is a challenge against individualism in mathematics education in advocating for group work that leads to quality mathematical discussion and collective understanding. On Day 2 of REALM youth were asked if they were in charge of their mathematics class what would they do, one youth wrote, "I'd make sure everyone understands the lessons even some of the kids who sit in the back," showing a desire for collective understanding in seeing the world outside of themselves. Another youth wrote "group projects are no longer prioritized but asking for helping each other is encouraged" which highlights that the blow up does not just mean group work is the best approach. Without a classroom space that welcomes collective understanding, group work will function as a continuation of individual learning in the classroom. For that reason, it is essential that the context of learning is engaging and connected to the real world. In REALM the Ratios of Power activity is an example that allowed youth to have a discussion and engage in real world data, where one young person reflected on the activity in writing "I feel like activities such as this ratio activity allow people to open their minds as they learn math." In the previous youth quote, we see that they (individually) believe the mathematics activity affect others, by opening their minds
(collectively) provide an example of the blow up and mathematics.
The blow up with respect to mathematics requires teachers to question their role in how students learn. Teachers are the gatekeepers to the content, context, and connections made due to pedagogical approaches to teaching (Martin, Gholson, & Leonard, 2010). During REALM one youth responded to the question, why do they think teachers do not utilize lessons with historical 130 content and real-world connections by writing "I think my math teacher doesn't use these b/c that's not how they taught them to teach." This shows an interesting take on the blow up by connecting the teachers teaching knowledge to how they were taught in that it shows the teachers need for understanding mathematics as a collective endeavor of self and societal becoming, which provides an example of why el compromison is so important in breaking the cycle of teaching how we were taught because the future is dynamic and always changing. A teacher should first ask themselves "How can math be used to understand the world?" so they may arrive to what the REALM youth learned to see that "Well math wasn't just found in one place. So, to understand math, we must open our minds to being more understanding and loving towards cultures." Ethnomathematics can be part of the blow up in that multiple forms of mathematics can help shape how an individual sees themselves as mathematical. In REALM ethnomathematics was central in youth, forming a collective in bridging history to the real world while doing mathematics activities. The blow up youth experienced due to REALM can best be seen in the follow two excerpts, "I learned more about myself… understanding more about understanding people's perspective" where we see learning about others leads to individual change and REALM activities "surprised me how it really connected to who I am and who we are" where the surprise (el arrebato) is not just I (individualism), it is we (collective understanding).
Shifting Realities: Counting our Acts of Resistance
The pull between what is and what should be. I believe that by changing ourselves we change the world … a two way movement – going deep into the self and expanding out into the world, a simultaneous recreation of self and reconstruction of society. (Anzaldúa, 2009 p.49)
Shifting realities and mathematics represent the external action taken upon the world to disseminate mathematical knowledge that keeps us from ourselves and others. On Day 2 of 131
REALM when two different young people asked if they could have a copy of the Patolli game board to share with their families, they committed to an action that shares mathematics with others. The worlds (nepantla) of the classroom and the home come together in shifting realities when students bring the mathematics they learned in school home with them – without being asked what they learned in school – when teachers allow for students to share their funds of knowledge in the classroom. Putting Coyolxauhqui together and the blow up provide context
(critical literacy) to ensure the mathematics allows for the formation of conocimientos. The stages of conocimientos are not linear, and the action of shifting realities does not have to be a direct action but can result from reflection of a previous action. For example, one youth wrote on a journal entry as follows,
I really hope middle school students can have the opportunity of learning how they can use math in the real world. I remember asking my sister how can fractions be used and she said for splitting foods like apples. But in truth fractions can be used for so much more.
By learning the importance of connecting mathematics to the real world, this young person was able to reflect to a time when they spoke to their sibling about the use of fractions, where they were engaging with their sibling is shifting realities. Another example from a different young person, "Now I do see myself as a mathematician I use it in my daily life when baking when shopping and the types of math we learned at school," shows how reflecting upon seeing one’s self as a mathematician leads the recollection of previously using mathematics in the real world.
Shifting realties links reflection to action and actions to reflections. Shifting realities does not only look back at previous actions but looks forward to future action. For example, one young person at the end of REALM wrote, “Because of this program I can tell my math teacher to teach us of our deep math's history to encourage us to use this skill to better our community and better the lives of those around us.” In this excerpt we can see two actions, the action of telling the 132 teacher what to teach and the action of bettering the lives of others, which represents a shifting realities. This excerpt also represents putting Coyolxauhqui together in that the young person mentions how the program is the reason for the action that they (an individual) can do; it represents the blow up in acknowledging making the community better for all of us (collective acknowledgement); it represents el compromison in showcasing the betterment of the community is possible along with mentioning math's history; and we see the multiple worlds (nepantla) referenced. Furthermore, this shows the multiplicity of the seven stages of conocimientos and the impact REALM had on the youth. As one person said about their experience, "Through our stories, we realize why we are who we are and [it] made me open my eyes to the connections in life" where the significance of shifting realities and mathematics is the ability to make connections in life through actions. Hence, critical literacy in mathematics education should learn from YPAR epistemologies and commit to understanding mathematical actions for social transformation.
Coatlicue: The Power and Fear of Knowing Mathematics
But now everything has meaning and is sacred – the people, the trees, you. There aren't some people who are more important than others, even though I love some people more. … The definitions, categories, and restrictions society has put on these activities are wrong, not the activities themselves. (Anzaldúa, 2009, p.86)
Multiple perspective come together in that the pain of knowing is a pain rooted in dissonance due to differences and unknown similarities. Harmony begins by healing mathematical trauma through experiences and the fostering of motivation comes from Coatlicue because "math is fun and at the same time sad." Mathematical trauma begins at school when "the teacher shows you and then you have to regurgitate it back to them," a form of banking education, a dehumanizing of mathematical learners (Freire, 1996). The pain of mathematics is not only found in school but is interwoven in society where school mathematics is a form of 133 status. You are smart if you know school mathematics and being smart comes with the privilege of seeming rational and having the right to make claims. Knowledge construction is key in understanding Coatlicue and mathematics because when young people do not believe they can contribute to their own education then they do not see education for them. Mathematics interwoven with YPAR allows for young people to know the injustices that limit them and give them an opportunity to recognize their own agency. As one REALM youth states "After
[REALM] I feel like math in school is missing something," the unsettling feeling of school mathematics missing something is powerful because post REALM youth will be aware that their high school math class makes them feel like they do not belong, and they will know that it is not their fault but that of the teacher (and school system as a whole). Both being at fault and being a victim are painful, but if a person believes they are at fault when they are not, it begins to stripe away their agency. Where if a person knows the system is the problem and not them, then they can engage other stages of conocimientos and commit to action in transforming the system. El arrebato is the shock that pushes a person to change and Coatlicue is the fuel needed to dive deeper into nepantla.
Coatlicue has a profound impact on mathematical learning in that it can cause barriers to student reflections, or it can motivate them to continue learning, highlighting the role of the teacher in being aware of Coatlicue. A teacher needs to recognize what they do and what they do not do can cause youth trauma in the mathematics classroom. When asked what is one thing you would want to tell you teacher one young person said: "why don't our teachers tell us the "truth" about our Ancestors and where math actually comes from" and a different young person wrote to teachers saying "Students wouldn't hate you if you weren't so condescending. Students wouldn't fall asleep if you gave them something to care about." This evidence shows students are acutely 134 aware of how teachers see them, and it negatively affects how they learn. In REALM, Coatlicue, is indicative of the lack of ethnomathematics and diversity in mathematics. The lack of acknowledging the mathematical histories of all people creates a hierarchy making those already part of the curriculum (white, heterosexual, Christians) feel welcomed and everyone else othered. As best stated by one of the REALM youth, "when I'm at school I don't feel like I'm as smart as other kids" which is a consequence of teachers not seeing/knowing that mathematics is already alive in each student, it just may look different.
Earlier it was mentioned Coatlicue is the fuel needed to travel nepantla, but Coatlicue must engage with other stages as well because it is the collective fuel of everyone that creates the energy for mathematical knowledge construction. Table 7.3 highlights how Coatlicue interacts with other stages.
Table 7.3
Coatlicue co-occurrence
Excerpt Code(s) It made me feel so small to the white population of our schools. Coatlicue & nepantla Using statistics and numbers we can use graphs and information to determine outcomes of games to understanding things like discrimination, depression, and anxiety in the real world. Nepantla & Coatlicue Coatlicue, nepantla, I think that inequality should be taught in math class because putting Coyolxauhqui people need to be informed about the truth in society. together & the blow up This is a real-world situation that women get treated less than men Coatlicue, nepantla and which is an issue waiting to be solved. shifting realities
The first excerpt shows that after learning about the demographics of student of color, it made the student feel small (Coatlicue). The second excerpt highlights nepantla in various types of math references, which are then connected to discrimination, depression, and anxiety. Knowing the system is acting against you is painful, and not doing anything about your situation can lead 135 to depression. The importance of conocimientos is that it is a collective process that helps to move through depression and anxiety with others through action. The third excerpt shows the collective spirit of transformation in that the young person shares what they think (putting
Coyolxauhqui together) because other people (the blow up) need to be informed about the truth of inequality (Coatlicue). Inherent in the last excerpt is shifting realities in that an issue waiting to be solved is an issue that requires action.
Conscious Raising Mathematics
Mathematical spiritual activism represents mathematical spiritual wisdom that seeks to build collective action in bringing about social transformation in and outside of mathematics. Mathematical spiritual activism reflects the mathematical and collective-cultural knowledge gained as and through conocimientos with others. (Martinez, Lindfros-Navarro & Adams-Corral, forthcoming)
As the seven stages of conocimientos are engaged, a new mathematical consciousness is formed through mathematical reflection (critical literacy) and mathematica action of shifting realities, which, once balanced, is a form of revolutionary praxis. Overall, REALM provided an answer of the question, “How can mathematics as a critical literacy and critical action lead towards a revolutionary praxis?” Understanding how the seven stages of conocimientos function with mathematics leads to see an answer of “How can mathematics be a conscious raising experience?” Below is what one young person wrote for their final reflection (Journal 10) of
REALM. I believe it is important to see how one individual collectively experienced the stages of conocimientos in reflecting on the entire experience to show how mathematics can be a consciousness raising experience. I included the entire journal entry in this section to provide an authentic discussion about mathematical consciousness that emerged from this project.
Journal 10 What is mathematics? Mathematics is a series of patterns and numbers. It can be found in plants, art and everywhere. Math is a creative outlet it can be used to bring out problems within our society the wage 136
gap, gender inequality, racism. Math is fun and at the same time sad. The series of patterns in life can be found through math. Math can be used to describe the world we live in and change it for the better. What is one word you would use to describe how you feel about learning mathematics when in school? Dull because when learning mathematics in school we barley on how math can be used in the real world. What is one word you would use to describe how you felt about learning mathematics over the last two weeks? Hopeful I feel hopeful because I learned how I can use math in the future. I really hope middle school students can have the opportunity of learning how they can use math in the real world. I remember asking my sister how can fractions be used and she said for splitting foods like apples. But in truth fractions can be used for so much more. Can you explain the difference between the math we have done over the last two weeks and what normally gets done in your math classes? The math we did over the last two weeks was more interactive instead of just learning the equation of a line and slope the numbers actually had meaning and related to our world today. The math we did lead me to think more about how math can be used in the world around us. Are you a researcher? Yes Are you a mathematician? Yes Are you a scientist? Yes What is one thing you would want to tell you math teacher about changing how they teach? Show us how math can be used in the real world with videos and make math more failure receptive it someone gets a problem wrong be more understanding. Here is where it went wrong but you can fix it. [by asking] What does this represent? [when looking at graphs] This is what I want to be changed instead of doing the same process and not exploring. Explain how I can use it in the future and why is it important. Give meaning to it.
The young person whose final reflection is shared above is one of the individuals that presented to the administration at the end of Day 9, where this young person did take part in each of the previously mentioned key turns (see Chapter 5). The first question in the journal, what is mathematics, shows mathematics connected to a wide range of topics (nepantla), from nature to creativity to raising awareness on societal issues (putting Coyolxauhqui together & the blow up) 137 like racism and gender inequality. Math is also described as fun and sad (el arrebato) in understanding the world we live in as we change it for the better (compromison). Furthermore, when I compared how this same person answered the same question before any REALM programming, I could see how this young person has changed in how they view mathematics.
The difference between school mathematics and mathematics as an experience in
REALM can be seen as the one word to describe school math is “dull” and the one word to describe REALM is “hopeful”. The disconnect between the two words is Coatlicue and nepantla because this person knows mathematics does not have to be dull. Furthermore, the blow up, shows this young person's desire for others to experience mathematics the way they did by connecting mathematics to the real world and learning how mathematics can be used for multiple things (compromison & nepantla). In explaining the difference between math in school and
REALM it is clear that context of a problem matters in that it allows learners to think about how mathematics is connected to the rest of the world. As the young person wrote, "the math we did lead me to think more about how math can be used in the world around us" a collective framing of mathematics (the blow up) impacting all of us emerges from the individuals reflection (putting
Coyolxauhqui together). In the final question, this young person not only states what they would say to their teacher, but they also provide an example (shifting realities) by encouraging teachers to ask questions based on the context of graphs. Whereby giving meaning to mathematics means no longer doing the same thing (compromison) in not exploring the real world with mathematics.
This one journal reflected all seven stages of conocimientos for this one individual who shared this experience with the entire group throughout REALM. REALM is an experience where YPAR can help return mathematical agency to youth and can best be see when we explore the question "are you a mathematician?" In journal 10 the young person above said yes, but if I 138 look at what they wrote in Journal 4, at the end of week 1, I see a different answer.
Where they wrote,
I do not consider myself a mathematician. I can do math, but it all depends on my idea of what a mathematician is. But the idea of a mathematician has been tainted by movies and the representation of math teachers at my school.
At the end of week 1, the idea of being a mathematician is not absolute. Yet at the end of day 9, the same question is revisited, and the young person writes,
Now I do see myself as a mathematician I use it in my daily life when baking when shopping and the types of math we learned at school. Seeing where math comes from makes me feel like a mathematician because math in everywhere.
On Day 9, mathematics is connected to daily life in and outside of school, and being able to see that mathematics is everywhere makes it easier to see that mathematics is inside of each of us. When knowing mathematics is part of self, it becomes natural to view the self as a mathematician and when mathematics are extended to the real world, conocimientos allow mathematics to be a consciousness raising experience. Mathematics as a shared experience is an opportunity to explore differences in learning about our interconnected selves.
Conocimientos with|in Mathematics
Conocimientos with|in mathematics is mathematics that is simultaneously scientific knowledge and spiritual knowledge in being a conscious raising experience. Furthermore, conocimientos with|in mathematics honors the multiple types of mathematics that have existed in human history. The implication of the seven stages of conocimientos within mathematical learning is an opportunity for unification of mathematics education that changes individual’s consciousness. Without special attention to how critical consciousness forms efforts to understanding identity in mathematics will always be incomplete. Conocimientos provides the key steps to view mathematics that lead to the development of critical consciousness. 139
Furthermore, conocimientos within mathematics education seeks to challenge injustice through collective action and YPAR EntreMundos provides an experience that addressed the root causes of societal injustice. Conocimientos with|in mathematics connects multiple areas of mathematical learning and research as seen in Table 7.4.
Table 7.4
Mathematics with|in conocimientos
Stage Mathematical Embodiment el arrebato, C1 Mathematics connected to emotions, from the surprise and joy of learning to the anxiety and fear created by mathematics. nepantla, C2 Mathematical wonder and the possibility of connections made while learning. Connections can be between learners, mathematical ideas or between a wide range of physical and metaphysical form of mathematics. Coatlicue, C3 The pain of learning mathematics and other strong feelings associated reflecting upon the mathematical world. compromison, C4 A mathematical imaginary between mathematical history, ethnomathematics, and future mathematics that are not fixed. putting Mathematical critical self literacy. The ability to use mathematics to Coyolxauhqui understand better the world and the process of transforming, the self, together, C5 that results. the blow up, C6 Collective mathematical reflection. Dialogue of mathematical learning in identifying that mathematics is already a part of each individual person. shifting realties, C7 Mathematical critical action. Living with mathematics in using mathematics when no one is looking. The action associated with sharing mathematical knowledge.
Table 7.4. can be used to ensure mathematical learning is providing the opportunity for people to experience conscious raising mathematics. Mathematics with|in conocimientos allows for mathematical learning to come alive as it makes a companion out of all of us. Mathematics that is alive, forges a relationship between learners as shared experiences revealing our 140 interconnected selves. The mathematics already present in any given individual already has the potential to connect to the mathematics in others and in self. A mathematics that is alive allows for mathematical learners to see themselves in the mathematical world, where there is no such thing as a non-mathematical world.
Conclusion
REALM showed how mathematics could thrive within YPAR, and PAR EntreMundos provided an epistemological playground to explore the seven stages of conocimientos with|in mathematical learning, that resulted in young people studying mathematics, themselves and the world. The youth who participated in this project, REALM, had a clear shift in their consciousness in living with mathematics as a collective with (Y)PAR EntreMundos, and most of the youth, individually, had all seven stages of conocimientos present in their journal entries.
REALM is transformative because it provides a space for both mathematical reflection and mathematical action. Without action, YPAR is not revolutionary and cannot ensure a transformative space for young people.
These results shed light on the importance of developing mathematical learning spaces space for providing an opportunity to engage with the seven stages of conocimientos.
Mathematics with|in conocimientos entails lesson planning that leads learners through rich mathematical experiences. The stages of conocimientos can also be used as a reflection tool for teaching by having a teacher ask themselves, “did my students engage in blank stage.”
Conocimientos, as a nexus of reflection, can be used by teachers and teacher educators alike.
Furthermore, the new conceptualization of conocimientos within the context of mathematics will continue to deconstruct and reconstruct what counts as mathematical knowledge. Conocimientos with|in mathematics embraces the multiple aspects of individual identity by connecting them with the numerous forms of mathematics connected to history and the future. I curated or created 141 the entire curriculum for the ten days’ program. Without the carefully designed curriculum, the outcome would not be as fruitful. The results and design of this research study provide mathematics teachers and educators a reminder of the importance of having a guiding epistemology behind mathematical learning. (Y)PAR EntreMundos ensured the curriculum was appropriate for the youth.
This study is not without limitations, such as the context-specific of being in the
Midwest, working with a community-based organization and a university college pathways program during a summer program. Wherein every activity we did could not necessarily be replicated to receive the same results. The complexity of identity and consciousness is a second limitation for this study, where it is impossible for me to accurately understand another person's consciousness. The most significant limitation of this study is that I am the sole author writing about youth when the only accurate way of capturing the youth embodiment of mathematical learning is for them to write with me. This study is additionally limited to summer programs where individual activities used can be applied to K-12 education school settings but as a whole program REALM may be difficult to replicate within traditional schooling.
A connection to revolutionary praxis and implications.
Future work in exploring conocimientos with|in mathematics will focus on four contexts.
The first is to continue the work and development of YPAR within mathematical learning in studying the formation of conocimientos. REALM ran in the summer of 2019, will run again in
2020, with plans of running the program indefinitely. Running the program multiple times will allows for a more robust understanding of conocimientos with|in mathematics. Additionally, youth will have the opportunity to be part of REALM for multiple years. Overtime a conceptualization of revolutionary praxis will emerge through and because of REALM. 142
Revolutionary praxis being the balance of both action and reflection in addressing the root causes of societal injustice has implications for mathematics for “social justice” at the K-20 level.
Whereby better understanding social issues during the teaching and learning of mathematics teacher and students alike can be empowered by extended the learning done during REALM to how future teachers are taught.
Future iterations of REALM lead to the second context of expanding the scope of conocimientos with|in mathematics & science. For example, REALM 2020 will focus on mathematics and microbes (biology), REALM 2021 will focus on mathematics and engineering and REALM 2022 mathematics and chemistry or computer science. The scope of a mathematics that is alive will not only impact mathematics education but will naturally inform STEM education as well. The third context, mentioned earlier, that will explore mathematical embodiment is taking what is learned through REALM and applying it to preparing future elementary teachers, where conocimientos will forefront teaching mathematics by creating rich experiences. The fourth and final context for future research represents my own teaching and mentoring of mathematics education graduate students by focusing on reconceptualizing how they come to engage with nepantla in seeing the multiple possibilities for themselves, others, the world, and mathematics itself. Endless connections exists for future work as it relates to mathematics for social justice, STEM education, curriculum and the philosophical/cosmological nature of mathematics 143
CHAPTER 8. FINAL REMARKS: A POETIC RETURN
This dissertation began with a poem written as my reflection after REALM had concluded. In the poem, I wrote “To begin with zero” to signify that I had no YPAR-Math template or playbook to follow. The process of creating the activities we did was a stressful but joyous task, as it reminded me that I was a teacher, and I do have the ability to develop rich mathematical learning opportunities. Furthermore, I was beyond happy with how all activities were received. A majority of the poem shares the same themes as the poem “Mathematics Me –
Therefore you” a poem written to reflect everything I wanted to accomplish, a year before
REALM took place. The most powerful line of the poem “To begin with zero” is,
A finite life – living forever with math
meaning we are never alone ... in that captures what I felt during and after REALM that we are not alone because mathematics is alive in each of us. Thus, I chose to end this dissertation with a poem as it embodies the start of the life-work captured in this dissertation. The poem utilizes the analogy of mathematics as obsidian to critique the current mathematics education field for having a lack of action upon the world while highlighting the interconnected nature of mathematics. Obsidian is a mineral, a rock that forms as magma cools where magma is lava from the earth's core that flows and erupts from cracks in the earth's mantle(surface). Obsidian naturally forms into rough, non-smooth rocks, that once polished, can be smooth into mirrors and various other tools. Furthermore, obsidian has spiritual properties in multiple cultures. Both poems echo Gramsci's (1971) comments on intellectuals, in that there is no such thing as a non-mathematical being, there are only those that have been rendered non-mathematical by society. 144
Mathematics Me – Therefore You
The poetic knowledge of mathematics is the story – of experiencing part of self the rate, the speed, the change the very viscosity of learning math – leads to this darken rock. Mathematics as obsidian Formed in the process of untamed magma, the lava ever-changing because we all learn at a different pace. Imagine a dance – now count the steps to learn to feel. Like math we must remember that lava comes from the very core of this Earth; all to make this rock grounded.
Question? So when can we stand tall?
The state of equity in math, is like this starless night sky mineral, intense heat that has cooled. Like obsidian we must still polish math to create a mirror – finding the spiritual self in the numbers. Math as an act of faith; as a way to see the world and as a way to see yourself in the world. Is a path where mathematics can be praxis; acting like a fulcrum in the balance between action and reflection. To change you and the universe in parallel. This is a call towards spiritual activism to see past the numbers and understand the power behind them. Thus, establishing mathematics as a relationship, an intimate experience, with critical pedagogies. Taking us from a mathematical identity to a mathematical consciousness A resulting resisting force; empowering student and teacher alike. To transform reality – with others; a commitment and goal of conocimientos. This is the want of a mathematical revolution and the steps needed towards liberation 145
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APPENDIX A. I AM MATH POEM TEMPLATE
I am Math
I am from ______’s (Your favorite shape) from where my ______taught me, (Family member) to add ______(An object form growing up) Celebrating the difference between me, my friend ______(The name of one of your friends) and ______is like math, (A cultural figure of your people) When trying to learn ______(A High School Math or college Class) or enjoying ______(A math concept that you enjoyed doing) and fighting to solve ______(A math concept you did not like doing) We are from math and ______(A game you like playing) Hidden numbers, I spy ______(A real-world object that matches your favorite shape) seeing patterns everywhere, especially at ______(Your favorite place growing up) and when I smell ______(something that smells nice) even when smelling ______(Something that smells bad) Math is me, ______(A positive characteristic that describes you) And math is you ______and ______(a characteristic that you like in others) (another characteristic that you like in others)
From negative to positive infinity (a characteristic that you like in others) math is us, as we become ______’s (Your future career) Fore, I am math
And
together we will change ______(Something in this world you want to change)
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APPENDIX B. DAILY JOURNAL PROMPTS
The following are suggested prompts given the participants at the end of the daily activities with expectation of Journal 0, which was given prior to any RELAM activities. Journal 0 What is mathematics? What are your expectations for the next two weeks? What is one word you would use to describe how you feel about learning mathematics when in school? Journal 1 Where does mathematics come from? What is one thing you enjoyed learning? What additional questions do you have, if any? What more do you want to learn about mathematics? What additional questions or comments do you have. Journal 2 What is one thing you learned today about how math can be shared out. What surprised you from todays activity What additional thoughts do you? What questions, if any do you have Journal 3 What is one thing you learned today about how math can be shared out. What surprised you from looking at the newspaper What additional thoughts do you have about the news What questions, if any do you have Journal 4 Share one thing you enjoyed about writing your poem Share one thing that surprised you by doing the I am Math Poem Do you consider yourself a poet? Do you consider yourself to be a mathematician Do you consider yourself to be a scientist? Journal 5 What is math? How can math be used to understand the world?
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Journal 6 Do you think this activity and activities like this one should be part of your school math classes? Please Explain How can math be used to understand the world? Journal 7 What did you learn after doing these math problems? Would you like to see math problems like this in your normal school? Why do you think your math teacher does not use problems like these? How could these problems be used in normal math class? Journal 8 What did the activity teach you? What math did you learn and what non-math did you learn? Did the percentages and slopes surprise you? If so how did it make you feel? What additional question do you still have? Journal 9 Do you see yourself as an activist? Please explain why or why not Do you want to be an activist? Please explain. Do you see yourself as a mathematician? Please Explain why or why not Do you see yourself as a researcher ? Please Explain Do you feel like you have grown closer to your peers over the last two weeks? Please Explain Journal 10 What is mathematics? What is one word you would use to describe how you feel about learning mathematics when in school? What is one word you would use to describe how you felt about learning mathematics over the last two weeks Can you explain the difference between the math we have done over the last two weeks and what normally gets done in your math classes? Yes or No question – Are you a researcher Yes or No question – Are you a mathematician? Yes or No question – Are you a scientist? What is one thing you would want to tell you math teacher about changing how they teach?
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APPENDIX C. IRB APPROVAL MEMO
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