Reflections about Mathematics, Education, and Systems

Miguel Rodriguez for Applied Systems Thinking 2021 (Collaborative Design)

Produced

Logic

Induction

Intuition

Reasoning

Inferential

Constructive

Creation

Innovation

It’s necessary to find or “construct” a specific example of a mathematical object to prove that an example exists.

Mathematical Constructivism Mathematical philosophies

Mathematical Platonism Mathematical objects are abstract.

They have no spatio-temporal or casual properties. They are eternal and unchanging. They exist regardless of humans.

Unchanged

Natural

Non-constructive

Knowledge

Innate

Observation

Acquired

Exists

Decoded

Miguel Rodriguez for Applied Systems Thinking 2021 (Collaborative Design)

differences between two

Life Cycle of a Mathematical Proof

An unverified proof is developed by a mathematican or group of mathematicians (known as a conjecture/hypothesis)

Proof is introduced to the mathematical community for verification

Revisions are conducted to save the proof

Discarded conjecture, hypothesis, theorem, corrollary, etc

Proof is validated and it becomes an accepted theorem

New information is introduced which affects the theorem

Miguel Rodriguez for Applied Systems Thinking 2021 (Collaborative Design) Based on a central theme in Imre Lakatos’ Proof and Refutations

Perceptions and actual practice of artists + mathematicans Perception of artists

Perception of Mathematicians

vs.

Independent thinker

Process follower

Free-form

Elegant

Free-spirited

Logic based

Creative

How things work

Aesthetics

Want things defined

Expression

Proving things

Reflective

Puzzle solvers

Feel things

Order and structure

Emotions

Connected variables

Order and structure Independent thinker How things work

Feel things Puzzle solvers Proving things

Connected variables

Logic based

Emotions

Actual Practice Expression Free-spirited

Aesthetics

Free-form

Process follower

Want things defined

Reflective Creative

Elegant

Miguel Rodriguez for Applied Systems Thinking 2021 (Collaborative Design) Based on a conversation with Kate McCallum, PhD

Reinvisioning mathematics in the K12 public school system Current System

Change Mechanism

Potential System

Unenthusiastic teachers

Increase teacher pay, decrease barriers to access grad school, and provide more power and influence in curriculum

Teachers are enthusiastic, which positively impacts teaching methods and styles

Overtly abstract mathematical concepts for students

Introduce more real world applications in curriculum

Increase in math affinity, and content interest and retention

Lack of student input in curriculum and classroom settings

Student-to-teacher feedback mechanisms and individualized learning plans

Math is seen as a challenge to overcome or a subject to explore, not imposed.

Untrained teachers

Regularly provide professional development and trainings to staff

Teachers are better equipped and responsive to student needs

Limited teaching and learning styles in classrooms

Modify classroom space and curriculum to include varying types of learning styles

More ways for students to engage with content, and it promotes equity and inclusion in the classroom

Student-to-teacher ratio is overwhelming

Hire more teachers to reduce high student-to-teacher ratio

Teachers have less students, which can mean more 1-on-1 support

Youth culture content relevancy

Share and highlight examples that are relevant to their interests

Relatability. Students can see themselves in it, and it’s interesting to them

Miguel Rodriguez for Applied Systems Thinking 2021 (Collaborative Design)

The Connection Between ORegon’s Mathematical Practice Standards + Student Math Identity Oregon’s Mathematical Standards

Mathematical Identiy as described by Rick Anderson

Oregon’s Department of Education released its most recent version of Mathematical Standards (v 5.2.1) in October 2021.

There are four faces of mathematical identity: engagement, imagination, alignment, and nature.

It provides a high level overview of mathematical concepts that public school teachers must provide to their students.

“Students must become mathematics learners—members of mathematical communities—if they are to have access to a full palette of future opportunities” (pg 8)

Standards are categorized by grade level from K-8th, and by a general 9-12th high school block.

imagination

engagement

Definitions & connections

nature

alignment

(1) Making sense of problems and persevere in solving them requires engagement + alignment

Engagement: “our direct experience of the world and our active involvement with others” Imagination: “the images we have of ourselves and of how mathematics fits into the broader experience of life” Alignment: “aligning our energies within institutional boundaries and requirements e.g. school, class, etc” Nature: “who we are from what nature gives us at birth, those things over which we have no control over”

(2) Reason abstractly and quantitatvely requires engagement + imagination + nature (3) Construct viable arguments and critique the reasoning of others requires engagement (4) Model with mathematics requires engagement + imagination + alignment + nature (5) Use appropriate tools strategically requires imagination (6) Attend to precision requires imagination + alignment (7) Look for and make use of structure requires imagination + nature (8) Look for and express regularly in repeated reasoning requires engagement + imagination + alignment + nature

Miguel Rodriguez for Applied Systems Thinking 2021 (Collaborative Design) Based on information from The Oregon Department of Education & Being a Mathematics Learner: Four Faces of Identity by Rick Anderson

Rodriguez 1 Miguel Rodriguez | Howard Silverman Applied System Thinking (COL551/1) 12/17/2021

Reflections about Mathematics, Education, and Systems with Kate McCallum I. MATHEMATICAL JOURNEY Mathematics has been a consistent companion—dare I say, a loyal friend—in my life. At a young age, my innate curiosity drew me to math problems and concepts: they were puzzles that needed solving. As a subject, mathematics helped me better understand my surroundings, as well as accelerating my reasoning and analytical skills acquisition. My parents and teachers, seeing my affinity for mathematics, actively encouraged me to explore and challenge myself with difficult math courses and resources. On a more fundamental level, they also knew it could positively impact my college-bound trajectory. After high school, I attended Umpqua Community College and completed my undergraduate studies with a BS in Mathematics from Portland State University, which was followed by employment with a college access nonprofit.

This work experience would hopefully answer some pressing questions: (1) will two years of grad school be worth it1, (2) what is a holistic view of a mathematics teacher’s day-to-day, and (3) what challenges do high school teachers and students face in our educational systems? I had a general intuition for the answers, but I intended to find out firsthand.

My work at McDaniel High School solidified my knack for teaching, tutoring, and mentoring, but it also affirmed my cynicism of the educational system and how mathematics is taught2. In particular, I became increasingly disillusioned with the high student-teacher ratios,

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Graduate Teacher Education Program (GTEP) in Mathematics from Portland State University College Possible + IRCO Sun Tutoring at McDaniel High School in Portland, OR (fall 2016 - fall 2018)

Rodriguez 2 overwhelming focus on standardized testing, lack of curriculum adaptability, and disparate understanding of how various social and cultural factors affect learning. My students—predominantly students of color—were experiencing eerily similar systems of oppression that I had to navigate throughout my K-12 and undergraduate education. Despite positive reinforcement from various teachers and family, I had a slew of negative experiences and associations with mathematics in my own educational journey. To this day, I remember being made fun of in middle school for being the smart, nerdy kid who was into math; I did not conform to the Latino and Mexican stereotypes my peers attempted to ascribe to my intellect. To this day, I remember the two undergrad professors that almost compelled me into changing my major, because they were baffled at my confusion with learning foundational set theory at a slower rate than my peers; these individuals, alongside other factors, almost made me drop out of school. The list goes on.

These experiences only made me more ambitious and determined to prove others wrong.

I strived to change the status quo with my cohort of 40 students, and I tried my hardest not to succumb to the same type of methods and techniques I saw my ineffective teachers and professors practice. I was continually combating the stigma associated with mathematics by addressing the following questions and sentiments with my students: Why are we learning this?

I hate math.

How does this relate to real life?

I’ve never been good at math.

When am I going to use this?

My teacher is so confusing.

Why is this so boring?

I don’t need math for my career or job

This is by no means an exhaustive list, but it resulted in spurts of creativity and spontaneous solutions. Bridging the gap between content and students required me to use alternative methods

Rodriguez 3 to engage and connect with them. Even then, usually in hindsight, I noticed how I still had perpetuated the aforementioned methods and techniques of my teachers and professors by frequently resorting to the banking model of education (Freire); the parallels were painstakingly obvious. I had many successes, but the failures were the most powerful and teachable moments.

However, being in community with my students was necessary to address them and refine my teaching styles and philosophies, which now lean heavily towards Seymour Papert’s Constructionism Learning Theory and Jean Piaget’s Theory of Cognitive Development. 3 Since leaving McDaniel High School, my professional career has taken an unprecedented and fulfilling trajectory alongside my personal interests, which have significantly evolved and transformed. The intersection of human resources, education, design, mathematics, community organizing, coding, videography, and music highlights my lifelong commitment to learning for the sake of learning. Amongst this lengthy but finite set, one stands out more than the others: mathematics. Besides design—which is a close second—it can describe or be applied as a tool for the other elements. This is the beauty of mathematics, and why I aspire to reference and incorporate as much of it as possible in my graduate studies; it will be a pillar of my future thesis work as I explore and design various techniques that make mathematical content more accessible. For example, using a coloring book as a mechanism to explore art, coding, and mathematics.

Generally speaking, our educational systems have skewed the public’s perceptions of mathematics, and have also introduced deficit-focused narratives in classrooms e.g. who uses it, why we need it, and how to read it (Aguirre et al. 5-6). Even more, how do race4, gender5, and 3

Piaget’s Constructivism, Papert’s Constructionism: What’s the Difference? By Edith Ackermann Disrupting Anti‑Blackness with Young Learners in STEM: Strategies for Elementary Science and Mathematics Teacher Education by Tia C. Madkins & Karisma Morton 5 Explaining the Gender Gap in Math Test Scores: The Role of Competition by Muriel Niederle and Lise Vesterlund 4

Rodriguez 4 other forms of identity6, affect students’ relationship with mathematics? We need a paradigm shift: we need a model in which students are excited to learn mathematics, and teachers are adaptive, well-resourced, and continuously practice cultural humility7 to foster stronger math identity and affinity.

II. INTERVIEW & MAPPINGS Upon hearing from Hafsa Aden, one of my Collaborative Design cohort peers, that their elective course professor, Kate McCallum, had explored pieces of the aforementioned topics for their PhD dissertation, I knew that I had to interview them! Kate is an artist, curator, and researcher born in the UK who now lives and works in Oregon. They teach at PNCA and Portland Community College, and they also develop community art projects in collaboration with local organizations. Kate’s dissertation focuses on how research mathematicians communicate their ideas, proofs, theorems, and many other mathematical objects amongst each other. Within their extensive and thorough research, I was most drawn to the analogous models and workshops they developed (McCallum 130-143, 300-324). For example, reimagining the mathematical paper: (1) with poetry; (2) with deconstruction; (3) with reconstruction; (4) pulling it apart within the LaTeX document environment. Kate’s efforts highlight the power of examining existing structures from a different perspective; it also demonstrates that even experts in the mathematics field can be shown new ways of thinking and engaging with content. The focus was on research mathematicians, which is a group I do not necessarily want to work with.

But, the impact potential gives me hope: art, creativity, and group facilitation can shift how

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Noticing Multilingual and Non-dominant Students’ Strengths for Learning Mathematics and Science by Salvador Huitzilopochtli, Julianne Foxworthy Gonzalez,Judit N. Moschkovich, Sam R. McHugh, and Maureen A. Callanan 7 3 Things to Know: Cultural Humility by Julia Sufrin

Rodriguez 5 people communicate and teach mathematics. What could a change in technique, format, and/or resource allocation provide youth in their mathematics education, especially for students from underresourced and underrepresented communities?

Kate and I met through zoom for an hour, and it was a flurry of excitement, resource sharing, and admiration of various intersections—with potential for systemic change—in the mathematics field. The conversation ran the gamut, but left me feeling inspired and wanting to reflect and learn more about my mathematical interests; during the last 2 years I have not been able to explore much, because I transitioned to a non-mathematics heavy field. The following are some of the highlights of our conversation, which inspired 5 mappings (pgs 10-14): ● Talking about code and how it’s organized. Easier and less easier, at the same time ● Mathematical notation is extremely important. Usually overlooked or forgotten ● Terrible learning experiences growing up, and people having huge walls of nope! Feeling and seeing all of these parallels is what inspired research ● Artist explorer in the world of maths. How people do maths together on human terms. ● Trying to also change the rules during experiments, and it being an artistic exploration ● Ended up using a lot of ideas from cognitive science and linguistics: ○ Situated cognition8, 4E Cognition9, and David Chalmers, Andy Clark, and Alva Noe 10 ○ Relevant theories about how people communicate ○ Graph theory, how people are processing and really digging in ○ Situational cognition. Instead you're going through this continual adjustment of your relationship with the world. You’re responding continuously

8

Situated Cognition and the Culture of Learning by John Seely Brown, Allan Collins and Paul Duguid Thinking avant la lettre: A Review of 4E Cognition by James Carney 10 The Many Bubble Interpretation, Externalism, the Extended Mind of David Chalmers and Andy Clark, and the work of Alva Noe in Connection with Experimental Philosophy and Dreamwork by John Yates 9

Rodriguez 6 ● Observing a research meeting, a wonderful process in which people are thinking their way through a problem in an intense entanglement ● Comparison of artists and mathematicians, and how they could be connected ● Ontology of mathematics11: ○ Mathematicians are Platonists12 on weak days, and whatever else they wanna be on the weekends. Mathematical constructivism vs. mathematical platonism ○ The idea of our world being in the shadows of another ideal realm ○ The idea that mathematics exists somewhere and that we are discovering it versus the idea of constructing it - it has a lot of cultural momentum ○ Unexamined platonism, a vague idea that it’s out there and that math is super perfect.

It contributes to the problem where people encounter maths, and end up feeling like they don’t get it. Big and all powerful, with feelings of disconnection. Drives her mad! It’s a shame! ○ Doing maths: the process is not inevitable. Smashing things together. How much further can you go? Especially if it links up to something unexpected. Immense playfulness to it. Doing things for the sake of it ○ Pure mathematics can have some nuggets of practical information, but people are just generally doing it for fun ○ Construction, and how you’re coming to your own understanding of concepts: ■ Instinct of how something works. How abstract things like numbers work.

■ Cme to us by metaphor. Manipulate things, abstract. Building it up from your experience. Objects and experiences of the world.

11 12

Philosophy of Mathematics by Leon Horsten and Ontology of Mathematics by Rafal Urbaniak Platonism in Metaphysics by Mark Balaguer

Rodriguez 7 ● The Bridges Organization13: annual conference, and its community of lovely nerds that are interested in mathematical aesthetics. Full spectrum, interesting profound work. Get weird stuff. A lot of golden spirals stuck on something. Sacred geometry. Perfection and where it sits on people ● Proofs and Refutations14: recursive structure, and how theorems and conjectures are not static, they are always in a state of dynamism ● Susan Gerofsky: ○ Associate Professor of Mathematics Education and Environmental Education in the Faculty of Education, University of British Columbia, Vancouver, BC, Canada. ○ She writes stuff about non-hierarchical learning in mathematics. Ball example: organize your body so that the ball is at the same angle relative to you. Moving backwards. Too far up or too far down. This reminds me of the bird flocking activity at the beginning of the semester

III. NEXT STEPS My conversation with Kate reignited my desire to explore mathematics content, learning theories, and a myriad of other interconnected topics. My thirst for knowledge is palpable, and I am super excited to introduce these concepts into my MFA thesis topic; I want to explore alternative mathematical teaching styles, methods, and techniques. The intersection of art, design, and mathematics is of particular interest. I firmly believe that the opportunity to create physical objects and interactive experiences with people—and not for them—will be essential for collective and community-based solutions. Additionally, I hope to further engage with Kate

13 14

The Bridges Organization founded by Reza Sarhangi Proofs & Refutations by Imre Lakatos

Rodriguez 8 as I embark on topic exploration during my second year, and I want to see how else we could possibly collaborate; I did notice that Kate has done work with the local nonprofit The Immigrant Story15, they are actively involved in the art scene, and they play a crucial role in the educator community. Intersecting paths within Portland’s social justice, community organizing, and educational circles are necessary to create community and challenge and change the status quo. This type of great work does not happen in silo.

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The Immigrant Story founded by Sankar Raman

Rodriguez 9 IV. WORKS CITED Aguirre, Julia, et al. The Impact of Identity in K-8 Mathematics Learning and Teaching: Rethinking Equity-based Practices. 1st ed., National Council of Teachers of Mathematics, Incorporated, 2013.

Freire, Paulo. Pedagogy of the Oppressed: 50th Anniversary Edition. Translated by Myra Bergman Ramos, 4th ed., Bloomsbury Academic, 2018.

McCallum, Kate. Situating Mathematical Communication: the Settings, Interactions and Material Practices of Contemporary Mathematical Research. PhD dissertation. January 2020. Kate McCallum, https://katemccallum.com/.