The Beauty of Mathematics: A Continuum

 “Mathematics reveals its secrets only to those who approach it with pure love, 

for its own beauty.”  Archimedes

 

 

I think most people who are reading this article have at one point or another made the following comment, “I don’t understand this ‘new-fangled’ (Common Core) math. This is not how I learned math when I was in school.” I know that I have repeated this comment more times than I care to admit. As a teacher of mathematics, I have had to learn math from a whole new perspective in order to teach it to my students. As a bibliophile, lover of books, I have never quite felt the same love for math that I feel for literacy. Even though I may not love math the same way I love literacy, I have grown to appreciate its beauty. I have come to understand math is also about storytelling with a language that is both universal and unique.

 

Literacy expert Louise Rosenblatt (1978) distinguishes between two types of reading: efferent and aesthetic. The stances represent opposite ends of a continuum. I believe the same is true for mathematics. When a student is viewing math from an efferent stance, he or she is engaging in math for pragmatic reasons. For example, completing a calculation using a traditional algorithm to efficiently solve a problem like 12 x 5 = 60. At the opposite end of the continuum, a student who is viewing math from an aesthetic stance finds joy, pure love, in curiously exploring the beauty of math.

 

What is beauty? Merriam-Webster defines beauty as, “the quality or aggregate of qualities in a person or thing that gives pleasure to the senses or pleasurably exalts the mind or spirit.” Beauty is in the eye of the beholder. Not sure who first said that famous quote, but I do know that sometimes we pass by beautiful people, or things, because we don’t take the time to stop, notice, reflect, and wonder. I have come to learn that “mathematics is not about numbers, equations, computations, or algorithms: it is about understanding”—William Paul Thurston. Math is orderly, sublimely pure, and perfect. The simplicity of math can be found everywhere (see pictures below). The University College London has used brain scans to show that complex mathematical equations, or formulas, can evoke the same sense of “beauty” as an artistic masterpiece or music from a great composer. My job as an educator is to help my students efficiently and accurately calculate numbers, equations, computations, or algorithms, as well as look for the universal beauty in math. My job as an educator is to help my students understand and appreciate that beauty, because mathematics is not just a worksheet, it inhabits the world all around us. 

 

According to Stahl and Nagy (2007), “Language is a key component in the learning process. People’s perceptions of the world are largely shaped by their vocabulary knowledge. Indeed, words are the tools we use to access our background knowledge, express ideas, and learn new concepts” (p. 4). Students can solve equations with little effort, or vocabulary knowledge, using a calculator (efferent stance). I need to also help my students develop the language they need to communicate their mathematical thinking beyond that of calculating a simple equation (aesthetic stance).

 

This “new-fangled” math is definitely not the math I learned in school. This “new-fangled” math reveals secrets that open a world of possibilities and opportunities far exceeding any numbers, equations, computations, or algorithms. It requires language, storytelling, processing skills, critical thinking, creativity, perseverance, and teamwork. All skills necessary for success in the global society we live in today. What a gift students have been given that I rarely had access to when I learned math. After being cooped up all winter due to the weather and COVID-19, take a moment to go outside with your child and explore the beauty of math in all its forms. What mathematical beauty do you notice? What do you wonder? What words can you use to tell the story of math? The real-world applications of math are endless!

Reflections While Walking

Where I live, we have been having periods of very hot and humid days, so instead of walking outside, I have walked on the treadmill in my basement. A couple of days ago, we got a little bit of a break from the heat and humidity, so I returned to the outdoors for my daily walk. I live in a subdivision that doesn’t have streets in a grid with sidewalks. We have streets and paths that wind around man-made ponds, parks, and playgrounds. 

As I was out walking, I realized there was a huge difference in my mental well-being, spirit, and energy being outside for a walk. Something about being outside allows me time to reflect, think, and ultimately calms my mind. Living in a pandemic means being confined to my house for long periods of time. I have rarely left my house for weeks at a time. So being outside—hearing the birds sing, seeing the blue sky, feeling the shade of the trees, and smelling the dampness of the air—fills my soul that has been emptied from the constant confinement.

I started to reflect on various periods throughout history and what it must have been like for others who were confined by circumstances beyond their control or because they were denied their freedoms. For example, how did slaves fill their souls? How did Jews survive in concentrations camps? Although I have been confined to my house for four months, it is not the same as those who survived confinement throughout history because I have still enjoy many of the comforts that my white privilege has afforded me. When I consider the atrocities that have occurred throughout history that caused some to be confined, my heart breaks for so many. There are times when I simply cannot understand how one human being can do the things that have been done to another human being. And, I am curious how those who have survive those confining atrocities filled their souls. We need those voices now—the world needs to fill, to heal, its soul. 

Then my mind wandered to what school would be like in the fall. Several weeks earlier, I completed an intensive week-long institute on reading. The learning was amazing, but sitting, confined, in a chair on my computer all day for 8 or 9 hours straight was really difficult—and I am an adult. How will kids sit for hours at a time confined to a chair in school? I could at least get up and go to the bathroom or get a snack when I needed a short break, but in school this fall that will probably not be an option for our kids. I was so relieved when the week of learning was over, because I wasn’t sure how long I could maintain that level of confinement (in my chair) and engagement. How will our kids be able to maintain this level of confinement and learn for months on end? As much as I believe that the best learning takes place face-to-face, and that remote learning was far from ideal, I cannot believe in-person school will afford us any of the benefits we typically enjoy with face-to-face learning. I believe teachers and students will feel to traumatized to feel safe OR to learn. 

Then I started thinking about how we help kids feel safe and engage in the process of learning. My brilliant friend, Ellin Oliver Keene, wrote a book Engaging Children: Igniting a Drive for Deeper Learning K-8. After spending hours observing children, Keene identifies how kids move from compliance, to participation, incited by motivation (What’s in it for me?), and fully absorbed in engaged learning (What’s in it for us?). Within engagement, she developed a theory that includes four pillars: intellectual urgency, emotional resonance, perspective bending, aesthetic world (think, feel, believe, act). I started thinking, how can educators use Keene’s theory of engagement and our current life situation to help kids feel safe despite the traumatic circumstances and learn? For me, when I am engaged in authentic learning (thinking, feeling, believing, acting), I feel like I have some control over a situation which affords me some level /feeling of safety. I think it can be done—instead of avoiding our life situation we can use it to teach content and provide kids with some sense of control and safety. 

But engagement, high levels of engagement, cannot be sustained all of the time. Engagement is kind of like bi-polar disorder, the higher the level of engagement, the hard the crash and burn. I have experienced this firsthand after completing my dissertation. Whoa! —talk about crash and burn! Something Keene did not address in her book but is incumbent upon educators to teach children is finding balance. We need to explicitly teach how to reach and sustain engagement, but we also need to instruct students in how to find balance and advocate for their needs. When we reach and sustain high levels of engagement, we also need to counter that with moments of rest and calm and sometimes those moments are found in simply being compliant. 

I was left with the question, how to I help children, and colleagues, find ways to advocate for themselves when they need balance—from the confinement of their chair in the COVID-19 classroom? If we are going to sustain teaching and learning, we are going to need to find ways to balance periods of intensity with periods of calm. I guess it is time to go for another walk. 

Who Lives, Who Dies, Who Tells Your Story – Hamilton, Reflection Part 1

“Who Lives, Who Dies, Who Tells Your Story?”—Hamilton

Last week, I participate in the Teachers College Reading and Writing Project (TCRWP) Reading Institute, which is based on the Calkins Units of Study. It was an intense week and powerful learning experience. It was a week of story. Even though I was SO excited to be a part of the institute, I was also sad that because of the pandemic, I had to participate virtually. I was very much looking forward to attending in person, so I could also spend some extra time in New York City. Friday after I completed the institute, instead of watching the Broadway show “Hamilton” live, I watched the made-for-television version on Disney Plus. Even though I was not in the theatre, the brilliance of the performance—the history, music, lyrics, choreography, staging, energy, diversity of the cast—took my breath away! The performance was so compelling that I wanted to know more about the actors, the historical figures, the time period, the perspectives—I wanted more of the story! 

The next morning while I was waiting for my coffee to brew, I began scrolling through my social media feeds. Of course, I saw others who had also watched “Hamilton” the night before and were commenting on the show. To my surprise, some of the comments were condemning the fact that the story had not been told from a different perspective, namely the perspective of a BIPOC (Black, Indigenous, person of color) historical figure. Since I have been actively working to listen, learn, and develop my liberatory consciousness, I put my educator hat on and took some time to reflect on these comments. I mulled around four ideas in my mind:  the deficit model, balance, character development, and compelling stories. 

One reason our educational model is oppressive and needs to be reimagined is because it is grounded in a deficit way of thinking. We often identify and label our students based on their deficiencies and then seek to determine how we as educators can “fix” those deficiencies. Instead, educators need to shift their perspective to celebrate cultural differences that can support critical thinking and academic achievement. So, when I read the comments criticizing “Hamilton,” my immediate thought was, are we going to find fault with every story? Are we going to view every story from a deficit perspective? Or, can we use stories, even some of the most racist stories, to support critical thinking and academic achievement? 

There needs to be a balance. Stories are not just racist or antiracist. Each story falls on a continuum and each person has a unique transaction (Rosenblatt, 1978) with story. I think of all of the training I have had that reminds me that I need to teach the reader (process) and not the book (product). As an educator, I can instruct my students on how to develop their liberatory consciousness by presenting a variety of stories (not the same as endorsing a story) and teaching them the critical process of becoming aware, analyzing each story, taking action, and being accountable (Love, 2010). My goal is to use story to meet students where they are on the continuum of their liberatory consciousness and move them one step closer to transformation and liberation. So, educating students on the process will serve them whenever they encounter story and where-ever they are in their process of becoming liberators. I worry that operating with a deficit model of story will lead to censorship and that is a slippery slope that leads all of us away from our goal of inclusive democratic ideals which include freedom of speech. 

The most compelling stories are like strong complicated characters—they are not black or white, right or wrong. They are grey, complex, and messy. They provide us with mirrors and windows (Bishop, 1990) into the best of humanity and the worse of humanity. We learn best, when we explore all the faucets of character and story. When we understand story within a particular setting or time. When we read stories from many perspectives to listen and learn instead of to pass judgment. Ultimately, that is how we will create a more inclusive society. When we acknowledge the devastatingly awful aspects of our history AND the brilliance that lies in each of us. When we learn from our mistakes, even the worst of them, and use those lessons to provide hope so we can take action for a better future. 

Who lives? Who dies? Who tells your story? Hamilton was a brilliant and compelling story! The danger is not that it wasn’t told from the perspective of a BIPOC historical figure, the danger is in letting that be the single story (Adichie, 2009). For me, part of the brilliance of Hamilton was in the fact that it compelled me to learn more about the various stories that are woven into the tapestry of our history. It compelled to write and tell my story—regardless of how imperfect a story it may be. Who will tell your story? And, when it is told, who will listen with an open mind and an inclusive heart?

Becoming Antiracist: Reflect. Reimagine. Act.

(Partially reprinted from my article published in the June, 2020, iCommunicate an eNewsletter of the Illinois Reading Council.)

Current events have raised our collective awareness of systemic racial trauma in the midst of a pandemic. In my mind, current events, while historic, unprecedented, and at times overwhelming; provide all of us with a rare opportunity to reflect, reimagine, and act to effect meaningful change. As we begin to process the physical, mental, and emotional trauma that is happening in our world, it is incumbent of all of us to support each other, our students, and their families by creating more inclusive communities. 

REFLECT. For me addressing issues of equity, inclusion, and working towards a more inclusive community really came to the forefront of my attention while working on my doctorate, specifically during a class on social equity and justice. The antiracist ideals of equity and inclusion, coupled with my biases and prejudices, are issues that I have been trying to reflect on and address in earnest for the past six years with the full understanding that such work is a lifelong journey with no final destination. As such, I have come to learn the importance of listening to understand and that our actions often speak louder than our words. 

REIMAGINE. I believe some of the biggest roadblocks to rooting out systemic racism and reimagining education include:  (a) educational funding (funding from property taxes is inherently inequitable), (b) the deficit learning model based on standardized testing (these types of tests were created with the intent of proving that black people and women were intellectually inferior to white men), (c) curriculum/resources (presenting multiple perspectives by curating multi-modal text sets), and (d) professional development (the majority of teachers are white). The first two are policy related roadblocks. Although it is imperative for educators to lobby for policy changes, policies are largely beyond our immediate individual control. Curriculum, resources, and professional development, however, are fully within our ability to control, so individual teachers can begin to reimagine a “new normal” and immediately effect change. 

ACT. After personally reflecting and reimagining, it is time to commit to action. I have created a public Padlet (digital bulletin board - see link below) with the top 20 equity and inclusion resources that have impacted my thinking and instruction. This is by no means exhaustive list, but it is a place to start. 

Reflect. Reimagine. However, never forget the ultimate goal is to commit to taking at least one actionable step toward a more inclusive community. Even the smallest step when taken consistently can effect BIG change. 

I have been reflecting on past practices in my classroom (I will continue to share ideas on this blog) and trying to reimagine the actions I can take to move toward creating an even more inclusive classroom where ALL learners blossom—both emotionally and academically. I have not taken the time to blog consistently for a while, but I am committing to blogging once a week. Writing about all that I am reading will help me process the information and begin to formulate plans for action. 

Becoming Antiracist Padlet

My Story of Imperfection: Embracing Brokenness, Embracing A Shared Humanity

I find myself drawn to the concepts of equity and social justice and all their messy complexities. We refer to the ideas of equity and social justice by so many names: diversity, inclusion, disruption, decolonization, etc. I strive to try to do as much as possible to learn about and understand the problematic power structures found in so many of our systems by reading, listening, asking questions, and engaging in deep ongoing reflection. As an educator, I grapple with the effects of equity and social justice on the kids and families I serve, especially after reading several books that have been weighing heavy in my mind:

·     The Gifts of Imperfection: Let Go of Who You Think You’re Supposed to Be and Embrace Who You Areby BrenéBrown

·     Just Mercy: A Story of Justice and Redemptionby Bryan Stevenson

·     Cultivating Genius: An Equity Framework for Culturally and Historically Responsive Literacy by Gholdy Muhammad

I have several friends who have experienced the repercussions of the problematic power structures found in so many of our systems. They have endured unthinkable life experiences and actively work to “disrupt” the status quo. When I hear their stories, I feel ashamed of my white privilege—ashamed of my story. I feel unworthy of entering into a conversation that will bring much needed change to so many inequitable power structures. I wonder, how I can contribute to the work of disrupting the status quo in a meaningful way when my white-privilege is the root cause of those power structures? In his book Just Mercy: A Story of Justice and Redemption, Bryan Stevenson addresses my feelings of shame when he states: “We all share the condition of brokenness even if our brokenness is not equivalent. … But our brokenness is also the source of our common humanity, the basis for our shared search for comfort, meaning, and healing. Our shared vulnerability and imperfection nurtures and sustains our capacity for compassion (p. 289).” In her book The Gifts of Imperfection: Let Go of Who You Think You’re Supposed to Be and Embrace Who You Are, BrenéBrown defines connection as “the energy that exists between people when they feel seen, heard, and valued; when they can give and receive without judgment; and when they derive sustenance and strength from the relationship (p. 19).”

With Stevenson’s and Brown’s words in mind, I began to rewrite the shame narrative playing in my head. I believe that all of us have something to contribute to the conversation of equity and social justice regardless of our past life experiences. Putting a measuring stick next to our brokenness is a nothing more than a value judgment of our past, and that distracts us from moving forward with a conversation that will affect change. I have also made the mistake of believing that my passion for a topic gives me the right to “call out injustice” in others. What I have learned is that being intensely passionate can be very off-putting to people, which often causes them to be defensive and shut down the conversation. Calling out others implies that I am better or that I have the right to judge. I have learned that, at its heart, advocacy and being a catalyst of change means actively listening to understand more than speaking out; and that building inclusive trusting relationships grounds the process. Trusting relationships provide us with the space to be vulnerable; allow us to feel seen, heard, and valued; and give us common ground to find our humanity.

But that is not enough. Advocacy is easy if I surround myself with people who share my beliefs. But, being a true advocate, or catalyst of change, also necessitates that I intentionally seek out people whose philosophical beliefs are in opposition to mine and continue the process of actively listening to understand. These beliefs are not easy to live. I continue to reflect on, and refine, my beliefs about equity and justice. Honestly, I am also overwhelmed by the messy complexities of equity and justice. There is so much work that needs to be done, especially in today’s world. So, I embrace my own imperfection in this process and take small steps every day to affect change. It is probably not enough, but it is a start—and we all need to begin by taking a step forward.

In the end, I remember the words of Brown, “Only when we are brave enough to explore the darkness will we discover the infinite power of our light (p. 6).” Together our light shines like the north star beckoning us toward the hope of connectedness. 

Listener + Speaker + Reader + Writer = Lifelong Learner.

Reflection

I just got back from the Boothbay Literacy Retreat in Boothbay, Maine. What an honor and a privilege it was to learn alongside some of my literacy mentors like:  Kylene Beers, Bob Probst, Penny Kittle, Linda Rief, Chris Crutcher, Teri Lesesne, Cornelius Minor, and Sara Ahmed. What an honor and a privilege to learn alongside many other incredible educators who attend the retreat from near and far. Five days of learning heaven. It filled my literacy heart and soul. As I continue to reflect on my educational values and philosophies this summer, I am left with a clearer picture of my professional identity and can set meaningful growth goals for myself.

One value I absolutely hold as a truth—I cannot be a teacher of literacy without first being a listener, speaker, reader, and writer. I forgot which one of my literacy mentors said this (I would like to give them credit for this analogy), but “would you bring your child to learn piano from someone who has never played the piano before?” My answer is, NO. Of course not, it seems silly to even ask that question. Similarly, I cannot expect the families of my students, or my students, to trust in me has a teacher of literacy, unless I first do the hard work that I ask my kids to do every day—listen, speak, read, and write. 

To build curious joyful learning communities, I need to begin by listening to my students, their families, and colleagues with an open mind and an open heart, not to pass judgment but to learn from new perspectives. I need speak to share my truth, and to advocate for those who are afraid to speak or are not being heard. I need to read to learn about myself and others in order to make meaning of this global world we live in. I need to write, because writing clarifies all the other aspects of literacy. Writing has always been the most difficult for me. Writing, good writing requires purpose, an understanding of the audience, and an intimate knowledge of craft. It requires an active engagement in the literacy process has a whole. 

Although the standards may change from grade-level to grade-level, and child to child based on where they are in their literacy journey, what remains constant is listening, speaking, reading, and writing are the mirrors, windows, and sliding glass doors to the world (Bishop, 1990). It is my job has a teacher of literacy to pull alongside my students to do the hard work of exploring the world together. Balanced literacy means:

·     Incorporating listening, speaking, reading, and writing into my classroom

·     Curating a diverse classroom library (genre, format, story, etc.)

·     Integrating multiple content areas through project-based learning/literacy experiences

·     Communicating within and beyond the walls of the classroom

·     Exploring the world through multiple modalities and formats

The journey begins in kindergarten. And when educators are passionate listeners, speakers, readers, and writers, that passion is contagious, so the journey continues through our adult lives. What an honor and a privilege it is to pull alongside learners of all ages.

Labels - Part 2

Reflection

I want to reflect further on how we, with the best of intentions, label children and how those labels impact their identity and self-efficacy.

Several years ago, I had the privilege of hearing Peter Johnston present. One thing he said, and I also read in his book, really resonated with me – we need to stop using value-laden language in our classrooms. One common example, teachers will often talk about reading strategies by saying, “this is a strategy good readers use…” In a child’s mind, they immediately think, “I use that strategy, so I must be a good reader;” or “I don’t use that strategy, so I must be a bad reader.” Either scenario is detrimental to a child’s identity as a reader. The “good” readers often establish a fixed mindset, slowing their growth; while the “bad” readers often internalize the comment and shut down. 

We are all readers! We are all at a different place on our reading journey! Our place as a reader during our lifelong journey, does not and should not, change the fact that we are all readers! My job as a teacher is not to judge you as a reader, but to learn about your and understand you has a reader, so I can help you move forward on your journey —with or without me! There is NO SUCH THING as a “good” or “bad” reader! (Can you tell by the exclamation marks that I am passionate about this topic?) When I adapted the mindset that everyone was a reader in my classroom, I found I could celebrate curious joyful reading with all of my students, which in turn helped them to build their identity, self-efficacy, and motivation as independent readers. That was a huge mind shift for me, and I believe with my whole heart that even though I did not always see tangible results in test scores, I saw tangible results in how my students became confident readers and began to love reading.

I know that is not a popular statement but, sharing reading levels with students is just another way of saying, you are a good reader or a bad reader. There is no need to share a reading level with a student. The level in and of itself does not provide a reader with specific feedback/information that will help them move forward on their reading journey. If I want to share the results of a reading assessment with a student, or their parent, I can do so by providing much more actionable feedback that a reading level.

Reflection to Practice

I believe I have done a good job at mitigating the use of labels in my classroom. I work hard to remove as much value-laden language in my classroom as possible. Teachers in the grade-level above me have mentioned how my mitigating the use of labels has made a noticeable difference in the kids and their mindset to learn. But, I also know that I am not doing enough to help kids grow has independent readers. 

My goal this upcoming year is to combine the work of Jennifer Serravallo and Kate and Maggie Roberts (see books below). This summer I am working on learning progressions related to text complexity. My hope is that by teaching readers through learning progressions on text complexity, it will improve their ability to independently preview and select texts that are just right for them—not because I say so, but because they can consciously make an informed choice on their own. I also believe that by naming and explaining what makes a text become increasingly complex, this practice will add in the kids’ ability to comprehend text as well as author text. 

Be sure to check back to the blog for updates on my progress.

Johnston, P. (2004). Choice words: How our language affects children’s learning. Portland, ME: Stenhouse.

Johnston, P. (2012). Opening minds: Using language to change lives. Portland, ME: Stenhouse.

Roberts, K. & Roberts, M. B. (2016). DIY literacy: teaching tools for differentiation, rigor, and independence.Portsmouth, NH: Heinemann.

Serravallo, J. (2018). Understanding texts & readers: Responsive comprehension instruction with leveled texts.Portsmouth, NH: Heinemann.