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    • StephenL: Tell me about yourself.

      @cbrownlmath: This question always gets me wanting to reply with questions...I will, however, not do so today.  A few personal details I am a...Husband, father of 3 grown women, no grandchildren, cancer survivor. I was born and raised in Central California, though it hasn't always been home, it is for now.

      I am a mathematics and STEAM educator with over 30 years of experience at this point. My first 14 years were in the High School setting where I taught all the courses a Secondary teacher might from General Math to AP Calculus. During those years I worked on a Master's of Arts Degree in Math Education and then changed roles and became a university faculty member at a small, faith-based, liberal arts university. In 2011 I went back to school and took a Leave of Absence from my teaching job at the university and earned my Ph.D. in Education, Policy & Practice with a focus in Mathematics.

      In 2019 I co-authored a book with Sunil Singh of Toronto, Ontario Canada; Math Recess Playful Learning in an Age of Disruption. Wherein we make a radical call for a change in how mathematics is viewed by society in general and the education community in particular. Mathematics is a vibrant, joyful and uplifting endeavor we as a species have engaged in for millennia. Only in the last 100 or so years have we turned it into this dull thing of drudgery that needs to be "gotten through," endured, tolerated. We need to recapture that sense, return again to the idea that mathematics is an adventure.

      I am an active researcher and professor of mathematics & STEAM Education, and creativity as well. I have several publications in-press now related to the intersection of creativities and mathematical understandings. I expect to see these having a broad impact on the teaching and learning sciences.

      StephenL: You have written several articles on creativity in mathematics.  That seems like a contradiction to the mathematics many older adults experienced during their  K-12 education. Can you provide an example of how a parent of a young child can create the conditions for a creative exploration of mathematics?

      @cbrownlmath: While thinking of mathematics as a creative endeavor may seem foreign to many who have been schooled in math in the last 50 years, it was never seen as such by those who have been busy making it over the past 8 millennia or so. The processes mathematicians follow to arrive at the end result we see are never as clear as what they publish in the public record. For instance, take the result we know as the Pythagorean Theorem. One of the earliest proofs of this idea is credited to a man named Euclid who wrote down his proof as the climax of his first book of the Elements. He wrote this down in a very neat and tidy was however not the manner in which Pythagoras thought of it. It came almost 150 years after Pythagoras had died. Now that refinement takes careful, considered, and often deep thought. This is creativity at its purist.

      First off parents, be curious yourself, never stop wondering, asking questions about how and why this or that idea in mathematics came to be or must be true??? The most critical thing parents can do to create conditions for creative exploration is to never answer a math question with an answer...but another question, even better one they don't know the answer to themselves. After this attitude of perpetual questioning is the family atmosphere, also provide your child with blocks, springs, gears, pendulums, puzzles, string, paper to draw on (few if any coloring pages where there is an expectation of "coloring in the lines")...let them entertain their fantasies, visions, and musings about reality.

      StephenL: What was the writing process like with a co-author?  Did you then work with a book editor on the manuscript draft?  How did you resolve differences of opinion?

      @cbrownlmath: Writing with a co-author was a fun experience for me. I enjoy collaborations, find renewed energy when bouncing ideas off a good friend. I don't recall having many differences but that was probably because we both scoped out various sections of the book to write from our own perspective. This format allowed us to stay out of each other's stories but to provide clarifying questions and suggestions to each other. Our publishers also provided some important guidance during the writing phase. He wanted a book that would bring our message to school administrators as well as math teachers; we had not been thinking big enough you see, and they worked with us, intensely, for more than a few meetings till we were able to incorporate their thoughts more fully.

      Yes, we did work with an editor, one that our publishers chose. Well really, several editors because we had imagery embedded within our text and as such some editors worked on our prose, others on our visuals. Finally, it came together under the watchful eye of an executive editor who gave final approval on all the little things here and there. Truth be told, that creative endeavor, messy and smudgy as it was, still went to press with errors. We found several in the first weeks after the printed versions were out and thanks to several kind friends we were able to rectify these and now in the second printing it is relatively error-free.

      StephenL: Blocks, springs, pendulums.  I get the sense that creativity in mathematics is about inquisitiveness on whatever it is one is learning.

      But aren’t you setting students up for disappointment when they enter the classroom and standardized tests and worksheets are more the norm?

      And what if that student shares their inquisitiveness by asking questions and gets shut down by the teacher who’s trying to just get through the curriculum?

      @cbrownlmath: Ohhhhh you are pushing all my buttons now, aren't you? ;)

      In short, yes I suppose I am setting children up for disappointment. Classrooms are inevitably disappointing though. In this era of accountability through relentless assessment, how can anyone expect schooling to be completely non-disappointing? So here my thoughts turn to parents. If you are genuinely interested in helping your child to enter school and not be disappointed after you have raised her to be the ultimate in creative person she can be...start now to get involved in the process of policy debate surrounding schools. It doesn't sound sexy or exciting but changing policy is where the hard work of changing school culture is going to take place.

      In the situation you describe where an inquisitive student is shut down by a harried and haggard teacher, first I would hope that the student could tell their parents about that happening and that they could go together and discuss the situation with the teacher. My initial reaction is to want to accuse the teacher of malpractice, but the truth is most teachers are doing their dead-level best to teach all the students in their give them some space to be human too. But never give up advocating for your students to have the opportunity to be fully creative within the classroom situation. Support teachers by working to improve their lot in life through greater autonomy not the overstructured, homogenized attempts at the one-size-fits-all curriculum that so many have to teach now.

      StephenL: You mentioned that your editors challenged you to broaden the scope of the book in order to make it more approachable to school administrators whose first language was not math.

      What changes did you make to the book as a result of that feedback?

      @cbrownlmath: Yes the publisher helped us to see that while we were seeking to be disruptive to the system of education regarding mathematics, we need to do so in ways that school administrators won't shut off from completely for fear of being “too radical.” So one important alteration we made was to link the problems, puzzles, conundrums, paradoxes etc. that are in the book to the "Common Core" Standards in some way. I did this by focusing on the most important, and least often thought of, Standards of Mathematical Practice (SMPs).

      The SMPs address the critical concept of purpose in the math curriculum. I see them as saying something like, "Look! Mathematics has within it so many sub-fields that we could never enumerate them all, many are still left out of the Content Standards...but really these eight statements are the PURPOSE for studying mathematics at all in the K-12 curriculum.”

      If you look at any single piece of the mathematical content you are likely to be hard-pressed to justify its artificial inclusion in the curriculum at the grade-level that it occurs, without connecting to the SMPs. They remind me of an excellent little article written by a couple of mathematics authors, Al Cuoco and Kenneth Levasseur, in which they describe what they have seen in mathematicians' ways of attacking problems and call for the curriculum to foster these positive habits of the mind.

      Sorry, I have gone afield short if you look in the book, after every puzzle, problem, or game you will see a parenthetical note that is meant to inform all who read the book that, what just preceded this spot is an example of the following Standards of Mathematical Practice. In this, we hope to give teachers a means by which they can "justify" the inclusion of such tasks in their classes. Because in this era of senseless accountability and high-stakes testing, EVERYTHING that is taught must be tied to some "standard." It is oppressive.

    • First off let me say unequivocally that I have the utmost respect for teachers and the work they do. Their job is often thankless, definitely undervalued economically, and challenging. Balancing all the groups and goals, Americans especially, have put on schools has gotten to be a near 24-hour gig, and that isn’t good for anyone. Now, about play, a good friend of mine gave this response to your question, “How can we NOT incorporate play into the classroom?” to which, I give a hearty YES! 

      Play is the child’s job, the teacher’s job is to provide environments, opportunities, structures and values that incorporate play. By opportunities, I mean using games and puzzles, conundrums and paradoxes that intrigue the mind. This morning while thinking about this question a video came across my Social Media feeds that show a man (Tadashi Tokieda) pushing a large circular cork coaster through a small square in a piece of paper. He reveals the magic without hesitation, and the mystery is not solved but rather is enhanced. No child who watches this could help but wonder what is happening. Watch this video for yourself and will know what he does...but you are likely to have more questions at the end than were answered. This is the heart of playful learning, where one idea draws out many many more. Look for opportunities like this.

      Create in your classroom, social structures that allow students to foster their creativity and to ask the questions that plague their minds. It is hard in the middle of a day where you know you have to get through certain aspects of the curriculum, resist that tyranny. Isn’t that the very goal of disruption? :) 

      Lastly, value laughter and joy in your classroom over everything else or well at least enough that you make daily decisions to be playful and allow playfulness. Shift the work to the child, let them accomplish their job description: Play.One other postscript to this question, I am reminded of the Latin meaning of “curriculum.” During the Roman Empire, horse-drawn carriages and carts were the means of moving goods and people. After dozens of years of trudging ruts would be worn into those famous “Roman Roads,” those ruts were called curriculae. Don’t fall into a rut.

    • Thanks Matthew for your great question. I'd like to say that my response would be definitive and declare that I hold the one true "THE first step," but will hopefully give you an idea for A first step teachers could take. Maybe start by committing to celebrating student questions more than student answers. When I made this shift in my teaching the joy and laughter in my classroom went upwards. Questions are more fun than answers because they are filled with fruitfulness and potential, while answers are pruned off thinking.

      As I said, this is just A possible first step, of course there are others that might be more fitting for other teachers, and for that I would like to ask the readers for their thoughts on this, collaboration is a great way to get at more truth.

    • Lil, your questions are always so insightful for your 12 years of experience, thank you for this one in particular.

      The U.S. is and has been suffering under a terrible misunderstanding of its place in the educational achievement world. This misunderstanding has caused many of us to spend way too much time in handwringing about our self-described mediocrity of late. Beginning in the hyper-paranoid Reagan era of public policy wherein the National Research Council coined the document "A Nation At Risk (ANR)," and continuing through to the Common Core State Standards response to the No Child Left Behind/Every Student Succeeds Act policies we have been frantically attempting to pull ourselves up by our bootstraps. We perceived ourselves as falling behind from our former glories and therefore in need of making ourselves over yet again.

      The data however does not support our misperceptions of it. The US is among the most diverse culturally, economically, and educationally populations in the world. Third largest in raw numbers behind China and India (which of these is the largest depends on which week you look), we boldly attempt to educate every single member of our population within a system originally designed to educate a very small, select subset of that population.

      I could go on and on about this but will try to focus more on your question of what prevents us from adopting a more playful learning model? In a word, FEAR. The ANR told us that we had in effect waged a war of mediocrity upon ourselves and those words have scared us so badly that we continue to flounder about for more and more control of our situation...not realizing that those fearful words were flat out WRONG.

      We as a culture need to come to grips with the real data that our achievements in education over time and populations is NORMAL (0,1) for one thing. Then look to other systems that are in fact achieving marginally higher in areas we would like to improve in, especially those that seem to graduate students who still enjoy learning after their 13 years of schooling.

      I said this recently to a group of educator friends and have decided to make it my schooling motto, "If a child graduates from High School less curious about the world than they entered it in Kindergarten, then schooling has failed her, we have failed her." If we set this as our standard to achieve and forego the relentless drive to over assess through over governing what happens in classrooms, we will remake our schools into places of joyful experience and exploration. This is what I think you are referencing when you speak of other countries who seem to have this attitude.

      Wow, sorry for being so wordy.

    • As I said, this is just A possible first step, of course there are others that might be more fitting for other teachers, and for that I would like to ask the readers for their thoughts on this, collaboration is a great way to get at more truth.

      I loved the reader chats you and @Mathgarden did this past spring and summer on Twitter. What a great use of “educhats” for teacher PD!

      Highlights from one you did last spring are here.

    • Thank you Drew for this question. I have pondered this often and hope that I reflect what I am about to say in my own classroom. In no particular order than the order they tumble from my mind:

      1. Always continue in your own curiosity. Never stop wanting to know more about stuff in general and things your students might want to know about in particular.

      2. Follows from 1. Share your curious thoughts with your students. (I suddenly hear Crosby, Stills, Nash, & Young..."Feed your children well...upon your dreams...").

      3. Be fearless in the face of not knowing the answer yourself. And if you can't be fearless, be brave enough to live with uncertainty yourself. Teachers are not required to be the source of knowledge, rather they should be the source of questions, ideas on how to get to the answers sure, but not the repository of the answers.

      4. (You asked about maths teachers specifically I will try to focus there) Love your subject, get lost in it. Right now my co-author Sunil is writing a book he is titling "Down the Rabbit Hole..." a phrase we chose for a chapter heading in Math Recess. He is doing so to allow himself the space to get lost in the maths. Be curiouser and curiouser about things you thought you once knew.

      5. About that last point in #4. I have found through my 30+ years that there are always new ways to see the things that I once thought I knew completely. For instance, when I met Dr. James Tanton and started to experience Exploding Dots I came to a much fuller and richer understanding of the ideas of polynomials in general and the specifics of arithmetic. Two subjects I could have argued I knew A LOT about before. Or like in the past few months, Dr. Po Shen Loh revealed a method he was clarifying for himself for the first time, regarding solving Quadratic Equations, you can watch this here. Both of these mathematicians found new ideas for themselves within what is arguably "Elementary" mathematics. Never diminish the power of mathematical thinking.

      6. Study the history of mathematics. This is far more critical than I once believed. But maths are a human creation, and therefore have human stories that surround them. Knowing those stories helps to contextualize and demystify mathematics. Also, studying the history of maths is a fabulous means to celebrate the diversity of cultures within your classroom. This process is known in some circles as re-humanizing maths and I am fond of that characterization.

    • This is also an interesting question. We spent a lot of time lamenting common problems such as the rise of reactionary factions in the math education community. In parts of Canada there are some serious conflicts between some vocal and contentious people as there are in the US. The struggle for equity and fairness is common to both our nations. Of course, you can imagine that his and my desire to see a more playful atmosphere in math classes are born from similar experiences, especially joys.

      The joy of seeing students find grandness and excitement in mathematics kept us both in this game. Children are the same the world over in this.

    • I have seen some absolutely amazing applications of VR and AR in the classroom. For instance there are folks, especially in Europe it seems, who are creating immersive geometric experiences for learning basic Euclidean ideas. Picture this, you have your Occulus on and step into a world where you construct a triangle with one fixed length edge and opposing vertex on a line parallel to that edge. With your gloved hand you can slide that vertex along the parallel line, step through the fixed and stable area that the triangle encompasses, or even "throw" the vertex along your "infinite" line and watch all the effects of this sheering on the area in "real-time." You can do this with your whole body involved. A project I am particularly fascinated with is being run by Henry Segerman at Oklahoma State University. He and some associates have created a VR experience wherein you enter a universe that behaves in a locally non-Euclidean manner. In this world you can walk through six rooms that surround a single, shared corner, each of them at right angles to their neighbors. This world is based upon the Hyperbolic Geometric Axioms of space. We now have the ability to help students to literally embody their understanding of abstract ideas.

      @Brzezinski_Math on Twitter is constantly publishing video of how Geogebra AR (an app you can run on any smartphone or tablet) can be used to model an 3D object you like (for a quick introduction watch this short video he made a few years ago, then imagine how much better things have gotten since) I wrote my dissertation a few years ago now and its focus was on teaching geometry from a transformation basis. So this is an area of keen interest to me. Tim argues quite potently for the idea that transformations should begin in three dimensions rather than two because it is so easy now to implement, thanks to AR being so readily available and easy to use, plus it is just so danged cool creating a model of your Spiderman coffee mug on your iPhone.

      Regarding Spatial Computing, could you clarify what you are referring to? I have an idea but am uncertain that it aligns with yours.

      Thanks for these super interesting questions. We live and work in wildly interesting times when it comes to the possibilities of doing teaching better. I like that I feel guilty when I fall back to a Lecture - Take Notes model of a classroom (which is not evil just not the best way to learn 100% of the time, small doses only please). I think these tools can be used, like the tools of a pencil and paper, to enhance learning. Let's NEVER shy away from implementing them when we are ready to use them in an enhanced manner. As my friend Alice is fond of saying, "A PDF of a worksheet is just that...a damned worksheet. Paperless is not a pedagogy." We get to keep learning ourselves, so we can model it for our students. This is the very essence of playful learning.