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    • 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.