Blue Tape Transformations (#TMC16)

I wrote a book. One major premise of the book is that changing the scale of a mathematical concept and/or exploration can help math learners make sense of the math at hand in new and helpful ways. This is a short story of how the blue tape I use to define the dancing space in Math in Your Feet brought unexpected new insights to me as both a teacher and a learner in a surprising, unexpected way.

I recently returned from Twitter Math Camp, a professional learning experience for math teachers, by math teachers. It’s tremendous. For my morning session (three days, two hours each morning) I chose Tessellation Nation (#tessnat). It was three days of open ended exploration and cross pollination of individual inquiry focused on making sense of a topic that often seems pretty straight forward but is actually NOT, in any way, shape, or form (ha!).

The first day I chose a random bag of shapes crafted by Christopher Danielson (of many fames) and see what might happen. Bryan joined me and we were completely engrossed in investigation for the entire morning.  We both left for lunch inspired and looking forward to the next day.

The second day I decided to act on my wondering about how pentominoes would tile. I also decided to challenge myself to work solely on graph paper. I spent a good 30 minutes with the F. People kept coming by to see what I was doing and every time I would say: “Visualizing transformations is not my strong suit. I really want to see if I can do this on paper. I don’t want to cut out a manipulative.”

Someone convinced me (probably Max, who was spiraling some kind of shape next to me at the table by the open window) to try a different shape. I picked W.  I was more successful at creating the W on graph paper, but was still really disoriented trying to tile one W with another visually.

Max, who has been my partner in crime on and off on matters of blue tape on the floor since TMC14 wondered if we might take this investigation to the lobby.  We taped out the W pentomino and I immediately thought of naming the pathway using spatial terminology. It sounded like this: “Start, down, over, down, over” as pictured below.


I walked and talked the pathway a few times but my bigger goal was to “nest” the Ws, so we built a larger grid and Max took colored squares of paper to keep track of our individual pathways (I’m pink, Max is green).


I immediately felt a rush of adrenaline and ran to the graph paper to draw it out. It was like the sun had come out after a week of rain. Partly because I FINALLY UNDERSTOOD the structure of the W pentomino and how it could tile with itself,  and partly because it was a spectacular example of what I know and have been writing about for the past couple years:

Changing the scale of a mathematical investigation has the potential to create new insights about previously understood and not-understood mathematics. In children AND adults.

Then Max and I tried standing on different corners of the grid to see what a double reflected or rotated W pathway might look and feel like. You can see the Start-Down-Over-Down-Over pathway with each of the colors. But it doesn’t end there.


The best part of putting blue tape on the floor is that we usually can only do it in open, common spaces which means folks walking by often get curious about what we’re doing.

During this time period one of the Tessellation Nation folks happened by (I’m sorry I don’t remember your name!! But I remember you!!)  This person had very bravely, not half an hour before, shared that she really didn’t understand reflections. We showed her the W pathway and then, again I think it was Max, asked her if she could reflect it. She got a lot of hints about using opposite feet, and she had some productive struggle but I think you can see how elated she was when she finally GOT IT. We were all so very happy. High 5s and hugs all around!


And THAT is the tale of two different kinds of transformations that happened during the TMC16 Tessellation Nation morning session — at once both personal and mathematical. I’m feeling verklempt just thinking about it but I have enough poise to share this one final thought:


#BlueTapeLounge at #TMC16

Perhaps my very favorite moment at Twitter Math Camp was setting up the Blue Tape Lounge for some informal after hours Math in Your Feet math/dance making. Brian Bushart, Max Ray-Riek and I set up our space. While we were waiting for folks to arrive we played around with the movement variables that help kids build their own foot-based dance patterns. Brian and Max both made new patterns and Max & I put them together into something quite terrific. Here’s the video evidence 🙂

As the night went on, and folks got used to learning, making and reproducing foot-based dance patterns, their physical spatial reasoning also kicked in which opened up possibilities for playing with transformations. Below, we’re all doing the same four-beat pattern but two of us are rotated 180 degrees and the pattern itself has two 90 degree turns.

The next day, someone new to my work told me that when he first walked into the room the activity seemed really random to him, but when he got inside the square and learned a pattern he realized that what we were doing was working within a set of mathematical constraints. Christopher Danielson has wondered

“whether a characteristic of a novice is an inability to distinguish noise from pattern”

and I certainly think this idea holds water.

One final thought: Max said, “Here, the math is the dance…” and that is really at the foundation of it all.

p.s. If you are curious about finding out more about this work of pairing the moving body with math learning, check out my new book which will be published by Heinemann in October 2016!

Gradients in the Wild

Found gradients

Spontaneous arranged gradients

Arranged gradients by my kid: Same objects, different categorizations. First up, not round to roundest.



Next, color gradient, most brown to least brown.


Largest to smallest.


And, finally, Part 1, NON GRADIENTS!!

And, finally, Part 2: A MOVING GRADIENT!! Courtesy of Lani Horn:

“Because of the outside I found an inside”

It was this piece that made me notice something very interesting about first grade thinking related to shape and structure.


When I asked the creator of the piece above, a first grade boy, about his work we started noticing together.  I said at one point, “You know what I notice? I see two triangles on one end, then three triangles, then two, then three more. It looks like you made a rectangle.” His teacher was chatting with him as well and the room was noisy so I missed the conversation that led to the insight, but it was a lovely phrase: “Because of the outside I found an inside.”

The activity was one of four loosely structured math/art stations. In this case there was a visual provocation, a sense of where we might be headed with the materials. At the same time I do make a point to structure and present the materials in a way that leaves them open to possibility and new ideas.  Continue reading

My NCTM Innov8 Workshop: Vote it up and see you there!

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It appears that my session proposal for the new Innov8 NCTM conference (St. Louis, MO, November 16-18, 2016) has been shortlisted and is awaiting your vote!  Here are some details about my 30 minute video interactive which I will deliver if voted up by the populous of #mathed land!

Beyond Mnemonics: The Body as an “Object to Think With”

We know kids love to move—in fact there’s a developmental imperative at play that cannot be ignored.  But how can we harness this innate playfulness in a way that moves students, literally, toward conceptual understanding of Pre-K to 2; 3 to 5 mathematics? Through multiple video examples learn how the whole, moving body can be a strong partner in the math learning process, not a break from it.

Fun Fact #1

The session title is also the title of Chapter 3 of my upcoming book, due out Fall 2016.

Fun Fact #2

To support educators in bringing whole body math learning into their classrooms I am building a Facebook group which, in conjunction with the new book, focuses on building a community of practice, support, and collegiality around this kind of  work. I haven’t fully launched the group, but if you want to get in on the fun please join us! Group members will start seeing previews from the book and other relevant articles and resources as the summer progresses.

You can vote for the session on the Presentation Picker. Thanks for the consideration and I hope to see you there!

Video: Are Mathematicians Creative?

From the YouTube description of the video (below):

“Is doing research in mathematics a creative process? When mathematicians talk of their subject as beautiful, what do they mean? What are their motivations? Their dreams? Their disappointments? These themes are explored in the collection of five short films produced as part of Mathematical Ethnographies project. The focus is not on mathematics, but on the people who create and teach mathematics – on mathematicians.”

Some of my favorite bits from the video focus on the process of doing math but the entire eight or so minutes are really revealing and relevant to math education:

“Like writing a sonnet, you have to conform to precise rules but having that structure there to constrain you somehow enhances the creativity.”

“To be good at mathematics you have to develop a new insight.”

“There are moments where things become clear, it comes out of nowhere.  I’ll be in the shower, cycling, and somehow the answer comes.”

“It’s hard to see how asking a question no one has ever thought of before is a logical process.”

I always think it’s nearer to architecture rather than to other arts; one is trying to build this formal structure up and there are supports and girders and there are connections…”

Are Mathematicians creative? If creativity is a process more than a product, I think then that the answer is yes. What do you think?

The rhythm of point, line, tri & quad

RN all 4

I have a brand new obsession. It’s called Rhythm Necklace. It’s an app described as a “musical sequencer for exploring the geometry of rhythm necklaces, and for experimenting with generating rhythms algorithmically.”  It’s got a deep conceptual base in the work of Godfried Toussaint as laid out in his book The Geometry of Musical Rhythm.

I see huge possibility for the classroom, but I’m not yet at the point of understanding what that would look like with real live kids. For now I am content exploring the tool and following my nose down a bunch of rabbit holes. On my fifth composition I hit upon something I really liked.  I call it “1, 2, 3, 4” or “Point, line, tri & quad”.

In the video, below, I add each rhythm one by one and then mute some of the rhythms so you can hear the relationship between the 1 and the other three, and also the 4 and the other three.

I’m curious other ways I can configure the same rhythm visually.

I’m curious how many different rhythms (not mathematically speaking and not counting tonal quality) I can create using a point, line, tri and quad.

I’m curious what might happen if elementary kids worked in teams of two with a series of tasks/challenges to investigate.

Just in writing out this little post I’m already thinking about what those tasks might be. For now, though, I am more than content to continue down my nonverbal rabbit hole. It’s just what I needed after 2.5 years of existing in the realm of words and book manuscripting. I am much happier here!

Some thoughts on “hands on” math learning

Note: This is a re posting of some thoughts I wrote down at my old blog on April 1, 2015, shared here in its entirety.  

Last night on Twitter Michael Pershan asked me to weigh in on hands-on math learning. The request stemmed from a conversation/debate about the various merits of different ways to learn math.

The minute I read the question I knew that my answer was going to be more detailed than a response on Twitter would allow. Here are some of my thoughts on the matter.

1. The discussion reminded me of the “concrete to abstract” conversations which, to me, seem like an especially frustrating example of recursion. They go round and round but we never really get anywhere new.

I think many connect the word “concrete” to Piaget and his discussions about children’s thinking moving from the concrete to the abstract. This in turn has led to many assumptions that take the term “concrete” quite literally. But, as Deborah Ball wrote in her article Magical Hopes,

“Although kinesthetic experience can enhance perception and thinking, understanding does not travel through the fingertips and up the arm.  And children also clearly learn from many other sources—even from highly verbal and abstract, imaginary contexts.”  

The best treatment of the concrete/abstract dichotomy comes from Uri Wilensky:

“The more connections we make between an object and other objects, the more concrete it becomes for us. The richer the set of representations of the object, the more ways we have of interacting with it, the more concrete it is for us. Concreteness, then, is that property which measures the degree of our relatedness to the object, (the richness of our representations, interactions, connections with the object), how close we are to it, or, if you will, the quality of our relationship with the object.”

I LOVE this treatment of “concrete” as simply the quality of your relationship to an idea. Seriously, read the whole piece. You’ll be glad you did.

2. Professional mathematicians utilize a multi-sensory approach to their work. Here is some perspective from researcher Susan Gerofsky:

“Movement, colour, sound, touch and other physical modalities for the exploration of the world of mathematical relationships were scorned … as primitive, course, noisy and not sufficiently elevated or abstract.  This disembodied approach to mathematics education was encouraged despite the documented fact that professional research mathematicians actually do make extensive use of sensory representations (including visual, verbal and sonic imagery and kinesthetic gesture and movement) and sensory models (drawings, physical models and computer models), both in their own research work and in their communication of their findings to colleagues in formal and informal settings.  These bodily experiences ground the abstractions of language and mathematical symbolism.”

3. Children think and learn through their bodies. We should use children’s bodies in math learning.

Known in the research world as embodied cognition (thinking and learning with one’s body) is something we begin developing from birth. Developmental psychologists have shown that in babies “cognition is literally acquired from the outside in.” This means that the way babies physically interact with their surroundings “enables the developing system [the baby!] to educate [herself]—without defined external tasks or teachers—just by perceiving and acting in the world.” Ultimately, “starting as a baby [as we all did!] grounded in a physical, social, and linguistic world is crucial to the development of the flexible and inventive intelligence that characterizes humankind.”

Understanding what embodied cognition and embodied learning looks like is the focus of a multidisciplinary group of cognitive scientists, psychologists, gesture researchers, artificial intelligence scientists, and math education researchers, all of whom are working to develop a picture of what it means to think and learn with a moving body.

Their research findings and theory building over the past few decades have resulted in a general acceptance that it is impossible to ignore the body’s role in the creation of “mind” and “thought”, going so far as to agree that that there would likely be no “mind” or “thinking” or “memory” without the reality of our human form living in and interacting in the world around us.

4. Finally, instead of sorting out the various merits of individual teaching/learning strategies what we really need to do is look at the bigger picture: Most student learn math best when provided with multiple contexts in which to explore a math idea.

A learner needs time and opportunity to experience a math idea in multiple ways before being able to generalize it and how it can be applied.  An idea, any idea, becomes “concrete” for the learner when the learner has had an opportunity to get to know it. Uri Wilensky said it best:

“It is only through use and acquaintance in multiple contexts, through coming into relationship with other words/concepts/experiences, that the word has meaning for the learner and in our sense becomes concrete for him or her.”

Pamela Liebeck, author of How Children Learn Mathematics, developed a useful and accessible learning sequence to help bridge the gap between a math idea and a meaningful relationship with that idea.  Based on the learning theories of psychologists such as Piaget, Dienes and Bruner, Liebeck’s progression is similar to how babies and young children learn to recognize the meaning of words, begin to speak, and then to first write and then read. It includes four different learning modes in which to interact and express mathematical ideas and includes:

  1. a) experience with physical objects (hand- or body-based),
  2. b) spoken language that describes the experience,
  3. c) pictures that represent the experience and, finally,
  4. d) written symbols that generalize the experience.

This sequence illustrates what many math educators already believe, whether or not they use this exact outline – that elementary students need active and interactive experiences with math ideas in multiple learning modes to make sense of math.

After a recent and particularly robust online discussion on the many different ways to support primary students in making sense of number lines, including a moving-scale line taped on the floor, Graham Fletcher said, “At the end of the day, it’s all about providing [students] the opportunity to make connections.”

Graham’s statement points to the importance of focusing on the child’s relationship to the math and the environment in which she learns that math. Hopefully it’s an environment where many different ways of thinking, expressing and applying mathematics are celebrated and nurtured.

“A different spatial outlook”

Exploring the differences in spatial reasoning between the dominant culture and indigenous cultures

Allocentric & egocentric coordinate systems

“Spatial navigation is a central human cognitive skill. Various scientific studies have concentrated on investigating differences in navigation performance and navigation strategies, in particular dividing egocentric and allocentric navigation. Navigation can be based on different reference frames using distinct coordinate systems to encode spatial information. An egocentric coordinate system is located within the agent and is conditioned upon his orientation in space, while an allocentric coordinate system codes relations between objects, independent of the observers’ orientation. [Source]

On the dominant culture’s conception of spatial skills:

“Spatial thinking, or reasoning, involves the location and movement of objects and ourselves, either mentally or physically, in space. It is not a single ability or process but actually refers to a considerable number of concepts, tools and processes.” (National Research Council, 2006).

On the spatial orientation skills of indigenous Australians:

“Australia’s Indigenous languages are rich in spatial terminology … Desert children’s ability to handle directional and spatial terminology in particular is taken as a sort of intelligence test similar to the counting prowess test among Europeans.”[Source]

“The ability to apply such knowledge is a product of nurture, not nature — it cannot be genetically transmitted any more than it is possible to transmit concepts about number and computation to other little Australians, except via processes of acculturation.”[Source]


“Not subject to the rotation of observers or reference objects

“Speakers of the Australian language Guugu Yimithirr (hereafter GY) at the Hopevale community near Cooktown, in far North Queensland, make heavy use in discourse about position and motion of inflected forms of four cardinal direction roots – similar in meaning to north, south, east, and west. The system of cardinal directions appears to involve principles for calculating horizontal position and motion strikingly different from familiar systems based on the anatomies of reference objects, including speakers and hearers themselves.

“Rather than calculating location relative to inherent asymmetries in local reference objects, or from the viewpoint of observers themselves characterised by such asymmetries, the GY system apparently takes as its primitives global geocentric coordinates, seemingly independent of specific local terrain and based instead on horizontal angles which are fixed, as it were, by the earth (and perhaps the sun) and not subject to the rotation of observers or reference objects. [Source]


“Left and right are not essential directions in your body”

“The Warlpiri children’s knowledge of compass terms was such that I preferred to use them in the classroom in some circumstances where left and right would normally be used. It is on the syllabus in the Space strand to teach young children to know left and right, but this is difficult for Warlpiri children. Really, left and right are not essential directions in your body. You know where your front is because that is the way you face. You know where your back is because that is behind you. You know where your head is and where your feet are because you can feel them in the orientation of your body.

“The point is that in Warlpiri language not only north, south, east and west, but also up and down are all in one united set of directional terms … The terms related to up and down are also included in the set of directional terms. So for a Warlpiri child left and right are really difficult; the compass terms are much more their way of doing things. ”  [Source]

yarning circle

Yarning (storytelling) Circle [Source]

Implications for education, in Australia and elsewhere:

“Educators contributing to, writing and implementing … [Australian] national curricula will be expected to “embed” literacy and numeracy strategies as well as Indigenous knowledge/s into diverse subject areas, including English and the arts.

“Such a cross-curricula approach means that into the foreseeable future Australian maths and science education will need to be conceptualised outside of what are often perceived as those disciplines’ own self-referential silos.

“The more difficult pathway will involve taking these ideas and shaping them into a curriculum that goes beyond inclusion of “Indigenous perspectives” but foregrounds “Indigenous knowledge” at the level of the episteme. [Source]

[Thanks to Carolyn Galbraith who started me down this path today by recommending I read the article It’s time we draft Aussie Rules to tackle Indigenous mathematics. It’s a great read. Go read it!]

A few thoughts on mathematical cognition

“…mathematical cognition is embodied in two senses: It is based in perception and action, and it is grounded in physical environment.” – Alibali & Nathan 2012.

Malke 4

short quote 1

Williams, 2012

CH3P9 cropped

Math knowledge is embodied

Alibali & Nathan, 2012

IMG_7144 - Copy



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