I joined twitter at the end of April. One thing I like to do is search through tweeted pictures of everyday objects that inspire mathematical thinking, excitement, and wonder. Examples of some of my favorite hashtags include: #diamondchat, #noticeandwonder, #wodb, #unitchat, and #tessellation. The teachers, parents, and mathematically curious that follow and use these hashtags clearly enjoy searching for and sharing shapes.
And for those who want to search for and share shapes with kids, Tana Hoban’s picture books are an excellent choice. The photographs in Hoban’s books remind me of a magical passage in a 2012 paper by Zalman Usiskin entitled “The Shapes of Geometry and their Implications for the School Geometry Curriculum.”
The only thing that keeps us from thinking that every physical object is a geometric object is that we do not always have names for the shapes of blades of grass, of lampposts, of tires, and so on. But we do have names for some of the classes of shapes; we recognize the shapes of animals, of leaves on trees, of fenders on cars, of scissors, of so many things that are either natural or created by us. In any room, the chairs, the knobs on the doors, the light fixtures, the patterns in the carpet or tile, the pens in our hands, the letters that are written or printed on pieces of paper in our pockets, the pockets themselves, our keys, our bodies…all are geometric. With the general definition of congruence and similarity in terms of transformations, we can speak mathematically of congruence and similarity with any of these objects. We can speak of the space they occupy, whether in two or three dimensions. For the 3-dimensional objects, we can ask about cross-sections and surface area and volume.
[To access the entire paper, readers are welcomed to contact Zalman Usiskin directly via email at email@example.com Additionally, an earlier version of this paper is available in the book We Need Another Revolution: Five Decades of Mathematics Curriculum Papers by Zalman Usiskin, Barbara J. Reys and Robert E. Reys (editors), Reston, VA: NCTM, 2014, pp. 329-344. This book contains 38 of Usiskin’s papers. Interested readers might want to look at this volume as well.]
Of course, it isn’t groundbreaking to say that Geometry can be observed in the world. Whether it’s symmetry in butterfly wings, triangles bracing a bridge, the similarity snapped in a photograph, or congruent shapes decorating a bathroom floor. So, why is this passage so magical?
Let’s just say that Usiskin got me at ALL. In reading this passage for the first time, I was struck by how limited I was being in my thinking and teaching of geometry. This idea that ALL physical objects can be viewed as geometric objects and could be studied as such was transformative for me. This new view enriched, deepened and stretched my understandings of shape, similarity, transformation, and congruence.
As you flip through the pages of Hoban’s books, the chalk drawings and their pavement canvas, a bunch of ripe, red tomatoes cradled in their wire baskets, ALL are geometric. Tana Hoban gets it. And thanks to Usiskin’s paper, I do too.
Now, let’s talk about these books.
Tana Hoban has won many accolades for her work as a photographer and filmmaker. Her photographs have been exhibited at the Museum of Modern Art in New York and in galleries around the world. In 1959, she was named one of the country’s top 10 female photographers.
Using her photography, Hoban created innovative illustrations for children’s books. Three of her books are featured in this post. They are:
- Hoban, Tana. Shapes, Shapes, Shapes. Greenwillow Books, New York, 1986.
- Hoban Tana. So Many Circles, So Many Squares. Greenwillow Books, New York, 1998.
- Hoban, Tana. Cubes, Cones, Cylinders, and Spheres. Greenwillow Books, New York, 2000.
With over 110 titles, Hoban’s books for children are well-known and loved throughout the world. [See this flyer for some of her books]. Most of her books are geared toward children 5 or younger, helping them to identify objects by sight and name. Her books help children explore the world by introducing concepts such as counting, shape, machines, colors, animals and textures.
Hoban describes her motivation:
“I try in my books to catch a fleeting moment and an emotion in a way that touches children and makes them want to respond. … Through my photographs and through open eyes I try to say, ‘Look!’ There are shapes here and everywhere, things to count, colors to see and always, surprises.” (Tana Hoban, 1986).
Tana Hoban’s books provide a window to observe the physical objects in the world through the lens of geometry. The pages of these three books are covered with vibrant, vivid images of shapes in a variety of creatively-captured contexts. Shapes by land, sea and air, shapes on playgrounds, on city streets, shapes in kitchens and market-places.
With every page turn, there is layer upon layer of shapes to notice. Tessellations, patterns and collections of shapes provide real world examples of similarity, congruence and transformation. In each book, Hoban’s carefully chosen photographs of bold-colored shapes shine brightly from beginning to end.
I’ve used these books with two groups of children.
First, after reading The Greedy Triangle by Marilyn Burns to my kids and my friend’s kids (ages 3-12), I placed the collection of Hoban’s books on the table and asked them to search, find, and describe a shape that they would like to add to a group scavenger hunt list.
After they shared their shapes and made their lists, they paired up and raced throughout the yard and house searching for shapes. Here are some pictures of shapes they spotted.
During the scavenger hunt, they debated (e.g., whether a saw was really a triangle or something else), shared their frustrations (e.g., scalene triangles are difficult to find), and their surprise (e.g., when an unexpected shape turned up, like a circle in a discolored patch of grass). It was exciting for them (and me) to see how many different shapes they were noticing. [There were many more shapes than are included in the slideshow. I wish I gave them cameras! These were just the pics I could capture when they showed me their discoveries.]
I continued using Hoban’s books with my children Liam (7) and Siena (5). We would flip through the pages and I’d ask them what they noticed and wondered. A few times, I asked them to be shape photographers like Hoban and take their own pictures of shapes on IPADs. On one particular occasion, Liam decided he was going to look for all the shapes on the first page of Hoban’s Shapes, Shapes, Shapes book.
Siena of course agreed that she was going to find all those shapes too. [Although later decided she didn’t want to find a trapezoid because she “didn’t feel like it.”]
Here is a slideshow of Siena’s shape photos.
And one of Liam’s shape photos. He proudly stated that he found all the shapes except the trapezoid.
If you peeked in at either of the slideshows from Liam and Siena, did you wonder which shape in the picture prompted them to snap the shot? The photos are full of shapes. Since I wondered what they were noticing as they snapped the picture, I sat them both down and we looked at the slideshows together.
First we looked at Siena’s photos. I asked Liam to share what he noticed about the first picture Siena took. After he shared, I asked Siena to share. I continued in this manner for the rest of her photos. Then we looked at Liam’s slideshow.
Here is an example of an interaction around a clock picture Siena took.
Liam: I see a circle and rectangles and more rectangles and a square.
Siena: I saw that if you clock your head (tilts her head to the left) this way it’s a diamond (talking about the white clock) and if you clock your head (tilts her head back so that it is upright) this way it’s a square. And it always a circle (talking about the clock face).”
I loved how she is making sense of both language (clock versus cock your head to the side) and her concept image of shape ( how for her, a shape is a diamond in one orientation and a square in another). As we looked through their pictures, they both noticed and named many different shapes. Liam even found his trapezoid in one of Siena’s pictures.
In Usiskin’s 2012 paper, he states “the more shapes that the geometry covers and the more ideas that relate to the shapes of figures, the better the shape of the geometry.” Tana Hoban’s books are a wonderful way to support children in noticing and describing the shapes around them. Her books inspire children and adults to peek in and see what they can find. As Hoban states in one of the tiles, there are “so many circles, so many squares.” Indeed in these books, and in the world, there are so many shapes to see and share.
Hashtag Help: I would LOVE to add to the hashtag list in the first paragraph of this post. If you have #suggestions to share, please leave them in the comments of this post or tweet me @KellyDarkeMath. Thanks in advance.
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Thanks and see you next Monday! #mathbookmagic