At the end of May, I had the pleasure of sharing a book entitled Which one doesn’t belong? with my son Liam’s 1st grade class. I have been thinking (and overthinking) this post for months, nervous whether I can capture the magic inspired by these books. But I can’t wait any longer. So here it goes.
Before I write about the books, I want to share a bit about the author. The Wonder-FULL Wizard of Math I speak of (or if you prefer magical wizard of math amazingness coined recently by Casey McCormick) is Christopher Danielson A.K.A @Trianglemancsd for twitter users. Christopher and I met as grad students at Michigan State University while working on the middle school math curriculum Connected Mathematics. Here I am (on the right) with the wizard himself (circa 2005 I believe), and the beautiful Sarah Sword between us. Christopher is one of the most thoughtful and creative teachers/ mathematical thinkers I’ve ever met. A kind and supportive colleague, I’ve left every conversation with him encouraged, valued, and inspired.
In the later books of the L. Frank’s Baum Oz series, the Wizard proves himself quite an inventor, providing devices that aid in various characters’ journeys. Since graduating grad school, Christopher has contributed many magical inventions that aid in the mathematical journeys of children, teachers and parents. As a former colleague, I know I’m a bit biased, but I encourage you to check out the list below and judge their magicality for yourself.
Some of his inventions include:
- TWO thoughtful and thought-provoking blogs Overthinking my teaching and Talking Math with your kids.
- Math-On-a Stick experiences at the Minnesota State Fair (Coming soon! August 24-Sept 4).
- Contributions to Desmos resources, an amazing teaching/learning tool I used often in my teaching.
- Summer of Math subscription program (see below for pics of my kid’s playing math with some of his materials. And I’m pretty sure he makes those wooden tiles himself.)
There’s more, but I think you get the point. See? Inventor! Wizard of math amazingness. Told ya!
And he is humble. He would rather we “pay no attention to the (@triangle)man behind the curtain!” ― L. Frank Baum, The Wonderful Wizard of Oz
When I tweeted this about his book, the man behind the curtain had this to say:
Of course, wise, humble wizard, what you say is true. But I think your books are magical too. And here are a few reasons why.
There are two Which one doesn’t belong? books. The Shapes Book and a Teacher (Parent) Guide to the Shapes Book.
The books were awarded the 2017 Mathical Book prize, an annual award organized by the Mathematical Sciences Research Institute (MSRI) for fiction and nonfiction books that inspire children of all ages to see math in the world around them. The Mathical group placed them in the 3-5 grade category, however these books can be used, and have, with kindergarten-aged children to Calculus students.
A Bit about the Shapes Book.
The design is simple. Each page of the Shapes Book is similar in design to the page shown below. Four shapes on a page along with the same question : Which one doesn’t belong?
Take a minute now to look at these four shapes and answer that question for yourself.
- Did you pick the shape in the top left corner? If you did, can you explain why it doesn’t belong in a way that makes sense to others? Great!
- If you picked the shape in the top right corner, can you explain your choice so it makes sense to others? Excellent.
- How about the bottom shapes? Did you choose either of those? Can you explain your choice so it makes sense? Perfect.
That’s the beauty of it! Every shape is a possible answer to the question, Which one doesn’t belong?. And the measure of what is right is what is true. You want to share your ideas about this set, don’t you? Please do! Post your ideas in the comments below and let’s see how many different ideas we can generate using this set of shapes. If you want to read more about Which one doesn’t belong? (WODB) prompts, go here to Danielson’s blog.
The book has 11 different sets of shapes offering opportunities for hours of discussion in the classroom and home. And if you want more examples, and I suspect you will, here is a website created by Mary Bourassa with more WODB prompts designed by Bourassa and others.
A Bit about the Teacher Guide
I learned a lot from this guide and I’m still learning. Christopher Danielson knows his stuff and how to communicate it. Inside the Teacher Guide you’ll find:
- A description of the Van Hiele model of Geometric learning providing insight into how students become geometers.
- Suggestions for using the book and guidelines to support you in “establishing and maintaining a safe space for sharing and learning from each others’ ideas.” [p. 19]
- A glimpse inside the minds of young geometers interacting with this book to help you “understand, appreciate, and wonder at the richness of children’s thinking” [p.37].
- Student-generated answers to each set give a range of possible answers. The mathematical ideas generated by these shape sets can not be bounded by a fixed answer key.
The Teacher Guide is the perfect blend of mathematical content, process, pedagogy, and wizardly insights. Christopher has walked the walk, noticed stuff, wondered stuff, figured stuff out, shared his wisdom, and left this reader curious. These books left me wondering: What will happen when I try this out? Could it be this magical? Will children be as delighted and excited by shape as I think they might? What will they wonder? What will they notice? What will they see?
These are book of shapes. There are Triangles, Diamonds, and Spirals, Oh my! But there is much, much more.
These books are about learning how to be a geometer (and more generally how to be a mathematical thinker). On pages 20-22 of the Teacher Guide, Christopher describes how he characterizes learning:
Learning is having new questions to ask.
(WOW! Pause. Ponder. Awesomeness.) Danielson asks, “What does it mean to teach mathematics in a way that it sparks new questions for students?”
He goes on: “Which one doesn’t belong? offers a partial answer to this last question. Each set of shapes offers students opportunities to ask questions, speculate and wonder about properties, relationships, and ways of seeing. ” [TG, p. 21]
“The important thing is not to stop questioning. Curiosity has its own reason for existing.” Albert Einstein
Here are some of the student-generated questions Danielson’s came across while using this book:
“What counts as shape?” [This question is following me and I love it! See MAGIC sub-section of my last post.]
“What is the difference between a corner and a vertex?”
“How can we measure the length of a spiral?”
“Is a square the lines going around the outside or the colored stuff inside those lines?”
“How do we know which of these properties matter?”
This is the math in this book—students asking rich mathematical questions, defining terms, categorizing and analyzing shapes, developing argumentation skills. Four shapes on a page. A simple concept inspiring authentic and profound mathematical work.
Now for the magic unleashed by these books during my visit to my son’s 1st grade class.
I introduced the book and some guidelines suggested in Chapter 3 of the Teacher Guide. For the next 25 minutes, students shared their ideas about three sets of shapes. I’ll share ideas from the first two pages. Each letter A-N is a different student. KD is me.
The first page of shapes
Here are some of the ideas the students shared about the page below and the question, Which one doesn’t belong?.
A: The rectangle (bottom right) doesn’t belong. Because all the other shapes are squares.
B: The diamond (bottom left). Because all of the other ones are faced the same. Because that one (bottom left) is facing that way.
H: The rectangle (bottom right). These two sides are bigger than these two sides (pointing to opposite sides in the rectangle), but the other three have this side and side the same (pointing to pairs of adjacent sides in each of the three other shapes).
I: I agree with “C” because if you turn this (bottom left) it will be square , this one is a square (top left), and no matter how big it is (top right), its still a square. And this one is a rectangle so its different.
A student noticed something about congruence and shared an idea about two shapes not belonging:
J: This (top right) and this (bottom right) don’t belong because these sides (sides of top right square) are longer than this (pointing to a side in the top left square). And this (pointing to the rectangle’s longer side) has way longer sides than these two.
J is noticing that the top left and bottom left shapes have sides that are congruent to each other. And perhaps that they are congruent as shapes.
Before I move on to the second shape set, I’d like to share my reflection on one more interaction involving this first set that occurred in the middle of this discussion.
D: I think it’s the diamond (that doesn’t belong) because rectangles are also squares and the diamond isn’t a square or a rectangle.
I want to take a minute to focus on the bit about “Rectangles are also squares.” In the moment, I interpreted this as related to an idea about subsets. In other words, the set of rectangles is a subset of the set of squares. I remember thinking, should I go here? [And to be clear, I don’t mean me throwing out the terms subset and set at this group of 1st graders.] A voice in my head screamed NO! Don’t do it. This can get messy. Whether all squares are rectangles or all rectangles are squares can spiral quickly into confusing conversation.
It isn’t just that students may be at a different Van Hiele level of reasoning than is necessary for understanding these ideas, but the language is confusing at any level really. I’ve often left my classroom thinking, did anything come out it other than students (and these are college students) memorizing this statement: A square is a special type of rectangle (or more formally The set of squares is a subset of the set of rectangles).
But, then what to do? I paused and asked:
KD: So rectangles are also squares? Can you say more?
D: They all have the same amount of vertices, edges and flat surfaces.
KD: So rectangles and squares have things in common. They have the same…
D: Amount of sides. Rectangles and squares have four vertices.
That’s it. Sameness and difference. That is where I should have been focusing all along when this whole business of subsets first comes up. It seems so obvious to me now. But it took reflecting on this interaction with D to get me to understand. Before when I resorted to only telling my students things like “Squares are a special type of rectangles,” I was sidestepping rich mathematical work. Work that this book (and these posters) support by giving a variety examples of a particular shape (e.g., squares) and asking what is the same and what is different about another type of shape (e.g., rectangles).
OK. Now we are 15 minutes in and they don’t want to leave the first set of shapes!, they have more to say, but we moved on.
The second set of shapes
This time, I asked them to share their ideas with a partner before the whole group (as suggested in the Teacher Guide).
Here is the second set of shapes and their ideas:
J: I think it’s this one (top right), because it doesn’t have any smooth sides like this (pointing to curves in the other shapes).
K: I think this one doesn’t belong (bottom right) because these are real (points to the other three). That one is a muffin, that one is an arrow and that one is a heart and I don’t know what this (bottom right) is.
L: I think it’s this one (top right) because it’s the only one that doesn’t have a part of it that’s round.
M: I think that this one doesn’t belong (top right). Because all of these ones are real. This one looks like a muffin, this one is kind of a fidget spinner, and this one is a heart.
Of course, with the mention of fidget spinner, there was a new level of excitement and interest in the bottom right shape. Then an objection.
N: This one isn’t a fidget spinner because a fidget spinner only has three sides. I think we shouldn’t have this shape (points to the bottom right), only these three. A fidget spinner doesn’t have a square on it. And it has 4 circles.
Leaving the “sort of a fight spinner” behind, we went on to the third set. While I won’t talk about that discussion, I will share one last thing. I displayed the third and last set and invited them to think deeply:
KD: “If you know one that doesn’t belong, I want you to find another one for a different reason. And if you find two that don’t belong, I want you to find another one for a different reason and another one for a different reason. You could think about this all day long.”
One excited geometer yelled out: “I could think about this for a year!”
This child is obviously hooked. They all seemed to be, with their “I’m almost hyperventilating I’m so excited to share, pick me” hand raising, grunts and groans. In 25 minutes, almost all of the approximately 25 students had shared at least one idea and explanation. A few even came up after I closed the book to share their thinking.
Here’s me, Liam and his teacher Ms. Burgoon (on the left). That night I emailed Ms. Burgoon asking her to reflect on the experience. Here is what she shared:
1. Did anything surprise you about the experience? I thought it was very interesting that one of the last things (in the first set) the kids pointed out was one of the most obvious- the shape that was a different color than all the others. I could tell they were really trying to apply their mathematical knowledge about defining attributes of shapes as opposed to non-defining attributes. I also did noticed they were really engaged and seemed to enjoy talking with one another. They came up with ideas I didn’t think of, which was pretty cool!
2. Any magical moments that you can recall from today? Honestly, the biggest “magical moment” was probably how engaged the students were, and they were able to sustain attention for a very long period of time. I think the open-ended conversations and allowing students to take risks, knowing there weren’t any “wrong” answers, was really meaningful for the kids.
3. Anything your found unique about this book? I loved that any of the shapes could be the one that doesn’t belong. It was really interesting to see which shapes the students thought of first, because it gave me an idea of whether they noticed the defining or non-defining attributes of shapes. This book is appropriate for people of all ages, which is hard to do with a book, especially about math!
4. Would you use this resource in your classroom?And if so, what purpose might it serve/why? I would absolutely use this resource! It would be a great warm up to start math lessons throughout the year, as class discussion to promote “math talk” or even a writing assignment. I think this book could be used in a lot of different ways. I’d love to use it to introduce our math topic on shapes and again at the end of the unit.
I was glad Ms. Burgoon answered the last question this way as I waiting to buy the books for her end-of-year teacher gift:) Here is the thank you card she sent for the books.
I emailed her immediately to say YES to a PD session! I’d love to share more about this book with other teachers and children.
Euclid is said to have replied to King Ptolemy’s request for an easier way of learning mathematics that “there is no Royal Road to geometry.”
One does not become a geometer by simply walking down a magical yellow brick road neatly laid out before you. You need to collect the bricks yourself, decide where they should be placed, and puzzle over how to lay them down. You need to build your own path, ask your own questions, wonder, try, fail, and try again. These first graders studied these sets of shapes carefully. They made connections. They described shapes (smooth, corners, vertices, points in, round.) They named shapes (circles, Squares, rectangles, diamonds, rhombus, muffin, sort of a fidget spinner). And they wondered about these names. They noticed equal side lengths (congruence), the tilts and turns of shapes (orientation), and invariance (e.g, A square is a square even if it is tilted or really big). With the help of Danielson’s books, these 1st graders built their own paths through these shape sets.
Lastly, there is no royal road to teaching geometry. Orchestrating these discussions is hard and complex work. There are so many things I wish I did (and some I wish I didn’t do). But I left this experience with many wonder-full questions. And thanks to these 1st graders, two of my questions were answered: What will happen when I try this (book) out? Could it be this magical? Yes, it was. It was indeed.
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Thanks and see you next Monday! #mathbookmagic