Uncovering the Complexities of Counting with a Magical Counting Book

The first day of my 1st graduate level math-ed course at Michigan State University, Professor Sharon Senk asked the class to write down their answer to a question similar to:

What mathematics from K-12 would you like to learn more about?

I remember smiling smugly thinking to myself: I can’t really think of anything. I pretty much know all of K-12 mathematics pretty well. Maybe I’ll say something about probability, that has always been tricky for me.

How naive I was! Fifteen years later, and I am still learning about K-12 mathematics. Thankfully, experiences like Senk’s course and my graduate assistantship with the Connected Mathematics project finally opened my eyes wide to the mathematical connections and complexities involved in learning and teaching K-12 mathematics.

It was at grad school at MSU that I met this week’s magical math book’s author, Christopher Danielson.  His new book beautifully illustrates the complexities and richness of one of the first topics taught in K-12 mathematics, counting.

The Book

Christopher Danielson’s book, How many?: A Counting Book, is the second book in a series. We blogged about the first book Which one Doesn’t belong? here.

How Many Bundle2

Published by Stenhouse in April 2018, you can purchase either Student Guide or a Teacher’s Guide Bundle. I highly recommend purchasing the bundle. There are so many wonderful things one can learn from this Teacher Guide.

Here is Stenhouse’s description of the Teacher Guide:

Throughout, he shares stories and excerpts from real classrooms where he facilitated How Many? discussions. Danielson helps teachers anticipate what students might notice and gives practical suggestions for facilitating rich conversations with students. Danielson’s interest in students’ ideas is infectious, and readers will soon find themselves seeking out opportunities to ask young mathematicians, “How Many?” [Stenhouse website]

Here is an example of a counting prompt from the book as described at Danielson’s blog:

How Many? is a counting book that leaves possibilities open and that seeks to create conversations. Creativity is encouraged. Surprises abound.

The premise is simple. Every page asks How Many? but doesn’t specify what to count. Each image has many possibilities.

An example. How many? [That’s your cue, if you are unfamiliar with this image, how would you answer?]



Maybe you say two. Two shoes. Or one because there is one pair of shoes, or one shoebox. Maybe you count shoelaces or aglets or eyelets (2, 4, and 20, respectively). The longer you linger, the more possibilities you’ll see. [see Danielson’s blog post for more]

I shared this book with my 5 (almost 6) year old and 8 year old. I would say between 5-8 are good target ages, however I can imagine this book being used in college courses for future/current teachers and K-12 classrooms as a way to encourage the sharing of ideas and providing evidence.


As children engage with How Many?, they offer a count, a unit of measure, and a description of how they counted. For example, there are 20 eyelets in the pair of shoes above. 20 is the count, and “eyelets” the unit.

Some sample counting strategies:

  • Count by twos on the first shoe ( 2,4, 6, 8, 10), then doubled that for the second shoe to get 20
  • Count by 5s, there are 4 groups of 5 eyelets on each “side” of the shoe
  • Count one by one

How many? conversations involve counting, number language, units, grouping, partitioning, place value, and vocabulary. They give students opportunities to see and share mathematical structure, notice units and relationships among units, question assumptions, make comparisons and make connections to their world and everyday counting experiences.

As parents and educators, How Many? conversations can open our eyes and ears to the subtleties, complexities, richness, and creativity involved in counting.

The Magic

Below is an image from the book. First of all, isn’t it gorgeous! Here’s a snippet of what my son Liam (8 years old) shared.

From https://christopherdanielson.wordpress.com/2017/04/09/the-new-basics/

L: There’s eight. The whole is eight. Wait… is it eight? It’s actually seven because they got cut in half.

Me: Wow. That was fast. So how many avocado are there? [Note: I was amazed how quickly he got this answer and I  forgot to ask him his unit. I assumed he was counting avocado when he counted seven.]

L: Seven.

Me: How do you count that?

L: One, two, three, four, five, six, seven (pointing to each stone).

Me: Oh, you don’t mean avocado, you mean avocado pits.

L: No. I was counting these (pointing to the stones), so that I would know how many there are. Because it is easier to keep track. See? (pointing to a pair in the top left corner)

Me: Oh, so these go together (pointing to a pair in the top left corner). Are there are any extra?

Liam concluded that there 7 1/2 avocados. 7 pits. He also noticed 8 stripes on the cutting board (something I never noticed before).  Each two page spread offered surprises (e.g., objects they noticed, strategies they used ) and insights into my children’s ideas about counting.

While the magic of this book truly lies in sharing this book with your own children/students, there are many inspiring stories/discussion snippets on Twitter (e.g., type in #howmany or #unitchat in Twitter’s Search engine). Recently I came across this lovely example of a book a kindergartener class created after being inspired How many?.

I never would have guessed 15 years ago that I would STILL be learning about counting.  But listening to children engage with simple prompts and carefully crafted images like in How Many? shows me there are deeper truths and things to be understood about counting and all K-12 mathematics learning.   Thank you Christopher Danielson once again for providing a resource to wander and wonder with.

counintin image
I tried to find the source for this quote, and came across an attribution to Barry H. Gillespie. Bonus, He has some cool math art on his website: http://www.barryhgillespie.com/Experiments_in_form.htm

Have a magical math book you’d like share? Please go to the Shared booklist to find out how.  If you’d like to receive these magical math book posts every other Monday, be sure to follow this blog in the side bar of this page.

Thanks and see you in two weeks!  #mathbookmagic

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