Art and Math · Arts in Education · Math with Art Supplies

Thinking Hands


Oh look, that’s me. For once, I’m not the one taking the picture!

Third and last post in this group of postings that are meant to help me clarify and remember where I am at in my thinking about the work that I do with students in schools.

After years doing bookmaking projects to make with children, I realized that many of the art and design skills I use every day align with some of the skills that mathematicians aim to develop. Part of the reason this alignment caught my attention is that I have a great affection for the mathematical thinking that I want to encourage.

Rotating Shapes after school with third graders
Rotating Shapes after school with third graders

In the relatively short time I am in schools with students, I hope to have a positive influence. My experiences and interests have led me to an unusual place where I can use colorful, artful materials to help kids create projects that enrich mathematical thinking. My place isn’t to teach art or teach math, but rather to plant seeds of engagement and excitement.

It seems to me that children intuitively understand concepts that are recognizably both artful and mathful. More and more, my thinking is centered around how to engage and encourage that which is already inside of students.

Spontaneous Scaling of Origami Pocket by First Grader. I see this happen frequently.
Spontaneous Scaling of Origami Pocket by First Grader. I see this happen frequently.

For instance, kids absolutely understand the idea of scale. They realize that their hands are the same, but smaller versions, of dad’s hand. Same with their shoes, their shirts, everything in their world.

Scaling shapes to make decorative border
Scaling shapes to make decorative border

There is no room in the school day for formal study of scale until the intuitive connection to it seems to have long disappeared. Turns out that scale doesn’t only have to do with making large models smaller, but it also is intrinsically connected to relationship thinking, predictive thinking and to the recognition of trends.

Symmetrical frame for a drawing of outer space by a first grade student

Discovering that children are naturally inclined to embrace symmetry has been another exciting area for me to explore with kids. When making books or other structures with students, there is nearly always symmetrical folding going on. I have choices when I teach folding: I can introduce what I do as step-by-step directions, or I can nudge the  students to see the symmetry of what’s going on so that they can predict for themselves what the next fold will be. The latter way gets them to see the project in a more global way, draws them in because they have understanding which includes them, rather than being like a little robot that is being programmed to do this then do that.

Doing some spontaneous symmetry during a few extra minutes at the end of a kindergarten session
Doing some spontaneous symmetry during a few extra minutes at the end of a kindergarten session

Symmetry is deeply embedded in math thinking, so I have been talking to children about connecting symmetry to what they are learning right now in math. Specifically, I talk to them about how when they are looking at an equality, such as 5+3 = 8, that this expression is balanced on both sides. It can also be understood as 5+3= 4+4. If I add 6 to one side of the equation, then I have to add 6 to the other side so that the symmetry of the equation remains true. Talking to students about equations as balanced forms just might help them, later on, when they will have to maintain balance in an equation to solve for x.

Symmetry, Scaling and Fractional Thinking all on display
Symmetry, Scaling and Fractional Thinking all on display

As far as I can tell, the only time symmetry is formally taught in elementary school it’s part of the examination of lines of symmetry in regular geometric objects. I like to be able to at least offer hints that symmetry has richer applications.

Making shapes from other shapes, parts and whole
Making shapes from other shapes, parts and whole

Children seem to have an innate sense of parts that make up the whole, which seems antithetical to the reality that teaching fractions is unfathomably difficult. Is it possible, though, to focus on having students work fractionally from a very early age, way before we introduce the numbers that describe the fractions?

Hexagon, Cube and Thing made from equilateral triangles
Hexagon, Cube and Thing made from equilateral triangles

Playing with blocks was one of my favorite activities as a kid. I certainly noticed halves, fourths, and wholes, but I didn’t make this connection between the blocks and fractions until I was much older. This makes me value not only exposing kids to artful mathematical thinking, but also, sooner rather than later, to help students connect their hands-on activities to the numbers.

Four Fold symmetry Pre-K
Four Fold symmetry Pre-K

There’s more I have to say about all this. but I reminding myself that I have to get to work getting ready for classes.

Am going to end with my list of ideas that I want to keep in mind, not all of which are explained here. Maybe I will get to writing about all these here and there through my teaching season. If not, at least I will have them here to keep me on track.

Art/Math concepts to explore with kids:

  • symmetry
  • Pattern recognition
  • Pattern Building
  • Scale
  • Inverses
  • Continuous magnitudes
  • Trends
  • Relationship thinking
  • Problem solving
  • Parts of a whole
A drawing of mine, that's artful/mathful
A drawing of mine, that’s artful/mathful

The first two posts in this set are here and here. 









Art and Math · Arts in Education

Magnitudes Count, a paper by Simon Gregg & Tali Leibovich-Raveh

Two, colored by my kindergarten daughter eighteen years ago

Back in October I saw a paper, Magnitudes Count in Math Education by Tali Leibovich-Raveh and Simon Gregg. Since I learn so much from Simon Gregg’s documentation of his work with students I was eager to devour this paper. It’s been on my mind as I consider the work I bring to the classrooms that I visit. This paper has given me new things to think about.

The discussion in the paper centers around examining counting and continuous magnitudes. Well, I know what counting is (1,2,3, etc) but what is continuous magnitudes? Good question. A straightforward way to express both of these ideas is by asking “how many,” which is counting, then “how much,” which is continuous magnitude. Think of two glasses of water after you’ve been running around on a hot day. You can pick one glass to drink from. Sounds straightforward. But one glass is larger, holds much more water. Now all of a sudden one glass of water versus the other has a different meaning. The larger glass has a larger magnitude of water, so volume of water in the glass becomes the priority concerning the choice of which vessel to choose.

The reason I am thinking about it is this: the authors are keenly interested in understanding and then building on that which children know intuitively, that which is “more intuitive and acquired earlier than the ability to understand numbers.” This is, in fact, something I’ve had sustained curiosity about: what concepts develop naturally in children, and what work can I do to help build on the foundations that are already in place? Also, what ideas do I not want children to internalize?

Magnitudes Count In Math Education - an interdisciplinary View by Tali Leibovich-Raven and Simon Gregg
Magnitudes Count In Math Education – an interdisciplinary View by Tali Leibovich-Raven and Simon Gregg

Once upon a time I was a child. I remember key moments, moments that marked paradigm shifts which impacted my relationship in the world. Here are two of these moments from early elementary school.

During art class I (nervously) asked Mr. Gunther if I could color the clouds in my drawing blue, and leave the sky uncolored. He lowered himself down to my little desk, sweetly but emphatically stated that this was my drawing, so it was my decision. This changed so much for me, I have no words to express it.

Another moment was during a dry 4 part lecture series that my school made all the students endure. We filed into the auditorium, lights dimmed. for a slide show that I remember nothing about other than one thing that shook my world. The lecture series was about the great religions of the world. I walked into those talks thinking my own sect of my own religion was the biggest religion in the universe, and, certainly, the only true religion. I walked out, shocked to know that I was wrong about this. I suppose I would have eventually figured this out, but knowing about this diversity in the world from a young age made a real difference to me.

These memories are enough to convince me that any amount of time with a child can make a difference to them. I want to be thoughtful and deliberate in what I bring to children.

I’m often on the lookout for paradigms that are infused into us from an early age that may not be broad enough to create a foundation for wider, wiser growth.  We learn numbers, we learn to count. Then comes adding, subtracting, multiplying and dividing. I dare say that this is about the place where most adults stopped retaining anything they learned in math class. In fact, this idea that numbers, counting, and arithmetic fully defines math is what I believed for many years. Where I have the opportunity, I want to let children know that math is something more than math facts.

The authors of Magnitudes Count have done three great favors for me.

  1. They’ve identified a mathematical concept that children innately understand, which they believe can be used as scaffolding to teach more complex concepts.
  2. While they state that more study needs to be done to investigate the relationship between an understanding of continuous magnitudes and deeper math understanding, the fact that they are pointing us in the direction of emphasizing mathematical thinking besides counting and arithmetic is exciting to me. I didn’t know that people were on the lookout for building on children’s intuitive understanding of math as a place to start
  3. The authors have provided me with a way for me to introduce my next post, which will be about my own observations about what children intuitively know, innate knowledge that I think about as being useful to use as scaffolding to teach more complex concepts. I won’t have the research or references to back up my thoughts, but I still want to get my thinking out here.

The one thing I would have liked to see in this paper (and it may be in there) is that when students are taught something that is mathematical I think it needs both to be identified as mathematical, as well as to be explained why it is mathematical. By doing this, students may, before too long,  understand that math, along with their math facts, can also be about seeing relationships, making comparisons, developing reasoning and all sorts of other good stuff.

One last note, the way I communicate with students is with hands-on materials, mostly paper that is cut, folded, and arranged. It may sound like all I am talking about is abstract, but it all gets translated into projects.

To be continued..

(this is part 2, Here’s Part 1)

Art and Math · Arts in Education

Paper, Books and Math Workshop

There’s this overlap of paperfolding, bookmaking and math that’s been in my sights for sometime now. Next month The Center For Book Arts has me on the schedule to share my interest with educators.

It’s a natural fit: Fold a piece of paper in half a couple of times and you’ve got a book. There you have it, all this things I’ve been thinking about in one sentence.

Bookmaking by Paula Beardell Krieg


Why math and book arts, you might ask? CBA asked me to propose a course for educators. Over the years I’ve taught classroom bookmaking dozens of times, though my focus during those years was literacy. In recent years it has occured to me, as I visit many schools and work with hundreds of different students each year, that teachers have loads of support for teaching literacy.

Teaching math, on the other hand, can be more challenging. In my desire to stay relevant, the hands-on projects I’ve been designing for classrooms had evolved towards supporting math curriculum and math thinking. And, oh yeah, I love this work.

Everyone folds paper, many people teach math, less people make books. Not too many people have a strong relationship with all three, Basically, I want to be teaching this workshop because otherwise I doubt it will exist.

Here’s what I know about offering a workshop for teachers:

  • They want content that they can use on Monday.
  • They do not have time to do special prep which requires more than the school copy machine.
  • They don’t have easy access to special materials.
  • They like having a handy resource folder.
  • They want their students to be learn and be happy.
Equivalent Fractions
Equivalent Fractions

Here’s how I know about math:

  • I play with math kind of obsessively….never got the memo about math being scary
  • I went through K -12 math three times, once as a student, twice as a parent.
  • I’ve been working in schools, discussing math projects with teachers and math coaches. We discuss standards and curriculum goals, and I talk to students about the math they are learning.
  • I’ve been rather passionately working on deepening my math knowledge in workshops, conferences, and connecting with math educators through their writing and through the #MTbos and #iteachmath communities on twitter as @PaulaKreig

In this one-day PD workshop I plan on focusing on deepening connections, doing hands-on, classroom friendly projects that address areas of math that will help students create strong foundation for future learning.

For instance, we’ll be working with number lines in a way that is both interactive and which illuminates patterns. I’ve been working out ways of presenting number lines in ways which delight kids. The sounds of discovery that come from students when they start seeing what I show them has been one of the most beautiful sounds of my career with kids. I will also have hands-on ways of showing the number line that moves from natural numbers to negative numbers and beyond.

Since the number line stays with children, evolving from finger counting through the coordinate plane (and beyond) my focus here in not only to use the number line, but also to elevate it as an important tool that they have reason to embrace. One of my favorite responses, which informs my work with number lines, was from a first graders who told me that they looked at number lines earlier in the year and now they were done with that.

How to make an origami Pocket by Paula Krieg
Click to enlarge for reading or printing

We’ll also be doing some work with perimeter and area. What frustrates me about students’ learning here is that they often mix up perimeter and area, not remembering which is which. One of the projects we’ll be making is a perimeter-pocket. We’ll make an origami pocket, which in itself is a wonderful lesson as is goes from being a square to a triangle to a trapezoid to a pentagon. But this perimeter pocket will have a ruler embedded in the structure and a string in the pocket to use to measure around things. Then there will be an area-rug book, with lift the flap peek-a-boo images hiding under the area rugs. Fun, easy, memorable!

I love that shapes are part of school curriculum at an early age. There are numerous projects I have in my toolbox that compose and decompose shapes.

I have some nice peek-a-boo projects that address the same composing and decomposing concept with numbers.

Symmetry will also get the attention that it’s due. Sure, we’ll talk about lines of symmetry by way of making pop-ups, but we’ll also look at the idea of symmetry as a it relates to equations, which can be seen as numerical symmetry. One of my exciting discoveries has been how naturally young kids grasp the idea of symmetry, and how well they they can connect it to equations.


A one hundred cents flower
A one hundred cents flower

We’ll even do some hands-on play with play money. I do these one-hundred cents designs with kids which gives then lots of practice with money, as well as practice counting by fives and tens and twenty-fives.

I could go on and on here, but I think you get the idea. And, actually, that’s part of my aim for the day, not just to present projects that can be immediately used by teachers in the classroom, but also that the idea of making the curriculum more hands on will inspire teachers to create their own simply made projects.

This is a rich, wide open inquiry into what we can do together to make math real.

The Center for Book Arts, NYC, Saturday, October 26



Arts in Education · Math and Book Arts · Math and Paper Folding · Math with Art Supplies

Peek-a-Boo Skip Counting for First-Graders

Peek-a-boo skip counting
What number is under the heart?

For weeks I’ve been burning through piles of papers and ideas trying to work out an engaging skip-counting project to make as part of a math-activities folder for first graders. Having just done a math activities folder with kindergarteners, which went really well, I’ve been wanting to do something similar for first graders. As I’ve also been doing math-with-art-supplies bookmaking projects with second graders, I’ve been keen to design something for the next grade up.

What I’ve  needed to get me going on this is a school to want me to create a project for them. A couple of weeks ago, late in the season, a school called me, asked if I had any time for them, and we struck a deal. We’re doing the project that I’ve been wanting to create.

There will be four hands-on projects in a folder that the students will be making. This post is about just one of the projects, one that supports skip counting, reasoning, and attention to numerical patterns.

Show me what under the butterfly, Oh, it's a 14.
Show me what’s under the butterfly, Oh, it’s a 14. Still not sure what’s under the heart.

Skip counting is a big deal in first grade. Not only does it set the stage to understand multiplication, it also is helps with learning to count money.

My work with second graders has piqued my intereste in skip counting. The projects we’ve been doing, which is making designs with “coins” that add up to $1.00, has been interesting in that I’ve noticed that even though a student can count by fives, you know, 5, 10, 15, 20, 25, 30, 35, 40….,, they have a really hard time doing this same counting by 5’s when you ask them to start at any number other than zero. So, if they have 25 cents plus two nickels they are at a loss as to how to proceed.

Show me what's under the flower. Oh, it a 6, Now can you tell me what's under the heart?
Show me what’s under the flower. Oh, it a 6, Now can you tell me what’s under the heart?


Maybe by now you’ve guess what is under the heart in the photo above. Maybe not. If you need more hints, I can reveal that there is an 8 under the star. This will likely finally be enough for you know know that there’s a 10 under the heart.

We’re not just counting by twos here. I’ve made a paper that slides under the windows that helps with counting by 2’s, 5’s, and 10’s.

I consider this to be an elegant design. One piece of folded paper for the holder, with a one piece of paper for four different number series. The little designs on the peek-a-boo doors are cut with paper punches, which I’ve collected over the years. The rhombus shaped window are made by folding the paper and cutting triangles on the fold.

One of my thoughts with this project is that  it can support students in practicing with going both forward and backwards with their skip counting. For instance, if they see two numbers, say 80 and 85, can they tell me the number that is before the 80 and after the 85? This takes some practice, some thinking, and reasoning, but if they can figure out what number is behind the hidden door, I anticipate the pleasure at solving this puzzle will delight them when the peek-a-boo door reveals the answer.

I do plan to share the template for this after I try it out with some real live first graders. To be continued.

Addendum 6/11/2019

Two classes of first graders made this with me. It went really well! To teach them to use it, I do the demonstration on the board, drawing doors that they could “look” behind for clues.

Here’s a video of what playing with this looked like:

Here’s a template so you can make these yourselves: skip counting first grade