My new book on the math of making comparisons! [#SocksPants]

My first new book of 2016 is out and it's gorgeous!

Do you want your students and children to feel like algebra is beautiful, playful, and intuitive? Come play, solve and make math with us!

Our new book is filled with a diverse collection of math games, puzzles, and activities exploring the mathematics of choosing, identifying and sorting. Teachers and parents have tested all activities in classrooms and living rooms. The activities are easy to start and require little preparation.

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Socks are Like Pants, Cats are Like Dogs, Really!!

Do you want your children and young students to feel like algebra is beautiful, playful, and intuitive? Come play, solve, talk, and make math with us! Support our book, reserve your copy, and make these math adventures available to children, parents, and teachers all over the world.

I've co-authored the book Socks are Like Pants, Cats are Like Dogs with Gordon Hamilton of Math Pickle. It's filled with a diverse collection of math games, puzzles, and activities exploring the mathematics of choosing, identifying and sorting. Teachers and parents have tested all activities in real classrooms and living rooms. The activities are easy to start and require little preparation.


There's even a pdf of sample activities from the book for you to try!

We’re almost done with our book; all that’s left is a few finishing touches. We’ve estimated the crowdfunding goal for this project to be $4,000. Any amount will help us reach our goal. Please help make this book a reality! Visit the crowdfunding site over at Natural Math for more images, and information about our project!

Thank you for your support!

New Year? New Books!

It’s official! I will spend 2015 writing about my approach to combining math and dance in the Math in Your Feet™ program! Meaning in the Making: Embodied Mathematics in the Classroom will be published by Heinemann in 2016.

Meaning in the Making is for any K-6 classroom teacher, music or P.E. specialist, or home educator interested in understanding how and why to bring dance or movement into their math teaching. The book describes, analyzes and illustrates the learning that happens at the intersection of math and dance in the Math in Your Feet program and usefully extends these methods to help bring the moving body into any math classroom in a meaningful way.

The book (in print and e-versions) will include links and QR codes that take readers directly from descriptions of students engaged in various aspects of their math/dance making to video clips of the work referenced in the text. These video clips are part of a larger online book companion website that will house even more resources for facilitating Math in Your Feet with your students.

In the long term, I hopes to create an online learning and peer support community for teachers and parents learning to facilitate math/dance and moving scale math work with their own students.

Also, my book, Socks are Like Pants, Cats are Like Dogs, co-authored with Gordon Hamilton of Math Pickle will be published in 2015! This book, for young children (up to about grade 3), their siblings, and their adults, explores the concepts of numerical and categorical variables through games, puzzles and making. A project of Natural Math and Delta Stream Media.

Sign up here to know when both books become available for purchase!

A Small Moment of Multiplication

Last school year I made my peace with numbers, but these days I am actually starting to feel a bit of mastery, albeit at a very elementary level.  It's a long story that starts with the seeds of math anxiety that were sown as a five year old trying to count and match bumblebees on a worksheet. Over the years, it turns out that geometry is really my thing, but numbers, for the most part, give me a general and unwelcome sense of dis-ease.  That is, until I became responsible for my daughter's math education.

Over the last year both of us have increased our capacity for mental  math, which basically means we are now able to see numbers essentially as flexible, interesting objects that shrink and expand as needed.

We have developed our addition and subtraction skills mostly through games and other play and narrative approaches.  For first grade we played hours and hours of UNO and many, many rounds of Shut the Box.  We rolled miles of dice.  We counted, earned, saved, and spent mountains of coins.  My mantra was 'find your tens' and for a long time, it seemed, tens were hard for her...until one day they weren't.  Now she's a pro at double and sometimes triple digit mental addition, and subtraction is not far behind.

She and I have different approaches as we work toward an answer, but it's frankly just awe inspiring that I can add two, three or even four 2-digit numbers before reaching for the calculator.  And my kid?  I wish I had had her skills at seven -- I would have probably loved math decades sooner if I had actually been supported in developing the numeracy she already has. 

Anyhow, this fall I tried to bump things up a little with a math version of the card came War using multiple cards to find sums and differences.  After a few weeks of this it's become obvious that addition/subtraction are not a big enough challenge for her any more and so I've been searching and searching for something else.  The kid has some basic understanding of multiplication and division through a lot of conversations and play with halves and doubles but that's not quite enough any more either.  So I wondered if there was a multiplication game out there that wasn't just about memorizing times tables...

...and then Denise at Let's Play Math wrote a fabulous post about multiplication including a section on multiplication models which included a card game she created!  (It's a free pdf download and about half-way into the post.) You can play the game in the form of Concentration, Go Fish or Rummy.  We spent some time sorting out and getting familiar with the different models (sets, measurement and array) and, after we played Go Fish a few times, all of a sudden the light bulb went off in my head about how I can bring those concepts of into our daily lives.

I may be more impressed with myself than need be, but I mean, just listen to me as I walk my daughter through using the juicer this morning -- I actually sound like I understand what I'm talking about!

Me:  Do you see the 4 on the measuring cup?  That means four ounces.  You're going to juice four ounces which is also one-fourth of cup.  When the juice gets to that line, you'll be done.

Juicing commences.

Me: Okay, you're done.

Kid: That's not enough! [As I pour it into a glass that is narrower than the measuring cup.] Oh, wait.  It looks like a lot less in the measuring cup.

Me: Why do you think that is?

Kid: Well, it spreads out more in the cup.

Me: Yep.  Hey, if you're drinking four ounces of juice and that is one-fourth of a cup, how many ounces make one whole cup?

Kid drinks and thinks.

Me: One-fourth is one of four parts.  Each part is four ounces.

Kid: [Done with juice and looking at her fingers.  She pulls up four fingers and starts skip counting...] Four, eight, twelve, sixteen!!

Me: Sixteen what?

Kid: Ounces.

Me: Excellent!

I feel successful just being able to have that kind of conversation.  And really, it's these kind of small steps that ultimately made last year's math work so successful.  So, score one for a small moment of multiplication!  We're on our way -- and I can't wait to get the Multiplication Models poster I ordered off of Amazon from Maria Droujkova and Natural Math.  It'll be great to have it around us as we find more small moments just like this one.

My Guest Post Over At Moebius Noodles!

I'm beyond thrilled and honored to have been asked to contribute a guest post at the Moebius Noodles blog.  If you've not heard of Moebius Noodles, here's their wonderful description:

Moebius Noodles is an off-the-beaten-path travel guide to the Math Universe for adventurous families. A snowflake is an invitation to explore symmetry. Cookies offer combinatorics and calculus games. Floor tiles form tessellations. The games in “Moebius Noodles” draw on these rich properties of everyday objects in ways accessible to parents and kids, even babies. As the world turns into a mathematical playground, it transforms, one family at a time.

Moebius Noodles is the brainchild of Maria Droujkova of Natural Math, both of whom/which have been instrumental to my growing understanding of mathematics and what math really is all about. Although I am quite comfortable with the math topics that connect to the math and dance we do in my program Math in Your Feet, I knew that there was a way of thinking and seeing math that I did not fully understand.

Over the last year or so following Maria's accounts of math clubs, math treks and, especially, the Moebius Noodles project I have found my way to math.  And, I am not ashamed to admit that, as an adult who is remediating herself from years of math anxiety, I have learned more about real math from an approach created for small children than I ever did in high school!  My new confidence and understanding is largely due to how Moebius Noodles models math and math learning for parents and families -- one small observation, question, inquiry or game at a time.

Here's the link to my Moebius Noodles post!  And, while you're there, don't forget to follow or subscribe to their blog.

Small Moments of Math

I often write about the times when mathematical inspiration hits but, most of the time, our daily lives are made up of smaller, less dramatic math moments. It's during these lovely transient events where I really get a good glimpse at how my daughter is thinking, mathematically speaking, and how she is applying her understanding in a number of different contexts.

My approach to math exploration at home has been hands-off, necessitated by a child who likes to captain her own ship.  This basically all boils down to the fact that math generally happens in bite-sized pieces around here.  It doesn't mean I am not influencing the process but it does mean I hardly ever make formal plans; instead, I am always looking for new games, thinking about what she might need or want to learn next and also how to introduce new things in a way that has the appearance of being at least 50% her idea.  I also leave stuff lying around to be 'discovered' or engage in my own pursuits, which inevitably leads to some curious inquiries from the wee bystander. 

In addition to all this stealth planning, these days I'm also preparing for some summer arts programming up in the city.  In this case, I'll have three groups of students (ages 6-7, 8-9, 10-12) an hour each day for a week or two at a time.  It'll probably be harder to orchestrate the kinds of gorgeous, in-the-moment discoveries we have at home in a windowless basement room of a run-down church building with lots of kids and no air conditioning, but I'm going to try.

Here are some examples of the small moments of math that have been happening around here lately, including some of my thinking and experimenting about the summer.  I'll start it off with half a lunch which actually started as a whole lunch, but when the kid saw the design with a whole tomato in the center she insisted in cutting it in half before eating:

A spontaneous, self-initiated design from the kid after breakfast, utilizing her rock and mineral collection.  I guess she understands reflection symmetry after all!

The kid was designing/drawing mazes while we were at the library but then got interested in a 2001 NCTM magazine on mathematics and culture I was reading, specifically an article was about a game played by the Fulbe children in Cameroon.  The game is essentially about creating designs by drawing line segments through a series of dots lined up in a grid. 

After a quick look my kid drew her own version  ("The same design equals two different things!") and then also started drawing the library ceiling (square tiles and long rectangles of lights).  "There are patterns everywhere, Mama...Oh, look a book about cats!"  And so it goes...

It's clear the kid likes functions, but the function machine she built is totally off limits to well-meaning but meddlesome adults.  The photo below is a bit blurry, but you'll see she really just likes the process of figuring out the equations using the same rule and different inputs.  The minute I said something about looking for a pattern in the answers the whole thing went south.  I've learned the hard way that, in this particular instance, there is no room for me to ask any questions about anything whatsoever.  However, I'm confident that she'll figure it out at some point like she did when made peace with the times tables last week.

She'd been multiplying on her fingers but finally realized how much time and effort that takes. "The multiplication table is like sight words," she told me, "you just have to memorize them."   Reading a book called Piaget for Teachers from the 1970's makes me realize that my approach is probably on target given her developmental age (newly operational) -- telling her that particular 'fact' myself will not help her thinking process.  The way to help her thinking process is, basically, to just let her figure it out through a variety of experiences (cleverly orchestrated in secrecy by her scheming mother, ha!).  

I made the kid a puzzle/game last week called "Make Them the Same" using elements of line, shape, design and color.  I didn't take a picture of the starting designs but if I had you would have seen that both designs in each pair were incomplete in different ways -- you had to compare the two to make them both a complete design.  I'm still playing around with this idea.  This version was too easy and my second try was too hard but I think I'll be able to make it work.  I really want to have it figured out by summer.

I spent a couple hours over the weekend looking through the math book lists at Love2Learn2Day and Living Math.  The theme for my summer program is tentatively called "What can you do with a square?" (or a pattern unit, I'm not sure which yet) and, in Three Pigs, One Wolf, Seven Shapes the answer is obviously 'tell a story with it'!  I made a sheet the kids could use while listening to the story, but when my kid discovered it she tried to put the tans on the outlines which are not to scale.  For the youngers I may have to give them each one or two real-size tan puzzle sheets instead (which I found at Mathwire), and use this sheet for the olders.

Chessboards are made up of lots of squares.  When I read this book to my daughter a few days ago to make sure it was something I wanted to use in the summer, I had the thought that I could make paper chess boards and give each kid a baggie of rice to follow along with the story. 

It might get messy with all that rice, but part of my thinking is that Math in Your Feet is about order and structure and pattern.  A chess board is so very orderly and there's a growing pattern in the story (doubling, my current favorite topic).  And, having the real rice at hand will hopefully make the story, well, more real.  Also, I'm thinking about doing more with paper quilts and tilings and the image of the chessboard would be great to refer back to when bringing up tessellations with the older kids.

Not surprisingly, more small math moments have happened during the in between times of writing and editing this post.  It seems that little jewels of math are all around us just waiting to be discovered at unexpected moments!  

As a postscript of sorts, I want to say that Maria Droujkova at Natural Math has been a huge inspiration to me over the last year and one of the main reasons I have learned to see math all around me.  I would not have grown my 'math eyes' if not for her Natural Math forum or reading about the math clubs and Math Treks she has orchestrated as well as the new Moebius Noodles project for young children. 

Happy Anniversary!

...and I nearly missed it.  Just this week I was thinking, "Gee, I think I've been blogging for about a year now.  I should go back and check the date."

I remembered just now, and good thing, too.  It's been exactly a year today since my first post.  I had just submitted my article for peer review to the Teaching Artist Journal and felt I had more to write.  And write I did.  When I started I didn't really know if anyone would read this blog; I haven't had a ton of readers, compared to other blogs, but am so grateful for the folks who have subscribed, followed, checked in, and commented. 

This blog has been a chance for me to illustrate and explain my work integrating percussive dance with elementary math topics, describe my work as a teaching artist more fully, and make connections between math, dance, and other similarly creative pursuits.  It's not really all over the 'map' but I do recognize that this might be categorized as a 'multi-topic' blog. That's fine with me -- I enjoy having multiple interests that intersect in sometimes fascinating ways over time.

This space has also been a way for me to connect with really interesting and smart folks in the mathed world, folks who have been really patient with me as I ask questions, share my ideas and generally expand my understanding of math thinking, topics and practices.  Sue VanHattum at Math Mama Writes, Maria Droujkova of Natural Math, Julie at Living Math, and Bon Crowder at MathFour have all provided wonderful support, forums, and conversations as I explore the world of math education.

It's been a whole year, but I feel like I'm just getting started.

Supporting 'Math Values in a Rich Context' / Origami as Math

I see a lot of geometry in origami but have always wondered what other math you can find in this kind of paper folding. 

Here is what Maria Drujkova had to say about origami during a recent blog post about the activities in her Natural Math Club:

Origami has the same values as mathematics, such as precision, modular reasoning (“bird base” as a group of folds), attention to detail, modeling, algorithmic thinking… Thus origami can be used to support math values in a rich context. The same goes for music.

I love the idea of supporting math values 'in a rich context.'  I love even just the idea of a 'rich context.'  I think all learning should happen in such a place.  Onward.

Conversational Math: Part Two

In trying to capitalize on the kid's penchant for 'talking math' I recently decided to try a game with her that I found in the booklet that came with our set of Cuisenaire Rods. 

The game is called Build What I Have.  One person describes a design they are making with their rods and others try and reproduce that design by listening closely.  One of the main points in this game is to introduce and/or reinforce math vocabulary.

The suggested age range for this activity is 2nd-8th grade; even though the kid is a young six I knew we could still get something out of it.  I decided that, to start, I would capitalize on concepts she already knew (parallel, points, edges, top, bottom, sides, etc.) and introduce some new ideas (perpendicular, horizontal, vertical). 

The rest we'd muddle through somehow, I figured, but she did surprise me by knowing her lefts and rights.  "We've been doing that in ballet class, Mama," she stated mater-of-factly.  Fabulous.

To start, we hid our designs from each other.


This is the first design.  I led and she followed, trying to make her design match mine by following my instructions. I started by saying: "Lay your blue rod parallel to the bottom of our work surface."  She already knows the concept of parallel really well, often times finding and noting examples of parallel lines when we are out and about.  "Then," I continued, "take your green rod and place in perpendicular [holding the rod in the air] up and down like this, and place the end in the middle of the blue rod."  Success!  Our designs matched!

This is the game she led.  To start she told me to put my orange rod parallel to the bottom of the workspace, but about an inch up.  The second orange rod was to be 'a couple' inches above the first one, but when she told me to put the blue rods on the sides to 'make a rectangle' I clarified the distance.  "Looks more like three or four inches, to me," I said.  I asked her to clarify the placement of the blue rods -- do they go on the outside ends of the orange rods, or inside?  Notice that this design is mostly made up of parallel lines, a concept she is most familiar with.

This is the second design I led.  I said, "Take your three light green rods and put them so they are together and vertical, up and down, in your workspace...Oh look!  They make a nice little cube!"  At first she thought she needed a fourth one to make it a square, but I clarified and said we're not making the outline of a square, but a solid shape.  When we revealed our designs to each other we saw some differences! 

This is how she recreated my instructions.  The white cubes are essentially in the right areas, but I had actually challenged her to put each white block 'point to point' with each corner of the light green square.  The dark green rods are essentially in the correct place; I knew that was somewhat complicated to execute.  And, I just noticed, the light green rods are horizontal, not vertical.

This is the last design in our session, which she led.  Perfect!  She wanted to use a bunch of different rods, but everything is still parallel here.
 
Here is what I find fascinating:  

My daughter's designs were much simpler today than normal and I think it might be because she had to describe what she was doing as she built them.  There is an equivalent experience that I find to be true in my work with 4th and 5th graders as well.  Often times I tell those kids that they are doing complex mathematics in their bodies and grade-level math on the page; they understand more math in their bodies than they can communicate through words or symbols.  Sometimes it is impossible for them to notate their Jump Patterns because they are just too complex for their current stage of symbolic mastery.

Often kids can do, know, and understand way more than they can communicate symbolically.  If we only judge a kid by her output on paper, we're not really seeing the whole child.  There are many ways represent comprehension: we need to listen and watch carefully for other indications of understanding as well. 

It wasn't too long ago when I brought the word 'parallel' into my daughter's universe.  It will be exciting to observe her body and conversations show me she's 'got' the concepts of perpendicular, horizontal and vertical. 

Multiplication Models Poster

Even though I don't do anything with multiplication in my program, I still thought this multiplication models poster from Natural Math was too good (not to mention too beautiful) to not share!

Update August 2013: This poster is available for purchase here. :-)

The Importance of 'Doing' Math: A Conversation with Maria

Maria Droujkova, from Natural Math, read and commented on my recently published article about the development and use of Jump Patterns in the classroom.  Here are some excerpts from an interesting exchange we had today on the Natural Math forum based on her questions after reading the article:

Maria: Great article, Malke - thanks for sharing! I loved the photos, and especially the cool graphic organizers and visuals you use. Do kids like to use the charts? Does it depend on the person?

Malke: What plays out again and again in this program is that teachers are really surprised when they see how enthusiastic their students are when it comes to writing about their experiences in Math in Your Feet.  Recording their patterns using the one best word to describe each category of each beat *is* challenging, but they are motivated toward accuracy because it is *their* pattern.  Also, it usually plays out that within each team of two, one person is more comfortable in the 2D realm of the page than the other, and one is more comfortable moving than the other -- it's a team effort, which makes it more comfortable for everyone. Once the kids do the tough work to record their pattern using the descriptive words, it's actually quite easy for them to plot their feet on the simple grid.  I still think there is a better, maybe more mathematically accurate way to do this, I just don't know what it is yet! 

There are, however, whole groups of kids who still just need the physical portion of the program (more and more, sadly).  These are kids who never had a chance to develop spatial reasoning in preschool, for instance. They don't have enough math, even in 4th or 5th grade, to use the program to take them further -- I find that they begin to understand the math concepts as if it's the *first* time they've ever seen or heard about them.  In these cases, I require just the minimum in their workbooks, and I purposefully stay in the physical realm.  It may be the only time they will ever have to just 'play' with math.

Maria:  Mathematics is "embodied" in that its grounding, basic metaphors come from bodily experiences and observations. You can't skip over that and go into formal math. Even working with adults, I find that you need to go through folding, building, mirroring, measuring and other physical activities and/or stories if math does not make sense to them.

Malke:  This is great to hear, and I believe it wholeheartedly based on what I see kids do in my program and in my personal math (re)learning...I just gave a very well attended 90 minute hands-on presentation at the NCTM [National Council of Teachers of Mathematics] annual meeting and it was surprising how many of these adults were really quite challenged.  It has nothing to do with being 'good' at dancing and everything to do with not having enough experience working with and within a physical realm.  I attended a session on the van Hiele Levels [for developing geometric thought] and realized that this probably applies to adults as well -- experience is key to understanding.

What is interesting to me is that my 'hunch' eight years ago, that there might be math in what I did as a percussive dancer, is now more true than I initially imagined.  At that point in my life I believed, as most of us probably do, that math is primarily symbolic.  I realize now that the math I bring to children in the form of rhythm and dance is some of the experiential math they may not have ever had, and that they need this kind of experience to move forward.  I've heard that only 10% of us will understand the symbolic realm of mathematics without needing to first have, as Maria says, "...bodily experiences and observations".  Just this fact alone makes DOING hands-on, experiential math that much more of an imperative.

Here's the link to our full conversation on the Natural Math forum.