The Map is Not the Territory

In the last few months I've been thinking, reading and having conversations about some (inter)related ideas in regards to math learning.

1. We should pay attention to building spatial reasoning skills in our students.  This newly published document from the Ontario Ministry of Education is a gold mine of ideas and conceptual support.

2. We should provide our students with diverse opportunities to experience a math idea in multiple modes and settings. Suzanne Alejandre at The Math Forum and I had a fabulous conversation about this idea in the comments of her post What's in a Touch.

3. The body can have many useful roles in math learning. One of those roles is to act as a "disruption of scale," a phrase I found reading Jasmine Ma's dissertation. Scale disruption generally means that we take familiar math off the page (exploration of polygons or work on a number line, for example) and make it body-scale which has the great potential for helping students build new insights about the math idea in question. I have LOTS of ideas for these kinds of lessons, but I also think that changing the scale of any lesson, even if it's not at body-scale, can have a positive impact on learning.

Not surprisingly, all this thinking and reading and conversing has influenced what's going on at home.

It turns out encouraging new ways to explore space and changing scale can lead to very interesting and inspiring outcomes!

One day I noticed that my kid, never one for block play but with a penchant for map making and for exploring the world with her whole body, started playing with blocks.  It was fun to hang out with her and interact within this new realm; I loved listening to her talk through her building -- definitely a new perspective for both of us!

So I got more blocks. And encouraged a generous uncle to buy even more. The most amazing thing was to watch her building from a set of visual directions; amazing because she's not usually one to follow someone else's patterns, preferring to make her own, and also because I was so happy she was having a completely, brand new experience in this realm.


Then there was the day I was walking through a big box store and passed the isle with paper and drawing supplies. All of a sudden I noticed a BIG pad of paper, much larger than her normal canvas. I thought, "I wonder what she'd draw if I got her THAT?"

Well, wonder no more.

Her drawings got BIGGER and the subject matter switched something other than very fine images of cats and girls in dresses. She drew Cahokia Mounds near St. Louis. She "sectioned off" one of the mounds with a GRID, to help her color.  Not sure of the logic, but this was the first time I had seen her draw a grid using long lines, instead of drawing out individual cells.


(Only a small aside: Maria Drujkova collects images of children's grid art. She'd be interested to know that the radiating lines of the circular grid, further down, were made section by section. I need to ask Maria more about this phenomenon of kids drawing individual cells vs. when and why they start making grids with intersecting lines.)

I don't think she actually finished the first picture because then she was inspired to draw this:


It's St. Louis and it's FULL of spatial concepts: grids, relative size, relationships (front, back, over, under), some perspective. A bigger pad of paper inspired her to draw a city, y'all. Cities are BIG.

Then some brand new doodling came into view.  A circular grid and some intricate star-like structures:



This last image was interesting to me. She took a quarter and traced around it on the big pad to create this design.


What ways can you think of to change the scale or the mode of exploration for a math idea in your children's life?  Try something and let me know -- I'd be fascinated to hear what new ideas, expressions and insights might emerge.

Last week was was a strange week of firsts for Math in Your Feet. I've been a teaching artist for about sixteen years and started exploring the connections between math and percussive dance in late 2003. Between 2004 and 2006 the program was piloted twice at all nine elementary schools in a large urban school district in Indianapolis, IN; prior to that I spent five years teaching clogging at many, many small, often isolated, rural schools across North Carolina, South Carolina and Kentucky. 

I've seen many schools and many students over the past sixteen years. But never anything like this.

In the first half of April I spent two weeks working at a public elementary school near Indianapolis, IN. This year they have six large classes of fourth graders, averaging 30 kids per class. I taught three of the classes the first week (one hour a day for five days) and three classes the second week. As is typical in my residencies, most of the kids were happy to be with me, worked hard, were proud of their work, and made progress but... During the second week I saw some startling things I've never seen before in all my years of teaching 4th and 5th graders.

This is the first time that... 

• kids made 3-beat patterns without noticing (they actually needed 4 beats)
• kids mistook their starting position as the first beat of their pattern 
• whole classes were still struggling to clarify footwork (directions, movement, foot position) by the time they created their second of two 4-beat dance patterns
•  on the fifth (and last) day of the program most kids were still not dancing fluently at a steady tempo...
• ...and, even more worrisome, a good amount of students in each class were still unable to reproduce their original dance patterns the same way every time.  Not surprisingly, they were also still working on dancing in unison (congruence) with their partners on the last day.  

I have honestly never seen this before and I have been puzzling over these observations for days. Here's what I'm thinking and wondering right now about all this:

1. It is definitely not about whether kids are 'good at dance' or not.
Some people might think that maybe part of these troubles are due to the fact that some kids are just not 'good dancers' but I do not agree.  My entire career has been focused on crafting meaningful learning experiences with my art form for students, no matter their dance backgrounds. This is the reason I developed the Jump Patterns tool in the first place. Jump Patterns provides a framework and basic feel of percussive dance for new dancers. (Interestingly, it provides an awesome challenge for more skilled movers as well.)  Also, I am a very flexible teacher of new dancers; I'm not looking for "good" dancing, just clarity of thought through the body whether dancing fast or slow.

2. I wonder if some of what I observed is about how much movement children are getting or not getting? 
Children think and learn through their bodies. Children develop spatial reasoning by moving their bodies. If their movement is severely limited due to a sedentary lifestyle, or a primarily sit-down education focused on test results, or school policies that use recess as a reward and/or punishment, then children are not getting the movement they need for developing their brains and bodies as a whole system.

The last time I saw difficulty like this was when I was working in very poor, rural parts of South Carolina in the late 1990s. I think the reason that I am writing this post is that only 20% of the children at the school last week qualify for free or reduced lunch. What's going on??

3. I also wonder if this is partly about how math is (generally) taught. 
I am teaching dance and math at the same time by facilitating a robust choreographic inquiry into the creation of multi-layered, three dimensional, moving patterns. Math in itself is inherently action-oriented which is why the body has so much potential in partnership with math learning. 

For example, in Math in Your feet we focus on the action side of math when we make, compare, compose/decompose, sequence, combine and discuss the patterns we are creating. This is mathematical activity. In addition, activities such as sorting, classifying, choosing, naming and comparing the attributes and variables that we use to build our patterns in the process of creating those patterns is mathematical activity.  This is what we do and how we think when we make percussive dance patterns AND when we do other kinds of math.

Because I've watched children think with their bodies for many years, most of that time in relation to mathematics, I think what I observed last week might be, quite literally, a visible deficit in experience with the process side of math, the part that builds conceptual understanding so that we know why and how we got an answer.

I think what I'm seeing is possibly a byproduct of math being taught as answer getting* rather than helping children build pattern-finding skills with numbers and in other mathematical situations. What I saw this week shows me that kids may know how to get answers, follow directions and learn procedures, but it is likely that many of the kids I saw in front of me have not had the chance to develop a strong conceptual understanding of mathematics, including:

- unitizing (the ability to compose and decompose shapes and numbers into smaller parts or larger new wholes)
- spatial language and concepts (built through the body and connected to math through language)
- pattern recognition beyond the (very basic) "red, blue, red, blue..." class of visual linear patterns

And I'm not the only one whose radar is pinging on this one. This blog post includes some of what I wrote on the Math in Your Feet Facebook page mid-week. A teacher who was part of the original pilot year with her students commented:

"We have noted that students need more concrete and visual / spatial experiences than they used to before they can move to abstract reasoning at the fifth grade level. We've wondered if our observations are correct and if they are, why?"

Abstract reasoning means we can take a math idea and use, apply and represent that idea in a number of different contexts. This cannot happen until the learner has built her/his own relationship with and understanding of that math idea. Abstraction itself is a process of coming to understand through conversations, observation, wondering, playing around with ideas, and noticing patterns and relationships. This is answer making.  Without this process an answer is essentially meaningless.

Ideally we should not have to remediate any of this. As a society we should provide our children with developmentally appropriate learning experiences at the time in their development that their brains and bodies need those experiences. In the case of spatial reasoning, unitizing, and pattern making/observing/identifying, this should start in preschool and increase in sophistication through elementary school. And, among many other tools, we should make a point of including the whole body in the math learning tool kit.

The reality right now is sadly quite short of this ideal. This post is simply intended to provide one educator's perspective on what seems to be happening as more and more children learn without their bodies.
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*Thanks to Tracy Zager for giving me the term 'answer getting' which ultimately helped me clarify my thoughts in the post. 

Math related conversations with my 8 year old tend to pop up unexpectedly. These are often around something she's making, and are often a gorgeous little gem of a surprise. Today was no different.

I found her creating a grid of tape on top of a piece of origami paper. She was coloring it in when she said, "You know, mama, patterns don't necessarily have to be colors in order."

Oh my gosh!

Me: What do you mean by that?

As you can see from the picture, below, she was coloring the whole page somewhat randomly, sometimes following the columns down, sometimes not. 


Her: Well, the colors don't have to be regular, they just need to be in the windows."

Me: You mean the windows are the places where the tape is not covering the paper, that's what you're looking at? It doesn't matter what color windows are, just that they're colored in?

Her: Yeah.

So here I pause to do a happy dance. My biggest discontent around patterns is that many kids grow into adults who think that "patterns" are only linear repetitions of colors. It is clear she has not internalized that particular reality.  The other reason I'm happy is that Prof. Triangleman once said to me:

"Math is when you say exactly what it is you want to pay attention to, focus only on that attribute and ignore everything else."

She was doing this!  Her pattern has 'windows' that are colored in, but the pattern is not defined by the colors themselves. She was consciously creating a pattern of windows (spaces between!) and consciously excluding the colors. I am thrilled to have caught a glimpse of this multi-layered attention in action.

This really goes to show you that it's worthwhile to keep your ears open while kid is focused on making or building something.  Even if it's after the fact, ask your kids to tell you about what they did, even if it doesn't look like much.  I mean, just look at the taped/colored piece again. It's pretty much a bunch of scribbles and it'd be super easy to pass it over, to think it was nothing special. In reality, though, there was so much thinking going on while she worked.

Here's what the piece looked like when she took the tape off:
Her: Oh.

Me: You seem surprised. Did that not turn out the way you expected?

Her: Why do you sound like a journalist?

In the past week or so I've had run-ins with very specific views on concrete-ness and abstract-ness. There are many people who feel very strongly about the notion that concrete objects create a bridge to more abstract ideas especially for young children. And, even if the discussions sound different, this viewpoint is not reserved exclusively for the little ones.

In any case, I used to be a strong believer in the concrete-to-abstract continuum but the more I read the more convinced I am of the pedagogical dangers that arise by taking this progression too literally. I also get the sense that 'abstract' in the context of these kinds of discussions is sort of code for 'hard to understand'.

This post is not intended to discuss or argue these points in full, only to share some thoughts I've run across in the last few months in order to illustrate why this topic is not as straightforward as some might believe.

First, from Uri Wilensky, a seriously brilliant take-down of the whole issue in his 1991 article Abstract Meditations of the Concrete and Concrete Implications for Mathematics Education. Here are just a few excerpts. I recommend reading the whole thing to fully understand his points.

"If we adopt the standard view [of concrete], then it is natural for us to want our children to move away from the confining world of the concrete, where they can only learn things about relatively few objects, to the more expansive world of the abstract, where what they learn will apply widely and generally.

"Yet somehow our attempts at teaching abstractly leave our expectations unfulfilled. The more abstract our teaching in the school, the more alienated and bored are our students, and far from being able to apply their knowledge generally across domains, their knowledge displays a "brittle" character, usable only in the exact contexts in which it was learned ...

"...I now offer a new perspective from which to expand our understanding of the concrete. The more connections we make between an object and other objects [not always physical, could mean ideas], the more concrete it becomes for us. The richer the set of representations of the object [idea], the more ways we have of interacting with it, the more concrete it is for us. Concreteness, then, is that property which measures the degree of our relatedness to the object, (the richness of our representations, interactions, connections with the object), how close we are to it, or, if you will, the quality of our relationship with the object."

Second, in all my reading there is a theme of caution: the adult can see the math in the materials but this may not be evident to the child. Piaget himself cautioned that the materials themselves cannot and do not hold mathematical meaning. As Kamii says in Children Reinvent Arithmetic (2nd Edition): 

"Base-10 blocks and Unifix cubes are used on the assumption that they represent or embody the 'ones,' 'tens,' 'hundreds,' and so on. According to Piaget, however, objects, pictures and words do not represent. Representing is an action, and people can represent objects and ideas, but objects, pictures, and words cannot." p31

In the introduction to Mathematics Their Way by Mary Baratta-Lorton (a classic activity-based math curriculum for K-2 first published in 1976) the author provides an interesting take on this idea:

At the same time, she points to the reality that "children learn easily and joyfully from real things":


And this is the conundrum in which we find ourselves, isn't it? Physical objects not only create sensory memories but also help mediate what's in our minds to expression through our bodies. However, it seems that things are further complicated: Our bodies are strong, capable and skilled but, for some reason which is fully outside my area of expertise, it seems that our physical selves require augmentation in order to fully realize and express our individual and collective visions. I think that's why it makes sense to think that physical objects could be an effective mediator in math learning, especially when math appears to be an extremely abstract subject...

...and now we are full circle, back again to Uri Wilensky, who said:

"...any object/concept can be become concrete for someone. The pivotal point on which the determination of concreteness turns is not some intensive examination of the object, but rather an examination of the modes of interaction and the models which the person uses to understand the object. This view will lead us to allow objects not mediated by the senses, objects which are usually considered abstract - such as mathematical objects - to be concrete; provided that we have multiple modes of engagement with them and a sufficiently rich collection of models to represent them."

In other words, it is the process of building multiple and varied relationships with an idea that makes something 'concrete' to the learner, not the properties of the physical objects themselves.

It's nice to have some of this thinking out of my head for now. Any thoughts?

I'm thinking about spatial reasoning: what it is, how its developed, and how it interacts with math learning. Also, why it is not spoken of much in my Twitter feed which happens to be a veritable goldmine of priceless thinking and information related to math education.

Yesterday, my 8yo and I went to the KidsCommons, a children's museum in Columbus, IN well known for it's larger-than-life toilet which resides inside an otherwise kid-scaled house. You climb in, there's a flushing sound, and then there's a slide to the bottom floor of the house.

This morning my kid woke up and immediately started describing all the little hidden passages inside that three-storied play house the adults can't really access. Then she said: "I'm going to draw you a map!"



So she did and, when finished, talked me through what she had drawn. Observing all this I was reminded, yet again, that she has been expressing her ideas through maps for as long as I can remember.

Drawing and reading maps is a key spatial reasoning skill.

This one is from our house to Grandma's house in Ohio, drawn one morning when she was newly six:



Half a year later, she was still drawing lots of maps.  In this one she mapped out a set of sequential instructions for sewing a dress:



We've made and played games with maps (this one is about capturing territory for our kitty clans):



There was her "map of angles"



And the time that she and her friends found a real x-marks-the-spot treasure map (on birch bark, no less) at the Farmers Market and spent 30 minutes investigating where that X was in real life. They found it.



Here's just one more map, where I wrote: 

"She used the new graph paper I had printed out for our graphing game and as she drew she described to me how each color, line and picture symbolized a landmark along our [walking] route downtown.  She even threw in a fractal tree to represent the campus woods."

Later on in that post (written over 1 1/2 years ago) I write:

"She seems really drawn to visual representation as a way to communicate to others what she knows, especially math concepts.  Over the last year she has shown me what she thinks and understands through spontaneous, unprompted creation of charts, maps, and diagrams; often these are private ruminations that I happen to unearth while tidying up after bedtime.  I view these self-initiated efforts to symbolize and quantify (in a way that makes sense to her) as a kind of bridge between the experience and her eventual use of standard mathematical notation and representation.

This is a particularly important to thing to re-read as it has led me to a very interesting realization: I have always assumed spatial concepts as being part of mathematics. But, maybe they're not?

There's lots of information out there about tests for determining spatial reasoning skills, many of them very specifically about rotating 2D and 3D shapes in one's mind. It also appears, based on some cursory reading, that one can improve one's spatial reasoning skills using activities that are similar to these tests.  

But there seems to be precious little documentation related to the intentional pedagogy of and reasoning around the development of spatial skills.  

Does the development of spatial reasoning and math reasoning interact? If so, how do we gracefully fold spatial skills into the daily mathematical lives of our students?  I have a feeling doing so could be very fun, as well as beneficial.

In a sense, these map artifacts from my daughter's life over the past 2+ years have become a treasure map of sorts for me as I try to figure this out. I always love company on my journeys -- if you have any insights, personal experiences, or aha! moments about what it looks and sounds like to include spatial reasoning in the classroom in an intentional way, please do get in touch.

In the course of one short day, all of a sudden I have an insatiable new curiosity.

Spatial reasoning. It's been so obvious to me that I don't even see it or talk about it any more. It seems intuitive to me that dancing and dance making have a role in building spatial reasoning skills, but now I'm really curious about how it all works.

What exactly is spatial reasoning, beyond an end-goal of being able to visualize positions and movement in 2- and 3-space? Why is that important? And, if it's so important to success in both math and science, why aren't we talking more about how to develop it in all our students?

It seems we hit it pretty heavily in ECE by infusing the preschool learning environment with spatial language (over, under, through, middle, forward, back, etc.) but starting at about 1st grade these efforts disappear.

And, honestly? I can tell. I've been working with 4th and 5th graders for over fifteen years. Every teacher I've talked to says that many/most of their students are lacking in spatial skills to some extent. This does not necessarily mean they are lacking ability to learn dance or are awkward movers.  It means they do not naturally or easily talk about how, where and why they are moving. As a result, I often feel like I am engaged in spatial remediation with ten and eleven year olds.

I shouldn't be the only one engaged in this activity. Nor should we be remediating this skill.

In this article on the subject (h/t to @msbjacobs on Twitter) the author emphasizes that:

Teachers (and parents) need to understand what spatial thinking is, and what kinds of pedagogical activities and materials support its development. Recall that spatial thinking involves noticing and remembering the locations of objects and their shapes, and being able to mentally manipulate those shapes and track their paths as they move. Because spatial thinking is not a subject, not something in which children are explicitly tested, it often gets lost among reading, mathematics, and all the other content and skills specified in state standards. Teachers need to be able to recognize where they can infuse it into the school day.

So, my focus is now:  

How are spatial reasoning skills developed?  What math subjects, beyond geometry, benefit from strong spatial reasoning skills?  How can/do we connect these skills to math learning? 

I have my own understanding of this topic based on years of actually doing it, but I am most curious about how this is being addressed in classrooms other than my own. I am also curious about how spatial reasoning and mathematics interact in the math classroom.

I will be working on this and reporting back from time to time. If you have any insights or experience with this topic I would LOVE to hear more!

Have you heard about Function Carnival created by Desmos, Christopher Danielson and Dan Meyer? The basic concept is this:

Students watch a video. They try to graph what they see. Then they play back the video and see how their graphical model would be represented as an animation. Does what they meant to graph about the world actually match the world?

Looking at it for the first time it appeared that the three graphing activities had very specific aims. But, for some reason, I was curious how it might play out with the elementary set.

I showed my eight year old the first activity, Cannon Man, after school on Monday and we did not stop for over 45 minutes. At first she tried to solve the graph and was working really hard, with great conversations between the two of us ('cause I'm still new to all this too, plus we're pretty good at learning stuff together sometimes). At one point she said, "Just show me the answer!" and I said, "Well, that's the point, it's you who figures out what the answer is."   

 

At some point the laptop touch pad made the line weird and it crossed, and all of a sudden there were two little men! Hilarity ensued as we started to move away from the script. She started drawing random lines (some crossing, some starting and stopping) just to see what would happen. 
 
Long story short, we spent the rest of our time (about 30 minutes) trying to make the graph do our bidding in terms of making the little guy move the way we want him too, and choreographing two, sometimes three guys at a time. (And, honestly, hundreds of him at times).

It was truly DE-lightful. 

And, even better, it's apparent she learned something.  I did my own version where I drew little bumps hopping across the bottom third of the graph.  Then I did one where the guy went up really fast and then tried to have his jumps be lower and lower. 

Frankly, my work was not subtle or effective. My girl took over the reigns, made some adjustments to my model and produced something truly sophisticated -- height, time, and speed merged to create a lovely, smooth choreography of bouncing Cannon Man who ultimately disappeared for a moment, and then glided back down. And, this is a child who spends very little time in front of the computer.  

So, the perfect tool to think with, yes? Clear goal, but open ended enough to produce unexpected learning. Intuitive and yet helpful in building skills. Accessible/meaningful to students younger and older than the target audience (MS/HS?).  Isn't the point of an "object to think with" that the student will learn how to use and think with the tool rather than simply work 'as instructed'?  It's clear there was a LOT of thinking happening 'round here this afternoon.

Go read more about their reasoning for making it and try it out for yourself!

Today was my daughter's third day of 3rd grade and the teachers had the kids write in their math journals about what they like about math.

We home schooled for 1st and 2nd grade and so, as you might imagine, I was very interested in her answer!

Here is what she wrote (spelling adjusted):

What I like about math is:
1) It's a challenge but not so much I hate it plus I'm good at it.
2) My mom is a math teacher.*

What I do not like about math is:
1) It can be too MUCH of a challenge.
2) and sometimes I am tired and sleepy.
I am not very good at division but that doesn't mean I don't like it.

What stands out for me about my daughter's assessment of her own math learning is the identification of that fine line between a good challenge and too much of a challenge. If half the battle of math learning is about keeping up a good attitude, then I'm very pleased to know it's not an either/or situation for her.  It sounds like she's felt successful in the work we've done together over the last few years.

Next week I get to start my weekly visit to the 3rd/4th grade classroom to work with small groups during math time.  My plan right now is to provide some fun, push ahead kind of challenge for certain kids and some just right challenge to help in attitude remediation for other kids. FUN.

It's shaping up to be a fun, fun year. Just this afternoon I found out that four parents at the school are programmers and university professors and they're going to start a Scratch club.  Did I mention FUN?
_________________

*I actually do not call myself a 'math teacher' but Yelena at Moebius Noodles did christen me a 'math explorer' last winter - and that's a title I really like.  I'm thinking about using it when I come in to work with the kids.  As in, I'm a math explorer which means there's lots I don't know but I'm always excited to learn more about math!

The same question has come up three times in the last three months, from three different sources.  Each time I have had no good answer.  When it happened again last week I knew it was absolutely time to figure this out.  The question, you ask?

"What is the difference between variables and attributes?"

Exibit A: The source of the confusion:


I hate giving answers I don't fully understand.  My answer has generally been: "I think I'm using the word 'variable' in the colloquial sense, you know, things that can change around -- I think that, mathematically, these are really what are called 'pattern attributes'."

You see? Totally unhelpful - to the person who doesn't understand math and to the person who does.

In my defense, up until just yesterday I didn't think the use of the word 'variable' in Math in Your Feet was mathematically accurate because I knew that variables are part of algebra and algebra is about inquiry into growing patterns. We make dance steps (pattern units) in our math/dance work but not growing patterns. When we evaluate sameness, similarity, difference and change we are focused more on the, well, attributes, that comprise each individual beat of a four beat percussive dance pattern.

After last week in Minnesota I knew I needed an answer, and I needed it soon! Luckily there was a meeting on the calendar with Gordon Hamilton and Maria Droujkova. We're working on a book that is currently titled, funny enough, "Variables and more". Luckily, I had warned both of my collaborators that this question was coming down the pike.

Maria's answer? Essentially, attributes and variables are almost interchangeable. There's the kind of variable used when looking at change (algebra) and the kind that is used when analyzing something that is set, or static (which we could call an attribute, if we wanted to) like in geometry (or statistics, apparently).

According to Maria, this class of variable is called a "categorical variable" and it is useful for things "that are not ordered".  I think I'm remembering correctly that ordered means, for example, a thermometer. You can compare differences in temperature - it can be hot or it can be cold or it can be in the middle, but the change is measured in a system that is already set.  With categorical variables there is much more freedom to analyze properties and make up your own categories, for example: Movies I Like, Movies I Hate.  Movies My Cat Likes, Movies My Cat Hates. In each of those four cases Movies have been sorted into different equivalent classes, meaning - every movie in the Movies I Hate category will be the same in some way.

The dance equivalent (ha ha!) would be that as students are building their first 4-beat pattern I often have us analyze, as a whole group, individual 'works in progress'.  We do this by focusing on our attention on one movement category at a time (e.g. identify only the directions in this pattern, or only the foot positions). The process of focusing their attention in this way makes their dancing much clearer and much more precise.

Clarifying direction.

But, as I found out, I was right to think that it is mathematical activity too, just a bit beyond my current elementary understanding of sameness, similarity and difference. When Christopher Danielson and I met after my Minnesota dance workshop last week he posited that perhaps a fundamental characteristic of mathematical activity is when you say exactly what it is you want to pay attention to (decompose), focus only on that attribute and ignore everything else, which is really what we are doing when we build and analyze our dance steps.

The only thing still in my mind is that instead of the activity of sorting these movement variables while they create moving patterns they are instead actively choosing them while they compose/design a dance pattern. It think it is probably a similar thinking process, as in "Hmmm, I don't like jumping on all four beats, what other movement can we use?" In that case, students really are focusing on one attribute/category at a time as they choreograph their steps.

This idea of decomposition and equivalence classes are a new conceptualization of 'sameness' for me, something well beyond the idea of congruence which we also use in our dancing. I've had hunches over the last year about how sameness and attributes are the mathematical ideas at the core of our dance work, and I've still got some thinking to do to integrate this new information about categorical variables but, in the end, I am just thrilled that my hunches been have validated in such a spectacularly specific manner.

Here are a few examples of how the math is going these days:

Scene 1: The grocery store

Seven-year-old is pushing cart around the store, narrating as she goes: "Go forward, now one quarter turn to the right, now go forward, parallel park.  Okay, now turn half way around, go straight, one quarter turn..."

She sounds like what I imagine kids are doing when they programmed their Logo turtles in Seymour Paper's classrooms. We've never discussed quarter turns or half turns but there they were, helping her guide our cart.

Scene 2: Rest room at City Hall

Her: Mama!  Look at the math on the wall!"

Me: What kind of math do you see?

Her: Look at the designs...let's figure out the perimeter!




















Me: What kind of designs do you see?

Her: [silence, moving away from the scene]

Me: When I look at it I see a large square with green tiles in each corner and one in the center.

Her: Why do you always have to take pictures?

Scene 3: The Park

My kid is playing with a younger kid.  Both are on a 1970's era climbing structure.  The other kid's mom calls to the younger child to be careful and climb down.  My daughter replies, "Don't worry, it's one hundred percent safe!"

We read a little about probability in G is for Googol which is where (I think) she first heard about this idea.  She's also been reading the daily weather reports in the newspaper which are full of both percentages and probability.  It's clear to me she's playing with these two ideas and trying to figure out how they work.

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I am always a little tickled when I overhear or observe my daughter applying or identifying math in new ways.  But over the last year I have come to wonder why she so clearly wants math to be all her own, separate from me.  After thinking about this on and off for the last year (over which time she has really come to see herself as "good" at math) I think it is partly that she is such a fiercely independent learner.  But I also think there is something more than personality at play here.

Based on my own math learning experiences these last few years, I can tell you that learning math is personal.  I'm reading Seymour Papert's The Children's Machine and he is brilliant at specifically and concretely illustrating how real learning is a series of personally relevant connections.  I think his is a theory that can be applied to many subjects, but it totally makes sense within the context of learning math.  I've also read in different places that we all have what I would call differently constructed math schemas -- we all see math differently.  The challenge of the math teacher is to teach from where the student is rather than require the student to assimilate the teacher's mind map of whatever math topic is being explored.

So, what I've ended up doing is creating situations for her to explore math within an actively hands-on, visual, and often narrative-based context.  This approach started out as my way of dealing with a learner who was resistant to instruction, but quickly became a wonderful opportunity for me to re-conceive what math is,  as well as where and how it can be learned.  Basically, I had to rearrange my concept of math to fit my particular learner.

In our first grade year the learning happened during conversations about the math we saw walking around town (which I started calling "sidewalk math"), reading living math books, playing lots of math-y games, and strategically placing math manipulatives around the house.  (I'm still sort of in awe at the independent work that went into this shape study using tangram pieces when she was six.)

In her second grade year we've done more math sitting down at a table using more recognizeable math manipulatives, but with the same approach as last year; our math has been hands-on, very visual, with a lot of room for personal aesthetic and narrative contexts.

I made a conscious choice to pause our math progression this year in about mid-March.  I knew we could have kept going, but my instinct (and my work schedule) told me she'd be fine.  And I was right.

What I've observed in my daughter over the last seven weeks or so is that within this 'void' of math lessons she has begun processing her learning by applying and using the math we've done together since mid-August.  There's been a veritable flood of daily self-initiated math activity, thinking and conversation. The three examples above are just a fraction of how she has been playing with the math she knows (or is trying to figure out) by applying and using it in a variety of different settings.

I did have some thoughts in March about seeing how far we could get into third grade math but now I'm glad I didn't push it.  It's been more than worth it to make the space and time for assimilation, that special kind of deep learning that happens unconsciously, below-the-surface.  You have to be patient for this kind of 'proof'' to bubble to the surface, but if you keep your eyes open, kids will show you every day what they know. Even though we could have gone further in math this year, it's clear that the math we did encounter and explore is now truly all her own.  And, I think this is the best possible outcome for my particular second grader.

I am looking for as many examples as I can find of contextually rich math and math art learning in elementary education.  I put the idea out on Twitter, but I'm hoping readers of this blog will be able to contribute any and all ideas you may have.

So far my list includes:

Froebel Kindergarten

Moebius Noodles (check out their brand new book!)

The work of Catherine Twomey Fosnot.  Here's a great video where she explains her approach:


What is not yet clear to me is how and where math learning and math art learning overlap.  Math Art, to my mind, can include all artistic media including dance, music and visual forms.  If you are able to help me with links or thoughts of your own, even including specific examples from your own teaching, I would be most grateful.

This is only the beginning of my inquiry to make sense of how math learning can happen in multiple contexts, including artistic and design settings, with all roads leading to real comprehension and mastery of mathematics heading into middle and high school.

I've been over in weaving land for a while (young children and grids here, inverse operations and multiples here, and Fibonacci here), but recently came back around to our star adventure.  As with most things, my seven year old learns best in an environment of exploration and self-directed learning, so I've been careful to present this star inquiry primarily as just that...exploration.  What's amazing to me is that although she and I are exploring different things about stars, we seem to be on parallel tracks, moving forward together.

A few weeks ago I built the skeletons for the "spiders who spin fancy webs" out of plywood and copper brads, but things were left there for a while.  Yesterday I decided I was ready to pull out all the beautiful embroidery floss we acquired for this project and try spinning some star webs. 

Below are some of my experiments with six, seven and eight pointed stars.  I do love how the different colors look but it was way too futzy for me to get the effect I wanted; mostly it was hard to keep the tension steady between posts.  I quickly determined that rubber bands work best in this activity, and that is now my official recommendation for when you try this at your house or classroom.
















The best part of trying to 'spin' these webs was that I saw the six pointed star next to the seven and eight pointed stars. That got me wondering: What it would look like if I drew the stars using the same color for each variety of star within the same circle?  Would I see anything new in the stars, especially compared to each other?

When you read about how to draw stars you often see the words "over two points" or "over three points."  If the number of points = n that means that for me (n, 1) was red, (n, 2) was purple, (n, 3) was green, etc. And, all that means is that you use the starting point as 0, and if you go directly to the next point, that's (n, 1).  If you skip the 1 and go directly to the second point, that's (n, 2), etc.

Here are the six and seven pointed stars (three colors, green is 'over three' points).  I loved the different flavors of even-ness, balance and symmetry between the even and odd numbered stars. 













































If math is about asking questions and exploring different ways to answer and understand the questions you pose, then I'd say we've been doing some math.

The absolutely fascinating thing for me is that through this kind of visual inquiry I am not only learning a lot about stars, I also appear to be finding my way to thinking and reasoning numerically.  As I figured how to divide the circle into fifths, sixths, sevenths, etc. with my compass and protractor, I started wondering if there was some kind of pattern that would emerge if I compared those measurements.  In a 5-pointed star, for example, the points are 72 degrees apart and a 6-pointed star is comprised of points that are 60 degrees apart.  I kept an informal chart and didn't come up with anything that seemed newsworthy, but it was very interesting to observe myself open up to this line of questioning.  As someone who is slowly but surely remediating herself in math, I am thrilled that this visual design approach is helping me begin to see numbers as interesting, useful and, perhaps, even friendly!

p.s. As always, a huge thank you to Paul Salomon of Lost in Recursion and the Math Munch blog.  He teaches math at St. Anne's school in Brooklyn but I was lucky enough to benefit from his help and feedback over the summer months via the magic of the inter-webs.

p.p.s. Check out my new Math in Your Feet Facebook page!  I'd love to see you there!

I recently discovered how important the practice of identifying and describing attributes really is to math learning, especially at the elementary level.  Learning to discern similarities and differences develops mathematical thinking skills that can be used at all levels and topics within mathematics. 

Despite the thrill of discovering how this kind of thinking is used in my program Math in Your Feet, I was left with lingering questions about the differences between identifying attributes and using attributes in a design process.  I suppose they are two sides of the same coin, but I can't help thinking that being able to choose from an inventory of possibilities is the preferable skill-building activity in the long run.

I tested my theory on myself and my six year old daughter by using the Symmetry Artist at Math is Fun.  Maybe you've seen this before?  Check out all the attributes you can choose from to make your symmetry designs:

You can choose between reflection or rotation symmetry, with six to eight choices in each category.

Your pen has five choices.

Seven thicknesses.

And many, many colors.

There's a lot to choose from here, but the tool is easy to use, so you can easily change your mind and start over, or take out your last move or series of moves. 

While you experiment and play around you are also noticing relationships between the lines and shapes you make and observing the structure of the final design.  The addition of color serves to increase the complexity and interest in both the process and the product.

I suppose the difference between identifying attributes and using attributes is that one is a more closed process than the other.  I'm thinking that with a set of attribute blocks, for example, although you are learning to discern differences and similarities, there are really only right and wrong answers.  The benefit of this kind of activity is that you are actively using math vocabulary: thick/thin, large/small, circle, square, triangle, edges, etc.  This is all very useful but, as I said, a somewhat closed process in terms of inquiry.

In contrast to 'compare/contrast/identify' there's the process like the one you use in Symmetry Artist. Instead of simply identifying attributes, you are using this skill in context while thinking mathematically in an active way -- you are actually 'doing' mathematics.  By this I mean you are asking questions ("What would happen if I started my circle here?  What would the same design look like with nine iterations instead of four?  How'd that pentagon get there?"), experimenting with and analyzing your 'answers' (designs), erasing your answers, starting over, printing out the answers you like, asking more questions.... This is the creative process in action; it is also mathematics in action.

I love Symmetry Artist because it is a beautiful and fun way to play around with, learn about, and compare how lines, shapes, and iterations interact within these two symmetries.  Below I've put just a few of the designs my daughter and I recently made. 

As a first grader, my kid uses this tool primarily for exploration.  I started by explaining the different categories, but not much more, and she jumped in from there.  I, on the other hand, being aware of just how many choices I had, jumped in but soon got overwhelmed with too many questions which led to too much erasing and/or starting over, resulting in pretty much nothing to show for my efforts!  That didn't stop me from noticing a few things, however.  Take a look at what we did, see what you think and then go try it yourself! 

Here's my daughter's first design.  Notice that she set it for rotation, 'eight', pen, one color, and a medium thickness.  She 'drew' with abandon, and I was thrilled with what resulted, mostly because I would have never thought to do it that way!  As you'll soon see, my initial approach was a bit more measured.