Vignette: Noticing, exploring, and discovering space with moving bodies

Today was my second of five days with three classes of 4th graders. Today Mr. C, whose class needs to be with me at 9am, smoothed out his morning routine and told them to come straight to the LGI room. Kids trickled in from buses, drop-off, breakfast, etc. In the absence of any particular activity the kids immediately started exploring the empty space. 

The activity mostly consisted of jumping from box to box and eventually led to discussion about which leap was the furthest and most difficult. 

I am a big fan of self-initiated explorations of space like this but after a while there were too many kids in the room to do this safely. Luckily one kid wanted to try out my dance board which led to a wonderful round robin of dancers, all being cheered on by their friends. So cute!

If you change a familiar space with tape kids will notice. When they notice they will explore this new structure with their bodies through movement. When they explore you will see how they are thinking. In this case their exploration was spatial and also mathematical in terms of magnitude and direction. 

New ways to know old math

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.

My Daughter the Map Maker [Spatial Reasoning]

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.

Spatial Reasoning: Ready, Set...Go!

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!