Some thoughts on "Hands-On" Math Learning

Last night on Twitter Michael Pershan asked me to weigh in on hands-on math learning. The request stemmed from a conversation/debate about the various merits of different ways to learn math.  

The minute I read the question I knew that my answer was going to be more detailed than a response on Twitter would allow. Here are some of my thoughts on the matter.

1. The discussion reminded me of the "concrete to abstract" conversations which, to me, seem like an especially frustrating example of recursion. They go round and round but we never really get anywhere new.

I think many connect the word "concrete" to Piaget and his discussions about children's thinking moving from the concrete to the abstract. This in turn has led to many assumptions that take the term "concrete" quite literally. But, as Deborah Ball wrote in her article Magical Hopes, 

“Although kinesthetic experience can enhance perception and thinking, understanding does not travel through the fingertips and up the arm.  And children also clearly learn from many other sources—even from highly verbal and abstract, imaginary contexts."  

 The best treatment of the concrete/abstract dichotomy comes from Uri Wilensky:

"The more connections we make between an object and other objects, the more concrete it becomes for us. The richer the set of representations of the object, 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."

I LOVE this treatment of "concrete" as simply the quality of your relationship to an idea. Seriously, read the whole piece. You'll be glad you did.

2. Professional mathematicians utilize a multi-sensory approach to their work. Here is some perspective from researcher Susan Gerofsky:

“Movement, colour, sound, touch and other physical modalities for the exploration of the world of mathematical relationships were scorned ... as primitive, course, noisy and not sufficiently elevated or abstract.  This disembodied approach to mathematics education was encouraged despite the documented fact that professional research mathematicians actually do make extensive use of sensory representations (including visual, verbal and sonic imagery and kinesthetic gesture and movement) and sensory models (drawings, physical models and computer models), both in their own research work and in their communication of their findings to colleagues in formal and informal settings.  These bodily experiences ground the abstractions of language and mathematical symbolism.”

3. Children think and learn through their bodies. We should use children’s bodies in math learning.

Known in the research world as embodied cognition (thinking and learning with one’s body) is something we begin developing from birth. Developmental psychologists have shown that in babies “cognition is literally acquired from the outside in." This means that the way babies physically interact with their surroundings “enables the developing system [the baby!] to educate [herself]—without defined external tasks or teachers—just by perceiving and acting in the world.” Ultimately, “starting as a baby [as we all did!] grounded in a physical, social, and linguistic world is crucial to the development of the flexible and inventive intelligence that characterizes humankind.”

Understanding what embodied cognition and embodied learning looks like is the focus of a multidisciplinary group of cognitive scientists, psychologists, gesture researchers, artificial intelligence scientists, and math education researchers, all of whom are working to develop a picture of what it means to think and learn with a moving body.  

Their research findings and theory building over the past few decades have resulted in a general acceptance that it is impossible to ignore the body’s role in the creation of “mind” and “thought”, going so far as to agree that that there would likely be no “mind” or “thinking” or “memory” without the reality of our human form living in and interacting in the world around us. 

4. Finally, instead of sorting out the various merits of individual teaching/learning strategies what we really need to do is look at the bigger picture: Most student learn math best when provided with multiple contexts in which to explore a math idea.

A learner needs time and opportunity to experience a math idea in multiple ways before being able to generalize it and how it can be applied.  An idea, any idea, becomes “concrete” for the learner when the learner has had an opportunity to get to know it. Uri Wilensky said it best:

“It is only through use and acquaintance in multiple contexts, through coming into relationship with other words/concepts/experiences, that the word has meaning for the learner and in our sense becomes concrete for him or her.”

Pamela Liebeck, author of How Children Learn Mathematics, developed a useful and accessible learning sequence to help bridge the gap between a math idea and a meaningful relationship with that idea.  Based on the learning theories of psychologists such as Piaget, Dienes and Bruner, Liebeck’s progression is similar to how babies and young children learn to recognize the meaning of words, begin to speak, and then to first write and then read. It includes four different learning modes in which to interact and express mathematical ideas and includes:

a) experience with physical objects (hand- or body-based),

b) spoken language that describes the experience,

c) pictures that represent the experience and, finally,

d) written symbols that generalize the experience.

This sequence illustrates what many math educators already believe, whether or not they use this exact outline – that elementary students need active and interactive experiences with math ideas in multiple learning modes to make sense of math.  

After a recent and particularly robust online discussion on the many different ways to support primary students in making sense of number lines, including a moving-scale line taped on the floor, Graham Fletcher said, “At the end of the day, it's all about providing [students] the opportunity to make connections.” 

Graham's statement points to the importance of focusing on the child's relationship to the math and the environment in which she learns that math. Hopefully it's an environment where many different ways of thinking, expressing and applying mathematics are celebrated and nurtured. 

Word-Mind, Body-Mind: Toward a More Balanced View of Learning and Knowing

I've just started a book project.  What's it about?  Well, for now, I think it's about digging into the whys and hows of learning math and dance at the same time. This may change, but that's where I'm at right now.

To organize and clarify my thoughts I've been doing quite a bit of background reading, having e-mail/online conversations with various wonderful people, and drafting some preliminary chapters in the blissfully serene Silent Reading Room at our local public library.  Tons of natural light, total silence except for the hum of the HVAC and the occasional cough.

One day last week I sat down to read and felt that old feeling coming on that I used to get in college. It signifies I am completely saturated and overwhelmed with thinking, reading and writing. It makes me a bit crazy, honestly. To combat it, I've learned to skim text and only read closely when I need specific information. Even better, go for a long, long walk with far away vistas.

I call that part of my brain my 'word-mind' and it is very clearly located in my head. 

Using my word-mind is a completely different experience from thinking with my body-mind.  I know, because after college I started dancing and was introduced to another part of the thinking equation -- the body thinks too, a phenomenon that is defined and described by studies in embodied cognition (a branch of cognitive science).

The body thinks, too.  This was truly a revelation to me.

Ultimately, there were years (and years) where I was required to use only my word-mind to learn. And then there were bunches of years (as an adult) where I learned and expressed myself solely through my body mind (dance and music performance, never wanted to get on the mic to talk). My body-mind thinks differently than my word-mind, but what I have come to realize is that, in addition to needing to find a balance between the two in my daily life, as a learner I learn best when I am using both at the same time. 

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I was in Minnesota this summer, sharing my work with a group of teachers and teaching artists. That's when I met Christopher Danielson in person. Yes, an educator of math educators came to my workshop. I was a wee bit nervous but I needn't have worried.  Even though in my workshops we make dance and math for a large proportion of the time, Christopher was game to dance and dance he did.  He also took copious notes by hand at various intervals.  I was again a wee bit worried, but it turned out our 90 minutes of dance and math making had gotten him thinking.  In a good way. By engaging with the work of Math in Your Feet from the inside of the experience instead of simply watching, he generated many new questions that moved his thinking forward in new ways.

The next day we met for a breakfast conversation during which, in relation to one of the questions he had while dancing, Christopher mentioned a now defunct unit in the Connected Mathematics series that investigates transforming squares in a series of combinations and is related to the thinking in modern algebra.  Later, he sent me the book.  I meant to get started on it but life took over.  And then my book took over.

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This week my word-mind gave out. I was literally a little dizzy from thinking in my head and was completely DONE with words. I do a lot with language these days.  I keep a blog, I'm active on Facebook and Twitter, and I edit a year-round, international online writing project for the Teaching Artist Journal.  So to have reached my limit with words is saying something.

I needed something completely different to do.

I pulled the Connected Mathematics book off my desk, sharpened my pencil and headed for the library.  I made my little square out of nice stiff paper. I labeled the vertices.  I combined series of turns and flips, filled in the table and investigated the patterns I saw.  For a good portion of the time I was completely frustrated and confused -- turns out I was turning clockwise when I should have been turning counter.  Apparently there is some convention among mathematicians that all rotations go counterclockwise.  They must have all been left handed and never learned analog (clockwise) time because this right dominated person kept seeing the "R" for rotate as meaning "turn RIGHT."

At some point, though, I was looking at my hands, turning, flipping, holding corners to orient them as I turned and flipped, and I realized I was in some kind of zone. I was talking to myself in words: specific conscious reminders to turn left when rotating, murmuring less conscious words while filling in the chart, a gentle narration as I talked myself through each new sequence of flips and turns and, eventually, the body and words merged and I was just doing.  I had fallen into a zone of concentration while making math that mirrored what often happens when I am learning something new in dance.

After making numerous combinations of transformations with my hands, I now have a new feel for what a square is and is capable of.  Both the word-mind and the body-mind had a part to play in this new understanding.

I hope it is obvious that this is not an either/or kind of story -- it's about both.  Understanding cognition and thinking and knowing is a vast endeavor and I don't know if we will ever fully understand the complexity of it all.  My focus is on what it means and looks like to learn math with the body as an equal partner and to do this I need to keep the body's way of knowing and thinking firmly in my sights.

Ultimately, what I am wrestling with is this:

We can see the body moving, but most of us have not yet learned how to look at and understand the knowing and learning happening while the body moves.  It's an inside process and not one that a child will necessarily be able to tell you about.  As more experienced learners the onus is on the adults to look for how children show us what they know and think with all their one hundred languages, including the body.

I'll leave you with a powerful paragraph found in the conclusion of the recent study Children's Gestures and the Embodied Knowledge of Geometry (bolding emphasis mine):

"Despite two decades of research on the embodied nature of cognition, constructivist perspectives continue to emphasize abstraction of knowledge from the physical engagement with the world ... In this way, constructivism orients our attention away from the body, concerned as it is with the construction of mental entities and representations...  

Once we acknowledge the body as the seat of knowledge itself rather than as a stepping stone to abstractions, it is possible to organize teaching differently.  In doing so, [this] brings forth the possibility of inquiring into such questions as, 'What might we do to recognize and understand children's knowledge expressed in modes other than speech or writing?'"

One thing you can do is to try learning math and dance at the same time yourself. I'm available to talk about how I can help, if you like.  Feel free to get in touch.

"Children Think and Learn Through Their Bodies"

Needless to say, as someone who harnesses movement as a teaching and learning tool, I am fascinated every time cognitive  science reveals another link between the body and learning, what is often called 'embodied cognition'.

I'm about to embark on a reading of Where Mathematics Comes From: How the Embodied Mind Brings Mathematics into Being, by George Lakoff and Rafael Nunez (thanks to a tip by Michael Paul Goldenberg.)  This is a book co-written by a linguist and a psychologist hoping to launch the discipline they would call "the cognitive science of mathematics." 

In a similar line of inquiry, the researchers who recently  published the article Children's Gestures and the Embodied Knowledge of Geometry concluded that "children think and learn through their bodies."

Here's another recent article on a similar topic: Thinking with your hands: A link between gesturing and intelligence:

"My colleagues at Germany’s Humboldt-Universität zu Berlin , Potsdam University and I have discovered differences between people who gesture frequently and those who gesture rarely. Our study shows that gesturing may be a function of, and may even contribute to, brain development... [Emphasis mine]

"We do not know yet whether gesturing facilitates the development of fluid intelligence or whether it is a by-product. But we do know that children who are asked to gesture in certain ways while learning new tasks learn better than children who are asked not to gesture. Considering that gesturing benefits children while learning, it is possible that gesturing plays a role in the development of fluid intelligence, perhaps by simulating action. If this proves to be true, children might be able to literally give themselves a hand in their own development by gesturing more."

...and thoughts about this article, from the Eide Neurolearning Blog:

"It's interesting to think that teaching children to problem solve certain types of problems should involve strategies that take into account that fact that one is trying to train the imagery of the students. Just verbally saying back the steps of a problem or even watching an explanation won't internalize the imagery. To really 'get' certain problems, we have to enter into the simulation and perceive the question and solution in a bodily way." [Emphasis mine.]

From the Edutopia blog of neuroscientist and teacher Dr. Judy Wills:

Students need to be explicitly taught and given opportunities to practice using executive functions to organize, prioritize, compare, contrast, connect to prior knowledge, give new examples of a concept, participate in open-ended discussions, synthesize new learning into concise summaries, and symbolize new learning into new mental constructs, such as through the arts or writing across the curriculum. [Emphasis mine -- one of the reasons 'the arts' are helpful in this context is that most artistic activities employ the body during the art making process.]

My only question is why, with research and thinking like this, many/most schools still insist that keeping kids at their desks, and lengthening the school day behind said desks, is the best way to insure student learning.  I know all schools don't think this.  I know a lot of really smart folks out there are connecting the dots, but our policies don't reflect the very real evidence that our bodies must be engaged and involved in the learning process.