Wednesday, April 1, 2015

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. 

Tuesday, February 10, 2015

Building Competence [#miyfeet Primary Project]

I've had some questions running around in my head about the Math in Your (little) Feet primary project/experiment. The one I'm pondering the most is: Why am I focusing more on the mapping aspect with the primary kids than in the upper ES version? After today, I think I might be closer to an answer.

Today, in all three of my sessions, the kids seemed to have settled into the dancing and by that I mean, compared to what I saw in the beginning, their bodies executed the (deceptively simple) dance moves with ease and accuracy, and we were moving more as a group. It truly was a beautiful, beautiful sight.

I believe this new-found ease with precision footwork has come about because I have also made a point to have them read and represent their foot work visually, even if it's challenging or not always "correct."  

Last week, instead of having the K-2 class build dance steps and then map them, I wanted to see what would happen if they could figure out how to dance a pattern starting with the mapIt was clear at that point that reading dance maps and making dance maps are two different processes. I wanted to see if I could move those processes closer together. I decided to make a game!

The "starter kit" included yellow cards for the movements Jump and Step. Blue cards were direction cues; arrows for split feet (2 feet on the ground at one time), dots for individual feet. You could also use the backs of the blue cards to draw in foot positions as needed.

It created an interesting interaction between what you want to do and how it's mapped out in front of you.  Were you doing what was on the card? No? Do you want to change the card to match what you were doing? Instead of committing your ideas to paper (which sometimes needs erasing) the cards were semi-permanent, flexible, and interactive. 

I also was hoping that the cards would scaffold teams of kids toward real collaboration, maybe help them share ideas more fluidly, in a way that, up to this point, hadn't really been happening. I think part of the issue is that the dance goes by so quickly they weren't catching each others' ideas. I'm hopeful that these cards can and will support the development of some teamwork and idea sharing in the next three weeks.

The class was as focused as I could have hoped for given everything. It was an incredibly pleasant 45 minutes. 

Just like everything else I've tried so far, some kids loved the cards and some didn't. Upper ES kids don't need this kind of reinforcement to be fully oriented to the math/dance making. For the primary kids, though, my efforts to help create a visible connection between the dancing and the maps seems to be supporting them in building competence in both realms.

Friday, February 6, 2015

Intent on Creation [Family Math Night]

Everywhere I looked last night I saw people intent on making math. It was glorious!

I mean, just look at these messes!

Just look at the focus!

Just look at all the ideas!


As a finale, I present three views of the interlocking cubes created by one participant. I also hasten to add that while there are lots and lots of materials provided, kids and adults are free to do with the materials what they will. This often leads to unexpected and marvelous personal discoveries and creations, like these:

Wednesday, February 4, 2015

A Small Challenge for the Adults: Please Report Back!

I’m going to give you a challenge. I want you to consciously reflect on how you are thinking while you interpret and reproduce a foot-based dance pattern.  You will get this pattern in two forms, one of them completely language based, the other in grid/map form. PLEASE try the word based one first and then, and only then, follow the link to the image.

Step One: Language based 
Note: Where you're facing when you begin is "front." All directions are set from this initial orientations, like a compass rose on a geographical map.

Beat 1: Jump, split right foot front, left foot back

Beat 2: Jump, split right foot front right corner, left foot back left corner

Beat 3: Jump, split left foot front left corner, right foot back right corner

Beat 4: (simultaneously) turn 90 right, slide left foot to meet right foot in back right corner

Step Two: Map based
Follow the link to the map image.

Step Three: Report back here in the comments section, PLEASE!
Your honest observations of your thinking/decoding/doing process will be incredibly helpful. If you have an opportunity to convince a child to try this out, I'd be even more grateful.

If you are having trouble leaving a comment, please visit the MiYF Facebook page and leave a comment there, or email me mjrbton at gmail.

I appreciate your time and comments!

Edited to add: It's really not scary. I'm happy with anything you want to report, including if you were confused. TRULY! It's all grist for the mill and extremely helpful. Give it a try!!

Tuesday, February 3, 2015

Side Center Window Door: Map Reading with the K-2s [Primary Project]

I met with my K-2 group on Monday. This was our fifth session together. They have experimented with different combinations of steps and jumps executed in a variety of directions inside their square dance spaces. Their self-made maps are showing burgeoning competency in many students to think precisely about the location of their feet while they dance. It's amazing to see their growth from one week to the next. 

This Monday five of the kids had been out sick the Friday before; many were back but still under the weather, and four others were absent.  Because of this I realized we needed something different than the high energy math/dance making we had done so far.  I decided to switch things up on them. 

Could they figure out how to dance a pattern starting with the map?

This is the pattern I gave them, fully notated.  Take a minute to try it out.  It’s not hard, but it will help you understand what happened next as I led them through transforming this map into a real-live dance step:

By giving them this map I was challenging them to use their current knowledge of “what makes our dance pattern” in the brand new context of decoding visual/spatial information. Instead of translating from action to notation, I wanted to see what would happen we moved in the opposite direction.

But I didn't want them to be overwhelmed, so I broke the process down into a number of steps.
Step 1: Some of the information, not all of it

Me: Look at the whole thing. We’re going to spend some time looking, some time talking to our partner about what we notice, and then we get a chance to try it out.

As kids started talking to each other, some of them gestured the foot positions with their hands as they talked to their partners.  One kindy girl came up to me and told me, “Malke, you missed a dot…” meaning the single dot in Beat 3. My response? “Well, this is for you to figure out.”  Here’s what they noticed:
  • Nothing in the front of the square 
  • In the third square there’s not a foot up in that corner, there’s no diagonal 
  • On the very last one there’s only one that’s colored in and one that is not colored in
  • In the center on the first one it’s uncolored but the second one they ARE colored

Me: What do you think this whole thing is about? What do you think I want you to do with this?
  •  It’s showing us how to write an example [of a math/dance map] 
  • It’s giving you an idea of stuff. Because if that’s empty then that’s where you were [gesturing the starting position with hands together] but when you colored it in that’s where you are [gesturing the foot position on Beat 1, hands apart] [WOAH.]

Me to the class: Can you tell me more? What does that mean I might want you to do with this?

Kindy kid: Start there [pointing toward the center of Beat 1]

Me: And then what do you think you would do?

Kindy kid: Go to the colored parts [gesturing with her right hand, 2 fingers split to make a ‘v’]

Me: Okay, let’s all stand up with our partners and see if you can figure out what to do.

Step 2: Let them come to an answer that makes sense to them.

As they worked on decoding the dance from the map my basic question at this point was to ask both partners, “Does that feel like a good solution to you?”  My biggest goal at this point was to see how they were interpreting the map.  Some of the answers came back looking like this:

This was a solution that showed up in a bunch of teams.  Jumping so far has meant that the feet hit the floor either “together” (side by side) or “split”.  Presented with a single dot on Beat 3 makes no sense if you’re jumping so the only logical solution at that point was to put both feet together in that position. Two Kindy girls (one of whom told me earlier that Beat 3 was “missing a dot”) came up with a solution that was the closest to the intended pattern:

Another fascinating interpretation by others in the K-2 class, and also in the 2nd grade class at my other school later in the day, was interpreting forward and backward from the map.  The original map shows two individual dots in the back L & R corners.  The team (whose work is pictured below) read that information and put themselves physically in the front two corners.  The reason this fascinates me is that most kids had oriented their physical selves within  the blue tape squares. They had also been successful writing out their own maps (which I could reproduce correctly). Nevertheless, many of these primary grade children moved themselves forward when the map read backward. Here's a representative example:

Step 3: Provide more details on the map

I wish I could say that by putting arrows facing “up” on the big math/dance map at the front of the classroom helped all the kids understand why they shouldn't move forward when the map read “back”.  Truthfully, what helped the most was adding one finally piece of information into the mix.

Up until this point not one kid had even thought to ask what movements they should use. They all just assumed they would jump.  So I said: “I on purpose did not give you all the information you needed in order to do this dance pattern exactly the way I made it. So, do you want to see what I'm going to put on now?  This is what the chart looked like:

I told the class “Take a minute to look at it, talk to your partner about what you noticed.” They noticed the Jumps and Steps. “But what does that mean?” I asked. “What does that mean is going to happen?

Kid: You jump and jump then step and step.

Me:  Okay. Does anyone else want to say that in a different way? [Silence] What does it mean when it says Jump under the 1 box?

2nd grader:  OH! So it means you're jumping to the sides! [Gesturing: hands splitting apart].

Me:  Ooooh!

1st grader:  And then we jump to the middle and then we step to the back corner and then you step to the other corner.

Me:  Okay! Let's stand up and try this with your partner! 

After a couple minutes I ask the class, “Let’s see if we came up with the same solutions for this pattern.”

A 2nd grader and kindy boy show their work.  The boy watches his partner’s feet very closely and gets that he needs to be stepping in a back corner, but goes left instead of right. I ask them to go slowly and on Beat 3 ask “Which foot goes first, the one toward the window or the one toward the door? Toward the window? Okay, so we'll say Sides, Center, Window, Door while they dance.”

Everyone says it with me while they dance: Sides, center, windoooowww...door.

And they got it!

And, finally, Part 4: Use the understanding of a some to synthesize the understanding of everyone

Me: Okay everyone. Put yourself in the center of your square facing toward the board.  Point toward the window?  Point toward the door! [Correcting facing of some kids while I move to the front of the room. Then I have them “mark out” the dance steps with their hands while saying] side, center, window door.

Me: We've got it! What are we going to call this pattern? Side center window door? 

Class: Yeah...!

Me: Okay, one last time. Let's hear everyone's feet together. SIDE CENTER WINDOW DOOR!  Excellent give us a round of applause!

At this point I decided it might be good for them to map out this pattern.  I turned the classroom map over (out of sight) because I was just curious what would happen.  While they waited for me to get the papers I hear that S, a first grader, has “made a song” out of the pattern. She sings it for us, “Side center window door!” and we sing along...twice.

And, although not everyone was able to map out their foot positions on the outline I gave them, many of kids wrote out the words we used: “sides, center, window,  door.” 

And everyone hummed S’s tune as they made their maps. 


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