Week 1: Embodied Mathematics and Measuring

I am going to go off-script right at the outset here. Hope that's ok.

First, I want to report and synthesize what surfaced for me in all three articles. Then, instead of measuring something using my body (something I have  have taught children to do over the course of my teaching), I am going to share my first "Body Ruler" video and write a script for my second video.

The readings this week really got me thinking about how much I use my body to develop, represent, and communicate my understanding of ideas. I have always identified as a dancer. Movement is beautiful and important to me. It helps me connect to ideas and allows me to share this understanding with others. I struggle to express myself if I cannot use my whole body. Cups of liquid often go flying when I get talking and thinking. So, yes, this idea of embodied learning is not a stretch for me. However, there are some ideas that really were new to me that I want to remember and chew on for a bit. Here they are:

Powerful Ideas About Gesturing (Goldin-Meadow, 2009; Gerofsky, 2011):

The power of combining gestures and mathematical actions reminded me to be explicit whenever I do so. I know that I gesture when explaining mathematical procedures, but I don't often teach students to use the gestures when solving problems on their own.  The combination of the Goldin-Meadow (2009) findings that students who learned helpful gestures successfully applied them to solving math problems, along with Gerofsky's (2001) findings that high-performing students use their whole bodies (as well as imaginative verbal metaphors) to represent their understanding of graphs causes me to revisit how I use my body to represent ideas. Do I always use body gestures? When do I lack body gestures? If I do, what does this mean? Does it mean that I do not really connect with the ideas I am relaying? In the case of me, probably. Finally, perhaps I should use some of these moves to choreograph some nice mathematical dance numbers, hey? Curious to know how students and teachers will respond! 

Powerful Grounding Metaphors of Mathematics (Nathan, 2021):

Collecting things (quantities) and object construction operations: combining, removing, sharing, repeating (basic arithmetic operations). I would actually say 'collecting objects' instead of 'collecting things', because it is nice to align the metaphors. I also prefer 'acting on objects' instead of 'object construction operations', because there is a lovely simplicity to the idea of collecting objects and then acting on them. Plus, acting is a verb and arithmetic operations represent actions. 'Object construction operations' are far too nouny for me. 

These metaphors are interesting to me because I have spent a couple of years trying to help teachers and students see the forest for the trees when it comes to quantities and mathematical operations. I regularly share the idea that much of elementary school mathematics is exploring variations of the the simple question, "How many?" In our Zoom discussion this morning, the toolkit metaphor was described as problematic, but I love the idea of a counting toolkit that can be used flexibly and skillfully when we wish to answer the question, "How many?" I made a film to help teachers and students make sense of this and it has been helpful to unite simple 1-1 counting with more complex operations. Further, it helps to demystify those more complex operations and bring them within reach of all learners. You are most welcome to check it out if you wish: Click here.

Locations along a path (number systems). Love this one. I have been doing a ton of work with numberlines from kindergarten to grade 8 for well over a year now. I love the idea of specific numbers being locations along a path. I have to think about how that might help me shape a film for teachers and students.

Motion (continuous quantities and direction). I haven't fully wrapped my head around this one, so I'm going to need to do some further exploring.

Fulcrum of a balance scale (=) and moveable objects (symbols). I've been using this one for a while, so I take it for granted. It always surprises me when teachers do not know it. Mathigon's Polypad has been very useful for showing and explaining this notion. I did a couple of short videos for parents, teachers and students last year (nothing fancy, just enough to support some remote learning activities): Click here.

Lengths of tangible things (measurement). The only thing that bothered me about this metaphor is that we also measure weight, volume, area, speed (etc) and it is helpful to develop personal referents for all qualities. 

Now, this is a nice segue to embodied (length) measurement. I love the idea that "tangible things" are things that you have with you always: your fingers, hands, arms, legs, feet. I promote the idea of using a "body ruler" with teachers and students regularly. Here is an example of work I have done along these lines: Click here and scroll down to Stump Stories Week 2.

Now, instead of embarking on my own measuring activity, I am going to write a script for a new film that will be better than my old one. I like the idea of adding history and using units such as "fathom", something I did not do in my first video. I picture the film taking place in my local park instead of inside my house (see picture above). I might have the video done in a couple of weeks. For now, here is the script:

Did you know that without even knowing it, you carry handy tools for measuring wherever you go? No, I'm not talking about measuring tapes or rulers or trundle wheels; I'm talking about your body. Ancient humans knew how to use their bodies as rulers and so can you.

First, know that all human bodies are similar, but not exactly the same. This means that you must be aware of two important things:

1. When you use your body ruler, you are estimating. 
2. Your body units are non-standard.

The ruler of your body is slightly different than your friend's and is also slightly different than what it will be when you are adult-sized. In other words, the units of your body ruler are non-standard. By comparison, a centimeter is a centimeter is a centimeter no matter how old you are or who is doing the measuring. A centimeter, therefore, is a standard unit of measurement.

Now that you are aware of these two very important ideas, let's begin, shall we?

It is particularly helpful to go somewhere with interesting things to measure. My local park will do quite nicely. We have the ocean over there, some old interesting ruins, lots of beautiful big trees, birds, rocks...all kinds of fascinating things of all different sizes. How big are they? How long? How tall? How deep? We are about to find out.

Let's start with small objects like these leaves here. How long are they? Now, you first need to decide what part of your body might be helpful to use as a measuring instrument. We need something small enough, that's for sure. Not a foot, or a hand, no...how about a finger or a thumb? Personally, I like to use my finger. Take a look: (shows measuring using the finger). How long is this lovely leaf? ___ fingers long...approximately of course. We could also use a half finger like this (shows half finger measurement of a small object) or a full finger (shows full finger measurement). And there you are: four different finger units for measuring small objects! But there's more...

How about medium-sized objects like this ____ here. It might be quite a chore to use small finger units to measure this. Might we have something larger? Of course! How about the "palm"? Simply tuck your thumb in, spread out your 4 fingers, and voila! You could also use your hand length, from the heel, to your finger tip, like so. Or a hand span...untuck that thumb and measure from it to the tip of your pinky. Handy, no? And yes, the pun was intended.

Onward to some larger units! Let's start with the cubit. Yes you have two of those! They are the distance from your elbow to your fingertip, very helpful units for measuring railings, I must say. Or how about fathoms? Sailors used fathoms--or the distance from the tips of the fingers of one hand to the fingertips of the other with arms outstretched--to measure ocean depth. Take a look: I take a nice long piece of rope and tie knots every time I measure a new fathom. Now I weight the end of the rope and lower it into the ocean until I feel a bit of slack like so. This part of the ocean is ____ fathoms! A little too shallow for an old sloop, but plenty deep enough for a paddleboard.

Next up is the yard. You have likely heard of this one. Yard sticks look a lot like meter sticks and are commonly found in old schools. What part of your body is a yard? Face straight ahead of you and measure from the tip of your nose to the tip of one arm fully stretched out to the side. Now, I'm not going to go sticking my nose onto a rock and measuring its length in yards, but I might measure this picnic blanket here. Ah ___ yards wide. Plenty wide enough for me and Georgie.

You can use your height to estimate the height of other objects. Is this building taller than me? Certainly. How tall? Double my height? Maybe triple? If I know my height in standard units, I have a pretty good estimate of a buildings height. If I am 1.68 meters tall and a building is three times my height, how tall is that building? Around 3 x 1.68 meters (or 5.04 meters...although let's round that to a clean 5 meters, shall we?)

Finally, we reach the feet. One foot is actually the length of your foot, or very close at least. How long is this sandbox? ___ feet. You can measure longer distances using steps. Simply count how many ordinary step you take between, say, the stairs and the edge of the shore. Or this big tree and the swings over here, or the water fountain the bathroom, or...well, you get the idea. You can also use a slightly larger distance measurement called a pace. Instead of counting every step, count 1 for every two steps like this (show walking along the shore and counting paces).

See what I mean? Your body is an amazing source of non-standard measuring units. I encourage you to go exploring some interesting place in your community--maybe your schoolyard or a park--and measure just like your great great great great great great great grandma did. Oh...and don't forget to take your body with you.