## Tuesday, February 18, 2014

### Identifying the Qualities & Actions of Mathematical Objects [Brainstorming Help Needed!]

Hey math friends, I could use some help brainstorming! I've recently been inspired by Anna Sfard's 1991 article on reification as well as her subsequent related writings on metaphor and the "embodied character of abstract thinking." [Sfard 1994]  As a result, I am experimenting with an idea that I hope will further deepen my understanding about how we can create meaningful overlaps between math and dance in the classroom. As part of this experiment I am hoping to benefit from the insights of kind hearted math teachers and mathematicians. Later, I will take these ideas to the dancers and work the process from the other direction.

Here's some background: In late January I took to FB and Twitter to develop a list of math terminology that implies action in some way. "Action" can mean a few things. For example, iteration directly implies an activity or process. On the other hand, gradient refers to an object, but there was some kind of process involved in its creation.

I sorted through all the ideas and created a list of math objects and actions (below) and added some cursory definitions. What I am looking for is a deeper sense of how these ideas work and how they are applied.  More specifically, what are the quintessential actions, functions, roles, movements, or personalities of these ideas related to the contexts in which they are used?

Ideally, your answers will be based on your own experiences with these ideas in the classroom and in your work in mathematics. All branches of mathematics are welcome and, if you can think of other math ideas/objects/concepts that should be on this list, please feel free to add more! Also, feel free to pick and choose from among the items on the following list.

I would be so grateful for your help! Please feel free to comment, below, or to e-mail me directly at malke dot rosenfeld at earthlink dot net. If I can further clarify anything, please let me know.
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Note: Any definition with a (MN) next to it comes from the Moebius Noodles book Adventurous Math for the Playground Crowd.

Iteration (MN)
Applying the same operation repeatedly.

Transformation (MN)
Isometric transformation – the result of the transformation is the same exact size (as opposed to stretching or shrinking)

Function
(MN)
A machine that converts values to other values or finds correspondences between values. Function machines work by rules people make up. The starting values are called input. The converted or corresponding values are called output. The rule must find a single output for each input.

Operation (MN)
Operation is another name for function. It is used when we focus on the rule of the function rather than the inputs, outputs, domain, range (all possible outputs) and other aspects.

Ordering by a change in quantity: size, volume, or number

Algorithm (MN)
A step by step description of actions; reuse again and again in similar situations and share them with others.

Pattern (MN)
Repetition or systematic change.

Circumscribe
Literally means "to draw around". A circumscribed circle of a triangle for example is the circle that passes through all three vertices. (http://www.mathopenref.com/circumscribed.html)

Dilate
A transformation in which a polygon is enlarged or reduced by a given factor around a given center point. (related to scale) http://www.mathopenref.com/dilate.html

Qualitative
How much? What type?

Inverse
Reversed in order, nature, or effect.

1. I don't know if this falls under what you are thinking for transformation, but we do rotations when finding the measurement of angles. Seems very applicable to dance and movement. We talk about the rotation of person to think about the interior and exterior angles. Also, reflection on a coordinate grid to talk about symmetry feels like something in dance as far as mirroring movements of others.
-Kristin

2. shear, simplify, rationalize, distribute, divide (fair share & measure), join, compare, reduce, map, represent, differential, integral, group action, commute, relation, measure/quantify, construct, compose... that's about it for me!

3. Our young calculus writing group is working on the chapter about fractals right now, so fractals are on my mind. Let me try to apply your question there.

"More specifically, what are the quintessential actions, functions, roles, movements, or personalities of these ideas related to the contexts in which they are used?"

Actions and functions: splitting, scaling, substituting, iterating
Roles: motif (the element that you repeat), application places (points or areas where you situate the motif)
Personalities: the total perspective vortex - infinitesimally small, infinitely large, and all levels in-between presented for your enjoyment or horror in one easy-to-grasp image. Feelings expressed by the survivors: this is weird, spooky, beautiful, creepy!