Metaphors for emergent learning

Re: Metaphors - light - for emergent learning

by Maria Droujkova -
Number of replies: 4

Mathematics is very creative, in that you are allowed and encouraged to make things up from scratch. There are no restrictions about your creations matching the physical world (as in natural sciences), or being useful (as in engineering and technology). What you make up is not even required to have an aesthetic value; it just has to follow its own logic, which you are also welcome to make up. You can search the web for "algebras" or "geometries" (plural) to see pretty exotic stuff! But there are certain values and traditions within math as a human endeavor, such as consistency, precision, elegance, and so on. 

Escher Stairs

You write, "Even though math seems to me to be closed into a world of rules and proper procedures it somehow allows itself to not pre-judge outcomes. I don’t know why that is." The seeming contradiction is resolved by viewing math as an open, creative world. It can be an allegory or analogy for something, but at its heart it's metaphoric. This brings me back to metaphors vs. analogies.

Both are two-parter structures with sources and targets. But analogies have pre-determined, pre-judged, pre-solved target. In contrast, metaphor is a tool for generating your own targets. I like the social links in your last paragraph (not seeing humans as individual processing units), because other people's suggestions, references, cultures, etc. mediate targets of our metaphors. But analogies not just mediate - they prescribe. Which is perfectly fine in many cases, for example, when bringing up a novel context. Say, to start playing with a non-commutative algebra, I might use the analogy with unrequited love. If I play with a toddler, I would use a sillier analogy, like the chair sitting on you not being the same as you sitting on the chair.

In terms of footprints, metaphor is a near-boundary tool, while analogy is a near-center tool.

In reply to Maria Droujkova

Re: Metaphors - light - for emergent learning

by Roy Williams -

Elegant analysis, thanks. 

So, mathematics is an infitite set of transformational metaphor processes /structures /tools, sometimes linked to other (less abstract, bu possibly equally complex) practices?

My first insight into metaphor was from my philosophy lecturer, who said they are 'deliberate category mistakes' - precisely a near-boundary / edge of chaos tools, and his favourite metaphor (and one of mine too) is a fruity one, see here ... 

In reply to Roy Williams

Re: Metaphors - light - for emergent learning

by Maria Droujkova -

Neat example. As a funny aside (the example's not about it), in affluent societies, food is easier to come by than sex, so food usually serves as the metaphor source, and the relationship as a fantacized target. It can be the other way around in different circumstances. 

Yes to math as an infinite set of human-made processes, structures, tools, and practices, with everyone invited to make their own. However, if you don't adhere to past practices, others may not care about the math you make. As usual, it's harder for some populations (kids, women...) to change practices. Ironically, it's very important for kids' learning and development to create their own math in their early years - the time the world typically gives kids the least opportunity to do so. And in many countries, it's still harder for a woman to publish a math article than it is for a man.

In reply to Maria Droujkova

Re: Metaphors - light - for emergent learning

by Scott Johnson -

Maria,

The idea of things being consistent to some sort of logic or structure makes sense. Without structure we have a nonsense of no relationships or interactions. From what reading “The Work of the Imagination” by Paul L. Harris, even children in a state of pretending have rules and are sensitive to falsehoods that are inconsistent with the “story” they are enacting.

Left to their own without the attraction of structure how do we discover meaning? Where would it be in the scattering of ideas that don’t somehow connect? Structure seems different from orderliness. To be in order is just an arrangement or collection of things. They don’t touch or need to interact for something emerge because without connection they are lifeless. Structured things call to each other, intermingle and create newness.

Have to think of an example of what I’m saying.

In reply to Scott Johnson

Re: Metaphors - light - for emergent learning

by Maria Droujkova -

Here is an example of how I understood what you are saying about emergent structure with consistent rules vs. pre-determined order with pre-determined rules.

Emergent structure. A kid makes up a function machine. A number goes in, the machine does something to it, and a number comes out. I give the kid who built the machine numbers I want to try, untill I can guess what the machine does. 

  • 5 in, 10 out. Does the machine add five? Oh.
  • 2 in, 4 out. Does it double the numbers? No?!
  • 1 in, 2 out. 0 in, 0 out. 10 in, 20 out. I say it doubles! No? I give up.
  • Oh, it adds three, then doubles, then subtracts six? How neat! It works like doubling, but it's much harder to guess. What other function machines work like doubling?

Pre-determined order. Write down times two tables.

Here are a few more function machine examples from our Moebius Noodles book, including iterations of the doubling machine. Because when kids can easily double, they do it again and again and again: an emergent but predictable behavior!

Function machine examples