Just getting people together with a prompt works great - with friends! I've tried it many times, with such mixed results that it's not something I would do again in a random group, not anymore. At least in math, many people are way too anxious for anything of that nature. And others simply prefer less openness. However, with a bit of scaffolding it may work. For example, you can tell people to write down three phrases that first come to mind when they look at the butterfly and larva, and think about learning. But you need to prepare some other scaffold for when they have the phrases. For example, they can tape their phrases to a whiteboard and create bridges to someone else's phrases.
Some quotations:
Chapter 4 The Development of Reasoning by analogy by Usha Goswami
From "The Development of Thinking and Reasoning" edited by Pierre Barrouillet and Caroline Gauffroy Psychology Press.
“Reasoning by analogy can be considered a core component of human cognition. Analogy is important for learning and classification, for thinking and explaining, and for reasoning and problem solving. The history of science demonstrates the role of analogy in key scientific discoveries. Indeed, many argue that reasoning by analogy is central to creative thinking (Holyoak & Thalgard, 1995). The contribution of analogical reasoning to cognition has been defined in various ways, including Holyoak and Thalgard’s concept of “mental leaps”. ...argue that even the simplest analogy involves a mental leap, as it requires seeing one thing as if it were another. The reasoner must make a ‘mental leap, between domains that may not previously have been connected. These mental leaps are often evident in famous analogies in the history of science.”
On to Keith Holyoak and Paul Thalgard 1995: “Mental Leaps: analogy in creative thought” MIT Press 1996
Chapter 1
“When we are young, before most of the familiar patterns of everyday life have been learned...Knowledge of the world awaits construction. For the child the pool of known situations is still small, and novelty is the norm; so much has to be understood with so little. But already the fundamental thought processes that guide the creation of understanding are hard at work. Perhaps it would be more accurate to say they are hard at play, since for a child (as for a scientist), understanding the world often becomes a game driven by natural curiosity.
Consider the following discussion between mother and her four-year-old son, Neil, who is considering the deep issue of what a bird might use as a chair. Neil suggested, reasonably enough it would seem, that a tree could be a bird’s chair. A bird might sit on a tree branch. His mother said that was so and added that a bird could sit on its nest as well, which is also a house. The conversation went on to other topics. But several minutes later, the child had second thoughts about what a tree is to a bird: “The tree is not the bird’s chair—it’s the bird’s backyard!”
In this conversation Neil makes a mental leap, exploring connections between two very different domains. He is trying to understand the relatively unfamiliar world of creatures of the air in terms of everyday human households. This small example conveys what we mean by reasoning by analogy, or analogical thinking. The child’s everyday world is the source analog: a known domain that the child already understands in terms of familiar patterns, such as people sitting on chairs and houses that open onto backyards. The bird’s world is the target analog—a relatively unfamiliar domain that the child is trying to understand. Analogical thinking is not “logical” in the sense of a logical deduction—there is no reason why birds and people should necessarily organize their habitats in comparable ways. Yet the analogy is certainly not haphazard. In a loose sense, there is indeed some sort of logic—call it analogic—that constrains the way the child uses analogy to try and understand the target domain by seeing it in terms of the source domain.” P.2
Scott, what are you thinking about the connections between these two quotes on analogies and the above discussion? I don't think I see any direct bridges. Please help me make that "mental leap" as the second story has it!
Hi Maria,
When I first noticed these quotes there was a very strong feeling in my mind that they were connected. Now that you ask I’m not sure about the relationship here. This often happens in thinking about how things relate to each other, the link, at first strong, turns out to be too weak to hold.
Been thinking a lot about how problems are solved and the tools needed to accomplish this. In the young boy I see the development of understanding that is impermanent and will develop as he develops. And this brings up the idea that decisions, assumptions, strategies, whatever are not end states but stops on a path. We start with the evidence as best we can from what we have, and build on that to finer and finer understandings.
At the beginning of this is my looking into diagnostic training for mechanics and how mental leaps can as often lead to miss-diagnosis as to correct diagnosis. I see a connection between emergence and diagnosis but struggle with there being more than one correct answer as in there being wrong’ish or right’ish but not perfect solutions.
The more I try to explain this the deeper in trouble I get:-)
Scott, the discussion of "right-ish" or "right-er" solutions sounds promising to me. In mathematics and sciences, even with open problems, you better have criteria for judging right from wrong. Which brings me to what I want to discuss: analogies vs. metaphors. I think the boy and the bird quote is more about metaphors than analogies. And here's one more quote:
“I cordially dislike allegory in all its manifestations, and always have done so since I grew old and wary enough to detect its presence. I much prefer history – true or feigned– with its varied applicability to the thought and experience of readers. I think that many confuse applicability with allegory, but the one resides in the freedom of the reader, and the other in the purposed domination of the author.” -J.R.R. Tolkien,
Even in emergent designs, we do need scaffolds and structures, common WHYs, and other ways to bring participants to a particular contextual neighborhood. But I think it's better not to exercise purposed domination all the time; and when we do, it has to be consensual ;-)
My hope: footprints and other mappings as tools of freedom and consent. These tools address one of the biggest pro-coercion arguments in education: that learners don't know pedagogy and design, don't know what's good for them. Learners who are aware that activities come in different types can make more informed choices about learning.
Maria ...
The JRR Tolkien piece on the 'degrees of freedom'/ agency in applications (high) and allegories (tends to zero) is very instructive. And from the footprints that we and many others have created so far, and taken into a range of conversations, its clear that learners have a keen, instinctual feel for pedagogy (or heutagogy - if you must), and a keen, instinctual aversion to the abstractions of pedagogical discourse.
In the Nested Narratives project (which overlaps with footprints to some extent - see the link to the BJELT paper, here, for details) we got students to engage in complex layers of reflective practice (using prompted narratives), which they loved, but only because they learnt reflective practice by doing, not by instruction. All we told them was that we wanted them to tell us stories about something they had learnt which was important to them - no mention of pedagogy, and we deliberately did not mention 'reflection', as they had told us they would tell us stories only on condition that it would involve no reflection, and no critical thinking - both of which they found deadly boring.
So ... why are we constructing a footprints tool that is immersed in abstract pedagogical discourse? - a good question. (Best answer I have, and its a poor one, is that we started off this developmental process as researchers, exploring the wonders of pedagogical, psychological, neurological, ecological, sociological, etc discourses - particularly me - I love the stuff!).
However ... tools of 'mapping and consent' (involving real choice) must ensure that there are some clear entry points into 'doing' footprints that are unencumbered by the requirement to first learn pedagogical abstraction - alongside entry points for the specialists who want to engage through (and with) pedagogical abstraction too, no?
Barb - if you are folllowing this thread, this might resonate with the concerns you raised about course designers who demand that participants 'learn (the technical, software skills) by doing' alongside learning the content and conceptual tools of the course itself. Food for thought.
The discussion of mapping in absract or lay language reminds me of "The hundred languages of children" from Reggio Emilia. There is no reason to ostracize a minority language, such as tech terms from a research field, as long as it's not mandatory for everyone. But we need some universal translators or Babel fish. Diagrams may be more universal than words.
Love the anatomy of Babel fish - and yes, we do need universal translators, and yes, no need to ostracise the strange language of academics.
Jutta (in Austria) will be running her next workshop using some of the images in the new mapping sheet - although I dont know in detail how she want to use them - it'll be interesting to see, and interesting to find out if the images 'travel' or not.
Maria,
Right’ish and wrong’ish suggest to me a spectrum of possibility that might be free of the need for being exact. Not an estimate but a range of working possibilities.
Not knowing much about math doesn’t help me here, though I do sense within the logic of math is the permission to approach problems from different, but equally valid solution strategies. Not that any old thing is right only that “right” has some signature characteristics of being got to that a procedure was followed. Not necessarily the “proper” procedure—more like a defendable series of decisions that lead to the answer. For instance the “birds backyard” statement is arrived at by a repeatable process that might yield clues to its reason for being an answer that is legitimate? As might emergent artifacts leave evidence of how they came to be?
My definition of allegory would be a completed object of thought that has a point to make. An allegory is a completeness beginning with a rhetorical problem that is “solved” by a culturally biased correct answer. I think of it as a resolution pre-made to prove the point of itself that exists in a closed logic. 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 beyond guessing that an open universe is more enticing to the intellect over a closed universe where everything has an unchanging final condition.
Do you think an analogy is more restrictive than a metaphor? That an analogy is an object of dominance, or at least an expected outcome that is correct at the level of approval by others over a metaphor being a raw outcome of the thinker’s mind? A decision unique to the thinker?
This is getting kind of ‘out there’ but thinking about it has brought up the realization (right, wrong or something else) that studying humans as individual processing units is not the path to understanding or explaining uniqueness. A social beings we have antenna finely tuned to a field of fertile suggestions evidenced somehow in our choice of what we mark as discovery or significance. We are referential, knowing and doing among the presence of others while also being consensual in having to convince ourselves to believe. As well as permitting or inviting others to teach us to the depth of our being.
Does any of this work? The idea of “the learner” is so complicated.
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.
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.
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 ...
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.
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.
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!
Scott, your question about emergence and diagnosis ...
Might it be that although the process of intervention - in mechanics - might be very specific, with little room for the luxuries of 'emergent thinking', the process of arriving at a diagnosis (and even tentative prognosis) might be of a different nature, and might involve tacit, fuzzy, decisions about not what specific interventions to employ, but what metaphors or homologies to employ in thinking about the diagnosis.
This reminds me (I hope this is pertinent) of a piece of research into mis-apprehensions in problem solving in science, in which the root of the mis-apprehensions was in the misapplication to either a calorific or a Brownian conceptualisation of heat, which led to 'bugs' in thinking through the problem, and that in turn led to an inability to think through a strategy to achieve the solution. It was only when the two frameworks were made explicit, and discussed openly, that both students and the teacher realised that the two frameworks / metaphors / models were appropriate to different kinds of context.
Roy,
Wonder if I'm substituting emergence for interruption in diagnosis. Many diagnostic strategies seem to rely on assumptions of causality that need to be questioned or 'interrupted' or displaced from their comfortable placing.
I like the idea of there being alternate successful paths or proceedures to reach a viable solution and agree that making them explicite and discussed openly is critical. Moving from one decided thing to anouther until a final decision is come to isn't diagnosis as far as I'm concerned. I've seen unsuccessful diagnosis done from a manual without thought because it was considered the right proceedure.
How would we force ourselves to openly observe the evidence without constantly coming to conclusions about it?
By learning to live on (and love) the edge of chaos.
No, that's too easy. Avoiding jumping to conclusions is a prerequisite for learning, and for many unusual diagnoses, no?
A favourite of mine: in my (brief) days as a medical student, I worked in a rural hospital, including a Leprosy unit - in which 'feeling no pain' is the key evidence for the disease (nerve degeneration leads to loss of sensation), which is why it is missed so often by doctors who don't normally see cases of Leprosy, and 'can't imagine' why NOT feeling pain would ever be a key symptom for disease.
So ... the rarer it is, the more difficult it is to diagnose.
Roy, jumping to conclusions is definately a bad habit in diagnosis. In your example 'feeling no pain' is also improprly placed in the brain as a category wellness and unnecessary of further investigation. I imagine it as a kind of switching phrase to relieve obligation to act.
My cardiologist's booking clerk likes 'no news is good news.' The problem is, she's busy and can't follow up to see why I never received the news that was bad news.
Excellent! Just the right kind of 'light-touch' scaffolding. Thank you so much for pointing the train of thought in this direction.
Let me see if I can take it further ...
Bringing their own associations and experience in, to add richness to the images is a very sound workshop technique. It might take longer, but it would be so much more grounded in their own sense-making, rather than asking them to 'detour' through the obscurity of pedagogical theory (which very few people have a taste for - even academics, lets be honest ...) and trying to use such abstract terms for their own sense-making.
In other words it might answer your and Nick's challenge to make the 'ivory tower thinking' more accessible - not by editing and reconfiguring it, but by destroying it, and getting the learners to build their own versions of the mapping sheet instead, just from the images. Wonderful idea!
So ...
If we go back to your design principle of always making a range of degrees of freedom available (and perhaps add a range of modes too), we could offer mapping sheets based on:
- Scoring (we started with a scale of 1-30 (and up to 40 for factors that fell off the edge of chaos). We later decided to move on, beyond 'scores', to emphasise the push and pull of the factors (as dynamic vectors, rather than discrete scores), but why not keep them available for people who want to use them?
- Mapping sheets:
- Our current 3.0 version, with graphics and text
- A new version, 4.0, with just graphics - but with collaborative exercises to generate associations and texts and more images (maybe a range of images for Risk, for example, from low to high risk, or comfortable to uncomfortable risk?) We could do this with graphics selected from the web, or created in collages, etc.
- A version 4.1, with graphics and factor titles only (perhaps a bit simplified, if possible)
That could be crazy and unmanageable, but it might allow precisely the range of degrees of freedom, and the range of light to heavy scaffolding which could provide an inviting entry point for just about everyone. It might increase the workshop team though, to support all these options. Mmmmm.
This might wreak havoc with comparisons between footprints, but perhaps not - if the factor points are kept in the same places, and if there is a reasonalbe commonality of meaning across various people's and various groups' self-generated factors, the footprint 'gestalt', comparisons between footprints, and collaborative conversations about them would still work.
And of course one of the underlying principles, namely that the footprints are visualisation tools for the learners (and designers and teachers) to explore their OWN experience of the learning, would be fully aligned with the practice of asking them to flesh out, and enrich their OWN sense-making of the factors.
Any thoughts?
Yes, people need to bring - or to make - their own content. As you explain, the main challenge in such designs is to integrate everybody's experiences into a coherent whole. We want to avoid the feeling of Babel - everybody speaking a different language!
How can some common analysis of big WHYs happen? How can I look at your footprint and you look at mine in ways that we can see similar big SO WHATs? Your footprint looks like a butterfly and mine has a big protuberance in the southwest - so what does it mean for the compatibility of our dreams and beliefs?
Most citizen science has citizens collect data, which is then analyzed back at the ivory tower. How can citizen scientists analyze data, too? Say, how can parents in my math clubs compare notes on how they help their five-year-olds with scavenger hunts for nonlinear functions? One parent helps by providing examples, another by asking questions, another by dispersing hugs. Different footprints; how can people analyze results together?
How can citizen scientists analyse data (or should we say create data) too? mmmm...Your examples of comparing notes (on asking questions, providing examples, and dispersing hugs) is a great start. Let me try and think it through a bit further ...
So ... if we can provide the tools (graphics, free association exercises - what words, what texts, what images, what sounds, what smells do you associate with this graphic, when you think about what happened when you were learning? -
... or vice versa: we could ask: What graphic would you associate with "risk", and the way it affected you and your learning? - and if we could provide an app for them to replace the label 'risk' with a thumbnail of their own graphic for risk, which they could double click to expand (and reduce) it, so that they could show it to others in a discussion, and tell the story of how and why they chose it, and what 'risk' means to them)
plus: ... we could provide different degrees of freedom for low, medium or high "risk': i.e. i) select from a given, small, set of images, or ii) find their own images, or iii) make their own images - in collages, in new graphics, etc.
and if ... participants (learners, designers, teachers) created footprints on that basis -
Then we could invite them to join in a conversation in which they could all compare notes: gestalts and icons and stories about how they arrived at their particular footprints.
They could do this on paper. It would be more fun and more interesting it they had ipads, or laptops, or mobile phones (?) or interactive tabletops, sure, but what a conversation it could turn out to be!
And ... we could add an app that allowed participants to i) superimpose, aggregate, etc, their footprints with other footprints or ii) play-back a series of footprints created at different stages of a learning event, to show the underlying narrative dynamics?
and so on ...