Yishay Mor,

Maths Lunch, IOE, Dec. 2008

A Narrative Epistemology for Mathematics; Genetic, Normative and Imperative

Agenda

• A model of narrative learning

• Three modes of epistemology

• Initial results and research questions

A possible path from experience to knowledge

Narrative :: something happened to someone in some circumstances

Narrative has..

• Context (setting, situation)

• Plot (sequence of events)

• Protagonist (subject, hero)

• Implicit Moral (endpoint, purpose)

• Genre

• VoiceMy focus is on narrative as a cognitive / social epistemic vehicle:A means for making sense of the world.

From body to story

PerceptionAttention

Selection

Sequencing

Linguistic & Neuro-cognitive evidence

• Readers retrieve personal memories while reading stories• Readers construct event-indexing models which track 5

dimensions: temporal, spatial, causal, motivational, and person/object

• Reading activates broad functional webs that are also activated when the referent is experienced.

• A number of identical areas appear to be involved in both narrative comprehension and narrative production.

• It is proposed that selection and causal-temporal ordering may underlie this construction and commonality.

Zwaan & Radvansky, 98; Mar, 04

Translated

Narrative construction / comprehension involves the reconstruction of experience as communicational nodes in a semantic model of events, which represents actors, objects, intentions and actions, and the temporal, spatial and causal relations between them.

Once experience is narrated, it is linked to a web of previous experience, and becomes an ever shifting concept.

Brain metaphors..

Fusing Narratives

My Swan Your Swan

Swancept

Paradigmatic

Imaginative Imaginative

Genre transposition

I saw a white swan

I saw a white swan

Swans are white

Ok, but what has this got to do with learning mathematics?

Learning = creating knowledge

• How does Johnny come to know?

• How do we know what Johnny knows?

• How do we increase the chance that Johnny will know?

Epistemology

/ipistimollji/

noun the branch of philosophy that deals with knowledge, especially with regard to its methods, validity, and scope. (Compact Oxford)

The branch of philosophy dealing with the study of knowledge; theory of knowledge, asking such questions as "What is knowledge?", "How is knowledge acquired?", "What do people know?", "How do we know what we know?". (Wiktionary)

Three Epistemologies

• How does Johnny come to know?

– Genetic epistemology (Piaget)• How do we know what Johnny knows

– Normative epistemology (for lack of better name. Aka scientific method)

• How do we increase the chance that Johnny will know?

– Imperative epistemology

Example: genetic

I predict that when I add up the totals they will start at 0.5 and increase by smaller and smaller amounts without getting to 1. I think this because the first number in my sequence is 0.5. This takes me half way to 1. To get to 1 I would need to add this again but I only add half of this, 0.25. This means that each time I only add half of what is necessary to get to 1 so I will never get there.

I think the line of the graph will look like the graph above upside down or the one underneath but not so wobbly!

Questions

• How do we read mathematical texts as narrative, and what do we gain?

• What neuro-cognitive evidence do we need to support the above model, and do we need it?

• What's missing?

Example: normative

19

Design patterns

[describe] a problem which occurs over and over again in our environment, and then describes the core of the solution to that problem, in such a way that you can use this solution a million times over, without ever doing it the same way twice(Alexander et al., 1977)

C o n t e x t

Problem Solution

Tell me a story Situation

Set the scene (I wasn't there)

Task What problem where you trying to solve?

Actions What did you do?

Results What happened?

Reflections

The three hats

Work in small groups One tells a story, second writes it down, third

presents it.

What do you see?

After a case story is presented, ask the audience to identify the primary points from their perspective.

What is the key message you take from this story?

Make it a pattern

C o n t e x t

Problem Solution

When, Where, Who

Collision of forces

Cookbook: ingredients, procedure, expected

outcomes

Questions: normative

• How do we leverage narrative and maintain validity, reputability, predictive power?

• Does the design pattern approach offer an answer?

• How do we turn design patterns into scientific narratives? Do we want to?

Example: Imperative

Example design pattern: objects to talk with

Learning activities involve the use or construction of artefacts. When providing tools for learners to discuss their experience ... allow them to include these artefacts in the scope of their discussion. [...]Whatever the nature of the objects, the medium should support a visual (graphical, symbolic, animated or simulated) 1:1 representation of these objects.

Questions: imperative

• What is specific to mathematics? (what is mathematics?)

• How do we leverage the imaginative without loosing the paradigmatic?

• How do we create conditions for the paradigmatic to emerge?

• How do we systematise it all?

http://xkcd.com/55/