Snips and snails and puppy dog tails: the need to preserve complexity in mathematics learning analysis

A Reply to “The Use of Digital Tools in Web-based Mathematical Problem Solving: different levels of

sophistication in Solving-and-Expressing” (Jacinto, Nobre, Carreira & Amado, 2014)

João Filipe Matos

jfmatos @ ie.ulisboa.pt

Instituto de Educação, Universidade de Lisboa

Conference Problem@Web | 2-4 May 2014 | Portugal

(mathematical) situation

mathematizing making explicit

an external representation

solving-and-expressing

solve & represent

solve & make explicit

expressiveness

•  do children tell the story within the

framework of the audience (school

mathematics, teachers)?

solve-and-express as mathematizing

•  is solving inherent to expressing?

•  what consequences do emerge from that

point of view?

•  how powerful is digital technology in

solving-and-expressing?

how digital technologies constitute human activity?

•  all artifacts enable and constrain human

activity

a need to enlarge the research object

•  consciousness of how digital technologies

constitute human being and activity

•  mathematics as human activity can not be

addressed without considering the ways how

digital technologies constitute practices

•  a need to consider a broader perspective

the system ‘human-computer’ should be extended

(Engeström, 2009)

I/we object

computer

power colective rules

a need to enlarge the research object

•  why isolate the system human-computer?

•  consider the whole activity system

…trends in HCI

a need to enlarge the research object

•  the tool mediating perspective but a need

to conceptualize 2 interfaces, not 1:

•  2 borders separating:

1st ) the user from the computer

2nd ) the user and the computer from the

outside world

PERSONAL

SPACE

co-action as 2 boundaries

CENTRAL

COGNITIVE

STRUCTURES IN

TE

RN

AL

PL

AN

E

OF

AC

TIO

NS

DIG

ITA

L T

OO

L

OUTER

WORLD

SUBJECTIVE

REALITY

a need to enlarge the research object

•  where is the boundary between the individual

who uses the tool and the external world?

•  does the boundary coincide with the individual-

tool boundary or with the tool-world one?

subject-technology technology-mathematics

a need to enlarge the research object

•  The answer from AT: concept of functional organ (Leont’ev)

functionally integrated, goal directed

configurations of internal and external resources

external tools: support and complement natural human abilities in

building up a more efficient system that can lead to higher

accomplishments

a need to enlarge the research object

external tools: support and complement natural human abilities in building up a more efficient system that can lead to higher accomplishments

efficient cutting organ improve human vision enhance memory

external tools integrated into functional organs are experienced as a property of the individual (improving the transparency of the interface)

why functional organs?

•  extension of the internal plane of actions (human

ability to perform manipulations with an

internal representation of external objects before

starting actions with the objects in reality)

PERSONAL

SPACE

co-action as 2 boundaries

CENTRAL

COGNITIVE

STRUCTURES IN

TE

RN

AL

PL

AN

E

OF

AC

TIO

NS

DIG

ITA

L T

OO

L

OUTER

WORLD

SUBJECTIVE

REALITY

why computerized systems / digital technologies?

•  computerized systems (digital technologies)

easily create controllable models of target objects

and provide opportunities to evaluate and

manipulate

•  what are the needs that require the development of a new functional organ (for problem solving)? what is the structure of students’ activity before the assimilation of the tool and what is the previous experience of students with the uncomputerized equivalent of the activity?

the symbiotic unit human-with-media and co-action

•  visible through co-action

•  but bringing in the context

… echoing the concept of digital habitat

… and distinguishing ‘context’ from ‘setting’

GeoGebra

(Jacinto, et al., 2014)

Example 1: humans-with-GeoGebra

•  3 scripts emerge from data

•  but do not unveil the associated cognitive

processes involved in mathematical

thinking for the production of the solution

Spreadsheet

(Jacinto, et al., 2014)

Example 2: humans-with-spreadsheet

•  connecting 2 forms of representation

•  the spreadsheet allows ways of expressing

functional relations but at the same time it

hides mathematical thinking (as it reifies

it)

co-action

•  conceptual models or forms of resolution

that are sophisticated are associated to a

stronger co-action with the spreadsheet

co-action

•  the relevance of co-action with the digital

artifact is the criteria used to define the

degree of robustness (visible in the level of

sophistication of solutions)

co-action

•  it is not enough to distinguish between different ways of addressing a problem and

solve it and to consider the emergent

conceptual models; we need to understand

how different conceptual models are

produced and materialized in relation to the form of use of he digital artifact

robustness of solutions/models

low

sig

nific

ant

/

fair

hig

h

visualization to observing

visualization to observing

and describing

visualization to imagining

de

gre

e o

f so

ph

istic

atio

n

robustness of solutions/models

observing describing imagining

degr

ee o

f sop

hist

icat

ion

weak sensible high

co-action

degree of co-action

•  Geogebra / Spreadsheet

•  is paper & pencil a zero co-action habitat?

•  what about web 2.0 tools?

criteria to assess co-action

… lack of criteria

•  robustness

•  simplicity

•  efficiency (metrics in space and time)

•  …

where to with geometric dynamic systems?

1) gaining insight and intuition,

2) discovering new patterns and relationships,

3) graphing to expose math principles,

4) testing and especially falsifying conjectures,

5) exploring a possible result to see if it merits formal proof,

6) suggesting approaches for formal proof,

7) computing replacing lengthy hand derivations, and

8) confirming analytically derived results

 

where to with spreadsheets?

•  Framing the problem,

•  Exploring the solution,

•  Conjecturing,

•  Justifying or investigating the conjecture

call for a conceptual framework for the analysis

“A conceptual framework is an argument including different points of view and culminating in a series of reasons for adopting some points – i.e., some ideas or concepts – and not others. The adopted ideas or concepts then serve as guides: to collecting data in a particular study, and/or to ways in which the data from a particular study will be analyzed and explained”

•  (Eisenhart, 1991, p. 209)

•  nature of the framework: structure and

justification

•  role of the framework: argumentative

network of analytical concepts

Learning

(as) Communication

Learning

(as) Communication (as)

Participation

(as) Transformation

micro level

institutional level

cultural-historical level

artefacts

Subject

Rules Community Division of labour

Object