snips and snails and puppy dog tails: the need to preserve complexity in mathematics learning...
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Plenary address in 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) Conference Problem@Web | 2-4 May 2014 | PortugalTRANSCRIPT
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
solving-and-expressing
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
powercolectiverules
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
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
resourcesexternal 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
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-mediaand 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
/fa
ir
hig
h
visualizationto observing
visualizationto observing
and describing
visualizationto 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 CommunityDivision of labour
Object