1. 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

3. (mathematical)
situation
mathematizing
making explicit
an external
representation
solving-and-expressing
solve & represent
solve & make explicit

4. expressiveness
• do children tell the story within the
framework of the audience (school
mathematics, teachers)?

5. 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?

6. how digital technologies constitute
human activity?
• all artifacts enable and constrain human
activity

7. 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

8. the system ‘human-computer’ should
be extended
(Engeström, 2009)
I/we object
computer
powercolectiverules

9. a need to enlarge the research object
• why isolate the system human-computer?
• consider the whole activity system
…trends in HCI

10. 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

11. PERSONAL
SPACE
co-action as 2 boundaries
CENTRAL
COGNITIVE
STRUCTURES
INTERNALPLANE
OFACTIONS
DIGITALTOOL
OUTER
WORLD
SUBJECTIVE
REALITY

12. 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

13. 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

14. 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)

15. 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)

16. PERSONAL
SPACE
co-action as 2 boundaries
CENTRAL
COGNITIVE
STRUCTURES
INTERNALPLANE
OFACTIONS
DIGITALTOOL
OUTER
WORLD
SUBJECTIVE
REALITY

17. why computerized systems / digital
technologies?
• computerized systems (digital technologies)
easily create controllable models of target objects
and provide opportunities to evaluate and
manipulate

18. • 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?

19. 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’

20. GeoGebra
(Jacinto, et al., 2014)

21. 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

22. Spreadsheet
(Jacinto, et al., 2014)

23. 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)

24. co-action
• conceptual models or forms of resolution
that are sophisticated are associated to a
stronger co-action with the spreadsheet

25. 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)

26. 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

27. robustness of solutions/models
low
significant/
fair
high
visualization
to observing
visualization
to observing
and describing
visualization
to imagining
degreeofsophistication

28. robustness of solutions/models
observing describing imagining
degreeofsophistication
weak sensible high
co-action

29. degree of co-action
• Geogebra / Spreadsheet
• is paper & pencil a zero co-action habitat?
• what about web 2.0 tools?

30. criteria to assess co-action
… lack of criteria
• robustness
• simplicity
• efficiency (metrics in space and time)
• …

31. 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

32. where to with spreadsheets?
• Framing the problem,
• Exploring the solution,
• Conjecturing,
• Justifying or investigating the conjecture

33. 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)

34. • nature of the framework: structure and
justification
• role of the framework: argumentative
network of analytical concepts

35. Learning
(as)
Communication

36. Learning
(as)
Communication(as)
Participation
(as)
Transformation

37. micro level
institutional level
cultural-historical level
artefacts
Subject
Rules Community
Division of labour
Object