Constructing Knowledge
through the
Perceiving of Affordances
of a Technology-Rich Teaching and
Learning Environment (TRTLE)
Jill P Brown
Constructing Knowledge through the Perceiving of Affordances of a Technology-Rich Teaching and Learning Environment (TRTLE)_jill brown_1
Affordances
Following
Gibson (1979) who made up the term to explain what is
perceived in a particular environment, and
Scarantino (2003), whose more recent elaborations of the
construct supports its interpretation in an educational
setting
Affordances of a TRTLE are
the offerings of the TRTLE for facilitating and impeding
teaching and learning.
Potential relationships, involving interactive activity,
between the teacher and/or students and the
environment.
Following Gibson, the environment includes both animate and
inanimate objects - in a TRTLE this includes technology.
Constructing Knowledge through the Perceiving of Affordances of a Technology-Rich Teaching and Learning Environment (TRTLE)_jill brown_2
Affordances
To take advantage of opportunities arising, the animate object (T or
S) need to perceive the affordances and act to facilitate learning.
Specific objects within the environment, (i.e., the TRTLE), that
enable an affordance to be enacted are the affordance bearers.
The manifestation of an affordance in a TRTLE in background
circumstances [C] involves an event [E] in which both the
affordance bearer [AB] and the actor [A] are involved.
the affordance, multiple represent-ability, is manifest when a
student investigates with a graphing calculator the information a
set of numerical data provides about the structure of the
graphical representation of the data using StatPlot, ZOOM and
WINDOW, given the relevant data set was entered into the
graphing calculator Lists from where it could be plotted.
The affordance is an opportunity that involves both the affordance
bearer and the actor. Although the technology always affords
multiple represent-ability, one must perceive the affordance in order
to act.
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Valsiner’s Zone Theory
“in the case of the developing child’s relationship to the
environment, the question of how effectivities [perceived and
enacted affordances] emerge in the child’s actions on the widely
affording environment is of special interest.” (Valsiner, 1984, p. 67)
He elaborated noting that what is possible “need not automatically
induce changes in the child’s actual behaviour [that is] need not
trigger the actualisation of these new opportunities.”
Expanding on Vygotsky’s ZPD, he proposed two additional zones
to describe the ever-changing structure of “the environment of
the developing child” (p. 186).
They describe the structure of the relationships between people in
the environment and also in terms of regulating an individual’s “own
thinking, feeling, and acting” (p. 188).
These zones “are always temporary, constantly changing structures
that organise the immediate construction of the future state out of a
here-and-now setting” (p. 319).
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Valsiner’s Zone Theory
The Zone of Free Movement (ZFM) - what is possible
Within the field of objects and affordances related to them in the
environment of the child, the zone of free movement (ZFM) is
defined for the child’s activities. The ZFM structures the child’s
access to different areas of the environment, to different objects
within these areas, and to different ways of acting upon these
objects. (pp. 67-68)
The ZFM describes what is possible at any given time in a
particular environment.
In TRTLE’s this includes affordances, allowable actions, and
technology and other learning artefacts available to a student
acting in the TRTLE at any given time.
Constructing Knowledge through the Perceiving of Affordances of a Technology-Rich Teaching and Learning Environment (TRTLE)_jill brown_5
Valsiner’s Zone Theory
The Zone of Promoted Action (ZPA)
“illustrates the direct efforts of the people around the
[student] to guide his or her actions in one, rather than
another, direction” (1997, p. 317).
“oriented toward the promotion of new skills” (p. 192).
Importantly, particularly in a secondary school environment
the ZPA is non-binding in nature.
Hence, the ZPA describes “the set of activities, objects, or
areas in the environment, in which the person’s actions are
promoted” (p. 192) by the teacher, or other students, in the
TRTLE as they attempt to direct others’ “actions in one,
rather than another, direction” (p. 317).
The two zone concepts should be considered as a complex
working together “by which canalisation of children’s
development is organized. This applies equally to the
development of children’s actions and thinking” (p. 195).
Constructing Knowledge through the Perceiving of Affordances of a Technology-Rich Teaching and Learning Environment (TRTLE)_jill brown_6
The Construction of Knowledge
The focus here [drawing on the theories of affordances and
zones] in on the construction of knowledge by one student
and the subsequent use of this student’s conjecture by the
teacher to provide opportunities for the co-construction of
knowledge by others in the class [n = 19, L21 of 35].
A ‘real’ classroom, Yr 11 Mathematical Methods,
Study of functions [linear, quadratic, non-example -circles],
Transformation of functions had not been a focus
Data presented will illustrate the construction of knowledge by
one student who when exploring a new function type was able
to recognise from given sets of numerical data that the
relationship between the two functions was g(x) = f(x) + 3,
that is the translation of the initial data set 3 units vertically up
was equivalent to the second data set and this subsequently
led to the student conjecturing, testing and correctly verifying
the algebraic representation for the second data set.
This knowledge was not directly connected to previous lessons.
Constructing Knowledge through the Perceiving of Affordances of a Technology-Rich Teaching and Learning Environment (TRTLE)_jill brown_7
The Construction of Knowledge
Peter: Okay I want you to look at this data, … Your task
is very simple for Part A. There is a piece of graph ZPA - promoting
investigative
paper. Would you see if, as a group, with an
whoever is beside you, if you can come up in the approach
next three or four minutes [with] what the graph
looks like. Have a look for A, what the graph
looks like. This is a situation where you haven’t
been given the formula first. You haven’t been
given just any point, you have been given a whole
lot of points there and obviously they are related,
in this case, to one particular function. Your task
is, what do these points tell you about the shape
or the structure of the graph? [L21, 11Mar05]
Constructing Knowledge through the Perceiving of Affordances of a Technology-Rich Teaching and Learning Environment (TRTLE)_jill brown_8
The Construction of Knowledge
Marc:
Peter:
…
Peter:
Marc:
Peter:
Can you use a calculator?
How are you plotting them?
Do you need every number?
No. You need an equation.
Why?
ZPA - Marc wants
clarification with
regard to exactly
what was being
promoted
What was Marc thinking?
Previously, on many occasions the students had
entered data and plotted this. [See table]
Affordance: Represent-ability [data plot-ability]
Constructing Knowledge through the Perceiving of Affordances of a Technology-Rich Teaching and Learning Environment (TRTLE)_jill brown_9
The Construction of Knowledge
Constructing Knowledge through the Perceiving of Affordances of a Technology-Rich Teaching and Learning Environment (TRTLE)_jill brown_10
The Construction of Knowledge
Peter:
Well come and show us what you have got,
put it on the board then …. What have you
found?
Tony: It would look like that [Draws hyperbola in
the air, then goes to the board and draws
it.]
Student: Two quadrants.
Peter: Someone said it is only in two quadrants.
How did you know it is only in two
quadrants?
Tony:
Because it only has negative-negative values
and positive-positive values.
Peter: You noticed it had a whole group of
negative-negatives to start with?
Student: [as if noticing for the first time] Oh yeah.
Peter then used
Tony’s input and the
numerical data
provided to
facilitate an
extended discussion
of the shape of the
graphical
representation of
the data.
The discussion concluded with the teacher noting: “So this
equation here, you are just dividing by the x’s”. He recorded on
1
the board, y 
.
x
Constructing Knowledge through the Perceiving of Affordances of a Technology-Rich Teaching and Learning Environment (TRTLE)_jill brown_11
The Construction of Knowledge
The class then considered the second set of data
At one point, Tony was again called upon to explain his thinking.
Tony: It kind of just looks like the same graph as before, just
moved up.
Peter: Is it just the same graph? What do you think?
Tony: Yes.
At this point the teacher used Tony’s
thinking of the class along similar lines.
ideas, by focussing on various pairs of
ensure other students were making sense
notion to promote the
Peter elaborated Tony’s
points, in an effort to
of Tony’s reasoning.
Peter: If it is the same graph as before, moved up, if it is this, this
is the statement he said. This is what we had before. If it is
just moved up and this graph is, how do you move the
whole graph up?
Although the discourse that followed saw Peter scaffolding the class it
appears that Tony was in his ZFM as he worked on the task without
teacher direction.
Constructing Knowledge through the Perceiving of Affordances of a Technology-Rich Teaching and Learning Environment (TRTLE)_jill brown_12
The Construction of Knowledge
Tony was again called upon to explain his thinking, first he explains
his actions in response to Peter’s telling of the function equation,
that is, he had earlier entered the algebraic representation of the
first function, given by data set A and checked that this curve
passed through all the points he had previously entered.
Tony: I just put in
x 1
to get your graph the first time.
Peter: Hold it, I want to write it down. You put in what?
Tony: y  x 1. That gave me the graph in the first part.

Verify-ability
 Subsequent dialogue showed that he not only looked at the
graphical representation of the entered algebraic function, but he
also checked numerically by observing the numerical values of the
function [in the TABLE] and compared some coordinate pairs, at
least for x = -1 and x = 1, to those provided by the teacher
initially.
Verify-ability [graphically]
Verify-ability [numerically]
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The Construction of Knowledge
Tony elaborates …
Tony: Then I looked at the -1 value, which is 2, and I looked at the 1
value, which is 4.
Peter: So you went to your TABLE and you checked which one.
Tony: When I checked the TABLE originally, I looked at the 1 and -1
values. I saw what they were.
Peter: [Speaking to whole class.] Go to the first graph, A, look at -1.
y = -1. This is what you are saying.
Peter draws classes attention to Tony’s thinking
- making
connections from one function to the next
Peter: He looked at (-1, -1) [on A] and saw what happens on B
Tony: On B, it has gone through (-1, 2). And here [data set A] it has
gone through (-1, -1).
Peter: Okay … between the 2 graphs, -1 went up to 2. So you said
there must have been a change of?
Tony: Three.
…
Peter: [addressing the class] Does it work for every point? Does
everything go up by three?
Constructing Knowledge through the Perceiving of Affordances of a Technology-Rich Teaching and Learning Environment (TRTLE)_jill brown_14
The Construction of Knowledge
Finally, Tony shares his conjecture …
Peter: Oh by the way Tony, let us keep going with what you are
saying. … then you said it was?
1
Tony: y  x  3 .
Tony verifies his conjecture …
Tony: And then I just checked the TABLE, checked to make sure I
was right.

Peter: So you went to Second TABLE, and checked the values?
Tony: Yes.
Subsequently, Peter used Tony’s conjecture to focus the attention
of the class on the differences between related pairs of values for
the 2 function with the intention of supporting their construction of
knowledge, namely that if from a graphical perspective g(x) can
be seen to have the same shape as f(x) only shifted up three units
in the vertical direction, or from a numerical perspective the value
of g(x) is always three more than the value of f(x) for any given
value of x, then from an algebraic perspective g(x) = f(x) + 3.
Constructing Knowledge through the Perceiving of Affordances of a Technology-Rich Teaching and Learning Environment (TRTLE)_jill brown_15
The Construction of Knowledge
The ZFM described here is in part a consequence of previous ZPA’s.
Although at times Peter was promoting a particular path [local ZPA],
Tony took the teacher’s initial challenge ‘to investigate’ to mean he
should follow whatever path his mathematical thinking took him, as
facilitated by the TRTLE of which he was a part.
What did Tony did do? He chose to check, rather than simply accept,
the mathematical knowledge shared by the teacher. This was possible
as a consequence of the current and past ZFM/ZPA complexes in this
TRTLE.
Hence, Tony was able to engage in ‘meaning making’ rather than
accepting an unjustified statement provided by the teacher.
The constancy of the affordances available to him during this lesson of
interest and previously during the year, enabled his perception and
enactment of several affordances thus allowing his construction of
knowledge.
The affordances of this TRTLE - and the consequent opportunities to
operate within his Zone of Free Movement - be this in or outside the
current ZPA – allowed this opportunity for Tony to construct his own
knowledge. Consequently, the knowledge of the teacher became his
own.
Constructing Knowledge through the Perceiving of Affordances of a Technology-Rich Teaching and Learning Environment (TRTLE)_jill brown_16
Thanks
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