Constructing Knowledge through the Perceiving of Affordances of a Technology-Rich Teaching and Learning Environment (

1. Constructing Knowledge through the Perceiving of Affordances of a Technology-Rich Teaching and Learning Environment (TRTLE) Jill P Brown

2. 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.

3. 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 studentinvestigates 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.

4. 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).

5. 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.

6. 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).

7. 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.

8. 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 paper. Would you see if, as a group, with whoever is beside you, if you can come up in the 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] ZPA - promoting an investigative approach

9. The Construction of Knowledge ZPA - Marc wants clarification with regard to exactly what was being promoted • Marc: Can you use a calculator? • Peter: How are you plotting them? • … Peter: Do you need every number? Marc: No. You need an equation. Peter: Why? What was Marc thinking? Previously, on many occasions the students had entered data and plotted this. [See table] Affordance: Represent-ability [data plot-ability]

10. The Construction of Knowledge

11. 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 the board, .

12. 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 notion to promote the thinking of the class along similar lines. Peter elaborated Tony’s ideas, by focussing on various pairs of points, in an effort to ensure other students were making sense 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.

13. 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 to get your graph the first time. Peter: Hold it, I want to write it down. You put in what? Tony: . 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]

14. 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?

15. 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? Tony: . 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.

16. 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.

17. Thanks