Temporal Analysis: Measures

To generate my own State Space Grid, I needed to prepare two behavioural streams from my data: gaze and didactic behaviours. Gaze behaviours related to student gaze—which included focused gaze students (i.e., more than four key frames) and scans of students (i.e., four key frames or less) and non-student gaze—which consisted of student materials, teacher materials and other miscellaneous behaviours (e.g., window). Gaze behaviour codes are represented on the x-axis of the State Space Grid (SSG) as represented in Figure 7.10.

Figure 7.10. The ‘state space’ of teachers’ didactic gaze. The x-axis consists of five gaze behaviours; the y-axis of five didactic behaviours. Each cell is a one didactic event. The present thesis focuses on rows A and B, row A being communicative gaze and row B being attentional gaze.

Didactic behaviours included address behaviour (i.e., directly instructing students to change their behaviour), interacting (i.e., student or teacher asking and answering questions; ‘question’ in Figure 7.10), lecturing (i.e., teachers talking; ‘straight talk’ in Figure 7.10), refer notes (i.e., teacher referring to presentation slides or students’

resources), logistics (e.g., teacher moving the presentation onto another slide). Didactic behaviour codes are represented on the y-axis of the SSG as represented in Figure 7.10. Together, gaze and didactic behaviours combined to form didactic gaze.

A

State space grids were constructed using GridWare 1.15a (Lamey, Hollenstein, Lewis & Granic, 2004). To do this, observational data files were created for each participant for generating these grids. A 5×5 grid was generated, yielding 25 possible states as my state space grid shows 25 cells in total. Gaze behaviours were plotted along the x-axis; didactic behaviours along the y-axis. Each axis thus represented one

behavioural stream of the same individual; each cell represented the co-occurrence of their gaze and didactic behaviours. On each axis, behaviours were plotted from the most to the least people-oriented, so that the intersection of the two axes was the most people-oriented state (i.e., focused gaze at students vs. address behaviour). It was in this way that I strived to plot categorical variables together so that two behavioural streams coincide to become ‘states’. Together, the multiple states form the ‘state space’ of teachers’ didactic gaze.

Each cell of the SSG (Figure 7.10) represents a didactic gaze state. Such a state consisted of a co-occurrence of gaze behaviour and didactic behaviour. Two didactic gaze types are of central interest in this paper: attentional gaze was inferred from gaze

behaviours that occurred during interacting (i.e., within-questioning gaze); communicative gaze was inferred from lecturing (i.e., within-talk gaze).

For each, attentional and communicative gaze, I exported mean duration per visit for analysis. Accordingly, the static measures that I analysed were as follows: mean attentional student gaze duration per visit, mean attentional non-student gaze duration per visit, mean communicative student gaze duration per visit, and mean communicative non- student gaze duration per visit.

Didactic gaze attractors among all the teachers in my sample were identified—that is, the most prevalent and stable didactic gaze used across both, the UK and Hong Kong. Attractors can be interpreted as the most relevant teacher gaze: the more teachers use these, the more they are sticking to the task-relevant gaze, and the more efficient they are.

Attractors were estimated visually at first, using state space grid images (e.g., Figure 8.1). Attractors were then derived systematically, using a ‘winnowing’ procedure (Lewis, Lamey & Douglas, 1999, see Table 7.5.

Table 7.5

An illustrative winnowing table

Step Duration (from short to long)

No. cells left (C)

Expected value (D/C) 𝑥1𝑦1 𝑥1𝑦2 𝑥1𝑦3 𝑥2𝑦1 𝑥2𝑦2 𝑥2𝑦3 𝑥3𝑦1 𝑥3𝑦1 𝑥3𝑦3 Total (D) 1 9 2 8 3 7 4 6 5 5 6 4 7 3 8 2 9 1

((Observed-Expected)2)/Expected Sum No. cells left H-Score H-Prop

1 9 2 8 3 7 4 6 5 5 6 4 7 3 8 2 9 1

Note. This is an example table for the winnowing process which a 3 × 3 state space grid would need (N.B. The thesis itself involves a 5 × 5 grid). The ‘H- Score’ is the heterogeneity accounted for by each cell: the H-score formula is in Appendix 5.

The aim of winnowing is to find the cell(s) that account for the most heterogeneity in the state space. The procedure involves listing the mean duration of each cell, from the smallest to the largest value. The heterogeneity accounted for (H-score) by each cell is calculated using observed and expected values for that cell, from the shortest to longest mean cell duration. From the H-score, the heterogeneity proportion accounted for is calculated (H-prop). When the H-prop decreases by .50 (i.e., 50%) or more, the cell responsible for this decrease is named as the scree (Figure 7.11). The cell(s) following this, which will have longer mean cell durations, are then named as the attractor(s; Lewis et al., 1999).

Figure 7.11. Winnowing process, with the cells following the scree identified as the attractor(s). Image from Lewis et al. (1999).

Once attractors (or efficient gaze types) were estimated, they were analysed in two ways. First, attractor presence was explored by examining mean cell durations. Attractor

presence reflects how much a teacher uses gaze the winnowing method has identified to be most relevant to teaching (i.e., how much teachers use an attractor): attractor presence will henceforth be referred to as rates of efficient gaze. Second, attractor strength was explored by examining mean cell return times—that is how long a teacher is away from the relevant gaze (i.e., attractor region) before returning to it. Attractor strength reflects how

prominent the teacher’s use of this specific gaze event is (rather than their general, strategic stability; see Section 7.6.4.1. Temporal Analysis: Measures). Attractor strength will henceforth be referred to as strength of efficient gaze.

To investigate teacher gaze flexibility, transitional entropy measures were used. Transitions are shifts between events. Entropy is a probability-based measure that

represents the complexity of an event sequence (Shannon & Weaver, 1949; see Appendix 5 for transitional entropy formula). That is, the more entropic a behaviour, the more complex it is. The more transitions made in teacher gaze, the more complex the gaze sequence is and therefore the higher the entropy. For example, entropy has been used to investigate the complexity of adolescent male friendships (Dishion, Nelson, Winter & Bullock, 2004), intelligent tutoring system use (Snow, Jacovinam, Varner, Dai & McNamara, 2014), and discourse during psychotherapeutic treatment (Lichtenberg & Heck, 1986). In the present research, entropy values related to teachers’ gaze transitions, as transitional entropy measures. The present transitional entropy values were obtained from GridWare (Lewis, Hollenstein et al., 2004) by identifying student gaze regions as the ‘origin’ and non-student gaze regions as the ‘destination’. This approach was taken for attentional gaze (i.e., questioning, Figure 7.10, Row B) and then for communicative gaze (i.e., straight talk, Figure 7.10, Row A). Didactic gaze flexibility therefore reflected the tendency for teachers to alternate between the specified regions of student and non-student

gaze (rather than within one attractor region, as in Section 7.6.4.3 Temporal Analysis: Measures).

To explore teacher’s strategic consistency in their gaze deployment, I examined the structural properties of teachers’ didactic gaze as a whole. To do this, I obtained whole- grid dispersion values (Hollenstein, 2013), a proportion metric indicating the range of cells occupied in a specified duration. Its whole-grid property meant that, in addition to attentional and communicative gaze, the dispersion value also accounted for gaze when teachers addressed students’ behaviour (i.e., address behaviour), when they referred to learning materials (i.e., refer notes) and when they were carrying out logistics (i.e.,

logistics). Thus, dispersion was a measure of overall strategic consistency in their didactic gaze: the higher the dispersion, the lower the strategic consistency. The dispersion

measure ranges from 0 to 1, with 0 representing no variation (i.e., high consistency) from one cell and 1 being maximum variation (i.e., low consistency), with every cell visited equally.