Table 1.
Characteristics and examples of the two dimensions in the codebook for the video and verbal data.
| Dimension | Characteristics | Description/Example | |
|---|---|---|---|
| Knowledge articulation | |||
| (1a) Knowledge content | |||
| Contribution (C) | Refers to utterances that are indicative of a student’s emerging proportional reasoning. | ‘My right hand has to move faster than my left hand to keep it green.’ | |
| Repetition (R) | Refers to repetition of previous contributions. | ‘When I move faster with my right hand, it remains green.’ [repetition of the utterance above] | |
| Null-content (N) | Contains no problem content at all. | ‘Can I start already?’ | |
| (1b) Solution strategy (from additive to multiplicative reasoning)a | |||
| Conceptual strategy | Motor action | ||
| Pre-additive (1)b | Comments are focused on the visual appearance of both bars. | ‘Right should be higher than left.’ | Random movements, green is being found based on chance. |
| Fixed interval (2)b | Students try to maintain a constant spatial interval between both hands/bars. | ‘There is a difference of two, so I have to go up two at both bars.’ | The difference between both bars is being held constant. |
| Changing interval (3)b | Students modify the spatial interval between both hands/bars in order to enlarge the distance. | ‘The higher I go, the bigger the distance needs to be.’ | The difference between both bars is being enlarged. |
| a-per-b (4)b | Student deploys sequential hand-movements, each hand moves up or down according to its respective quota. | ‘For every unit left, I go up two unit’s right.’ | Both bars descend or ascend at respective constant values. |
| a-per-Δc (5)b | Student deploys a strategy that attends to the interval between the left- and right-bar as it changes with respect to the height of the lower bar. | ‘1–2 is one line apart, 2–4 is two lines apart, 3–6 is 3 lines apart.’ | When the left bar rises, the right bar rises by one unit more than the previous difference between both bars. |
| Multiplicative (6)b | Quantitative statements about the numerical location of one of the bars directly as a product of the numerical location of the other bar. | ‘The right bar is twice as high as the left bar.’ | A value is determined for the left bar, which is continuously doubled to find the value for the right bar. |
| Speeds (7)b | Statements are about the relations between both bars in terms of their respective velocity. | ‘My right hand has to go faster than my left hand, in order to keep both bars green.’ | Both bars ascend and descend at different constant velocities. |
aThe given examples are based on pre-set proportion 1:2. bUsed ordering of the strategies in brackets, the ordering of the used solution strategies was based on the literature into proportional development (e.g., Misailidou, 2007; Van Dooren et al., 2010; Abrahamson et al., 2014). Furthermore, following Abrahamson and Sánchez-García (2016), ‘speeds’ was interpreted as a simultaneous enactment of the a-per-b strategy while at the same time can be interpreted as a qualitative indication of the multiplicative solution strategy. cΔ = Magnitude of interval between hands.