I apologize that I have not yet read this paper, so perhaps this is noted, but the same problem occurs with geometric diagrams. To illustrate the Pythagorean theorem you have to draw _some_ right triangle. If you draw a 3-4-5 right triangle, then you are being over-specific, if you draw an isosceles right triangle you are similarly over-specifying.

The mystery, it seems to me, is not that the diagrams are over-specific (that is necessary) but the stability of the particular over-specific example that was chosen (perhaps for aesthetic reasons?)

Professor Carman wants to claim that the overspecification in the diagrams is an accidental result of the copying process. I am unconvinced because I see overspecification in medieval diagrams as far to frequent to be explained as an accidental effect. It does not seem likely to me that all this overspecification arose simply as an accident in copying the early manuscripts. And even if overspecification did arise in the copying process, why should it then be preserved and recopied and repeated in so many manuscripts?

Incidentally, Professor Carman has recently published a follow-up article on his hypothesis in the newest issue of Centaurus — in case you would like to follow up on his idea.

I think that your first point is related to Gem’s comment, and my reply is similar. In a formal sense, you are right, no real diagrams conform to our idealized specification. However, in practice a teacher would assess a diagram as correct if it were “close enough” to the desired diagram, incorrect if it is outside of tolerance.

The main point of our paper is that it is possible to imagine a collection of representations systems all being used with a single diagram. Once concerning the actual lengths of all of the bars of a bar chart, another concerning the relative lengths of the bars, a third concerning only the first bar, and so on. We show how to formalize this idea, in terms of representation systems that are “derived” from the basic underlying point-by-point semantics. So as you say, reader approach the diagram with different issues in mind, and the representation system that they choose to use is determined by those issues. If they can choose a representation system which makes the diagram true for their purposes, then they can interpret the diagram appropriately.

Thanks again for your comment.

Thank you for the feedback. Replies are as follows:

1. By “avoiding inaccurate and ineffective properties does not necessarily ensure effectiveness”, we mean that even if all of these properties are avoided, it is still possible that a diagram is not effective due to other factors, such as curve smoothness and curve-edge closeness. So, when a diagram has undesirable properties, then the diagram is likely to be ineffective, but the opposite (if no properties then effective) does not necessarily hold.

2. The studies are likely to focus on the categories of tasks most commonly performed on these types of diagrams, based on existing literature. The key variables to consider would be accuracy and speed with which the users will perform these tasks. The results of such studies will either reinforce the results in this paper, or will provide new insight with respect to the relationship between properties that were used in the evaluation and effectiveness of diagrams.