In my definition of diagrams, the Hyperproof diagrams and pictures in which people are depicted on paper (which are “realistic diagrams”) are kinds of “concrete” diagrams because extra information that is not related to the inference required (e.g., face, hair and hand so on) is included. In contrast, such information was excluded from the black or circle diagrams in my study. When students draw diagrams by themselves, the dimension of abstraction is important as abstract diagrams are easier for students to draw and easier to mentally manipulate.  (However, understanding and using abstract diagrams require some preparation of students – particularly students in the early stages of learning. For example, it is necessary for them to understand what is represented and how to use such representations.)

I do not mean that more concrete diagrams are not useful or effective. In some cases, realistic diagrams work in the same way as more abstract diagrams (e.g., picture of a length of road works just as effectively as line diagrams in solving some math word problems). Semi-abstract diagrams can also bridge the transformation (and necessary perceptual apprehension) between concrete and abstract. When we consider students’ spontaneous use of diagrams, the cost to construct effective diagrams is important (Uesaka & Manalo, 2011).

Uesaka, Y., & Manalo, E. (2011) Task-related Factors that Influence the Spontaneous Use of Diagrams in Math Word Problems. Applied Cognitive Psychology.

And yes, I agree with your idea that the dimension of the abstractness is on a continuum!

I wonder what you think about the situation where the diagram consists of realistically rendered objects, as opposed to abstractions on the one hand, and actual physical objects on the other. I am thinking, of course, of Hyperproof diagrams, an example of which can be seen here https://www.gradegrinder.net/Products/lrds-index.html.

The obvious conjecture is that the more “realistic” the diagram, the more students would benefit. I wonder though, whether there is a smooth increase from abstract to concrete, or whether physical objects are genuinely different, and provide a discontinuous effect. Do you have thoughts on this?

Thanks again for the presentation.