Duval explains his theory of “registers” — proposed mental processes for interpreting mathematical symbols. This becomes a construct for analyzing cognition in students developing mathematical thinking. Duval’s theory promises a way to model attempted comprehension by the student — whether successful or not — of mathematical representations.

Obvious questions for a reader include: Is this model accurate? Does this model point to an improved pedagogy? Duval includes direct observations of pre-teens learning geometry over a period of months, documenting the applicability of his model. The concluding material in Chapter 4 recommends mixing verbal explanation with pictorial representation and paying attention to the basic ideas that need to be gleaned first from a geometrical arrangement before secondary concepts can be gotten at. Sometimes a comedian talks about a “long walk” to get to a punchline. This book feels like a long walk to get to actionable advice. I am interested, however, in hearing about more detailed applications of this theory. This work is largely one of exposition rather than praxis; only about a fifth of the content speaks to real-world application in strategies to teach geometry toward the end of primary education.

Typographical and illustration decisions and imperfections unfortunately get in the way of some exposition. Excessive use of italics and all-caps dilutes emphasis into distraction. In one example:

From the only coding rule, *which can never lead beyond* series* of points*, to these two operations there is a DIMENSIONAL JUMP IN THE VISUAL CONTINUUM…

Duval’s book *Sémiosis et Pensée Humaine*, even more theoretical in content, may be the source for some included illustrations. These have French labels with no translation, putting them at odds with the English text. I have no explanation for the stray Greek labels, however. Then there is Figure 4.7, in the climactic chapter covering application, where text accompanying an unlabeled image only makes sense if the vertices were labeled.

Mathematics educators can benefit from considering Duval’s cognitive theory. Deconstructing the relevant cognitive procedures is a worthwhile exercise for the teacher. A fundamental thesis of the book impresses like a thought-disrupting koan: “There is no noesis without semiosis.” This is expounded at different times in ways worth considering for any teacher of mathematics, especially with respect to elementary topics rich in graphs and images. For instance, “…the learning situation focuses on problem-solving. From a cognitive point of view, let us first focus on the organization of recognition tasks…” Duval reminds us that students must decipher symbols and shapes in an expected and logical manner to get at core mathematical concepts. As teachers, our obligation is to keep the symbols from occulting comprehension. As Duval warns, “The visualization of geometric shapes has nothing in common with the Gestalt recognition of shapes and objects in the images, or even with the graphs that visualize functions. This invisible divide is not taken into account when teaching mathematics.”

It is interesting to me that these issues, when considered in the context of computer presentation, summon for Duval less certainty and even the appearance of contradiction. On one hand, “there is little actual research on this issue, considering that the computer is a powerful tool, which is both fun and simple to use…” Fun, simplicity, and not knowing does not preclude assertions such as

Seeing any geometric configuration on the screen requires the same dimensional deconstruction and visual discrimination of the merological transformations possible than seeing it on the paper.

The statement causes two reactions to arise with me: First, “merological transformations” is one of the many ideas that could be expressed more simply in this text. Second, I offer computer animations of some geometric configurations as helpful to guide a student through an organized set of recognitions tasks. Printed works that attempt this with series of images often fail to do anything more than overwhelm the student.

Duval directs us to consider the difficulties present for students attempting to understand the basis of visualizations of mathematical notions. Even more so, the focus on problem-solving itself may distract from deeper comprehension.

Understanding in mathematics that first consists in recognizing the mathematical object represented and the possible transformations of their semiotic representation depend on their *semiosis*. As long as they cannot achieve this cognitive activity, students have a block in their learning of mathematics that is interpreted as ‘incomprehension’…

Literary theorist Kenneth Burke famously defined humanity by stating that “Man is the symbol-using (symbol-making, symbol-misusing) animal…” Philosopher Susanne K. Langer observed in *Philosophy in a New Key* that “The assignment of meanings is a shifting, kaleidoscopic play, probably below the threshold of consciousness, certainly outside the pale of discursive thinking….” There may be no more economical way to illustrate a circle than the image of the closed curve and an indicated center. Yet I have encountered many students competent to, say, go between graph and equation, yet unaware of any connection to a set of coplanar points equidistant from a fixed point of the same plane, no matter how many images they have produced or interpreted. Duval challenges us to make the transmission of the base concept as the measure of success, not superficial figurative problem-solving. As Duval sums up,

Contrary to what has been always postulated in mathematics education, discrimination of the relevant units of meaning in different representations does not result from the acquisition of concepts, but it is the prerequisite for this acquisition.

Tom Schulte, who teaches mathematics as an adjunct lecturer, is a software architect at Plex Systems, drawn to computer science from mathematics by seeing the common symbols of basic functions beautifully expressed in the poetry-like indented code blocks of FORTRAN77.