This evolution, this movement forward and upward, is only possible if the path in the material world is clear, that is, if no barriers stand in the way. This is the external condition. Then the Abstract Spirit moves the Human Spirit forward and upward on this clear path, which must naturally ring out and be able to be heard within the individual; a summoning must be possible. That is the internal condition.

To destroy both of these conditions is the intent of the black hand against evolution. The tools for it to do so are:

(1) fear of the clear path;

(2) fear of freedom (which is Philistinism); and

(3) deafness to the Spirit (which is dull Materialism).

Therefore, such people regard each new value with hostility; indeed they seek to fight it with ridicule and slander. The human being who carries this new value is pictured as ridiculous and dishonest. The new value is laughed at as absurd. That is the misery of life.

I was brought to Kandinsky's remarkable essay via a surprising source: Scott McCloud's brilliant and heuristic book, UNDERSTANDING COMICS: The Invisible Art. I've been thinking a lot about the medium of comic books as a way to teach mathematics (not an original thought, and one most recently planted in my head by Fred Goodman), who is also responsible for my looking at McCloud's work.

One point that struck me intensely that McCloud spends an entire chapter looking at is that of "closure" and the role of the gutter in comics (I will refrain from the more specific "comic books" or "comic strips" to keep things as open as possible). For those unfamiliar with the term, McCloud defines the gutter as the space between borders. He calls the gutter one of the most important narrative tools in comics, invoking as it does the procedure he defines as closure.

What, then, is "closure," and what is its importance for mathematics education, if any? In UNDERSTANDING COMICS, McCloud says, "This phenomenon of observing the parts but perceiving the whole has a name. It's called CLOSURE." But I'm not sure that particular definition quite does justice to how McCloud develops this notion in the book. What comes through is that between any two panels there is a gap in space/time, and into that gap, represented literally by the empty space of the gutter, each reader pours his/her imagination to create closure, thereby determining their own connections between separate moments in the sequential visual narrative that is comics. No two readers can conceivably do this identically for a host of reasons not unlike what undergirds constructivist learning theory.

While I'm not claiming that this is somehow unique to comics, McCloud makes a persuasive argument for it being sharply defined in this medium in ways that offer no choice for readers but to engage in the closure process dozens of times. And it is that notion that grabbed my attention as I read his book, not just because I come from a literature background with a deep interest in narrative and in the relationship between author/text/reader, director/movie/viewer, etc., but because I am a committed mathematics educator with an abiding passion for how teachers craft lessons and how students engage (or fail to engage) in them.

And so I found myself thinking about the implications for mathematics education in McCloud's notion of closure. In particular, I have thought about some of the best teaching I've witnessed or experienced, what to me made the lesson and the execution of it compellingly successful and remarkable, in the radical sense of that word. In all cases, there was a delicate hand at work: in the choice of problems and examples, in the mathematical questions students were asked to engage in, in the scaffolding process, in the conducting of discourse between teacher and students, student and student, and between individual students and the whole class. And always there were gaps, spaces that appeared to be intentionally left blank into which students were asked to engage their thinking, employ their prior knowledge, strategies, methods, and understanding, in order to make connections and move ahead towards solving a problem, deepening their understanding of a previously-considered concept, etc.

I contrast this with some much of the more mundane mathematics teachings I've seen, experienced, and been responsible for in my own practice. And what inevitably is lacking is the sort of things that persuade students to engage as deeply as seems necessary for students to carry away meaningful mathematical residue. How many times have teachers presented what at least to another reasonably knowledgeable math person would appear to be a clear, logical, organized lesson on some standard topic in the curriculum - an arithmetic operation on the integers, comparing fractions, converting from decimals to percents, long division, graphing linear equations, etc., only to discover shortly thereafter that a sizable number of the students are clueless during, immediately at the end of, or on the day following the lesson (if not throughout all three)? What has gone wrong? Was there something wrong with the teacher's explanation? Were the examples unclear, poorly chosen, badly explicated? Were the practice problems too hard, too dull, too disconnected from the lesson?

Of course, it's a commonplace amongst far too many teachers to conclude that the fault likes not with ourselves and our lessons but with our students (and of course, the poor job LAST year's teachers did in getting the students up to speed, though we wouldn't repeat that to the teachers from the previous grade, at least not to their faces). But let's pretend just for a moment that most students might just learn mathematics more effectively if our lessons were better. What would have to be the nature of those lessons? What would be necessary in the lesson content, structure, and presentation for it to be a rarity for a student to, barely seconds after "experiencing" the lesson, or barely seconds after having gotten some help with a particular problem related to that lesson, come right back to ask for help on essentially the same idea, concept, procedure, etc. in a nearly identical problem?

And it is here that I believe McCloud's idea about closure can tell us a great deal. Because it is my belief that most of our students who are doing poorly in mathematics and who evince deadly passivity as they sleepwalk from math course to math course, are not engaging in any sort of meaningful closure during lessons. They are operating as if math class were television, or some other medium that does not invite closure, rather than comics, which demands it. And their passivity does not lead to the sort of mathematical learning most teachers would like for students.

I suggest that a key question we need to be asking in mathematics education is how to build lessons that effectively get students to engage in the closure process. How do we craft lessons that leave a reasonable number of reasonably-sized "gutters" that will get students to engage in the closure-process, filling in the gaps with their best ideas, actively making connections rather than sitting back waiting for someone to magically implant understanding and mastery in their heads. Who knows? Maybe the very medium of comics itself is part of the answer, something I'm increasingly thinking about. One thing I'm sure of: the main methods we've been using in US mathematics classrooms aren't cutting it for a huge percentage of our students, and if we want to change that, it seems worth thinking about the nature of the media we're using to present and deliver instruction.

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