Bongo-Bongoism in Education

In time I will argue--successfully, I hope--that the no doubt confusing philosophizing going on in my previous post on this topic was neither a hypocritical misapplication of good principles of information organization nor a faithful application of bad principles.

For now, though, the confusing post remains, and I shall do my best in this one and the next few to try to make my thoughts clearer while expanding on them a bit from what I hope to be a firmer foundation.

Because of (among other things) education's enormous social and institutional complexity, even reasonable, mainstream discussions within mathematics education tend to suffer from a kind of bongo-bongoism, "the venerable but ultimately sterile anthropological practice of countering every generalization with an exception located somewhere at some time":

When a generalization is tentatively advanced, it is rejected out of court by any fieldworkers who can say: 'This is all very well, but it doesn't apply to the Bongo-Bongo.'*

This is not so much a bug as it is a feature of the discourse among educators who directly and daily influence the learning of students. After all, these are people who traffic in exceptions rather than rules--for whom bongo-bongoism is a necessary, valuable, critical orientation toward "outside" ideas (from research, standards, professional development gurus, and textbooks) that often miss the trees for the forest (or, rather, the students for the statistics).

Yet it is one thing to be naturally suspicious of generalization and quite another to treat it as a thought crime. On this view, Eratosthenes's investigations concerning the circumference of the Earth should never have begun given that they were based on wild assumptions about the shape of the planet, the direction of the Sun's rays, and the locations of cities in Egypt. Better to wait for a 25,000-mile-long tape measure. Darwin's belief-shattering ideas, too, are ridiculous in this light. Were you there? is a question indoctrinated children are taught to ask of those who present the evidence for the theory of evolution by natural selection. This is much worse than missing the forest for the trees. It is missing the forest for the missing trees.

Mathematics education desperately needs a good generalization--one that, at the very least, can keep its denizens somewhere between a healthy bongo-bongoism and a destructive solipsism.

Finding the Circumcenter of a Triangle

Teaching as Shaping and Organizing an Information Environment

The principles of precision, clarity, order, and cohesion are clearly applicable to situations involving intentions to communicate carefully.

Thus, one way I can attempt to argue for the extension of these principles to mathematics education is to outline the similarities between careful communication on the one hand and mathematics education on the other. To borrow from the syntactical structure of the Transitive Property: if these principles apply to careful communication, and careful communication is what mathematics education is all about, then the principles should apply to mathematics education as well.

Reaching this kind of abstraction about mathematics education--that it is, at bottom, about careful communication (or, rather, a careful shaping and organizing of an information environment)--seems either impossible or inadvisable given its enormous institutional complexity. There are standards, unions, professional organizations, research, students, publishers, national news reports, technological innovations, blogs, assessments, philosophies, markets, buzzwords, and on and on. Only an inept or mendacious observer could find just one story or one solution among all of this disparity.

Yet, why not make use of mathematical thinking to analyze mathematics education? Mathematics derives its power as a cognitive tool because of abstraction. It tosses away pesky and unhelpful and irregular incidentals that stand in the way of useful generalizations about the world. We can at least try to do the same.

A good way to start, I think, is to ask, What is teaching? What image or images come to mind when you read or hear the word teaching?

• a person giving another person directions to a location?

• someone reading a newspaper?

• a group of finches from the Galapagos Islands?

• a painting?

I would expect that none of the above would be your answer. For me, the strongest image conjured by the word teaching is a classroom with a teacher and students, doing . . . something. I can't presume to think that this is the image everyone would have, but I would expect it to be the number one answer on Family Feud.

In truth, though, the only difference between the image of a classroom with a teacher and students and, say, that of a person giving another person traveling directions is the content of the message being delivered. In both situations there is a sender, a receiver, and an educational message.

What about the image of a person reading a newspaper? Is this teaching? Here again we have a sender, a receiver, and an educational message, though the sender is not present, may be more than one person, and has had his or her educational intentions filtered through a number of limitations outside of his or her control. With the exception of the presence or absence of the sender, this description could just as well describe any school teaching environment.

To consider the painting and the Galapagos finches as examples of teaching is to move much further along the abstraction ladder. Whether or not either have identifiable or even extant senders or receivers is irrelevant. Both represent information environments that have been carefully organized--in the case of the former, by a perhaps long-dead painter; in the case of the latter, by the learner himself. A lesson plan that sits on a desk (or in the cold of space) for possibly all of eternity is still an example of teaching, and the cosmos itself, though lacking a "sender" altogether as far as we can tell, is a teacher to the extent that a creature can organize and shape its informational content.

A line of inquiry seems naturally to follow the description of teaching as a careful shaping and organizing of an information environment: are there criteria we can apply to this process to maximize its efficacy? This is the question the principles of precision, clarity, order, and cohesion attempt to answer in the affirmative.

Precision, Clarity, Order, and Cohesion

It would be best for me, I think, to introduce the four principles I will talk about--precision, clarity, order, and cohesion--with a mundane example followed by a question.

So, let's start with this nonsense, which is a jumble of something that is not nonsense:

n u p i k u e o c x e e g c z t e t o g d i r n o h g

First, I need to put these bits of information in the correct order. For people with background knowledge in reading English text, this re-ordering will help out quite a bit, although there will still be confusion. For those without this background knowledge, the text below is likely just as confusing as that above:

x e n u z i c k e d t h e p u c e t o g o r i g o n g

So, now we need to package this information. Bits that belong together should appear together, and bits that belong apart should be separated from each other. That is, we correct the information to give it cohesion:

Xenu zicked the puce to gorigong.

Note that placing a capital letter at the beginning of the sentence and a period at the end also helps to give the information more cohesion--we clearly identify the sentence as a separate 'chunk' of information, a complete thought.

For clarity, we at minimum should probably define 'zicked' and 'puce' and 'to gorigong." (Or did you think that 'gorigong' was a name?) But note that addressing clarity here requires us to revisit order and cohesion also: Should we define the terms before using them in instructional sentences or after? Should we define them all together or separately? Etc.

Finally, to apply the precision principle here simply requires us to ask if the resulting clear, cohesive, and well ordered information is actually true--although, as we will see, this is only a starting question.

With those basic concepts introduced, I ask the question: In what way can attentiveness to these principles shine light on how to teach mathematics?

Teachers Should Write Their Own Textbooks

Jason Rosenhouse at ScienceBlogs writes about a few problems he sees with mathematics textbooks:

If I were to express my major objection in the most charitable possible way, it is that most textbooks are written like reference books. They are usually very good at recording the basic facts of a subject and proving them with admirable rigor. If you just need to look up some elementary theorem or formal definition, then by all means consult a textbook. The trouble, though, is that textbooks are seldom written from the perspective of a student encountering the material for the first time.

Mr Rosenhouse inadvertently overstates the case by leaving it to the reader to infer that he is likely talking about college-level textbooks. It would be difficult, for example, to describe elementary and middle-school textbooks such as Everyday Mathematics and Connected Mathematics--which are among the most commonly used texts nationally--as primarily referential. These texts, instead, are highly programmatic, containing activity-based and discovery-oriented instruction with more or less prescribed lesson progressions.

Still, though, Rosenhouse is generally correct. Referential texts, in contrast to programmatic texts, continue to enjoy greater market shares at the elementary, middle-school, and high school levels--a difference that increases as the grade level increases. So, when we talk about "mathematics textbooks," in general, we are talking about books that function primarily as reference material.

Yet, teachers, it seems, are not as disappointed as Jason (or mathematics education experts) about this state of affairs. Roughly 40% to 45% of teachers, regardless of grade level, rate their mathematics textbooks as being very good or excellent, while only between 5% and 10% rate them as poor or very poor.

More interesting, however, is to consider these attitudes in light of the table below (also from the report linked above):

Roughly 60% to 70% of teachers, in their most recently taught unit, supplemented their mathematics textbooks with material from other sources. About half took what was important and skipped the rest. The most often cited reason (shown in Table 6.13 in the report) for skipping textbook material was that teachers "have different activities for those mathematical ideas that work better than the ones . . . skipped." And very large percentages of teachers supplemented their textbooks to provide for practice or differentiated instruction.

How do we reconcile these schizophrenic results? A semi-strong plurality of teachers seem to be happy with their more referential math textbooks, yet at least three out of five of them skip or supplement the material in important ways.

What I would suggest is that the majority of teachers have come to accept and expect that mathematics textbooks and classroom teachers play different roles in education. Classroom teachers have become resigned to the notion that mathematics texts can be nothing better than reference books and that it is their job to bring the material to life for their students.

But what if teachers could write their own textbooks?

Imagine, for example, a partnership between a small publisher (or a division of a large publisher) and a school or district, formed to create classroom materials for mathematics instruction specific to that district or school. The publisher's resources would include access to printing capabilities, market research, and writing/editorial talent. The schools' resources would include content expertise, classroom experience, and cultural connections to the audience served by the instruction.

Instead of working construction or being a lifeguard over the summer, teachers could work in tandem with editorial staff to produce lessons, units, workbooks, full textbooks, technology applications, intervention materials, etc. Maybe the copyright stays with the publisher with the school or district earning a percentage from sales. Maybe the copyright is handed over to the teachers so that they can sell their product, with a marketing assist from the publisher, elsewhere in the state. The devil may be in the details, but it's something that seems feasible.

To the extent that we want to continue to rely on some sort of guiding text for classroom instruction, it is likely worthwhile to put its development into the hands of the people who know best--teachers.