Lesson Two: Mathematical thinking is not bookable.There's an important point here but I don't think Kilpatrick quite nails it. For starters, speaking as someone who has recently been working with a number of students coming of different courses, grades and schools, I can attest that being familiar with the ideas doesn't always help and sometimes hurts (see Feynman for details). Until you've done it, you have no idea how distracting it can be to teach out of a text where the authors don't really understand the material.

"Bookable" is a term used by publishers to describe the capability of a concept or mental process to be captured in print in a form that teachers will accept and can use. Many of the new math projects concentrated on bringing about reform primarily through the production of innovative materials. In particular, by providing sample textbooks for mathematics courses, the SMSG attempted to influence the commercial textbook publishing process, which would presumably then change how mathematics was taught. The SMSG author teams, by providing fewer problems for students to work, expected that those problems would be treated by teachers in greater depth and detail. The problem-solving process, however, proved not to be bookable. Books are not good at handling tentative hypotheses, erroneous formulations, blind alleys, or partial solutions. Teachers misunderstood what they were to do and called for more problems instead.

Max Beberman, in the new math materials he designed for the University of Illinois Committee on School Mathematics, attempted to incorporate into many lessons what he called guided discovery. Students would be led to see patterns in mathematical expressions and thus to arrive at generalizations that would not need to be made explicit in the materials (until a later lesson). The materials apparently worked well when restricted to teachers who had been trained in their use. When they were later published in the form of commercial textbooks, however, the guided discovery feature was greatly attenuated in order to capture a larger market.

Several recent curriculum projects have run into the same phenomenon. When efforts are made to encourage students to think about mathematical ideas rather than having them enshrined in a text, teachers who are not familiar with how those ideas might be handled criticize the books as incomplete and unsatisfactory. Textbooks are expected to contain authoritative rules, definitions, theorems, and solutions. Consequently, asking students to think about and formulate their own versions of these things rather than providing them ready-made can make a textbook unusable for many teachers.

In order to get get the concept into more usable form, let's introduce a couple of subcategories:

Semi-bookable. These are concepts or mental processes that can be partly but not entirely captured in print in a form that teachers will accept and can use. For example, lots of people teach themselves a foreign language from a book, but the process is almost always supplemented by conversing and by listening to live or recorded speakers;

Readily bookable. Some ideas and processes are difficult to convey in print, either because they can be hard to express or because most textbook authors have a weak grasp of them and miss the subtleties. Feynman's critique hit this point heavily;

Ebookable. An idea or process that becomes much more readily or completely bookable with the introduction of other media.

I'll come back to this later, particularly with regard to why attempts to incorporate Pólya into math textbooks often go so badly.

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