This study investigates 3 hypotheses about proof-based mathematics instruction: (a) that lectures include informal content (ways of thinking and reasoning about advanced mathematics that are not captured by formal symbolic statements), (b) that informal content is usually presented orally but not written on the board, and (c) that students do not record the informal content that is only stated orally but do if it is written on the board. The authors found that (a) informal content was common (with, on average, 32 instances per lecture), (b) most informal content was presented orally, and (c) typically students recorded written content while not recording oral content in their notes.

### Timothy Fukawa-Connelly, Keith Weber and Juan Pablo Mejía-Ramos

### Angela L. E. Walmsley and Aaron Hickman

In the *Paideia Program*, Mortimer Adler (1984) states that “the activity of the mind is occasioned or initiated by wonder, sustained by interest and excitement, and reinforced by the pleasure inherent in the activity itself and by delight in its success” (p. 47). Yet, in many mathematics classrooms, the teaching methods used are contrary to each piece of Adler's statement. More specifically, teachers insist on note-taking strategies that bind and inhibit the curiosity and creativity of students. Most of these strategies involve copying main points word for word from the board, copying theorems from the book word for word, or possibly matching words with their definitions on a worksheet. One might question whether these styles of note taking are effective in fostering genuine understanding and prolonged retention of the material being taught.

### Paula Maida

How can students achieve mathematics literacy? One of the goals set by the NCTM to develop such literacy in students is the implementation of mathematical communication. Mathematical communication allows students to express their thought processes and is fundamental to the comprehension of mathematics. Mathematical communkation also forces students to be active learners as opposed to passive learners who simply accept and memorize procedures. For students to learn to communicate mathematically, NCTM's *Curriculum and Evaluation Standards for School Mathematics* recommends that teachers foster a classroom environment that encourages reading, writing, and verbal communication of students' thoughts. “Ideas are discussed, discoveries shared, conjectures confirmed, and knowledge acquired through talking, writing, speaking, listening, and reading” (NCTM 1989, 214).

### Rob Wieman

After many years of teaching mathematics, I still fall into the trap of assuming that my students think as I do. Indeed, this failure to recognize my own assumptions and to acknowledge that others may not share them is at the root of most of my teaching problems.

to dig into is note taking. At my school, students do not take notes. They are given printouts with a few blanks that need to be filled in. As I go through the lesson, they just want to know what word goes in the blank. I’m thinking of beginning the