Forming,
Firming And Fitting-In-Place:
The use of student writing assignments, both to promote student learning and as a means of assessment, has been an integral part of my teaching for over a (baker's) dozen years now. These assignments have ranged from "letters to friends" and "problems of the whenever" write-ups to weekly concept papers and written explanations of true-false responses to more formal course topic essays and research papers. As appropriate for the audience, I have used these assignments (and combinations thereof) in courses as diverse as College Algebra, Mathematics for the Liberal Arts, Introductory and Intermediate Calculus, Elementary Concepts of Mathematics for pre-service teachers, Linear Algebra, History of Mathematics, and Advanced Calculus.
Over the years, I have
increasingly come to realize the value of first draft writing as a means to
promote both concept formation and student ability to communicate. By (temporarily) relaxing our usually
strict standards of "mathematical correctness", as well as formal
standards for grammar, spelling and composition, first draft writing places an
emphasis on making sense of the mathematics in question. In turn, students are provided with an
opportunity both “to form” and “to firm" their conceptual understanding of
new ideas, as well as to "fit" these new concepts into place among
previously understood mathematical concepts and structures. Further steps towards the development of
robust conceptual understandings can then take place through revisions in
response to instructor feedback, or by way of follow-up assignments that
emphasize more formal written communication styles.
This talk will focus on my
experience with two quite different uses of first - draft writing. The first of these is the use of journal
writing in a course for pre-service elementary teachers. The prospective teachers in this course are
provided with writing prompts about specific mathematical ideas, often in the
context of a question posed by an elementary - aged student, to which they are
expected to write a thorough (but not necessarily completely correct) response. Mid - stream corrections to a response are
encouraged, and often take place, as students clarify their ideas about the
concepts in question through their writing.
Journals are collected on a regular basis for instructor feedback, that
sometimes include invitations to revisit and revise incorrect or incomplete
responses. Exams and other writing
assignments provide a more formal assessment of student progress with respect
to the mathematical content in question.
The second assignment that this
talk will highlight is the use of Course Reading Notebooks in upper division
courses such as Linear Algebra and Advanced Calculus. In this instance, the primary objective is to promote students'
ability to communicate formal mathematics by supporting their efforts to engage
in active and critical reading of a formal mathematics text. In their reading notebooks, students are
expected to record (in first draft form) their responses to Study Guide
questions about specific definitions, examples and proof details. Exercises that complement the reading are
also assigned in these Study Guides, with the expectation that students will
arrive in class ready to discuss their first draft solutions to these
exercises. Reading notebooks are
reviewed by the instructor on a regular basis, in addition to the more formal
proof writing that is required as part of homework and exams.
In addition to providing
details concerning the logistics of the two assignments described above, this
talk will elaborate on the advantages that first draft writing has to offer to
students. Time permitting, I will
also share examples of assignments for first year calculus students in which
revision and audience play a role analogous to that played by first draft
writing in these two assignments.