Using journals for formative assessment encourages students to explore topics they might have been intimidated by if they were being graded.
Background and Purpose
A journal is a personal record of occurrences, experiences and reflections kept on a regular basis. In my mathematics classes students keep a journal of their mathematical experiences inside and outside of class. The purpose of journals is not to assign a grade for each entry but to help students find their own voices and to be reflective about the subject. Allowing more informal tentative writing into the classroom encourages students to think for themselves as opposed to only knowing second hand what others have thought before them.
Mathematics is sometimes perceived as stark and unbending. This may be caused by presentations which are strictly definition-theorem-proof, or lack a sense of historical evolution and excitement. "Mathematics has a public image of an elegant, polished, finished product that obscures its human roots. It has a private life of human joy, challenge, reflection, puzzlement, intuition, struggle and excitement. The humanistic dimension is often limited to this private world. Students see this elegant mask, but rarely see this private world, though they may have a notion that it exists." (D. Buerk , page 151)
Some professors insist that all answers to homework and exam problems be in full sentences, with severe penalties for violations. The students are given a sheet with instructions and illustrative examples and are left to sink or swim. There is little effort to convince students of the merits and advantages of this demand. The perception is that the professor is not "student friendly."
If the object is to help students learn to express themselves in writing, journals offer a more natural approach, and the perceived relationship with the professor is not so confrontational. When the task is to "solve a problem," students who are already writing in their journals may approach the solution in a more expansive and discursive fashion. Mathematics may thus be elevated above memorization of facts and formulas.
When I asked my 20 to 30 students in Calculus and in Linear Algebra to keep journals that would be read and commented upon by me, they remarked that they were also keeping journals in their humanities, social science and biology courses. Harvey Mudd College is a small (650 students) college of engineering and science with a demanding workload. One-third of the curriculum is in the humanities and social sciences. A Writing Center exists to help students improve their abilities. All first year students take a writing intensive course that is limited to eighteen students per classroom. This course is taught by writing specialists who are part of the Humanities and Social Science Department.
Journals are a form of self-assessment, an opportunity for students to think about their knowledge of the subject and to strengthen their confidence. The journals are not graded; they afford an opportunity for dialogue between each student and the teacher. Grades are based on homework, class participation, quizzes and exams. The journal dialogues allow the students and the teacher to know each other beyond the anonymity of the classroom activities.
I ask my students to keep journals about their encounters with mathematics in or out of class. These may include insights or puzzlements. Some students may be interested in pursuing the history or the implications of some aspect of the subject. Entries should be made at least once a week. The journals are collected every two or three weeks and returned within a week. No length or other guidelines are offered. Students are encouraged to express themselves in writing. The exercise of writing may encourage further discussion among students and between students and teacher. Some entries have nothing to do with mathematics, but come from other concerns of young people.
An early entry for my students is "A Mathematical Autobiography." Students seem to enjoy writing about their experiences with good or not so good teachers, their awards and successes, sometimes starting in the third grade. Other entries may be responses to questions or topics that arise. There may not be enough time to reflect on those ideas in class. Students may also express their puzzlements in their journals. The journal becomes part of the conversation between students and teacher.
A colleague periodically asks his students to turn in a looseleaf binder that contains returned, corrected homework, personal responses to readings in the text and classroom notes. These portfolios may or may not be graded, but will be commented upon in the margins.
A successful alternative or addition to journals that I have tried is to ask students to write a short (3-5 pages) essay on anything related to mathematics:
Is there some subject that you are curious about? Learn something about the subject and write about it.
Many students seize the opportunity to learn independently and to write about their knowledge. Grades are not mentioned. (The activity might be part of class participation.) All students, with rare exception, participate enthusiastically. Some of the responses have been inspired. An Iranian student could not think of an appropriate topic. I suggested that he learn and write about the Persian poet-mathematician Omar Khayyam. He was so excited by what he learned that he insisted on writing a second paper about his countryman. The activity offered an independent, self-directed task that contributed to knowledge and self confidence. A follow-up for that last activity might be duplicating all of those papers and distributing to the class (and beyond) a stapled or bound set.
Use of Findings
Grading journal entries might inhibit the free exchange of ideas and the viewing of the journals as part of an extended conversation. One might assume that not formally grading them would lead many students to neglect keeping journals. Paradoxically, the absence of grades seems to liberate students to be more adventurous and willing to explore ideas in writing and in the classroom. The usual subjects of the journals are the ideas that are studied and discussed in the classroom or from supplementary reading done by all or part of the class. The students flourish from the intrinsic rewards of expressing themselves about things that matter to them, and having a conversation with the teacher.
Although I do not grade the students' journals, I do correct grammar and spelling, and comment on the content with questions, references and appropriate praise. Students respond to the conversations via journals by speaking in class with more confidence and a greater variety of ideas. Entries made before discussions are rehearsals for class participation. Entries made after class are opportunities for reflection.
How do students acquire knowledge of mathematics? Memorization and solving problems are two routes that may be followed. Constructing personal meaning by reflection and conversation is another route. These routes are not mutually exclusive. Asking students to discuss in their journals some of the questions that were touched on or discussed in class extends the intellectual agenda in a meaningful way. The journal is an opportunity for each student to be personally involved in the agenda of the class.
Students will not be anonymous members of the class if they participate in writing and reflecting as well as speaking.
 Buerk, D. "Getting Beneath the Mask, Moving out
of Silence," in White, A., ed., Essays in
Humanistic Mathematics, MAA Notes Number 32,
The Mathematical Association of America, Washington,