Practical experience and applied research have shown me that one way to promote ownership of mathematical learning among college students, a part of the process of self-regulation, that works well with my teaching style is the use of mathematical autobiography. A mathematical autobiography is an essay assignment wherein students must reflect, at length, on their past experiences with mathematics, painful or joyful as they may be (see Figure 1 for the assignment page).
As one pre-service teacher in her final semester at university put it,
I always try to find the answer right away and if I can't find it, then I say those three words, ``I hate Math.'' Sure, I'll eventually find the solution, but I'll be frustrated and upset the whole time.On the other hand, disappointment and a feeling of betrayal may take precedence over hatred. Consider the view of a 27 year-old student returning to university, and to her struggle with pre-calculus mathematics, after a six year absence from school,
I went through a very messy divorce in the sixth grade: mathematics and I separated, citing irreconcilable differences. In the fifteen years since then we have only grown further apart.
First, the construction of a coherent narrative from poorly organized recollections of past mathematical selves and stressful events allows for the repackaging of these experiential keepsakes into streamlined memory structures. The reformulated memory may be stored and regulated more efficiently than its disorganized and intrusive predecessor [King2002]. Through the expressive writing in a mathematical autobiography narrative an otherwise mathematically disabled student may be enabled. Working memory that had been used to suppress or deal with intrusive cognitions and emotions triggered by interaction with mathematics may be freed for use in current cognitive demands [Klein & Boals, 2001].
Second, the writing process for the essay brings into conscious consideration those habits of mind and of behavior which may be influencing a student's success in mathematics. Note that the word ``success'' in the previous sentence includes not only success as defined by the student (perhaps in terms of marks in class) but also success as I define it as their teacher: demonstrable understanding of the mathematics they can do in addition to a conception of what kinds of ideas lie just beyond their grasp.
When in elementary school, children engage enthusiastically in mathematical problem solving. They become emotional participants in mathematics, constructing meaning and establishing personal, intimate, connections with the process of thinking mathematically [DeBellis1998]. Over the years of schooling, teachers may invade the personal mathematical spaces of children. It is quite common to observe an elementary school teacher interrupt a child at work on mathematics to question or prompt her. One rarely sees such interference when a child is reading or writing a paragraph. Even if the intimate engagement is not interrupted, a positive relationship with mathematics may not develop. As DeBellis and Goldin have remarked, a student working on mathematics
...may feel disappointed, angry, or betrayed by unexpected mathematical outcomes, failures, negative reactions from loved ones, rebuke from a trusted teacher, or scorn from peers. Such `intimate betrayal' does not distinguish between the mathematically talented and the mathematically challenged individual. Even among professors, graduate students, and professional scientists - and certainly in mathematically gifted children - one finds a great deal of pain in relation to mathematics [DeBellis & Goldin, 1999].
Also, as noted in research done in the late 1980's in Canada on mathematical autobiography, the power of self-story telling is that it provides both student and teacher with ``a mirror to see ourselves, not only the aspects we want to see, however, but ones we need to see'' [Brandau, 1988]. Thus the mathematical autobiography assignment is intimately connected to my own philosophy that learning is part self-discovery and part synthesis of other's discoveries.
The mathematical autobiography project is something I have included in every first-year college mathematics course I have taught. In my teaching of general education mathematics, rather than challenge long and dearly held beliefs and attitudes about mathematics, I encourage students to build an additional layer of engagement, meta-affect,. To make what I mean clear, consider the following analogy: imagine for a moment a person buckled into a cramped car seat who believes herself in danger; she is feeling fear or anxiety, even screaming. The affect here is clear: terror. However, if that person is on a roller coaster, willingly, the meta-affect is joyful anticipation (of being scared senseless, perhaps). I encourage students to accept their fear or anxiety and help them construct viable meta-affective layers for working in and with mathematics. On several occassions students have written comments in their end-of-term evaluations of a course which refer to this; for example:
While I don't usually like math at all, Dr. Hauk allowed me to feel that way while constantly offering her aid in and outside of class. I feel I have a better understanding of math because of her.