Beyond Questions and Answers: Prompting Reflections and Deepening Understandings of Mathematics Using Multiple-Entry Logs

There is little discussion in mathematics education about me~hods of effective tutoring; in particular, in so-called o~hce hour or tutmial sessions Nevertheless, these sessiOns are as much occasions for interactive instruction as classr?oms are, and deserve theoretical and pedagogical attentwn By initiating discussion in this area we invite others to contribute critically to these beginnings Clearly, d1scusstons m th1s area will inform and be informed by classroom interactions .. For this reason, though we focus on office-hom or tutorial interactions, we do indicate how t~e pedagogical tool we describe infmms classroom practice During office-hour or tutorial sessions, students customarily set specific goals that, with the tacit agreement of an mstructm or tutor, govern the interactions of these sessions . These goals typically center on questions or problems for which students seek immediate answers or solutions. Two problematic features often characteiize interactions of these sessions. First, matters of mechanics or techniques constrain the mathematical dialogue and second in this "dialogue," the instructm or tutor d~es most of ~he talking or explaining Stemming conspicuously from the first feature IS the state of affairs that instmctors and tutors operate rather like machines, pmducing worked-out solutions each time students pose questions: "How do you do problem 15 on page 35?" or, on rare occasions, "I was able to solve this problem to here but cannot see how to complete it Would you show me how to continue?" Curricular emphases on content over process and answers over understandings contribute to fmming academic cultures in which students respond to mathematics learning in these ways ~o.r~over the pace and superficiality of many courses mh1b1t efforts to engage students in justifying techniques, ~n demonstrating derivations of formulae, or in intenogatmg and negotiating meanings of mathematical objects and processes Students either ~gnore these gestures or plead, Just show me how to do 11. After the exam, I may think about why it works." The second problematic feature of office-hour or tutorial sessions concerns who performs most of the actions does most of the talking, and therefore, acquires most ~f the learning Usually, we, instructors and tutors, dominate these sessions with our action and talk Consequently, we, ~ot stud~nts, do most of the cognizing: thinking, explainmg, solvmg, and so forth. That is, we appropriate available opportunities to re-experience and re-conceive mathematics. Fm~e~~ore, ~is appropriated time and space offer us the possibility of mcreasing our insight into a piece of n:athematics while students, in awe, mostly listen to our display of knowledge and, when it occurs, witness om mathematical growth On the whole, then, the benefits of office-hom· or tutorial sessions mostly accme to us How can instructors and tutors go beyond question-andanswer sessions and provide time and space fm students to reflect deeply on, gain insights into, and increase their understandings of mathematics? What pedagogical vehi~les can prompt these behaviors and cognitive acts? Leammg, contrary to the famous adage, does not occur from expetiences alone. For it possibly to occm, learners must reflect on their experiences. They must connect and make sense of their experiences in the context of knowledge they already possess. Instructors and tutors, therefore, need pedagogical vehicles that explicitly engage students in reflecting on mathematics. One such powerful vehicle is writing It can prompt students to reflect critically on their mathematical experiences and respond to mathematical situations and questions that are personal and of their own choosing. [2] Perhaps, we could address both problematic features of office-hour and tutorial sessions if we were to hav~ ~tudents write about their reflections on a problem an~ If we were to use their reflections as starting points fOr office-hour or tutorial discussions. Below, we, an instructor and a student, describe such an approach in the context of an office-hour session