## Christine Johnson and Professor Janet Walter, Department of Mathematics Education

As a tutor in the BYU Math Lab, I attempted to model correct vocabulary to be consistent with the students’ textbooks and professors. However, at times it seemed as though I needed to use more creative language to effectively communicate mathematical concepts. I began to wonder if it was possible to model correct mathematical vocabulary and convey my intended meanings at the same time. As an undergraduate in the Mathematics Education program, I knew that studying mathematical communication would make me a more effective teacher.

I began working with Dr. Janet Walter in September 2005. She and Dr. Hope Gerson had been teaching mathematics content courses to a group of elementary school teachers for three semesters, and were beginning the fourth semester. I joined the team of researchers, and was given the opportunity to create and describe videos of the semester long course in calculus for elementary school teachers.

As I began to research the topic of vocabulary with respect to mathematics education, I came upon “linguistic invention,” a term used by Tony Brown (1997, p. 76) to describe the act of relating one’s personal experience to a mathematical situation. I observed the elementary school teachers using various forms of linguistic invention to describe the problems that they were working on. I wanted to determine through observation how I could use linguistic invention as a teacher. Therefore, I developed three research questions, (1) How do the participants use linguistics invention? (2) How do the participants react to others’ uses of linguistic invention? and (3) What are optimal uses of linguistic invention?

I chose a segment of video data during which the participants worked on “The Reservoir Task” (Connally, et al., 1998, p 53). This task asked the participants to describe changes in the volume of water in a reservoir, given a graph of the rate of water entering the reservoir. In a traditional calculus course, this task would be similar to exercises that ask a student to describe a function given a graph of its derivative. I analyzed not only video data, but also researcher field notes and the participants’ notes and solution that they submitted for the task.

To analyze vocabulary, I first created a transcript for the video that I had chosen. This took a lot of time, but other members of the research team were willing to help me out. As I created the transcript, I coded the language that the participants used as either “conventional language” or “linguistic invention.” “Conventional language” was characterized as uses of conventional mathematics terminology such as “rate,” “rate of change,” “increasing,” “constant,” and “volume.” “Linguistic invention” was any time that the participants related the graph to a personal experience. Some related the graph to a reservoir that they had seen in real life, others related the graph to taking a bath, and one even related it to driving. I next analyzed how the participants reacted to their peers’ use of linguistic invention. I found that the participants continued to return to the bathtub experience while the other interpretations were questioned or forgotten.

My next step was to determine why the bathtub experience seemed to be more useful to the participants than the other personal experiences. I analyzed the participants’ explanations in more detail. I developed more sophisticated codes to characterize the participants’ vocabulary and their reactions to each other’s vocabulary. I identified some linguistic invention as more personal than other linguistic invention. I also began to code uses of units of measurement, numerical values, and references to the graph that the participants were given.

I found that the presentation and development of the bathtub experience differed from the other linguistic invention experiences in five ways. First of all, the participants collaborated to form the bathtub “story,” or the way in which they related the experience. Second, the participants identified key mathematical concepts that guided their interpretation. Third, the participants continually altered the bathtub experience as they rehearsed it various times. Fourth, the nature of a bathtub allowed the participants to separate and relate the concepts of rate and volume. Finally, the key concepts of increasing, decreasing and constant rate were identified in both the graph and the bathtub “story” through the use of conventional language.

My findings support many of my current practices as a middle school mathematics teacher. I make an effort to provide opportunities for my students to collaborate and share their mathematical ideas. As I speak to them about math, I seek to identify key mathematical concepts that are present in the things that we are discussing. I realize that different personal experiences allow my students to emphasize different concepts in different ways. When my students participate in linguistic invention, I realize that it is not just important for them to connect mathematical situations to personal experience, but that I must be vigilant in pressing them to investigate and communicate why they see similarities between the two situations. These instances of identifying connections between mathematical situations and personal experiences are great opportunities to introduce and practice conventional mathematical vocabulary.

I believe that I am a much more effective and confident teacher as a result of my qualitative research experiences. Through the support of the ORCA grant and the Honors Program, I was able to travel to Roanoke, Virginia, where I presented my findings at the annual conference of the North American Chapter of the International Group for Psychology of Mathematics Education. My interest in graduate school increased as I had the opportunity to associate with professors and graduate students from various universities. Most importantly, I had the opportunity to work with and observe a number of people who exhibit scholarship and compassion in the profession of teaching.

### References

- Connally, E., Hughes-Hallett, D., Gleason, A. M., Cheifetz, P., Flath, D. E., Lock, P. F., et al. (1998). Functions modeling change: A preparation for calculus (preliminary ed.). New York, NY: John Wiley & Sons, Inc.
- Brown, T. (1997). Mathematics education and language: Interpreting hermeneutics and

post-structuralism. Dordrecht, Netherlands: Kluwer Academic Publishers.