Reflective writing in preservice content courses.

Abstract

Students enter preservice content courses for elementary school teachers with preconceived ideas about the K-8 curriculum that influence their views of the content and methods of these courses. Written reflections on journal articles chosen to reinforce a mathematical concept or method can help increase student awareness of the K-8 curriculum and is readily extended to other disciplines.

Introduction

Preservice elementary school teachers enter content courses with beliefs and attitudes about teaching and mathematics that can profoundly influence their future teaching. Brown and Borko (1992) explored various facets of teacher education programs and report that students begin teacher preparation and content course with attitudes and beliefs that developed during their own childhood with "simplistic expectations" about their future as teachers (p. 222). Some of the beliefs held by preservice teachers include the expectation that problems should take little time to complete and a single method of approach is sufficient (Karp 1991). Another common belief is that instruction should be based on developing procedures (Stuart and Thurlow 2000). Many preservice teachers believe that liking children and maintaining a motivating environment are adequate for successful teaching (Lasley, 1980 and Veenman 1984). Preservice students assume that the concepts in mathematics and science that they struggle with will be difficult to explain to their future students at a conceptual level (Stevens and Wenner 1996). Other students in these preservice courses have preconceived notions about the alignment of the mathematics curriculum in K-8 that are inconsistent with National Council of Teachers of Mathematics (NCTM) and state standards. Still others see the emphasis on multiple representations as unnecessary, with the view that one method should work for all (Fennema and Franke, 1992). Similar findings concerning beliefs and attitudes by Stuart and Thurlow support the need for preservice teachers to voice and examine their beliefs about teaching.

In discussing the implications of beliefs for teacher educators, Thompson (1992) urges teacher educators to assist students in reflection on their attitudes and beliefs, though recognizes that there is insufficient time in a methods course to alter deep rooted conceptions of mathematics. I suggest that this reflection on beliefs and attitudes can begin prior to methods courses. In the mathematics content courses for preservice teachers, I complement content with reflective reading and writing. Through these reflective activities, the mathematical and pedagogical foundation around which the preservice courses are structured is tied to personal reflection about teaching and mathematics.

Activity Design

Students are given an article to read, purposefully selected to elicit reflections on content standards and pedagogical themes such as multiple representation. Each article is accompanied by a set of prompts to which students must respond. The prompts also form a basis for assessment. When choosing papers or journal articles for students to read and respond to, I have three criteria in mind. First, the paper needs to have examples of actual student work or classroom dialogue. The voices of practicing teachers and elementary school students are much more convincing of who, what, how, and when mathematics is taught in an elementary classroom than a teacher educator or a text. For this same reason, a second criterion I use is that the reading chosen must align a curricular topic from K-8 with the content of the preservice mathematics courses. Finally, I choose articles that either illustrate multiple representations of a concept or discuss different teaching methods.

Since the substance of the papers is aligned with their preservice course content, students are required to complete their reflection in a week or less. This allows for a timely class discussion of student reactions. The responses must be a page in length and minimally address the specific prompts identified when the papers are distributed. Assessment of reactions is holistic, with a score of four assigned for those reactions which speak to each prompt and elaborate on at least one additional aspect of the paper. A score of three corresponds to a reaction which addresses each prompt but does not include any further expansion. Lower scores are assigned based on the number of prompts addressed. Typically eight paper reactions are assigned during a semester.

Sample Assignment and Student Reaction

The article selected for the following assignment, while quite brief, did engage the students in an elementary classroom dialogue focused on a discussion of subtraction of rational numbers and involved multiple representations. The article also illustrated the need for teachers to be adaptable and able to adjust a lesson plan, an important aspect of teaching analyzed by Ball and Bass (2000).

   Read the passage "When the Wrong Way Works" from
   Mathematics Assessment: Cases and Discussion Questions for Grades
   K-5, pages 24-25 (Bush, 2001). Your reaction should minimally
   consider the following:

   a) Besides the number line method, what else might the teacher have
   used to help the students?

   b) Choose another method and construct a dialogue that may have
   occurred if that model was used instead of the number line.

   c) Comment on your ability to think-on-you-feet and find other ways
   to look at a problem to enable learning.

The proposed dialogues typically exhibited a teacher-as-coach model and involved praise for correct answers, as seen in sample response that follows.

   In the article, "When the Wrong Way Works," the students displayed
   the tact that many children in school today learn the methods of how
   to do things without learning why they work. If they had thought
   about what the problem was asking for they would have seen that
   simply changing the denominator changes the problem completely ...
   The teacher could have taken some fraction strips with forths [sic],
   tenths and twentieths to show what the problem was asking for.

   "Okay we have 3 of the one-tenths strips and we need to take away 1
   one-forth strip"

   "Now, these sizes do not subtract nicely, so what do we need to do?"

   "Find a common denominator." "It's 20!"

   "Very good Johnny, so we take the 20th strips and match them with
   the 10ths & 4ths."

   "How many 20th strips match 3 10th strips?" "6"

   "Good, and how many 20th strips match 14th strips?" "5"

   "Good, so we now have 6/20 minus 5/20 leaving us with ...?" "1/20"

   ... Thinking on your feet is critical for teachers because the best
   teaching opportunities are reactions to students' questions while
   the thoughts are still flowing. (Student response, April 2003.)

The dialogue constructed by this student mimicked that of the original paper. Some students had very little dialogue, with the teacher providing a short lecture. While this student chose fraction strips, others utilized circle graphs or fraction bars. This variety was the basis for a whole-class discussion of the necessity for teachers to have a ready repertoire of concept representations. It was observed in the class discussion of the varied teacher roles, that for some students the focus was on developing a correct procedure rather than mathematical thinking. This important contrast of procedural and conceptual knowledge emerged unexpectedly through class discussion, and as such, may have a more lasting impact on their beliefs about mathematics teaching, particularly when reinforced in a subsequent methods course.

For this assignment, the students' contemplations of their ability to think-on-their-feet were quite revealing. Many recognized the importance of developing an inventory of representations. Some also acknowledged their own short-comings in sticking with one way that works. Others expressed a sense of apprehension for such a scenario in their future classroom. These reactions and beliefs are consistent with research (Fennema and Franke, 1992). Through this assignment students reflected on these beliefs, and connected the learning of content in their future classrooms with the content of this mathematics course.

Reflection

Reading students' reactions leads me to the conclusion that these exercises are productive in (a) making students aware of the K-8 curriculum, (b) illustrating the importance of multiple representations and (c) providing an understanding of why their preservice content courses cover particular topics and focus on understanding rather than algorithms. The following sample response to a reading from The Mathematical Education of Teachers (MAA, 2001) suggests that the activity successfully challenges students to contemplate the curriculum and the need for teachers to have a deep understanding of the topics they will teach.

   I am totally blown away after reading this article, to discover
   that there are teachers out there that have trouble understanding
   the material they teach to their students. As a future teacher I
   couldn't imagine myself in a similar situation, and hope it's one
   I'll never have to experience. To me it's a scary thought. I want
   to be totally prepared and totally skilled enough to be able to
   explain to my students why concepts are the way they are, and be
   able to provide proofs and verify them.... While reading this, I
   noticed a lot of the content mentioned we covered in class, ...
   (Student response, April 2003.)

The following student reflections on the use of paper reactions from December 2003 reinforce my commitment to continue this reflective practice in future courses.

   I had never realized that their [sic] were cultural differences in
   mathematics until I took this class. The handouts that we read and
   reacted on were great eye openers for me. I have learned to become
   a more open-minded person and in turn, I will be a more open-minded
   teacher. (Student response, December 2003.)

   The papers were probably my favorite part of this course. I liked
   being exposed to different teaching resources and read of ways to
   improve instructional methods. I believe that this contributes to
   making me a better teacher before I even step foot in a classroom.
   (Student response, December 2003.)

Not all students looked favorably on the required readings and reactions, though none were emphatically negative. The reading and comprehension level of particular students may have influenced their feelings about these activities as seen with the following reaction.

   Some of the papers I read helped show me new ways of teaching
   mathematics, but some I found more confusing than anything. Some of
   the ones that tried to talk about three different things at once
   became jumbled in my head and I lost track of what I was supposed
   to be focusing on. But for the most part they did help. I got to
   look at other ways of looking at how to bring about a concept in
   the classroom. (Student response, December 2003.)

Through the use of these reflective reading assignments, students are exposed to resources beyond the typical texts used in preservice content courses, prior to their methods course. This early exposure may validate the content and methods employed in the preservice courses. Although attitudes and beliefs are difficult to change (Tatto, 1998), these activities immerse students in actual classroom curriculum and pedagogy and therefore are powerful complements to the content of preservice courses.

Extensions

Although the criteria for selection of articles are framed in the context of a mathematics course, the central ideas of actual student work, curricular alignment with standards, and alternative teaching strategies are relevant for other disciplines. Reading selections primarily have been found in publications of the NCTM, such as the journal, Teaching Children Mathematics, or the publication, Reflecting on Practice in Elementary School Mathematics (Teppo, 1999). Interested teacher educators in other content disciplines will find similar resources from national organizations. For example, the National Science Teachers Association (NSTA) publishes the journal, Science and Children, which includes articles written by teacher educators and elementary school teachers that include classroom vignettes and illustrate alignment with NSTA standards. Likewise, an excellent source for standards-based classroom activities involving social science can be found in the journal, Social Studies and the Young Learner, a publication of the National Council for the Social Sciences.

References

Ball, D.L. & C Bass, H. (2000). Interweaving content and pedagogy in teaching and learning to teach: Knowing and using mathematics. In J. Boaler (Ed.), Multiple Perspectives on the Teaching and Learning of Mathematics (pp. 83-104). Westport, CT: Ablex.

Brown, C.A. & Borko, H. (1992). Becoming a mathematics teacher. In D. Grouws (Ed.), Handbook of Research on Mathematics Teaching and Learning (pp. 209-239). New York: Macmillan.

Bush, W. S. (Ed.). (2001). Mathematics Assessment: Cases and Discussion Questions for Grades K-5. Reston, VA: National Council of Teachers of Mathematics.

Fennema, E. & Franke, M. L. (1992). Teacher knowledge and its impact. In D. Grouws (Ed.), Handbook of Research on Mathematics Teaching and Learning (pp. 209-239). New York: Macmillan.

Karp, K.S. (1991). Elementary school teachers' attitudes toward mathematics: The impact on students' autonomous learning skills. School Science and Mathematics, 91 (6), 265-270.

Lasley, T. (1980). Preservice teacher beliefs about teaching. Journal of Teacher Education, 31(4), 38-41.

Mathematical Association of America. (2001). The Mathematical Education of Teachers. Issues in Mathematics Education, Vol. 11. (2001). Washington: Author. Retrieved January 25, 2004 from, http://www.cbmsweb.org./MET_Document/index.htm

National Council of Teachers of Mathematics. (2000). Principles and Standards for School Mathematics. Reston, VA: Author.

Stevens, C. & Wenner, G. (1996). Elementary preservice teachers' knowledge and beliefs regarding science and mathematics. School Science and Mathematics, 96 (1), 2-9.

Stuart, C. & Thurlow, D. (2000). Making it their own: preservice teachers' experiences, beliefs, and classroom practices. Journal of Teacher Education, 51 (2), 113-121.

Tatto. M.T. (1998). The influence of teacher education on teachers' belief about purpose of education, roles, and practices. Journal of Teacher Education, 49 (1), 66-77.

Teppo, A. R. (Ed.) (1999). Reflecting on Practice in Elementary School Mathematic. Reston, VA: National Council of Teachers of Mathematics.

Thompson, A.G. (1992). Teachers' beliefs and conceptions: a synthesis of the research. In D. Grouws (Ed.), Handbook of Research on Mathematics Teaching and Learning (pp. 127-146). New York: Macmillan.

Veenman, S. (1984). Perceived problems of beginning teachers. Review of Educational Research, 54 (2), 143-178.

Sherrie J. Serros, Western Kentucky University 

Sherrie Serros, associate professor of Mathematics, is interested in preservice elementary teachers' motivation and learning of mathematics.