Reflecting on the Teaching of Integer Addition and Subtraction via Knowledge Quartet: The Case of Mr. Tunç

Author(s):

Ramazan Avcu(presenting / submitting)Seher Avcu

Conference:

ECER 2014

Network:

24. Mathematics Education Research

Format:

Paper

Session Information

24 SES 02 A, Issues in Mathematics Teacher Education (Part 1)

Paper Session: to be continued in 24 SES 03

Time:

2014-09-02

15:15-16:45

Room:

B113 Sala de Aulas

Chair:

Javier Diez-Palomar

Contribution

It is widely accepted that pupil achievement is dependent to a large extent on the quality of teaching (Stronge, Ward, & Grant, 2011). Besides, mathematical content knowledge of teachers is regarded as an important factor in the teaching and learning of mathematics (Williams, 2008). Nevertheless, researchers identified that teachers have limitations in their mathematical content knowledge (e.g., Ball, 1990; Ma, 1999). Therefore, mathematics educators around the world attempted to develop measures or generate theories for deepening teachers’ mathematical content knowledge (e.g., Ball, Hill, & Bass, 2005; Rowland, Huckstep, & Thwaites, 2005). Ball et al. (2005) developed items to test both common and specialized content knowledge of teachers. According to Rowland, Turner, Thwaites, and Huckstep (2009), Ball et al.’s questionnaire might give some clues about teachers’ pedagogical content knowledge but might not reflect how teachers act in practice. Rowland et al. (2009) added that in order to assess teachers in their actual practice, we need to observe those teachers while they are teaching. By adopting this idea, in this paper we attempted to reflect on a middle school mathematics teachers’ teaching of integer addition and subtraction by using the Knowledge Quartet framework generated by Rowland, et al. (2005). This framework includes four broad categories: foundation, transformation, connection and contingency (Rowland, Turner, 2007; Turner, 2012). In particular, we used the transformation category of the Knowledge Quartet. This category is about “teacher demonstrations; use of instructional materials; choice of representation; choice of examples” (Turner, Rowland, 2011, p.200).

We selected the topic of integer addition and subtraction as the unit of analysis of the current study for several reasons. First, integers and integer operations are conceptually difficult for students (Gregg and Gregg, 2007; Janvier, 1983; Vlassis, 2004). Second, the topic of integer addition and subtraction strongly displays the teaching practice under scrutiny (Mitchell, Charalambos, & Hill, 2014). Third, since there is no single representation that can be used to teach integers and its operations (Stephan & Akyüz, 2012), teachers usually need to utilize several representations in order for students to make sense of the underlying ideas of integers.

We asked ourselves the following questions while reflecting on teacher’s teaching of integer addition and subtraction. Does the teacher use appropriate equipment; select appropriate forms of representation; choose appropriate examples; clearly explain ideas or concepts possibly by using an analogy; and demonstrate clearly and accurately how to carry out procedures? By focusing on the transformation category we tried to delve into teacher’s ability to transform his knowledge of integers to students in an accessible and appropriate way. Thus, the research question of this study was formulated as follows: How do middle school mathematics teachers transform their knowledge of integer addition and subtraction to their students?

Method

To find answers to the above mentioned research question, qualitative case study methodology was used. In particular, single case design with single unit of analysis (Yin, 2003) was employed to carry out the study. The context of the study was a middle school mathematics classroom, the case was a middle school mathematics teacher with 13 years of teaching experience, and finally the unit of analysis was middle school mathematics teacher’s teaching of integer addition and subtraction. Since our aim was to investigate experienced middle school mathematics teachers’ ability to transform their knowledge of integers to their students, we used purposeful sampling strategy when selecting the participant that best suited the research purpose. The participant of this study, Mr. Tunç (a pseudonym), was an experienced middle school mathematics teacher, who has been teaching at a public middle school located in a city center at the inner region of Turkey. Mr. Tunç is thirty six years old and completed his undergraduate education in mathematics in a public university of Turkey. He also completed a pedagogical formation certificate program during his undergraduate years since this certificate is compulsory for teaching at schools governed by Ministry of National Education. The data were gathered by the first author through direct observations and field notes. In total, five mathematics lessons taught by Mr. Tunç were observed and videotaped. However, since “much of the qualitative research involves observation by multiple observers as at least on the part of the data collection” (Wiersma, 2000, p.211), the second author watched the recordings of the videotape. Before coding the data, videotapes were transcribed verbatim. Then, the actions of Mr. Tunç in the videotaped lessons were analyzed by means of pre-existing codes generated by Rowland, Huckstep, and Thwaites (2005). For internal consistency we independently analyzed the whole data and reached approximately 90% agreement for all cases. Moreover, we tried to enhance the validity of our findings by stimulated recall interviews with Mr. Tunç when his actions were ambiguous.

Expected Outcomes

Direct observations on integer addition and subtraction suggested that metaphors, examples, explanations, and representations abounded in Mr. Tunç’s lessons. Mr. Tunç started introducing integers by having students feel the need for negative numbers. Particularly, Mr. Tunç let students think which buttons to press when they enter the elevator in the ground floor (0th floor) if they want to go down. Besides, he asked students to think how to express temperature below 0. Similarly, Mr. Tunç had students ponder how to express numerically the depth of a swimmer under the sea. In addition to introducing these examples verbally, he drew pictures on the white board to help students make better sense of these examples involving negative numbers. When generating examples, Mr. Tunç chose numbers less than 10 and took account of sequencing such as providing the addition of integers with same signs first and integers with opposite signs later. In teaching integer addition, Mr. Tunç used three different models: the real-world money context (i.e., debt and asset model); static pictures of counters that are non-manipulatable; and the diagram-based number line model. Finally, Mr. Tunç gave the following rules for integer addition: “If the two numbers have same signs, we first overlook the signs and add them as in natural numbers and then we put the common sign in front of the sum. However, if the signs are opposite, we again overlook the signs and then we subtract the smaller from the larger number and put the sign of larger number in front of the difference. Although Mr. Tunç used several different models or representations in the teaching of integer addition, he only used a single model in the teaching of integer subtraction. Actually, he invented the ‘blind man-smiling man” metaphor which is in fact rooted in the rules for subtracting integers.

References

Ball, D. L. (1990). The mathematical understandings that prospective teachers bring to teacher education. Elementary School Journal, 90(4), 449-466. Ball, D. L., Hill, H. C, & Bass, H. (2005). Knowing mathematics for teaching: Who knows mathematics well enough to teach third grade, and how can we decide? American Educator, 29(1), p. 14-17, 20-22, 43-46. Gregg, J., & Gregg, D. U. (2007). A context for integer computation. Mathematics Teaching in the Middle School, 13(1), 46-50. Ma, L. (1999). Knowing and teaching elementary mathematics: Teachers’ understanding of fundamental mathematics in China and the United States. London: Lawrence Erlbaum. Mitchell, R., Charalambous, C. Y, & Hill, H. C. (2014). Examining the task and knowledge demands needed to teach with representations. Journal of Mathematics Teacher Education, 17, 37-60. Rowland, T., & Turner, F. (2007). Developing and using the knowledge quartet: A framework for the observation of mathematics teaching. The Mathematics Educator, 10(1), 107-123. Rowland, T., Huckstep, P., & Thwaites, A. (2005). Elementary teachers’ mathematics subject knowledge: The knowledge quartet and the case of Naomi. Journal of Mathematics Teacher Education, 8, 255-281. Rowland, T., Turner, F., Thwaites, A., & Huckstep, P. (2009). Developing primary mathematics teaching. London: SAGE. Stephan M. & Akyüz D. (2012). A proposed instructional theory for integer addition and subtraction. Journal for Research in Mathematics Education, 43(4), 428-464. Stronge, J. T., Ward, T., & Grant, L. (2011). What makes good teachers good? A cross-case analysis of the connection between teacher effectiveness and pupil achievement. Journal of Teacher Education, 62, 339-347. Turner, F. (2012). Using the knowledge quartet to develop mathematics content knowledge: The role of reflection on professional development. Research in Mathematics Education, 14(3), 253-271. Turner, F., & Rowland, T. (2011). The knowledge quartet as an organizing framework for developing and deepening teachers' mathematics knowledge. In T. Rowland & K. Ruthven (Eds.), Mathematical knowledge in teaching (pp.195-212). New York: Springer. Vlassis, J. (2004). Making sense of the minus sign or becoming flexible in ‘negativity’. Learning and Instruction, 14, 469-484. Wiersma, W. (2000). Research Methods in Education: An introduction (7th ed.). Allyn & Bacon. Williams, P. (2008). Independent review of mathematics teaching in early years settings and primary schools. London: HMSO. Yin, R. K. (2003). Case study research: Design and Methods (3rd ed.). Thousand Oaks, California: Sage Publications.

Author Information

Ramazan Avcu (presenting / submitting)

Aksaray University

Aksaray

Seher Avcu

Aksaray University

Elementary Education