The errors and imprecisions in teaching mathematics are part of mathematics classrooms and might negatively influence student learning (Ball & McDiarmid, 1990; LMT, 2011). The errors and imprecision dimension is one of five in Mathematical Quality of Instruction (MQI). The other domains are the classroom work is connected to mathematics, the richness of mathematics, working with students and mathematics, and common core-aligned student practices.  MQI (2014) defined the errors and imprecision dimension as "teacher errors or imprecision in language and notation, or the lack of clarity/precision in the teacher's presentation of the content"(p.19). Most studies in the literature are interested in possible errors, misconceptions, or difficulties in learning statistics (e.g., Batanero et al., 1994; Capraro et al., 2005). To illustrate, certain studies conducted on revealing possible errors or difficulties made by students during interpretations of graphs (e.g., Aydın-Güç et al., 2022 for scatterplots; Capraro et al., 2005 for bar, line, and circle graphs; Edwards et al., 2017 for boxplots). On the other hand, there are also studies showing pre-service teachers’ errors or imprecisions on graphs as well (e.g., Işık et al., 2012 for line graphs; Ulusoy & Çakıroğlu, 2013 for histogram). However, there is a lack of research on what kinds of teacher errors and imprecision are present in teaching statistics. It is essential to explore teacher errors and imprecision to learn from them for not to transfer inaccurate information teachers possess to the students (Ball & McDiarmid, 1990). By exploring teacher errors and imprecisions in teaching practices, it is possible to identify areas of improvement, leading to more effective and engaging instruction and better student outcomes. Ultimately, understanding teacher errors and imprecisions is crucial in promoting high-quality mathematics education (LMT, 2011). In light of this gap in the literature, the purpose of this case study is to explore two 7th-grade mathematics teachers' statistics teaching with regard to errors and imprecision. The central research question guiding this study is: What types of teacher errors and imprecisions are present in 7th-grade mathematics teachers' statistics teaching?

This study is a part of a qualitative case study that allows for an in-depth examination of teaching practices within the real-life classroom context (Creswell & Plano Clark, 2007). Two middle school mathematics teachers were selected through purposive sampling with the following criteria: teaching 7th grade, having an undergraduate degree in a middle school mathematics education program, having at most 12 years of experience teaching, and working at the current school for at least two years. I observed teachers' instruction while teaching statistics. Fourteen hours for the Cem teacher and 13 for the Esra teacher were video and audio-recorded in order to explore the quality of instruction, specifically the errors and imprecisions dimension for this proposal. I analyzed all videos with three elements in this dimension. Mathematical Content Errors (MCE), Imprecision in Language or Notation, Lack of Clarity in Presentation of Mathematical Content, and Overall Errors and Imprecision are the codes for the dimension. This dimension only considers the errors not corrected during the segment. I assigned Not Present (1), Low (2), Mid (3), and High scores (4) for the codes to 7.5-minute segments determined by the MQI instrument. In this dimension, Not Present (1) means that the segment is free from errors and imprecisions, and high (4) showed that the segment consists of a significant amount of errors and imprecision.

The results showed that the teachers' statistics instructions did not include errors and imprecision in most segments for all dimensions (59% for Cem, and 66.7% for Esra). The instructions included brief errors and imprecision (11.5%for Cem and 8.8% for Esra). To exemplify the Mathematical Content Error code, both defined the mode as the most frequent number instead of the value. They did not focus on the data type in their lessons and mostly worked on examples with quantitative data while teaching average. The definition does not obscure statistics in those examples. However, students made errors in the examples with categorical data; they reported frequency numbers as the mode of the data set. Teacher definitional error might lead to student error. Also, some segments included high content errors due to the inconsistencies between the graph's aims and the context in constructing a graph. Both teachers used ordinal data on the x-axis in a line graph task similar to the study of Işık et al. (2012). All in all, detecting and learning from teacher errors and imprecision might prevent possible misconceptions in student learning (Ball & McDiarmid, 1990). I will provide the results related to other codes with further discussion.

Aydın-Güç, F., Özmen, Z. M., & Güven, B. (2022). Difficulties scatter plots pose for 11th-grade students. The Journal of Educational Research, 115(5), 298-314. Ball, D. L., & McDiarmid, G. W. (1990). The subject matter preparation of teachers. In R.Houston (Ed.), Handbook of research on teacher education (pp. 437-449). New York: Macmillan. Capraro, M. M., Kulm, G., & Capraro, R. M. (2005). Middle grades: Misconceptions in statistical thinking. School Science and Mathematics, 105(4), 165-174. Creswell, J. W., & Plano Clark, V. L. (2007). Designing and conducting mixed methods research. Thousand Oaks, CA: Sage Edwards, T. G., Özgün-Koca, A., & Barr, J. (2017). Interpretations of boxplots: Helping middle school students to think outside the box. Journal of Statistics Education, 25(1), 21-28. Işık, C., Kar, T., İpek, A. S., & Işık, A. (2012). Difficulties Encountered by Pre-Service Classroom Teachers in Constructing Stories about Line Graphs. International Online Journal of Educational Sciences, 4(3), 644-658 Learning Mathematics for Teaching Project. (2011). Measuring the mathematical quality of instruction. Journal of Mathematics Teacher Education, 14(1), 25-47. Ulusoy, F., & ÇAKIROĞLU, E. (2013). İlköğretim matematik öğretmenlerinin histogram kavramına ilişkin kavrayışları ve bu kavramın öğretim sürecinde karşılaştıkları sorunlar. İlköğretim Online, 12(4), 1141-1156.