Teachers’ Developing Talk About the Mathematical Practice of Attending to Precision Samuel Otten, Christopher Engledowl, & Vickie Spain University of Missouri, USA

Rationale  Mathematical practices, such as reasoning, problem solving, and attending to precision, are important for students to experience but difficult for teachers to enact successfully.  The Common Core (2010) Standards for Mathematical Practice explicitly include attending to precision (SMP6).  Precision of computations and measurement  Precision of communication and language (Koestler et al., 2013)  In order to support teachers in enacting SMP6, we need to understand how they interpret this mathematical practice. 2

Research Question  How do middle and high school mathematics teachers talk about the mathematical practice of attending to precision?  Initially – based on the Common Core paragraph description  Over time – based on extended experiences with the SMPs 3

Project Overview  Participants: Eight mathematics teachers (grades 5-12)  Five Summer Study Sessions centered around the Standards for Mathematical Practice from Common Core (15 hours)  Data Sources  Audio/Video recordings  Teacher written work  Focus on Attending to Precision (SMP6)  Session 1 – brainstorm, discussion based on Common Core paragraph  Session 3 – reading, task, transcript, and related discussions 4

Analysis  Sociocultural/Sociolinguistic perspective (Lave & Wenger, 1991; Halliday & Matthiessen, 2003)  Lexical chains and thematic mappings (Herbel-Eisenmann & Otten, 2011; Lemke, 1990) 5

Analysis 6 TOPIC 1TOPIC 2TOPIC 3 XXX TERM relation

Preliminary Results 7

Initial Discourse about SMP6  Precision as appropriate rounding within a problem context  Emilee: Knowing when to round versus when to truncate. Like, if you need 8.24 gallons of paint, what’s an acceptable answer for that? Nine’s a great answer but what about 8 gallons and one quart? And that could get into the discussion.  Teachers provided other examples  $  3 and a half people  Negative kittens 8

Initial Discourse about SMP6  Precision as correct use of vocabulary / mathematical language 9 Unofficial Vocabulary Official Vocabulary rootszeros x-intercepts MARF factoring by grouping xy-plane coordinate plane Examples SYNONYMS

Initial Discourse about SMP6  Precision as correct use of the equal sign (=) 10 2x – 5 = 13 2x = 18 = x = 9 2(8) = = 21 ÷ 7 = 3 2x + 5 7

Later Discourse about SMP6  Vocabulary comes up again with regard to precise communication, but it is connected to precision in reasoning.  E.g., carefully formulated argument  Precision with symbols are discussed with regard to possible misinterpretations.  E.g., 2a in the denominator of the quadratic formula  Using parentheses to clarify expressions 11

Later Discourse about SMP6  With regard to number/estimation, precision as an awareness of exactness vs. inexactness  E.g., 1/3 vs  “If you round in step one, and then you round in step two, and round in step three, each time you’ve gotten further and further and further…”  Dilemma about how to push students toward precision without turning them off. Which students should be pushed and when? 12

Discussion  Initial talk focused on student errors and a desire for more correctness (as opposed to precision, per se).  The distinction between precision and correctness may be important to make explicit as we support teachers in enacting SMP6.  Initial talk did involve both rounding/measurement and language, but these became more nuanced and comprehensive in later discussions.  Discussions of classroom examples where SMP6 occurred seemed helpful in promoting new ideas in the teacher’s discourse. 13

Acknowledgments  Thank you for coming  Funding provided by the University of Missouri System Research Board and the MU Research Council  We appreciate the participation of the teachers and students who made this study possible 14

References Halliday, M., & Matthiessen, C. M. (2003). An introduction to functional grammar. New York, NY: Oxford University Press. Herbel-Eisenmann, B. A., & Otten, S. (2011). Mapping mathematics in classroom discourse. Journal for Research in Mathematics Education, 42, Koestler, C., Felton, M. D., Bieda, K. N., & Otten, S. (2013). Connecting the NCTM Process Standards and the CCSSM Practices. Reston, VA: National Council of Teachers of Mathematics. Lave, J., & Wenger, E. (1991). Situated learning: Legitimate peripheral participation. Cambridge, England: Cambridge University Press. Lemke, J. L. (1990). Talking science: Language, learning, and values. Norwood, NJ: Greenwood Publishing. National Governors Association Center for Best Practices, & Council of Chief State School Officers. (2010). Common Core State Standards for Mathematics. Washington, DC: Author. 15