Examining Interactions Between Reasoning and Attending to Precision in Secondary Classrooms

• In Event: 14.043 - Roundtable Session 1
  In Roundtable Session: 14.043-14 - Analyzing the Common Core Standards for Mathematical Practice

Abstract

The mathematical practice of constructing arguments and critiquing reasoning (SMP3 in the Common Core State Standards, 2010), is central to both mathematics and mathematics education (Hanna, 2000; Hersh, 2009), in part because it connects to many other mathematical practices. For example, reasoning involves noticing regularities to make claims (SMP8) and attending to the precision of one’s communication (SMP6). These practices, though important for students to experience first-hand, are often dominated by teachers or textbooks (Bieda, 2010; Author, YEAR). The present study examined the following question: In what ways do students engage in SMP3, SMP6, and SMP8 in middle and high school mathematics classrooms?

Data consisted of multiple classroom observations of eight middle and secondary mathematics teachers in a Midwestern school district. We framed our work as a study of enacted curriculum (Remillard & Heck, 2014) from the perspective of socioculturalism (Gee, 1999) because we focus on students coming to legitimately participate in the disciplinary practices of mathematics. Analysis involved coding the classroom data for mathematical practices, using a scheme adapted from Fennell, Kobett, and Wray’s (2013) “look for’s,” and conducting thematic discourse analysis (Author, YEAR) on the instances of SMP3 and SMP6, specifically.

We briefly share results involving the interactions between SMP3 and SMP6 to illustrate the complexities of analyzing these mathematical practices. In one lesson, students in Mr. Forrest’s advanced algebra class explored several different function transformations and their graphs. Figure 1 depicts the coding of SMPs in this lesson wherein SMP3 occurred in long stretches and SMP6 also occurred as students refined conjectures and clarified language.

Figure 1. Coding for instances of SMPs.

For example, Mr. Forrest asked about the effect of scaling a function by real-valued a:

Rebecca: It affects the steepness of the graph.
Forrest: What do you mean by steepness?
Rebecca: If you did 2 times the f of x, then it would be more steep. Then if you did half, it would widen out.
Forrest: Are we ready to accept Rebecca’s conjecture as fact? (pause) Did anyone describe it something other than steeper?
Kyle: The y values increase.
Forrest: So how can we categorize this. How do I know when it's going to be steeper or narrower?
Jenny: If a is between 0 and 1, the graph will get wider.
Forrest: Does the evidence support that?

In this excerpt, Mr. Forrest questions Rebecca’s conjecture and invites the class to critique it and provide evidence for agreement or disagreement. This is another example of how SMP3 and SMP6 work together to form a more coherent and clear description of Rebecca’s observation. Thematic discourse analysis reveals how, in some cases, the language (SMP6) and the underlying idea being expressed (SMP3) both shifted in the classroom discourse and, in other cases, the language was refined while the underlying idea remained constant.

Overall, this study reveals complexities and interrelationships of SMP3 and SMP6 that are relevant to researchers who are analyzing these important practices and also to mathematics educators who are promoting their enactment with students.

Authors

• Samuel Otten, University of Missouri - Columbia

• Christopher Engledowl, University of Missouri