Sociomathematical Norms
Sociomathematical norms describe normative ways for students and instructors to interact in the classroom. Compared to other classroom social norms, such as respecting one's peers and asking questions in class, sociomathematical norms are specific to creating and justifying mathematics.
Two general principles guide the establishment of sociomathematical norms (Cobb, Wood, & Yackel, 1991).
- The intructor should explicitly establish norms early in the semester, ideally in the first week.
- The instructor should reinforce these norms, in action, regularly throughout the semester.
In the video shown below, we see how an instructor might explicitly establish norms in the classroom. Then, for each of the specific norms listed below, we share video clips of the instructor reinforcing those norms--and students enacting those norms--at various times throughout the semester.
The instructor regularly asked students to work on proofs and logical reasoning tasks, in groups, at their tables. Research shows that groups collaborate more effectively when each group member is assigned a role (Johnson & Johnson, 1987; Johnson et al., 1998). Therefore, the instructor set expectations for engagement by assigning roles to students at each table: recorder, reporter, and facilitator. Students rotated roles each day.
Recorder. The recorder makes sure the group's ideas and solutions are written down.
Reporter. The reporter is responsible for sharing the group's ideas and throughts during whole class discussion. They speak for the group, not just their personal viewpoint.
Facilitator. The facilitator makes sure the group stays on task and that all members contribute their ideas.
In the video shown below, a group/table of students discusses an application of Rolle's theorem (see task document linked below). Note how multiple students share their ideas, ask each other questions, and ultimately come to agreement on the validity of a logical argument.
References:
This norm demonstrates value for students own reasoning, and for them to express that reasoning--not what the student thinks their teacher or peers want to hear.
Students were encouraged to think aloud in whole-class discussions, as well as during group discussions. In the following video, a student reasons aloud about the logical equivalence of a logical implication (P implies Q) and its contrapositive (not Q implies not P).
Mathematical statements often include unfamiliar symbols, as well as terms that might seem familiar but have unfamiliar and precise definitions. Making personal meaning for symbols and terms is one way students become part of the mathematical community in which those symbols and terms are defined.
In the next video clip, the instructor introduces "short-hand" symbols for representing terms in logical arguments.
And here, the instructor engages the class in a discussion about how logical arguments can be worded in different ways. She refers to statements included in the document linked below, wherein words like "whenever" and "provided" are used to express "if."
Classroom discussion continues with a student sharing her interpretation of the word "necessary."
In the document linked below the video, you can see results from one group's work in interpreting each of the statements listed in the prior document.
Students often view their teacher as the final authority for making judgments about mathematical arguments. This view can be harmful to students' development into mathematicians who can rely on their own logical reasoning.
As part of his Teaching for Robust Understanding (TRU) Framework, Alan Schoenfeld (2014) has identified five dimensions of mathematically powerful classrooms. One of these dimensions is agency, ownership, and identity which Schoenfeld (2014) defines as:
The extent to which students have opportunities to conjecture, explain, make mathematical arguments, and build on one another’s ideas in ways that contribute to their development of agency (the capacity and willingness to engage mathematically) and authority (recognition for being mathematically solid), resulting in positive identities as doers of mathematics.
In the next video, the instructor shares a proof generated by one of the groups and then asks the rest of the class to evaluate its validity.
Reference:
Schoenfeld, A. H. (2014). What makes for powerful classrooms, and how can we support teachers in creating them? A story of research and practice, productively intertwined. Educational researcher, 43(8), 404-412.
Students should refine colloquial language into conventional language for communicating mathematical logic. This entails understanding the distinctions between how colloquial uses and conventional uses may not correspond to the same logic.
This refinement should be bidirectional:
- Students should refine the logical meanings of terms they encounter in mathematical statements.
- Students should refine their own use of everyday language to form precise mathematical statements.
In the video shown below, the instructor is explaining how a "for all" statement can be disproven with a single counterexample, and a student asks about the use of the common term "for some" in the statement. The student seems to be disambiguating the use of "some" to describe a single number, rather than a plurality of numbers.
In this final video, the instructor explains why the word "opposite" does not fit with the negation of a statement. For example, the negation of "hot" is not "cold" (its opposite) but rather "not hot."
This particular use of common language--the word "opposite" to describe a counter-argument--might contribute to a potential epistemological obstacle in which students conflate the negation of a logical implication (P implies Q) with the opposite implication (P implies not Q).