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Running head: ACTIVE LISTENING
Students Actively Listening: A Foundation for Productive Discourse in Mathematics Classrooms
Samuel Otten, Beth A. Herbel-Eisenmann, & Michael Steele
Michigan State University
Michelle Cirillo
University of Delaware
Heather M. Bosman
Michigan State University & Mason High School, Mason, MI
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Students Actively Listening: A Foundation for
Productive Discourse in Mathematics Classrooms
Engaging students in rich, subject-specific discourse can increase students’ opportunities
to learn that subject (Gibbons, 2009; Schleppegrell, 2004), and mathematics appears to be no
exception (Lampert, 2001; Pimm, 1989). Recent national standards documents have advocated
for discourse-rich environments in mathematics classrooms to foster student development of
reasoning and communication (National Governors Association, 2010; National Council of
Teachers of Mathematics, 2000). Inquiry into mathematics classroom discourse has typically
focused on written and oral communication (e.g., Esmonde, 2009a; Herbel-Eisenmann, 2007;
Staples, 2007) rather than a foundational component of productive discourse—listening. When
listening has been addressed in the literature, the topic is typically teacher listening (Davis, 1997;
Jacobs, Lamb, & Philipp, 2010; Peressini & Knuth, 1998). The present study, however, focuses
on student listening as a basis for meaningful mathematical discourse.
Just as it is important for teachers to attend carefully to student thinking as a prerequisite
to orchestrating productive discussions, it is equally important for students to attend to one
another’s thinking for productive discourse to occur. Because investigations of this aspect of
classroom discourse are scarce, we focus on student listening as a basis for meaningful
collaborative development of mathematical ideas. In what ways can active listening by students
support discussion and move mathematical ideas forward? We address this question by
qualitatively examining classroom interactions that exhibit potential benefits of active listening.
Theoretical Framework
This work is based on the broad notion that participating in the practices (Lave &
Wenger, 1991) and discourses (Gibbons, 2009; Lemke, 1990) of an academic subject is
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inseparably linked to the learning of that subject. From this perspective, one of the teacher’s
roles is to act as a more knowledgeable other, enculturating students into the discourse of the
discipline. Additionally, it is important to provide students with rich opportunities to discuss
mathematical ideas. But what characterizes “rich opportunities”?
With respect to mathematics, rich discussions build toward learning goals such as
connections of mathematical ideas or formalizations of mathematical concepts (Staples, 2007;
Stein, Engle, Smith, & Hughes, 2008). With respect to discourse in general, the nature of rich
discussions can be illuminated by distinguishing between univocal and dialogic interactions.
Drawing from Wertsch and Toma (1995), Peressini and Knuth (1998) explained that univocal
interactions function to convey information between individuals with the goal of having the
receiver’s final construed meaning closely resemble the sender’s original meaning. Dialogic
interactions, on the other hand, function to create new meanings among individuals using
utterances and texts as thinking devices around which to develop shared meaning. Although any
instance of discourse is both univocal and dialogic because individuals must decipher language
and also construe meaning, we follow Peressini and Knuth (1998) in recognizing the value of
identifying particular interactions as more univocal or more dialogic in nature.
In traditional mathematics classrooms, univocal discourse is dominant with the teacher as
the transmitter and arbiter of mathematical knowledge (Smith, 1996). Students may only be
required to engage in passive listening, attentively receiving the speaker’s meaning.
Alternatively, in mathematics classrooms where teachers and students are expected to make
sense of one another’s ideas, active listening is required for individuals to build meaning together
and collaboratively develop ideas (Hufferd-Ackles, Fuson, & Sherin, 2004). This involves
working to understand a speaker’s meaning by asking questions (either internally or aloud) about
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the ideas being shared, seeking clarification when needed, and placing those ideas into dialogue
with the listener’s own ideas and experiences. Moreover, if broad participation in the classroom
is a goal, then students must be prepared to shift fluidly between listening and speaking, which
would seem more likely to occur when active listening is taking place.
An additional point is that, within dialogic interactions, communication is not
unidirectional because listeners actively work to make sense of the speaker’s ideas while the
speaker simultaneously attends to listeners’ progress in understanding these ideas. As Esmonde
(2009b) explained, “[i]f students focus on their own mathematical explanation rather than the
hearers’ understanding, they may be hampering the cooperative learning process” (p. 1013–
1014). In other words, opportunities for productive learning within mathematical discussions are
increased when all participants are actively listening to one another. One teacher talk move that
has been identified as playing a role in the cultivation of rich mathematical discussions (Chapin,
O'Connor, & Anderson, 2009) is asking students to revoice or rephrase an idea shared by another
student. Echoing Hufferd-Ackles, Fuson, and Sherin (2004), who viewed students “[explaining]
other students’ ideas in their own words” (p. 89) as evidence of listening for understanding, we
conceptualize a connection between active listening and student revoicing. In particular, student
revoicing may serve as evidence of active listening and asking students to revoice one another
may promote norms of active listening.
Method
Classroom observation data (video and field notes) gathered from a multi-year study
(Herbel-Eisenmann, PI; for details, see Herbel-Eisenmann, Drake, & Cirillo, 2009) is currently
being used as the basis for creating professional development materials related to teacher
discourse moves (Herbel-Eisenmann, Steele, and Cirillo, co-PIs). Because of the theoretical
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connection between active listening and student revoicing mentioned above, we examined
instances of student revoicing to identify classroom excerpts that seemed to contain active
listening. The first excerpt included here comes from the mathematics classroom of a middleschool teacher who was involved in a five-year research and professional development project
focused on purposeful discourse practices (see Herbel-Eisenmann & Cirillo, 2009). Ms. K, a 14year veteran teacher in a suburban Midwest district, joined the project motivated to improve the
quality of discourse in her classroom. The second data excerpt comes from a calculus course at a
large public Midwestern university in which the first author was an instructor (Otten, Park,
Mosier, & Kaplan, 2009).
Ms. K’s data excerpt involves a whole-class discussion, transcribed in a script style with
bracketed descriptions of significant gestures and actions. Because the calculus data excerpt
involves two students working as partners, an audio recording of the pair was used in addition to
the whole-class video. As mentioned above, the presence of student revoicing led to the initial
selection of these excerpts. To illuminate active listening and dialogic interaction, further
analysis included topic tracking (Schiffrin, 1994), to trace mathematical ideas as they moved
between students, and thematic analysis (Lemke, 1990), to identify how the mathematical ideas
developed in the discourse over time. We also highlighted the ways students checked back with
one another after rephrasings and the emergent ideas distinct from ideas initially present in the
discourse, which suggest dialogic development of meaning.
Findings
The analysis of these classroom episodes suggests that active listening can play an
important role in making mathematical ideas available in the public discourse and, ultimately, in
the mathematical ideas students may take away from the conversation. The following two
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excerpts illustrate the role of listening in the discourse and sense-making of the mathematical
ideas. In the first example, we show how a teacher promotes active listening in the classroom
discourse and show how active listening seemed to help students understand another student’s
solution. The second excerpt shows how active listening seemed to support the actual solution of
a calculus problem. One important difference between these two excerpts, which we return to in
the final section of the paper, is that the first example is orchestrated by a teacher, whereas the
latter example occurs between two students working together in a small group.
Example 1
The first example is from Ms. K’s classroom. After having students find the intersection
of the lines in Figure 1, Ms. K facilitated a whole-class discussion in which she prompted
students to consider multiple solutions. One of the first students to share his approach was
Wheeler (Table 1).
Figure 1. How might one determine the intersection point of these two lines?
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Table 1
Wheeler’s explanation for his solution to the intersecting lines task (see Figure 1)
Student
Explanation
Wheeler: [at the board] OK, well, these two lines are two squares away [points to the xaxis], and so it's one, two, three, four, five, six up [counting up to the top of the
grid] until the next one where it's only one square away. And then, since at six,
right there [points to the top of the grid], it'd have to be--, you'd have to go six
more to get one square close, so it'd be twelve to the next one.
Ms. K elicited from Wheeler the solution (6,12) before moving on to ask other students for
“summaries” of Wheeler’s method. As students attempted to revoice Wheeler’s explanation
(Table 2), Ms. K asked Wheeler to remain at the board to “answer questions” from the students.
Table 2
Revoicings of Wheeler’s explanation for the intersecting lines task
Student
Revoicing
Sage: [from her seat] He said that he went up six, and that was how many until
another point. And then it went over one, or something. I don't know.
Carly: [at the board] I think he said this is six [points to the top of the grid], this is two
[points to the gridline at x=2]. And he--, I think he guessed how he got six. This
is from six [points to top of the first line segment], this is from the six [points to
the top of the second line segment], and--, I forgot. Sorry.
7
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Madlin: [at the board] Uh, where he drew the line here earlier [on the x-axis between
the two lines], there is two squares in between the points. And the next two
points that they got closer together [points to the top of the grid], they got six up
and one apart. And so he thought if he went up another six, they would meet.
Although the first two revoicings fell short of capturing Wheeler’s full explanation, they
do show evidence that students were actively listening to the ideas being shared. By attempting
revoicings, Sage and Chaney provided Ms. K (and Wheeler) with information about how they
were making sense of Wheeler’s method. This sense-making process continued as Sage, several
minutes later, said the following: “Well, I don't know if this is the same like Wheeler said, but if
you added another six squares up there, then it'd be zero [apart].” Although she seemed unsure of
how this new idea related to Wheeler’s original explanation, Sage seemed to recognize how the
pattern of horizontal distances led to the solution.
Madlin successfully revoiced Wheeler’s explanation, even mimicking some of his
gestures at the board, and a shared meaning seemed to have been produced. Yet one might
wonder whether passive listeners attempting mere repetition would produce similar rephrasings.
It is important, therefore, to consider the teacher’s role in these interactions. Ms. K asked
Wheeler to give the initial explanation and then asked students to revoice it. Following the two
incomplete revoicings, Ms. K mentioned to Wheeler that she was not sure the ideas were “clear
yet,” thus providing Wheeler with feedback regarding how his original communication might
have been heard by the class. After Madlin’s successful revoicing, Ms. K asked Wheeler if
Madlin had summarized “accurately.” Wheeler responded that she had, forming a feedback loop
between Wheeler and Madlin (i.e., Madlin’s contribution allowed Wheeler to hear how someone
Active Listening
else interpreted his explanation, and Wheeler’s confirmation allowed Madlin to know that her
rephrasing had been understood by Wheeler). Active listening and such shared meaning-making
may often be linked, but because of Ms. K’s scaffolding, this case may be viewed as a teacher
attempting to demonstrate and set norms for active listening and dialogic interaction. In the next
excerpt, students appear to engage in active listening without a teacher’s intervention.
Frank and Joe, undergraduate calculus students, were assigned as partners by their
instructor to solve an optimization problem (Figure 2). Their interactions around this problem
seemed to involve active listening (Table 3).
Figure 2. Where should D be located to minimize the time it would take a dog, starting at A, to
fetch a ball at B?
Table 3
Frank and Joe realize that Joe has labeled AD as x while Frank has used x to label DC
Student
Utterance
Frank: Hold on a second. [Looks back and forth between his work and Joe’s] Oh,
your time for the land and the water in the last one is wrong. ‘Cause if you
have seven meters divided by four, plus…
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Joe: I took it as--, you said seven meters was left, and so I only did one meter [on
land]. So these [entries in Joe’s table] are the answers for if you did only one
meter on land and then swam the rest. I didn’t think you meant where you
took, um…
Frank: So…
Joe: What you’re saying is that he ran seven meters [Frank: Uh huh.] and then
swam the last one?
Frank: Right.
Joe: Right. So, then this whole one--, then if this [DC] was one, then it’d be right.
Frank: Hold on, let me get my mind around this. [Joe: Alright.] OK, you’re saying
this [DC] is seven, and this [CB] is five, so then this [DB] is eight-point-six,
right?
Joe: Yeah.
Frank: Ahhh. OK. Alright.
Following this interaction, Frank and Joe filled in their tables, arrived at an estimated solution,
and then used the generating function for their table to proceed toward an analytic solution.
Joe began to work through the confusion pointed out by Frank by revoicing Frank’s
earlier utterance (“you said…”) about which distance on land he was calculating. Joe continued
by clarifying his interpretation of Frank’s land choice (“I didn’t think you meant…”). In Joe’s
next turn, he further explicated Frank’s position by verbally tracing the entire route that Frank
was using to fill in his table. Frank confirmed this interpretation twice (“Uh huh,” “Right”), thus
providing feedback to Joe about his understanding of Frank’s position at the same time that Joe
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was providing feedback to Frank about Frank’s past communications. Having established this
feedback loop, Joe validated Frank’s approach (“…then it’d be right”), squaring it with his own.
Frank then revoiced Joe’s land choice (“OK, you’re saying…”), checking with Joe about the
accuracy of this rephrasing (“Right?”), thus maintaining the feedback loop. Frank’s final
utterance in this excerpt was a proverbial “Ah ha” moment as, through a simple change of
perspective (looking at DC instead of AD), shared meaning was made.
Frank and Joe listened actively to one another, repeating back what they believed the
other to be saying, forming feedback loops by checking for and giving validation to those
revoicings. This dialogic interaction allowed them to avoid the potential confusion of their
arbitrary labeling choices (Arcavi, 2004) and continue in their problem-solving.
Discussion
This study contributes to the work related to mathematics classroom discourse and
student learning by focusing explicitly on the role of student listening. In particular, a distinction
is made between passive listening, which may be all that is necessary in traditional mathematics
instruction, and active listening, which is foundational for rich mathematical discussions.
Building from past work (e.g., Hufferd-Ackles, Fuson, & Sherin, 2004), this study attempts to
provide images of what active listening may look like in mathematics classrooms and what role
it might play in constructing shared meaning between students. By listening actively to one
another—which involves revoicings, verifications, questions, and clarifications—mathematical
reasoning can be made more explicit and more accessible. Additionally, more students can
become involved in articulating mathematical thoughts and developing shared meanings, thus
supporting learning.
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Many would agree that it is desirable to have students independently engaged in active
listening, as was seen in the second excerpt. Such interaction, however, may be unlikely to arise
spontaneously, suggesting that teachers have a role in fostering active listening as with Ms. K. In
fact, research that focuses on the development of norms in mathematics classrooms does so in
order to better understand the ways in which students become enculturated into valued practices
that support students’ learning of mathematics. Yet, a valuable part of that work that deserves
more attention is the ways in which these shifts—from teachers acting as more knowledgeable
others who model discourse practices to students appropriating the discourse practices in order to
further their own learning—take place. This study can serve as a basis on which to ground future
work that continues to explore the nature of active listening with respect to dialogic discourse, its
cultivation by classroom teachers, and its link to student learning.
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Acknowledgments
This work was supported by the National Science Foundation (NSF) under award numbers
0347906 and 0918117. Any opinions, conclusions, or recommendations contained herein are
those of the authors and do not necessarily reflect the views of the NSF. We thank the rest of our
project team for their help in the continual development of these ideas: Kate Johnson, Jen Nimtz,
Shannon Sweeny, Alexandria Theakston, and Rachael Todd as well as Lorraine Males and Faith
Muirhead.
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