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Abstract

Learning to orchestrate class discussions that are based on students' mathematical thinking is one of the most difficult aspects of learning to teach in ways that build on students' mathematical experiences. Based on a research project in which student teaching was restructured so as to focus on using student thinking, we describe the steps of a process that teachers move through when using students' mathematical thinking. We also identify some roadblocks that keep student teachers from listening to and understanding, from recognizing, and from effectively using student mathematical thinking for classroom discussion. We discuss how understanding this process and these roadblocks can be useful to mathematics teacher educators in their work with preservice mathematics teachers.
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Learning to Use Students’ Mathematical Thinking to Orchestrate a Class Discussion
Blake E. Peterson
Keith R. Leatham
Brigham Young University
Abstract
Learning to orchestrate class discussions that are based on students’ mathematical thinking is
one of the most difficult aspects of learning to teach in ways that build on students’
mathematical experiences. Based on a research project in which student teaching was
restructured so as to focus on using student thinking, we describe the steps of a process that
teachers move through when using students’ mathematical thinking. We also identify some
roadblocks that keep student teachers from listening to and understanding, from recognizing,
and from effectively using student mathematical thinking for classroom discussion. We
discuss how understanding this process and these roadblocks can be useful to mathematics
teacher educators in their work with preservice mathematics teachers.
Keywords: Orchestrating Classroom Discourse, Preservice Teacher Education, Secondary
Student Teaching, Student Mathematical Thinking, Teacher Knowledge, Teaching Practice
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Learning to Use Students’ Mathematical Thinking to Orchestrate a Class Discussion
Mathematics classrooms wherein the teacher promotes mathematical discussion based
on students’ mathematical thinking
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, and then orchestrates that discussion in ways that
facilitate yet deeper mathematical thinking, provide opportunities for students to
meaningfully “struggle with important mathematics,” (Hiebert & Grouws, 2007, p. 387)
something that research has shown is critical for students to learn with understanding
(Hiebert & Grouws, 2007; Hiebert et al., 1997; National Council of Teachers of Mathematics,
1991, 2007). Orchestrating such discussions, however, seems to be one of the most difficult
aspects of this approach to teaching (Sherin, 2002a), particularly for novice teachers. This
paper reports on the results of a research project that tried to put student teachers (STs) in
situations where they were trying to elicit students’ mathematical thinking, and then studied
how they navigated the road to using that thinking in their teaching.
USING STUDENTS’ MATHEMATICAL THINKING
The NCTM (2007) recommended that mathematics teachers “orchestrate discourse
by… listening carefully to students’ ideas and deciding what to pursue in depth from among
the ideas that students generate during a discussion” (p. 45). It is this careful listening to, and
pursuit of students’ ideas that we refer to as using students’ mathematical thinking; we refer
to such opportunities to use students’ mathematical thinking as teachable moments. Although
teachable moment is not a well-defined construct in the literature, the idea of teachers
capitalizing on students’ mathematical thinking “in the moment” is frequently discussed in
the literature on mathematics classroom discourse (e.g., Doerr, 2006; Manouchehri & St.
John, 2006; Schoenfeld, 2008). In this section we describe a conceptualization of the process
of using students’ mathematical thinking that both informed and evolved over the course of
our study. In order to effectively use students’ mathematical thinking, teachers need to 1)
listen to and understand student thinking, 2) recognize the thinking as a teachable moment,
and 3) use the thinking for a mathematical and pedagogical purpose.
Listening and Understanding
Effective teaching “requires careful listening” (Erickson, 2003, p. ix), in large
measure because effective teaching builds on what and how students (particularly those
present) think. In order to use students’ mathematical thinking, teachers need to listen with
the intent of using that thinking in order to build the classroom understanding of the
mathematics at hand. Such listening must occur both when teachers explicitly elicit their
students’ ways of thinking as well as in the myriad moments that arise unexpectedly.
Although careful listening creates teachable moments that serve meaningful purposes beyond
content (Schultz, 2003), the listening to which we refer in this paper is content-specific. We
are speaking of listening to students’ mathematical thinking with the intent to use that
thinking in order to further the learning of mathematics for all students in the classroom.
From our countless daily interactions, each of us can attest to the fact that it is
possible to listen yet not understand. Thus, teachers may listen to students explain their
thinking but may not understand that thinking. In order to understand students’ mathematical
thinking, teachers themselves must have an understanding of the mathematical concepts at
hand (Ball, Lubienski, & Mewborn, 2001; Ma, 1999; Sherin, 2002b). Although we find it
valuable to consider listening and understanding as separate steps in the process of using
students’ mathematical thinking, it is often difficult to distinguish between the two in
practice.
Listening to students has long been both advocated and studied by educators (e.g.,
Confrey, 1993; Davis, 1997; Paley, 1986; Schultz, 2003). Davis, for example, considered
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By “students’ mathematical thinking” we mean students’, among other things, their solution
strategies, their justifications and reasoning, and their models and representations.
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different types of listening that led to different types of teacher actions. Listening was
characterized as evaluative when it was somewhat superficial and as interpretive when it
sought merely to understand; with both types of listening there was no apparent intention by
the teacher that the results of listening and understanding students’ thinking would inform or
redirect the lesson. Such listening could be classified as funneling (Wood, 1998), wherein a
teacher listens for student thinking that will lead toward a preconceived “best solution” and
away from alternative and wrong strategies.
In contrast to evaluative and interpretive listening, Davis’ (1997) hermeneutic
listening had at its very core the notion that students’ thinking would in large measure
determine the direction of the lesson. Wood’s (1998) focusing pattern encompasses Davis’
(1997) hermeneutic listening as well as other types of pedagogically-sound listening. In the
focusing pattern, the teacher listens for alternative and incorrect strategies as a means of
elevating (and focusing) students’ mathematical thinking toward important mathematical
ideas. We thus adopt focused listening to describe the listening in our process—listening with
the intent to understand and then to meaningfully use students’ mathematical thinking in
order to further mathematical objectives.
Recognizing the Teachable Moment
Once a teacher has listened to and understood a student’s mathematical thinking, he
must then recognize this moment as a teachable moment; he must recognize the potential
pedagogical and mathematical value in pursuing that thinking in order to be able to
eventually use it. Although good intentions and common content knowledge are likely
sufficient to allow a teacher to listen to and understand students’ thinking, recognizing such
thinking as a teachable moment takes a great deal of specialized knowledge. Recognizing
such moments may also depend on the goal of the lesson, unit or course and may depend on
how the student thinking that was shared fits with the flow of the lesson. Lewis & Tsuchida
(1998) quoted a Japanese teacher as saying, “A lesson is like a swiftly flowing river; when
you’re teaching you must make judgments instantly” (p. 15). Recognizing shared student
thinking as a teachable moment is one of the instantaneous judgments that are made during
lessons. Recognizing the pedagogical and mathematical value in students’ thinking in the
moment is a difficult step in the process of using students’ thinking, even for experienced
teachers (Chamberlin, 2005).
Using Students’ Mathematical Thinking Effectively
Once a teacher has listened in a focused way to students’ mathematical thinking, has
understood that thinking, and has recognized the mathematical and pedagogical value of that
thinking, he is in a position to use that thinking. In our conceptualization, the purpose of such
use is to help all students to gain a better understanding of the concept at hand. Thus the
thinking that is used may be correct or incorrect and this use is more than just sharing
different methods for solving the same problem. Effective use of students’ mathematical
thinking requires the teacher to orchestrate a discussion about the connections between the
different methods or discuss why some methods work and others do not. Effective use
involves more than explanations of the methods or thinking; it involves making explicit the
reasoning behind the thinking.
The Influence of Teacher Knowledge on the Process
We have found Hill, Ball & Schilling’s (2008) conceptualization of teachers’
knowledge to be a useful way to think about our process conceptualization—how one listens
to, understands, recognizes as valuable and uses students’ mathematical thinking. In their
conceptualization, Hill et al. consider subcategories of Shulman’s (1986) content knowledge
and pedagogical content knowledge. These subcategories begin to delineate some of the
specialized knowledge that teachers have and that others do not. Thus, although mathematics
teachers have mathematical knowledge shared with those who are not teachers (common
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content knowledge – CCK), teachers also have specialized mathematical knowledge; this
knowledge, such as knowledge of the affordances and constraints of various mathematical
representations and models, is not commonly held by the general public or even by working
mathematicians, but is integral to the work of a mathematics teacher (specialized content
knowledge – SCK). Hill et al. (2008) also consider different types of pedagogical content
knowledge: knowledge of content and students (KCS), knowledge of content and teaching
(KCT), and knowledge of curriculum.
We now consider how these types of knowledge aid us in viewing the process of
using students’ mathematical thinking. For example, in reflecting on our own experience as
mathematics teachers, a common scenario often gives rise to teachable moments—students
often make mathematical statements that are slightly incomplete or incorrect. Various types
of mathematical knowledge for teaching help us to take advantage of such teachable
moments. Our KCS might prompt us to listen very closely to the response to a particular
question because we know that students often have misconceptions in this area and that an
incomplete answer might be masking just such a misconception. Our knowledge of
curriculum might help us to recognize that the incompleteness of this response is likely
connected to a bit of mathematics that is just on the mathematical horizon (Ball, 1993), and
thus one that would be valuable to pursue. Our specialized content knowledge (SCK) might
help us use the incompleteness of this response as just the right motivation to discuss a
different representation or model for the mathematics at hand. We thus view mathematical
knowledge for teaching as critical for teachers to be able to carry out the process of
effectively using students’ mathematical thinking.
Illustrating the Process Conceptualization
In order to illustrate the process of using students’ thinking that we have just
described, we present here an episode from our data wherein a ST uses her students’
mathematical thinking quite effectively. We then point to the evidence within the episode and
from the STs’ reflections that allow us to use the process conceptualization in order to
interpret the episode.
ST Emily
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began her pre-algebra lesson by having students work individually for
several minutes on three true-false questions regarding similarity of geometric figures. She
then began a class discussion by asking for volunteers to share their decision on the first
statement: “All squares are similar.” Christopher volunteered and stated that he thought that
the statement was true because, in a given square, all of the sides are equal, all of the angles
are equal, and opposite sides are parallel. ST Emily followed up briefly and then asked
whether anyone thought the statement was false. Brandon volunteered, but also argued that
the statement was true, although for a different reason:
Brandon: I think it’s true because, no matter what, they both have parallel lines, and if
you just draw two sets of parallel lines you get a square.
ST Emily: Okay. What do parallel lines tell you about them being similar?
Brandon: You have to have two sets of parallel lines to be a square, so obviously, if
it’s a square they all have parallel lines so that’s what’s similar.
ST Emily: So, if something has parallel lines, that makes it similar?
Brandon: Well, to be a square they have to have parallel lines. If they’re a square,
they all have parallel lines. So if you have like four squares, they’re all similar
because they all have two sets of parallel lines.
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All names for STs and students are pseudonyms.
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By this point in the conversation there were many students in class who were shaking their
heads, saying “no, that’s not right,” and raising their hands in hopes of responding to
Brandon’s thinking.
ST Emily: Okay. What do you think about that Kayla?
Kayla: I think that’s not true—it is true, but—.
ST Emily: Okay. What’s not true and what is true?
Kayla then went to the board, drew two squares (one with side length 1 unit and one with side
length 2 units), and then argued that there was a common scale factor of 2 between the two
squares. Although it is never stated explicitly, apparently Kayla agreed with Brandon about
the pairs of parallel lines always existing in squares, but not in his use of this reasoning to
conclude that all squares are similar. Samantha then asked Kayla why she found the scale
factor as part of her explanation, to which Kayla responded, “Because that’s what we found
for similarity last time.” Samantha then continued the conversation:
Samantha: You could also kind of verify with a square and a rectangle. Rectangles,
they have to have parallel sides, but squares, all their side lengths have to be
the same.
ST Emily: Okay. So how is that different from what Brandon was saying?
Samantha: Brandon was saying that they just had to be parallel.
ST Emily: So can you show us a rectangle that’s not similar—that has parallel sides?
Samantha went to the board, drew a skinny rectangle and a square, and pointed out that each
figure has pairs of parallel sides. Then ST Emily asked the class whether the square and the
rectangle are similar:
Terry: Yeah, they have parallel lines. Because that one on top is the same as that
[apparently comparing the top and bottom sides of the square].
ST Emily: Okay, so we know that they are parallel, but does that make them similar?
[Various students say “No” and “Maybe.”] What do we have to know in order
for something to be similar? [Numerous students, including Brandon, start to
share their responses, some mentioning scale factor.] Let’s listen to Brandon
for a second. What did you say?
Brandon: Similar means they have something in common. The thing that they have in
common is parallel sides. [Lots of murmuring amongst the students]
ST Emily: It’s true that when we’re talking about something being similar they have
to have something in common. But when we’re talking about something being
mathematically similar—
Alex: It has to— You have to find the scale factor.
Cody: Doesn’t it have to have the same angles and the same sides?
ST Emily then orchestrated a discussion with Cody and others about the two criteria for
figures to be mathematically similar (i.e., scale factor and equal angles). Having done this, ST
Emily turned the conversation back to Brandon.
ST Emily: So, Brandon, is it enough for the lines to be parallel for it to be similar?
Brandon: No.
ST Emily: No. Why not?
Brandon: Because they have to have corresponding sides and then corresponding
lengths and stuff like that. Just parallel sides wouldn’t be enough.
ST Emily: Okay. Do we all understand that? It’s true that they do have something in
common, Brandon, but they’re not mathematically similar.
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We now analyze this episode according to the process of using students’ mathematical
thinking. We find evidence that ST Emily listened to Brandon’s initial contribution in her
first follow-up question, which also represents ST Emily’s attempt to better understand
Brandon’s thinking. Her second question demonstrates that she understood his claim. She
later reflected on the beginning of this conversation in this way: “I continued to push him to
explain until I understood that his definition of similar was having something in common.”
At least in part, it was ST Emily’s KCS that contributed to her ability to listen to and
understand Brandon’s thinking.
That ST Emily recognized this situation as a teachable moment and felt as though she
had taken advantage of it was also revealed in her reflection: “Then I was able to validate his
thinking and talk about what it means to be mathematically similar.” We find this statement
to be somewhat understated. ST Emily allowed the class to respond to Brandon’s thinking
and that thinking did not always directly address the underlying issue Brandon had raised. ST
Emily brought the conversation back to Brandon’s thinking on several occasions, thus
helping to focus Brandon and the rest of the class on the distinction between the common and
the mathematical definitions of similar. Her SCK provided her the ability to draw this
distinction and her KCT helped her to direct the class discussion in that direction. This
episode thus demonstrates the process of listening to and understanding, recognizing, and
effectively using students’ mathematical thinking and the role that mathematical knowledge
for teaching played in supporting ST Emily in carrying out that process.
METHODOLOGY
This study took place in the context of a larger project wherein we altered the
structure and purpose of student teaching in an attempt to emphasize the elicitation and use of
students’ mathematical thinking while deemphasizing survival and classroom management.
In this student teaching project, a pair of STs was placed with one cooperating teacher and
two or three pairs of STs at different schools formed a cluster. The STs taught at most one
lesson per week during the first 5 weeks of their 15-week student teaching internship. These
lessons were planned in their pairs but were taught individually and observed by the other
STs in the cluster, the cooperating teacher and the university supervisor. Following the
teaching of the lessons, a reflection meeting was held in which the STs who taught the lesson
would reflect on the goals of their lesson and on how the tasks of the lesson were intended to
meet those goals. The observers then had an opportunity to ask questions and to make
comments. As part of the altered structure, the STs also conducted directed observations and
student interviews, and wrote weekly reflection papers (all focused on students’ mathematical
thinking) as a means of processing and synthesizing what they were learning.
For this study six female STs (Emily, Christina, Holly, Megan, Jennifer and Ashley)
and three cooperating teachers were purposefully selected to participate. The STs were
chosen based on feedback from their past professors, who were asked to recommend students
who they felt were primed to excel during student teaching. Emily and Christina were placed
in a middle school and Holly and Megan were placed in a junior high school. These four STs
were placed with teachers who were approaching their instruction from an NCTM Standards
perspective and were using a reform-based curriculum. Jennifer and Ashley were placed in a
high school setting with a new teacher who taught fairly traditionally but was open to
learning new ideas and who supported the STs in implementing such ideas.
The data for this study consisted of video recordings of all ST lessons and the
accompanying reflection meetings as well as the reflection papers that the STs wrote
regarding these observations and reflection meetings. To identify candidates for teachable
moments, the reflection meetings were analyzed for specific comments made by the person
who taught the lesson or by observers. The comments of interest were those that made
reference to student thinking observed in the lesson. Once these comments were identified,
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the lesson was analyzed to locate the episode that was being referenced. The episodes were
then analyzed to determine whether they were teachable moments. This determination was
based on whether both researchers felt that they might have pursued the students’
mathematical thinking had they been teaching the class. In addition, the reflection papers
written by the STs who taught the lessons were analyzed to identify any additional thoughts
they had on the identified episode.
Once these teachable moments, as well as any comments or reflections about them,
were identified an analysis of how the teacher used the student thinking ensued. In that
analysis we viewed the episode through the lens of our conceptualization of the process of
using students’ mathematical thinking, seeking evidence as to whether each step was
accomplished by the ST. Having identified the stopping points in the process, we looked for
evidence of why the ST stopped the process where she did. This assessment was done by
evaluating comments made during the lesson, during the reflection meeting or in the
reflection paper. From these various data sources, we identified and describe here a variety of
roadblocks that inhibited these STs from effectively completing the process of using student
mathematical thinking.
ROADBLOCKS TO EFFECTIVE USE OF STUDENTS’ MATHEMATICAL THINKING
All the STs believed to some extent that their lessons would be more productive if
their students were given opportunities to make comments or to share their solutions to
problems. Therefore, there were many times during their lessons when they elicited students’
mathematical thinking; often these instances could be viewed as teachable moments. As was
expected of novice teachers, and regardless of whether the STs were using reform curricula in
a middle school or traditional curricula in a high school, they ran into similar issues when
trying to conduct a whole class discussion that used their students’ thinking in order to assist
all students to come to a deeper understanding of the underlying mathematics. We describe
here a collection of roadblocks that hindered the process of effectively using student thinking
for classroom discussion. Although we use the term “roadblocks,” we view such instances in
a positive light. As mathematics teacher educators we were pleased to see our STs grappling
with these important issues—bumping up against important dilemmas of teaching. The
identification of these roadblocks informed us as to where we needed to go as teacher
educators in our efforts to help the STs continue their development as mathematics teachers.
Roadblocks to Listening and Understanding
The literature is replete with examples of novice teachers (e.g., Borko et al., 1992;
Cooney, 1985; Schultz, 2003) and experienced teachers (e.g., Ball, 1993; Davis, 1997;
Lampert, 1990; Schultz, 2003) who struggle to attend to the complexities of teaching. It is no
small task for teachers to balance attending to what students are saying with attending to what
they will do or say next. Thus, one major roadblock to listening seemed to be the inability to
attend to student thinking and attend to other aspects of teaching. In addition, even when the
STs were listening to the substance of their students’ thinking, they sometimes struggled to
understand what was being said. When leading a class discussion where students are
encouraged to share their thinking and methods for solving a problem, a ST’s knowledge or
experience may not allow her to understand a student’s thinking. Because the student strategy
is unique or different, the ST may not understand the point the student is trying to make, even
though she is listening. Student thinking that is not understood cannot be used to enrich the
class discussion for the benefit of all students.
An example of a roadblock to listening and understanding occurred as ST Jennifer
taught a pre-calculus lesson that she had planned with her partner ST Ashley. The task that
they had created was a set of cards containing different linear functions represented using
words, graphs or equations. The students were asked to classify the cards according to their
attributes, such as increasing or decreasing slopes. The STs wanted the students to reflect on
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the attributes of parallel, perpendicular, vertical and horizontal lines by looking at the
similarities and differences among the various representations of a line. However, the STs
had created this task by adapting an activity they had done in a university class, where each
of several different kinds of functions was represented in four different ways – numerically,
graphically, verbally and algebraically (see Cooney, Brown, Dossey, Schrage, & Wittmann,
1996, pp. 41-45). One of the primary purposes of this original task was to help preservice
teachers review many different types of functions while simultaneously considering the
attributes of these multiple representations. The STs adapted the original task to have three
different representations of each of several linear functions that varied according to their
slope or y-intercept. Because this “matching” characteristic of the original task still existed,
many of these pre-calculus students attempted to group their cards only according to the three
different representations of the same function, rather than considering classifications based on
more general characteristics such as increasing or decreasing slopes.
Although much of the expressed student thinking was focused on grouping the cards
according to the different representations of the same function, some of the thinking was
clearly related to slope, one of the intended foci of the activity. However, as ST Jennifer
elicited her students’ thinking as part of a class discussion, she mostly just commented,
“Okay, okay. Yeah, that’s interesting.” and then moved on. A similar behavior was seen as
ST Jennifer moved from group to group prior to the class discussion. She asked one group
how they were classifying their cards but did not ask any follow up questions about what they
were looking for or why. After looking at another group’s work, she said “Oh, [you are
classifying] by y-intercept. Good job.” This student thinking seemed to meet the lesson goal
and yet she had no further discussion beyond this comment. Thus, ST Jennifer was not
listening to her students’ thinking, even though some of it could have helped her to meet her
mathematical goals.
In the reflection meeting, ST Jennifer was asked to explain the classification she was
hoping to see. She responded by saying,
We didn’t really expect them to say, “Okay, well, this is the graph, it matches this
equation, it matches this story.” We didn’t think they’d do that right off the bat….
Every single group did that…. The first thing that they came up with every time is,
“We just matched them up.”
Throughout this discussion ST Jennifer continually returned to the problematic nature of the
unanticipated responses. We believe that she was so preoccupied with the number of
unanticipated solutions that she did not listen for the thinking that might lead to her goal. In
this case, ST Jennifer’s attention to the seeming failure of her task hindered her from listening
to her students’ thinking.
Concerns about classroom management also functioned as roadblocks to listening and
understanding. With respect to this same card-sorting lesson, ST Jennifer said the following
in her reflection paper:
I was noticing that some students were finished with the activity, so I asked them to
write their answers on the board to give them something to do. Had I been more
aware of their answers, I would not have had them present. Their answers were not
beneficial to the class discussion.
In this case, ST Jennifer did not listen to her students’ thinking before she attempted to use it.
Her attention to issues of classroom management hindered her from listening to the student
thinking that she was observing in the class.
In summary, two of the roadblocks to listening and understanding are the challenge of
keeping the classroom running smoothly when the lesson feels like it is falling apart and
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using student thinking for a management purpose instead of using it to better understand the
mathematics. As mentioned previously, effective teaching is a complex endeavor. Learning to
use students’ mathematical thinking requires learning to attend to that thinking while
attending to many other aspects of the class and of the lesson.
Roadblocks to Recognizing the Teachable Moment
It is one thing to understand what a student says. It is quite another thing to recognize
that thinking as a teachable moment—to understand the significance of what the student has
said and to see value in that thinking from a mathematical and pedagogical stance. We have
identified a number of roadblocks to such recognition.
Assumption of Understanding
STs often work on the assumption that their students already understand the
mathematics at hand. Our data revealed two variations of how these assumptions play out.
Fill in the blanks. Novice teachers have a tendency to implicitly “fill in the blanks”
when their students are talking about mathematics rather than asking the students to do so (cf.
Ball, 2001). Students often use imprecise language when answering questions or sharing their
work. The STs frequently forgave this imprecision, assuming that the student understood
what they were superficially or inadequately describing. They failed to recognize such
moments as important opportunities to push the student to clarify their statements and
thinking.
An example of this roadblock occurred in a lesson taught by ST Jennifer. One of the
homework questions had asked the students to find a line through the point (6, 5) that is
perpendicular to . As a class the students had arrived at the equation .
ST Jennifer then asked what they needed to do next for the line to pass through (6, 5). ST
Jennifer described the student response and her thinking as follows:
I got the chorus answer “you plug in (6,5).” I then assumed that most students knew
how to do this and moved on. After the reflection meeting, I now see this situation
differently. If I was in this situation again, I would ask why I can’t plug in any value I
want to. This discussion probably could have deepened students’ understanding of an
equation for a line. I hope that next time, I can be more aware of little situations like
this that could strike up a mathematically engaging discussion.
It is clear from ST Jennifer’s comments that she assumed the students understood the
underlying mathematics when they said “you plug in (6, 5)” and filled in the blank about why
plugging in (6, 5) yields the desired results.
Simply remind. When students display incorrect or incomplete thinking about
mathematics that has been recently talked about in class or that they have learned in the past
or that was written in the instructions, the STs tended to assume that the students actually
understand it and that they “just need to be reminded” about it (thus equating learning with
memorizing). Such incomplete or incorrect student thinking was often viewed by the STs as
“mistakes” rather than “misunderstandings.” In these instances the STs tended not to question
students’ understanding of that mathematics. Instead they either reminded the students of the
time or place where the concept had been addressed before or rephrased the student
comments by correcting or completing the response.
An example of this roadblock occurred when ST Holly was teaching a lesson wherein
the students were asked to complete a table describing the distance from a motion detector at
time t as a person walks toward it. (Although this task was carried out without a motion
detector, the students had been involved in an activity where they had used the motion
detector earlier in the week.) The students were given tables with several values of t already
included and were asked to complete the tables and to graph their results. Some of these
values of t could be interpreted to mean that the person had walked past the motion detector.
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The table was labeled with time as the independent variable and distance as the dependent
variable. The STs wanted the students to enter negative distances once the person walked past
the motion detector even though this solution was not clear from the context. There were
many students who did not use negative numbers in their solutions. As ST Holly interacted
with those groups or students she gave a variety of little hints about how they might fill in the
paper. In her reflection paper she commented that she had tried to resolve the issue that
students were not using negative numbers in their solutions by encouraging “the students to
read the problems carefully and to make sure what they were saying.” She also said, “I know
the students just didn’t read the problem correctly.” ST Holly did not see the situation as
being problematic for the students and assumed that they would have understood if they had
just read the instructions more carefully. Her tendency to view the students’ alternate
solutions as the result of mistakes rather than as attempts to make meaning of the task
hindered ST Holly from recognizing this thinking as a teachable moment.
In each case here, the STs’ assumption of understanding inhibited them from
recognizing the moment as a teachable moment. They assumed that when a student provided
a simple response that was correct, the student had the desired depth of understanding the STs
were seeking. They also assumed that when a student made an incorrect statement, they had
“just forgotten” but they really understood the concept at hand. Both of these types of
assumptions kept the STs from recognizing the potential rich conversation that could occur if
only they dug a little deeper.
Funneling
The other main roadblock to recognition that we identified in our data was referred to
earlier in this paper: “funneling” (Wood, 1998), or looking for a particular response and, in
the process, failing to recognize the mathematical and pedagogical significance of other
responses. It may seem as though funneling could be categorized earlier in the process, as a
roadblock to listening, but we do not think this is the case. In order to funnel, one must
actually listen to student thinking and understand it enough to recognize that it is not the
thinking being sought. Thus, STs who funnel have at least listened and understood. What
they fail to do is to recognize the mathematical and pedagogical significance of the response.
Because they have a preconceived notion of the response that will lead to a teachable
moment, they fail to recognize other responses that may lead to similar or even different but
still valuable teaching moments.
An example of funneling that prevented a ST from recognizing a teachable moment
occurred when ST Christina was launching a lesson wherein students were going to input
equations into a calculator and look at the tabular outputs to make decisions about the
situation. In anticipation of this approach, ST Christina asked the students to identify the
independent and dependent variables in the equations and . This activity was
meant to be a quick review so that the students would be able to input equations properly into
the calculator. ST Christina wanted to hear that r was the independent variable and that A and
C were the dependent variables in their respective equations. After a student had shared his
response that r was the independent variable and A was the dependent variable in the former
equation, Morgan said that she thought C was the independent variable and r was the
dependent variable in the latter equation. Because this response was not what ST Christina
wanted, she began a funneling process:
ST Christina: Morgan, come up and explain to us what you have here.
Morgan: I did the circumference because the radius depends on how big or small the
circle is. So I said the circumference is independent and the radius is
dependent on the circumference.
11
ST Christina: OK. Thanks Morgan. Who has the same thing as Morgan? Who has
something different? Who doesn’t know? [pause] Who said they have
something different? Brian’s the only one who has something different?
Sage: I don’t know.
ST Christina: Nobody else? Katherine, do you? Do you want to explain?
Katherine: I just said the independent would be the radius and the dependent would be
the circumference.
ST Christina: OK, Why?
Katherine: Because… the circumference is the—. Wait, no, I agree with her
[Morgan].
ST Christina: Are you sure? You were going good there. Do you want to keep
explaining what you were saying?
Katherine: I was going to say that the circumference would change if the circle gets
smaller. But um, you can find the circumference without the radius I think.
ST Christina: You can find the circumference without the radius? How would you do
that?
Katherine: Um. I don’t know.
.
.
.
ST Christina: Brian, what do you think?
Brian: Couldn’t kind of both of them go both ways? Because like in area. Like as the
area gets smaller so does the—. Oh, never mind.
ST Christina: So let’s look back at this one. How did these equations relate with each
other with the independent and dependent, um, with both of them and how can
we think this through? Anybody have some ideas besides Brian? Brian, thanks
for your help though. Abe, what do you think?
In this episode Morgan presented a solution that was not what ST Christina had
anticipated, so she asked the class if someone had approached the situation differently and
this is where the funneling began. Katherine started to say that she disagreed with the first
student and then changed her mind. ST Christina tried to pursue Katherine’s initial thinking
because it was what she was looking for, namely r as the independent variable and C as the
dependent variable. Brian then suggested that it could go either way and then backed off as
did Katherine. In this case, however, the ST Christina did not pursue Brian’s comment. With
both Katherine and Brian ST Christina funneled toward her preconceived correct solution and
away from the divergent thinking of the students. This funneling had ST Christina trying to
get Katherine to explain her original thoughts because they supported her goal and yet the
funneling hindered ST Christina from recognizing the richness of Ben’s thinking as a
teachable moment.
Roadblocks to Effectively Using Students’ Mathematical Thinking
Because STs are in the process of learning how to teach, it comes as no surprise that
they might listen to student thinking, understand what students have said, recognize that
thinking as a teachable moment, yet still not be able to use the student thinking in a way that
furthers their mathematical learning goals—that capitalizes on the teachable moment. Our
analysis of the data revealed a number of roadblocks to effective use of student thinking. In
our categorization of these roadblocks, we make a distinction between roadblocks to trying to
use student thinking and roadblocks to effective use. The former roadblocks inhibited the STs
from even attempting to use their students’ thinking. With the latter roadblocks, the STs
attempted to use their students’ thinking but fell short.
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No Attempt to Use
First, we located a number of instances in our data where the STs were able to
recognize student thinking as a teachable moment and yet they made no attempt to use that
thinking in their lesson (i.e., they did not pursue the thinking with the class). When such
instances occurred we could trace the reason to a lack of knowledge, usually a lack of either
SCK, PK or CK. We use several episodes to illustrate such roadblocks:
Lack of SCK. ST Holly had given her pre-algebra class the table shown in Figure 1,
which gives the number of people out of 100 surveyed who would go on a bike tour for the
given total prices. The students were asked the following question: “To make a graph of these
data, which variable would you put on the x-axis? Which variable would you put on the y-
axis? Explain.” The students were also asked to “make a coordinate graph of the data on the
grid paper” (Lappan, Fey, Fitzgerald, Friel, & Phillips, 2006, p. 32).
ST Holly had anticipated “that students would have problems with the independent
and dependent variables,” so during the first few minutes of the task she “went around to the
different groups… looking for students that had their independent and dependent variables
labeled correctly and also students that had their variables mixed up.” Having noticed that
many students seemed to be struggling a great deal with the identification of the independent
and dependent variables, ST Holly decided to bring the class together for a class discussion.
In order to initiate the class discussion she polled the class:
ST Holly: How many of you guys think that the total price is the independent? [Quite
a few students raise their hands.] How many of you think that the number of
customers is the independent? [Some (but fewer) students raise their hands.]
Okay. Someone who thinks that the number of customers is independent, will
you tell me why?
Spencer: I’d say that it’s the number of customers because the customers depend on
their opinion of what the price should be.
ST Holly: Okay. So the number of customers depends on what the price will be?
Spencer: No. The money depends on what the number of customers should be.
ST Holly: Okay. Is that what it says in the prompt? That’s the one I did. Is there
another idea why that one would be the independent one?
Breanna: You can’t, because it’s how many people want the price.
ST Holly: How many people want that price?
Breanna: Yeah. So, like, it’s kind of hard to explain.
ST Holly: It’s kind of hard to explain? Okay, someone who picked the total price to
be the independent one, do you want to give me an explanation?
George: Yeah. I think it’s the total price because, like, just because they want to go
doesn’t mean they can change the price, so the price stays the same. [ST
Holly: Uhuh.]. So the number of people changes depending on if they want to
pay that much or not.
ST Holly: Okay. Does everyone understand what he said? A little bit? Okay, let me
give you another example.
ST Holly then proceeded to give an example that she and ST Megan had developed
while planning their lesson, one that they had hoped would help students gain a better
understanding of how the analysis of the context of a situation helps you to identify the
independent and the dependent variables:
ST Holly: Ariel, let’s say I come up to you and I’m like, “Do you want to buy my
ipod?” What do you say?
Ariel: Um, no.
ST Holly: No? Why not? Why don’t you want to buy my ipod?
13
Ariel: Well, probably because you may have used it already and everything and you
already have your own songs that you already had on it.
ST Holly: Uhuh. That’s a good thing. So it kind of depends on different things, right?
Ariel: Yeah.
ST Holly: So I can’t just, like come up to you and be like, “Do you want to go on the
bike tour?” You probably want to know how much it is first. Right?
Ariel: Uhuh.
ST Holly: So that’s something to think about.
ST Holly then asked the students to get into pairs and return to the task.
In reflecting back on this class discussion ST Holly recognized that the discussion
“never came to closure” on the issue of determining the independent and dependent variables.
She then shared this important bit of insight into her thinking at the time:
People would argue both sides and I could see both sides, but I didn’t know how to
justify them. And I just kept trying to bring it back to the context. But every time I’d
bring it back to the context they’d come up with something different and I was like,
“Oh, I didn’t think of that.” And so I really didn’t know how to—. So I don’t know.
In this episode ST Holly asked questions that allowed her students to share their thinking
about determining the independent and dependent variables in this situation. There is
evidence that she was listening to and understanding what they said (i.e., “I could see both
sides”). She also recognized this discussion as a great opportunity to talk about the very thing
she wanted to talk about—deciding which variable should be independent and which should
be dependent. What seemed to keep ST Holly from using the students’ thinking in this
situation was a lack of SCK. ST Holly had developed a strong enough knowledge of
independent and dependent variables to know that such classification was highly dependent
on the context. She did not have sufficient understanding of the underlying mathematics,
however, to allow her to see how to justify or refute the various responses that she received.
Without this knowledge she was unable to push on that thinking, to point out what was
correct and what was incorrect in the students’ thinking. Rather than use the student thinking
that had been proffered, her lack of SCK forced her to retreat from that thinking and to
introduce her own thinking. In this episode ST Holly’s lack of SCK was a roadblock to her
using her students’ mathematical thinking.
Lack of Pedagogical Knowledge. Lack of pedagogical knowledge (PK) also impeded
the STs from using their students’ mathematical thinking. One of the most common ways this
lack of knowledge revealed itself was when the STs would notice a productive conversation
within a small group. The STs would often listen to such conversations, recognize them as
valuable, and have the desire to use that thinking in a class discussion. They soon learned,
however, that recreating such individual group discussions was not easy. Their approach
usually took the form of asking the group to share with the class the conversation they had
just had; this approach never succeeded. As Ashley put it, “It was really fun to sit in on their
argument, but for me, it’s hard to recreate that argument in the classroom discussion because
they feel like they’ve already had the conversation; they don’t want to have the conversation
again.” In our experience, although students may indeed be reluctant to reproduce such
conversations, such reproduction is practically impossible for them. Students tend to focus on
the results of their conversations, seldom on the process or on pitfalls of those conversations.
Thus, successful reproduction of valuable small-group conversations must be facilitated by
the teacher, and usually entails involving the rest of the class in the issue that was being
discussed, thus recreating the situation for the whole class that caused the productive
conversation in the small group. This pedagogical knowledge of how to use students’
mathematical thinking was not yet available to the STs in our study, and the lack of this
14
knowledge was a roadblock to their use of that thinking. We note, however, the important
progress the STs were making in that they were beginning to recognize their lack of this
knowledge and to find ways to acquire it.
Lack of Curricular Knowledge. The STs’ lack of curricular knowledge often impeded
their ability to use students’ mathematical thinking. For example, in the motion detector task
described earlier, many students questioned the notion of negative numbers in the context of
the problem, because “the students were unsure about the motion detector being able to read
the person if they were behind the motion detector” (ST Holly). Students who focused on the
context of the problem questioned the use of negative numbers and many started to develop
solutions that were building toward the notion of absolute value. The STs recognized that the
solutions were headed in that direction and chose not to pursue them. As ST Holly stated, “I
stayed away from that idea because I didn’t want to talk about absolute value functions. I
didn’t even know if it was ok to talk about absolute value functions this early.” In this
situation ST Holly’s lack of curricular knowledge impeded her from using the student
thinking that was elicited by this task. She did not have a sufficient understanding of the
connections between the current day’s mathematics topic and the underlying mathematical
ideas of absolute value.
Thus, in general, the STs in this study were often inhibited from using student
thinking because they did not understand the mathematics enough to pursue the thinking with
their students (lack of SCK), they did not know how to carry out that pursuit (lack of PK), or
they did not know whether they “should” pursue it (lack of curricular knowledge). In each
case the STs had sufficient knowledge to listen to, understand and recognize the value in their
students’ thinking; what they they lacked was the knowledge to use that thinking.
Naïve Use
The STs who participated in this study often tried to use their students’ mathematical
thinking. This use, however, was not always effective. In analyzing their attempts to use
students’ mathematical thinking, we found a number of times when it was clear that the STs
believed that they were indeed using the students’ thinking effectively, although from our
observation this was not the case. We classified such usage as naïve use—the STs were
“technically” using their students’ thinking, but such use was based on a naïve assumption
about how students learn and did not really capitalize on the mathematical thinking of the
students. The following sections describe the various types of naïve use that emerged from
our analysis of the data.
Student Thinking as a Trigger. Using students’ mathematical thinking as a trigger is
somewhat akin to funneling. With funneling, however, the STs basically pass by all student
thinking until they hear the thinking they are looking for—they then pursue or validate that
thinking but fail to recognize the value in the other mathematical thinking that was shared. In
the case of a trigger, the STs did recognize some value in what students’ said, but the value is
that they see a way that they can take some portion of what the students said (often not
necessarily related to what the student meant) in order to redirect the conversation toward
where they intended it to go. In terms of Woods’ (1998) funneling and focusing constructs,
when STs use students’ mathematical thinking as a trigger they funnel but they think they are
focusing—they think they are effectively using their students’ thinking.
An example of using student thinking as a trigger comes again from the motion
detector task and revolves around the issue of whether it was okay in this situation for the
output numbers (distances) to be negative. In ST Megan’s class she invited several students
to put their answers on the board. Marcus then explained how they found the values in their
table:
15
Marcus: He went to six, one, and then—it would usually be negative four, but it didn’t
say it was in front or behind. So we just thought it was four feet away from the
[motion detector] because it went back up again.
ST Megan: Okay. Was that confusing to anyone else, whether or not you could go
into the negatives? I saw a couple of papers where there was some argument.
Let’s talk about this idea of whether or not you can go into the negatives.
It appeared that ST Megan has listened to the student thinking that had been presented
and that she recognized this thinking as worth pursuing. Rather than pursuing the reasoning
of this pair of students, however, ST Megan chose to use their explanation as a trigger to talk
about why it does make sense to use negative numbers in this situation. The students’
explanation contained their reasoning for using positive numbers, not negative numbers. In
fact, the mathematics of their explanation is focused on arguing that the context of the
problem calls for the use of positive numbers. ST Megan viewed this as an incorrect answer
and chose to use the statement “it would usually be negative four, but it didn’t say it was in
front or behind” as a trigger to first focus on the difficulty in deciding and then to focus on
putting forth arguments that the values should be negative. Although the student’s
explanation was technically used in this situation, the mathematics of that explanation was
neither used nor valued. Instead, a phrase about the problematic nature of the decision was
taken up and used in order to redirect the focus of the discussion. We categorize such use of
students’ mathematical thinking as a trigger as naïve use. ST Megan seems to believe that she
is indeed using the students’ thinking, but her use is for her own purpose, which in this case is
actually to try to make a point that is opposed the point the student was trying to make.
Mere presence of the correct solution. It was fairly common for the STs to elicit
students’ mathematical thinking and then fail to do anything with that thinking. In some
cases, we concluded that the student thinking was merely elicited, but never listened to, let
alone recognized as valuable and then used. In other instances, however, analysis revealed a
variation on this phenomenon that we felt clearly should be classified as using (although
naively) student thinking: the STs clearly believed that they were using the student thinking
that had been elicited, although this use was at best implicit.
An episode from ST Megan’s classroom illustrates this naïve use of student thinking.
ST Megan had engaged her students in the Bicycle tour task (see Figure 1). For this part of
the task, the students were asked to respond to the question, “Based on your graph, what price
do you think the tour operators should charge? Explain” (Lappan et al., 2006, p. 32). After
the students had worked on the task for some time, ST Megan initiated a class discussion. She
asked the students to share their answers for part (c)—how much did they think should be
charged and why. One pair of students said that they should charge $150 because it was at
that price that the most customers had indicated that they would participate. ST Megan then
asked for others to share their solution and a pair of students volunteered $350 (the correct
solution) and explained that they used a revenue table to come up with their solution. A brief
discussion followed about this latter answer, in which ST Megan implied that this latter
answer was correct, and then the class moved on.
In this episode ST Megan elicited two different solutions with different solution
methods and justifications. The first solution was incorrect; the second was correct. We
classify this situation as a teachable moment because there are two reasonable solutions on
the board and the class is primed to make arguments about their validity. Such a conversation
would bring up the important mathematical ideas of revenue and maximization. ST Megan
listened to the students’ solutions and ensured that she understood them. Now, it is tempting
to characterize ST Megan as not having recognized the teachable moment, and in terms of the
explicit discussion and comparison of the two solutions, that might be correct. However, we
16
believe that ST Megan actually thought that she was using both students’ thinking. She
facilitated the presentation of both solutions and she implicitly endorsed the latter (correct)
response. We believe that ST Megan was operating under the following naïve assumption
regarding student learning: the presence of the correct solution clears up the misconceptions
underlying the incorrect solutions. It is by looking at ST Megan’s teaching here through the
lens of this assumption that we classify it as naïve use. Such naïve use of students’
mathematical thinking seems to be based in a lack of KCS. This lack of knowledge allowed
ST Megan to use here students’ mathematical thinking, but only naively, thus hindering her
from effectively (in this case, explicitly) using that thinking.
Mere presentation of multiple solutions. We briefly highlight a different but related
naïve use of students’ mathematical thinking that occurred fairly often in the STs’ lessons.
Sometimes the STs managed to elicit significant student thinking, had students record at the
board and explain this thinking, and then the STs moved on. The goal of the lesson seemed to
devolve into “student sharing,” rather than developing mathematics from what the students
were sharing, a phenomenon that has been previously noted in the literature (e.g., Ball, 2001;
Silver, Ghousseini, Gosen, Charalambous, & Font Strawhun, 2005). Again, like before when
the presence of the correct solution was interpreted as having cleared up the misconceptions
underlying incorrect solutions, in these situations the STs seemed to be operating under the
assumption that the connections between the multiple presented solutions, and the
mathematics that could be derived from exploring those connections were evident to the
students—that the students had learned important mathematics simply from being exposed to
multiple solutions or solution strategies. Again it is under this assumption that we classify
such an approach as naïve use of student thinking.
Incomplete Use
As we have mentioned previously, the STs often did manage to use the student
thinking that they listened to and recognized as valuable. The extent to which they were able
to use it effectively, however, even when they tried, was often limited by their limited
knowledge. Consider the following episode:
The students in ST Megan’s class were considering some data that reported the
amount of time certain students spent watching TV and those students’ GPA. ST Megan
asked the class whether TV Time or GPA would be the independent variable. One student
answered TV Time and ST Megan follows up:
ST Megan: What helped you decide it was TV Time?
Jamie: Because it was related to time.
ST Megan: Because it’s related to time. [She notices another student with their hand
raised.] Yes?
Nate: Time usually ends up as the independent variable and so it should go on the x-
axis.
ST Megan recognized this student thinking as worth pursuing, as she had noticed that her
students appeared to be choosing time as the independent variable almost automatically if it
showed up as one of the variables. This recognition is evidenced by what ST Megan did next:
ST Megan: In some cases could it end up being the dependent variable? Do we have
to be careful with that sometimes? What makes it hard to tell in this
circumstance if its going to be independent or dependent? [pointing to a
student who looks eager to contribute] Did you have an idea?
Jenna: Well, because there could be different situations where they could both be
either independent or dependent.
ST Megan: Exactly. So, what kind of situation would we be thinking of if we said that
time was dependent?
17
Taken together, Nate and Jenna’s comments provided excellent student thinking on
which ST Megan built toward a nice question that pushed her students to think more deeply
about the mathematics of this situation. In particular, the class was poised for a discussion on
determining the dependent and independent variables based on context. In the end, however,
this conversation did not lead anywhere. The students were not able to think of a situation
where time could be considered the dependent variable; most critically, neither was ST
Megan able to construct such a situation. Because ST Megan lacked the SCK that would
allow her to create such examples, she was not able to use effectively the student thinking in
order to move the mathematical conversation towards her big mathematical idea—that the
independent and dependent variables depend on the context (not just on whether time is one
of the variables). ST Megan definitely recognized the shared student thinking as an
opportunity to pursue important mathematical ideas, and she made a noble effort to do so.
Her lack of SCK, however, hindered her ability to do so effectively and resulted in
incomplete use of that thinking.
DISCUSSION AND CONCLUSION
Our analysis of the data revealed numerous roadblocks to the steps in the process of
using students’ mathematical thinking. By and large, these roadblocks can be characterized
by a lack of teacher knowledge; the STs often lacked the SCK, PK, KCS or knowledge of
curriculum to capitalize on teachable moments. At times, the STs recognized the value of
their student’s mathematical thinking and either did not pursue it or struggled to use it
productively when they did pursue it because they did not have adequate knowledge of the
representations or connections that would allow them to do so. At other times the STs heard
and recognized student mathematical thinking that could be used to help all students better
understand the topic at hand but did not have the PK to have a productive discussion about
that thinking (like in the case of how to use three incorrect solutions in order to get at
important mathematics).
Lack of curricular knowledge also often inhibited the STs from using student
thinking. We hypothesize that the further removed a mathematical concept is from the lesson
at hand, the less likely it is that STs will have the curricular knowledge to capitalize on a
teachable moment that gets at that concept. The curriculum can be viewed as a set of
concentric circles (see Figure 2), where the topic of the day is in the center. The goals of the
unit, the course and mathematics in general are related to and often underlie the day’s goal
but often feel far removed to STs. STs’ limited view of the curriculum (i.e., knowledge of
curriculum) thus inhibits them from pursuing worthwhile mathematics.
In many of the roadblocks to effective use we have described in this paper, the STs
recognized the teachable moment but lacked the knowledge (SCK, pedagogical, curricular)
prevented them from using student thinking or only allowed minimal use. We identified a
number of roadblocks, however, to the recognition of teachable moments. It seems as though
KCS, in particular a knowledge of how students think about and learn mathematics, was the
primary type of knowledge that inhibited this recognition. A common roadblock to
recognition was an assumption of understanding, which comes from a lack of KCS. The STs’
knowledge of students and how they learn led them to believe that once a concept had been
“covered” (either by them or in a previous class) the students knew and understood that
concept. If students seemed shaky with that knowledge, the STs tended to assume that the
students had “just forgotten” what they had learned.
Another place where KCS had an influence on the productive use of student thinking
was the naïve use. When the STs used the mere presentation of a correct solution as a way to
clarify the flawed thinking that led to an incorrect solution, they exhibited their limited KCS.
In this case, their KCS led them to believe that a student who had incorrectly solved a
18
problem could resolve their misunderstanding by simply observing a correct solution, without
having an explicit conversation about the approach. Similarly, the STs’ KCS drove their
approach of merely having the students share multiple solutions without any discussions of
the connections between those solutions. Because of their lack of knowledge about how
students learn, the STs assumed that the students would be able to see the similarities and
differences between the different solutions without an explicit conversation about them.
It is interesting, however, to contrast this confidence in students’ previous learning or
their ability to make connections with the surprise (and doubt) that STs often express at the
kinds of problems their students are able to solve. Thus STs tend to have high confidence in
their students’ previous learning abilities, but low confidence in their current learning
abilities. Deeper KCS would likely foster quite the opposite set of assumptions about student
learning, namely, a confidence in students’ abilities to learn mathematics through solving
problems on their own, but a healthy skepticism of their current mathematical understanding.
Such skeptical optimism fosters an approach to teach wherein the teacher is very inquisitive
about their students’ thinking, always seeking to push on that thinking in the belief that such
pushing will lead to great strides in student understanding.
The results of this study have important implications for mathematics teacher
educators. First, our data demonstrate that STs are capable of learning to teach through
focusing on their students’ mathematical thinking. The revised student teaching structure in
which these STs participated supported and encouraged their efforts to both elicit and use
their students’ thinking. These results add to a growing body of literature (e.g., Feiman-
Nemser, 2001; Sowder, 2007) that refutes the logical fallacy that because novice teachers
tend to begin with somewhat self-centered concerns, that teacher education programs should
explicitly focus on addressing those concerns (Fuller, 1969). Our research demonstrates that
STs are capable of focusing on and learning from their students’ mathematical thinking.
Second, STs could benefit a great deal from an understanding of the steps of the
process of effectively using student mathematical thinking: 1) listen and understand student
thinking, 2) recognize the thinking as a teachable moment, and 3) use the thinking for a
mathematical and pedagogical purpose. Although this process is certainly not the only way to
teach in a way that is responsive to students’ needs and thinking, the process does represent a
tangible learning objective for novice teachers. Once the overall process is understood, STs
are better able to reflect on their own teaching and that of others. They can focus on the
points at which the process of using student thinking breaks down and on the type of
knowledge that might help them to move further along in the process in the future. Also, as
novice teachers better understand and value the process of using their students’ mathematical
thinking, they will realize that they need to plan into their lessons the time needed to pursue
that thinking. It is difficult to discuss the mathematics of teachable moments if the time has
not been allotted to do so.
Finally, teacher educators need to evaluate the degree to which their teacher education
programs are designed to help novice teachers gain the knowledge needed to overcome these
roadblocks. The content and structure of mathematics teacher education programs either
afford or constrain the construction of this knowledge. Learning to teach activities, including
open discussions with novice teachers about the pitfalls of an assumption of understanding
could help them to begin to develop the skeptical optimism needed to recognize teachable
moments. These discussions with novice teachers about how students learn mathematics
could also help them to see the importance of making connections between solution strategies
explicit through class discussions. Further activities designed to strengthen mathematical
knowledge for teaching are then needed for novice teachers to develop the knowledge
necessary to use those moments effectively.
19
Although much has been said about the importance of using significant mathematical
tasks in order to elicit students’ mathematical thinking, relatively little is known about the
factors involved in using that mathematical thinking effectively. These results illustrate the
complexity of this issue. More work needs to be done on designing and researching the
effectiveness of “learning to teach” activities that can help novice teachers learn how to listen
to, understand and recognize the value of their students’ thinking, and then be able to use that
thinking in order to orchestrate meaningful mathematical discussions. A good first step in this
direction would be to discuss the process conceptualization and roadblocks presented here
with preservice teachers as part of their teacher preparation program.
20
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Figure 1. Table from Connected Mathematics 2 (Lappan et al., 2006, p. 32)
Figure 2. A representation of the broadening layers of the mathematics curriculum.
... In mathematics education, research has focused on teachers' attention as an object for both investigation and development, under the premise that it shapes teachers' actions and practices (Ayalon, 2019;Mason, 2008;Miller, 2011). Findings of research focusing on teachers' attention to classroom situations have indicated that teachers, especially novices, often prioritize management of learning above other aspects, such as the mathematical activities taking place in the classroom (Jacobs, Lamb, & Philipp, 2010;Peterson & Leatham, 2009;Sherin & van Es, 2005;Stockero & Van Zoest, 2013). Where actual teaching is concerned, this means that they might miss opportunities to build on students' mathematical ideas which could be used to advance instruction (e.g., Mason, 1998). ...
... Two exceptions to this general trend were found in two items-using multiple models and clarity-possibly because during the first activity, participants solved the problem algebraically, while during the second activity, they realized the efficiency of graphic solutions, leaving out other representations. The finding that MC was the component most attended to by this study's participants contradicts findings from other studies that have indicated that teachers often attend to aspects of management of learning above other aspects such as the mathematical activity (Jacobs et al., 2010;Peterson & Leatham, 2009;Sherin & van Es, 2005;Stockero & Van Zoest, 2013). A possible explanation for these differences may be attributed to the nature of the mathematical problem presented to participants; the problem focused on solving equations, a task typically associated with using known algorithms for its solution. ...
... The fact that the teacher participants were responsive to students' mathematical thinking so as to promote their understanding cannot be taken for granted. Research has found that often opportunities to use student thinking to further mathematical understanding either go unnoticed or are not acted upon by teachers, especially novices (Peterson & Leatham, 2009;Stockero & Van Zoest, 2013). ...
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This study explores secondary mathematics teachers' attention to the three components of Jaworski's (1992) 'teaching triad' (i.e., mathematical challenge, learning management, and sensitivity to students) as they planned a lesson revolving around a rich mathematics problem and assessed pre-designed student solutions for that problem. Written work was gathered from 17 cohorts. Qualitative analysis generated categories. Quantitative analysis revealed that some components of the teaching triad were attended to in both activities, some were not attended to in either activity, and some were attended to in one activity but not in the other. Findings are interpreted in light of theory.
... In video-based interventions, researchers generally use classroom videos that include information about multiple dimensions of a classroom environment such as the students, the teacher, management, climate, pedagogy, and mathematical thinking (e.g., Sherin et al. 2011; van Es and Sherin 2008). Although analyzing and discussing classroom video cases help prospective teachers to improve their noticing skills (Santagata et al. 2007;Schack et al. 2013), they also have various difficulties in noticing critical instances in classroom videos (Peterson and Leatham 2009;Stockero et al. 2017b;Superfine et al. 2015). Due to the complex nature of classroom environments, students' mathematical thinking is not always clearly distinguishable in the video cases (Freese 2006;Mitchell and Marin 2015). ...
... When such cases are used in teacher education, teachers tend to notice instances that are less relevant to students' mathematical thinking such as classroom management or climate, especially in early video sessions (Santagata et al. 2007;Sherin et al. 2011;Star and Strickland 2008). Thus, noteworthy instances related to student thinking may go unnoticed in the analysis of raw classroom videos (Peterson and Leatham 2009;Stockero and Van Zoest 2013). Moreover, the presentation of a raw video to prospective teachers is not usually effective without making certain edits (Seago et al. 2018;Ulusoy 2020). ...
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... 10). The teacher's role in meaningfully incorporating SMT into classroom discourse is critical (e.g., Hunter et al. 2016;NCTM 2014), but challenging (e.g., Hunter 2008Hunter , 2012Peterson and Leatham 2009). Effective incorporation requires teachers to elicit and attend to, make sense of, and appropriately respond to SMT-actions widely recognized as the three components of teacher noticing (Jacobs et al. 2010). ...
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Teacher responses to student mathematical thinking (SMT) matter because the way in which teachers respond affects student learning. Although studies have provided important insights into the nature of teacher responses, little is known about the extent to which these responses take into account the potential of the instance of SMT to support learning. This study investigated teachers’ responses to a common set of instances of SMT with varied potential to support students’ mathematical learning, as well as the productivity of such responses. To examine variations in responses in relation to the mathematical potential of the SMT to which they are responding, we coded teacher responses to instances of SMT in a scenario-based interview. We did so using a scheme that analyzes who interacts with the thinking (Actor), what they are given the opportunity to do in those interactions (Action), and how the teacher response relates to the actions and ideas in the contributed SMT (Recognition). The study found that teachers tended to direct responses to the student who had shared the thinking, use a small subset of actions, and explicitly incorporate students’ actions and ideas. To assess the productivity of teacher responses, we first theorized the alignment of different aspects of teacher responses with our vision of responsive teaching. We then used the data to analyze the extent to which specific aspects of teacher responses were more or less productive in particular circumstances. We discuss these circumstances and the implications of the findings for teachers, professional developers, and researchers.
... Not all SMTs always offer high potential for achieving central instructional goals, and many teachers, especially beginning teachers, fail to recognize and grasp the opportunities to build on SMT to develop or extend student mathematical understanding (Peterson & Leatham, 2009). Leatham, Peterson, Stockero, and Van Zoest (2015) developed an analytical framework for supporting teachers in identifying most productive in-the-moment SMT to pursue-Mathematically Significant Pedagogical Opportunities to Build on Student Thinking (MOSTs). ...
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Recent research on teachers’ use of student mathematical thinking (SMT) and recommendations for effective mathematics instruction claim that how teachers respond to SMT has great impact on student mathematical learning in the classroom. This study examined some Chinese mathematics teachers’ responses to student in‐the‐moment mathematical thinking that emerged during whole class discussion. The findings of this study revealed that the majority of Chinese elementary mathematics teachers in the data involved the whole group of students to make sense of in‐the‐moment SMT. They either invited students to digest SMT involved in the instance or provided an extension of the instance to further develop student mathematical understanding.
... Because how students view feedback is critical to their learning, much emphasis has been placed on the quality of teachers' feedback (Sadler, 1998) as well as a way for them to understand the state of each student's knowledge by employing strategies to elicit student understanding and methods to interpret students' products (Bishop et al., 2014;Leatham et al., 2015). Thus, formative assessment feedback is related to the effectiveness of teachers' questioning and more generally to the teacher's role as a facilitator of learning, which depends on his/her ability to observe students' learning and attend to their thinking (Lee, 2018;Lee & Cross Francis, 2018;Peterson & Leatham, 2009;Stockero, Rupnow, & Pascoe, 2015). ...
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