Responsive Teaching and the Instructional Reasoning of Expert Elementary Mathematics Teachers

Abstract

This study examines instructional reasoning in an approximation of practice that simulates a teacher sitting down after class to examine students’ written work. The participants were prompted to attend to, interpret, and decide how to respond to student thinking contained in a piece of written work. Our purpose was to capture the additional cognitive work that teachers engage in. Using qualitative content analysis, we identified the most frequent types of instructional reasoning used by expert teachers just prior to engaging in a responsive deciding action about how to respond. We used the results of our analysis to present three illustrative cases of responsive instructional reasoning.

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Citation: Lindstrom, D.; Selmer, S.
Responsive Teaching and the
Instructional Reasoning of Expert
Elementary Mathematics Teachers.
Educ. Sci. 2022,12, 350. https://
doi.org/10.3390/educsci12050350
Academic Editor: Federico Corni
Received: 8 April 2022
Accepted: 3 May 2022
Published: 18 May 2022
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education
sciences
Article
Responsive Teaching and the Instructional Reasoning of Expert
Elementary Mathematics Teachers
Denise Lindstrom * and Sarah Selmer
Curriculum and Instruction/Literacy Studies, West Virginia University, Morgantown, WV 25606, USA;
sarah.selmer@mail.wvu.edu
*Correspondence: denise.lindstrom@mail.wvu.edu
Abstract:
This study examines instructional reasoning in an approximation of practice that simulates
a teacher sitting down after class to examine students’ written work. The participants were prompted
to attend to, interpret, and decide how to respond to student thinking contained in a piece of written
work. Our purpose was to capture the additional cognitive work that teachers engage in. Using
qualitative content analysis, we identified the most frequent types of instructional reasoning used by
expert teachers just prior to engaging in a responsive deciding action about how to respond. We used
the results of our analysis to present three illustrative cases of responsive instructional reasoning.
Keywords: approximations of practice; professional noticing; teacher reasoning
1. Introduction
Effective mathematics instruction involves teachers engaging with and taking up student
ideas and then deciding how to respond in ways that develop that thinking [
1
–
4
]. There are var-
ious constructs used by mathematics educators to refer to teaching that centers student thinking,
including cognitively guided instruction [
5
], formative assessment focused on disciplinary
thinking [
6
], professional noticing of children’s mathematical thinking [
7
], and responsive teach-
ing [
2
,
8
,
9
]. While these practices are nuanced, they all require a dynamic interplay between
students and teachers that is difficult to facilitate even for expert teachers [8,10].
In this study, we draw on research related to professional noticing of children’s
mathematical thinking [
7
] and responsive teaching [
2
,
8
,
9
] to create and study teachers
engaged in a practice space that simulates a teacher sitting down after class is over to
examine students’ written work. To prepare for their participation in this practice space,
the teachers were asked to bring pieces of written work from their classrooms. They were
then prompted to attend to, interpret, and decide how to respond to the student thinking
contained in the piece of written work. When asked to do this, teachers first attended to and
interpreted student thinking directly on the piece of written work; they often then made
sense of the student thinking by reasoning in ways removed from the written work and
prior to deciding how to respond. Our purpose was to capture this additional cognitive
work, referred to by Dyer and Sherin [
2
] as instructional reasoning, that involves the
ways teachers interpret what they notice about student thinking that are instructionally
relevant. We were particularly interested in the ways that teachers reasoned about student
thinking just prior to deciding how to respond in ways that align with responsive teaching
practices. Therefore, we seek to answer the following research question: What types of
instructional reasoning support responsive teaching practices as a teacher examines student
written work?
2. Literature Review
2.1. Teacher Cognition: Pedagogical and Instructional Reasoning
Teacher cognition is complex. It involves the processes teachers work through as they
make instructional decisions. This invisible cognitive work is often referred to as peda-
Educ. Sci. 2022,12, 350. https://doi.org/10.3390/educsci12050350 https://www.mdpi.com/journal/education

Educ. Sci. 2022,12, 350 2 of 16
gogical reasoning. Loughlin et al. [
11
] described pedagogical reasoning as “the thinking
that underpins informed professional practice” (p. 4). According to Loughlin et al. [
12
],
understanding how pedagogical reasoning develops and the way it influences practice
is critical for teacher development. Building on the concept of pedagogical reasoning,
Dyer and Sherin [
2
] introduce the term instructional reasoning to describe the additional
thinking that goes beyond describing what a student said or did, to making interpretations
of student mathematical thinking that “
. . .
help teachers make sense of student thinking
in ways that are instructionally-relevant” (p. 70). For example, teachers may connect
student thinking to broad disciplinary thinking [
13
], synthesize and compare across in-
terpretations of student thinking [
14
], hypothesize links between classroom experiences
and interpretations of student thinking, or consider individual student characteristics [
15
].
While many of the ways that teachers might engage in instructional reasoning are known,
more research is needed on how teachers engage in instructional reasoning in ways that
align with responsive teaching.
2.2. Responsive Teaching
Responsive teaching is grounded in a sociocultural theory in which learning is viewed
as an active process situated in authentic activities and discourses of communities of
practice [
16
–
18
]. It is both a teaching stance and a practice enacted in classroom settings
in which teachers recognize the importance of using the substance of student thinking to
guide instructional decisions [
19
]. To engage in this kind of learning, teachers must view
students as active learners with prior knowledge developed in and outside the classroom
and draw on that knowledge to develop student ideas. For example, responsive teachers
ask questions in a way that draws out student emergent ideas and helps students connect
and build off each other’s ideas during classroom instruction [
18
] or encourages students
to provide alternative explanations of natural phenomena [20].
Dyer and Sherin [
2
] identified three responsive teaching practices during classroom
discussions that involve: (1) a substantive probe of student ideas; (2) an invitation for
student comment; and (3) a teacher uptake of student ideas. To characterize the instructional
reasoning that led to responsive teaching practices, Dyer and Sherin [
2
] identified video
clips of whole class instruction that captured these practices. Then, they analyzed the
video clips alongside the interview transcripts to identify the instructional reasoning just
prior to an instance of responsive teaching. Dyer and Sherin [
2
] were able to identify
three types of instructional reasoning that lead to responsive teaching. These involve
a teacher (a) connecting specific moments of student thinking across different points in
time and situated student thinking in relation to two or more students; (b) considering
the relationship between student thinking and the structure of a mathematical task; and
(c) developing tests of student thinking. In this study, we identified instructional reasoning
that leads to responsive teaching practices in a different context. Rather than using teacher-
selected video clips, we used teacher-selected pieces of student work. In addition, we
drew on literature related to the design of approximations of practices in which responsive
teaching practices can be rehearsed in a setting of reduced complexity [8,21,22].
2.3. Modeling Responsive Teaching
To capture how expert teachers make instructional decisions that are responsive to
student thinking, there is a growing movement toward practice-based teacher education.
Practice-based teacher education is a form of teacher education that focuses using scaffolded
experiences to prepare teachers to enact high-quality instruction [
10
,
23
,
24
]. One approach
to practice-based teacher education includes the design and facilitation of approximations
of practice [25].
There are numerous studies that have examined the affordances of approximations
of practice that include teachers reviewing video of their own teaching [
21
] and coached
classroom discussions [
26
]. These studies can be organized into two categories, the first
of which are studies that explore the teachers learning a set of instructional moves and

Educ. Sci. 2022,12, 350 3 of 16
responses that center student thinking and work to press on student thinking. For example,
researchers have identified discussion moves that work to elicit student thinking and pro-
vide concrete exemplars for teachers to adapt and use during mathematics instruction [
14
].
These studies help educators to conceptualize and enact responsive teaching practices.
The second category of studies that examine approximations of practice involve
teacher educators shifting their focus from a set of behaviors or actions to a focus on
the teacher as an instructional decision maker who can leverage core practices that keep
learner thinking central to teaching and learning [
8
,
10
,
26
,
27
]. One finding from these
studies involves the idea that because teaching is dynamic, practice spaces should not be
scripted, allowing educators to have time and space to examine learner thinking and to
respond and act spontaneously [
8
,
25
,
28
]. A second characteristic involves the idea that
approximations of practice should be authentic in terms of its setting (i.e., using video
of an actual classroom), its artifacts (i.e., the use of student work), or in the nature of the
activity (i.e., writing specific questions to ask students) [
8
]. These characteristics of effective
approximations of practice helped to conceptualize the approximation of practice used in
this study, which was designed to engage participants in professional noticing of children’s
mathematical thinking.
2.4. Professional Noticing of Children’s Mathematical Thinking
Professional noticing of children’s mathematical thinking, hereafter referred to as pro-
fessional noticing, involves three skills: (1) attending to children’s strategies, (2) interpreting
children’s understanding, and (3) deciding how to respond based on children’s understand-
ing [
7
]. According to prior research, professional noticing skills do not happen in isolation;
they are interrelated and often cyclical [
7
]. Research exploring the interrelated nature of
professional noticing skills often consider attending and interpreting together and explore
the relationship of these two skills with the skill of deciding how to respond [
7
,
29
,
30
].
For example, Monson et al. [
30
] discovered that after participation in a learning module
designed to support prospective teachers in attending to student strategies and deciding
how to respond they not only showed growth in deciding how to respond but also showed
gains in attending and interpreting.
Researchers also suggest that there are components embedded within professional
noticing skills. For example, Luna and Selmer [
31
] examined the skill of deciding how to
respond and identified the components of time, focus, and action. In this study, we examine
expert teacher instructional reasoning when their deciding actions and purposes align with
responsive teaching practices. We engage participants in an approximation of practice that
involves three expert teachers in the professional noticing of student written work outside
the active classroom.
3. Methods
3.1. Approximation of Practice
The approximation of practice used in this study (see Figure 1) involves participants
teaching a typical math lesson (A), choosing pieces of written work from that teaching event
(B), then participating in a semi-structured interview (C–F). The semi-structured interview
includes the questions: (1) How would you describe your students’ work? (2) What does
that tell you about your students’ thinking? (3) How do you make sense of/interpret
what you have noticed? (4) How did you respond to this student? (5) Can you explain
your thinking regarding your decision about how to respond? These questions were used
to prompt the participants to engage in professional noticing and served to make their
instructional reasoning explicit. Instructional reasoning occurred at two points in this space,
first, when the participants interpreted what they noticed about student thinking on the
written work, and second, when they shared their purposes for a deciding action. This
latter type of instructional reasoning is used as a method to identify and verify whether
participant decisions about how to respond demonstrated responsive teaching practices.

Educ. Sci. 2022,12, 350 4 of 16
The instructional reasoning that occurred as participants interpreted student thinking not
directly shown on the written work is the focus of the current study.
Educ. Sci. 2022, 12, x FOR PEER REVIEW 4 of 16
explain your thinking regarding your decision about how to respond? These questions
were used to prompt the participants to engage in professional noticing and served to
make their instructional reasoning explicit. Instructional reasoning occurred at two points
in this space, first, when the participants interpreted what they noticed about student
thinking on the written work, and second, when they shared their purposes for a deciding
action. This latter type of instructional reasoning is used as a method to identify and verify
whether participant decisions about how to respond demonstrated responsive teaching
practices. The instructional reasoning that occurred as participants interpreted student
thinking not directly shown on the written work is the focus of the current study.
Figure 1. Approximation of practice space (A–F).
3.2. Study Context
The three participants for this study all worked at Hill Elementary, a public school
located in a suburban, ethnically diverse neighborhood in a medium-sized city in the
Southern Appalachian region of the United States. Over 650 students attend the school
and 43% of the students qualify for free or reduced-price lunch, and 33% of the students
identified as a minority. At Hill Elementary school, 65% of the students scored at or above
the proficient level for mathematics, well above the state and county average. Hill Ele-
mentary is part of a professional development school partnership with a research univer-
sity and faculty in a college of education that is focused on identifying best practices for
teaching science and mathematics. The participants for this project were chosen because
they were active in professional development leadership initiatives. Two of the partici-
pants had previously participated in research projects that specifically focused on profes-
sional noticing of student thinking.
Ingrid has five years of teaching experience in a fifth-grade classroom. She earned
her national board certification as a Middle Childhood Generalist and an Elementary
Mathematics Specialist Certification. As part of her National Board Certification, she en-
gaged students in math conversations and had students self-assess, peer-assess, and
teacher-assess the accuracy of their mathematical language and thinking. Ingrid also par-
ticipated in several partnerships activities that involved video clubs to support the pro-
fessional noticing of students’ mathematical thinking.
Kendall has seven years of teaching experience in a fourth-grade classroom. She is a
certified elementary and general science (grades 5–9) teacher and earned a Master of Arts
in Instructional Design and Technology. At the time of the project, Kendall was working
on her Elementary Mathematics Specialist certification and National Board Certification.
Kendall also participated in partnership activities that involved a workshop on noticing
student thinking in the context of an integrated science and mathematics garden-based
learning project.
Figure 1. Approximation of practice space (A–F).
3.2. Study Context
The three participants for this study all worked at Hill Elementary, a public school
located in a suburban, ethnically diverse neighborhood in a medium-sized city in the
Southern Appalachian region of the United States. Over 650 students attend the school
and 43% of the students qualify for free or reduced-price lunch, and 33% of the students
identified as a minority. At Hill Elementary school, 65% of the students scored at or
above the proficient level for mathematics, well above the state and county average. Hill
Elementary is part of a professional development school partnership with a research
university and faculty in a college of education that is focused on identifying best practices
for teaching science and mathematics. The participants for this project were chosen because
they were active in professional development leadership initiatives. Two of the participants
had previously participated in research projects that specifically focused on professional
noticing of student thinking.
Ingrid has five years of teaching experience in a fifth-grade classroom. She earned
her national board certification as a Middle Childhood Generalist and an Elementary
Mathematics Specialist Certification. As part of her National Board Certification, she
engaged students in math conversations and had students self-assess, peer-assess, and
teacher-assess the accuracy of their mathematical language and thinking. Ingrid also
participated in several partnerships activities that involved video clubs to support the
professional noticing of students’ mathematical thinking.
Kendall has seven years of teaching experience in a fourth-grade classroom. She is a
certified elementary and general science (grades 5–9) teacher and earned a Master of Arts
in Instructional Design and Technology. At the time of the project, Kendall was working
on her Elementary Mathematics Specialist certification and National Board Certification.
Kendall also participated in partnership activities that involved a workshop on noticing
student thinking in the context of an integrated science and mathematics garden-based
learning project.
Hannah is a certified k–6th grade elementary teacher with additional certifications in
English as a second language and gifted education (K-12). At the time of the project, she was
obtaining a mentoring certification and was mentoring prospective teachers. Hannah also
participated in several university and school partnership projects that involved noticing
student thinking in the context of an integrated science and mathematics garden-based
project. Ingrid, Kendall, and Hannah were considered teacher leaders because they all had
five or more years of teaching experience and were active participants in a professional
development school partnership. They all co-authored presentations and/or articles with

Educ. Sci. 2022,12, 350 5 of 16
university faculty focused on improving the teaching and learning of math and science in
their school context [32].
3.3. Data Sources
The data set for this study involved the participants’ engagement with the approxima-
tion of practice across four cycles and consisted of 12 interview transcripts and 37 pieces of
student written work. We coded the transcripts by identifying and separating out evidence
of each professional noticing skill identified as attending, interpreting, and deciding. Next,
we coded each attending and interpreting segment as either written work or not written
work. A segment was coded as written work if the participant was attending to or interpret-
ing student thinking directly on the piece of learning work. If the participant was attending
to or interpreting student thinking not observable on the written work, the segment was
coded as not written work. We separated the not written work segments, focusing on
the written work segments. We conducted a preliminary analysis focused on the written
work segments to confirm that the participants had noticed the important mathematical
elements in 36 out of 37 pieces of written work, a prerequisite to teaching responsively [
9
].
Across the remaining 36 pieces of written work, participants shared 59 deciding actions
and related shared purposes, hereafter referred to as deciding sequences. Each deciding
sequence was identified as responsive based on whether the given action aligned with
a responsive teaching practice and whether the shared purpose was rooted in the idea
of picking up and pursuing student thinking [
2
,
9
]. This resulted in the identification of
35 responsive deciding sequences. Each of these deciding sequences contained an identified
responsive deciding action and a related purpose. The four responsive deciding actions are
displayed in Table 1.
Table 1. Responsive deciding actions.
Deciding Actions
The Teacher . . . Transcript Excerpt
. . . asks the student to elaborate on and/or
clarify their thinking “explain why you would use meters” (K1)
. . .
prompts the student to reread the problem
situation and consider their related strategy
“I would tell him to re-read it and see what
he does”. (I2)
. . . asks the student to use a different strategy “I would encourage her to solve a
second way”. (H1)
. . . asks the student to work on a new task “I would give her another one (task)”. (K4)
The four deciding purposes (see Table 2) identified as responsive were that the teacher
wants: to test student understanding, to understand additional student thinking, the
student to make mathematical connections, and the student to address a conceptual error.
Table 2. Responsive deciding purposes.
Purpose Codes
The Teacher Wants . . . Transcript Excerpt
...to test student understanding “ . . . to make sure she grasped this concept of
exactly what kind of division we’re doing here” (I3)
. . . to understand additional
student thinking
“I would ask him to explain how he figure out the
six, because I want to know what he
was thinking”. (I1)
. . . the student to make mathematical
connections
“I want him to think about the actual relationship of
the numbers”. (H3)
...the student to address a
conceptual error
“to make him look at the bigger picture (the problem
context) of how it all fits together”. (J2)

Educ. Sci. 2022,12, 350 6 of 16
Each of these 35 deciding sequences were identified as responsive in the context of
this study. It is important to note that the deciding sequences were not enacted in a live
classroom setting. Next, for each of the 35 identified responsive deciding sequences, we
returned to the transcripts and combined the chunks of transcript coded as not written
work. We then separated the transcripts each time the participants shifted in their in-
structional reasoning focus; hereafter referred to as an instructional reasoning turn. For
example, a participant might start out by reasoning about their students’ thinking using
big mathematical ideas (e.g., this student understands the properties of quadrilaterals) and
then shift to reasoning about the source of a student’s understanding (e.g., learned this in
a previous classroom lesson). Across the four cycles, we examined 36 pieces of written
work and identified 35 responsive deciding sequences and 85 instructional reasoning turns
(see Table 3).
Table 3. Instructional reasoning turns.
Cycle Pieces of Written Work Responsive
Deciding Sequences
Instructional
Reasoning Turns
1 8 7 16
2 8 9 19
3 11 10 34
4 9 9 16
Total 36 35 85
3.4. Instructional Reasoning Coding Scheme
To develop the codes for the instructional reasoning turns, we conducted a qualitative
content analysis using “theme” as the unit of analysis [
33
]. For example, we noticed that
our participants were comparing students’ strategies with each other and then generalizing
about what needed to be addressed in the next lesson [
2
]. At other times, our participants
connected what they had noticed about student thinking to the source of that thinking, such
as a previous lesson [
15
,
34
]. At other times, our participants considered the relationships
between the curriculum and the development of students’ ideas [
35
]. The final coding
scheme (see Table 4) resulted in the following codes in which a teacher (1) uses previous
classroom experiences to reason about student thinking, (2) considers the relation between
student thinking and the structure of a mathematical task [
2
], (3) situates a student’s
idea in relation to two or more other students’ ideas, (4) considers student characteristics,
and (5) situates student thinking in relation to mathematics according to (a) conceptual
understanding, (b) procedural understanding, and (c) content standards.
Table 4. Instructional reasoning codes.
Instructional Reasoning
A Teacher . . . Transcript Excerpt
Uses previous classroom experiences to
reason about student thinking
“They worked with a partner, and we did a
strategy which is that one person solves the
problem, and the next person writes down what
they did”. (I3)
Considers the relation between student
thinking and the structure of a
mathematical task
“I think that the problem asking them to come up
with a definition helped them better understand it
because it is their words”. (K2)
Situates a student’s idea in relation to two
or more other student ideas
“I was just kind of flabbergasted at the variety of
responses, I mean I had three fourths the class get
the right answer”. (H3)
Considers student characteristics
“I definitely think she’s probably not confident and
that is why she couldn’t finish the problem”. (H3)

Educ. Sci. 2022,12, 350 7 of 16
Table 4. Cont.
Instructional Reasoning
A Teacher . . . Transcript Excerpt
Situates student thinking in relation to
mathematics: conceptual understanding
“ . . . she needs a better understanding of the
metric system and how a meter grows into a
kilometer or how it shrinks into millimeters, so
that she can have better understanding of size and
its relationship to place value”. (K1)
Situates student thinking in relation to
mathematics: procedural understanding
“I think she understands the basic concept of
multiplying all the numbers, the procedural aspect
of finding volume”. (H3)
Situates student thinking in relation to
mathematics: content standards
“I don’t know that want them to know how to
switch from metric to customary units because that
is not a part of the fourth-grade standard”. (K1)
The researchers first coded separately and then met to discuss and modify the emerg-
ing codes. Once the final codes were developed, the researchers coded the remaining
data independently, met and discussed differences, and modified the codes until a 100%
consensus across all data points was achieved.
4. Results
4.1. Instructional Reasoning
The participants engaged four times in the approximation of practice, which resulted
in 35 responsive deciding sequences and 85 instructional reasoning turns prior to the 35
responsive deciding sequences. Table 5shows the frequency of each type of instructional
reasoning that occurred during the four cycles.
Table 5. Instructional reasoning results.
Instructional Reasoning Cycle 1 Cycle 2 Cycle 3 Cycle 4 Total
Using previous classroom experiences to
reason about student thinking. 3 4 9 2 18
Considering the relation between student
thinking and the structure of a
mathematical task.
5 5 3 1 14
Situating individual students’ thinking in
relation to two or more other
student ideas.
2 4 6 4 16
Considering student characteristics. 1 0 6 1 8
Situating student thinking in relation to
mathematics: conceptual understanding.
3 4 6 7 20
Situating student thinking in relation to
mathematics: procedural understanding.
10203
Situating student thinking in relation to
mathematics: content standards. 12216
Total Moments 16 19 34 16 85
Our results show that the most frequent type of instructional reasoning involved
teachers situating student thinking in relation to mathematics with a focus on conceptual
understanding (20 instances). The second most common type of instructional reasoning
involved teachers using previous classroom experiences to reason about student thinking
(18 instances), followed by situating the individual students’ thinking in relation to two or

Educ. Sci. 2022,12, 350 8 of 16
more other student ideas (16 instances) and considering the relationship between student
thinking and the structure of a mathematical task (14 instances).
4.2. Responsive Deciding Sequences
Recall that these 85 instructional reasoning turns occurred prior to the participants’
sharing of a deciding sequence identified as responsive. The most common of these
deciding sequences involved a participant sharing the deciding action of the teacher
asking the student to work on a new task because the teacher wanted the student to make
mathematical connections (nine instances). The second most common deciding sequence
involved the deciding action of the teacher asking the student to elaborate on and/or clarify
their thinking for the purpose of testing student understanding (eight instances). The third
most common deciding sequence (seven instances) involved the deciding action of asking a
student to elaborate on and/or clarify their thinking for the purpose of having the student
make mathematical connections. The frequency of all the deciding sequences, including the
three most common, and the associated number of instructional turns are shared in Table 6.
Table 6. Deciding actions and instructional reasoning.
Purpose
The Teacher Wants . . .
Deciding Action
The Teacher . . .
To Test Student
Understanding
To Understand
Additional
Student
Thinking
To Have the
Student Make
Mathematical
Connections
For the Student to
Address a
Conceptual Error
. . .
asks the student to elaborate on
and/or clarify their thinking
Educ. Sci. 2022, 12, x FOR PEER REVIEW 8 of 17
Situating student thinking in relation to
mathematics: conceptual understanding. 3 4 6 7 20
Situating student thinking in relation to
mathematics: procedural understanding. 1 0 2 0 3
Situating student thinking in relation to
mathematics: content standards. 1 2 2 1 6
Total Moments 16 19 34 16 85
Our results show that the most frequent type of instructional reasoning involved
teachers situating student thinking in relation to mathematics with a focus on conceptual
understanding (20 instances). The second most common type of instructional reasoning
involved teachers using previous classroom experiences to reason about student thinking
(18 instances), followed by situating the individual students’ thinking in relation to two
or more other student ideas (16 instances) and considering the relationship between stu-
dent thinking and the structure of a mathematical task (14 instances).
4.2. Responsive Deciding Sequences
Recall that these 85 instructional reasoning turns occurred prior to the participants’
sharing of a deciding sequence identified as responsive. The most common of these de-
ciding sequences involved a participant sharing the deciding action of the teacher asking
the student to work on a new task because the teacher wanted the student to make math-
ematical connections (nine instances). The second most common deciding sequence in-
volved the deciding action of the teacher asking the student to elaborate on and/or clarify
their thinking for the purpose of testing student understanding (eight instances). The third
most common deciding sequence (seven instances) involved the deciding action of asking
a student to elaborate on and/or clarify their thinking for the purpose of having the stu-
dent make mathematical connections. The frequency of all the deciding sequences, includ-
ing the three most common, and the associated number of instructional turns are shared
in Table 6.
Table 6. Deciding actions and instructional reasoning.
Purpose
The Teacher
Wants…
Deciding Action
The Teacher…
To Test Stu-
dent Under-
standing
To Understand Addi-
tional Student
Thinking
To Have the
Student
Make
Mathemati-
cal Connec-
tions
For the Stu-
dent to Ad-
dress a Con-
ceptual Error
…asks the student
to elaborate on
and/or clarify their
thinking
8 (24) 2 (3) 7 (17) 3 (6) 20 (50)
… prompts the stu-
dent to reread/re-
consider the prob-
lem situation and
strategy
2 (7) 2 (7)
…asks the student
to use a different
strategy.
1 (2) 2 (3) 3 (5)
…asks the student
to work on a new
task
1 (0) 9 (23) 10 (23)
2 (3)
Educ. Sci. 2022, 12, x FOR PEER REVIEW 8 of 17
Situating student thinking in relation to
mathematics: conceptual understanding. 3 4 6 7 20
Situating student thinking in relation to
mathematics: procedural understanding. 1 0 2 0 3
Situating student thinking in relation to
mathematics: content standards. 1 2 2 1 6
Total Moments 16 19 34 16 85
Our results show that the most frequent type of instructional reasoning involved
teachers situating student thinking in relation to mathematics with a focus on conceptual
understanding (20 instances). The second most common type of instructional reasoning
involved teachers using previous classroom experiences to reason about student thinking
(18 instances), followed by situating the individual students’ thinking in relation to two
or more other student ideas (16 instances) and considering the relationship between stu-
dent thinking and the structure of a mathematical task (14 instances).
4.2. Responsive Deciding Sequences
Recall that these 85 instructional reasoning turns occurred prior to the participants’
sharing of a deciding sequence identified as responsive. The most common of these de-
ciding sequences involved a participant sharing the deciding action of the teacher asking
the student to work on a new task because the teacher wanted the student to make math-
ematical connections (nine instances). The second most common deciding sequence in-
volved the deciding action of the teacher asking the student to elaborate on and/or clarify
their thinking for the purpose of testing student understanding (eight instances). The third
most common deciding sequence (seven instances) involved the deciding action of asking
a student to elaborate on and/or clarify their thinking for the purpose of having the stu-
dent make mathematical connections. The frequency of all the deciding sequences, includ-
ing the three most common, and the associated number of instructional turns are shared
in Table 6.
Table 6. Deciding actions and instructional reasoning.
Purpose
The Teacher
Wants…
Deciding Action
The Teacher…
To Test Stu-
dent Under-
standing
To Understand Addi-
tional Student
Thinking
To Have the
Student
Make
Mathemati-
cal Connec-
tions
For the Stu-
dent to Ad-
dress a Con-
ceptual Error
…asks the student
to elaborate on
and/or clarify their
thinking
8 (24) 2 (3) 7 (17) 3 (6) 20 (50)
… prompts the stu-
dent to reread/re-
consider the prob-
lem situation and
strategy
2 (7) 2 (7)
…asks the student
to use a different
strategy.
1 (2) 2 (3) 3 (5)
…asks the student
to work on a new
task
1 (0) 9 (23) 10 (23)
3 (6) 20 (50)
. . . prompts the student to
reread/reconsider the problem
situation and strategy
2 (7) 2 (7)
. . . asks the student to use a
different strategy. 1 (2) 2 (3) 3 (5)
. . . asks the student to work
on a new task 1 (0)
Educ. Sci. 2022, 12, x FOR PEER REVIEW 8 of 17
Situating student thinking in relation to
mathematics: conceptual understanding. 3 4 6 7 20
Situating student thinking in relation to
mathematics: procedural understanding. 1 0 2 0 3
Situating student thinking in relation to
mathematics: content standards. 1 2 2 1 6
Total Moments 16 19 34 16 85
Our results show that the most frequent type of instructional reasoning involved
teachers situating student thinking in relation to mathematics with a focus on conceptual
understanding (20 instances). The second most common type of instructional reasoning
involved teachers using previous classroom experiences to reason about student thinking
(18 instances), followed by situating the individual students’ thinking in relation to two
or more other student ideas (16 instances) and considering the relationship between stu-
dent thinking and the structure of a mathematical task (14 instances).
4.2. Responsive Deciding Sequences
Recall that these 85 instructional reasoning turns occurred prior to the participants’
sharing of a deciding sequence identified as responsive. The most common of these de-
ciding sequences involved a participant sharing the deciding action of the teacher asking
the student to work on a new task because the teacher wanted the student to make math-
ematical connections (nine instances). The second most common deciding sequence in-
volved the deciding action of the teacher asking the student to elaborate on and/or clarify
their thinking for the purpose of testing student understanding (eight instances). The third
most common deciding sequence (seven instances) involved the deciding action of asking
a student to elaborate on and/or clarify their thinking for the purpose of having the stu-
dent make mathematical connections. The frequency of all the deciding sequences, includ-
ing the three most common, and the associated number of instructional turns are shared
in Table 6.
Table 6. Deciding actions and instructional reasoning.
Purpose
The Teacher
Wants…
Deciding Action
The Teacher…
To Test Stu-
dent Under-
standing
To Understand Addi-
tional Student
Thinking
To Have the
Student
Make
Mathemati-
cal Connec-
tions
For the Stu-
dent to Ad-
dress a Con-
ceptual Error
…asks the student
to elaborate on
and/or clarify their
thinking
8 (24) 2 (3) 7 (17) 3 (6) 20 (50)
… prompts the stu-
dent to reread/re-
consider the prob-
lem situation and
strategy
2 (7) 2 (7)
…asks the student
to use a different
strategy.
1 (2) 2 (3) 3 (5)
…asks the student
to work on a new
task
1 (0) 9 (23) 10 (23)
10 (23)
Totals 10 (26) 2 (3) 16 (40) 7 (16) 35 (85)
These three deciding sequences accounted for 24/35 of the responsive deciding sequences and 64/85 instructional
reasoning turns.
We used the common deciding sequences in Table 6, starting with the most common
decide sequence, to serve as illustrative cases of responsive instructional reasoning.
4.2.1. Case 1: A Teacher Asks a Student to Work on a New Task for the Purpose of Having
the Student Make Mathematical Connections
Across the three participants, there were nine instances of asking a student to work
on a new task because the teacher wanted the student to make mathematical connections.
Prior to these deciding sequences, the participants engaged in 23 instructional reasoning
turns (see Table 7).
To illustrate this case, we present an example from Kendall’s third cycle in the approx-
imation of practice. Kendall began her engagement in the practice space by examining a
piece of written work that involved students identifying angles on the playground (see
Figure 2). Kendall stated, “I think that she’s got a very good handle on all of them” (finding
angles on the playground) and highlighted that a student identified the monkey bars as
an example of an acute angle, an affirmation that the chosen task was effective in helping
Kendall determine the student’s understanding of angles.

Educ. Sci. 2022,12, 350 9 of 16
Table 7. Case 1: Instructional reasoning turns.
The Participants’ Instructional Reasoning Turns Frequency
Using previous classroom experiences to reason about
student thinking. 5
Considering the relation between student thinking and the
structure of a mathematical task. 6
Situating individual students’ thinking in relation to two or
more other student ideas. 3
Considering student characteristics. 0
Situating student thinking in relation to mathematics:
conceptual understanding. 5
Situating student thinking in relation to mathematics:
procedural understanding. 0
Situating student thinking in relation to mathematics:
content standards. 4
Total 23
Educ. Sci. 2022, 12, x FOR PEER REVIEW 10 of 17
Figure 2. Piece of student work (Cycle 3, Kendall).
Kendall then considered the relation between student thinking and the structure of
the mathematical task, explaining that it might be beneficial to give the students a task in
which the students first choose an angle and then determine the measurement of the other
angles without using a protractor. She explains: “I might have them choose a specific an-
gle and find the measurement for that one using a protractor and then try to figure out
what all of the rest would be without using the protractor (Transcript 3K)”. Kendall con-
tinues to engage in instructional reasoning by situating student thinking in relation to the
mathematical content standards:
“I would also like to connect to our next standard, with shapes and how she has
an acute angle for the monkey bars ……so what type of shape would that be so
taking what you know about this and translating it into our next standard (Tran-
script 3K).”
Kendall then shifted her instructional reasoning back to considering the relation be-
tween the mathematics of student thinking and the structure of a mathematical task: “I
think maybe another thing that I could have them do is to disconnect it from school, have
them look at their home for these shapes (Transcript 3K).” Then, Kendall decided that she
was going to have the student work on a new task involved finding angles and shapes at
home, for the purpose of having students connect what they know about angles to shapes
found in their houses. This deciding sequence was coded as the teacher asking the student
to work on a new task for the purpose of having the student make mathematical connec-
tions.
4.2.2. Case 2: A Teacher Asks the Student to Elaborate on and/or Clarify Their Thinking
for the Purpose of Testing Student Understanding
Across the three participants, there were eight instances of a teacher asking students
to elaborate on and/or clarify their thinking in order to test student understanding. Prior
to sharing these deciding sequences, the participants engaged in 24 instructional reason-
ing turns (see Table 8).
Table 8. Case 2: Instructional reasoning turns.
The Participants’ Instructional Reasoning Turns Frequency
Using previous classroom experiences to reason about stu-
dent thinking. 4
Considering the relation between student thinking and the
structure of a mathematical task. 4
Figure 2. Piece of student work (Cycle 3, Kendall).
Kendall then considered the relation between student thinking and the structure of
the mathematical task, explaining that it might be beneficial to give the students a task
in which the students first choose an angle and then determine the measurement of the
other angles without using a protractor. She explains: “I might have them choose a specific
angle and find the measurement for that one using a protractor and then try to figure
out what all of the rest would be without using the protractor (Transcript 3K)”. Kendall
continues to engage in instructional reasoning by situating student thinking in relation to
the mathematical content standards:
“I would also like to connect to our next standard, with shapes and how she has
an acute angle for the monkey bars
. . . . . .
so what type of shape would that be
so taking what you know about this and translating it into our next standard
(Transcript 3K)”.
Kendall then shifted her instructional reasoning back to considering the relation
between the mathematics of student thinking and the structure of a mathematical task:
“I think maybe another thing that I could have them do is to disconnect it from school,
have them look at their home for these shapes (Transcript 3K)”. Then, Kendall decided
that she was going to have the student work on a new task involved finding angles and
shapes at home, for the purpose of having students connect what they know about angles
to shapes found in their houses. This deciding sequence was coded as the teacher asking

Educ. Sci. 2022,12, 350 10 of 16
the student to work on a new task for the purpose of having the student make mathematical
connections.
4.2.2. Case 2: A Teacher Asks the Student to Elaborate on and/or Clarify Their Thinking
for the Purpose of Testing Student Understanding
Across the three participants, there were eight instances of a teacher asking students
to elaborate on and/or clarify their thinking in order to test student understanding. Prior
to sharing these deciding sequences, the participants engaged in 24 instructional reasoning
turns (see Table 8).
Table 8. Case 2: Instructional reasoning turns.
The Participants’ Instructional Reasoning Turns Frequency
Using previous classroom experiences to reason
about student thinking. 4
Considering the relation between student thinking and the
structure of a mathematical task. 4
Situating individual students’ thinking in relation to two or
more other student ideas. 5
Considering student characteristics. 2
Situating student thinking in relation to mathematics:
conceptual understanding. 6
Situating student thinking in relation to mathematics:
procedural understanding. 2
Situating student thinking in relation to mathematics:
content standards. 1
Total 24
To illustrate Case 2, we present an example from Ingrid’s third cycle in the approxima-
tion of practice as she examined a piece of student written work that involved the following
problem: Stella bought an aquarium with a rectangular base that measures 18 inches wide
by 14 inches long and a height of 10 inches. One fish needs 40 cubic inches of space in the
aquarium. What is the volume of the aquarium? How many fish can live in the aquarium
and how do you know?
The student thinking shown in Figure 3involved a student-constructed strategy for
finding the volume.
Educ. Sci. 2022, 12, x FOR PEER REVIEW 11 of 17
Situating individual students’ thinking in relation to two or
more other student ideas. 5
Considering student characteristics. 2
Situating student thinking in relation to mathematics: concep-
tual understanding. 6
Situating student thinking in relation to mathematics: proce-
dural understanding. 2
Situating student thinking in relation to mathematics: content
standards. 1
Total 24
To illustrate Case 2, we present an example from Ingrid’s third cycle in the approxi-
mation of practice as she examined a piece of student written work that involved the fol-
lowing problem: Stella bought an aquarium with a rectangular base that measures 18
inches wide by 14 inches long and a height of 10 inches. One fish needs 40 cubic inches of
space in the aquarium. What is the volume of the aquarium? How many fish can live in
the aquarium and how do you know?
The student thinking shown in Figure 3 involved a student-constructed strategy for
finding the volume.
Figure 3. Piece of student work (Cycle 3, Ingrid).
Ingrid began by describing and interpreting the procedures and operations the stu-
dent used to solve the problem: “I can see that she was able to multiply 14 by 8 got 112
and then 112 by 10 so she was able to find her volume and knew it was 1120 (Transcript
3I).” Notice how she interpreted the procedures and operations the student used to solve
the word problem and provided evidence of the student’s conceptual and procedural un-
derstanding for finding volume:
“I think that she grasps what math goes into solving this problem…that you
need to multiply to find the volume and that she needed to take that amount of
volume and divide it up to find out how many fish needed to go in it. So overall
she understands the process and the operations behind what fit with the sce-
nario (Transcript 3I).”
This transcript excerpts provide evidence of two different, but often related, types of
instructional reasoning, both involving mathematics. The first is that the instructional rea-
soning focuses on a student’s procedural understanding, which was one of the least used
Figure 3. Piece of student work (Cycle 3, Ingrid).

Educ. Sci. 2022,12, 350 11 of 16
Ingrid began by describing and interpreting the procedures and operations the student
used to solve the problem: “I can see that she was able to multiply 14 by 8 got 112 and then
112 by 10 so she was able to find her volume and knew it was 1120 (Transcript 3I)”. Notice
how she interpreted the procedures and operations the student used to solve the word
problem and provided evidence of the student’s conceptual and procedural understanding
for finding volume:
“I think that she grasps what math goes into solving this problem
. . .
that you
need to multiply to find the volume and that she needed to take that amount of
volume and divide it up to find out how many fish needed to go in it. So overall
she understands the process and the operations behind what fit with the scenario
(Transcript 3I)”.
This transcript excerpts provide evidence of two different, but often related, types
of instructional reasoning, both involving mathematics. The first is that the instructional
reasoning focuses on a student’s procedural understanding, which was one of the least used
types of instructional reasoning by our participates (three total instances). For example,
notice how Ingrid focused on detailing the accuracy of the processes and operations
the student used to solve the problem. For example, she explained: “she needed to
take that amount of volume and divide it to find out how many fish needed to go in
it, so she understood the process
. . .
. the operations behind what fit with the scenario
(Transcript 3I)
”. This type of instructional reasoning differs from instructional reasoning
that situates student thinking in relation to students’ conceptual understanding, which was
the most often used type of instructional reasoning (20 total instances). Ingrid engaged in
this instructional reasoning just prior to the deciding action of asking a student to elaborate
on and/or clarify their thinking for the purpose of testing student understanding. Ingrid
explained: “I would have her further explain her thinking about the problem to make sure
she grasped the overall idea of the problem and the way to solve it (Transcript 3I)”.
4.2.3. Case 3: A Teacher Asks the Student to Elaborate on and/or Clarify Their Thinking
for the Purpose of Having the Student Make Mathematical Connections
Across the three participants, there were seven instances of the deciding sequence
involving the deciding action of the teacher asking students to elaborate on and/or clarify
their thinking for the purpose of having the student make mathematical connections. Prior
to sharing this deciding sequence, the participants engaged in 17 instructional reasoning
turns (see Table 9).
Table 9. Case 3: Instructional reasoning turns.
The Participants’ Instructional Reasoning Turns Frequency
Using previous classroom experiences to reason
about student thinking. 4
Considering the relation between student thinking and the
structure of a mathematical task. 0
Situating individual students’ thinking in relation to two or
more other student ideas. 2
Considering student characteristics. 4
Situating student thinking in relation to mathematics:
conceptual understanding. 6
Situating student thinking in relation to mathematics:
procedural understanding. 0
Situating student thinking in relation to mathematics:
content standards. 1
Total 17
To illustrate this case, we present an example from Hannah’s third cycle in the approx-
imation of practice when she examined a piece of written work that involved a student
finding the volume of a cube and explaining their solution (see Figure 4).

Educ. Sci. 2022,12, 350 12 of 16
Educ. Sci. 2022, 12, x FOR PEER REVIEW 13 of 17
Figure 4. Piece of student work (Cycle 3, Hannah).
As with the other pieces of written work, Hannah began her engagement in the ap-
proximation of practice by noticing the student thinking on the piece of written work,
stating: “The shape is four blocks up and across and four blocks back, so she didn’t nec-
essarily use the mathematical term but definitely she has an understanding of up, across,
and back, and that’s where she is getting her measurements (Transcript H3).” Hannah
continued to instructionally reason by situating individual student thinking in relation to
two or more other student ideas (16 total instances), “so I felt like her description there
was very fifth grade student like, and it fit perfectly with what we were doing (Transcript
H3).” Hannah’s instructional reasoning about her students’ thinking in relation to others
is important as teachers need to focus simultaneously on individual students and on the
entire class. Hannah also instructionally reasoned using previous classroom experiences
that the student had perhaps drawn on to find a solution to the current problem:
“We practiced these types of problems, I gave them graph paper, I used this
great really neat interactive volume site where you can like put layers up and
then click the sides…so we spent I mean a whole day just on that and figuring
out ok the length the width and the height and then practiced multiplying them
together (Transcript H3).”
Shifting her noticing back to the student, Hannah stated that she would respond to
this student by asking the student “to add to her answers for the purpose of having the
student use more formal mathematical vocabulary and to connect her written explanation
more clearly to the figure and her use of units, and then connect what she wrote to the
actual shape.” This response was coded as the teacher asking the student to elaborate on
and/or clarify their thinking so that the student can make mathematical connections.
5. Discussion
5.1. Instructional Reasoning
Our data analysis helped us to identify and distinguish between two types of instruc-
tional reasoning. The first type of instructional reasoning is novel and involves a teacher’s
purpose for a deciding action. By capturing a teacher’s purpose for a deciding action, we
prompted additional instructional reasoning which, in turn, provided additional evidence
of responsive teaching. For example, a deciding purpose that involved a teacher wanting
a student to make mathematical connections is focused on student thinking, in contrast to
a teacher whose purpose is for a student not to make a procedural error. We suggest that
this type of instructional reasoning is empirically important as it could be used in future
research on responsive teaching to identify and understand teachers’ intentions behind
teaching practices that appear responsive.
The second type of instructional reasoning involves the instructional reasoning that
occurs just prior to engaging in a responsive deciding action. Like Dyer and Sherin [2], we
Figure 4. Piece of student work (Cycle 3, Hannah).
As with the other pieces of written work, Hannah began her engagement in the
approximation of practice by noticing the student thinking on the piece of written work,
stating: “The shape is four blocks up and across and four blocks back, so she didn’t
necessarily use the mathematical term but definitely she has an understanding of up, across,
and back, and that’s where she is getting her measurements (Transcript H3)”. Hannah
continued to instructionally reason by situating individual student thinking in relation to
two or more other student ideas (16 total instances), “so I felt like her description there was
very fifth grade student like, and it fit perfectly with what we were doing (
Transcript H3
)”.
Hannah’s instructional reasoning about her students’ thinking in relation to others is
important as teachers need to focus simultaneously on individual students and on the
entire class. Hannah also instructionally reasoned using previous classroom experiences
that the student had perhaps drawn on to find a solution to the current problem:
“We practiced these types of problems, I gave them graph paper, I used this great
really neat interactive volume site where you can like put layers up and then click
the sides
. . .
so we spent I mean a whole day just on that and figuring out ok the
length the width and the height and then practiced multiplying them together
(Transcript H3)”.
Shifting her noticing back to the student, Hannah stated that she would respond to
this student by asking the student “to add to her answers for the purpose of having the
student use more formal mathematical vocabulary and to connect her written explanation
more clearly to the figure and her use of units, and then connect what she wrote to the
actual shape”. This response was coded as the teacher asking the student to elaborate on
and/or clarify their thinking so that the student can make mathematical connections.
5. Discussion
5.1. Instructional Reasoning
Our data analysis helped us to identify and distinguish between two types of instruc-
tional reasoning. The first type of instructional reasoning is novel and involves a teacher’s
purpose for a deciding action. By capturing a teacher ’s purpose for a deciding action, we
prompted additional instructional reasoning which, in turn, provided additional evidence
of responsive teaching. For example, a deciding purpose that involved a teacher wanting a
student to make mathematical connections is focused on student thinking, in contrast to a
teacher whose purpose is for a student not to make a procedural error. We suggest that
this type of instructional reasoning is empirically important as it could be used in future
research on responsive teaching to identify and understand teachers’ intentions behind
teaching practices that appear responsive.
The second type of instructional reasoning involves the instructional reasoning that
occurs just prior to engaging in a responsive deciding action. Like Dyer and Sherin [
2
],

Educ. Sci. 2022,12, 350 13 of 16
we found that our participants engaged in this type of instructional reasoning when con-
sidering the relation between student thinking and the structure of a mathematical task
and when they situated student thinking in relation to two or more students. However,
our findings characterize additional types of instructional reasoning that include a teacher
(1) using previous classroom experiences to reason about student thinking, (2) considering
student characteristics, and (3) situating student thinking in relation to the mathemat-
ics by focusing on (a) conceptual understanding, (b) procedural understanding, and/or
(c) content standards.
The ways that teachers instructionally reasoned about student thinking prior to en-
gaging in a responsive deciding sequence was clearly connected to the kinds of work
these teachers conducted in their classrooms. To support this claim, we revisit the cases to
demonstrate how instructional reasoning by expert teachers enhances the work they do in
their classrooms.
5.1.1. Case 1: A Teacher Asks a Student to Work on a New Task for the Purpose of Having
the Student Make Mathematical Connections
In Case 1, recall that Kendall began by examining a piece of student work that involved
students identifying angles on the playground. Prior to asking students to work on a new
task, she engaged in instructional reasoning that involved considering the relation between
student thinking and the structure of a mathematical task. For example, she thought about
how a future task involving her students using a protractor in a real-world context would
develop her students’ understanding of angels as well address critical content standards,
because “
. . .
students can only learn what they have the opportunity to think about,
student thinking is bounded by the tasks they are assigned”. [
36
] (p. 606). Choosing
and/or designing the right task to frame student thinking and connect these tasks to
content standards requires strong pedagogical knowledge and the ability to attend deeply
to student thinking as they reason about future task structures. Additionally, this type
of instructional reasoning results in tasks that draw out student thinking, which in turn
enhances responsive teaching practices. This suggests that teachers would benefit from
professional learning experiences that center task structures and mathematical content
standards to develop responsive teaching practices.
5.1.2. Case 2: A Teacher Asks the Student to Elaborate on and/or Clarify Their Thinking
for the Purpose of Testing Student Understanding
In Case 2, Ingrid examined a piece of student written work that involved a student-
constructed strategy for finding volume (see Figure 3). Ingrid’s instructional reasoning
focused on connecting student thinking to disciplinary ideas [
7
]. It is not surprising that
our participants (teacher leaders) primarily reasoned about conceptual mathematical un-
derstanding over procedural understanding or the content standards (20 times vs. 3 times
vs. 6 times, respectively). This finding suggests that teacher leaders are intentional about
developing students’ conceptual mathematical understanding and focus on substantive
rather than surface-level understanding of students’ ideas [
6
]. This finding also suggests
that approximations of practice should be designed to provide novice teachers with oppor-
tunities to practice connecting students’ mathematical thinking to broader mathematical
ideas and standards.
5.1.3. Case 3: A Teacher Asks the Student to Elaborate on and/or Clarify Their Thinking
for the Purpose of Making Mathematical Connections
In Case 3, Hannah discussed a piece of student written work that involved a student
finding the volume of a cube (see Figure 4). In this deciding sequence, Hannah engaged in
the deciding action of asking the student to elaborate on and/or clarify their thinking for
the purpose of having the student make mathematical connections. Hannah first situated
her students’ thinking in relation to other fifth-grade students. This type of instructional
reasoning suggests that responsive teachers should unpack and hold on to the substance of
individual students’ thinking in ways that are connected to patterns of thinking in groups

Educ. Sci. 2022,12, 350 14 of 16
of students in connection to a whole class. This finding suggest that approximations of
practice should be designed to help prospective teachers develop efficient ways of attending
to the breadth and substantive details of an individual student’s thinking while connecting
that thinking to groups of students within a whole class.
Hannah also connected her students’ thinking by reasoning about a previous class
experience. We, like others, find the idea of teachers referencing specific moments from
classroom teaching events notable as it suggests that responsive teacher thinking and
reasoning is situated in the context in which that thinking appeared [
2
]. Therefore, as
teachers reason about the source of students’ knowledge, they might be recalling and
synthesizing several past classroom learning experiences, suggesting that their ability to
easily reference and make sense of these specific moments may be a necessary condition
for teaching responsively. These findings support the idea that approximations of practice
should be implemented in ways that provide prospective teachers with the time and space
to develop the practice of recalling specific moments of classroom learning in connection
with student thinking [8,25].
6. Conclusions
This study confirms and expands our understandings about how teachers engage in
instructional reasoning in ways that have pedagogical value and empirical implications
for teacher education. Our research distinguished between two types of instructional
reasoning. The first focused on teachers’ purposes for their deciding actions. This type of
instructional reasoning is empirically important because it provides additional evidence
of the often-hidden instructional reasoning that supports responsive teaching practices.
The second type of instructional reasoning focused on the ways participants made sense of
the student thinking noticed in pieces of written work. The second type of instructional
reasoning was illustrated through three cases that captured the different ways expert teach-
ers instructionally reasoned about students’ thinking prior to engaging in a responsive
deciding sequence. These findings demonstrate the importance of capturing responsive
teaching practices taking place in and out of active classroom settings. In doing so, teacher
educators can design approximations of practices to develop prospective teachers’ profes-
sional noticing skills and ways of instructional reasoning that support responsive teaching
practices. The authors acknowledge that the small sample size involving three teachers
from the same school setting means that there are limitations to the generalizability of the
findings. Future research should include a larger sample size of teachers from diverse
school settings to confirm and/or identify the various ways mathematics teachers reason
about student mathematical thinking.
Author Contributions:
Conceptualization, D.L. and S.S.; methodology, D.L. and S.S.; software,
D.L. and S.S.; validation, D.L. and S.S.; formal analysis, D.L. and S.S.; investigation, D.L. and S.S.;
resources, D.L. and S.S.; data curation, D.L. and S.S.; writing—original draft preparation, D.L. and S.S.;
writing—review and editing, D.L. and S.S.; visualization, D.L. and S.S.; supervision, D.L. and S.S.;
project administration, D.L. and S.S.; funding acquisition, D.L. and S.S. All authors have read and
agreed to the published version of the manuscript.
Funding: This research received no external funding.
Institutional Review Board Statement:
The study was conducted in accordance with and approved
by the Institutional Review Board (or Ethics Committee) of West Virginia University, IRB Code:
1602999970. Date of Approval: 2 November 2016.
Informed Consent Statement:
Informed consent was obtained from all subjects involved in the study.
Data Availability Statement: Not applicable.
Conflicts of Interest: The authors declare no conflict of interest.

Educ. Sci. 2022,12, 350 15 of 16
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