Professional Noticing of Children's Mathematical Thinking

Abstract

The construct professional noticing of children's mathematical thinking is introduced as a way to begin to unpack the in-the-moment decision making that is foundational to the complex view of teaching endorsed in national reform documents. We define this expertise as a set of interrelated skills including (a) attending to children's strategies, (b) interpreting children's understandings, and (c) deciding how to respond on the basis of children's understandings. This construct was assessed in a cross-sectional study of 131 prospective and practicing teachers, differing in the amount of experience they had with children's mathematical thinking. The findings help to characterize what this expertise entails; provide snapshots of those with varied levels of expertise; and document that, given time, this expertise can be learned.

Full text

Professional Noticing of Children’s
Mathematical Thinking
Victoria R. Jacobs, Lisa L. C. Lamb, and Randolph A. Philipp
San Diego State University
The construct professional noticing of children’s mathematical thinking is introduced
as a way to begin to unpack the in-the-moment decision making that is foundational
to the complex view of teaching endorsed in national reform documents. We define
this expertise as a set of interrelated skills including (a) attending to children’s strate-
gies, (b) interpreting children’s understandings, and (c) deciding how to respond on
the basis of children’s understandings. This construct was assessed in a cross-sectional
study of 131 prospective and practicing teachers, differing in the amount of experience
they had with children’s mathematical thinking. The findings help to characterize what
this expertise entails; provide snapshots of those with varied levels of expertise; and
document that, given time, this expertise can be learned.
Key words: Children’s strategies; Early childhood, K–4; In-service teacher education;
Pedagogical knowledge; Planning, decision-making; Preservice teacher education;
Professional development; Teaching practice
The range of what we think and do
is limited by what we fail to notice.
And because we fail to notice
that we fail to notice,
there is little we can do
to change
until we notice
how failing to notice
shapes our thoughts and deeds. (Goleman, 1985, p. 24)
Noticing is a common activity of teaching, but, as Goleman suggested, noticing
effectively is both complex and challenging. For many years, psychologists have
studied how we attend to stimuli in our environments, and researchers have learned
not only that we have focusing and capacity limitations (Schneider & Shiffrin, 1977;
Shiffrin & Schneider, 1977) but also that instead of perceiving the world objectively,
we construct what we see (Gibson, 1979). Therefore, individuals looking at the
same thing may see it in different ways. In addition, studies on inattentional blind-
ness and change blindness have highlighted what we do not see even when it is
present (Most et al., 2001; Simons & Chabris, 1999). Consider Simons’s (2000)
Journal for Research in Mathematics Education
2010, Vol. 41, No. 2, 169–202
An earlier version of this article was presented at the 2009 annual conference of the American
Educational Research Association. This research was supported in part by a grant from the National
Science Foundation (ESI0455785). The opinions expressed in this article do not necessarily reflect the
position, policy, or endorsement of the supporting agency. The authors thank Bonnie Schappelle for her
critical work on the project and early drafts of the manuscript.

170 Professional Noticing
popular study showing that a large percentage of individuals watching a video of
two teams playing basketball completely miss the unexpected arrival of a man in a
gorilla suit! Studies dating back to the classic study by Bartlett (1932) also have
shown that individuals’ knowledge, beliefs, and experiences influence what is seen
and in what ways a stimulus is processed. More recently, Erickson (2007) compared
viewing (in this case, video) to completing projective tests in which individuals are
asked what they see in ambiguous stimuli (e.g., inkblots); their reactions extend
beyond the information available to reflect the viewer as much as the stimulus
itself.
Given the subjective nature of noticing, researchers have found that distinct
patterns of noticing have evolved for groups of individuals who hold similar goals
and experiences, such as groups of professionals. Goodwin (1994) used the term
professional vision to capture how members of a profession develop perceptual
frameworks that enable them to view complex situations in particular ways. For
example, archeologists develop sensitivities to variations in color, texture, and
consistencies of sand, and attending to these details is a critical component of their
abilities to reason about a landscape. Similarly, Stevens and Hall (1998) used
disciplined perception to describe the visual practices characteristic of particular
professions (or disciplines), and Mason (2002) focused on the idea of intentional
noticing, contrasting this type of noticing that is characteristic of a profession with
everyday noticing (what everyone does). In short, learning to notice in particular
ways is part of the development of expertise in a profession. In the next section, we
explore research on the noticing expertise of a particular group of professionals,
mathematics teachers.
NOTICING OF MATHEMATICS TEACHERS
Researchers have explored the noticing of mathematics teachers to understand
how they make sense of complex classroom environments in which they cannot be
aware of or respond to everything that is occurring. Sherin and van Es have offered
the most extensive work on noticing in mathematics education (Sherin, 2001, 2007;
Sherin & Han, 2004; Sherin & van Es, 2005, 2009; van Es & Sherin, 2002, 2006,
2008; van Es, in press). They have used Goodwin’s (1994) concept of professional
vision to examine teacher learning, most often in video clubs in which teachers
watch and discuss video from their own classrooms. Their three-part learning-to-
notice framework includes (a) identifying noteworthy aspects of a classroom situ-
ation, (b) using knowledge about the context to reason about the classroom interac-
tions, and (c) making connections between the specific classroom events and
broader principles of teaching and learning (van Es & Sherin, 2008). In a series of
studies, they have found that teachers can improve their noticing by changing what
they notice (e.g., moving from a focus on teachers’ actions to students’ conceptions)
and how they reason (e.g., moving from mere reporting of events to synthesizing
and generalizing, and moving from evaluative comments to interpretive comments
based on evidence) (Sherin & Han, 2004; van Es & Sherin, 2008). They have also

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demonstrated that the teacher learning that occurs in video clubs can extend to
classroom instruction in similar ways (Sherin & van Es, 2009).
Of particular interest is the critical role that interpretation plays throughout the
Sherin and van Es work—they have argued that how individuals analyze what they
notice is as important as what they notice. In contrast, in the noticing work of Star
and Strickland (2007), the focus was exclusively on identifying noteworthy aspects
(Sherin and van Es’s first component). Star and Strickland argued that their interest
in what teachers do and do not attend to in classroom lessons is foundational for
future instruction. Their results showed that, after a methods course, prospective
secondary teachers were better able to notice classroom events in four observation
categories (classroom environment, tasks, mathematical content, and communica-
tion). Thus, similar to Sherin and van Es, Star and Strickland found that noticing
expertise could improve with support.
Other researchers have engaged in a variety of work closely related to noticing.
For example, Santagata, Zannoni, and Stigler (2007) have offered a lesson-analysis
framework as a means for helping prospective teachers gain expertise in observing
and reasoning about classroom events. Their three-part framework highlights the
identification of (a) learning goals, (b) student learning in relation to those goals,
and (c) alternative teaching strategies to accomplish those goals. In their work, they
asked teachers to analyze classroom videos that were collected as part of the video
studies in the Third International Mathematics and Science Study (TIMSS) (Hiebert
et al., 2003; Stigler & Hiebert, 1999). Because of the cultural nature of teaching,
using videos from other countries can make instructional routines more visible and
thus available for analysis. Miller and Zhou (2007) also used international work to
explore teachers’ noticing. In a comparison of what U.S. and Chinese elementary
school teachers noticed in classroom videos, they found striking differences
between the two cultures. For example, U.S. teachers were more likely to comment
on pedagogical issues and the videotaped teachers’ personalities, whereas Chinese
teachers were more likely to comment on the mathematical content of the classes.
These differences were consistent with the documented training and beliefs of U.S.
and Chinese teachers.
Throughout all these studies, researchers define noticing in a multitude of ways,
but the connecting thread is making sense of how individuals process complex
situations. Furthermore, their findings all serve to underscore the idea that teachers
see classrooms through different lenses depending on their experiences, educational
philosophies, cultural backgrounds, and so on and that particular kinds of experi-
ences can scaffold teachers’ abilities to notice in particular ways. We build on the
noticing research both inside and outside of mathematics education, and we have
chosen to enter this dialogue by selecting a particular focus for noticing—children’s
mathematical thinking. In our study, we document groups of teachers’ expertise
with a specialized type of noticing, what we call professional noticing of children’s
mathematical thinking. By identifying a focus for noticing, we attend less to the
variety of what teachers notice and more to how, and the extent to which, teachers
notice children’s mathematical thinking.

172 Professional Noticing
Professional Noticing of Children’s Mathematical Thinking
During the past few decades, mathematics educators have gained substantial
knowledge about both children’s mathematical thinking within specific content
domains and the power of teachers’ regularly eliciting and building on children’s
thinking (Grouws, 1992; Kilpatrick, Swafford, & Findell, 2001; Lester, 2007;
NCTM, 2000). Instruction that builds on children’s ways of thinking has been
linked to rich instructional environments for students (Clarke, 2008; Cobb et al.,
1991; Gearhart & Saxe, 2004; Schifter, 1998; Sowder, 2007; Wilson & Berne,
1999) and documented gains in student achievement (Bobis et al., 2005; Carpenter,
Fennema, Peterson, Chiang, & Loef, 1989; Fennema et al., 1996; Jacobs, Franke,
Carpenter, Levi, & Battey, 2007; Villaseñor & Kepner, 1993). In addition, research
has shown that teachers who, through professional development, learn how to learn
from the thinking of the individual children in their classrooms can continue
learning even after formal professional development support ends (Franke,
Carpenter, Levi, & Fennema, 2001). Given these documented benefits for both
students and teachers and the instructional vision of building on children’s thinking
suggested in national reform documents (Kilpatrick et al., 2001; NCTM, 2000),
we have chosen to target teachers’ expertise in professional noticing of children’s
mathematical thinking. We conceptualize this expertise as a set of three interrelated
skills: attending to children’s strategies, interpreting children’s understandings, and
deciding how to respond on the basis of children’s understandings.
Attending to Children’s Strategies
Terms such as highlighting (Goodwin, 1994) or making call-outs (Frederiksen,
Sipusic, Sherin, & Wolfe, 1998) have been used to describe how professionals
attend to noteworthy aspects of complex situations. We are interested in the extent
to which teachers attend to a particular aspect of instructional situations: the math-
ematical details in children’s strategies. Research has shown that these strategies
can be complex and that the strategy details are important because they provide a
window into children’s understandings (Carpenter, Fennema, Franke, Levi, &
Empson, 1999; Carpenter, Franke, & Levi, 2003; Lester, 2007). Similar to those
studying expertise in other areas, we hypothesize that teachers with more expertise
in children’s mathematical thinking will be better able to recall the details of children’s
strategies because they have developed meaningful ways to discern patterns and
chunk information in complex situations (Bransford, Brown, & Cocking, 2000).
Interpreting Children’s Mathematical Understandings
We are also interested in how teachers interpret children’s understandings as
reflected in their strategies. On the basis of a single problem, we do not expect a
teacher to construct a complete picture of a child’s understandings, but we are
interested in the extent to which the teacher’s reasoning is consistent with both the
details of the specific child’s strategies and the research on children’s mathematical

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development. Mason (2002) contrasted this type of productive (evidenced-based)
interpretation with unproductive (snap evaluations based on minimal evidence)
interpretation. We have chosen to exclude snap evaluations from our construct of
interpretation, and our decision to separate interpretation and evaluation is consis-
tent with the work of others (Blythe, Allen, & Powell, 1999; Seidel, 1998; van Es
& Sherin, 2008).
Deciding How to Respond on the Basis of Children’s Understandings
Our third component skill of interest is the reasoning that teachers use when
deciding how to respond. We are not arguing that there is a single best response,
but we are interested in the extent to which teachers use what they have learned
about the children’s understandings from the specific situation and whether their
reasoning is consistent with the research on children’s mathematical development.
Note that we do not include the execution of the response in our conceptualization
of professional noticing, so, in a sense, we are focusing on intended responding.
With this focus, we join others interested in how potential instructional responses
link to the other component skills of attending and interpreting. For example,
Erickson (in press) has argued that teachers’ selective attention is determined by
consideration of next instructional steps in that teachers judiciously direct their
attention to what is necessary to take action, and the lesson-analysis framework of
Santagata and her colleagues (2007) included making sense of the teacher’s actions
and then proposing alternative instructional strategies to accomplish the same
goal.
Investigation of Professional Noticing of Children’s Mathematical Thinking
In investigating professional noticing of children’s mathematical thinking, we
explored these three component skills to begin to unpack the in-the-moment deci-
sion making that is often hidden, but foundational to the complex and challenging
view of teaching endorsed in national reform documents (Kilpatrick et al., 2001;
NCTM, 2000). We focus on a particular type of decision making—decision making
that occurs on a daily basis in the classroom when a child offers a verbal- or written-
strategy explanation. This type of in-the-moment decision making is in contrast to
the long-term decision making (or planning) that teachers do after school when
they are not interacting with children.
Because teachers cannot preplan in-the-moment responses, this improvisational
part of teaching requires teachers to constantly analyze and connect specific situ-
ations to what they know about children’s mathematical development (Franke,
Kazemi, & Battey, 2007; Heaton, 2000; Lampert, 2001). We suggest that, before
the teacher responds, the three component skills of professional noticing of chil-
dren’s mathematical thinking—attending, interpreting, and deciding how to
respond—happen in the background, almost simultaneously, as if constituting a
single, integrated teaching move. Thus, our conceptualization of the construct of
professional noticing of children’s mathematical thinking makes explicit the three

174 Professional Noticing
component skills but also identifies them as an integrated set that provides the
foundation for teachers’ responses.
Our assessment of professional noticing of children’s mathematical thinking built
on the scenario methodology used in a series of expert/novice studies conducted in
the 1980s (for a summary, see Berliner, 1994). In these studies, researchers used
scenario methodology to assess teachers’ recall and analysis of classroom situa-
tions. For example, participants were shown slides or video and were asked, in
structured interviews, to recall as much as possible of what they had seen and then
to comment on the situation. Of most interest to researchers was the participants’
recall of general pedagogical techniques such as management and organization. In
our study, we also used scenario methodology but moved beyond a focus on recall
of general pedagogical techniques to instead focus on the specific expertise needed
to teach mathematics effectively by building on children’s mathematical thinking.
We also extended the earlier methodology by moving beyond comparison of only
two groups (experts versus novices) in which only limited conclusions can be drawn
about how teachers come to acquire skills (Peterson & Comeaux, 1987). Instead,
we investigated the professional-noticing expertise of four groups of participants
with differing amounts of experience with children’s mathematical thinking. In the
next section, we describe these participant groups and give further details about
our measures and analysis approach.
METHOD
The data were drawn from a larger study entitled “Studying Teachers’ Evolving
Perspectives” (STEP), in which our overall goal was to map a trajectory for the
changing needs and perspectives of teachers engaged in sustained professional
development focused on children’s mathematical thinking. In this cross-sectional
study, we explored the professional noticing of children’s mathematical thinking
by 131 prospective and practicing teachers, and we used two written measures
designed to assess the three component skills of attending, interpreting, and
deciding how to respond.
Participants
Participants included three groups of practicing K–3 teachers and one group of
prospective teachers who were beginning their studies to become elementary school
teachers (see Table 1). Consistent with the population of K–3 teachers, our group
of participants was overwhelmingly female (119 females and 12 males).
Participant groups differed in their experience with children’s mathematical
thinking. Specifically, Prospective Teachers, by virtue of their lack of teaching
experience and professional development, had the least experience with children’s
thinking, followed by Initial Participants who had teaching experience but no
professional development, and then by Advancing Participants who had teaching
experience and 2 years of professional development. Emerging Teacher Leaders

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had the most experience with children’s thinking because they not only had teaching
experience coupled with 4 or more years of professional development but also
engagement in at least a few leadership activities to support other teachers.
Examples of these formal and informal leadership activities include mentoring
other teachers by visiting their classrooms, sharing mathematics problems with
their grade-level teams, and presenting at faculty meetings or conferences.
Note that the average number of years of teaching in the three groups of prac-
ticing teachers was similar (14–16 years). We did not choose to group teachers by
the number of years they had been teaching because, although we recognized that
many important aspects of teaching improve with experience, we hypothesized that
teachers needed more than teaching experience alone to learn to teach mathematics
by building on children’s mathematical thinking in ways suggested in reform docu-
ments. Our larger study is specifically designed to explore this hypothesis by
comparing the knowledge, beliefs, and practices of teachers at three points during
sustained professional development. The prospective teachers were included as an
anchor point for this trajectory in which additional experience with children’s
mathematical thinking is hypothesized to be connected with enhanced expertise.
In this article, we focus on the participants’ professional noticing of children’s
mathematical thinking, both to investigate the hypothetical developmental trajec-
tory related to this expertise and to characterize the range of expertise in each of
the three component skills of professional noticing.
Practicing teachers were drawn from three districts in Southern California that
were similar in demographics, with one third to one half of the students in these
Table 1
Participant Groups
Participant group Description
Prospective Teachers
(n = 36) Undergraduates enrolled in a first mathematics-
for-teachers content course
Experienced practicing teachers
Initial Participants
(n = 31) Experienced K–3 teachers who were about to
begin sustained professional development focused
on children’s mathematical thinking
Advancing Participants
(n = 31) Experienced K–3 teachers engaged with sustained
professional development focused on children’s
mathematical thinking for 2 years
Emerging Teacher Leaders
(n = 33) Experienced K–3 teachers engaged with sustained
professional development focused on children’s
mathematical thinking for at least 4 years and
beginning to engage in formal or informal leader-
ship activities to support other teachers
Note. All practicing teachers had at least 4 years of teaching experience (with a range of
4–33 years), and the number of years of teaching experience in each group of practicing
teachers averaged 14–16 years.

176 Professional Noticing
districts classified as Hispanic, about one fourth classified as English Language
Learners, and one fourth to one half receiving free or reduced-cost lunch.
Prospective teachers were undergraduates, generally in their first 2 years of study,
in a nearby comprehensive urban university, and they had just begun their first
mathematics content course for teachers.
Professional Development
The Initial Participants, Advancing Participants, and Emerging Teacher Leaders
were all volunteer participants in sustained professional development focused on
children’s mathematical thinking (Lamb, Philipp, Jacobs, & Schappelle, 2009),
although the Initial Participants had yet to begin.1 The professional development
occurred prior to the study and was almost always facilitated by the same experi-
enced mathematics program specialist. It included about 5 full days of workshops
per year (in either half- or full-day increments) and drew heavily from the research
and professional development project Cognitively Guided Instruction (CGI)
(Carpenter et al., 1999; Carpenter et al., 2003). CGI is based on the idea that instruc-
tion can be improved by providing teachers access to research-based knowledge
about children’s thinking and by helping them to explore instruction that builds on
children’s thinking. CGI has documented gains in student achievement and teacher
learning (Carpenter et al., 1989; Fennema et al., 1996), including in urban class-
rooms (Jacobs et al., 2007; Villaseñor & Kepner, 1993).
The overarching goals of the professional development were to help teachers
learn how children think about and develop understandings in particular mathe-
matical domains and how teachers can elicit and respond to children’s ideas in ways
that support those understandings. Children’s mathematical thinking served as the
focus for interactions, and conversations were informed by research on children’s
thinking, but the goal was not to provide teachers with a completed framework that
summarized children’s reasoning. Instead, teachers worked together to create their
own frameworks that made explicit the similarities and differences in children’s
strategies. Teachers were given opportunities to recognize the power of attending
to the subtle details in individual children’s strategies—details that reflected math-
ematically relevant differences in the understandings children bring to their problem
solving. Teachers also engaged in conversations about how mathematical tasks,
classroom interactions, and classroom norms could be used to support and extend
children’s understandings within particular mathematical domains.
1We initially worried that, because of participant dropout, those teachers who persisted 2 years as
Advancing Participants or 4-or-more years as Emerging Teacher Leaders were different from the teach-
ers who chose to leave the professional development early. If so, the Initial Participants (who had yet to
begin professional development) might differ in important ways from the Advancing Participants and
Emerging Teacher Leaders for reasons other than the number of years of professional development.
We examined the enrollment records and concluded that dropout was not an issue in our study because
fewer than 10% of the Advancing Participants or Emerging Teacher Leaders dropped out by choice,
for programmatic reasons. Furthermore, even though some individuals were forced to discontinue their
participation for reasons outside their control (e.g., funding), they overwhelmingly chose to re-enroll
when the opportunity became available.

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Jacobs, Lamb, and Philipp
During professional development sessions, teachers engaged by solving math-
ematics problems, reading research, and analyzing video and written student work
derived from their own classrooms as well as from artifacts provided by the facili-
tator. Between professional development sessions, teachers were asked to pose
problems to their students and to bring the written student work to the next session.
With the help of their colleagues, they then worked to make sense of their students’
thinking in ways that highlighted core mathematical ideas and their developing
frameworks of children’s understandings related to those ideas.
Measures
In this article, we share a compilation of results from two assessments that we
developed to capture participants’ professional noticing of children’s mathematical
thinking. Each assessment was structured around an artifact of K–3 classroom
practice focused on problem solving involving whole-number operations: a class-
room video clip (Lunch Count) or a set of written student work (M&M’s®).
Participants were asked to watch the video clip or examine the set of written student
work and then to respond, in writing, to prompts about attending, interpreting, and
deciding how to respond.
Artifacts of Practice
The Lunch Count and M&M’s artifacts provided the core of the two assess-
ments.
Lunch Count video clip. The participants watched an edited 9-minute video clip
of a 40-minute lesson taught in February in a combination class with Grades 1 and
2. This lesson was selected because of its complexity, including on-task and off-task
behaviors, extensive teacher questioning, and sharing of various strategies reflecting
a range of understandings. Additionally, a child (rather than the teacher) posed the
problem for all children to solve. This lesson included many characteristics of
instruction recommended in reform documents.
In the video clip, a child posed the problem “We have 19 children, and 7 are hot
lunch. How many are cold lunch?”2 The teacher then allowed the children to work
on the problem individually or in pairs, solving the problem in any way that made
sense to them. Various tools were available. While the children were solving the
problem, the classroom was noisy, and when the children came together to share
their solutions, the teacher reminded them to put their tools aside and listen to their
classmates. Three correct strategies were shared. The first pair of children (Katie
and Sam) wrote 19 – 7 = and shared that they had counted back 7 from 19 on
their fingers. The second pair of children (Annette and Maureen) drew 19 individual
tally marks (not grouped in 5s) and erased 7 to find the number of cold lunches.
2Hot lunch was the classroom terminology used to refer to students who were buying their lunches,
and cold lunch referred to students who had brought their lunches that day.

178 Professional Noticing
The third strategy was shared by Sunny, a boy who used a counting frame (10 rods
with 10 beads on each rod). He first built 19 lunches by isolating 2 beads on 9 rods
and 1 bead on 1 rod. Next, he removed 7 (hot lunches) by removing 1 bead from
each of 7 rods so that what remained was 2 rods with 2 beads and 8 rods with 1
bead. Finally, he counted the remaining beads to find the number of cold lunches;
he subitized 4 (the 2 rods with 2 beads) before counting-on by 1s to reach the correct
answer of 12. During the sharing, the teacher questioned the children about their
reasoning and corrected some off-task behavior.
M&M’s written student work. The participants were provided three samples of
written student work in response to the problem “Todd has 6 bags of M&M’s. Each
bag has 43 M&M’s. How many M&M’s does Todd have?” (See the Appendix for
the three samples.) The set of written student work came from a second-grade class
and was selected because the strategies reflected a range of base-ten understand-
ings. Alexis’s strategy provided the least evidence of base-ten understanding in that
she drew six groups of 43 tally marks (grouped by 5s), but exactly how she counted
the tallies (by 1s, 5s, 10s, or 40s) to arrive at the correct answer of 258 is unclear.
Cassandra’s strategy provided the most evidence of base-ten understanding in that
she decomposed numbers in several ways to combine quantities. After writing six
43s, she worked with the 43s in pairs, and by combining the tens and then the ones,
she arrived at three 86s. She then combined two (of the three) 86s by combining
the tens to get 160 and the ones to get 12 for a total of 172. She then took 20 from
the 70 (in 172) to add to the 80 (in the third 86) to make 100. After adding this 100
to the 100 (in 172) to get 200, she added the 52 (remaining from 72 – 20) to arrive
at her answer of 252, but she made a minor error, forgetting to add the 6 from the
third 86. Josie’s strategy was between the other two strategies in terms of providing
evidence of base-ten understanding. She f irst represented six groups of 43 by
decomposing each 43 into the numeral 40 and 3 tally marks, and although her
counting strategy is not completely clear, she appears to have skip-counted by 40s
to arrive at 240. She then counted-on from there (but whether she counted-on by
1s or 3s is difficult to tell) to correctly arrive at 258.
Writing Prompts and Coding Schemes
For each artifact, participants were asked to write in response to three prompts
related to the three component skills of professional noticing of children’s mathe-
matical thinking. We coded the responses on scales indicating the extent to which
we had evidence for participants’ engagement with children’s mathematical
thinking.
Attending prompt. To assess participants’ expertise in attending to children’s
strategies, we requested, “Please describe in detail what you think each child did
in response to this problem.” The specific names of the children whose strategies
were shown in the video clip or in the written work were listed after the prompt to

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Jacobs, Lamb, and Philipp
ensure that participants commented on each strategy. Coding responses was a three-
step process. First, for each of the six strategies, we identified the mathematically
significant details. For example, how children counted was considered mathemat-
ically significant, whereas whether children shared their strategy standing at the
board or sitting on the floor was not. Second, for each of the six strategies, we
determined whether the response demonstrated attention to most of these mathe-
matical details or only a few. Third, for each artifact, we aggregated the three
strategy codes to identify whether we had evidence for each participant’s attention
to children’s strategies: evidence (1) or lack of evidence (0). Participants who
provided most details for at least two of three strategies were considered to have
provided evidence of attention to children’s strategies for that artifact. Note that we
did not require most details for all three strategies because demonstrating expertise
does not require that individuals always recall and understand everything; even
teachers who have acquired expertise in attending to children’s strategies can lose
focus and miss a strategy or not fully understand a particular aspect of a strategy.
Interpreting prompt. To assess participants’ expertise in interpreting children’s
understandings, we requested, “Please explain what you learned about these chil-
dren’s understandings.” We coded responses on a 3-point scale that reflected the
extent of the evidence we had of participants’ interpretation of children’s under-
standings: robust evidence (2), limited evidence (1), or lack of evidence (0). Prior
to the study, we determined our focus on interpreting children’s understandings,
but the number of categories and their characterizations emerged from the data.
Deciding-how-to-respond prompt. To assess participants’ expertise in deciding
how to respond on the basis of children’s understandings, we asked, “Pretend that
you are the teacher of these children. What problem or problems might you pose
next?” A part of the response space was labeled “Problem(s)” and another part was
labeled “Rationale” to ensure that participants provided both a next problem and
their reasoning.3 We coded responses on a 3-point scale that reflected the extent of
the evidence we had of participants’ deciding how to respond on the basis of chil-
dren’s understandings: robust evidence (2), limited evidence (1), or lack of evidence
(0). Similar to our coding of interpreting data, our coding of these data reflected
our prestudy focus on children’s understandings, but the number of categories and
their characterizations emerged from the data.
3We recognize that selecting a next problem is only one of the many ways that a teacher can respond
to a child. Other types of responses include probing existing strategies, facilitating comparison of
strategies, purposefully pairing children to share ideas, and so on. In this study, we chose to focus on
participants’ reasoning when selecting a next problem as a way to extend children’s understandings
after a correct answer was given (Jacobs & Ambrose, 2008). This focus seemed appropriate because
the six strategies presented were all valid (even though Cassandra’s answer was incorrect, given her
minor error of forgetting the last 6). See Jacobs, Lamb, Philipp, and Schappelle (in press) for a discus-
sion of similar results when participants’ noticing was focused on a situation in which a child needed
support to solve a problem correctly.

180 Professional Noticing
Participants watched the Lunch Count video clip before being provided with the
prompts, but they were given the M&M’s written student work and the prompts
together and were able to refer to this written work while writing their responses.
In the Lunch Count assessment, we chose to delay the presentation of the prompts
and allow the participants to view the video clip only once because we wanted the
video to serve as a proxy for actual instructional situations in which children often
share their thinking verbally and a rewind button does not exist. Similarly, in
the M&M’s assessment, we allowed participants to view the written student work
while writing their responses because in many instructional situations, teachers
have access to written work while they are trying to make sense of and respond to
children’s thinking.
Analyses
We began our analyses by coding the three professional-noticing skills of
attending, interpreting, and deciding how to respond for each of the two assess-
ments. Because the three skills each had unique prompts in the assessments, we
first examined the written response linked to the relevant prompt and then reviewed
the responses on the entire assessment to see whether related information was
included elsewhere. Data from all four participant groups were mixed and blinded
so that group membership was hidden during coding. All data were double-coded
by the first two authors, and interrater reliability for each set of coding was 80%
or more. Discrepancies were resolved through discussion.
In this study, we combined the results from the two assessments to achieve a more
stable measure of professional noticing of children’s mathematical thinking—one
that reflects participants’ noticing of children’s thinking when interacting with both
their verbal- and written-strategy explanations. By focusing on these two common
and important teaching activities, we addressed participants’ abilities to articulate
some of the types of noticing that teachers typically do in classrooms on a daily
basis. Specifically, we constructed an overall score for each of the three component
skills for each participant, and in creating each score, we wanted to capture what a
participant could do—the highest level of expertise demonstrated by that partici-
pant. Thus, we used the participant’s score that showed the greater engagement with
children’s thinking on the two assessments as his or her overall score for that skill.
This decision stemmed, in part, from our desire to minimize underrepresenting
participants’ expertise, given the limitations of written assessments without follow-
up questioning.4
Means were then calculated for these overall scores for each participant
group, and we used our cross-sectional design to capture the development of these
professional-noticing skills. Group differences were tested with four planned
comparisons: a monotonic trend reflecting increased experience with children’s
4We felt comfortable combining the two assessments because, for each assessment, the patterns
of results across participant groups generally mirrored those for the overall scores. (We describe the
overall-scores patterns in the following section.)

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Jacobs, Lamb, and Philipp
mathematical thinking and three pairwise comparisons of adjacent groups
(Prospective Teachers vs. Initial Participants, Initial Participants vs. Advancing
Participants, and Advancing Participants vs. Emerging Teacher Leaders). One-tailed
tests were conducted because we hypothesized that more experience with children’s
mathematical thinking should bring gains in professional-noticing expertise. The
Type I error rate of .05 was split among the four comparisons using Holm’s proce-
dure.
FINDINGS AND DISCUSSION
We examined the professional noticing of children’s mathematical thinking
across our four participant groups. We begin with an overview of our findings about
the differences between groups, and then we examine each component skill indi-
vidually to characterize in more detail the various levels of expertise.
Overview of Participant-Group Differences
One of our major goals in this study was to identify group differences among the
four participant groups to capture the development of professional-noticing exper-
tise. Thus, means were calculated for each participant group for the overall scores
of each component skill, with higher numbers indicating more evidence for engage-
ment with children’s mathematical thinking (see Table 2). Monotonic trends for all
three component skills were significant, indicating that increased experience with
children’s thinking was related to increased engagement with children’s thinking
on the professional-noticing tasks.
To better understand the development of professional-noticing expertise, we also
looked at the pairwise comparisons of adjacent groups for the three component
skills. For attending to children’s strategies, the pairwise comparisons of Prospective
Table 2
Participant Group Means (Standard Deviations) for the Overall Scores of the Component
Skills
Component skill Scale Prospective
Teachers Initial
Participants Advancing
Participants
Emerging
Teacher
Leaders
Attending to
children’s
strategies 0–1 0.42 (0.50) 0.65 (0.49) 0.90 (0.30) 0.97 (0.17)
Interpreting
children’s
understandings 0–2 0.47 (0.51) 0.94 (0.63) 1.19 (0.54) 1.76 (0.44)
Deciding how
to respond on
the basis of
children’s
understandings 0–2 0.14 (0.35) 0.29 (0.53) 0.84 (0.73) 1.45 (0.79)

182 Professional Noticing
Teachers versus Initial Participants and Initial Participants versus Advancing
Participants were significant with effect sizes of 0.58 and 0.66, respectively. These
findings provided evidence that expertise in attending to children’s strategies grew
with teaching experience and continued to grow with 2 years of professional devel-
opment. The pairwise comparison between Advancing Participants and Emerging
Teacher Leaders was not significant and therefore did not provide evidence for
additional gains with more years of professional development and opportunities to
engage in leadership activities, perhaps because performance was already at a high
(almost ceiling) level.
For interpreting children’s understandings, all three pairwise comparisons were
significant with effect sizes ranging from 0.49 to 1.06. Thus, expertise in inter-
preting children’s understandings grew with teaching experience and 2 years of
professional development. Furthermore, unlike expertise in attending to children’s
strategies, expertise in interpreting children’s understandings continued to grow
significantly when teachers had engaged in 4 or more years of professional devel-
opment and leadership activities.
For deciding how to respond on the basis of children’s understandings, the pair-
wise comparisons of Initial Participants versus Advancing Participants and
Advancing Participants versus Emerging Teacher Leaders were significant, with
effect sizes of 0.88 and 0.99, respectively. Because the comparison of Prospective
Teachers versus Initial Participants was not significant, we found no evidence that
expertise in deciding how to respond on the basis of children’s understandings
resulted from teaching experience alone. We did find evidence, however, that
expertise grew with 2 years of professional development and again when teachers
had engaged in 4 or more years of professional development and leadership
activities.
In summary, we have begun to construct a picture of the development of profes-
sional-noticing expertise.5 Teaching experience seems to provide support for
individuals to begin developing expertise in attending to children’s strategies and
interpreting children’s understandings, but we did not find similar evidence for
expertise in deciding how to respond on the basis of children’s understandings. In
contrast, professional development seems to provide support for developing exper-
tise in all three component skills. Furthermore, when professional development is
sustained beyond 2 years and coupled with leadership activities, teachers continue
to gain in their abilities to interpret children’s understandings and to use those
understandings in deciding how to respond. In the next three sections, we further
characterize professional-noticing expertise by sharing a range of sample responses
for each of the component skills.
5We recognize the limitations of written assessments in that individuals have different proclivities
(and abilities) to articulate their ideas in writing, but we had no reason to believe that writing expertise
differed systematically across our four participant groups. Thus, we argue that the patterns we identified
were not simply a reflection of writing ability.

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Attending to Children’s Strategies
Professional noticing of children’s mathematical thinking requires the ability to
attend to the mathematically important details of children’s strategies—details that
could inform a teacher’s instruction. In the sections that follow, we share sample
responses that provided evidence of attention to children’s strategies and responses
that did not.
Evidence of Attention to Children’s Strategies
When responses provided evidence of attention to children’s strategies, the
descriptions included mathematically significant details such as how children
counted, used tools or drawings to represent quantities, or decomposed numbers to
make them easier to manipulate. For example, consider the following description
of Cassandra’s strategy in the M&M’s written student work (see the Appendix for
Cassandra’s strategy):6
I think that Cassandra made 6 circles with the number 43 in each one. Then she
combined every 2 circles by adding the 10s together and then adding the 1s together
for each pair. Next, she added the 10s (80 + 80) and the 1s (6 + 6) for the first 4 circles.
After adding 160 + 12 to equal 172, she needed to add 86. Knowing that 80 + 20 =
100 (a familiar #), she took 20 from the 70 to get to 100. Then she figured she needed
to add the 52 left from the 172. What she forgot about was the 6 left from the 86. That’s
why her answer is off by 6.
This response captured the mathematical essence of the strategy. Specifically, the
participant articulated the decomposition of numbers into place values to combine
the pairs of 43s and the two 86s; the decomposition of 172 to create a familiar
number of 100; and the omission of the final 6 (from the third 86), which led to the
incorrect answer. Responses demonstrating evidence of attention to children’s
strategies were phrased in many ways, but they all tracked the entire strategy with
substantial detail about the mathematically important aspects of that strategy.
Lack of Evidence of Attention to Children’s Strategies
When responses did not provide evidence of attention to the details of children’s
strategies, comments tended toward general features of the strategies, such as
identifying a tool or mentioning that the problem was solved successfully, but
omitted details of how the problem was solved. For example, consider this vague
description of how Cassandra solved the problem: “Wrote 43 down 6 times, then
added them together in groups of 2. Then added those answers together to come up
with her final answer.” This description is general, missing all references to place
value and decomposition, and leaving open the question of whether the participant
6To provide a sense of the data in both assessments and for ease of comparison among scores within
a component skill, we have chosen to provide examples from the M&M’s assessment when discuss-
ing attending and deciding how to respond and examples from the Lunch Count assessment when
discussing interpreting.

184 Professional Noticing
fully understood Cassandra’s approach. At times, these lack-of-evidence responses
also included information that was inconsistent with the written work provided or
showed the participant’s confusion about Cassandra’s work, often implying that it
was problematic because of its complexity:
Cassandra’s work is very practical and simple too, but it’s not understandable. Why
did she subtract 20 and where did she get the 70 from? Her work was not very clean,
and she probably lost herself with too many numbers and lots of adding.
Developmental Patterns
Almost all the participants in the professional development groups (90% of the
Advancing Participants and 97% of the Emerging Teacher Leaders) provided
evidence of attention to children’s strategies. In contrast, only 65% of the Initial
Participants and 42% of the Prospective Teachers did so. Thus, expertise in
attending to children’s strategies is neither something adults routinely know how
to do nor is it expertise that teachers generally develop solely from many years of
teaching. Given the high level of performance of the Advancing Participants and
Emerging Teacher Leaders, we suspect that they may have been better able to recall
details of the strategies because they had learned to chunk strategy details in mean-
ingful ways, and these findings corroborate those in the expert/novice studies in
that experts tend to chunk information in ways that significantly enhance later recall
(Bransford et al., 2000).
A final note. We initially wondered whether the Prospective Teachers’ and Initial
Participants’ performance was lower than that of the other groups simply because
they did not understand the task of describing children’s strategies. However, we
concluded that misunderstanding the task was not a sufficient explanation for the
differences among participant groups. Although the participant-group patterns for
each of the six strategies generally mirrored those of the overall pattern, there was
one exception. In the Lunch Count assessment, all groups described well the Pair
2 strategy (the most basic of the six strategies), with more than 80% of each group
providing most details of the strategy. Given that all participants (not just those in
professional development) could successfully describe strategy details for certain
strategies, we concluded that all groups understood what was expected. Thus, the
participant-group differences on the overall attending score were reflective of
differing expertise in attending to children’s strategies rather than of some groups’
misunderstanding of the task.
Interpreting Children’s Understandings
Professional noticing of children’s mathematical thinking requires not only atten-
tion to children’s strategies but also interpretation of the mathematical understand-
ings reflected in those strategies. When identifying the extent of the evidence
participants demonstrated in interpreting children’s understandings, we were not
seeking a single best interpretation but were instead interested in the extent to which

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Jacobs, Lamb, and Philipp
participants’ reasoning was consistent with the details of the specific children’s
strategies and the research on children’s mathematical development. In the sections
that follow, we share sample responses for each level of the scale: robust evidence,
limited evidence, and lack of evidence of interpretation of children’s understand-
ings.
Robust Evidence of Interpretation of Children’s Understandings
We begin with a sample response showing robust evidence of interpretation of
the understandings of children shown in the Lunch Count video:
The first pair understands the problem is a [subtraction problem] by writing a number
sentence that showed 19 – 7 = .They did not need to count out 19 and take away 7
to get 12. They simply used their fingers to count backwards from 19. They seem to
have good number sense.
The second pair has a simpler strategy than the first because they have to count out
19 tallies and then take away 7. They still need to make the amount. They can’t hold
it in their head yet like the first pair. Also they did not group their tallies into 5’s which
[would] allow them to keep better track of their numbers.
The last boy has good number sense and understands different amounts. He was able
to count by groups of 2’s and switch to a group of 1 to make 19. He then took away 7
and counted what remained. He was able to start with 4 and count on by 1’s which
shows he has some understanding of amounts. He still needs to make 19 and so I think
the 1st pair has the best number sense because they were able to start right at 19 and
count down.
This participant interpreted the children’s understandings in several ways. First, she
made sense of the details of each strategy and noted how these details reflected
what the children did understand. For example, when discussing Sunny’s (“the last
boy’s”) understandings about quantities, this participant recognized Sunny’s ability
to count by 2s, his ability to switch between counting by 2s and 1s, and his ability
to subitize an amount of 4 and count on from that quantity. These comments all
point to mathematically relevant details that reflect Sunny’s understandings.
Second, the participant also recognized what strategies and understandings the
children did not demonstrate. For example, when discussing Pair 2’s understand-
ings, this participant recognized that they did not group their tallies into 5s, which
would have been a more efficient strategy, perhaps less prone to error. Finally, the
participant compared the strategies by recognizing that the ability to mentally
abstract a quantity was a required understanding only for Pair 1’s counting-back
strategy, which meant that this strategy reflected better “number sense” than the
strategies of Pair 2 and Sunny. Responses demonstrating robust evidence of inter-
pretation of children’s understandings focused on making sense of strategy details
in a variety of ways, but these interpretations were all consistent with the strategy
presented and the research on children’s mathematical development.

186 Professional Noticing
Limited Evidence of Interpretation of Children’s Understandings
The middle level of the scale included responses in which participants maintained
a focus on interpreting children’s understandings but with less depth than responses
that demonstrated robust evidence. The following is a sample response that
provided limited evidence of interpretation of children’s understandings:
The first set that shared had computational and representational understanding of the
problem. They knew what the algorithm would be to solve the word problem.
The second set was very one sided. They were at the one-to-one correspondence
picture stage and could have easily miscounted or made a computational mistake.
The third boy seemed to have very good number sense. He was able to group beads,
skip count, and explain his thinking very clearly. I would say that he would be able to
solve much more complex problems.
This participant described the children’s understandings, but often in broad terms
that were sometimes undefined (e.g., Pair 1’s having “computational and represen-
tational understanding” and Pair 2’s being at the “one-to-one correspondence
picture stage”). Specific connections to the children’s strategies existed, but they
were more limited than in responses with robust evidence, and conclusions were
sometimes overgeneralized, going beyond the evidence provided. For example,
another participant with a limited-evidence response wrote, “These children under-
stand subtraction and addition—and which to choose when presented with a
problem. . . . They know how to write a number sentence.” These broad conclusions
about addition and subtraction are difficult to justify on the basis of the children’s
performance on a single problem in which they all used a separating action.
Furthermore, only Pair 1 wrote a number sentence, but this participant seemed to
imply that Pair 1’s understandings were necessarily shared by the other children.
Thus generality, sometimes coupled with overgeneralization, characterized the
limited-evidence responses, but unlike responses described in the next section,
these limited-evidence responses were still focused on interpreting the children’s
understandings.
Lack of Evidence of Interpretation of Children’s Understandings
Some of the responses did not provide any evidence of interpretation of children’s
understandings, even though participants had been explicitly prompted to do so
(“Please explain what you learned about these children’s understandings”). These
responses had alternative foci, such as something learned about mathematics
teaching and learning in general, as in the following response:
I learned that it’s important to allow students to use different tools to come up with
mathematical problem solutions. Of course with this, it’s vital to provide lessons on
how to use several different tools. Only after that, can students decide what’s easiest
for them, and in turn choose tools which best work for the individual. I also learned
that a math lesson can be so much more than just math. This teacher invited the
students to a lesson in communication, listening, and respect in addition to subtraction
(no pun intended).

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Jacobs, Lamb, and Philipp
Other responses with these alternative foci included a positive evaluation of the
teaching in the video (e.g., “. . . I was glad that the teacher allowed her students to
use multiple ways of arriving to the correct answer”) or suggestions for improving
that teaching (e.g., “. . . I would have liked to hear the word difference and would
have liked to have seen a way of checking or proving answers were correct”).
Finally, some responses included commentary on the children but not on their
understandings (e.g., “I noticed all eager to try . . .”).
A lack of focus on individual children was another characteristic of these
responses. Fewer than half (43%) of the participants with an overall interpreting
score of lack of evidence differentiated their comments about the various children
who shared their work. Thus, the majority of these participants shared nothing—not
even something unrelated to the children’s understandings, such as behaviors or
affect—that provided evidence that they had noted anything about the individual
children on either assessment. This result is in contrast to participants whose overall
interpreting scores demonstrated limited or robust understanding: 66% of partici-
pants with an overall score of limited evidence and 100% of participants with an
overall score of robust evidence differentiated their discussions to explicitly address
individual children on at least one of the assessments. This differential approach to
our request for what was learned about the children’s understandings—commenting
on individual children versus discussing children only as a group—is also reflected
in the sample responses shared earlier. This distinction is critical because when
participants view groups of children only as a group, identifying the understandings
reflected in specific strategies becomes challenging, if not impossible.
Developmental Patterns
About half of the Prospective Teachers and three fourths of the Initial Participants
provided analyses with some evidence of interpretation of children’s understand-
ings, although no Prospective Teachers and only about one sixth of the Initial
Participants provided robust evidence. In contrast, every Emerging Teacher Leader
and all but two Advancing Participants focused on interpreting children’s under-
standings in their responses (see Table 3 for a comparison of participant groups on
Table 3
Percentage Within Each Participant Group Demonstrating Each Level of Evidence of
Interpreting Children’s Understandings
Prospective
Teachers Initial
Participants Advancing
Participants
Emerging
Teacher
Leaders
Robust evidence 0% 16% 26% 76%
Limited evidence 47% 61% 68% 24%
Lack of evidence 53% 23% 7% 0%

188 Professional Noticing
their overall interpreting scores). Thus, like expertise in attending, expertise in
interpreting children’s understandings is neither expertise that adults routinely
possess nor something that teachers generally develop solely from years of teaching.
Furthermore, providing robust evidence is particularly challenging, and this exper-
tise takes years to develop; almost three times the percentage of Emerging Teacher
Leaders compared to Advancing Participants generated responses that demon-
strated robust evidence. Thus, these results underscore the importance of profes-
sional development that extends beyond 2 years.
Deciding How to Respond on the Basis of Children’s Understandings
Professional noticing of children’s mathematical thinking requires not only
attending to children’s strategies and interpreting the understandings reflected in
those strategies but also expertise in using those understandings in deciding how
to respond. Although teachers’ responses could be of many types, we chose to focus
on the reasoning involved in selecting the next problem. When identifying the
extent of evidence that participants demonstrated in deciding how to respond on
the basis of children’s understandings, we were not seeking a particular next
problem or rationale but were instead interested in the extent to which participants
based their decisions on what they had learned about the children’s understandings
from the specific situation and how consistent their reasoning was with the research
on children’s mathematical development. In the sections that follow, we share
sample responses for each level of the scale: robust evidence, limited evidence, and
lack of evidence of deciding how to respond on the basis of children’s understand-
ings.
Robust Evidence of Deciding How to Respond on the Basis of Children’s
Understandings
Consider the following sample response showing robust evidence of deciding
how to respond on the basis of children’s understandings in relation to the M&M’s
written student work:
Problems and Rationale
For Alexis, I would use more round numbers to see if she could use a more efficient
strategy, like 6 bags of 50.
For Cassandra, I would try similar numbers again to see if she could perform her
calculations without error. Maybe 8 bags of 48—8 would leave even numbers of bags
to add up.
For Josie, I’d go with a larger number of M&M’s and more bags. Maybe she would try
a different strategy. 13 bags of 77. I’d be interested to see if she’d use the same strategy
and if so how would she break the numbers up?
This participant customized her suggestions for each child, explicitly considering
the child’s existing strategy and, in some cases, anticipating a possible next strategy.

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Jacobs, Lamb, and Philipp
Each suggestion reflects not only expertise in interpreting a child’s understandings
as reflected in the strategy used on the M&M’s problem but also, based on these
understandings, knowledge about next steps that research on children’s mathemat-
ical development has shown are likely to further this child’s understandings. For
example, on the M&M’s problem, Alexis showed evidence of grouping her repre-
sentation of 43s into 5s, but she also made all 258 tallies. Providing an opportunity
to add groups of 50 (instead of 43) might encourage Alexis to move away from
representing all 300 M&M’s. Fifty is a familiar number that children often know
in combination (e.g., 50 + 50 = 100), and the inclusion of a decade number can
facilitate use of mental strategies or strategies that do not involve counting by 1s
(or 5s). For each child, this participant suggested not that she would force or even
show a particular strategy but instead that she would strategically provide a next
problem to invite use of a more sophisticated strategy as a reasonable next step for
that child. Note that we were not evaluating whether the suggested moves were the
best moves (if that assessment is even possible). Instead, when responses demon-
strated robust evidence, we tracked the participant’s consideration of the children’s
understandings reflected in the strategies already used and how those understand-
ings could be building blocks when proposing a new problem.
Limited Evidence of Deciding How to Respond on the Basis of Children’s
Understandings
As in the scale for interpreting, the middle level of this scale included responses
in which participants used children’s understandings in their reasoning but in a more
general way. The following is a sample response that provided limited evidence of
deciding how to respond on the basis of children’s understandings:
Problem: I think I would give them the same problem using a 100 number, such as 6 bags
of 154 M&M’s.
Rationale: All of these kids know how to break numbers into 10’s plus 1’s. I think they are
ready to go to the next step and look at 100’s, 10’s, plus 1’s.
This participant shared a problem with a rationale that accounted for the children’s
past performance (“these kids know how to break numbers into 10’s plus 1’s”) and
anticipated next strategies (“they are ready to go to the next step and look at 100’s,
10’s, plus 1’s”) but did so in a general, somewhat vague, fashion. Furthermore, all
three children were assumed to have similar understandings and to need a similar
next step, even though their strategies on the M&M’s problem showed mathemati-
cally important distinctions, such as their different ways of breaking numbers into
10s plus 1s. Nonetheless, despite the minimal specificity and lack of customization
in the reasoning, this participant clearly considered these children’s strategies and
understandings in deciding how to respond, a characteristic missing from the
responses described in the next section.

190 Professional Noticing
Lack of Evidence of Deciding How to Respond on the Basis of Children’s
Understandings
Some responses provided no evidence of deciding how to respond on the basis
of children’s understandings; these responses included little or no reference to
building on the children’s understandings or anticipating future strategies for the
proposed problem. Sometimes these responses did identify the operation used or
the children’s success in solving the given problem but not their thinking on that
problem. In fact, the proposed next steps often seemed as if they could have been
generated without the participants’ having seen the children’s strategies, and
reasoning other than the children’s existing understandings were offered to justify
them.
For example, some lack-of-evidence responses identified other multiplication
problems (“The zoo field trip requires 4 buses. If each bus can hold 33 students,
how many students can go?”) so that the students could practice (“I would definitely
try a similar problem but with different wording. This is until I knew that the
students have become pretty familiar with those problems.”), but without reference
to existing understandings or anticipated strategies. Others focused on problem
difficulty, again without any specific link to these children’s existing understand-
ings. Instead, they included problems that would generally be considered more
difficult regardless of which children were solving them (“I would continue with
the same type but give more difficult numbers such as 110 in each group . . .”).
Finally, responses such as the following identified problems that introduced a new
mathematical topic (often a related operation) but did not specify how future work
related to this new focus could link to the children’s existing understandings:
Problem: Johnny has 56 blocks. He can put 7 blocks into each toy chest. How many toy
chests does Johnny have?
Rationale: This is obviously a lesson on an intro to division and multiplication. The students
are learning how to divide quantities and how to add up these quantities.
Therefore, since the students just learned how to add up certain amounts they
should learn to divide the amounts next.
Lack of discussion about number selection was another characteristic of these
responses in which participants used reasoning other than children’s understandings
to determine their next steps. Only 38% of participants with an overall deciding-
how-to-respond score of lack of evidence included any discussion of number selec-
tion in relation to their proposed problem(s) on either assessment. Thus, the
majority of these participants included no reasoning about number selection (e.g.,
specific numbers or classes of numbers such as larger numbers or decade numbers).
This result is in contrast to participants whose overall deciding-how-to-respond
scores demonstrated limited or robust evidence: 85% of participants with an overall
score of limited evidence and 96% of participants with an overall score of robust
evidence discussed the reasoning underlying their number selection for at least one
of the assessments. We argue that attention to the details of number selection,

191
Jacobs, Lamb, and Philipp
similar to attention to the details in children’s strategies, reflects an orientation to
teaching mathematics that is critical for providing tailored instruction that builds
on children’s existing understandings.
Developmental Patterns
Note that about two thirds of the Advancing Participants and more than four fifths
of the Emerging Teacher Leaders provided some evidence of using children’s
understandings, whereas only about one fourth or fewer of the Initial Participants
and Prospective Teachers did so (see Table 4 for a comparison of participant groups
on their overall deciding-how-to-respond scores). Furthermore, although almost
two thirds of the Emerging Teacher Leaders exhibited robust evidence, fewer than
one fifth of participants in each of the other groups did so. Thus, these results again
provide evidence that professional development, especially professional develop-
ment focused on children’s mathematical thinking that extends beyond 2 years, can
help teachers increase their engagement with children’s mathematical thinking.
Table 4
Percentage Within Each Participant Group Demonstrating Each Level of Evidence of
Deciding How to Respond on the Basis of Children’s Understandings
Prospective
Teachers Initial
Participants Advancing
Participants Emerging
Teacher Leaders
Robust evidence 0% 3% 19% 64%
Limited evidence 14% 23% 45% 18%
Lack of evidence 86% 74% 36% 18%
CONCLUSIONS
Evidence from our cross-sectional study indicates that the construct of profes-
sional noticing of children’s mathematical thinking merits attention from teachers,
professional developers, and researchers working toward the vision of successful
classrooms put forth in national reform documents (Kilpatrick et al., 2001; NCTM,
2000). On the one hand, the Prospective Teachers’ minimal engagement with chil-
dren’s thinking in all three component skills showed that professional noticing of
children’s mathematical thinking is challenging and not something that adults
routinely know how to do. On the other hand, the accomplished performance of the
Emerging Teacher Leaders and the consistently significant monotonic trend
(capturing increasing experience with children’s mathematical thinking) showed
that this expertise can be learned and that both teaching experience and professional
development support this endeavor.
We suggest that our study provides three types of resources for educators working
toward this vision: (a) In our conceptualization of professional noticing of children’s
mathematical thinking, we identify three specific skills—attending, interpreting,
and deciding how to respond—worthy of consideration; (b) our cross-sectional data

192 Professional Noticing
provide a nuanced story of the development of this expertise; and (c) our assess-
ments and results can serve as useful tools for professional developers.
Conceptualization of Professional Noticing of
Children’s Mathematical Thinking
Noticing is a promising construct that contributes to efforts to make explicit the
work of teaching, and researchers in mathematics education are just beginning to
mine this construct to explore how teachers process complex instructional situa-
tions.7 Thus, in characterizing a range of noticing expertise, we contribute to the
growing research base on how prospective and practicing teachers see and make
sense of classrooms in different ways and how particular types of experiences can
support the development of their abilities to notice in particular ways (see, e.g.,
Santagata et al., 2007; Sherin & van Es, 2009; van Es & Sherin, 2008).
We chose to focus on a specialized type of noticing—professional noticing of
children’s mathematical thinking—and a particular slice of teaching—the hidden
practice of in-the-moment decision making when teachers must respond to chil-
dren’s verbal- or written-strategy explanations. In these situations, if instruction is
to build on children’s thinking, teachers must be able to attend to children’s strate-
gies, interpret their understandings, and use these understandings in deciding how
to respond. Furthermore, they must execute these three skills in an integrated way,
almost simultaneously, while they are making these in-the-moment decisions.
We believe that the ability to effectively integrate these three component skills
is a necessary, but not sufficient, condition for responding on the basis of children’s
understandings, a core tenet of the vision of instruction promoted in reform docu-
ments (Kilpatrick et al., 2001; NCTM, 2000). In other words, attending to children’s
strategies, interpreting children’s understandings, and deciding how to respond on
the basis of children’s understandings collectively provide a foundation for
responding on the basis of children’s understandings. We recognize that effective
integration of these three skills—professional noticing of children’s mathematical
thinking—precedes a response and thus does not necessarily translate into effective
execution of the response, because execution requires yet another set of complex
skills. However, we argue that teachers are unlikely to base their responses on
children’s understandings without purposeful intention to do so, and it is this
purposeful intention that we have tried to capture with our construct of professional
noticing. In the next section, we summarize what we have learned about the devel-
opment of this expertise.
Developmental Trajectories of Professional-Noticing Expertise
Our cross-sectional design enabled us to capture each participant group’s patterns
of engagement with children’s mathematical thinking on all three component skills
7A forthcoming book, Mathematics Teacher Noticing: Seeing Through Teachers’ Eyes (Sherin,
Jacobs, & Philipp, in press), will present a compilation of the noticing work in mathematics educa-
tion.

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Jacobs, Lamb, and Philipp
of professional noticing. Although variability existed within each participant group,
we found the consistent patterns across groups to be convincing and worthy of
attention. Through these patterns, we can begin to paint a developmental trajectory
of professional-noticing expertise, highlighting which component skills developed
with teaching experience and which seemed to require the support of professional
development.
Prospective Teachers struggled with all the component skills, and for each skill,
fewer than half demonstrated evidence of engaging with children’s thinking. In
contrast, more than half of the Initial Participants showed evidence of attending to
children’s strategies and interpreting their understandings, but fewer than one fifth
provided robust evidence of interpreting children’s understandings. Thus, teaching
experience alone seemed to provide support for at least the initial development of
expertise in attending to children’s strategies and interpreting children’s understand-
ings. It did not, however, provide much support for development of the expertise
needed in deciding how to respond on the basis of children’s understandings; only
about one fourth of the Initial Participants demonstrated any evidence of deciding
how to respond on the basis of children’s understandings, and only one participant
demonstrated robust evidence.
For teachers who had engaged in professional development, the group patterns
changed substantially, in that engagement with children’s thinking in attending,
interpreting, and deciding how to respond was the norm. Nonetheless, the extent
of professional development was important, and potential for growth remained for
both groups. Specifically, almost all the Advancing Participants and Emerging
Teacher Leaders demonstrated evidence of attending to children’s strategies.
Furthermore, although almost all these teachers also provided some evidence of
interpreting children’s understandings, about three fourths of the Emerging Teacher
Leaders provided robust evidence, whereas only about one fourth of the Advancing
Participants did so. Thus, all gains from professional development in attending to
children’s strategies seemed to come within the first 2 years, but additional gains
were found in interpreting children’s understandings when professional develop-
ment continued through 4 or more years and included opportunities to engage in
leadership activities.
Emerging Teacher Leaders also showed superior expertise in deciding how to
respond on the basis of children’s understandings. Although a large majority of both
professional development groups demonstrated some evidence of deciding how to
respond on the basis of children’s understandings, about two thirds of the Emerging
Teacher Leaders offered robust evidence, whereas fewer than one fifth of the
Advancing Participants did so. Because this deciding-how-to-respond expertise is
essential for achieving instruction consistent with the reform vision, this study
provides strong evidence of the need for professional development that is sustained
over not only months but many years.
Teaching is a learning profession (Darling-Hammond & Sykes, 1999), and, as
such, teachers need opportunities to learn, with support, throughout their teaching
careers. Unfortunately, professional development has typically been short term and

194 Professional Noticing
fragmented (Hawley & Valli, 1999; Hill, 2004; Sowder, 2007; Sparks & Hirsh,
1997); however, our study context provided a unique opportunity for investigation
of the effects of professional development that was sustained over 4 or more years.
We found that sustained engagement was valuable, and in the next section, we
provide specific suggestions to support the development of professional-noticing
expertise in professional development.
Supporting Professional Noticing of Children’s
Mathematical Thinking in Professional Development
Our conceptualization of professional noticing of children’s mathematical
thinking and the findings from our cross-sectional study provide several resources
for professional developers who want to support the development of professional-
noticing expertise for prospective and practicing teachers. We highlight two of these
resources: discussion prompts and growth indicators.
Discussion Prompts
Our specific prompts to assess participants’ expertise in attending, interpreting,
and deciding how to respond could be useful discussion prompts during profes-
sional development. Not only would these discussions provide the facilitator with
valuable information about participants’ perspectives but participants would also
have targeted opportunities to explore these important instructional skills. In the
sections that follow, we offer some additional considerations for discussions
focused on each component skill.
Attending to children’s strategies. We suspect that the skill of attending to chil-
dren’s strategies is the skill that is most likely to be overlooked. Professional devel-
opers may assume that all adults possess this skill and that everyone sees the same
details in children’s strategies. We found otherwise, in that a substantial number of
Prospective Teachers and Initial Participants struggled to provide evidence of
attending to children’s strategies. In a related study, we found that even when asked
to watch a video that depicted only one child engaged in problem solving in an
interview setting, some adults still struggled to recall the mathematically important
details of the child’s strategy (Jacobs, Lamb, Philipp, & Schappelle, in press). Thus,
the challenge associated with attending to children’s strategies is not simply that
teachers need to pay attention to many things in a busy classroom with numerous
distractions. Given that some individuals still struggled to attend to a child’s strategy
in an interview setting, these challenges extend beyond a processing-capacity issue;
attending to children’s strategies requires not only the ability to focus on important
features in a complex environment but also knowledge of what is mathematically
significant and skill in finding those mathematically significant indicators in chil-
dren’s messy, and often incomplete, strategy explanations.
These results are consistent with other studies that have underscored the impor-
tance and challenge of description (Blythe et al., 1999; Rodgers, 2002), especially

195
Jacobs, Lamb, and Philipp
for teachers whose work orients them to immediately consider their next moves
(Sherin, 2001). Given the role of attending to children’s strategies as a foundational
skill for interpreting and deciding how to respond, professional developers should
consider the development of expertise in attending to children’s strategies to be a
worthwhile goal that will require time and targeted support to learn.
Interpreting children’s understandings. Although the majority of practicing
teachers in our study provided some evidence of interpreting children’s understand-
ings, fewer than half of the Prospective Teachers did so. Furthermore, the Emerging
Teacher Leaders were the only group to have a substantial number of participants
who provided robust evidence of this expertise. We were intrigued with the range
of issues addressed when responses did not focus on interpreting children’s under-
standings, especially given that one prompt explicitly asked participants to do so:
“Please explain what you learned about these children’s understandings.” We
suspect that the task was too challenging for some; to interpret children’s under-
standings, one must not only attend to children’s strategies but also have sufficient
understanding of the mathematical landscape to connect how those strategies reflect
understanding of mathematical concepts. When participants—especially Prospective
Teachers—did not focus on children’s understandings, they instead discussed other
issues such as the child’s affect, what they learned more generally about mathe-
matics teaching and learning (e.g., the importance of tool use or the idea that
multiple ways exist to solve a problem), or their evaluation of the teacher’s actions,
sometimes suggesting improvements. We encourage facilitators to be watchful for
the appearance of these alternative topics—topics that are important to discuss but
that may also derail an explicit conversation about children’s understandings.
Avoiding discussion of children’s understandings may be reflective of a lack of
attention to children’s strategies or a lack of mathematical knowledge for how to
make sense of those strategies, and professional developers will need to address
these challenges.
Deciding how to respond on the basis of children’s understandings. In this study,
we chose to focus on the reasoning behind one type of instructional response:
selecting the next problem. This expertise requires not only expertise in attending
to children’s strategies and interpreting children’s understandings but also knowl-
edge about children’s mathematical development to identify a reasonable next step
and the ability to facilitate that next step by selecting a problem that will be acces-
sible yet also challenge children’s thinking. Thus, selecting a problem that builds
on children’s existing understandings is complex and worthy of discussion.
However, selecting the next problem is only one of many ways that teachers can
build on children’s understandings after they have solved a problem correctly. Other
effective ways to respond include probing for the reasoning underlying children’s
strategies, asking children to compare strategies, encouraging the symbolic repre-
sentation of mental or tool-based strategies, and so on. In other situations, unlike
those reflected in the artifacts used in this study, children may struggle to even solve

196 Professional Noticing
a problem, and teachers can again choose from an array of responses including
clarifying the problem, probing children’s initial solution attempts, pairing children
so that they can help each other, and so on. All these types of responding are part
of the complex work of teaching (see Jacobs & Ambrose, 2008, for a more extended
discussion of the range of potential responses that build on children’s understand-
ings).
We argue that the reasoning behind all these types of responses is similar in that
the reasoning is more likely to be productive when it includes consideration of the
children’s existing understandings. Therefore, we suspect that many of the issues
and developmental trajectories identified in this study, in which the focus was
exclusively on selecting next problems, would also apply to other types of teachers’
responses. We have, in fact, explored teachers’ reasoning when deciding how to
respond to a child who is struggling to solve a problem, and the patterns of results
across participant groups were similar (Jacobs et al., in press). Nonetheless, profes-
sional developers (and researchers) need to address the professional noticing of
children’s mathematical thinking across the range of responding in which teachers
engage.
Growth Indicators
Our coding schemes and cross-sectional findings provide growth indicators that
can help professional developers identify and celebrate shifts in teachers’ profes-
sional noticing of children’s mathematical thinking. Specifically, we encourage
attention to the following shifts:
• A shift from general strategy descriptions to descriptions that include the math-
ematically important details;
• A shift from general comments about teaching and learning to comments specif-
ically addressing the children’s understandings;
• A shift from overgeneralizing children’s understandings to carefully linking
interpretations to specific details of the situation;
• A shift from considering children only as a group to considering individual chil-
dren, both in terms of their understandings and what follow-up problems will
extend those understandings;
• A shift from reasoning about next steps in the abstract (e.g., considering what
might come next in the curriculum) to reasoning that includes consideration of
children’s existing understandings and anticipation of their future strategies;
and
• A shift from providing suggestions for next problems that are general (e.g., prac-
tice problems or harder problems) to specific problems with careful attention to
number selection.
Note that some of these shifts may be minimal at first. Thus, professional devel-
opers need to be patient and initially expect limited, rather than robust, evidence of
shifts. Expertise in professional noticing of children’s mathematical thinking is

197
Jacobs, Lamb, and Philipp
complex, and, as our cross-sectional results illustrate, may require years to develop.
We conclude with a caution. In conceptualizing the construct of professional
noticing of children’s mathematical thinking, we envisioned the existence of a
nested relationship among the three component skills such that deciding how to
respond on the basis of children’s understandings can occur only if teachers inter-
pret children’s understandings, and these interpretations can be made only if
teachers attend to the details of children’s strategies. Given this nested relationship,
one could conclude that professional development should focus exclusively on
attending before interpreting and interpreting before deciding how to respond. We
worry that an approach that addresses these skills only independently and sequen-
tially may seem too removed from teachers’ everyday work. Instead, we argue that
professional developers can focus on all three skills in integrated ways but be aware
of the component skills and their growth indicators.
Next Steps
Through this study, we have become further convinced of the complexity of the
expertise needed to teach in ways that are consistent with the reform vision. We
focused specifically on the expertise underlying the hidden practice of in-the-
moment decision making that is needed to respond to children’s verbal- and written-
strategy explanations. Our theoretical conceptualization of professional noticing
of children’s mathematical thinking proved useful for characterizing this expertise;
providing snapshots of those with varied levels of expertise; and documenting that,
given time, this expertise can be learned. We recognize that our results are tied to
the particular type of professional development experienced, and future studies will
need to confirm the generalizability of our findings to other professional develop-
ment in which learning about children’s mathematical thinking is central. We also
recognize that the ultimate utility of this construct will depend on the ways in which
future studies can connect teachers’ professional noticing of children’s mathemat-
ical thinking with the execution of their in-the-moment responses. Nonetheless, we
hope that our study provides a starting point for researchers and professional devel-
opers seeking to identify teachers’ existing perspectives and productive next steps
for their learning in much the same way that frameworks for children’s thinking
have assisted teachers in identifying children’s existing understandings and produc-
tive next steps.
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Authors
Victoria R. Jacobs, School of Teacher Education and Center for Research in Mathematics and
Science Education, San Diego State University, San Diego, CA 92182; vjacobs@mail.sdsu.edu
Lisa L. C. Lamb, School of Teacher Education and Center for Research in Mathematics and Science
Education, San Diego State University, San Diego, CA 92182; Lisa.Lamb@sdsu.edu
Randolph A. Philipp, School of Teacher Education and Center for Research in Mathematics and
Science Education, San Diego State University, San Diego, CA 92182; rphilipp@mail.sdsu.edu
Accepted November 1, 2009

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APPENDIX
M&M’s Written Student Work

202 Professional Noticing
M&M’s Written Student Work (continued)