Teachers’ Mixed Implementation of Mindset Mathematics Practices During and After a Novel Approach to Teacher Learning

Abstract

Supporting teachers to transfer their learnings from innovative professional development settings to the classroom is challenging. In this paper, we investigate a novel approach to teacher learning, in which teachers from seven US districts taught in a mathematics summer camp using a research-based curriculum centered on student reasoning and mindset messages. We examined the practices teachers did and did not implement in their camp and school-year classrooms, as well as the possible reasons for the greater or lesser changes in practice. Through analysis of classroom video, teacher artifacts, and teacher interviews, we found that teachers implemented several important practices in both camp and school-year classrooms, namely posing open tasks, giving students ample time to collaborate, and asking questions that pushed students to reason. Interview analysis revealed that the act of centering students’ reasoning and witnessing their subsequent engagement seemed to motivate teacher uptake of these practices. At the same time, however, teachers less frequently integrated mindset messaging directly into their teaching and gave space for the exploration of students’ mistakes and struggles. These findings suggest implications for innovative professional development efforts outside of the school year, as well as incremental approaches.

1. Introduction

Decades of research in mathematics education has demonstrated the need to support all students to engage in mathematics, as opportunities to learn are not equitably distributed across subgroups of students [1], and many students do not have opportunities to actively engage in mathematics [2,3]. Narratives about mathematical ability as limited to the gifted few [4,5] and disproportionately held by white and Asian males [6] permeate our society and our schools. Problematically, many teachers have lower expectations for students who are from racially marginalized backgrounds [7], have lower socioeconomic statuses [8], and/or have learning differences [9].

Researchers have found that teachers tend to teach in line with these expectations. Low income and marginalized students are less likely to be taught in ways that encourage student agency, thinking, and collaboration [2,10] or to have choice in lessons [11]. Additionally, some teachers believe that rich mathematical tasks are only appropriate for high-achieving children and limit which students have access to inquiry-based mathematics [12]. This is especially concerning given that the use of narrow questions rather than rich tasks has been shown to contribute to student underachievement, thus perpetuating this inequality [13,14,15,16].

A related area of concern in mathematics education is the over-emphasis on procedures and answer-finding rather than the development of deep and connected mathematical understanding through inquiry-based mathematics. Decades of research have communicated the importance of supporting students’ development of mathematical reasoning [17,18,19,20,21]. In 2000, the National Council of Teachers of Mathematics (NCTM) released the Principles and Standards for School Mathematics, calling for an emphasis on not only procedural fluency but on conceptual understanding [22]. In 2013, the Common Core State Standards for Mathematics reinforced this message with the release of the Standards for Mathematical Practice [23].

Boaler and Greeno (2000) argued that supporting students’ development of conceptual understanding goes hand in hand with supporting their agency in their learning [24]. Drawing on Holland and colleagues’ (1998) notion of the “space of authoring”, Boaler and Greeno (2000) noted that different mathematics classroom environments create different possibilities for student engagement, which may range from authoring their own ideas as agentic actors to passively receiving content [24,25]. Importantly, the authors found that when students are given opportunities to create, explore, and reason about their own ideas—as well as those of their peers—they can construct and own their mathematical understandings [24]. Multiple studies across grade levels have found that when students generate their own ideas and engage with those of their peers, this can lead to powerful individual and collective learning [26,27,28].

While there is a strong consensus from research that students need opportunities to reason, engage actively, and use their agency, less is known about effective ways of supporting teachers to make and sustain these shifts in practice. Studies of professional development in student-centered and justice-oriented pedagogies have shown the efficacy of the following structures: providing concrete lesson analysis tools [29], anchoring equity discussions in school data [30], looking closely at students’ work [31], watching classroom video [32], and engaging in mathematics tasks [33]. These structures appear to support teachers to engage in rich discussions about students’ thinking and about equity, and to begin to enact more student-centered teaching approaches, to some extent.

At the same time, other work has revealed that supporting teachers to instantiate (and sustain) these ideas into their practice remains a challenge [34,35] and may be impeded by teachers’ ideas about student engagement and mathematics [36]. More broadly, transformational professional development, which has ambitious instructional goals and requires an intensive commitment from participating teachers, often does not gain traction beyond a small group of teachers [37], especially given the practice-based constraints [38,39] and discourses about mathematics [35,40] with which teachers must contend.

The challenge of designing effective learning opportunities for teachers, along with the need to center students’ reasoning and to change ideas of who can be successful in mathematics, motivated the initiative and study that is the focus of this paper. In this paper, we share findings from our study of a novel approach to teacher learning, in which teachers from seven US school districts taught in a mathematics summer camp, the format of which seemed to encourage more teacher learning than may be typical, while leaving some important practices unlearned or unenacted.

2. Conceptual Framework: Teacher Learning Approaches

To examine teachers’ learning, we draw on several approaches, all of which recognize that teaching is a complex practice, as expert teachers know how to respond, in the moment, to issues that take place at the intersection of students, content, pedagogy, and community [41,42]. The situated and relational nature of the knowledge teachers need to be effective [42,43,44] has led researchers to propose and study different models of teacher learning that go beyond learning from books and university courses, to learning in and from the practice of teaching [45].

Cochran-Smith and Lytle (1999) helpfully offered a framework for understanding teacher learning, proposing three forms of knowledge that can be developed in different models of teacher learning [45]. “Knowledge for practice” describes the type of knowledge that comes from sites outside of teaching, such as university departments, often used with an assumption that teachers can learn about practice outside of the act of teaching. This idea was famously challenged by Ball and Cohen (1999), who proposed that learning to teach outside of classrooms is analogous to learning to swim on a sidewalk [42].

Ball and Cohen (1999) proposed instead that teachers, even in university courses, learn in and from records of practice, such as student work, videotapes of lessons, curriculum materials, and teachers’ notes [42]. This is described by Cochran-Smith and Lytle‘s (1999) second form of knowing, “knowledge in practice”, which is the knowledge teachers learn from experience, the practical knowledge that is embedded in teaching, and teachers’ reflections on that knowledge [45]. Whereas knowledge for practice centers upon knowledge shared by others, knowledge in practice is usually assumed to be knowledge that a teacher develops, sometimes referred to as the “craft knowledge” of teaching. Cochran-Smith and Lytle‘s (1999) third conception of knowledge, “knowledge of practice”, blurs the distinction between the first two. It is knowledge that is often constructed collectively, with those involved holding critical views of education and the power entailed by it. Knowledge of practice, similar to knowledge in practice, is learned as a pedagogical act, relevant to immediate use, but, different to knowledge in practice, it is also learned through a process of theorizing. In this way, the knowledge is not bound by the situation in which it develops, and it helps to shape the “conceptual and interpretive frameworks teachers develop to make judgements, theorize practice, and connect their efforts to larger intellectual, social and political issues” [45] (p. 273).

In the field of teacher learning, there is widespread consensus that teachers should learn in and of practice. Researchers have worked to increase opportunities for such learning, including lesson study [46], video clubs [32], and teacher groups studying artifacts of students’ work [31,47]. Franke and colleagues (2001) found that years after participating in a professional development program focused on records of students’ mathematical thinking, teachers continued to engage in “generative growth” [48]. They found that the act of teaching—combined with opportunities for collaboration and reflection—became an ongoing learning opportunity that supported teachers to shift their practice, even after the professional development program had ended. Franke and colleagues’ (2001) findings are particularly striking given the potential of environmental factors—such as lack of teacher prep time and administrative support—to limit, rather than encourage, teachers’ transfer of their learnings from professional development to the classroom [38,39]. All these initiatives, as well as valuing the opportunity for learning of practice, are based on the theoretical assumption that teacher learning, like all learning, is situated within particular contexts [44], co-constructed in community with others [31,32], and accomplished through engagement with tools and artifacts, especially mathematics tasks [33].

Relatedly, researchers have reached some consensus on the content of teacher learning that is important, particularly the types of mathematics knowledge that support teaching and that help teachers to understand and formatively assess student thinking [49,50,51,52]. Additionally, studies have shown that teachers need to engage students actively, giving them opportunities to think, problem-solve, and reason, rather than passively receive knowledge [21,24,26]. The extent of the field’s belief in the importance of mathematics knowledge and active student engagement is revealed by the number of times this list of effective components of teacher learning has been reproduced:

(a) focus on specific content, (b) engage teachers in active learning and inquiry, (c) facilitate interaction and collaboration among teachers, and (d) be aligned with state standards and school goals [53,54].

• This set of characteristics has been repeated often, though with variation [55]. Increasingly, mathematics educators have paid more attention to the need to support teachers in implementing culturally relevant and equitable pedagogies [29,34,56,57], which may be worth adding to this frequently cited list of effective components of teacher learning.

Although researchers have coalesced around the importance of infusing mathematics content and active learning pedagogies into professional development, there is some disagreement about whether teacher beliefs or practices should be the focus of a particular effort, as it is unclear whether beliefs influence practices or practices influence beliefs. Research has shown, however, that these two concepts are interconnected [36,58,59]. For example, Maher and colleagues (2023) found that teachers whose beliefs about mathematics education aligned with the NCTM standards were more successful in recognizing students’ mathematical reasoning than those whose beliefs were less aligned [60]. The connection between teacher beliefs and pedagogy suggests that professional development efforts must attend to both teachers’ practices and their perceptions of students and their engagement with mathematics.

As such, we designed, implemented, and studied a novel approach to teacher learning that sought to support teachers in shifting both their practices and their perceptions of students. Through the act of teaching in a mathematics summer camp with a research-based curriculum, teachers had opportunities to develop “knowledge of practice” [45]. In the next section, we describe the mindset mathematics pedagogy used in the camp and our approach to studying teachers’ learning of it.

3. The Mindset Mathematics Approach

The “mindset mathematics” approach [5,15] has similarities with other initiatives in mathematics education, such as “ambitious math instruction”, “active learning”, “inquiry-based learning”, and “student-centered learning.” All of these pedagogies involve students engaging in cognitively challenging mathematics tasks and generating their own mathematical ideas, such that teachers can then draw on that thinking to support the class to develop new understandings. Variations of these approaches have been shown by different studies across grade levels to be effective [61,62,63]. What the mindset mathematics approach adds to previous initiatives is the sharing of ideas about growth mindset and the importance of times of struggle.

3.1. Mindset Messaging Through Open Tasks, Encouraging Struggle and Formative Assessment

In the mindset mathematics approach, teachers first engage students in open tasks that enable them to reason, problem-solve, and struggle. In particular, Yankelewitz and colleagues (2010) have shown that the design of the mathematics task itself can shape the forms of mathematical reasoning in which students engage [64]. As students work on the tasks, teachers highlight that students can learn and grow, and that struggle is important for learning. As such, teachers communicate mindset messages both implicitly, by providing time for students to collaborate on and discuss open tasks, and explicitly, by sharing that all students are capable and that being challenged is good, as it provides time for an important ‘brain workout’. The encouragement of times of struggle, drawing from multiple research studies [65,66,67] and policies in mathematics education [22], is a relatively novel aspect of the mindset mathematics approach, which was emphasized in the professional development that accompanied the camp curriculum.

Steuer et al. (2018) conceptualized the importance of struggle and what they call ‘positive error climates’ in mathematics classrooms through eight interrelated features [65]. They propose the following:

• Error tolerance by the teacher;
• Irrelevance of errors in assessment;
• Teacher support following errors;
• Absence of negative teacher reactions;
• Absence of negative classmate reactions;
• Taking error risks;
• Analysis of errors;
• Functionality of errors for learning.

The most significant of these features for this study of teacher learning are the seventh and eighth features, which describe classrooms in which teachers spend time analyzing mistakes and errors (7) and center mistakes as starting points for learning (8). These practices can be challenging for teachers to implement, given dominant views of mathematics that emphasize right answers [35]. In their study of pre-service teachers’ skills around eliciting students’ thinking in response to an incorrect answer, Shaughnessy and colleagues (2020) found that teachers were more likely to elicit the students’ process rather than their conceptual understanding and more likely to elicit their revised ideas rather than explore why the mistake had been made [66]. Analyzing errors and centering mistakes, though challenging, can support student learning [65,66].

The mindset mathematics approach also recommends changes to assessment, as traditional assessment practices of grading and testing often give fixed messages to students, suggesting that students are defined by a particular number or grade—undermining growth mindset messages [15]. The recommended alternative is for teachers to assess formatively, during a course, consistent with Black and Wiliam’s (1998) proposed “assessment for learning” approach [68]. When assessing formatively, teachers observe and make use of information on students’ understanding gained from classroom activities and discussions, to guide their teaching decisions [69]. Consequently, research shows that when formative assessment is integrated into professional development through teachers’ engagement with formative assessment routines and classroom artifacts, this may lead to enhanced teacher reflection [70] and increases in student achievement [52]. With this research in mind, formative assessment routines were incorporated into the training for teachers of the summer camp, which we explain in more detail in the next section. Teachers engaged in formative assessment throughout the camp, as these routines were embedded in the summer camp curriculum, and the only test given was an end-of-intervention assessment administered by the researchers.

3.2. Studying Teachers’ Learning of the Mindset Mathematics Approach

This study of teachers’ learning during and after a mindset mathematics summer camp drew from the wisdom of the teacher learning studies preceding it but departed from previous work in two important ways. First, the study focuses upon a novel approach to teacher learning, by centering learning from teaching, through a curriculum that was designed by university researchers, to empower teachers to envision what is possible. In engaging with this research-based curriculum and attending trainings on it (which we describe below), teachers had the opportunity to build their understanding of the mindset mathematics approach, developing “knowledge for practice” [45]. In teaching the curriculum themselves, teachers had the opportunity to learn directly from the act of teaching, developing “knowledge in practice” [45]. Taken together, the project is captured by Cochran-Smith and Lytle’s third conception, “knowledge of practice”, which blurs this distinction, as the teachers learned research-based pedagogical practices through collective implementation with their colleagues, with an eye to challenging inequitable outcomes [45]. The approach was also novel in that the teaching took place outside of the regular school year, such that teachers were not constrained by district tests, mandated curricula, or pacing guides, and instead had the freedom to focus on student learning.

Second, our initiative departed from the well-circulated list of teacher learning characteristics by working to offer meaningful learning opportunities to students who may have experienced marginalization in mathematics, such as those from racially and linguistically marginalized communities [2,3] and those with special education needs [9,71,72]. The curriculum took seriously the recognition that many mathematics teachers (and students and parents) subscribe to dominant discourses about mathematics and mathematical ability [35], which dictate that students either have a math brain or they do not [4,5] and that mathematical excellence is available only to some students, particularly white or Asian males [6]. To combat these discourses, the curriculum was specifically designed to enable student thinking to be visible to teachers, to share representations of mathematicians of color, and to offer opportunities for students to see themselves as mathematicians.

In this study, we examined teachers’ learning of transferable practices through the act of teaching in a mindset mathematics summer camp, which was centered around a research-based curriculum. Through analysis of classroom video, artifacts, and interviews, we explored which practices from the approach teachers did and did not implement both during and after the camp. Further, we examined teacher interview data to understand how teachers made sense of their learning, uncovering the mechanisms of the novel learning experience of teaching a summer camp. In particular, we ask the following research questions:

• Which practices from the mindset mathematics approach did teachers implement during and after the summer camp? Which practices did they not implement?
• How did teachers make sense of their learning of mindset mathematics practices during the summer camp?

4. Methods

4.1. Origins of the Camp

Prior to this study, researchers developed and taught a “mindset mathematics” four-week summer camp in 2015, which was designed to actively engage middle school students, with prompts and tasks that encouraged them to ask their own questions, come up with conjectures, and investigate ideas focused upon algebraic reasoning [73]. Mindset messages were not only shared with students—centralizing the ideas that all students can achieve at high levels, that struggle is an important goal, and that speed is not valued—they were infused into the teaching of content [74]. This meant that students were given tasks that gave them the opportunity to struggle, and when they found the work difficult, teachers shared that it was because their brains were working so hard. They were also given tasks that encouraged brain connections [75], as students connected numbers with visuals, physical objects, and movement. The students took a pre- and post-assessment of algebraic thinking before and after camp, which revealed that they had improved their performance by an average of 50 percent across the students, with an effect size of 0.91 standard deviation, equivalent to 2.8 years of growth [73].

As a follow-up to the 2015 study, teachers from ten school districts were invited to teach the same summer camp in their local areas in 2019, and they were given access to a range of professional development opportunities, described below. A different study investigated the student learning and achievement from the ten camps that were conducted across the US, finding that students who attended the camps significantly increased their achievement at the end of camp, as measured by MAC/MARS performance tasks, and achieved significantly higher math GPAs in their school districts in the following school year [76]. This paper reports upon the learning opportunities that the teachers received, through the act of teaching in the camp, to learn knowledge of practice in new ways [45], and to learn new teaching practices that could be used in their regular school year.

4.2. Student and Teacher Demographics

In the summer of 2019, ten districts in five US states implemented the summer camp. The districts recruited students who were diverse in terms of ethnicity, gender, and socioeconomic status, focusing on students who are Black, Latine, and/or experiencing poverty to ensure that camp attendees reflected these groups. This study shares findings from 20 teachers, across seven of those ten districts, who provided multiple sources of data. Student data for these seven districts are shown in

Table 1. Student demographics of participating school districts.

School District	State	Urban/Rural Classification	Total Students	% Black	% Latine	% Free/ Reduced Lunch
District 1	Michigan	Suburb: large	3000	24%	3%	48%
District 2	Alaska	City: small	15,000	5%	9%	31%
District 3	Illinois	Suburb: large	20,000	5%	38%	41%
District 4	Illinois	Suburb: large	40,000	7%	54%	59%
District 5	California	City: small	5000	1%	38%	41%
District 6	California	Suburb: large	1500	1%	61%	53%
District 7	New Mexico	Rural: fringe	2000	0%	79%	62%

and teacher data for these seven districts are shown in

Table 2. Teacher demographics of participating school districts.

School District	Number of Camp Teachers	Gender		Years of Teaching Experience
		Male	Female	0–2	3–5	6–8	9–11	12–15	15+	N/A 1
District 1	19	6	13	3	2	2	1	3	3	5
District 2	4	1	3						4
District 3	10		10	1	3	1		2	3
District 4	4		4	1	1				2
District 5	2	1	1			1				1
District 6	5	2	3
District 7	5	1	4

¹ N/A refers to teachers who did not report their years of teaching experience.

. Teachers’ experience from these seven districts ranged from first-year teachers to veteran teachers with more than 15 years of experience.

4.3. Support for Teachers

Different forms of support were offered to participating teachers before and during the camp (see

Table 3. Forms of support for teachers.

Type of Support	Specific Offerings
Classroom resources	Summer camp curriculum Mindset videos to share with students Manipulatives for math tasks
Teacher learning opportunities	Three one-hour webinars Online Mindset Mathematics class Reading material

). In terms of classroom resources, teachers were given a curriculum, mindset videos, and physical manipulatives. The curriculum included a flexible lesson plan for each day and focused on several big ideas: number sense, algebra as a tool for problem solving, generalization, and mathematics as pattern seeking [77]. A lesson plan included a “number talk” to build number flexibility and a short video with growth mindset messages. The remaining time was dedicated to instruction organized into big ideas [78], with students participating in open-ended mathematics tasks that encouraged them to engage with agency and authority [79,80], time to work in groups, and often a whole-class discussion. The tasks all had the feature of being “low floor and high ceiling”—the low floor meant that all students could access the tasks with multiple entry points, and the high ceiling meant that the tasks extended to high levels and in multiple directions/areas of mathematics [74,81,82]. In conjunction with these open tasks, the curriculum emphasized whole-class discussions in which students shared their reasoning from the task with their classmates. Various structures for discussion were offered, including students presenting their ideas to peers via group posters, or the approach of “convince yourself, convince a friend, convince a skeptic” in which students practiced justifying their thinking to peers taking on a “skeptic” perspective.

Consequently, the summer camp curriculum did not prescribe ways to teach but instead offered teachers multiple structures for collaboration and discussion from which they could choose, and shared rich tasks that students could take to different levels. This flexibility invited teachers to act with agency and to actively iterate on the curriculum in the moment, based on their students’ thinking [77]. As such, teachers had opportunities to develop their “knowledge in practice” by engaging with, implementing, and continually revising the curriculum, learning directly from their teaching [45]. Further, because many teachers co-taught their camp classes with colleagues (resulting in lower student-to-teacher ratios), they also had opportunities to collectively make sense of the mindset mathematics approach, which is critical to the co-construction of “knowledge of practice” [45].

In addition to the curriculum, teachers were given videos to share with students on mindset messaging. Physical materials and manipulatives (e.g., sugar cubes, Cuisenaire rods) were also provided to (1) remove barriers teachers might face in accessing supplies and (2) encourage students to explore different ways of representing their thinking.

To build teachers’ understanding of the mindset mathematics approach, several trainings were offered. In the spring of 2019, all participating teachers were required to take part in three one-hour webinars and were offered additional learning opportunities, including reading materials and an online class. These webinars were hosted by teacher educators and involved supporting the attendees to (1) understand the mindset mathematics approach and (2) engage with some of the curriculum that would be taught during the camp. Teachers learned about growth mindset and mindset messages, the value of a struggle and mistakes-friendly culture [65], and the concept of open tasks that enable students to reason, problem-solve, and struggle. Teachers were also encouraged to take their time with the curriculum and be flexible with timing (i.e., take more time on tasks that students seem particularly excited or intrigued about). The online class offering (in addition to the webinars) showed videos of these mindset mathematics practices in action during the original offering of the summer camp and encouraged teachers to reflect on how these practices impact student reasoning, mindset, and learning. These learning opportunities not only trained teachers on how to use the curriculum but also introduced teachers to research on the mindset mathematics approach. In engaging directly with research and then enacting a research-based curriculum, teachers were encouraged to build both theoretical and practical understandings of the mindset mathematics approach. As Cochran-Smith and Lytle note, this “knowledge of practice” can help shape “conceptual and interpretive frameworks” to guide teachers’ instructional decisions [45] (p. 273).

4.4. Data Sources

We investigated teachers’ learning through three data sources: videos of their teaching, teacher timeline artifacts, and teacher interviews. To consider which practices of the approach teachers implemented during camp, seven classroom videos across four sites were analyzed. All teachers had been asked to record and submit a classroom video of the same task, “Painted Cube” (Figure 1). Across all districts, nine classroom videos were submitted and seven were determined to be of sufficient length and audio quality for analysis. The two excluded videos each consisted of several short clips: the first included eight clips (ranging from eight seconds to two minutes and 45 seconds) and the second included three clips (ranging from 20 seconds to two minutes). Both videos were excluded for analysis because (1) it was impossible to contextualize the individual clips in relation to each other and the broader lesson, (2) the total time recorded across the clips was less than half of the duration of each lesson, and (3) audio quality was an issue in several of the clips. For these reasons, the research team agreed that we did not have enough context or audio input to accurately code these two videos.

The research team did, however, watch both excluded videos, noting the potential presence of several mindset mathematics teaching practices in the first one and the appearance of several less-aligned practices in the second one. While excluding these videos from our analysis may have impacted our findings, the alternative of attempting to code these videos without proper context or audio would have been problematic, as it would have required coders to make inferences beyond what was directly observable in the clips. Further, because one eliminated video likely contained mindset mathematics teaching practices and the other seemed to contain less-aligned practices, we argue that these two videos were likely to be similar to the other seven videos in terms of which practices were implemented. We expand on these limitations in our discussion section.

To consider teachers’ shifts in their practice from before camp to after camp, teachers submitted timelines or logs of the typical set of practices and routines they would conduct during class time. The goal of the timelines was to understand how teachers structured their class time and what kinds of activities they engaged in with students, both before and after teaching in the summer camp. An example of a pre-and post-timeline (from the same teacher) is shown in

Table 4. Pre- and post-timeline example.

Pre-Timeline

Post-Timeline

. Pre-timelines were collected in spring 2019 and post-timelines were collected in fall 2019. Thirty-eight teachers completed at least one timeline, with 28 teachers completing both pre- and post-timelines.

To dive more deeply into and make sense of teachers’ uptake of practices, researchers conducted semi-structured interviews [83] with teachers during the 2019–2020 school year, following the implementation of the camp. All participating teachers were invited for interviews. The 20 teachers who accepted the invitation to be interviewed came from seven districts, representing a range of camps that differed in their achievement impact, based on MARS effect sizes [76]. During the interview, teachers were asked about their experiences teaching in the summer camp (i.e., how they used the curriculum, how their students engaged in it) and their experiences teaching in the current school year (i.e., which practices, if any, they were currently using and how they were using them). These interviews lasted approximately 45 min, were conducted and recorded via Zoom, and were transcribed.

4.5. Analytic Approach

4.5.1. Video Analysis

The videos were analyzed in two different ways. Researchers created content logs [84] of approximately seven hours of video, outlining the events on each video and conducting a time analysis of how many minutes were spent on each segment of the lesson (i.e., task launch, work time, and whole-class discussion) for each teacher and across the teachers. One of the seven videos was excluded from the time analysis because it only captured one segment of the lesson.

In the second form of analysis, researchers examined the seven videos using the Mindset Mathematics Teaching Guide (MMTG) (https://www.youcubed.org/mathematical-mindset-teaching-guide-teaching-video-and-additional-resources/) as a tool for coding classroom practice. An initial video was selected to consider in depth, based on the teacher’s implementation of several mindset mathematics teaching practices, which was determined during the content logging process. A team of three researchers then re-watched this video, independently identifying 7–10 “critical moments” in which one of the mindset mathematics teaching practice categories (Growth Mindset Culture, Nature of Mathematics, Challenge and Struggle, Connections and Collaboration) was enacted. The fifth category, Assessment, was excluded because it was not feasible to identify the teacher’s range of assessment practices in one lesson. After discussing these moments and the extent to which each aligned with the proposed practice category, the combined critical moments were developed into a list of indicators for each dimension of each practice category. This initial draft was then tested on two contrasting cases from two different sites, for which researchers identified evidence that either validated an indicator or suggested a need to refine an indicator (e.g., re-wording, clarifying, adding). These pieces of evidence were discussed until consensus was reached, which led to refinement of the indicators. These revised indicators (

Table 5. Video analysis tool.

Practice Category	MMTG Dimensions	Indicators (Micro Practices)
1.Growth Mindset Culture	1A. Mindset messages	Teacher gives explicit messages about brain growth, challenges, struggle, etc.
	1B. Praising the learning process	Teacher provides space and time for students to (i) grapple with the task on their own or with peers and (ii) to share their thinking with each other. Teacher elicits, engages with, and praises student thinking and ideas.
	1C. Students’ mindsets	(i) Student shares their thinking even if it differs from the rest of the class; (ii) multiple students volunteer to share their answers; (iii) students come up to the board to physically model their ideas; (iv) students feel the freedom and ownership of the physical space to share their ideas (e.g., going to the board or standing up, without asking for permission).
2. Nature of Mathematics	2A. Open task (task as written)	Does the task allow for multiple approaches?
	2B. Reasoning and multiple perspectives	(i) Students approach the task in multiple ways (e.g., student work shows different approaches, student participation shares different ideas); (ii) these various approaches are given attention, valued, and explored by the teacher (who makes space for students to do so too); (iii) students are given access to a range of resources and materials that afford multiple approaches; (iv) students are expected and explicitly invited to bring multiple ideas to the task and to justify or reason through their ideas (in writing and/or verbally).
	2C. Depth over speed	(i) Teacher encourages and makes time for students to be curious about the task and their peers’ ideas; (ii) teacher encourages the class to come to a consensus via justification (convincing the class), rather than focus on the answer; (iii) consensus comes from the students, not the teacher.
3. Challenge & Struggle	3A. Mistakes	Space and time are provided for students’ thinking to evolve, without the undue pressure of a finished correct answer. Mistakes are valued and explored.
	3B. Struggle and persistence	Teacher provides space and time for students to (i) grapple with the task on their own or with peers and (ii) to share their thinking with each other. Teacher elicits, recognizes, and celebrates students’ struggles.
	3C. Questioning	At various times in the lesson, teacher asks deep thinking questions that (i) center students’ current thinking, such as, “What do you notice?” “What do you wonder?” “What are you trying to figure out?”, before pushing them in new directions; (ii) encourage multiple ways of thinking and require justification, such as, “Do you agree with _____?” “Does anyone else think it’s something different than ___?” “How can you prove it?” “How do you know?”
4. Connections & Collaborations	4A. Mathematical connections	(i) Students have the resources and time to try out different methods; (ii) in class discussion or small groups, there are connections being made between methods and representations.
	4B. Connecting in small groups	(i) Students collaborate and build off each other’s ideas in small groups; (ii) all students within a group are involved in the task.
	4C. Connecting as a whole class	Students talk directly to each other in math discussions, instead of relying on the teacher to mediate.

) were then used to code the remainder of the data set, after which the team constructed visuals summarizing the distribution of practices across teachers. While “indicator” was a useful term in the context of our analysis, we refer to the indicators as “micro practices” throughout our findings to emphasize the ways in which these micro teaching moves are nested within the broader teaching practices reported in timelines and interviews.

4.5.2. Teacher Timeline Analysis

To consider potential shifts in teachers’ practice from before camp to after camp, timelines from the 38 teachers who had returned at least one timeline were analyzed. Four researchers first open coded a subset of timelines and generated a list of reported practices, which the team then consolidated into a codebook of practices (see

Table 6. Descriptions of practices reported in teacher timelines.

Practice	Description
Practices Less Aligned with MM
Review/assign homework	Teacher goes over answers to students’ homework from the night before or assigns students to work on their homework during class time.
Direct instruction	Teacher introduces and/or explains new content via direct instruction or lecture with limited or no student discourse.
Practice	Teacher assigns students to work on practice problems. Sometimes referred to as the last part of the “I do, we do, you do” style of instruction, in which students independently work on sets of short problems at their desks.
Practices More Aligned with MM
Mindset messages	Teacher explicitly communicates that all students can learn and grow and that struggle is key to learning and brain growth.
Classroom discussion	Teacher facilitates a whole-class discussion, during which multiple students share their mathematical reasoning and ideas with the class.
Pose an open task	Teacher poses an open, low-floor, high-ceiling task for students to investigate during class. Sometimes referred to as the first part of the “launch, explore, summarize” style of instruction, in which the teacher launches an open task for students to then explore.
Number talk	Teacher facilitates a number talk, a classroom routine in which teachers pose a problem for students to solve mentally and then facilitate a discussion in which multiple students share their strategies while the teacher scribes it on the whiteboard. Because of their relatively short length (approximately 15–20 min), number talks are often used as warm-ups.

) and applied to all the timeline data. Practices were chosen as the unit of analysis, which resulted in a total count of n = 126 individual reports of practice in the pre-timelines and n = 104 individual reports of practice in the post-timelines. Researchers then constructed visuals to compare the distribution of reported practices in the pre-timelines to those in the post-timelines.

4.5.3. Teacher Interview Analysis

To analyze the interview data, a team of three researchers open coded the 20 transcripts [85], constructed a codebook based on emergent themes and a priori codes [86] from the MMTG, and refined and validated the codebook on a subset of interviews through a process of adjudication. Researchers applied the revised codebook to the entirety of the data set and then analyzed code application and co-occurrence across the coded data, selecting the four highest codes for deeper theme analysis. These themes were considered alongside the timeline and video analyses, which enabled triangulation of several key findings.

5. Findings

Multiple forms of data showed that teachers implemented several practices of the mindset mathematics approach in both their summer camp and school-year classrooms. While video and timeline data revealed teachers’ implementation of these practices, interview data triangulated this finding and clarified how teachers made sense of these shifts themselves, highlighting the importance of centering students’ reasoning and witnessing their high engagement. At the same time, however, other practices proved challenging for teachers to take up in both settings, as the following sections will share.

5.1. Implementing Practices of the Mindset Mathematics Approach

5.1.1. Practices Implemented During Camp

Applying the video analysis tool to the seven videos revealed that teachers implemented some of the micro practices at high levels and others at lower levels. The task that was the subject of analysis (see Figure 1, Painted Cube) prompted students to consider the faces of the small 1 × 1 × 1 cubes that comprise a 3 × 3 × 3 cube. The task is “low-floor” in that all students can build with cubes and think about patterns and is also “high-ceiling”, as the upper ends of the task involve forming different expressions, linear, quadratic, and cubic. Figure 2 shows teachers’ observation scores using the video analysis tool.

As shown in Figure 2, the majority of teachers scored “developing” or “expanding” on the following dimensions: praising the learning process (1B), students’ mindsets (1C), reasoning and multiple perspectives (2B), teachers’ questioning (3C), making mathematical connections (4C), connecting in small groups (4B), and connecting as a whole class (4C). Below, we discuss each of these dimensions with evidence from the videos.

Across the videos, teachers explicitly encouraged multiple perspectives (2B) in several ways. First, all teachers launched the task by inviting students to build different-sized cubes using sugar cubes—beginning with a 3 × 3 × 3 cube and then extending to a 4 × 4 × 4 cube—and to engage in three-dimensional visualization and drawing. To help them draw different-sized cubes, students were provided notebooks with squared paper. These resources enabled students to experience math physically, see it visually, and think about generalization. Throughout the lesson, teachers offered students opportunities to experience these ideas in multidimensional ways: they saw a 2-D representation of the cube, built a 3-D model, drew different sized cubes, collected and recorded patterns, organized their thinking, discussed ideas with each other, and considered generalization of different-sized cubes.

To praise the learning process (1B), teachers not only made explicit comments like, “Thank you for persevering with us”, but put that statement into action by giving students significant time to grapple with the task in their groups. Time analysis showed that teachers launched the task for approximately five minutes on average and then allowed students to grapple with the task in groups of two to four students—building cubes, collaborating with peers, and recording in their journals or on chart paper—for an average of 53 min. Nearly an hour is a significant amount of time for students to work in small groups, talking with each other and physically constructing different-sized cubes using the sugar cubes (4B). As students worked, teachers supported students to connect in small groups (4B) by circulating and asking questions (3C) like, “Do you agree with your partner?” and “Can you prove it to her? Show her how you know it is eight”.

To support students in finding, extending, and reasoning about patterns (2B), teachers in six out of seven classrooms created a table on the whiteboard to document the number of cubes within each type of cube that would have each amount of their faces shaded. This table, as written in a student’s journal, is shown in Figure 3.

After most students had moved beyond the original question to work on the 4 × 4 × 4 cube or generalized even further, teachers facilitated whole-class discussions (4C) to synthesize and connect the ideas from student groups (4B). These discussions were noted in five of the seven videos and lasted approximately nine minutes on average. In these discussions, teachers asked questions that pushed students to justify their thinking (3C), such as, “How do you know it is one?” and “Does anyone else think it is something different than…?” Teachers also asked questions to support students to make sense of and connect each other’s ideas (4C), such as asking students to predict what a 5 × 5 × 5 cube would look like based on the patterns discussed for 3 × 3 × 3 cubes and 4 × 4 × 4 cubes.

Across the videos, students displayed positive mindsets (1C) by sharing their thinking even if it differed from previous answers given and coming up to the whiteboard to physically model their ideas. For example, in one lesson, four different students came up to the board to draw their cube and write and explain their thinking. These actions suggest a sense of student agency and a comfort with sharing their ideas with each other.

At the same time, as shown in Figure 2, teachers implemented some of the micro practices at a lower level: mindset messages (1A), role of mistakes (3A), and supporting struggle and persistence (3B). We return to these practices later in this section.

5.1.2. Practices Implemented After Camp

Timeline and interview data revealed that not only did teachers implement many mindset mathematics practices during the camp, but that they reported using many of these practices in their school-year classrooms after teaching in the camp.

Figure 4 shows the practices teachers reported using in their school classrooms before they taught the summer camp (pre-camp) and the practices they reported using in their school classrooms after teaching the camp (post-camp). Note that the unit of analysis displayed in this figure is practice, rather than teacher, such that 126 counts of reported practice were recorded in the set of 32 teachers’ pre-camp timelines and 104 counts of reported practice were recorded in the set of 34 teachers’ post-camp timelines. To allow for comparison of pre-camp and post-camp reports of practice, each practice is displayed as a percentage of the whole set of reported practices for that timeline.

As shown in Figure 4, the teachers reported that they used more open tasks, more number talks, more classroom discussions, and shared more mindset messages in their post-camp timelines. Consequently, their reports showed a decrease in reported time spent reviewing homework, introducing content through direct instruction, and engaging students in individual practice time. Figure 4 suggests that the teachers took up new practices during their teaching of the summer camp that they continued to use once they returned to their regular classroom settings.

Interview data supported the timeline findings, as teachers shared more details of how they implemented a mindset mathematics approach once they returned to their regular classrooms. Of the 20 teachers interviewed, 15 returned to elementary and/or secondary mathematics classrooms in the academic year following the camp, while the remaining five took on roles as coaches (three people) or taught other subjects (two people). In this section, we focus on the 15 educators who taught mathematics in the following school year, though it is worth noting that coaches discussed working with teachers on the mindset mathematics practices and non-mathematics teachers reflected on using growth mindset messages in their classrooms.

All 15 of these teachers discussed implementing one or more mindset mathematics practices in their school-year classrooms. The practice with the highest uptake was posing open tasks, as 10 of the 15 teachers shared that they had implemented open tasks from the summer curriculum (and beyond) during the school year. For example, Laura (all teacher names are pseudonyms) noted that she regularly implements open tasks during a specific portion of her math block:

We have a half hour “WIN” time, which is “What I Need”, and that’s where they absolutely have to differentiate if they haven’t already in their math time. So that was a great place to incorporate some of these tasks, especially since, yes, they’re getting the practice underneath, but when you challenge them with “Find the Pattern” or “Find the Relationships”, they keep going and they keep going back and back and back until they figure it out. (Laura)

• In this quote, Laura noted the affordances of open tasks in offering students an opportunity to practice discrete skills, to experience intellectual challenge, and to build persistence, simultaneously.

While Laura discussed implementing open tasks in the elementary grades, Richard shared that he used open tasks with his high school students:

I think it was called “[Curved] Shapes Task”, but it’s basically giving them...You give them different regions and they have to try to figure out a way to find the area under a curve, which is the intro to Riemann sums, second semester calculus. So it was cool because they kinda came up with their own ways to approximate the area under the curve before we actually did Riemann sums and now we’re starting to do some calculus methods with anti-derivatives. But it was really powerful and it was trying to do some of the stuff that I did over the summer. (Richard)

• Here, Richard appreciated the way that this open task supported his students to construct their own methods for approximating the area under a curve, eliciting student-generated ideas from which he could then build.

In addition to using open tasks across grade levels, teachers described implementing student-centered participation structures that are embedded in the mindset mathematics approach. For example, nine teachers reported their use of classroom discussions and/or small-group collaboration, which created opportunities for students to interact with each other and to explain their reasoning. The same teachers reported that their students were engaging in productive struggle and generating their own ways of solving problems, as Laura and Richard both described above. Nate explained the shift in his pedagogical approach:

So we take that conversation, we take those underlying messages in math, and then we teach the content, and we find that the kids, I’ve been finding anyway, using it quite a bit, that the kids are more prepared for those curve balls. They’re thinking more about what’s happening, ‘cause they have some just genuine understanding of that content versus, we’re just gonna go over this worksheet and we’re gonna do several example problems…So it’s been really, really kind of a wonderful experience this year. (Nate)

• Here, Nate noted that his shift away from direct instruction and worksheets to tasks and discussions supported students to construct their own knowledge and to think critically.

Finally, teachers noted implementing instructional strategies and messaging from camp in their school-year classrooms. Eight of the teachers shared that they were using manipulatives and games from the summer in their classrooms. Additionally, six of the teachers noted that they were implementing specific routines from camp, such as number talks (as Nate mentioned above), “Notice and Wonder”, and “Which One Doesn’t Belong?”. Further, ten teachers discussed sharing messages about growth mindset, mistakes, and struggle with their students, through watching videos, prompting students to reflect in writing, or facilitating discussions. Taken together, analysis of video, interview, and timeline data revealed that teachers implemented many mindset mathematics practices during and after the summer camp.

5.2. Making Sense: Centering Students’ Reasoning and Witnessing Engagement

When sharing their implementation of open tasks, student-centered pedagogies, instructional routines, and/or mindset messages in their school-year classrooms, many teachers also self-reflected on their journey into making these shifts. Specifically, teachers discussed the ways in which implementing these practices enabled them to elicit and draw on students’ reasoning in ways that promoted high student engagement. Witnessing this engagement supported teachers to see their students in new ways, motivating them to continue to try out these new approaches.

5.2.1. Centering Students’ Reasoning

One of the most frequent codes that emerged from the interviews was that of shifting the cognitive load to students. Sixteen of the 20 teachers interviewed discussed students’ reasoning, and they described making a pedagogical shift from guiding the students to mathematical answers to asking them questions to further their thinking, as Nate noted:

A lot of the time, traditionally, we teach, and then have them [students] learn our method, and then explain back to us what our method was. This takes a little bit of a non-traditional approach, a better approach in my opinion… [to allow] the kids to form meaning, and from that meaning, we can kind of guide their learning. Because they might be noticing things that are true, but maybe not traditional, or they might be noticing things in a different way that might impact their education better. (Nate)

• Here Nate explained that by supporting students to generate their own noticings, rather than memorize traditional approaches to problems, students could construct their own understandings. Nate also contrasted direct instruction with the approach of the summer camp, which was to engage students in tasks before teaching methods, allowing new strategies and ideas to emerge from students and from whole-class discussions.

5.2.2. Witnessing Engagement

Analysis of interviews revealed that as teachers shifted the cognitive load to students, they witnessed high engagement. Seventeen of the 20 teachers shared that they were excited to see students from various subgroups engaged, excelling, and developing confidence in themselves during the camp. These teachers reported that the students’ motivation came from the tasks in which they were deeply engaged, as Laura shared:

They want to do it, they don’t want to stop. And some of the smaller activities that you gave us were “Close to 100” or “Tic Tac Toe Products.” I’ve never seen fifth graders wanna practice multiplication…To see the kids say, “No, we don’t wanna go to recess. Can we keep playing?” Like, “We’ll play after lunch, okay?” That’s super exciting. Because when they have that drive, they’re going to succeed. (Laura)

• Laura’s surprise at students’ engagement when they were able to practice multiplication facts through a conceptual game, rather than worksheets, was shared by many teachers.

A subset of teachers specifically expressed surprise at seeing students typically labeled as “low-achieving” reasoning at high-levels and making mathematical connections during camp. Teachers spoke of the “incredible” engagement of students who had not been previously successful in classrooms, as Theresa noted:

We’re drawing from kids on an IEP, native, migrant, homeless…And so, I would say kids a lot of times maybe get labeled as not always being the greatest math students …But I think we’ve seen the biggest gain in our kids that were on—it’s usually labeled as not great math students, on an IEP for math, and then they are able to do grade level math and make these connections and realize anybody is capable, right? (Theresa)

• Here, Theresa shared that students who had been labeled as not “the greatest math students”—via IEP labels and other marginalized identities—were able to not only complete the mathematics but also, importantly, realize their own capabilities.

Similarly, other teachers described the learning of students who had previously been unsuccessful, which they witnessed in the post-assessment, as Nate shared:

One of the most powerful things for me was during our summer program, I had a student who really, really struggled. We had the entrance exam for her in the [research center] program and she did not do very well at all, I mean very close to zero. What I found was by the end of the summer, she was more confident to answer those questions, she had no fear about this test, she had no remorse about this test, she put answers on paper, she thought about nontraditional ways, she put a lot more time and energy into it and she did exceptional on it regarding her first score. She went from zero to a passing score, which for somebody like that is a really, really important thing, it builds that confidence huge. So I can just... I’ll never forget this one student who really had no way to access that information when she came into the camp, but only five weeks later, she could develop that into some really, really solid thinking. And she wasn’t necessarily always right, but you could see her thinking progress, and that was a beautiful thing. (Nate)

• In this quote, Nate noted the increased confidence and achievement of a previously low-achieving student through the camp’s emphasis on thinking and reasoning.

Some teachers expressed surprise that previously high-achieving students who were usually bored in class became highly motivated. Richard shared that one of his students he considered to be “gifted” was usually bored and uninterested in his regular math class, yet his engagement in the summer camp was so noteworthy that he contacted the student’s parents to share his changed motivation:

And there was another kid too. So, so unbelievably gifted in math and he would...Same thing, I think he was bored, I think his thing was more that he was bored with math, and when they got together in groups and they were doing the activities, like, he was really shining. It was incredible. I emailed his parents, and I told them how impressed I was and they were shocked by how engaged he was and how big of an asset he was to the summer program. Yeah, I could keep going, there’s so many students that really shined with that. (Richard)

• Like Richard, many teachers reported that students from varied academic backgrounds improved their achievement significantly, which was confirmed by student data [76]. Indeed, the success of the summer teaching to mixed-achievement groups of students disrupts a widespread belief that students have to be divided into tracks in order to be appropriately challenged or supported [87].

Importantly, teachers’ excitement about students’ engagement and capabilities appeared to be linked to their uptake of these practices in the following school year. Eight of the 15 math teachers reported that not only did seeing their students’ high engagement during the summer camp “surprise” them, but also that seeing the approach in action in their own teaching shifted their sense of what was possible in their teaching moving forward. It was the witnessing of engagement and learning for such a broad range of students—which was not typical and therefore a surprise for some—that seemed to prompt teachers’ uptake of these practices in their school-year classrooms.

5.3. Struggling with Struggle: Practices Implemented Less Frequently

In interviews, one of the practices that teachers discussed implementing was that of sharing growth mindset messages with students. Video analysis, however, revealed a complicated picture. As shown in Figure 2, the following micro practices were implemented at lower levels, with the majority of teachers falling in the “beginning” level: mindset messages (1A), role of mistakes (3A), and supporting struggle and persistence (3B). Although teachers were likely sharing growth mindset messages in conjunction with showing mindset videos and prompting students to reflect on them (as described in interviews), these messages appeared to be rarely integrated directly into mathematics tasks, as was the case in the Painted Cube lessons.

As noted earlier, Steuer et al. (2018) conceptualized positive error climates in mathematics classrooms as the interrelation of eight classroom features [65]. Although many of the features (e.g., error tolerance, teacher support following errors, absence of negative teacher reactions) were observed in the analyzed videos, two features were notably absent: analyzing student errors and using them as a starting point for learning. Despite positive overall messaging to students, when working inside mathematics problems most teachers in the videos steered students to correct answers if they were incorrect, rather than facilitating collective sense-making of mistakes.

Video analysis showed that mistakes (3A) were rarely explored or valued during small group work-time and whole-class discussion. For example, one teacher said to a student as they were circulating to their small group that they should fix their mistake so “it doesn’t throw you off”. Similarly, although students were given ample time to explore the task, at times teachers’ interventions when circulating to small groups interfered with a potentially productive struggle (3B). For example, one teacher directly instructed students who were working in a group how to paint the bottom of the sugar cube and then counted the cubes for the students. To consider the challenges inherent in giving space for struggle and cultivating respect for mistakes, we discuss one classroom in detail below.

5.3.1. An Example: Adriana’s Lesson

Adriana’s lesson offers a representative example from the video data set of a teacher implementing some of the micro practices, but not others. In her launch of the task, Adriana encouraged students to take the time to think before sharing, saying: “I want you to think first, visualize in your mind, hands down” (1B). While students were working on the task, Adriana circulated, telling one group who shared an answer with her that they needed to “show it to me, prove it to me” and asking another group probing questions, such as, “what did you use?” and “tell me your strategy” (2B). For their part, some students in the class physically built 3 × 3 × 3 cubes with their partners during the work time and eagerly raised their hands at the beginning of the whole-class discussion (1C, 4B, 4C). A poster in the classroom read, “Mistakes Allow Teaching to Happen” (3A). As such, Adriana scored “developing” on these indicators, offering students opportunities to reason throughout the lesson.

In other moments in the lesson, however, Adriana steered students to the correct answer, rather than allowing time and space to explore mistakes (3A). When students called out multiple answers during the discussion, Adriana called on the student who was correct and asked them to explain their thinking, without addressing or returning to those who shared incorrect answers. Adriana then validated that student’s answer, saying, “okay it’s eight”, rather than asking other students in the class what they thought of the shared answer. Later in the discussion, Adriana rephrased the thinking of the student who was correct and praised them with “good job”. Multiple times throughout the discussion, Adriana asked students with the right answers to explain their thinking and wrote down their answers on the whiteboard as they shared and did neither when students shared incorrect answers.

Similarly, while circulating, Adriana intervened as students worked in ways that may have limited a potentially productive struggle (3B). With one group of students, she offered explicit scaffolding, telling them to “color it” and pointing to which faces of the small cubes should be colored. While this kind of direct instruction may be useful for some students at some point during a task, this directive was given four minutes into the lesson and only a few minutes into group work. With another group, when Adriana verbalized the task directions to them (perhaps because they were building a rectangle, rather than a cube) and they looked at her with potentially confused looks, she responded by saying, “here’s what you’re going to do, you’re going to color it…” Throughout the work time, Adriana interacted with groups that were obtaining correct answers by asking them to explain their reasoning (as noted above) but interacted with groups that were obtaining incorrect answers and/or were confused by telling them what to do and asking yes or no follow-up questions.

Adriana did implement several mindset mathematics practices in this lesson (e.g., praising the learning process, reasoning and multiple perspectives, supporting students to connect in small groups, etc.); however, these practices may have been undermined by a focus on right answers and correct reasoning, rather than an encouragement of struggle or exploration of mistakes. Analysis of all Painted Cube videos similarly revealed that giving students space to struggle without teacher intervention and exploring students’ mistakes continued to be a challenge for six out of seven teachers in that lesson.

At the same time, all teachers took up some mindset mathematics practices in their videos, which begs the question of why some practices might have been “easier” than others for teachers to implement. Additionally, nearly all teachers implemented mindset mathematics practices alongside traditional practices, such as focusing on right answers. For example, analysis of videos frequently showed a teacher asking a probing question in one moment and then a narrow question in the next. The latter seemed to undermine the former for some teachers in some moments. We return to this point in the discussion.

5.3.2. Teacher Reflections on Struggling with Struggle

In interviews, teachers similarly noted that supporting their students to struggle productively on the tasks was a challenge for them at times. Several teachers specifically pointed to the Painted Cube task, as Theresa noted: “We really struggled with the Painted Cube activity. I think...Partly I think we probably should have just really done fully from start to end, done it ourselves, all the way through”. Theresa and others shared that they found the Painted Cube task to be challenging mathematically, in contrast to the other tasks. Consequently, it is possible that discomfort with the task itself may have hindered teachers’ capacity to explore students’ mistakes or encourage their struggles in this lesson.

Regardless of the task, however, several teachers noted that their students sometimes experienced frustration while engaging with the mathematics and that they grappled with how to respond to that frustration. Dolores reflected:

So I think the only thing I will say that I struggled with is for some of those activities they would lose stamina. Some of them just would get frustrated. And trying to keep them working on... I had some that had no problem pulling that out. They wanted to keep working on that, but others that quickly just felt defeated like, “I can’t figure this out.” But I think overall, their perception changed.

• Here, Dolores explained that her students experienced the tasks differently, with some enjoying the challenge and others feeling defeated by it. Given that direct instruction is common in most classrooms, it is likely that students may have been used to teachers intervening to offer a hint or a procedure when they are struggling (as was observed in Adriana’s video), such that being prompted to figure things out for themselves or with their peers, even when stuck, may have felt frustrating. At the same time, Dolores also noted that over time, her students’ perception of struggle and how to engage with it productively may have changed during their experience at the camp.

5.4. Summarizing Teachers’ Learning

When teachers returned to school, under the typical constraints and pacing pressures, it is highly possible that the practices they reported implementing in their school-year classrooms morphed or became infrequent. Here, we discuss the barriers to implementation that teachers experienced upon returning to their classrooms and then reflect on which aspects of the camp’s design seemed to support the learning that did happen.

5.4.1. Barriers to Implementation

In interviews, when discussing challenges to implementing the mindset mathematics approach in their school-year classrooms, teachers coalesced around several specific barriers (Figure 5). Pressure to cover their mandated district curriculum emerged as the most frequently reported barrier to implementing these practices as fully or as regularly as they had at camp, as reported by six teachers. Melanie noted:

So I’ve been struggling this year just because I have 30 minutes with the kids, but yet I’m expected to cover all of this curriculum. So if we get into a good discussion and my lesson goes on to the next day, I get kind of sunk... Sometimes it’s hard to let those discussions happen just because I’m under the crunch of getting through curriculum and I only get them for 30 minutes. (Melanie)

• Here, Melanie explained that her short math block, combined with curriculum pressures, made it challenging for her to extend a “good discussion” beyond its allotted time, as this would mean also extending her lesson into the following day.

Relatedly, four teachers reported system-level constraints, consisting of the following: (1) district grading practices that contradicted the mindset mathematics emphasis on mistakes as learning opportunities, (2) convincing administrators to allow teachers to use open tasks, and (3) teaching in isolation, as compared to co-teaching during the camp, which allowed for observation and support of students, as well as collaboration among teachers. In terms of being allowed to use open tasks, Laura reflected: “But a question that a lot of us took to our math director is how do we get more of these [tasks] in our pacing guides”. In sharing these concerns, teachers seemed to imply that district curricula, pacing guides, and grading practices had different philosophies than the mindset mathematics approach and the camp curriculum, such that integrating the latter within the former was a challenge.

Additionally, teachers cited their colleagues’ lack of knowledge of this approach as a barrier. Isaac shared: “The lack of understanding by admin, principals, colleagues, grade-level friends. It’s really hard, especially if you’re not supported”. Similarly, Melanie shared: “And I think there’s a lot of teachers who are stuck on fact fluency and time tests and procedures. And they have to teach the procedures so they can get through the curriculum”. Specifically, four teachers expressed a need for more training for their colleagues in how to implement open tasks, how to continue to hold a growth mindset, and how to avoid the trap of low expectations of their students. Additional barriers included the lack of time to create open activities or adapt open activities to their grade level (two teachers) and lack of access to manipulatives (one teacher).

Despite these challenges, teachers expressed a desire to implement mindset mathematics practices when possible:

The biggest struggle I have is balancing the standards that I have to cover and the approach that I wanna take, and that’s the approach that we did in the summer. So, for me, it’s tough because I wanna always have them explore. I wanna have them take ownership in the mathematics we’re doing, but at the same time I feel like it’s tough because I’m told I have to cover all this content. (Richard)

• Here, Richard summarized the tension between what he wanted to do in his classroom and what he felt he had to do. Despite these challenges, our analysis showed that teachers did implement some features of the approach during the school year, which suggests that some aspects of the camp were still useful to their learning.

5.4.2. Design Features of the Camp

As teachers described implementing mindset mathematics practices during and after camp, as well as the ways in which these practices supported their students’ reasoning and engagement, they commented on the collective impact these elements had on their own professional growth. Theresa explained:

The power of working at the summer math camp and seeing that in action when it can be messy and we don’t have these bell schedules and we don’t feel that we have all this curriculum we have to cover, right? We can just be like, “Oh, we didn’t get there today. Oh, well. It doesn’t matter.” That kind of freedom that the math camp kind of gives in that, I think that it would really be beneficial for other teachers to be able to come in and see that and experience that, right? Look, when you let kids have a little bit of freedom, they’ll surprise you. (Theresa)

• In one sense, Theresa explicitly commented on the “freedom” that the summer camp gave students—the freedom to generate their own thinking and to engage with mathematics in new ways. In another sense, however, Theresa implicitly commented on the “freedom” that the summer camp gave teachers—the time that she and the other teachers had to explore students’ ideas, given the lack of pressure to cover content standards.

Theresa’s suggestion that other teachers might shift to more exploratory work if they saw and experienced it themselves, as that would lead to surprise at students’ engagement and reasoning, is important to consider. Throughout interviews, teachers spoke about the ways in which longer lessons, smaller class sizes, and lack of curricular and district pressures supported them to use exploratory tasks and to elicit and focus on their students’ thinking. About half of the teachers noted that co-teaching or collaborating with colleagues during camp also supported their learning and eventual uptake of the practices. These design features of camp—combined with the task-based curriculum—seemed to support teachers to implement the mindset mathematics approach in the summer, experience students’ wonder and engagement for themselves, and then bring some of these practices into their school-year classrooms.

Taken together, our analysis revealed one potential mechanism for this learning: teachers’ experiencing the mindset mathematics approach themselves in camp and witnessing their students’ reasoning. We discuss the implications of this mechanism below, especially how it might be applied to both transformational and incremental PD efforts.

6. Discussion

Mathematics education researchers and policymakers have long called for a focus on problem solving, reasoning, and student thinking [18,19,20,21,22,23]. Yet, the mathematics curricula that are used in most classrooms are filled with short questions that do not invite student reasoning or problem solving. The summer camp was designed as an antidote to this broader context. The teachers in this study engaged in an unusual model of teacher learning that involved teaching a task-based curriculum designed by researchers, in an unusual setting of a summer camp, free from pressures of the regular school year.

Through analysis of classroom video, timelines, and interviews, we found that teachers implemented several important practices in both their camp and school-year classrooms, namely, posing open tasks, giving students ample time to collaborate, and asking questions that pushed students to reason. Further, interview data suggested that it was teachers’ witnessing of students’ reasoning—rather than engaging in a coaching cycle or in a professional development session—that motivated these shifts. Teachers reported that they believed other teachers would shift their practice if they had the opportunity to teach in these ways and to observe students’ high engagement and reasoning during open tasks. Teachers’ learning appeared to come from building their “knowledge of practice” [45], as they connected research ideas to their own expertise through a task-based curriculum that left space for them to make their own decisions.

Although we found that several mindset mathematics practices were taken up, we also found that others were not. Teachers less frequently integrated mindset messaging directly into their teaching and less frequently gave space for the exploration of students’ mistakes and struggles. We hypothesize that it may have been easier for teachers to share growth mindset messages during the separate activity of showing a mindset video than to integrate these messages into their teaching, which requires shifts in potentially ingrained pedagogical habits, such as asking narrow questions (as observed in Adriana’s video). In Louie’s (2017) study of teachers working to implement inclusive pedagogies, she found that although teachers employed the inclusive pedagogies that they had learned, they did so alongside the traditional ones that they had been accustomed to using, such as centering procedures [35]. In other words, the teachers in Louie’s study may have had an easier time adding new pedagogies to their practice, rather than subtracting ingrained pedagogies [35]. In sharing mindset messages through isolated activities but less frequently exploring mistakes, we wonder if the teachers in our study similarly added in a new practice without changing prior practices that emphasized right answers.

Further, we hypothesize that exploring mistakes and struggles may have been less familiar to teachers than practices like encouraging groupwork and emphasizing multiple perspectives, which are present in the Common Core State Standards [23] and in CCSS-aligned curricula. It may have been easier for teachers to implement these practices during camp, if they were already somewhat familiar with them, and to bring these practices into their school-year classrooms, if they were also reinforced by their schools’ curricula. Research has similarly shown that teachers are more likely to transfer their learning to the classroom when their schools’ systems and philosophies are aligned with those of their PD experience [38,39]. Additionally, even if a particular practice was not present in teachers’ curricula, it may have still been more feasible for teachers to occasionally pose open tasks, integrate collaboration into their lesson plan, and ask probing questions, while still following their mandated curriculum and pacing guides, rather than offering students extended time to productively struggle or engage in a lengthy class discussion.

6.1. Limitations

Given that the interview and timeline data are self-reported, it is possible that teachers over-stated their use of these practices. As noted previously, teachers cited specific barriers to implementing the mindset mathematics approach in their school-year classrooms. On the one hand, the existence of these barriers necessitates continued discussion in the field of how to create the conditions in schools and districts that enable teachers to sustain their learnings from PD efforts. On the other hand, however, as teachers shared these barriers in their interviews, they simultaneously discussed ways that they were implementing this approach despite the barriers, which might bolster these findings.

Another limitation relates to our sample and setting. It is possible that teachers who consented to participate in interviews and to turn in timeline data had a higher rate of uptake than the entire summer camp teacher population. Similarly, it is also possible that those who turned in video recordings of the Painted Cube lesson had a higher rate of uptake than the entire summer camp teacher population. Relatedly, it is possible that among the nine teachers who turned in video recordings, the seven teachers who turned in recordings of sufficient length and audio quality to be analyzed had a higher rate of uptake than the two teachers who turned in sets of short clips. For example, if these two sets of short clips contained minimal implementation of the mindset mathematics teaching practices, then eliminating them from our analysis would have skewed our findings. Based on our watching of these two sets of clips, however, it appeared that several mindset mathematics practices were implemented in one of the videos and several less-aligned practices were implemented in the other video. Given the mixed implementation that we found in our analysis of the seven videos, we contend that these two excluded clips were not significantly dissimilar from the rest of the data set. While there is always a possibility for bias when excluding parts of a data set, the alternative of attempting to code videos with limited context and poor audio quality would have been problematic.

More broadly, it is also possible that teachers who choose to teach in a math summer camp in the first place comprise a small, unique sample that is not representative of the broader teaching force. Consequently, we cannot argue that our findings are generalizable to the broader teaching force or that they are generalizable to the school-year context, as opposed to the unique context of a summer camp. Instead, we contend that this is a first result that reveals the potential efficacy of a novel approach to teacher learning with a small group of teachers. Given the low transfer rate of many teacher learning initiatives [38,39], however, teachers’ potential transfer of practice from the summer camp to their teaching in the regular school year is particularly noteworthy.

6.2. Implications for Research and Practice

Our findings illuminate several implications for the field of teacher learning. By teaching a research-based curriculum and engaging in associated professional learning opportunities, teachers in this study were able to integrate new research ideas with prior expertise to construct “knowledge of practice” in and through their teaching [45]. This blending of theory and practice is an important form of knowledge, as a focus only on theory or only on practice is not sufficient. As researchers and practitioners design professional learning experiences that support teachers to learn in and of their practice, they might provide teachers with opportunities to draw on and experiment with research-based approaches in classroom environments free from the typical constraints. While other teacher learning initiatives have also supported teachers to experiment with research-based approaches of some kind [31,32,55], here we offer a novel approach to professional development in that teachers could engage in and focus on this experimentation without having to simultaneously juggle testing, grading, and curricular pressures. Although this is a transformational rather than incremental approach, our study shows that it has efficacy in supporting some teachers to take on some practices.

Building on these results, future studies might investigate teachers’ learning in specialized settings at a larger scale and with a particular focus on longer-term impacts. Researchers might closely follow teachers during their teaching of the camp, collecting video recordings of multiple lessons, teacher reflections, and student work artifacts, to more extensively capture the implementation of mindset mathematics practices and to do so among a larger group of teachers. Similarly, researchers might follow teachers for several years after participating in the camp, observing teachers’ school-year classrooms to investigate which mindset mathematics practices they are continuing to implement, which they are not, how they might be adapting the practices for their school-year contexts, and what external barriers impede implementation. Given the limitations mentioned previously, observing teachers’ school-year classrooms might triangulate shifts in practice that teachers self-report in an interview. Additionally, studying the learning of cohorts of teachers from the same schools may also illuminate specific site-based barriers and supports that they experience when transferring these practices to their schools.

Further, scaling the intervention such that teachers participate in the camp for multiple summers could allow for multiple rounds of data collection across both camp and school contexts. Researchers could analyze how teachers continue to grow their practice across several years through multiple iterations of teaching in the camp and multiple opportunities to try out the practices in their school-year classrooms. For example, what might teaching in the camp for a second year look like? And how might teachers’ implementation of these practices in their regular classrooms inform how they approach a second year at camp? Teachers’ learning of mindset mathematics practices would likely travel back and forth between these two sites of practices in specific ways that could inform future interventions. Moreover, a multi-year, multi-site study may afford opportunities for collaboration between camps and schools, which may begin to ameliorate the disconnect that many teachers experienced upon returning to their schools.

At the same time, our finding that a subset of practices did not seem to be transferred to classrooms raises important questions about the efficacy of the transformational teacher learning design embedded in this study. An argument might be made that the mindset mathematics approach involves too many components to learn at once and/or that it is too challenging to implement in a school-year setting with mandated curricula and assessments that may limit teachers’ time and capacity to try out new approaches. It is possible that an incremental, practice-based approach [37] might more effectively nudge teachers toward integrating mindset messaging into mathematics tasks directly or responding to incorrect answers with probing questions. In particular, the practice of supporting students to struggle, which requires teachers to actively resist reducing the cognitive challenge, may be an especially challenging practice that takes time for teachers to develop, perhaps after engaging in multiple incremental shifts over time.

Our finding about teachers’ observations of students’ engagement suggests that an incremental approach [37] may be fruitful for learning mindset mathematics practices. As noted previously, implementing open tasks that made space for students to author their own ideas enabled teachers to witness engagement from students of all achievement levels and to appreciate students’ mathematical ideas. Consequently, teachers’ implementation of student-centered pedagogies that made students’ ideas (and by extension, their competence) visible to themselves and to their teachers emerged as a critical component of teacher learning [53,54,55]. If teachers’ witnessing of students’ engagement can motivate uptake of student-centered practices after a summer camp, then teachers’ witnessing of students’ engagement and reasoning when trying out new practices in their regular classroom may motivate them to sustain that practice and try out similar ones. For example, imagine a teacher is working on responding to incorrect answers with probing questions rather than leading students to the right answer. In asking probing questions, they will likely elicit students’ reasoning, potentially making students’ thinking and competence visible to them. As we found in our study, observing students’ engagement may motivate this teacher to continue to make small shifts. As such, the notion that small tweaks in practice may yield student engagement and reasoning, which then may motivate continued shifts, may be critical to teachers’ learning of mindset mathematics practices.

We contend that both incremental and transformational approaches to teacher learning are needed to improve mathematics education for students in the US. The shifting of teachers to a mindset mathematics approach through an immersive learning experience is only one piece of this challenge. Many factors, outside of teachers’ control, work against change. It is our hope that this study pushes the field to think more expansively about teacher learning and to consider professional learning opportunities beyond the typical school setting.