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A wide range of evidence points to the need for students to have a growth mindset as they approach their learning, but recent critiques of mindset have highlighted the need to change teaching approaches, to transfuse mindset ideas throughout teaching. This shifts the responsibility from students themselves to teachers and schools. This paper shares a mixed methods study conducted across the US, that measured the impact of a “mathematical mindset teaching approach” shown to be effective when taught by the authors, scaled to teachers in 10 US districts. The effectiveness of this novel mathematics approach was measured using pre and post assessments during a summer intervention followed by measures of GPA change when students returned to schools. Both measures showed that a mathematical mindset approach to teaching significantly improves students’ mathematical achievement, and changes students’ beliefs about themselves and their approach to learning. Accompanying analyses of teaching and of teacher interviews give insights into the ways students change, highlighting the need to bring about shifts in students’ mindsets through a changed approach to mathematics teaching and learning.
In recent years there has been considerable attention paid to the idea of mindset, a construct developed and researched by Carol Dweck and teams of other researchers (
Multiple research studies have demonstrated the positive impact of having a ‘growth mindset’ in mathematics and other subjects.
Finally, a study with college students looked at the impact of growth mindset on overall grade point average compared to two control groups, a multiple intelligence intervention and a no-treatment control (
Taking the growth mindset message beyond the traditional boundaries of classrooms and schools,
These different studies all suggest that when students change their minds about what is possible, and they are released from ideas of fixed intelligence, they achieve at higher levels, whether or not the teaching they receive changes. Despite this,
A key part of a mathematical mindset teaching approach (
While traditional narrow questions communicate to students that mathematics is about recalling and applying a procedure, open tasks provide opportunities for students to engage in what
Some of the growth mindset information that is most powerful to students draws upon the evidence from neuroscience, showing the potential of the brain to grow and develop connections (
Brain network underpinning mathematics knowledge, lang chen in
The dorsal visual pathway has reliably been shown to be involved when both children and adults work on mathematics tasks. This area of the brain particularly comes into play when students consider visual or spatial representations of quantity, such as a number line. A number line representation of number quantity has been shown in cognitive studies to be particularly important for the development of numerical knowledge and a precursor of children’s academic success (
The different studies on mindset and on teaching with a growth mindset suggest that while mindset interventions aimed at changing students’ ideas can be powerful, the biggest improvements can be brought about when students’ ideas change at the same time as teaching is designed to encourage a growth mindset (
The Mathematical Mindset teaching approach was first developed and studied in a youcubed summer camp implemented in the summer of 2015 and detailed in
While the results of the original youcubed camp were promising, several important questions remained. Could the mathematical mindset approach to teaching only be done by this particular teaching team at this particular university? Could it be specified well enough to scale this approach to other summer programs? If so, would students at other programs experience increased achievement and shifts in mindset as a result? The remainder of this paper will communicate the results of a study monitoring the impact of the youcubed summer camp, taught in ten districts across the United States, considering any potential improvements in mathematics achievement after the camp and when the students returned to their mathematics classes in the following school year.
Over several years, workshops designed by the research team were offered for teachers to learn about the mathematical mindset teaching approach. During the workshop teachers were given the curriculum and trained with mathematical mindset pedagogical practices, and multiple additional resources were shared with them to support their learning on this teaching approach. In 2019 a partnership between ten school districts and youcubed enabled a study of the learning of students who participated in the camps in their districts, which is the focus of this paper.
In the summer of 2019, ten districts in five states implemented the youcubed summer camp, agreeing to provide data on their students’ mathematics achievement at the beginning and end of the camp and later when the students returned to school. The districts recruited students to attend the youcubed camps who were diverse in terms of ethnicity, gender, and socioeconomic status. Additionally, district recruitment focused on students who are Black, Latinx, and/or experiencing poverty, to ensure that camp attendees reflected these groups. Overall district data is shown in
Characteristics of participating school districts.
School district | State | Urban/Rural classification | Total students | % Black | % Latino | % Free/ Reduced price lunch |
---|---|---|---|---|---|---|
District 1 | Michigan | Suburb: Large | 3,258 | 24 | 3 | 48% |
District 2 | Alaska | City: Small | 13,780 | 5 | 9 | 31% |
District 3 | California | Suburb: Mid-size | 9,494 | 3 | 78 | 80% |
District 4 | Illinois | Suburb: Large | 20,973 | 5 | 38 | 41% |
District 5 | New Mexico | Rural: Fringe | 2,026 | 0 | 79 | 62% |
District 6 | Illinois | Suburb: Large | 6,827 | 3 | 42 | 58% |
District 7 | Illinois | Suburb: Large | 38,934 | 7 | 54 | 59% |
District 8 | Illinois | Suburb: Large | 8,580 | 2 | 16 | 15% |
District 9 | California | Suburb: Large | 1,531 | 1 | 61 | 53% |
District 10 | California | City: Small | 4,637 | 1 | 38 | 41% |
Source: NCES Common Core of Data, most recent available school year (2017–18).
Data source for this statistic is Michigan Student Data System (NCES data not available for that year).
Camp attendees differ from overall student population of district, with at least 50% experiencing poverty.
Camp implementation characteristics, by district.
School district | Days of instruction | Hours of math instruction | Student enrollment | Share of students who attended 75 (%)or more days of camp |
---|---|---|---|---|
District 1 | 20 | 60 | 316 | 65 |
District 2 | 12 | 72 | 47 | 85 |
District 3 | 10 | 30 | 78 | 86 |
District 4 | 16 | 80 | 66 | 100 |
District 5 | 16 | 36 | 40 | 95 |
District 6 | 15 | 52 | 126 | 87 |
District 7 | 28 | 84 | 107 | 60 |
District 8 | 23 | 40 | 70 | 66 |
District 9 | 12 | 36 | 24 | 83 |
District 10 | 10 | 30 | 45 | 42 |
Different forms of support were offered to participating teachers before and during the study summer camps. In the spring of 2019, all participating teachers were required to take part in three 1-h webinars and were offered additional learning opportunities, including a book detailing the approach (
Teachers of the multiple camps were all given detailed curriculum that described the objectives and activities for each lesson during the camp. Two sequences of the curriculum were shared with teachers to account for the variation in instructional days across the sites: one for two-week camps expected to include 30 h of instruction, and one for four-week camps expected to include 60 h of instruction. The mathematical topics included in the curriculum were number sense, algebra as a tool for problem solving, generalization and mathematics as pattern seeking. Additionally, specific structures and activities were provided. A typical day 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,” with students engaging in an orientation activity, open-ended mathematics tasks that encouraged then to engage with agency and authority (
Given the goal of understanding the impact of a mathematical mindset approach taught within summer camps, scaled to ten districts, a mixed methods approach was implemented, drawing from both quantitative and qualitative methods. A matched comparison analysis was employed to assess the effect of the approach on students’ achievement. School districts provided a variety of achievement measures of both participant and non-participant students (GPA and MARS scores, before and after camp participation; and a baseline math standardized test score), and a battery of control variables (race, ethnicity, gender, free and reduced-price lunch status, English learner status, and special education status). To examine the enactment of the mathematical mindset approach, a subset of classroom videos from the camps were collected and analyzed using qualitative methods. Finally, to investigate students’ mindsets, interviews with a subset of teachers were conducted and transcripts of these interviews were analyzed.
Two data sources were used to examine student achievement: a standardized assessment of conceptual mathematics– MARS tasks–was administered at the start and end of the camps to measure changes in students’ mathematical understanding. Participating sites administered four MARS performance tasks at the beginning of camp and on the final day, with the same tasks used for pre and post camp across all grade levels. Each task was scored by an external partner Silicon Valley Mathematics Institute (SVMI) on a point-score analytic rubric for numerical responses and mathematical reasoning. There was a total of 36 possible points across the four tasks. MARS scores were analyzed for all students who met the following criteria: (i) they were in a district that had submitted MARS assessment papers by November 4, 2019; (ii) they had both pre and post camp scores available, and; (iii) they had completed at least two of the four MARS tasks.
To consider change in mathematical understanding, measured through MARS tasks, a composite score was achieved by summing students’ scores across the four tasks with the pre-score subtracted from the post-score, to give a measure of growth and enable the calculation of main effect sizes, following the same method as the original camp study (
To measure the program’s impact on student achievement when students returned to their school classrooms after the conclusion of the summer camps, mathematics grade point average (GPA) were collected for the school year following the mathematics summer camp (2019–2020). A matched comparison analysis (
The analysis was conducted through the creation of a uniform GPA variable across the 10 districts by mapping standards-based grades (which have four levels) onto a standard, 4-point GPA scale (i.e., “advanced” was coded as 4, “proficient” was coded as 3, “below proficient” was coded as 2, and “basic” was coded as 1, equivalent to a D letter grade in the standard GPA scale). Neighbor matching (
Multiple model specifications were used to assess whether the overall impact estimates were robust. Among models that included the key baseline variables of mathematics GPA and test score, all impact estimates were positive and of a similar magnitude, and the model with the richest set of matching variables (adding race, ethnicity, gender, free and reduced-price lunch (FRPL) status, English language learner (ELL) status, and special education status as matching variables) yielded a very similar impact estimate (0.14 GPA points). The chosen model included prior GPA and math score, both to avoid reducing the sample size (thus making the findings as broadly applicable to camp participants as possible) and to prioritize identifying matched comparison students with the most similar prior academic achievement.
To capture the teaching that was implemented in the different camps, not only the intended teaching approach, 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” (shown in
Painted cube task.
The teaching was analyzed in two different ways. Researchers created content logs (
In the second form of analysis, researchers examined the seven videos using the Mathematical Mindset Teaching Guide (
To consider changes in students’ mindsets and engagement with mathematics, semi-structured interviews (
Analyses revealed that students’ mathematics achievement both at the conclusion of camp and in the following school year significantly increased, as measured by MARS scores and mathematics GPA. To better understand the mechanism for this change, teachers’ enactment of the mathematical mindset teaching practices was analyzed. This analysis of teaching revealed that students were given significant time to grapple with open tasks in summer camp. Additional analyses of teachers’ interviews showed that students’ experiences with open tasks was a significant factor in students’ changed mindset and engagement with mathematics.
Students who attended the youcubed camps achieved at significantly higher levels at the conclusion of the camps, as evidenced by a significant difference in pre/post MARS assessments. The average gain score for participating students across all sites was 0.52 standard deviation units (SD), equivalent to 1.6 years of growth in math. On average, at baseline, camp participants received 6.6 points out of a total of 36 on the 4 MARS tasks, whereas the mean score after the camp was 8.8, a gain of 2.2 points that was statistically significant with a 99% degree of confidence. There was variation across ten districts in the size of gains students demonstrated, with gains ranging from 0.24 SD to 0.96 SD (i.e., 1.02 to 4.16 points, respectively). In nine out of the ten camps, gains were statistically significant with a 90% of confidence.
Pre/post design results by district, listed by effect size.
Sample | Obs | Pre-test mean | Post-test mean | Difference (post - pre) | MARS effect size |
---|---|---|---|---|---|
All School Sites | 825 | 6.6 | 8.8 | 2.2 | 0.52 *** |
District 8 | 25 | 4.16 | 8.32 | 4.16 | 0.96 *** |
District 5 | 45 | 6.09 | 8.91 | 2.82 | 0.65 *** |
District 6 | 34 | 5.09 | 7.76 | 2.68 | 0.62 ** |
District 2 | 234 | 6.71 | 9.32 | 2.62 | 0.61 *** |
District 10 | 20 | 6.25 | 8.7 | 2.45 | 0.57 *** |
District 1 | 289 | 7.88 | 9.93 | 2.06 | 0.48 *** |
District 9 | 34 | 3.24 | 4.97 | 1.74 | 0.40 *** |
District 3 | 42 | 6.02 | 7.67 | 1.64 | 0.38 * |
District 4 | 52 | 5.63 | 6.69 | 1.06 | 0.25* |
District 7 | 50 | 5.38 | 6.4 | 1.02 | 0.24 |
Source: MARS scores provided by camps.
Note: Effect sizes are calculated using the same approach adopted in previous research on the youcubed mathematics camp, dividing the difference in raw score by the in-sample standard deviation of the raw pre-test score. ***
To consider the impact of the teaching time in different camps, investigation of correlations between the amount of instruction provided by a camp (in days of camp and hours of instruction) and the growth in learning students demonstrated (the effect sizes of the learning gains) were conducted. These showed moderate, positive correlations in the total number of days of camp duration (r = 0.65) and total number of hours (r = 0.58) each site devoted to the youcubed camp approach.
The MARS gains showed that camps who offered the mathematical mindset approach for more days and hours, achieved significantly larger gains. There was little evidence of a difference in MARS gains based on recruitment approach (see
When the students returned to their regular mathematics classes in their school district they experienced a variety of forms of instruction. The students who had attended the camps were compared with students in their districts who were at similar levels of achievement but had not attended the camps. This analysis showed that at the end of the first term or semester back at school, the students who attended the youcubed summer camp achieved a significantly higher mathematics GPA (
Estimated effects of the MMSP on student math grades.
Outcome measure | Camp participants | Matched non-participants | Effect |
---|---|---|---|
Math GPA (4-point scale) | 2.62 | 2.46 | 0.1579*** |
Percentage with grade of B or better | 54.3 | 48.5 | 5.83** |
Percentage with grade of D or worse | 15.7 | 20.9 | −5.23*** |
Notes: These estimates consider fifth, sixth and seventh graders in all school sites. In all estimates, sample size consists of 536 camp participants and 1,881 matched non-participants. ***
Among the seven sites that shared science GPA data, a matched-comparison analysis indicated that camp participants also had slightly higher science GPAs than similar nonparticipants, but that difference–0.11 GPA points–was not statistically significant at the 5% level.
Overall, exploratory subgroup analyses suggest that the mathematical mindset intervention had a similar impact for students with different demographic characteristics including students of different racial groups, English Learners, and students who received low or average grades at baseline (
Estimated impacts on math GPA and gains on MAC/MARS among subgroups of program sites.
Math GPA | Sites | MARS Gain | Sites | |
---|---|---|---|---|
District Recruitment | ||||
Remediating (score below avg.) | 0.2058*** | 3 | 0.5239*** | 3 |
Regular (score at median) | 0.1045 | 7 | 0.4965*** | 7 |
Camp Dosage | ||||
Low (30–36 h) | 0.2683*** | 4 | 0.4328*** | 3 |
Medium (40–60 h) | 0.0934 | 3 | 0.5341*** | 3 |
High (72–84 h) | 0.0404 | 3 | 0.6302*** | 3 |
Note: Impact on math GPA results come from matched comparison estimates while MARS gain results come from t-tests. The GPA analysis includes one district that did not provide MARS scores, and the MARS analysis similarly includes one district that did not provide GPA records, so the samples are not identical between the two analyses. Districts were classified as “remediating recruitment” if the standardized difference of the average test score between camp and comparison students was below 0.4 standard deviations. Otherwise, they were classified as “regular recruitment.” ***
At camps that targeted recruitment to students with substantially lower math test scores than the district average, impacts on math GPA were larger (
Video analyses of teaching were conducted to consider the aspects of mathematical mindset teaching practices that brought about these positive gains in mathematics achievement, and that stand in contrast to more typical forms of mathematics teaching (
Drawings of cubes from student notebook.
To support students in finding and extending patterns during their exploration, 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. Of these six classrooms, two teachers constructed partial tables, which documented the number of cubes with each number of faces shaded for solely a 3 × 3 × three cube or both a 3 × 3 × three and 4 × 4 × four cube. The other four teachers constructed tables that extended to 5 × 5 × five and n x n x n cubes. An example of this table, written by a student in their notebook, is shown in
Table of patterns from student’s notebook.
Surprisingly perhaps, teachers rarely shared explicit growth mindset messages during this task, but they frequently supported a growth mindset culture in implicit ways, as evidenced by critical moments in which teachers pushed students to justify their thinking, invited students to come up to the board to share their thinking, gave students time to grapple with the task on their own before intervening, and praised students’ thinking and struggle.
Time analysis showed that teachers afforded students ample time to grapple with and persist through the task, supporting the students in encountering challenge and struggle–a key aspect of a mathematical mindset teaching approach. Teachers launched the task for approximately 5 minutes on average and then allowed students to grapple with the task--building cubes, collaborating with peers, and recording in their journals or on chart paper--for an average of 53 min. The open nature of the task meant that even as students figured out one part of the question there were still other areas to explore. After students had sufficient work time and most students had moved beyond the original question to work on the 4 × 4 × 4 cube or generalized even further, the teacher then facilitated a whole class discussion to synthesize the ideas from multiple student groups. This was noted in five of the six videos and lasted approximately 9 minutes on average. Analyses revealed that students were afforded significant time to work on the task in groups and that teachers pushed students to justify their thinking and to connect to each other’s ideas in whole-class discussions.
The teaching analyses revealed that teachers offered students multiple opportunities to experience mathematical 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. Painted Cube was one of many open tasks in the summer camp curriculum, which afforded students a new mathematical experience, through which they could experience important brain connections, as they saw and experienced mathematics in different ways. An absence of any tests or grading practices during the weeks of the camps was also an important feature designed to avoid the fixed messages associated with such practices (
The main focus of the study reported in this paper was the relationship between a mathematical mindset teaching approach and student understanding and achievement, but teacher interviews conducted with 20 teachers also enabled consideration of the students’ shifts in engagement and mindset, as observed by the teachers. All of the interviews were coded and the three most common codes that emerged, as teachers discussed the students’ experiences, were: tasks, student engagement, and student beliefs. Theme analysis across these three codes showed that teachers reported that a significant factor in the students’ engagement in the mathematics in camp came from the openness of tasks, which also helped support students’ changes in mindsets.
All 20 of the teachers interviewed shared details of how the summer camp curriculum impacted students’ engagement and mindsets positively. Sixteen of the teachers (80%) reported the importance of the tasks allowing students to develop and share their own thinking and reasoning--rather than share a single method or answer--and the ways this shifted the dynamics of the classroom. The 16 teachers differed in the particular aspect(s) of the tasks they foregrounded in their interviews: 30% foregrounded the opportunities for multiple approaches to the tasks, 25% foregrounded the open and explorative nature of the tasks, 25% foregrounded the focus on students’ explaining their thinking, and 20% foregrounded the opportunities for students to experience mathematical ideas physically. The teachers explained that when their students shared their thinking with one another, they saw that there were multiple ways to think about the same problem, shifting students’ ideas about what it means to solve a mathematics problem. The teachers noted the multiple entry points for students to participate and engage in tasks and the multiple ways students could find success. These features of tasks resulted in an overall increase in student excitement and engagement and a decrease in anxiety and fear of making mistakes.
Interview analyses also revealed that the majority of teachers observed shifts in students’ engagement throughout the summer camp. Fifteen of the teachers (75%) commented on two types of shifts in student engagement: shifts at the whole-class level and shifts for particular groups of students or individual students. For both types of shifts in engagement, teachers shared stories of students increasing their participation in groupwork, sharing and showing their thinking more readily, persisting on problems rather than shutting down, and building confidence in thinking mathematically. Eight of the teachers (40%) shared that their students displayed excitement while doing mathematics tasks (to the point of not wanting to go to recess or lunch).
Additionally, many teachers connected the task not only to increased engagement but to students’ changed beliefs about themselves as mathematics learners. Sixteen (80%) of the teachers commented on shifts in two types of student beliefs: beliefs about the nature of mathematics and the ways students could engage in the subject (
Many of the teachers shared stories of student transformation, particularly highlighting students who had previously been unsuccessful, shedding negative ideas about the nature of mathematics and their potential, and engaging in new ways. We close this section of the paper with one of the teacher’s reflections:
“I had a student who really, really struggled. We had the entrance exam for her in the Youcubed 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.”
The summer camp intervention built upon research from psychology, neuroscience, and mathematics education, in designing new ways for students to experience mathematics (
An important feature of a growth mindset approach to learning is a comfort with struggle and the belief that struggle is good for learning. Studies of beginning college students, who were asked to engage in complex tasks, found that students were uncomfortable with struggle and their lack of awareness of the value of struggle caused them to avoid complex tasks (
The mathematics tasks, that emerged clearly as pivotal in the students’ experiences, from the teacher interviews, and that we have described as low floor and high ceiling, also had another important feature–they were mathematically interesting to students. Other studies have highlighted the value of students working on tasks that are based in realistic contexts, and that give students opportunities to consider and tackle social injustices (
This study had several limitations and some unanswered questions. Missing data, due to student turnover and the difficulties of following students who changed districts, was one challenge. One unanswered question is why some camps raised students’ achievement significantly more than others, a question that could be investigated in a further study.
Despite these limitations and unanswered questions, the data reported shows that a mathematics approach that is based on mindset and neuroscience, that enables students to embrace struggle and to encounter mathematics in multiple ways, can have a transformative impact on students. This approach is not one that is typically used in schools, partly because of the pressure teachers feel to “cover” the curriculum, and to prepare students for narrow tests, as well as the textbooks on offer to teachers, usually filled with narrow questions. For these reasons the teachers who took part in the study believed that a summer camp is needed, free from these constraints, to unlock students’ potential (
The study reported in this paper adds to this important evidence–showing that a two-to-four-week summer camp sharing a mathematical mindset approach can have a transformative impact on student mathematics achievement. We hope that these different forms of evidence, from summer camp and from school teaching, will prompt policy makers to reconsider the mathematics approaches they encourage in schools–that contravene mindset messages and have resulted in widespread under achievement across the US. When students are released from negative ideas about mathematics and themselves, they learn and approach mathematics differently, and begin a changed, mindset infused, pathway.
The original contributions presented in the study are included in the article/Supplementary Material, further inquiries can be directed to the corresponding author.
The studies involving human participants were reviewed and approved by Institutional Review Board, GSE, Stanford University. Written informed consent to participate in this study was provided by the participants’ legal guardian/next of kin. Written informed consent was obtained from the individual(s), and minor(s)’ legal guardian/next of kin, for the publication of any potentially identifiable images or data included in this article.
JB designed and directed the study, GP-N, JD, and MS-A ran statistical analyses, TL and ML led the analysis of teaching. The full team interpreted results and contributed the writing and revising process.
This report is based on research funded in part by the Bill and Melinda Gates Foundation. The findings and conclusions contained within are those of the authors and do not necessarily reflect positions or policies of the Bill and Melinda Gates Foundation.
The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.
All claims expressed in this article are solely those of the authors and do not necessarily represent those of their affiliated organizations, or those of the publisher, the editors and the reviewers. Any product that may be evaluated in this article, or claim that may be made by its manufacturer, is not guaranteed or endorsed by the publisher.
The authors wish to recognize Cathy Williams, Executive Director of youcubed at Stanford as a co-designer of the summer camp curriculum and member of the research team. The team also would like to acknowledge Gregory Chojnacki, senior researcher at Mathematica, for his invaluable assistance with this research.