Status & Mindset Interventions
In her book Strength in Numbers: Collaborative Learning in Secondary Mathematics, Ilana Horn writes: “Judgements about who is smart based on prior achievement or social categories violate a fundamental principle of equity and are consequential: learning is not the same as achievement” (2012, p.20). The resources below were curated to help you redefine "smarts" in math, disrupt status divisions, develop growth mindsets, and foster a collaborative math community.
Anticipatory Planning
How often does your planning for math involve searching for the "best" problem and then thinking about how you want to teach the problem? It's safe to say this is how most of us approach(ed) lesson planning. The problem with this approach, however, is that it is teacher focused and neglects to consider how students might perceive and respond to the problem. Conversely, anticipatory planning focuses planning efforts on imagining how students might respond to a problem and using that information to plan questions that will push and clarify student thinking and build understanding by sequencing and connecting approaches students are already using.
If you've ever tried to facilitate constructivist math learning in your classroom and it fell short of your expectations, it's likely because the key factor, anticipatory planning, was missing! The template below can help you prepare to facilitate constructivist math learning in your classroom. Grab a planning buddy and give it a try!
Anticipatory Planning
How often does your planning for math involve searching for the "best" problem and then thinking about how you want to teach the problem? It's safe to say this is how most of us approach(ed) lesson planning. The problem with this approach, however, is that it is teacher focused and neglects to consider how students might perceive and respond to the problem. Conversely, anticipatory planning focuses planning efforts on imagining how students might respond to a problem and using that information to plan questions that will push and clarify student thinking and build understanding by sequencing and connecting approaches students are already using.
If you've ever tried to facilitate constructivist math learning in your classroom and it fell short of your expectations, it's likely because the key factor, anticipatory planning, was missing! The template below can help you prepare to facilitate constructivist math learning in your classroom. Grab a planning buddy and give it a try!
Mathematical Agency Improvement Community
The Mathematical Agency Improvement Community (MAIC) – a diverse network of K-12 schools in Southern California supported by the Center for Research on Equity and Innovation (CREI) at the High Tech High Graduate School of Education – came together to work on the persistent problem of improving students’ mathematical agency and outcomes in middle- and high-school mathematics.
Network teachers used an improvement science framework to test, adapt, refine, and spread student centered math practices in their respective contexts.
The Context
Student achievement in mathematics is now recognized as a critical equity issue in the United States. Success in math predicts future social mobility and economic equality (Rose, 2004). However, significant gaps in math achievement persist for traditionally underserved students, typically students of low socioeconomic status and students of color (Bohrnstedt, et al., 2015; Lee, 2002). These students are also most likely to experience conventional teacher-centered math pedagogies that promote a lack of engagement and hinder the formation of mathematical identities. As early as elementary school, many students see math as something they cannot do (Boaler, 2014). Due to this lack of mathematical agency (i.e. the belief that I can do math), many students interpret struggle as a lack of ability, and fail to persist. Unfortunately, math classrooms often reinforce this lack of agency by rewarding speed and procedural fluency over deep understanding of mathematical concepts and problem solving ability.
The goal of the study was to improve student outcomes and to understand 1) what student-centered practices supported mathematical agency and success for traditionally underserved students and 2) what factors/conditions supported their effective implementation/adaptation in diverse contexts.
Daisy Sharrock and Stacey Caillier discuss the MAIC network in this Research Collaborative webinar.
Network teachers used an improvement science framework to test, adapt, refine, and spread student-centered math practices in their respective contexts. The network defined mathematical agency as follows:
Mathematical Agency
Growth
Mindset
" I can learn if I put in the effort"
Sense
of Belonging
"I am part of a mathematical community"
Purpose
& Value
"I feel intellectually engaged and challenged"
The Network
32
Teachers
12 Schools
Teacher Demographics
19% Asian
6% Black
9% Hispanic
65% White
1722
Students
Student Demographics
12.5% Asian
5.6% Black
59.7% Hispanic
21.6% White
6
Convenings
-
Launch, explore, discuss lesson
-
Anticipatory planning
-
Group work norms
-
Group roles
-
Status interventions
-
Accountability quizzes
-
Participation quizzes
The Mathematical Agency Improvement Community (MAIC) was modeled after the Institute for Healthcare Improvement’s (IHI) Breakthrough Collaboratives (2003), which were designed to help healthcare organizations close gaps by creating a structure where organizations could easily learn from each other and experts, as they collaboratively made improvement toward a common goal.
The Center for Research on Equity and Innovation (CREI) at the High Tech High Graduate School of Education acted as the “hub” for the network. Following the IHI model, CREI faculty and MAIC teacher leaders co-facilitated three in-person “Learning Sessions/Convenings” per year, each of which was followed by an “Action Period” where school teams tested student-centered math practices in their own contexts. Between each convening, CREI faculty hosted virtual improvement reviews to provide support and share emerging learnings across the community. The hub also provided practical measures for assessing mathematical agency and online access to student survey responses.
IHI Breakthrough Collaborative Model
Network Theory of Action
To focus their work, MAIC teachers explored the following essential question:
How can we create learning cultures that nurture mathematical agency and success?
To launch the network, CREI faculty worked with teachers to identify promising student centered practices from research and participating classrooms to put together a preliminary menu of practices for teachers to test during the first action period. To determine which student-centered practices to include, the planning team created a driver diagram – a working theory of action to guide their decisions. Driver diagrams are considered living documents and as such, networks return to them periodically and update them to reflect new learning. The current driver diagram for the MAIC network can be seen on the left.
Network Measures
To understand what student-centered practices support mathematical agency and success, MAIC teachers had students complete a Student Agency Survey three times a year.
Improvement science also places an emphasis on “practical measures” that teachers can use to know if they are making progress toward their aim and to guide their next steps (Yeager et. al., 2013). As MAIC teachers implemented the student centered practices in the MAIC change package – practices such as the launch, explore, discuss lesson structure – they also had students complete the Purpose of Whole Class Discussion Survey and the Purpose of Group Work Survey to gain insight into how their students were experiencing these shifts in instructional practice.
Aggregate data from the student agency survey and the other two surveys were shared with teachers for group analysis using a data protocol during each convening. Teachers explored the data and captured their noticings, wonderings, and possible next steps.
Student Agency
(1585 students)
Student Agency Survey developed and validated by the Carnegie Foundation’s Student Agency Improvement Community (SAIC)
Instructional Environment
(1350 students)
Purpose of Whole Class Discussion developed and validated by the Middle school mathematics and the Institutional Setting of Teaching project (MIST) from Vanderbilt University
Mathematical Success
Final Math Grades of students from participating classrooms
CAASPP data
Findings: Classroom Practices & Mathematical Agency
In an effort to move away from teaching math as a set of procedures to be memorized, MAIC teachers were adopting more student-centered practices, such as a launch, explore, discuss lesson structure that forefronts student problem solving strategies and utilizes whole class discussion as an instructional method. As teachers explored how to implement student discussions of different problem solving strategies into math class, they used the Purpose of Whole Class Discussion Survey to determine the degree to which (1) their students felt that the purpose of math class was to listen and make sense of other students thinking, and (2) to learn multiple ways to solve a problem. Two statistically significant correlations between these measures and the growth mindset measure on the Student Agency Survey were found:
The more students perceive that the purpose of classroom discussion is to learn different strategies to solve a problem, the more likely they are to believe that anyone can be a math person. (p= 0.017, n= 23 teachers, representing 1585 students)
The more students perceive that the purpose of classroom discussion is to understand another’s thinking, the more likely they are to believe that anyone can be a math person. (p=0.014, n= 23 teachers, representing 1585 students)
These two correlations suggest that classroom practices that emphasize learning different ways to solve a problem and practices that support students in understanding each other’s thinking support a growth mindset with respect to math in students. Recent findings from a longitudinal study on identity by Bohrnstedt, et al., (2018) suggest that having a mathematics identity – i.e. student belief that they were “a math person” – was an important predictor of high school mathematics achievement. Taken together, these findings have important implications for the way we teach math. Conventional classrooms often focus on ‘one right way’ of doing problems (i.e. the way taught by the teacher) which leaves little room for students to share and develop their own mathematical thinking and problem solving methods. Without the ability to build on their own understanding and problem solving strategies, many students do not find math relevant and do not develop a mathematical identity.
Shifts in Teacher Practice
Over the second year of MAIC, students in seven of the seventeen classrooms that used the Purpose of Whole Class Discussion survey in a pre/post format responded more favorably to the survey measure: “Listening to my classmates helps make my thinking better.”
Nine of the seventeen teachers also saw a decrease over the year in students’ belief that: “In math class we need to solve problems using the steps the teacher showed us”, seven of which were strongly statistically significant (n=96, p=0.0008; n=47 p=0.005; n=85 p=0.004; n=30 p=0.004; n=55 p=0.007; n=98, p=0.00003; n=40, p=0.047), suggesting that these teachers had shifted to more student centered instruction.
Teacher interviews corroborated these findings.
“MAIC has shown me that students can learn just as well from each other as from a teacher and that when students share ideas and even correct each other in their misconceptions, that is much more powerful and received better then when it comes from a teacher.”
~
“I feel like I have become such a better math teacher. I know that my students expect to be pushed to think hard, work together, and justify their thinking.”
~
"This program was an anchor and a reminder of why it is important to do what we are doing. In the past, I had not considered the importance of taking into account the experiences our students bring with them."
~
Student Centered Practices Adopted by MAIC Teachers (n=29)
MAIC teachers actively adopted and adapted student-centered math practices and collected data on student participation, academic language use, and mathematical understanding, through ongoing testing cycles. Based on this evidence, and teacher self-reports, the practices teachers found most useful for supporting student agency include: the launch, explore, discuss lesson structure, anticipatory planning, group work norms, accountability quizzes, participation quizzes, and status interventions.
Practice Frequency
From teacher responses it is clear that some practices were found to be more helpful when used frequently – such as students working in groups, students presenting their thinking, and whole class discussions – while other practices were useful when used more infrequently – such as revisiting group roles or norms, and participation and accountability quizzes.
Student Centered Practice
Number of Teachers Using Practice 1+ Times a Week
(n=32)
Number of Teachers Using Practice 1-3 Times a Month
(n=32)
28
27
23
20
17
11
8
4
3
3
4
5
8
11
9
17
12
17
20
18
Conditions that Supported Testing & Adaption
"By 'doing the math' in the student role; discussing, and sharing out in different ways, made it easier to incorporate new practices into my own. The most impact I've seen in my classroom is in the productivity of group work, increase in participation, and richness in discussions."
"Anticipatory planing with colleagues has helped me clarify what my goals are for students in a math classroom - what types of thinking I'm hoping for and how they might grapple with ideas. I can now clearly articulate what I hope students might say during a lesson that would demonstrate they are grappling with the ideas I hope they will!"
Participating in MAIC as part of a school team, experiencing practices first as students, and planning lessons and collecting data with colleagues all supported teachers in implementing and adapting more student-centered practices.
Another important finding related to the uptake of student-centered practices and the correlational findings from the student perception data, is that the teachers who adopted the student-centered practices as a suite of practices and integrated them consistently – i.e. daily instruction included a launch, explore, discuss lesson structure where students shared their thinking in groups and with the whole class vs. trying these practices in an ad-hoc fashion less frequently – were the classrooms where more students believed that anyone could be a math person.
In light of this, we hope to explore further how teachers’ beliefs about math as a set of procedures to be memorized (i.e. how they/we were taught) shift over time, and which conditions/network experiences contribute to the integration of beliefs with more student-centered classroom practices.
Experiencing the practices, anticipatory planning, and practice mingles
An important activity at each of the MAIC convenings was for teachers to experience the practices as learners first. All of the practices were modeled while teachers did math together as students first. Teachers shared that this experience helped them understand how the practices felt different from conventional math practices, and also provided a model of the “teacher moves” required to test it out in their own classrooms the next day.
Teachers also participated in “practice mingles” during each convening, where teachers testing a particular practice would meet and share how the practice was working in their context and any adaptations they had made.
In addition, the opportunity to engage in anticipatory planning with colleagues, and observe and collect data in other teachers’ classrooms as they tested MAIC change ideas, supported the spread of practices between teachers. Sixty five percent of teachers shared that the lesson studios hosted during MAIC convenings were helpful for adopting the practices highlighted in the MAIC change package.
Being part of a school team
“My math partner and I both attend MAIC and have been able to share in our growth as educators. We work closely to implement new change ideas, and discuss ways to increase student agency regularly.”
“We have worked together to plan and implement MAIC change ideas to increase the level of mathematical engagement tailored to our students.”
Increased Teacher Confidence (n=29)
MAIC teachers felt confident using the tools of improvement science to test student-centered practices in their own contexts and were instrumental in sharing the student-centered practices with non-MAIC colleagues.
97% Participating in MAIC helped me create a learning culture where students grapple with mathematical ideas
93% Participating in MAIC helped me become more confident using open ended math problems in my class
93% Participating in MAIC helped me create a learning culture where multiple approaches are expected
93% Participating in MAIC helped me create a learning culture where mistakes/misconceptions are valued
90% Participating in MAIC helped me become more confident facilitating whole class discussions
90% Participating in MAIC helped me create a learning culture where students work productively in small groups
Access to tools, resources, and new practices
“My participation with MAIC has transformed my classroom teaching practices. Prior to my participation in MAIC I had a vision for teaching and learning in my classroom that was aligned with the vision of MAIC, but I did not necessarily have the pedagogical tools, mathematical knowledge for teaching, or collaboration partners I needed to accelerate my learning and improve my practice. MAIC has introduced me to tools, structures, and ways of thinking that have helped me facilitate student-centered learning in my classroom.”
“How valuable it is to work with people who have similar goals and hopes for what math education should be like. Being given thoughtfully curated resources that I can try in my classroom has been so helpful and testing many of these ideas.”
“MAIC has helped me become a better math teacher because I am a better facilitator. I feel more prepared to anticipate student thinking and have access to protocols and structures that will increase student participation and rigor of activities.”
Using the tools of improvement
MAIC participants captured their planning and learning from testing cycles on a shared slide deck, which was accessible to all and updated prior to and during each in-person convening. CREI faculty also hosted virtual convenings, where MAIC teachers could share and analyze findings from their ongoing testing cycles.
“Engaging in PDSA cycles has resulted in me becoming a more reflective teacher generally. I tend to look at student learning outcomes more often to determine the effectiveness of various interventions. I am also more willing to try new things!”
“By reviewing data, I reflect on my practice and am willing to try new interventions.”
Broader impact: the spread of MAIC practices at school sites
In addition to feeling increased confidence in their own ability to use student-centered practices, MAIC teachers also felt empowered to share practices with their colleagues at their respective school sites. In response to the question, “Have you shared change ideas with colleagues at your schools? If so which ones?” twenty three MAIC teachers indicated that they had.
“Yes - I have shared MAIC findings with the math discipline group at my school. I have also shared resources for the accountability and participation quiz with a number of colleagues who have asked for suggestions to get students to be more engaged in their learning.”
“Yes, I have shared agency warm-ups, group roles, and participation/accountability quizzes. Lesson studios has become a presence within our practice as well.”
“All of the above! But especially status treatments, accountability quizzes, and launch/explore/discuss.”
Lessons Learned About Being a Hub for a Network Improvement Community
The following include a series of lessons learned about how to support networked improvement science in schools, and the role of a “hub” in supporting this work.
In order to generate momentum and establish a communal vision early, we launched MAIC around a clearly identified aim, and provided research-backed change ideas that were aligned to the network goals for teachers to test out immediately.By creating a short menu of change ideas to test, participants could try something quickly and share learning with others in the network, generating “early wins” that increased motivation and interest.
Start with clearly identified change ideas and data collection methods to get moving quickly
MAIC teachers were more likely to report that student-centered practices were spreading at their school sites when multiple teachers from the same site participated in network convenings consistently, and when their work was explicitly supported and understood by their school leaders.
Teams matter
In improvement science, there are lots of tools people can use. The tools are not enough. For this work to be transformative, networks need to attend to relational dynamics and help people construct their thinking together. We’ve learned to not front load improvement content and then ask teams to create a Fishbone Diagram or Driver Diagram. Instead we have them use a protocol to guide them through the process, so they are doing the work of improvement, but also learning how to work with one another in a way that feels inclusive and purposeful.
Protocols are liberating structures that support equitable dialogue and collaborative work
We were sensitive to the baggage associated with “data” – especially data for accountability – and worked to broaden people’s conceptions of what “data for improvement” was by anchoring it to student mathematical understanding and participation. PDSA level data was also a challenge for participants. We worked to find simple ways of collecting/tracking data that could be easily integrated into the daily routines of teaching. Several teachers involved students as collaborators and data collectors for ongoing PDSA cycles.
Data is essential, and tricky
If we want to build people’s improvement capacity, they need authentic opportunities to participate in - and facilitate - improvement work. As MAIC progressed, we increased our efforts to have teachers leading sessions for the whole NIC, communicating that improvement expertise was not located solely in the Hub. During in-person meetings, participants also experienced and helped facilitate protocols, which we believe made them more likely to use them in their own contexts.
Grow NIC participants’ improvement skills and facilitation by sharing leadership
Some people assume that improvement and innovation are in opposition. We’ve found that if you want to create shared investment – especially with your most creative teachers and leaders -– you have to allow room for agency and innovation when testing and refining change ideas. This helps network participants learn from variation and also surfaces promising adaptations that others can benefit from.
Emphasize adaptation, not replication
Implications for Teachers
One challenge of developing pedagogical knowledge for more equitable student outcomes is the isolated nature of teaching. Most teachers in the United States experienced math through an ‘I do, we do, you do’ lesson structure, where math was taught as a subject of discrete procedures modeled by the teacher and then memorized. Conceptual connections, if attended to at all, were explained by the teacher, and students rarely engaged in collaborative problem solving or discussion of the underlying mathematical ideas, two practices shown to support student achievement (Boaler & Staples, 2008; Carpenter et al., 1996). Despite prior experiences, many teachers wish to create a classroom where students are actively engaged in exploring and discussing mathematical ideas. However, without concrete experiences and models of the teaching practices that support this type of classroom culture, teachers often struggle with how to enact their vision. In addition, teachers rarely have the opportunity to watch each other teach, or collaborate in a focused way to improve their teaching practice (Brill & McCartney, 2008).
Through MAIC, teachers had the opportunity to experience student-centered practices themselves as learners first. These experiences were credited for creating a communal vision of what student-centered mathematics instruction looks and feels like. In many schools teachers have regular collaboration time through Professional Learning Community structures (PLCs) in which they could engage in this type of experience. The practices and associated video deliverables from this study may be of particular interest to instructional coaches and teacher leaders who facilitate these school based communities.
Implications for School Leaders
MAIC teachers repeatedly brought up the benefits of seeing their colleagues test out practices in their own classroom. Teachers appreciated doing anticipatory planning together, observing each other teach and test out student-centred practices, and debriefing together. By focusing on student thinking during the lesson study events, teachers gained important insights into how students make sense of mathematics, and came to see this thinking as a valuable source of data to drive instruction. School leaders are perfectly positioned to create structures and find opportunities for teachers to plan together, observe each other’s classrooms, and help facilitate debrief sessions.
Implications for Teacher Education Programs
The student-centered practices tested by MAIC teachers differ markedly from the way that most of us were taught math. Findings from this study suggest that practices that focus on exploring multiple strategies to solve a problem and listening and making sense of each other’s thinking support student mathematical agency. As such, teacher education programs need to focus on the twin goals of shifting incoming teacher beliefs about mathematics, as well as developing teaching skills such as anticipating student thinking, setting up positive group work structures, and facilitating whole class discussions based on student problem solving strategies. Most importantly, they need to model such practices for pre-service and in-service teachers so that they have the opportunity to experience the practices as learners and see them facilitated well.
Implications for State & District Policy
Findings from this study also have implications for district and state education departments. If we want teachers to cultivate student agency and focus on student thinking, the curriculum matters. Districts need to provide a “what” that is aligned with more student-centered practices so teachers can focus on the “how” of teaching. Without good open-ended problems (where many solutions are possible) or open-middle problems (where many strategies to reach a particular solution are possible) it is difficult to have rich mathematics discussions. Narrow curriculum focused on following procedures undermines students mathematical agency and has yet to result in robust student achievement for traditionally underserved students.