Executive Summary

Problem-Based Learning

This is a summary of CPM’s 2023 research base on Problem-Based Learning. For more information and for references, please see the full report, available from https://cpm.org/research-base (PDF copy).

Mathematics education researchers have largely accepted that problem-solving is an essential part of doing mathematics. They now focus on how the distribution and delegation of power in mathematics classrooms influence students’ opportunities to learn (Agarwal & Sengupta-Irving, 2019; Schoenfeld, 2016). While there are many types of authority at play in classrooms, the most relevant type of authority for mathematics learning is mathematical authority, or “who possesses mathematical knowledge that is taken as true” (Langer-Osuna, 2017, p. 238). This research base article focuses on the influence of mathematical authority on mathematics sensemaking in problem-based tasks.

CPM’s Pillar of Problem-Based Learning refers to:

A. using tasks that cannot be solved by a simple algorithm and that are often embedded in complex real-world contexts;

B. supporting students in collaboratively constructing knowledge through productive discourse practices (i.e., active, student-centered learning); and

C. sharing mathematical authority with students. In problem-based learning, students construct knowledge by making connections to form big mathematical ideas.

Problem-based learning develops students’ problem-solving skills, content knowledge, and eventually, the ability to be self-directed, independent learners (Duch et al., 2001; Hmelo-Silver & Barrows, 2015; Hammond, 2014). For example, empirical research has demonstrated that problem-based tasks are better than traditional tasks for supporting students in applying their knowledge to novel problems (i.e., transfer) and being self-directed in their learning (Hmelo, 1998; Hmelo & Lin, 2000; Schmidt et al., 1996). According to Langer-Osuna (2017), a leading scholar on mathematical authority, agency, and identity, fostering shared authority between teachers and students supports increased student ownership of mathematical ideas (Bianchini, 1999; Ehrlich & Zack, 1997; Lotan, 1997), conceptual understanding (Hiebert et al., 1997), and positive identification with mathematics (Hand, 2012; Lotan, 2003).

CPM infers from this research that…

Problem-based learning supports students in learning mathematics in ways that will be useful to them in their future mathematics classes, in their careers, and in their lives outside of school and work. When students understand how to solve problems rather than how to correctly select and implement procedures, they develop mathematical authority and thus are more likely to enjoy and see the value in mathematics.

Over half a century of cognitive and sociocultural research on learners’ mathematical thinking indicates that it is more effective (i.e., it better supports understanding and problem solving), although perhaps less efficient (i.e., it may not allow for “covering content” as quickly), to let students work with their peers to struggle through making sense of a problem and invent ways to solve it, rather than showing students how to solve problems (Choppin, 2022; Schoenfeld, 2016). In fact, to do mathematics, the problem has to actually be a problem for the problem-solver (Schoenfeld, 2016). In sum, if we wish for students to develop productive mathematics problem-solving dispositions that follow them into adulthood, it is essential that students’ own ideas become more central in instructional activities (Boaler & Greeno, 2000, Boaler & Selling, 2017).

CPM infers from this research that…

Problem-based learning supports students in authentically engaging in mathematics. Doing mathematics inherently requires struggle since problem-solving only happens when the path to the solution is unclear. Problem-based learning is thus riskier for teachers than other forms of instruction because it requires putting students’ ideas at the center of instruction—including their uncertainties and misconceptions—rather than the de facto correct ideas of the teacher and the textbook. Using heterogeneous grouping by randomly assigning students to teams can expose students to many different ideas, which then supports them in making decisions about which ideas to pursue. Though problem-based learning may take more time than other ways of teaching, it pays off as students develop productive mathematics problem-solving dispositions that stick with them for life.

Unfortunately, many commercially available mathematics curricula today do not support problem-solving (Larson, 2014). A 2022 study by Choppin et al. characterized the curricula of six mathematics textbook publishers—Math in Focus, Holt, Prentice Hall, Glencoe, CMP, and CPM—and found that all but CPM and CMP were substantially composed of tasks that served as delivery mechanisms for content rather than tasks that functioned as thinking devices that support student sensemaking.

CPM infers from this research that…

Problem-based learning is essential for supporting mathematical sensemaking, and CPM is one of the few secondary mathematics curricula that function as a “thinking device” for students rather than as a “delivery mechanism.” Because students do not come into the classroom as blank slates who will integrate everything they encounter exactly as the teacher and textbook intend, it is essential for curricula to support students in developing problem-solving skills as they persevere in making sense of mathematical tasks. Engaging in productive struggle during problem-solving is much more aligned with the mathematical practices of mathematicians than consuming a curriculum of discrete facts and skills derived from experts’ knowledge. In CPM materials, tasks are designed to engage students in mathematical practices that make learning the knowledge and skills found in traditional curricula much more sensible, enduring, and relevant.

Although it is common practice to argue that students with learning and intellectual disabilities need explicit instruction instead of problem-based learning, research has shown that these students can develop conceptual understandings through problem-solving by (a) using mathematics tasks that are connected to real-world applications; (b) collaborating with peers and using manipulatives; and (c) inventing their own strategies rather than using standard algorithms (e.g., Lambert & Sugita, 2016). For more information, see the NCSM publication, “Inclusion and Intervention: Understanding “Disability” in the Mathematics Classroom,” by Jasien and Hayes (2022).

CPM infers from this research that…

Problem-based learning is appropriate for all students in inclusive classrooms.

Mathematical authority is not a binary (e.g., students do not either have or lack mathematical authority), but rather occurs along a spectrum, and is dynamic (rather than static), expanding (rather than finite), and situational (rather than stable across time) (Bishop et al., 2022). When students have a high degree of mathematical authority, they “are ‘authorized’ to solve mathematical problems for themselves, are publicly credited as the ‘authors’ of their ideas, and develop into local ‘authorities’ in the discipline” (Stein et al., 2008, p. 332).

CPM infers from this research that…

Problem-based learning requires inviting students to make mathematical decisions and contribute substantive mathematical ideas. In other words, in a classroom where problem-based learning is flourishing, students exercise a high level of mathematical authority by making conjectures, explaining their work, justifying their reasoning to one another, and building on each other’s ideas. Of course, this is unlikely to happen at the beginning of the school year in any classroom because it requires building a classroom culture that supports students in taking on more and more mathematical authority over time. Mathematical authority can shift from moment to moment and day to day depending on student engagement and the mathematical ideas of the lesson, but over time, teaching for problem-based learning should support all students in exercising mathematical authority.

In too many classrooms, the teacher and textbook are the primary sources of claims about what is right and how mathematical thinking should unfold (Engle & Conant, 2002; Herbel-Eisenmann & Wagner, 2007; Litke, 2015, 2020; Mehan, 1979; NCTM, 2014). In classrooms where the teacher and textbook are the primary sources of mathematical authority, students’ mathematical sensemaking is backgrounded in favor of behavioral compliance, with demands like giving the teacher complete attention and always being on task without socializing (Cazden, 2001; Hand, 2012). Teaching in ways that support students in becoming problem solvers who enjoy mathematics requires making space for students to be their full selves, including by attending to students’ relationships with each other (e.g., their social and academic status), attending to their emotional and physical needs, and allowing goofiness and social talk to occur with mathematical discourse (Joseph, 2021).

For many students, including high-achieving students, traditional classroom practices that do not make room for problem-solving activities like exploration and justification can lead students to disaffiliate with mathematics (e.g., the common phrase, “I’m not a math person;” Boaler & Greeno, 2000; Pope, 2001).

CPM infers from this research that…

Problem-based learning may require teachers to significantly shift their pedagogy, moving from a teacher-centered approach toward a student-centered approach. While traditional teaching methods may work for some students in terms of performing well on assessments, they do not work well for most students. Even more, there are too many students earning good grades or taking advanced mathematics classes who achieve in mathematics not because they enjoy or see intrinsic value in it, but because they know how to “do school” and see its importance for their future (e.g., collegiate gatekeeping). Problem-based learning invites more students to see themselves as belonging in mathematics class and creates more meaningful mathematics learning for those who already see themselves as belonging.

Sharing mathematical authority does not mean that teachers should withdraw or abdicate their own authority. The idea that authority is finite (i.e., there is only so much to go around) comes from the widespread misconception that mathematical authority is a zero-sum game (Bishop et al., 2022). When teaching problem-based learning, teachers act as expert learners by using open questions to scaffold students toward mathematical understandings, thus offering students a kind of apprenticeship in mathematical thinking (Hmelo-Silver & Barrows, 2015).

CPM infers from this research that…

Problem-based learning expands the mathematical authority within a classroom as teachers share, rather than give away, their intellectual authority with students. In fact, when teachers abdicate their mathematical authority, students’ mathematical reasoning can suffer. By modeling the practices of expert learners rather than acting as a source of finished knowledge, teachers support students in reasoning mathematically.

Teachers can support students in developing mathematical authority by eliciting and probing student thinking (Arnesen & Rø, 2022; Ellis et al., 2019; Hamm & Perry, 2002), building on students’ ideas (Arnesen & Rø, 2022; Drageset, 2014; Sherin, 2002), supporting students in engaging in evaluative work by making connections (Arnesen & Rø, 2022; Drageset, 2014; Kazemi & Hintz, 2014; Stein et al., 2008), assigning competence (Esmonde, 2009; Hand, 2012; Jilk, 2016), and facilitating peer-to-peer mathematical relationships (Langer-Osuna, 2015).

CPM infers from this research that…

Problem-based learning requires teachers to exercise their mathematical authority while still sharing mathematical authority with students. Teachers can do this by questioning in ways that:

- probe process before probing content (e.g., encouraging the student(s) to explain their work and justify their reasoning to one another, or using study team and teaching strategies to help students learn from each other);
- model mathematical practices (e.g., asking for detailed explanations (how) and justification (why), or asking for connections between multiple students’ thinking and between multiple representations); and
- foster mathematical discourse amongst students (e.g., asking open-ended questions that make space for students to share and build on each other’s ideas, strategies, representations, and explanations).

Langer-Osuna et al. identified a small set of student actions that frequently preceded shifts in who exercised mathematical authority in teams, with all of these actions involving students publicly naming aspects of their teamwork: asking for clarifications (e.g., “So what are we working on?”), stating the work to be done (e.g., “We are supposed to get a new card”), and making their own process the object of consideration (e.g., by counting manipulatives aloud).

CPM infers from this research that…

Problem-based learning reflexively supports and requires students to develop a growth mindset, persistence with an expectation of difficulty during mathematical problem-solving, and willingness to take social and academic risks, all essential aspects of developing mathematical authority. For example, students engaging in problem-based learning will often need to ask for clarifications from their teammates, direct their team members, find resources for problem-solving, and expose their own uncertainties. CPM materials are designed to support students in engaging in these activities and more by incorporating team roles, which are used to their fullest potential when they support students in making their mathematical ideas (including their uncertainties) public.

###### Used throughout CPM middle and high school courses

###### Concrete, geometric representation of algebraic concepts.

###### Two-hour virtual session,

###### Learn how students build their conceptual understanding of simplifying algebraic expressions

###### Solving equations using these tools.

###### Determining perimeter,

###### Combining like terms,

###### Comparing expressions,

###### Solving equations

###### Use an area model to multiply polynomials,

###### Factor quadratics and other polynomials, and

###### Complete the square.

###### Support the transition from a concrete (manipulative) representation to an abstract model of mathematics..

This professional learning is designed for teachers as they begin their implementation of CPM. This series contains multiple components and is grounded in multiple active experiences delivered over the first year. This learning experience will encourage teachers to adjust their instructional practices, expand their content knowledge, and challenge their beliefs about teaching and learning. Teachers and leaders will gain first-hand experience with CPM with emphasis on what they will be teaching. Throughout this series educators will experience the mathematics, consider instructional practices, and learn about the classroom environment necessary for a successful implementation of CPM curriculum resources.

Page 2 of the Professional Learning Progression (PDF) describes all of the components of this learning event and the additional support available. Teachers new to a course, but have previously attended Foundations for Implementation, can choose to engage in the course Content Modules in the Professional Learning Portal rather than attending the entire series of learning events again.

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The Building on Instructional Practice Series consists of three different events – Building on Discourse, Building on Assessment, Building on Equity – that are designed for teachers with a minimum of one year of experience teaching with CPM instructional materials and who have completed the Foundations for Implementation Series.

In **Building on Equity**, participants will learn how to include equitable practices in their classroom and support traditionally underserved students in becoming leaders of their own learning. Essential questions include: How do I shift dependent learners into independent learners? How does my own math identity and cultural background impact my classroom? The focus of day one is equitable classroom culture. Participants will reflect on how their math identity and mindsets impact student learning. They will begin working on a plan for Chapter 1 that creates an equitable classroom culture. The focus of day two and three is implementing equitable tasks. Participants will develop their use of the 5 Practices for Orchestrating Meaningful Mathematical Discussions and curate strategies for supporting all students in becoming leaders of their own learning. Participants will use an equity lens to reflect on and revise their Chapter 1 lesson plans.

In **Building on Assessment**, participants will apply assessment research and develop methods to provide feedback to students and inform equitable assessment decisions. On day one, participants will align assessment practices with learning progressions and the principle of mastery over time as well as write assessment items. During day two, participants will develop rubrics, explore alternate types of assessment, and plan for implementation that supports student ownership. On the third day, participants will develop strategies to monitor progress and provide evidence of proficiency with identified mathematics content and practices. Participants will develop assessment action plans that will encourage continued collaboration within their learning community.

In** Building on Discourse**, participants will improve their ability to facilitate meaningful mathematical discourse. This learning experience will encourage participants to adjust their instructional practices in the areas of sharing math authority, developing independent learners, and the creation of equitable classroom environments. Participants will plan for student learning by using teaching practices such as posing purposeful questioning, supporting productive struggle, and facilitating meaningful mathematical discourse. In doing so, participants learn to support students collaboratively engaged with rich tasks with all elements of the Effective Mathematics Teaching Practices incorporated through intentional and reflective planning.

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