Key Qualities of a Problem-Based Learning Approach

WHAT'S ALL THE BUZZ ABOUT PROBLEM-BASED LEARNING?

WHAT DOES ALL THIS HAVE TO DO WITH PBL?
Though research on PBL lacks a single, universally accepted definition, most uses of the term describe a learning environment in which students are actively engaged in collaboratively solving complex problems with multiple solution pathways. PBL uses motivating tasks to drive learning. Although this approach puts students at the center of building new understanding through inquiry and discovery, teachers play an important role in facilitation. Eliciting evidence of student thinking and posing purposeful questions prompts students to advance their thinking and unpack key connections and relationships while engaging with the task.  

Considering the achievement backdrop seen in California and across the United States, a learning environment that fosters a robust understanding of mathematics through active, task-based learning experiences that also meets individual students where they are in the learning process is certainly compelling. Is it possible that Problem-Based Learning could hold the key to helping all students learn mathematics in a deep and meaningful way, allowing them to reach and exceed the levels of mathematics performance seen pre-pandemic?  

Let’s take a look at what guidance related to PBL is provided by the California Mathematics Framework. 

WHAT DOES THE 2023 CALIFORNIA MATHEMATICS FRAMEWORK SAY ABOUT PROBLEM-BASED LEARNING? 

It may be surprising in light of the challenges reported with student mathematics achievement in California that the California Mathematics Framework doesn't reference Problem-Based Learning by name. Does this mean PBL is not a recommended curricular or pedagogical approach in California? 

Not exactly. 

The 2023 California Mathematics Framework clearly, and in great detail, describes the importance of curricular materials and teaching strategies that promote a rich and connected understanding of the big ideas of mathematics while fostering students’ beliefs in themselves as capable doers of mathematics.  

To achieve these results, the Framework encourages teachers to establish learning environments with key qualities, including:  

• Engaging, open tasks to expand access to rigorous mathematics for all students 
• Student-centered exploration leading students to ask questions, make conjectures, and engage in discovery 
• Engagement in the Mathematical Practices to promote habits of mind and habits of interaction leading to productive mathematical interactions 
• Collaborative learning experiences emphasizing justification and reasoning to make and strengthen a shared understanding of mathematics 
• Productive struggle fostered through learning environments where students see the value of their perseverance through authentic tasks 

Problem-Based Learning is one of many approaches that can support learning environments consistent with recommendations within the California Mathematics Framework. Rather than focusing on a specific approach, the Framework reminds us that our focus should be on the qualities of mathematical learning experiences fostered by the pedagogical approaches we use. Using the qualities of mathematical learning experiences as our guideposts, we can ensure that any pedagogical approaches, including Problem-Based Learning, are interpreted and implemented in ways that lead to meaningful and lasting student learning.  

KEY QUALITIES OF LEARNING EXPERIENCES CALL FOR BY THE CALIFORNIA MATHEMATICS FRAMEWORK

The key qualities of learning experiences called for by the California Mathematics Framework work together to provide a learning environment where all students can flourish as confident doers of mathematics with a strong understanding of mathematics as a connected web of big ideas. 

Let’s dive into each quality in more detail. 

It is hard to overstate the importance of mathematical tasks used as a catalyst for student learning. For decades, it has been recognized that “there is no other decision that teachers make that has a greater impact on students’ opportunity to learn and on their perceptions about what mathematics is than the selection or creation of the tasks” (Lappan & Briars, 1995, p. 139).  

In Principles to Actions, The National Council of Teachers of Mathematics (NCTM) recognizes task selection and implementation as one of its eight Effective Mathematics Teaching Practices. Further guidance in Taking Action outlines the important relationship between lesson learning goals, building procedural fluency from conceptual understanding, and the mathematical tasks used within a lesson, with these three elements working together to determine the direction of student learning through lesson implementation.

Engaging and open tasks can expand access to rigorous mathematics for all students. Engaging tasks are student-centered (versus teacher led) and provide contexts that students can relate to, spurring them to ask questions and seek solutions. Open tasks afford multiple solution pathways at varying levels of sophistication, ensuring that all students have an entry point to begin engaging with learning linked to the lesson learning goals. 

STUDENT-CENTERED EXPLORATION

If the tasks students engage with are vehicles of learning, then student-centered exploration reminds us to ensure students are in the driver’s seat by asking pertinent questions, making conjectures, and engaging in discovery.  

In this type of learning environment, the teacher plays an important role as a facilitator, listening, watching, and asking purposeful questions to elicit evidence of student thinking. The teacher uses assessing questions to make students’ thinking visible and advancing questions to help prompt students to move toward the lesson learning goals. But ultimately, the conjectures and conclusions are driven by the students’ thinking, not dictated by the teacher. This helps to ground new student learning in experience and connections, helping students to build the rich and connected understanding of mathematics called for in the California Mathematics Framework. 

ENGAGEMENT IN MATHEMATICAL PRACTICES

The Standards for Mathematical Practice (SMPs) describe how students should engage with mathematics while learning and the proficiencies that they should develop as a result. 

The California Mathematics Framework describes the important role of Mathematical Practices in describing how students should engage with mathematics. Engaging students in learning content while using the Mathematical Practices leads to “powerful math experiences centered on exploring, discovery, and reasoning” (CMF, Chapter 4, p.1). A classroom centered around engagement in the Mathematical Practices requires meaningful tasks and a student-centered learning environment. 

COLLABORATIVE LEARNING EXPERIENCES 

The California Mathematics Framework also explains that students should work collaboratively to build a shared understanding of mathematics. The potential of open tasks that facilitate the use of the SMPs is recognized when students have opportunities to share multiple approaches, using justification and reasoning to explain and support their strategy while sharing with classmates.  

This also means students are active listeners as their classmates share, critique reasoning, ask questions, and ultimately build a shared understanding of mathematics by bringing together their thoughts and approaches in meaningful ways. 

Again, the teacher plays a critical role in coordinating collaborative learning experiences, particularly when it comes to supporting students in building a shared understanding centered around the multiple approaches used to solve a meaningful problem.  

In 5 Practices for Orchestrating Productive Mathematical Discussions, Peg Smith and Mary Kay Stein provide a pathway for supporting a shared understanding via classroom discourse, as called for the in the California Mathematics Framework, that includes:

• Beginning with a rich math task with multiple solution pathways (step 0) 
• Anticipating student pathways before instruction (step 1) 
• Monitoring student solutions as they work (step 2)  
• Selecting student solutions to be shared as part of the classroom discussion (step 3) 
• Sequencing the order in which solutions are shared to foster the shared understanding that is sought (step 4), and  
• Connecting the various solution pathways to build the rich and connected understanding of big mathematical ideas (step 5) 

PRODUCTIVE STRUGGLE

The qualities of learning outlined above put a great deal of expectation on students. Students are expected to be actively engaged, to use the SMPs as they make sense of problems and persevere in solving them, while engaging collaboratively through meaningful discourse with their classmates as they work together to build a shared understanding.  

The SMPs tell us that students should persevere in problem solving, continuing to work even when solution pathways aren’t immediately available to them. But the onus of perseverance is not only on students; teachers also play an incredibly important role in supporting students through the learning process. The term productive struggle reminds teachers of the importance of allowing students to do the heavy lifting while ensuring they aren’t suffocated by the weight of the task.  

Teachers are active facilitators of learning, eliciting evidence of where students are in their thinking journey and using questions and prompts to help advance toward the lesson learning goals. Fostering productive struggle requires maintaining high expectations for every learner in the classroom and expecting all students to actively engage in the dialogue and building of shared understanding when various approaches to solving problems are shared. The result of these efforts is not only seen in terms of mathematical learning, but also in how students begin viewing themselves as capable doers of mathematics. 

HOW DOES BIG IDEAS LEARNING SUPPORT THE KEY CHARACTERISTICS OF PROBLEM-BASED LEARNING?

Big Ideas Learning is committed to providing a curriculum that helps teachers build a learning environment that embraces the learning experience qualities outlined in the California Mathematics Framework. California Math & YOU provides a learning design in the Student Experience and a robust Teaching Experience to help teachers bring these qualities of learning into their own daily classrooms.  

STUDENT EXPERIENCE 

The California Math & YOU Student Experience provides quality learning experiences focused on building students’ rich, connected understanding of the big ideas of mathematics as described in the California Mathematics Framework. Here are some highlights of how you see this emphasized in the California Math & YOU Student Experience. 

• Big Ideas Framework: Students begin every chapter of California Math & YOU by exploring a Big Idea, an activity designed to help students connect key mathematical concepts to real-world applications and interdisciplinary topics. This structure helps students become involved in engaging, student-centered tasks while building an understanding of mathematics as a connected web of big ideas. Students engage in a Connecting Big Ideas task four times per course, in which they apply the big ideas from across multiple chapters to explore and solve engaging tasks in authentic contexts. 
• Investigations: Students begin every lesson of California Math & YOU with an Investigate activity. These activities put students at the center of a mathematical task that engages students in working collaboratively while asking questions, making conjectures, and engaging in discovery. These tasks build conceptual understanding by connecting new ideas to students’ prior knowledge and experiences, providing a strong building block for new learning. 
• Connections to Real Life & Data: Engaging and authentic applications are embedded through Connect to Real Life and Connect to Data tasks in each lesson. Students apply the SMPs as they analyze, model, and interpret real situations using mathematics. The meaningful contexts of these problems support student engagement and foster productive struggle.
• Performance Tasks: Students synthesize the concepts of every chapter with a Performance Task, applying their learning to solve complex, real-world problems related to the chapter Career. These tasks promote higher-order thinking and use of the SMPs to solve challenges occurring in real-world contexts. 

TEACHING EXPERIENCE

Big Ideas Learning recognizes the important role of teachers in bringing quality learning experiences to life in the classroom and is dedicated to empowering teachers with tools and resources to support learning in their unique classroom environments. The following features of the California Math & YOU Teaching Experience help empower teachers to build learning experiences steeped in the qualities described by the California Mathematics Framework. 

• Author’s Notes: The Teaching Experience includes notes at point of use that offer prompts, suggested questions, and examples of common student approaches. These guiding notes assist the teacher in facilitating discussion, encouraging reasoning, and supporting productive struggle. 
• Discourse Notes: While the Student Experience ensures students have plenty to talk about through engagement in meaningful learning experiences, the Teaching Experience provides support to empower the teacher to facilitate meaningful, collaborative discourse to build a shared understanding of mathematics. Talk About It, Math Talks, and Data Talks are found in the Teaching Experience and provide specific prompts to encourage students to explain and justify their reasoning, critique the reasoning of their classmates, and refine their strategies and shared understanding through rich mathematical discourse.
• SMP Support: California Math & YOU provides detailed support to help teachers enact instruction that keeps the SMPs at the center of how students engage with mathematics. Every lesson of California Math & YOU provides a Supporting the Mathematical Practices: Lesson Facilitation Guide which highlights opportunities to fully engage students in one or more SMPs with detailed notes and prompts to help teachers support students in their use of the Mathematical Practices. Each chapter Performance Task and Connecting Big Ideas task includes a Scoring Rubric to provide guidance for the teacher while assessing student work to the open-ended tasks. These rubrics offer clear, task-specific guidance to help teachers assess how well students are applying the SMPs. Teachers might choose to share rubrics with students in advance, providing support to encourage them to think about how they engage in solving the tasks through the lens of the SMPs. 

WHERE CURIOSITY MEETS CLARITY: BUILDING MEANINGFUL MATH EXPERIENCES WITH CALIFORNIA MATH & YOU

Together, the Student and Teaching Experiences offered in California Math & YOU provide meaningful tools for students and teachers to engage in a quality learning environment that supports students’ deep and connected understanding of the big ideas of mathematics. Read more about how you can leverage California Math & YOU in a Problem-Based Learning environment. 

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About the Author

Dr. Courtney Nagle joined Big Ideas Learning in 2023 as the Vice President of Teaching & Research. Dr. Nagle has nearly two decades of teaching experience, including both mathematics and mathematics education courses at the undergraduate level at Penn State Behrend in Erie, PA. Her teaching accolades include the 2022 Penn State University Eisenhower Award for Distinguished Teaching and the 2018 MAA Allegheny Mountain Section Award for Distinguished College or University Teaching of Mathematics.

Dr. Nagle holds a B.A. and M.A. in Mathematics, a Ph.D. in Mathematics Education, and her Pennsylvania Secondary Mathematics Teaching certificate. In addition to teaching, Dr. Nagle has established a strong record of research, serving as PI or Co-PI on four NSF grants totaling nearly $2.5 million and publishing her work in journals such as Educational Studies in Mathematics, The Journal of Mathematical Behavior, The International Journal of Science and Mathematics Education, and The Mathematics Teacher.

Dr. Nagle is passionate about helping students develop their confidence in mathematics while building a deep conceptual understanding of the mathematics they study. She works with preservice and inservice teachers on a variety of professional development topics including rich mathematical tasks, NCTM's 8 Effective Teaching practices, and building conceptual understanding.

About California Math & YOU 

California Math & YOU is a comprehensive, pedagogically rich K-12 mathematics program that builds a strong conceptual foundation for students through an immersive digital experience. Written by renowned author, Ron Larson, and his expert authorship team, California Math & YOU features engaging and relevant material that sparks students' curiosity and learning, while empowering teachers to successfully meet the needs of all learners. California Math & YOU is built on the California Mathematics Framework and is completely aligned to the California Common Core Standards (CA CCSS).

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