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Proficiency-Based and Problem-Based Instruction: Mutually Supporting Moves or

Contradictions?

Nicole Rigelman, Portland State University, Oregon

Proficiency-based and problem-based teaching and learning are two instructional approaches that can appear to be at odds with one another in the mathematics classroom. For instance, while the instructional move of clarifying and sharing learning intentions with students (Black and Wiliam 1998) that is advocated in a proficiency-based approach may improve student achievement, the way a teacher employs this move may diminish the cognitive demand of the problem-based task, a key component of a problem-based approach. Students’ dispositions can also influence the implementation of proficiency-based and problem-based instruction. If a student’s focus is on meeting the criteria in a proficiency-based approach merely to address “What do I need to do to get an A?” and not on “How can I show you what I know?” it can be challenging to draw out mathematical thinking and problem solving—important components of a problem-based approach. This chapter suggests that teachers who use proficiency-based and problem-based learning as mutually supporting are able to engage students with problem-based tasks to support learn-ing, and that they are also able to use tools, such as rubrics, not only for grading but also for supporting teaching and student learning. Providing students with examples of a range of student work shows what quality work looks like and supports students in understanding clear, high expectations. When students use these tools to guide self- and peer-assessment prior to the teacher providing growth-focused feedback, the tools support students during learning as opposed to merely being used summatively. Thus students understand that assessment and learning are ongoing and the student is actively involved in both processes.

■ Background“Proficiency-based learning refers to a system of instruction, assessment, grading and reporting that is based on students demonstrating mastery of the knowledge and skills they are expected to learn as they progress through their education” (Great Schools Partnership 2014). The goal of proficiency-based teaching and learning is to ensure that students learn what they are expected to learn. Many educational stakeholders find proficiency-based teaching and learning attractive in that it can provide a more fine-grained analysis of a student’s strengths and areas for improvement. It can also provide the opportunity for more detailed feedback on a student’s work. Instead of receiving a holistic grade or score on a

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paper, communication can be about progress toward specific learning standards. In mathematics, the NCTM Standards (National Council of Teachers of Mathematics [NCTM] 1989, 2000) and Common Core State Standards (National Governors Association Center for Best Practices and Council of Chief State School Officers [NGA Center and CCSSO] 2010) inform learning stan-dards development in states, and ultimately districts and schools.

Problem-based learning refers to a student-centered approach to instruction and assessment where learning of content and cognitive skills occurs through problem solving (Hiebert et al. 1997; Kazemi and Stipek 2001; NCTM 2000; National Research Council 2001, 2005). Since the release of Agenda for Action (NCTM 1980), mathematics educators have been focused on increasing students’ opportunities to engage in problem solving with some agreement about what this can look like in the classroom (c.f., Brannan and Schaaf 1983; Bruni 1982; Kantowski 1980). Because problem-based learning is student centered, it is critical that students engage with “a question that cannot be immediately answered; [problem solving] requires some effort in making appropriate use of previously learned concepts and skills for its solution” (Bruni 1982, p. 10). Real problems typically can be approached in many ways. “Occasionally, the resulting investigations are nonproductive. Sometimes they are so productive that they lead to many different solutions or sug-gest more problems than they solve” (Brannan and Schaaf 1983, p. 43). Additionally, Pólya (1945) states that problem solvers “should solve problems, chose problems which are in his line, meditate upon their solution, and invent new problems” (p. 206). Similarly, the Mathematical Teaching Practices set forth in Principles to Actions (NCTM 2014) convey the importance of both estab-lishing clear goals that focus learning, as well as implementing tasks that promote reasoning and problem solving. I have found that these high-leverage practices are critical for teachers interested in combining proficiency-based and problem-based instruction in mutually supportive ways.

Table 13.1 summarizes key characteristics of proficiency-based and problem-based teaching and learning. A review of this list leads one to see ways in which the approaches are mutually sup-portive (e.g., student-centered learning, students engaged in self-monitoring and self-assessment). However, contradictions may also arise between proficiency-based and problem-based instruc-tion. Black and Wiliam (1998) call attention to several contradictions— places where the typical assessment practices in schools are weak and do not support high-quality teaching and learning. In mathematics classrooms, these practices include assessment tools that encourage rote learning rather than mathematical thinking and problem solving, an emphasis on quantity or presentation of written work rather than quality, and using tests to compare students to one another rather than personal improvement. My experience with proficiency-based learning is that school policies require teachers to articulate their learning targets to students, a strategy shown to positively impact student achievement (Black and Wiliam 1998). A contradiction arises when the timing of sharing of the target with students diminishes the cognitive demand of a task by communicating a solution strategy. For example, while “solve quadratics by completing the square” may be an appropriate target, if shared at the beginning of problem-based lesson, it would mean that students would not have to ponder an appropriate solution strategy for the given task. If students already know a solution path they are not problem solving; instead, they are completing exercises.

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Table 13.1. Key features of proficiency-based and problem-based teaching and learning

Proficiency-Based Teaching and Learning Problem-Based Teaching and Learning

• Students progress at their own pace with the teacher providing the necessary supports.

• Students progress based on their mastery of competencies.

• Students are active in the learning process. – reflecting on their performance. – taking responsibility for their learning• Instruction based on students’ needs.

• Students direct their own learning with teacher as facilitator.

• Students are active investigators. – organizing and accessing prior knowledge – acquiring new knowledge through investigation in

small group settings• Students self-monitor and seek meaning and

understanding.• Instruction centered on challenging open-ended

problems.

Both proficiency-based and problem-based approaches offer benefits to mathematics teach-ing and learning and should not necessarily be seen as contradictory. In the following sections I discuss ways to incorporate both approaches in classroom practices related to student learning and assessment.

■ Posing Worthwhile Tasks and QuestionsAnother assessment issue identified by Black and Wiliam (1998) relates to the types of tasks students engage with. According to Smith and Stein (1998), “The day-in and day-out cumulative effect of classroom-based tasks leads to the development of students’ implicit ideas about the nature of mathematics—about whether mathematics is something about which they can personally make sense and about how long and how hard they should have to work to do so” (p. 269). If students are interacting with rote tasks, they are unlikely to experience struggle with knowing what they should do, and they come to believe mathematics is a discipline where they memorize and apply procedures. Often, because it is easier to assess a single target with a single task, proficiency-based assessment is characterized by narrowly defined tasks or those tasks where students are expected to apply a partic-ular procedure. The implication of such an approach is that unproductive beliefs about mathematics and the students’ role as a mathematician develop (NCTM 2014).

Alternatively, with problem-based learning, teachers pose worthwhile tasks designed to encour-age students’ inquiry and invention. Interacting with such tasks using questioning and discourse draws out student thinking and reasoning. However, worthwhile tasks are more cognitively demand-ing than rote tasks and have been shown to be more difficult for teachers to implement well (Stein et al. 2009). In their work through the Silicon Valley Mathematics Initiative, Foster and Noyce (2004) chose worthwhile tasks that “were of sufficient scope to allow for deep student thinking, and that could be reasonably assessed on a performance exam that lasted just a single class period” (p. 2). The tasks are characterized as ones that “require students to evaluate, optimize, design, plan, model, transform, generalize, justify, interpret, represent, estimate, and calculate their solutions” (p. 4). During these open-ended tasks students are engaged in the habits of mind of a mathematician aligned with the Common Core Standards for Mathematical Practice (NGA Center and CCSSO 2010), and students’ successful performance with the tasks became a predictor of success on the

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state’s multiple-choice exam (Foster and Noyce 2004). This problem-based approach supports stu-dents with developing more productive beliefs and experiencing more success as a mathematician.

To shift students’ focus away from “What do I need to do to get an A?” and toward the perspec-tive of “How can I show you what I know?” teachers need to pose challenging activities that promote thinking and discussion. Figure 13.1 illustrates this point with a patterning task. This task provides a context to examine a visual representation, a tile pattern, and its associated table of values. This task was posed to third graders with one of two different prompts: (1) “How many squares will be in the 25th arrangement?” or (2) “Investigate and report all you can about this pattern.” The resulting student work demonstrates a contrast in the thinking elicited by each prompt. In figure 13.2, the stu-dent focuses on “answer getting” (Daro 2013), and the attention is on finding the number of squares, which reveals more about the student’s computational abilities and persistence than about algebraic thinking beyond pattern recognition. On the other hand, the work shown in figure 13.3 reveals a student’s more thorough investigation of the tile pattern with an extended table of values and a description of the characteristics of the tile pattern. In slowing down to investigate and make sense of the tile pattern and how it is growing, the student reveals more robust algebraic thinking than that shown in figure 13.2. The student’s response in figure 13.3 provides insight into where the student is seeing the constant and the change from one figure to the next as well as the number of tiles in sever-al larger figures through the table. Finally, the student expresses a verbal generalization, “Three plus 2 times the figure,” that can be used to find the total number of tiles in any figure.

Figure Number

Total Number ofSquares

1 2 3

5 7 9

Fig. 13.1: Patterning task

From a proficiency-based learning perspective, determining the learning targets associated with this later version of the task is more complex. With the first version of the task, determin-ing the targets prior to implementing the task is more straightforward, as a teacher lists “analyze a pattern represented with a visual model and table” and “find a value for a larger figure in the pattern.” With the second version of the task, anticipating what students might do with the task is necessary. Both targets listed previously are possible, but with the openness of the task a student may provide additional representations of the pattern (i.e., words, tables, or graphs), describe ways the pattern is staying the same and changing from one figure to the next, and make conjectures and generalizations about larger figures and/or any figure. A teacher may generate this initial list as potential targets while preparing to see evidence of additional targets when reviewing student work. These targets may even include aspects of cognitive skills such as those in the Standards for Mathematical Practice. A teacher may look for evidence of “make sense of problems and persevere

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in solving them” as the student explains correspondence among the various representations or “use appropriate tools strategically” as the student makes decisions about which representations of the task are most useful when investigating the pattern (NGA Center and CCSSO 2010).

Fig. 13.3. A third-grade student’s response to “Investigate and report all you can about this pattern.”

■ Establishing Norms for Doing MathematicsStudents hold differing views of what it means to know and do math. The two versions of the pat-terning task and the associated student work provide an example of the how the openness of a task influences students’ interaction with that task. To develop the mathematically proficient students described in the Standards for Mathematical Practice, it is helpful to consider what it looks like and/or sounds like to think and work like a mathematician. Once articulated, these habits of mind can be highlighted for students, thus helping to guide their work on open-ended tasks in mathe-matically productive ways.

Communicate Expectations about Student PerformanceIf teachers desire explanations that include mathematical arguments (such as in fig. 13.3) and not simply procedural summaries of the steps taken to solve the problem (such as in fig. 13.2),

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they need to support students in seeing the difference between the responses. Specific ways that teachers can establish these norms are described below.

For example, collectively examining exemplars for areas of strength and improvement, a common practice in a proficiency-based approach, can help students develop a critical eye for the difference between explanation and argumentation in their work (Orsmond, Merry, and Reiling 2002; Schoenfeld 1992; Wood 1999). Such a move provides students with a better understand-ing of the expectations, promoting a vision of what to strive for. The opportunity to examine the mathematical thinking of others helps students see relationships among multiple strategies which can deepen their understanding and improve their work.

Another proficiency-based strategy to support students with understanding of what it looks like to engage in doing mathematics is to provide a “rubric” or “scoring guide” that communicates clear and high expectations for exemplary work and encourages the student to bring richness to the task. Table 13.2 includes an example of such a scoring guide. The top portion of this table is a “study guide” that conveys to students the learning targets addressed in the exam. Students use this to focus their preparation. The bottom portion is the “scoring guide” that is used to score each item on the exam. The Comparing Quadrilaterals task (shown in fig. 13.4) is a task from an eighth-grade exam. Note that this task, like the second version of the patterning task, is an open task that encourages investigation. A difference is that the students, in this case, were provided with the “study guide,” the top portion of table 13.2, and were encouraged to use these ideas to prompt their thinking as they investigated this pair of quadrilaterals. This move positions the scoring guide as a teaching tool as it reinforces the expected detail for a top-level response. That is, students understand that they need to explore multiple strategies, justify their thinking, and generally show all that they know as they explore this task. A teacher reviewing the student work in figure 13.5 could consider both the learning targets addressed through the student’s response as well as the level of thinking revealed in the work. It communicates to students a concept articu-lated in a recent TED Talk by Messerman (2012), “Show me what you know and I will assess it, I will help you learn.”

Table 13.2Exam Study/Scoring Guide

Measurement Geometry

• Understand measurable attributes and the idea of a unit—observe and predict the effects of changing the unit on given measurement.

• Develop and use formulas to determine circumfer-ence of circles, and the area of triangles, rectangles, parallelograms, trapezoids, and circles.

• Develop strategies to find the area of more complex shapes.

• Identify, compare, classify, and analyze attributes of two-dimensional shapes.

• Make and test conjectures about geometric proper-ties and relationships, and justify conclusions.

• Identify, describe, and prove that objects are congruent and/or are symmetrical.

• Use terminology accurately and precisely.

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Exceptional• Reflects all the criteria of Quality and goes above and beyond in one or more of the

following ways…– Provides multiple examples and/or approaches to solving the problem.– Justifies by providing mathematical reasoning behind the process and solution.– Shows relationships to prior learning. – Poses and investigates “what if” questions and “I wonder” statements.

QualityShows understanding of the problem’s mathematical concepts and principles.

• Reflects an appropriate and systematic strategy for solving the problem that is complete and correct.

• Is easy to follow, giving a clear indication of the solution process through diagrams/sketches, words, and/or numbers and math symbols.

• Provides a complete response with clear, unambiguous explanations and/or descriptions.

In Development• Shows nearly complete understanding of the problem’s mathematical concepts and

principles.• May contain computational errors.• Provides clear evidence of the solution process, and the process is complete or nearly

complete and systematic. • Provides a fairly complete response with reasonably clear explanations and/or

descriptions.

Rethink• Misunderstood major portions of the problem.• Contains major mathematical errors.• Provides incomplete, ambiguous explanations and/or descriptions.

No Evidence• Major portions of the problem are partially complete or not attempted.

Fig. 13.4. Comparing Quadrilaterals task

Encourage Student Reflection and Goal SettingA teacher who provides students with tools to support productive reflection and self-assessment also contributes to the development of classroom norms and habits of mind for students (Orsmond, et al. 2002; Schoenfeld 1992; Wood 1999). Expectations, such as those communicated in the scoring guide example from table 13.2, could be regularized behaviors as students engage in classroom tasks, group activities, and discussions. It serves to encourage a student to interact with the goals, use them to motivate their performance, and evaluate their work before submission. Also, once students receive feedback about their performance, they may have an opportunity to revise their work or set goals for their future work. The intention would be to use student reflection and self-assessment along with feedback to encourage continued growth. Whether the teacher is reporting progress relative to learning targets that are content-focused, cognitive-skill-focused, or both the student is receiving feedback to grow on.

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Fig. 13.5. Student response to Comparing Quadrilaterals task

■ ConclusionThe importance of the selected tasks is specifically highlighted in this chapter as the task influences each of the other phases in the teaching and learning process. If the task is not an open task, it may not provide the opportunity for the students to show all that they know. It is critical to recognize the essential nature of this planning decision; mathematical tasks influence not only how students come to view mathematics but also what they have the opportunity to learn through the task. In addition, these tasks and the tools that teachers use to communicate expectations—such as student reflection tools, scoring criteria, and samples of exemplary work—further develop the norms for doing mathematics, but they should not reduce the cognitive demand of the task.

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While proficiency-based and problem-based instructional moves may have features that may appear contradictory to one another, there are also ways that the two approaches can support one another. Attention to clear learning goals is present in a proficiency-based environment. This focus can provide guidance for a teacher facilitating a problem-based lesson. Students actively reflecting on their performance relative to exemplars or scoring guides, supports continued growth in knowledge and habits of a productive problem solving. When teachers recognize the potential con-tradictions that exist between proficiency-based and problem-based approaches and purposefully plan to avoid such contradictions, they can support students with reconsidering their role in the classroom and move them toward the reality of engaging as the mathematically proficient students described in the Standards for Mathematical Practice.

ReferencesBlack, Paul and Dylan Wiliam. “Inside the Black Box: Raising Standards Through Classroom Assessment.” Phi

Delta Kappan 80, no. 2 (1998): 139–48.

Brannan, Richard and Oscar Schaaf. “An Instructional Approach to Problem Solving.” In The Agenda in Action. edited by Gwen Shufelt and James R. Smart, pp. 41–59. Reston, Va.: NCTM, 1983.

Bruni, James V. “Problem Solving for the Primary Grades.” Arithmetic Teacher 29, no. 6 (1982): 10–15.

Daro, Phil. Against Answer Getting. Strategic Education Research Partnership. (2013). http://serpmedia.org/daro-talks/.

Foster, David and Pendred Noyce. “The Mathematics Assessment Collaborative: Performance Testing to Improve Instruction.” Phi Delta Kappan 85, no. 5 (2004): 367–74.

Great Schools Partnership. Proficiency-Based Learning. The Glossary for Educational Reform. Accessed May 10, 2014. http://edglossary.org/proficiency-based-learning/.

Hiebert, James, Thomas P. Carpenter, Elizabeth Fennema, Karen C. Fuson, Diana Wearne, Hanlie Murray, Alwyn Olivier, and Piet Human. Making Sense: Teaching and Learning Mathematics with Understanding. Portsmouth, N.H.: Heinemann, 1997.

Kantowski, Mary Grace. “Some Thoughts on Teaching for Problem Solving.” In Problem Solving in School Mathematics, edited by Steven Krulik and Robert E. Reys, pp. 195–203. Reston, Va.: NCTM, 1980.

Kazemi, Elham and Deborah Stipek. “Promoting Conceptual Thinking in Four Upper Elementary Mathematics Classrooms.” The Elementary School Journal 102, no. 1 (2001): 59–80.

Messerman, Craig. Standards Based Grading and the Game of School. TEDxMCPSTeachers. (2012). https://www.youtube.com/watch?v=bn_sCLoQNVs&feature=youtu.be.

National Council of Teachers of Mathematics (NCTM). An Agenda for Action, Reston, Va.: NCTM, 1980.

________. Curriculum and Evaluation Standards for School Mathematics, Reston, Va.: NCTM, 1989.

________. Principles and Standards for School Mathematics. Reston, Va.: NCTM, 2000.

________. Principles to Actions: Ensuring Mathematics Success for All. Reston, Va.: NCTM, 2014.

National Governors Association Center for Best Practices and Council of Chief State School Officers (NGA Center and CCSSO). Common Core State Standards for Mathematics. Washington, D.C.: NGA Center and CCSSO, 2010.

National Research Council (NRC). Adding it Up: Helping Children Learn Mathematics, edited by Jeremy Kil-patrick, Jane Swafford, and Bradford Findell. Washington, D.C.: National Academies Press, 2001.

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________. How Students Learn: History, Mathematics, and Science in the Classroom. Committee on How People Learn, A Targeted Report for Teachers, edited by M. Suzanne Donovan and John D. Bransford, Division of Behavioral and Social Sciences and Education. Washington, D.C.: National Academies Press, 2005.

Orsmond, Paul, Stephen Merry, and Kevin Reiling. “The Use of Exemplars and Formative Feedback when Using Student Derived Marking Criteria in Peer and Self-assessment.” Assessment & Evaluation In Higher Education 27, no. 4 (2002): 309–23.

Pólya, George. How to Solve It. Princeton University Press, 1945.

Schoenfeld, Alan H. “Learning to Think Mathematically: Problem Solving, Metacognition, and Sense-Making in Mathematics,” In Handbook for Research on Mathematics Teaching and Learning, edited by Douglas A. Grouws, pp. 334–70. New York: Macmillan, 1992.

Smith, Margaret Schwan, and Mary Kay Stein. “Selecting and Creating Mathematical Tasks: From Research to Practice.” Mathematics Teaching in the Middle School 3, no. 5 (1998): 344–50.

Stein, Mary Kay, Margaret Schwan Smith, Marjorie A. Henningsen, and Edward A. Silver. Implementing Standards-Based Mathematics Instruction: A Casebook for Professional Development. 2nd ed. New York: Teachers College Press, 2009.

Wood, Terry. “Creating a Context for Argument in Mathematics Class.” Journal for Research in Mathematics Education 30, no. 2 (1999): 171–91.