Procedural Skills and CCSSM
The NCTM standards documents and CCSSM all call for students’ development of fluency in the basic facts. In addition to the expectation that students will recall the facts fast and accurately, Kling (2011) clarifies that the standards also expect the development of number sense in the learning of
FIGURE 2.1 Content Overlap across Grade Levels
FIGURE 2.2B Fluency Recommendations: High School
FIGURE 2.2A Fluency Recommendations: K-8
Source: Adapted from http://www.louisianabelieves.com/docs/common-core-state-standards-resources/guide-math-ccss- self-learning-module-3.pdf?sfvrsn=4.
the basic skills: “Fluent students use the facts they have memorized in flexible, mathematically rich, and efficient ways to derive facts they do not know” (82). Figures 2.2a and 2.2b show the fluency expectations for grades K-8 and high school as summarized by the Louisiana Department of Education.
Standards for Mathematical Practices
The Standards for Mathematical Practices (SMP) describe the ways that mathematically proficient students approach and think through problems and are exactly the same through grades K-12. What differs are the tasks used to increase the depth of understanding as students master new and more advanced mathematical ideas. A close look at the SMP show that they are founded on the NCTM process standards of problem solving, reasoning and proof, communication, representation, connections—as well as on the strands for mathematical proficiency from National Research Council’s report (2001), Adding It Up—adaptive reasoning, strategic competence, conceptual understanding, procedural fluency, and productive disposition. The following details include the connection to the NCTM process standards in parentheses. The practices recommend that students be provided tasks that help them to:
- 1. Make sense of problems and persevere in solving them mathematically (NCTM: Problem Solving). “Mathematically proficient students start by explaining to themselves the meaning of a problem and looking for entry points to its solution. They analyze givens, constraints, relationships, and goals . . . make conjectures about the form and meaning of the solution, and plan a solution pathway rather than simply jumping into a solution attempt”—CCSSM
- 2. Reason abstractly and quantitatively (NCTM: Reasoning and Proof). “Mathematically proficient students make sense of quantities and their relationships in problem situations”—CCSSM
- 3. Construct viable arguments and critique the reasoning of others (NCTM: Reasoning and Proof; Communication). “Mathematically proficient students . . . justify their conclusions, communicate them to others, and respond to the arguments of others”—CCSSM
- 4. Model with mathematics (NCTM: Representation). “Mathematically proficient students can apply the mathematics they know to solve problems arising in everyday life”—CCSSM
- 5. Use appropriate tools strategically (NCTM: Connections; Representation). “Mathematically proficient students consider the available tools when solving a mathematical problem”—CCSSM
- 6. Attend to precision (NCTM: Communication). “Mathematically proficient students try to communicate precisely to others” —CCSSM
- 7. Look for and make use of structure (NCTM: Representation; Reasoning and Proof; Communication). “Mathematically proficient students look closely to discern a pattern or structure”—CCSSM
- 8. Look for and express regularity in repeated reasoning (NCTM: Reasoning and Proof). “Mathematically proficient students notice if calculations are repeated, and look both for general methods and for shortcuts”—CCSSM
Mathematical practices 1-3 and 5 are about students developing the ability to engage in problems that are new to them and for which they have no immediate algorithm. This requires that they have a positive disposition toward mathematics and are willing to persist toward a solution by trying different strategies or conceptual pathways. Persistent students are willing to engage in challenging tasks because they accept false starts and struggles as a by-product of learning; take time to reflect on their process so that they can redirect or justify their thinking; and attend to precision in their computations and in communicating their understanding of the problem in writing and verbally.
Mathematical practices 4 and 5 on modeling and using tools strategically focus on students’ ability to use tools to model situations in the world and mathematics. Students are able to use appropriate tools that include diagrams, tables, graphs, manipulatives, and estimation. To use them strategically, students must have access to several tools, and the students, not the teacher, must decide which to use for a given problem. Bill McCallum, one of the lead writers of CCSSM, created a structural diagram (see Figure 2.3) to cluster the SMP It shows how SMP 1 and 6 are overarching practices if mathematics is to be taught within a problem-solving framework. Note that it is unreasonable to expect that all of SMP be incorporated in a lesson.
In his article, Debunking Myths about the Standards for Mathematical Practice, Mateas (2016) discusses five myths about the SMP Myth 1: Every lesson must incorporate all eight SMP Myth 2: Students can engage in only one mathematical practice as they work on a task. Myth 3: The mathematics task alone determines which mathematical practices students will use. Myth 4: Only specialized tasks can be used to develop mathematical practice. Myth 5: Mathematical practice can be taught separately from mathematical content (93-96). Awareness that comes from debunking the myths is critical to effective implementation of the SMP
FIGURE 2.3 Organizational Structure of the Mathematical Practices, 1996
Source: From Bill McCallum, http://commoncoretools.me/wp-content/uploads/2011/03/practices.pdf.