The Structure of the Common Core State Standards for Math (CCSSM)
|Overarching habits of the mind of a productive mathematical thinker|
|Make sense of problems and persevere in solving them|
|Attend to precision|
|REASONING AND EXPLAINING|
|Reason abstractly and quantitatively|
|Construct viable arguments and critique the reasoning of others|
|MODELING AND USING TOOLS|
|Model with mathematics|
|Use appropriate tools strategically|
|SEEING STRUCTURE AND GENERALIZING|
|Look for and make use of structure|
|Look for and express regularity in repeated reasoning|
The CCSSM call for mathematical practices (MP) and mathematical content (MC) to be connected as students engage in mathematical tasks. These connections are essential to support the development of students’ broader mathematical understanding—students who lack understanding of a topic may rely too heavily on procedures. The MP standards must be taught as carefully and practiced as intentionally as the Standards for Mathematical Content. Neither should be isolated from the other; effective mathematics instruction occurs when the two halves of the CA CCSSM come together as a powerful whole.
The higher mathematics standards specify the mathematics that all students should study in order to be college and career ready. In California, the CCSSM for higher mathematics are organized into both model courses and conceptual categories.
The model courses for higher mathematics are organized into two pathways: traditional and integrated. The traditional pathway consists of the higher mathematics standards organized along more traditional lines into Algebra I, Geometry, and other higher math courses, such as Advanced Placement Probability and Statistics and Calculus. An integrated pathway is an optional path that presents higher mathematics as a connected subject, in that each course contains standards from all six of the conceptual categories.
The standards for higher mathematics are also organized into conceptual categories:
- Number and Quantity
- Statistics and Probability
The conceptual categories portray a coherent view of higher mathematics based on the realization that students’ work on a broad topic, such as functions, crosses a number of traditional course boundaries.
Connecting the Standards for Mathematical Practice to the Standards for Mathematical Content
The Standards for Mathematical Practice describe ways in which developing student practitioners of the discipline of mathematics increasingly ought to engage with the subject matter as they grow in mathematical maturity and expertise throughout the elementary, middle and high school years. Designers of curricula, assessments, and professional development should all attend to the need to connect the mathematical practices to mathematical content in mathematics instruction.
The Standards for Mathematical Content are a balanced combination of procedure and understanding. Expectations that begin with the word “understand” are often especially good opportunities to connect the practices to the content. Students who lack understanding of a topic may rely on procedures too heavily. Without a flexible base from which to work, they may be less likely to consider analogous problems, represent problems coherently, justify conclusions, apply the mathematics to practical situations, use technology mindfully to work with the mathematics, explain the mathematics accurately to other students, step back for an overview, or deviate from a known procedure to find a shortcut. In short, a lack of understanding effectively prevents a student from engaging in the mathematical practices.
In this respect, those content standards which set an expectation of understanding are potential “points of intersection” between the Standards for Mathematical Content and the Standards for Mathematical Practice. These points of intersection are intended to be weighted toward central and generative concepts in the school mathematics curriculum that most merit the time, resources, innovative energies, and focus necessary to qualitatively improve the curriculum, instruction, assessment, professional development, and student achievement in mathematics.