The Mathematics Common Core State Standards for Mathematics (CCSSM) are not a dichotomy when it comes to content and practice. While the Standards for Mathematical Content are a balanced combination of procedure and understanding, the Standards for Mathematical Practice (MP) describe ways in which students should approach and engage in mathematics as they grow and progress through the elementary, middle, and high school years. Together they provide focus and depth and emphasize understanding of and connections among topics within and across grades on the pathway to achieving college and career readiness.
Prior to the release of the CCSSM, “mathematics practice” would equate to students doing practice problems demonstrating their knowledge and skills. But the MPs are not just about having knowledge and skills, but more about using knowledge and skills. They are comprised of processes and proficiencies, often referred to as habits of mind that are skills specific to mathematics. Providing every student the opportunity to learn mathematics promoting these habits of mind, enhances their likelihood of being successful in learning and doing mathematics.
The foundation of the MPs is based upon two important sources. They rest upon the National Council of Teachers of Mathematics process standardsi including problem solving, reasoning and proof, communication, representation, and connections. In addition, they are based upon the five strands of mathematical proficiency - adaptive reasoning, strategic competence, conceptual understanding, procedural fluency, and productive disposition – outlined in the National Research Council’s report Adding it Up.
The MPs have a structure Youtube:Video about them in that MPs 1 and 6 are the overarching habits of a productive mathematical thinker, while MPs 2 and 3 relate to reasoning and explaining, MPs 4 and 5 relate to modeling and using tools, and MPs 7 and 8 relate to seeing structure and generalizing.
Mathematical Thinking Overarching Habits of Mind
1. Make sense of problems and preserver in solving them.
6. Attend to precision.
Reasoning and Explaining
2. Reason abstractly and quantitatively.
3. Construct viable arguments and critique the reasoning of others.
Modeling and Using Tools
Seeing Structure and Generalizing
7. Look for and make use of structure.
8. Look for and express regularity in repeated reasoning.
Just as the content standards call for a focus on fewer topics at a deeper level, the MPs call for shifts in practice and instructional behavior in the classroom. Take for example the following problem:
“Sasha, Sanko, Sid, and Sheena were all scheduled to work together on a project but their teacher changed her mind and only two students could work on a project. To determine which two students would work together, all the names were placed in a basket and two names drawn out. What is the probability that Sasha and Sidney will work together on the project?” What is the probability of them working together if there were n students to choose from to work on the projects?
Traditionally, students would solve the problem using the formula for solving a dependent events problem that the teacher had shown them.
Probability (A and B) = Probability (A) x Probability (B given A)
x = probability of Sasha and Sid working together
In order for students to develop conceptual understanding of important mathematics content and become proficient in the MPs, they must be given the opportunity to be actively involved in learning and engage in mathematics in various ways as described in the eight practices. Rather than the teacher showing students a formula to use, a classroom in which the MPs are being implemented would include opportunities for students to determine the formula by modeling the problem as outlined in MP4 and looking for and making use of structure as outlined in MP7.
What student and teacher roles, behaviors, and actions would you see evidenced in a mathematics classroom where the MPs are being implemented? Even though content changes throughout the K – 12 mathematics pathway, the characteristics of any classroom where the MPs are being implemented would be the same in in terms of the following. The teacher sets the tone for the class by creating an environment where mathematical thinking and communication are supported. Students are given time to think and opportunities to communicate their thinking and ask questions without being judged for incorrect which can often prevent students from even attempting to respond to a question. The teacher conveys to students that it’s acceptable to struggle and make mistakes, but continually guides them in keeping focused on thinking and reasoning.
For more specific instructional and learning components aligned to the MPs, a Mathematical Practices Monitoring Form is available by clicking below. This form can be used by teachers for self-monitoring purposes, for peer monitoring by colleagues, or by a supervisor for evidence of the MPs. For classroom videos modeling the MPs visit Inside Mathematics.