President, Colorado Council of Teachers of Mathematics
Catherine Martin, CCTM President
As we conclude our focus on the Standards for Mathematical Practice, this message focuses on Math Practice 8: Look for and express regularity in repeated reasoning. According to Math Practice 8, mathematically proficient students:
- -Notice if calculations are repeated,
- -Look both for general methods and for shortcuts,
- -Maintain oversight of the process, while attending to the details, and
- -Continually evaluate the reasonableness of intermediate results,
This Practice supports students in developing their skills and abilities to abstract and generalize mathematical concepts as they strive to determine efficient methods for procedures.
Searching for patterns is at the core of Practice 8 and is embodied in many of the CCSS content standards that begin with “apply and extend.” For example, students “apply and extend” their previous understandings of: 1) multiplication–to multiply a fraction by a whole number (fourth grade); 2) multiplication and division with whole numbers–to multiply and divide fractions (fifth grade); 3) numbers–to the system of rational numbers and previous understandings of arithmetic to algebraic expressions (sixth grade); and 4) operations with fractions–to operations with rational numbers (seventh grade). In all of these, students are focusing on structure, to extend their previous understandings through repeated reasoning. Further this notion of apply and extend, even when not explicitly called out in the Common Core, supports students in applying and extending their understanding of addition and subtraction with whole numbers based on place value to adding and subtracting decimals, again based on place value. Another example of applying and extending addition and subtraction of whole numbers to decimals might also be extended to addition and subtraction of polynomials, with like terms being the analog of place value.
Important in students’ development of their expertise of Practice 8 is the opportunity to make sense of and look for regularity in calculations. A rush to provide students in earlier grades with standard algorithms for arithmetic computations, or students in upper grades with algorithms for algebraic computations, denies students their opportunity to make sense of repeated reasoning and to develop a deep conceptual understanding that will support their procedural understanding and fluency.
How, then, do we support students in developing this expertise? First and foremost, we must create classrooms with norms and expectations that allow students to notice patterns, to make conjectures about the patterns they notice, and then to justify the reasoning behind those conjectures. Teachers in these classrooms select tasks that provide students with rich opportunities to look for repeated reasoning and then to generalize this repeated reasoning. Teachers support students in generating rules from repeated reasoning—rather than teaching steps or rules—and pose such questions as:
- -What do you notice?
- -Why does your shortcut work?
- -How might you generalize your pattern?
- -How can your justify your generalization or conjecture?
Students’ engagement in tasks that develop their expertise in Math Practice 8 further support their development of expertise in other math practices. For example, they must make sense (Practice 1) of such tasks in order to search for repeated reasoning, often make use of structure (Practice 7) in searching for repeated reasoning, and will need to construct an argument (Practice 3) to justify their conjecture or generalization. Rich tasks, then, are at the heart of all of these standards and at the heart of classrooms where students are empowered to become mathematical thinkers and problem solvers.
As we conclude our study of the Standards for Mathematical Practice, I hope that you have deepened your understanding of them and have gained many insights into how you might support your students in developing their expertise with each. Consider rereading the previous CMT messages and articles that focused on each Practice as you reflect on your year with students AND anticipate your work next year.
The next President’s Message will begin a focus on the Mathematics Teaching Practices in Principles to Actions: Ensuring Mathematical Success for All (NCTM, 2014). If you don’t yet have a copy of this seminal book, I highly recommend that you purchase a print or eBook copy (http://www.nctm.org/store/Products/Principles-to-Actions--Ensuring-Mathematical-Success-for-All/). It will be great summer reading!