Classroom Sneak Peek #8 - Look for and express regularity in repeated reasoning

The past couple of weeks I began blogging about the 8 Mathematical Practices from the Common core.  I have finished Mathematical Practice # 1, Mathematical Practice # 2, Mathematical Practice #3,  Mathematical Practice #4, Mathematical Practice #5, Mathematical Practice #6, and Mathematical #7.  This week the focus is on CCSS Mathematical Practice #8 - Look for and express regularity in repeated reasoning.  I'll address what this looks like in the classroom, what students will be doing, what teachers will be doing, and the most important, the type of questions teachers will be asking.

8. Look for and express regularity in repeated reasoning.

Mathematically proficient students notice if calculations are repeated, and look both for general methods and for shortcuts. Upper elementary students might notice when dividing 25 by 11 that they are repeating the same calculations over and over again, and conclude they have a repeating decimal. By paying attention to the calculation of slope as they repeatedly check whether points are on the line through (1, 2) with slope 3, middle school students might abstract the equation (y - 2)/(x - 1) = 3. Noticing the regularity in the way terms cancel when expanding (x - 1)(x + 1), (x - 1)(x² + x + 1), and (x - 1)(x³ + x2 + x + 1) might lead them to the general formula for the sum of a geometric series. As they work to solve a problem, mathematically proficient students maintain oversight of the process, while attending to the details. They continually evaluate the reasonableness of their intermediate results.

What does this really look like?  The chart below is a work in progress.  I've designed this with the expertise of many classroom teachers.  If you have other ideas, please don't hesitate to email me and share your expertise as well.  If you are interested in using this process with your staff, read What Do The Common Core Standards Look Like in the Math Classroom.

Mathematical Practice: Look for and express regularity in repeated reasoning.

Student Actions:	Teacher Actions:	Open-Ended Questions:
Engage in similar activities over many weeks through a constructivist approach. Several weeks dividing fractions from a conceptual stand point. Through many exposures of a concept, discover rules on their own without being told to memorize. Discover connections between the procedure and the concept.	Intentionally and purposefully help students make connections between mathematical concepts and procedures. Provide real world problems for students to discover rules and procedures through many exposures. Slow down to speed up. Purposefully design lessons for students to make connections. Many adding ten, followed by adding nine. What do you notice? Discovery lessons Allow time daily for students to discover the concepts behind the rules and procedures. Pose a variety of similar type problems.	Explain why that makes sense? How would you describe your method to us? How would you explain why it works? Does your answer seem reasonable? What have you learned about ....? How would this work with other numbers? Does it work all the time? How do you know? What do you notice when ...?