Standard 8: Look for and express regularity in repeated reasoning

The Standard:
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.

Classroom Observations:
Integrating Standard Eight into classroom practice is not only a matter of planning for lessons that occasion students to look for general methods and shortcuts. It also requires teachers to attend to and listen closely to their students’ noticings and “a-ha moments,” and to follow those a-ha moments so that they generalize to the classroom as a whole. The video clips included here are intended to show as a composite how teachers create the conditions for students to look for and express regularity in repeated reasoning, and follow and elaborate students’ thinking when they begin to make these connections.

Mathematically proficient students notice if calculations are repeated, and look both for general methods and for shortcuts... They continually evaluate the reasonableness of their intermediate results.

Stephanie Letson works with her 2nd grade students in a daily number talk routine called “Number of the Day.” In this number talk, Letson encourages her students to find multiple ways to arrive at the total of 170. In this clip, she engages the class in a discussion of one student’s approach, noting that several students have tried to extend and elaborate their equations by multiplying by 1 or adding 0. One student observes that these are functionally equivalent, leading to a group realization about the meaning of the identity property. Letson connects this student’s observation to work she had previously noted in her mathematics journal.

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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.

In Katie Arrillaga’s 2nd grade Spanish bilingual classroom, she engages her students in a number talk, discussing whether or not their approaches to solving a two-digit addition problem (59 + 37) are reasonable. In this clip, her student Cesar begins with sharing one strategy for solving and addressing the challenge of how to use the 16 that results from adding 9 and 7. He begins, catches himself in the “intermediate results” stage, and changes his approach to use the “Australian Method” that the class had used elsewhere (and which is explained more fully in Arrillaga’s debrief.)

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See this video in the context of an entire Number Talk.

Mathematically proficient students notice if calculations are repeated, and look both for general methods and for shortcuts... They continually evaluate the reasonableness of their intermediate results.

In Elysha Pesseggi’s 4th grade classroom, students work on a number talk asking them to evaluate the reasonableness of a two-digit problem and solution—to “see if they might be true or false without thinking.” As her students discuss their rationales for determining why 62 + 78 is or is not equal to 238, various students offer responses that the numbers are “too small” or “would have to be bigger.” Early on, a student observes that “to get 200 you’d need at least 100 and 100.” Later in the discussion, a student advances this idea, stating that “two 2-digit numbers cannot…no matter what… if it’s addition…cannot be more than 198.” Other students then affirm that they agree with this student’s rule.

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Mathematically proficient students notice if calculations are repeated, and look both for general methods and for shortcuts... They continually evaluate the reasonableness of their intermediate results.

Passeggi works with her 4th grade students in a daily number talk routine called “Can This Be True?” In this number talk, Passeggi challenges her students to look at the relationships between numbers to determine whether or not a given answer can be correct, without calculating the result. In this clip, after initial discussion by the class, Passeggi asks the students to generate ideas for testing the reasonableness of their solutions with their "shoulder partners." One student then proposes that "if there's one number on one side, and another number on the other side, and this number is bigger, then this [other] number has to be bigger." Passeggi then challenges the class to test out "Maddy's way" to see if the strategy is effective. In so doing, the students generate and develop their understanding of general methods for strategic solving.

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See this video in the context of an entire Number Talk.

Mathematically proficient students notice if calculations are repeated, and look both for general methods and for shortcuts. Noticing the regularity…might lead them to the general formula…. 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.

In the closure section of a number talk, Fran Dickinson works with his 5th/6th grade students to identify many different ways of generating a rule to govern the inputs/outputs of a chart: various students offer x3 – 3, times 3 minus 3, 3x – 3. The students discuss the rule and the best way to represent it, making connections to their mathematics textbook in their conversations. They discuss the various ways of representing the rule and evaluate the reasonableness of the parameters of the problem. This clip is also indicative of standard 3 (construct viable arguments and critique the reasoning of others), standard 6 (attend to precision), and standard 7 (look for and make use of structure).

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Mathematically proficient students notice if calculations are repeated, and look both for general methods and for shortcuts... 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.

Jesse Ragent asks the students to do a "matching game." He passes out sets of tables and equations to the students, and challenges the students to "find a triple"-- an equation, a table, and a graph that all make up a "family, triple, or set." He asks students to consider distinctions, characteristics, attributes as they make their grouping decisions. Students jump right in and start the matching quite eagerly. They are justifying their matches in different ways. Some are using the math vocabulary from the prior whole class discussion, while others are just matching points from the graph with the t-chart, or checking to see if an equation will generate the points on the chart or graph. Either way, students are making connections among these different representations in ways that are meaningful to them. In the clips excerpted it seems that the students are listening to one another and letting the mathematics sway the argument rather than the force of a personality.

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See this video in the context of an entire lesson.

Mathematically proficient students notice if calculations are repeated, and look both for general methods and for shortcuts... 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.

Jesse Ragent asks the students to do a "matching game." He passes out sets of tables and equations to the students, and challenges the students to "find a triple"-- an equation, a table, and a graph that all make up a "family, triple, or set." He asks students to consider distinctions, characteristics, attributes as they make their grouping decisions. Students jump right in and start the matching quite eagerly. They are justifying their matches in different ways. Some are using the math vocabulary from the prior whole class discussion, while others are just matching points from the graph with the t-chart, or checking to see if an equation will generate the points on the chart or graph. Either way, students are making connections among these different representations in ways that are meaningful to them. In the clips excerpted it seems that the students are listening to one another and letting the mathematics sway the argument rather than the force of a personality.

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