# MATHEMATICS—MRS. JESSICA MANGINI

The problems we encounter in the “real world”—our work life, family life, and personal health—don’t ask us what chapter we’ve just studied and don’t tell us which parts of our prior knowledge to recall and use. In fact, they rarely even tell us exactly what question we need to answer, and they almost never tell us where to begin. They just happen. To survive and succeed, we must figure out the right question to be asking, what relevant experience we have, what additional information we might need, and where to start. And we must have enough stamina to continue even when progress is hard, but enough flexibility to try alternative approaches when progress seems too hard.

The Mathematical Practice Standards teach students how to do this.

To refresh, here are the 8 Standards for Mathematical Practice:

1. Make sense of problems and persevere in solving them

2. Reason abstractly and quantitatively

3. Construct viable arguments and critique the reasoning of others

4. Model with mathematics

5. Use appropriate tools strategically

6. Attend to precision

7. Look for and make use of structure

8. Look for and express regularity in repeated reasoning

*This month I will highlight the 8th Mathematical Practice Standard- Look for and express regularity in repeated **reasoning*

The standard states:

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…. 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. —CCSS

WHAT DOES IT REALLY MEAN?

A central idea here is that mathematics is open to drawing general results (or at least good conjectures) from trying examples, looking for regularity, and describing the pattern both in what you have done and in the results. For example: the recognition that adding 9 can be simplified by treating it as adding 10 and subtracting 1 can be a discovery rather than a taught strategy. In one activity children start with 28 and respond as the teacher repeats only the words “ten more” (38), “ten more” (48), “ten more” (58), and so on.

They may even be counting, initially, to verify that they are actually adding 10, but they soon hear the pattern in their responses (because no other explanatory or instruction words are interfering) and express that discovery from their repeated reasoning by saying the 68, 78, 88 almost without even the request for “ten more.” When, at some point, the teacher changes and asks for “9 more,” even young students often see it as “almost ten more” and make the correction spontaneously. Describing the discovery then becomes a case of “expressing regularity” that was found through “repeated reasoning.” Students then find it very exciting to add 99 the same way, first by repeating the experience of getting used to a simple computation, adding 100, and then by coming up with their own adjustment to add 99.

Teachers ask:

Will the same strategy work in other situations?

Is this always true, sometimes true or never true?

How would you prove that…?

What do you notice about…?

What is happening in this situation?

What would happen if…?

Is there a mathematical rule for…?

What predictions or generalizations can this pattern support?

What mathematical consistencies do you notice?

Source: Institute for Advanced Study/Park City Mathematics Institute

May 2016 Tiger Times