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# MATHEMATICS—MRS. JESSICA MANGINI

Common Core mathematics is a way to approach teaching so that students develop a mathematical mindset and see math in the world around them. We are making problem-solvers and the eight mathematical practice standards support students in understanding and solving complex problems.
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 Standard 5- Use appropriate tools strategically.
Mathematically proficient students consider the available tools when solving a mathematical problem. These tools might include pencil and paper, concrete models, a ruler, a protractor, a calculator, a spreadsheet….
Proficient students are sufficiently familiar with tools appropriate for their grade or course to make sound decisions about when each of these tools might be helpful, recognizing both the insight to be gained and their limitations. —CCSS

Classroom Observations
Teachers are always developing students’ capacity to “use appropriate tools strategically” by making it clear to students why the use of manipulatives and other tools will aid their problem solving processes.
Counters, base-10 blocks, Pattern Blocks, measuring tapes or spoons or cups, and other physical devices are all, if used strategically, of great potential value in the elementary school classroom. They are the “obvious” tools. However, this standard also includes “pencil and paper” as a tool, to include such tools as diagrams, two-way tables, graphs, etc. The number line and area model of multiplication are two more tools—both diagrammatic representations of mathematical structure—that the CCSS Content Standards explicitly require.
So, in the context of elementary mathematics, “use appropriate tools strategically” must be interpreted broadly to include many choice options for students.
Essential, in this practice standard, is the call for students to develop the ability “to make sound decisions about when each of these tools might be helpful, recognizing both the insight to be gained and their limitations.” This requires that students gain sufficient competence with the tools; that their learning include opportunities to decide for themselves which tool serves them best; and that students develop a competency that would allow them to make “sound decisions” about which tool to use.

The number line is sometimes regarded just as a visual aid for children. It is, in fact, a sophisticated image used by all mathematicians. For young children, it helps develop early mental images of addition and subtraction that connect arithmetic with measurement. Rulers are just number lines built to spec! On a number line, students can see “the distance from 1 to 6” or “how much greater 6 is than 1.”

Children who see subtraction that way can use this model to see “the distance between 79 and 128” as 49, and to do so without crossing out digits and borrowing and following a rule they may only barely understand. In fact, many can learn to see this model in their heads, too, and do this subtraction mentally.  This is essentially how clerks used to “count up” to make change.
The number line model also extends naturally to decimals and fractions by “zooming in” to get a more detailed view of that line between the whole numbers. It thereby unifies arithmetic, making sense of what is otherwise often seen as a collection of independent and hard-to-remember rules. The number line remains useful as students study data, graphing, and algebra: two number lines, at right angles to
each other, label the addresses of points on the coordinate plane.

These versatile tools build mental models that last. What makes a tool like the number line truly powerful is that it is not just a special-purpose trick or temporary crutch, but is faithful to the mathematics and is extensible and applicable to many domains. These tools help students make sense of the mathematics; that’s why they last. And that is also why the CCSS mandates them.
When helping your child complete homework here are some great questions you ask them to tap into this mathematical practice:

• What tool did you use to help you solve?
• Why did you use that tool?
• How does it work?
• Did you get the results you were expecting? Do the results make sense?
• Is there another tool you could use?

Tools are meant to make sense of mathematics and the world around us. They are meant to improve efficiency and support accuracy. They are part of modern life – as is our responsibility to understand them, use them strategically, and develop those proficiencies in all students.
I hope to see you all at Math Night on February 9th where we will explore these practice standards together!

Tiger Times February 2016