The Common Core State Standards list Standards for Mathematical Practice at each grade level. These practices are

Mathematical Practices

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.

These same eight
expectations are listed in the descriptions for every grade level and for every
advanced (a.k.a. high school) course.
But carrying out the Mathematical Practices will look different at
different grade levels.

The practices that may require the most clarification are probably #4 (Model with mathematics), #7
(Look for and make use of structure) and #8 (Look for and express regularity in
repeated reasoning).

Modeling with
mathematics typically involves using and perhaps even creating mathematical
objects (such as algebraic expressions, equations, inequalities, graphs, etc.)
to capture key aspects of a situation to be explored. A kindergartner might use 2+3 to represent
the number of people involved if two people are joined by 3 more; a sixth
grader might describe a relationship
between the numbers of tables to chairs in a room by the ratio 1:4; an Algebra
I student might use the expression 10x
to represent the value (in cents) of x
dimes.

But not all word
problems involve mathematical modeling.
It is

*not*mathematical modeling to use a contrived algebraic expression such as a quadratic expression obtained by curve-fitting bi-variate data without any plausible a priori reason for believing that the two variables should be related quadratically.
Mathematical
Practice #7, seeking and using
structure, is key to both pure and applied mathematics. A first-grader begins to recognize that an
addition fact such as 2+3=5 carries with it a family of related arithmetic
facts, e.g., 3+2=5, 5-2=3, 5-3=2, etc. ; a seventh-grader can see that because
a+0.05a = 1.05a, increasing a quantity by 5% is equivalent to scaling the
quantity by 1.05; a geometry student recognizes and introduces structure by
adding an auxiliary line to a geometric diagram. Mathematical Practice #7 is definitely not
about memorizing or plugging into formulas--both practices, when applied inappropriately, can allow
students to ignore the underlying structure .

I personally need
further explanation of Mathematical Practice #8. Here is how it's first described in the CCSS:

*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 \( \frac {y – 2}{x – 1} = 3\) . Noticing the regularity in the way terms cancel when expanding (x – 1)(x + 1), (x – 1)(x^2 + x + 1), and (x – 1)(x^3 + x^2 + 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.*

The phrase
"repeated reasoning" presumably refers to the "R" in
Guershon Harel's DNR. Some of Harel's work is listed in the mathematics CCSS
references, but there do not appear to be any direct attributions cited.

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