Implications for Modeling and Reasoning from Common Core State Standards for Mathematics

The Council of Chief State School Officiers (CCSSO) and the National Governors Association Center for Best Practices (NGA Center) recently released the draft Common Core State Standards (CCSM) for English/language arts and mathematics for review and feedback. The website for accessing those standards is http://www.corestandards.org/. These standards define what students should know and be able to do from K – 12 with the hope that high school students graduate with the knowledge and skills needed to be productive in the 21st century.

Kentucky was one of 50 states who participated in the development along with national organizations, educators, and other content experts.

For mathematics, there are eight Standards for Mathematical Practice that provide a process of thinking mathematically about concepts and applications of those concepts. They include:

  1. Make sense of problems and persevere in solving them.
  2. Reason abstractly and quantitatively.
  3. Construct viable arguments and critiques 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.

In my review of the interface of these mathematical practice standards interfaced with the content of number, algebra, geometry and measurement, probability and data there are several things that occurred to me as I connected recent experiences as a consultant working with teachers in a hybrid algebra research project and mathematics/science statewide project. They include how we provide opportunities for students to model with mathematics for making sense of the problem solved and to reason quantitatively.

The number line is one of the best models to use in mathematics in that it allows for students to get an image in their head as to where the numbers are, how that number compares to other numbers, and understanding operation with numbers and variables. Additionally, movement on the number line supports the kinesthetic learner by allowing manipulations while solving the problem. The use of various applets assists in these investigations.

For modeling with mathematics, the use of the number line to model operations, solutions, and Real Numbers is a standard of practice from grades K – 12. No matter what grade, the writers of the Standards support the use of the number line to assist students in visualizing relationships (size, distance from zero, equivalence, inequality), identifying quantities, and operating with numbers and variables.

For showing multiple representations of rational numbers, the number line serves as a model for locating ½ and displaying the various forms of that fraction: ½ = .50 = 50% = two quarters, five dimes = rational number set model

Also, an illustration from http://www.green-planet-solar-energy.com/fractions.html provides another way to use the number line for conceptual understandings of our rational number system.
rational number linear model

For showing operations with rational numbers, adding whole numbers such as 5 + 4 and to be able to illustrate that the commutative property allows one to change the order of the numbers being added 4 + 5 and get the same answer.
rational number linear model 2
(NCTM, Illuminations)

For visualizing and modeling the multiplication of 4 X 3 where the student shows movement along the number line of four groups of three.
rational number linear model 3
(NCTM, Illuminations)

Investigations with the model of three groups of four, or 3 x 4 to determine if the answer is the same as 4 X3 is the logical extension? Is the operation commutative?
number line 1
(NCTM, Illuminations)

For multiplication of integers, such as 3 X (-2) one could show three groups of moving two units in a negative direction.

number line multiplication

Whereas the model for testing the commutative property for this same problem (-2) X 3 is different and requires students to understand the negative being opposite of. For this example the model is the opposite of two groups of three units to the right.
number line multiplication 2

The question remains, if the number line provides a visual model for students to display and operate with numbers and variables, why do we not see a number line in every classroom where it is prominently displayed, used, and referred to at all time?

Our next discussion will focus on expanding the use of the number line to help students create a visual schema for solving equations and inequalities.

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