These Standards are not intended to be new names for old ways of doing business. They are a call to take the next step. … It is time to recognize that standards are not just promises to our children, but promises we intend to keep. CCSS-M, p. 5
The Core Curriculum State Standards in Mathematics (CCSS-M) emphasize the integrity of mathematics as a discipline—a language, a collection of related, progressively inter-dependent concepts, and a system of procedures and tools to learn more mathematics and to be able to solve problems.
The CCSS-M demand a shift in our teaching and learning mindset so that our students are fully prepared for higher education and for the world of work. The aim is that the students are college and career ready. The content of mathematics, as envisioned in CCSS-M, if carefully taught and well learned, provides sound preparation both for the world of work and for advanced study in mathematically based fields ranging from natural and physical sciences to social sciences.
Mathematics is the study of patterns in quantitative, spatial, and ideas of change. In other words, at each grade level, the standards statements are meant to be mathematically sound, and the progression from topic to topic is logical, systematic, and coherent. In the absence of a viable “national curriculum”, our schools have followed a de facto curriculum dictated by program choices (textbook series), standardized and state tests, and placement examinations (e.g., SAT, ACT, AP courses, etc.). In contrast, the nations where children achieve higher in mathematics have national curricula that identify (a) nonnegotiable skills at each grade level, (b) common definitions of knowing across grade levels for key developmental milestones, and (c) best practices for instruction.
The CCSS-M framers have tried to meet these three criteria for mathematics curriculum and instruction. The CCSS-M fulfill the first two conditions—the focus of the content and standards of knowing and the SMPs provide standards for instruction. The mathematical practice standards of the Common Core do not introduce new knowledge to be learned but the mathematical actions used by mathematicians and that are needed for successful work and living in the new technological age. These include such important actions as problem solving, making sense of mathematics, persevering, looking for patterns and structure of mathematics, reasoning and communicating mathematics in different ways.
The CCSS-M address multiple limitations in the de facto national curriculum. I want to take an example from each level (elementary, middle, and high school) to illustrate the difference between de facto mathematics teaching and instruction in the context of CCSS-M.
Elementary School Example:
Addition and subtraction facts, such as 8 + 6 or 17 – 9, in the early grades are, generally, derived by “counting up” and “counting down” using concrete objects, number line, hash marks, or fingers. And, sometimes they are just memorized by rote by using flash cards. CCSS-M, on the other hand, emphasizes strategies and instructional materials using decomposition/ recomposition and properties of numbers and operations. E.g., 8 + 6 = 8 + 2 + 4 = 10 + 4 = 14; 8 + 6 = 4 + 4 + 6 = 4 + 10 = 14; 8 + 6 = 2 + 6 + 6 = 2 + 12 = 14; 8 + 6 = 8 + 8 – 2 = 16 – 2 = 14; or, 8 + 6 = 7 + 1 + 6 = 7 + 7 = 14. Similarly, children should see the problem: 17 – 9 as the answer to questions as:
- 17 is how much more than 9?
- 9 is how much less than 17?
- What is difference between 17 and 9?
- What should be added to 9 to get 17?
- What should be subtracted from 17 to get 9?
- What is left when we take away 9 from 17?
Each of these statements gives a strategy to arrive at the answer using decomposition/recomposition.
- 9 + 1 + 7
- 17 – 10 + 1
- 7 + 10 – 9 = 7 + 1 = 8; 17 – 9 = 8 + 2 + 7 – 9 = 8;
When children arrive at answers using effective and efficient strategies, they can extend to higher numbers. Strategies provide understanding and then with a little practice, they result in fluency and applicability.
Middle and High School Example:
There is a general misunderstanding about something as basic as what it means to solve an equation—most textbooks and teachers teach it as a collection of sequential steps; solving an equation is presented as an isolated procedure without any purpose and reasons for each step employed in the procedure. The CCSS-M, however, emphasize that students “understand solving equations as a process of reasoning” and as a conceptual tool and define what needs to be taught about this process (see Standard A-REI 1, p.65 in High School Algebra CCSS-M). Students should know that equations are tools for solving problems in mathematics, sciences, and even social science. An equation is developed by the conditions of the problem or brought to bear to model the problem. The following illustrates the point.
Solve the equation for x: 3(x − 4) + 12 = 27
Before we give the standard procedures to students to solve this equation, they need to understand how equations emerge (what is an equation?) and the reasons behind each step in the procedure as the procedure is derived. They need to know that this equation has been obtained by performing a series of transformations to an unknown x.
The number of bacteria was reduced by 4 million in the first hour, but tripled in the next hour. It was augmented by combining a culture with 12 million, the result was 27 million strong.
This unknown could represent a physical state, condition or parameter. Here x represents the number of bacteria initially in a dish.
We begin with the variable x and observe the role of transformations on the variable
The student’s goal is to find what value of the unknown “x” is before it was transformed into this equation. By representing the transformations by this flow diagram, one can see how to arrive at the standard procedure for solving linear equations—the transformations are reversed and “x” is obtained back.
A solution is now derived and organized in the standard procedural form as we observe the flow diagram and organize the procedural steps. The formal procedural steps for solving the equation can easily be derived by following the flow diagram.
3(x − 4) + 12−12 = 27−12 (applying subtraction property of equality)
3(x − 4) + 0 = 15 (12 −12 = 0, inverse property of addition)
3(x − 4) = 15 (zero property of additive identity)
3(x − 4) ÷ 3 = 15 ÷ 3 (division property of equality)
1(x − 4) = 5 (property of multiplicative inverse)
x − 4 =5 (property of multiplicative identity)
x − 4 + 4 = 5 + 4 (additive property of equality)
x + 0 = 9 (property of multiplicative inverse)
x = 9 (property of additive identity).
The initial amount of the bacteria was 9 million strong.
Students do not have to solve every equation by providing these reasons nor the flow diagram, but they should know where the standard procedure comes from. They should have the ability to demonstrate and communicate the reasoning behind each step in the procedure. On closer examination, one finds that the procedural steps in solving an equation are in reverse of applying the order of arithmetic operations: Grouping (both transparent and hidden), Exponentiation, Multiplication, Division, Addition, and Subtraction (GEMDAS, with multiplication and division, and then addition, and subtraction being applied in order of their appearance in the expression). When they have practiced it and are able to demonstrate this reasoning, they can then follow and focus on the procedure and apply it to solving real problems.
Once the procedure for solving equations has been arrived at with understanding, students should practice it to achieve fluency and competence in applying it.
Similarly, irrational numbers are taught in middle school and high school as a collection of arithmetic procedures (e.g., simplifying radical numbers), and students miss the importance of the completeness of the real number system. Most students do not even think of or know the difference between rational and irrational numbers, except in their appearance and difficulty. For example, many textbooks and teacher instructions ask:
Students just apply the procedure to find the prime factorization using the “Factor Tree” method and their solution is:
It is posed as a simple calculation problem and the focus is only on applying a procedure, just to simplify the radical. There is little discussion about whether this is an irrational number or not. Why is this an irrational number? What is the difference between an irrational and rational number? Can we construct irrational numbers? Can we locate them on the number line? Are there more irrational numbers than rational numbers? If it is an irrational number, where is the location of this number on the number line? How do we locate it? Why is the number line called the real number line? There should be a discussion of the “richness” and “completeness” of the number line as a result of these new numbers.
As another example, when our previous state standards asked that the concept of congruence be taught in middle school, students learned that congruence means the same size and same shape. It is only an intuitive description of the relationship of congruence—a starting point. By contrast, the CCSS-M explains that the relation of congruence should be understood as the outcome of a sequence of transformations. For example, when rotations, reflections, and translations (grade 8, Standard 8.G 2, pp 55-56, and High School Geometry GCO, p. 57 and 76 CCSS-M) are applied to an object and if we reverse those transformation, we will get the original shape, then the object and its new image are congruent to each other. Therefore, the intuitive starting point should be followed by constructions using rigid transformations. As a result of these constructions, they should arrive at conclusions as conjectures such as that all congruence relationships are outcomes of a string of transformations. These constructions should then be extended to define congruence formally. It is a relationship between objects that satisfy certain properties (reflexive, symmetric, and transitive). They should then formally prove or disapprove the conjectures derived from constructions by logical reasoning. They should understand the role of transformations not only in geometry but also in algebra and coordinate geometry. Students should understand that the congruence relationship plays the same role in the collection of spatial objects and in spatial reasoning as the relationship of equality (it is reflexive, symmetric, and transitive) plays in quantities and in quantitative reasoning. And they should see that the process of measurement connects spatial and quantitative objects and reasonings.
Students should also see that certain other transformations (stretching, shrinking, scaling, etc.) do not preserve congruence of shapes, figures, and diagrams but instead define a range of other results relating to similarity. Similarly, they should know not only the relationship between the two-dimensional representation (nets) and the corresponding three-dimensional objects but also how 3-dimensional shapes are related to 2-dimensional objects. For example, a cylinder is a 3-dimensional object whose every section perpendicular to its axis are concurrent, concentric circles, and a cube is a 3-dimensional object whose every section perpendicular to any axis are congruent, concentric squares. They should know how a 3-dimensional object is derived from or related to 2-dimensional objects.
In learning mathematics ideas, intuitive arguments, metaphors, and analogies are good starting points but not sufficient for the development of comprehensive conceptual schemas. For example, the concept of fractions begins as (a) part-to-whole, as an intuitive and concrete concept, but students should also see fractions as (b) comparison of quantities (e.g., ratio and realize that although every fraction is a ratio, but not every ratio is a fraction), (c) comparison of a quantity with a standard (e.g., a decimal number and percent), (d) comparison of comparisons (e.g., proportion), and finally, (e) the idea that some fractions lead to the idea of rational numbers—an expansion of the set of integers, whereas some other fractions extend the number system to more comprehensive. Ultimately, for a true understanding of fractions and rational numbers, they should realize that not all fractions are rational numbers (e.g., are fractions, but not rational numbers) and every rational number is a ratio of two integers (a ratio, , is called a rational number, where a and b are integers, b ≠ 0, and a and b are relatively prime). And then the teacher should relate rational numbers and fractions into their decimal representations (rational: terminating decimals, repeating, not-terminating decimals, and irrational: non-repeating, non-terminating decimals).
Similarly, teaching operations on decimal numbers by appealing to the analogy with whole numbers (when multiplying decimal numbers, just multiply them as whole numbers and then count the number of digits after the decimal point in the multiplicands and place the decimal at the appropriate place in the product—same number of digits after the decimal point) does not give students the ability to (a) estimate the outcome of operations on decimal numbers and fractions, (b) apply the concept of decimals to problem solving, (c) the role of decimal numbers as representations for rational numbers as either terminating (.457 or .444) or repeated non-terminating (,45454545…) and, (d) represent irrational numbers as non-repeating, non-terminating decimals (.23223222322223…). Without this kind of rigor, students learn all of these as isolated ideas and a collection of mindless procedures.
The objective of CCSS-M is to have a focus on a body of meaningful mathematics at each grade level, to be taught with rigor by integrating the mathematics language, concepts, and procedures in teaching, and the teacher has a perspective on mathematics at several grade levels in order to provide coherence to the curriculum and instruction. In this process, teachers help students also to appreciate the structure and nature of mathematics with focus, coherence, and rigor that helps them to see the depth and breadth of mathematics.