Most of mathematics and mathematics activities at the school level are goal oriented, and every vocabulary term, concept, procedure, or skill should have a purpose and place in the curriculum and in the overall development of mathematical thinking. According to CCSS-M, there should be focus on particular aspects of mathematics at each grade. Focus requires that in order to learn mathematics meaningfully, we significantly narrow the scope of content, both in a lesson and at the grade level, particularly in the formative earlier grades in order to build a strong foundation for future mathematics learning.
Focus: The Importance of Non-Negotiable Skills
The focus of arithmetic teaching in the K—5 standards is to build an important life skill and to develop mathematical ways of thinking—to observe patterns, make conjectures, generalize, and abstract—and to have mastery of numeracy skills with understanding and processes that make higher mathematics accessible to all children. Mastery of the content at the elementary level (numeracy skills) should prepare them to be able to apply these skills and concepts to learn mathematics and solve problems not only in other parts of mathematics and other disciplines but also in real-world situations.
In essence, such a focus means mastering certain non-negotiable skills at each grade level.
Non-negotiable skills form the basis for other concepts, procedures and skills at that grade level and build the foundation for future mathematics learning. For example, in CCSS-M, learning the concept of “decomposition and recomposition” is identified at the Kindergarten level, but it should be used extensively in deriving the addition and subtraction arithmetic facts at the first and second grade levels. Similarly, multiplication is introduced in the second grade as repeated addition, equal groups of objects, and arrays. By the end of third grade, however, children should have mastered the concept as repeated addition, groups of, arrays, and the area of a rectangle so that they understand the distributive property of multiplication over addition and subtraction better and have automatized multiplication facts. At the same time, the multiplication as the area of a rectangle should be developed from repeated addition, groups of, and arrays. Such an understanding of multiplication prepares student to multiply fractions and decimals in fifth grade using the area model of multiplication and arriving at the standard algorithm of multiplying fractions. Then they can apply multiplication of fractions to master the standard operations of addition and subtraction on fractions with efficiency and understanding.
Using the same principles, they should automatize division facts by the end of fourth grade and division of fractions by sixth grade along with the mastery of operations on integers and operations of rational numbers in the seventh grade. Similarly, concepts of fractions, e.g., adding and subtracting fractions with same denominators are introduced in the third and fourth grades, and by fifth grade, children should have mastered the operations on fractions with deeper understanding. This is possible because the mastery of multiplication and division facts and the additive and multiplicative reasoning was developed as focus areas in the third and fourth grades.
In the early grades, the standards rightly concentrate on the mastery of arithmetic skills (number concept, numbersense, and numeracy) along with spatial orientation/space organization, and measurements relating quantitative and spatial reasoning. As we expect all children to read fluently with comprehension by the end of the third grade, we should expect all children to have mastery of numeracy by the end of fourth grade so that they can learn higher mathematics easily, effectively, and efficiently in later grades.
Numeracy means a child can execute the four whole number operations correctly, consistently, fluently, in standard/efficient forms with understanding. They may begin with idiosyncratic methods and explorations of number operations, but ultimately they should be fluent in standard algorithms that are the distillation of centuries of efforts by mathematicians from many cultures.
The objective of mathematics throughout the middle and high school grades is to deepen and broaden their mathematics understanding and competence and to prepare students for college and meaningful careers. As a result of many surveys, it is clear that postsecondary instructors value greater mastery of prerequisites skills for higher mathematics over shallow exposure to a wide array of topics with doubtful relevance to postsecondary work. At present, the shallow exposure in our curriculum to mathematics concepts is true for all students, including exceptional students at institutions with high expectations. For instance, according to the director of undergraduate studies in mathematics at Johns Hopkins University, high performing students “have the grades and the test scores to be there” but lack “a deep understanding of why the techniques they’ve been taught work, the actual underlying mathematical relationships. They walk into to my classroom in September and don’t have the study habits or proper foundation to do the work.”
Just like number concept, numbersense, and numeracy are expected to be mastered during the early elementary school and are used to learn future mathematics. Similarly, appropriate algebraic concepts and skills (generalization of arithmetic facts, concepts, and arithmetic operations and procedures on algebraic expressions, the idea of functions and operations on functions, and modeling problems by algebraic equations and solving them) are acquired and used during the middle and high school years. However, aspects of these concepts are acquired throughout the curriculum and not reserved only for the middle grades. They are introduced and expanded appropriately throughout the grades, but they are expected to be mastered by the end of eighth grade.
Mastery of non-negotiable skills at each grade helps maintain continuity of learning from grade to grade and provides for a seamless transition from arithmetic to algebra in the middle grades and preparation for higher mathematics in later grades. The mastery of the eighth grade algebraic and geometric concepts helps students to learn and master high school algebra and geometry comprehensively and provides access to higher mathematics such as calculus and discrete mathematics. Students leave high school with the ability of applying arithmetic, geometrical, algebraical, probabilistic, discrete and continuous models to problems in college and at work.
Mastery of non-negotiable skills also ensures that the next grade level teacher can begin the teaching of mathematics concepts at the grade level rather than endlessly review work from previous grades. Because of lack of mastery of non-negotiable skills (focus concepts and procedures), every grade has such a diversity of skill and mastery levels that teachers end up teaching several groups at different grade levels, thereby devoting limited time on grade level tasks. The reinforcement and practice of skills, going deeper, making connections, and providing special care for needy children (on both ends of the spectrum—from slow learners to gifted and talented) should be done in smaller groups or at the individual level after the main grade level concept has been presented to the whole class or taking meaningful but brief digressions to review previous concepts and skills and providing extensions.
Mastery of non-negotiable skills avoids the over-reliance on review of concepts and procedures in the beginning of every grade, causing many high achieving children to lose interest in mathematics. It sends students the message that what is learned can be forgotten by the end of each grade because it will be reviewed in the next grade anyway. The over emphasis on pre-tests and review also indicates that the claim the previous grade teachers made that “they taught the material and children learned it” is not true and sends teachers the message that their colleagues cannot be trusted. In any case, it indicates that the material was not learned in the first place.
In the process of developing these non-negotiable skills, teachers should keep in mind that concepts and procedures are not taught in isolation but need to be interwoven with linguistic, quantitative, algebraic, and geometric reasoning. Children should see the fundamental nature of that concept or operation. They should see the trajectory of the development of the concept—how does it begin and how does it transform over the years? Teachers should make these connections transparent. For example, taxonomy of actions involving quantitative reasoning and numeracy skills shows that operations on numbers can be matched with operations on objects that numbers describe:
- Addition models begin as putting “things” together, combining and joining collections and then become shifting and translating forward (a transformation),
- Subtraction models begin as take away from a collection, comparisons of collections and quantities and then become shifting and translating backwards (a transformation), or recovering an addend,
- Multiplication models begin as repeated addition, grouping, organizing and area of a rectangle and then become size change (stretching and shrinking) (a transformation), or use of a rate factor, and,
- Division models begin as repeated subtraction, grouping, organizing and then become ratio, rate, rate division, size change (a transformation), or recovering a factor.
Similarly, when students have learned operations on fractions, finding the perimeters, areas, and volumes should involve figures and shapes with measurements in terms of fractions and decimals. Formulas or memorizing interesting facts is fine in context but only after students have understood the concept. Memorizing formulas and facts has a very important place, just as learning vocabulary does— however, the presence of rich schemas for concepts together with automatized essential skills helps students to create richer, more nuanced work, and the possibility of applying it to problem solving with ease and flexibility.