CCSS-M: Non-negotiable Skills at the Middle School: Grades Seven and Eight

The goal of elementary school (K through 6) arithmetic is for students to master additive reasoning—the inverse relationship of addition and subtraction (achieved K through 2), multiplicative reasoning—the inverse relationship of multiplication and division (achieved 3 through 4), and proportional reasoning—a relationship between two quantities (achieved 5 through 6).

With additive reasoning we can compare only two quantities by their sizes (smaller than or greater than, or how much smaller or greater) whereas multiplicative reasoning helps students to compare (beyond relative sizes) two quantities in terms of each other, e.g., one quantity can be expressed in terms of the other, such as it is twice as much or is going to be three times larger than now. The transition from additive reasoning (a one dimensional linear concept) to multiplicative reasoning (a two-dimensional concept—represented by an array or the area of a rectangle) is cognitively an important milestone.

Multiplicative reasoning provides the ability to understand and apply proportional reasoning—ability to make comparisons and define relationships between objects and quantities in multiplicative form rather than additive form (5 through 6). It provides the ability to predict the state of the two quantities in a future state, if the relationship persists. Having multiplicative reasoning and extending it to proportional reasoning is a great achievement on the part of children and is the beginning of algebraic thinking in the true sense. The understanding and competence in proportional reasoning facilitate students’ transition from arithmetic to algebra.

Many topics in upper elementary and middle school mathematics and science curricula require proportional reasoning, including problem solving with fractions, decimals, percentages, scale, transformations of objects/shapes, probability, trigonometry, conversions of measurements, geometry of shapes, density, molarity, speed, acceleration, force, mapping, scale drawing, etc.

At the end of elementary school, children should have strong numeracy skills (number concept, numbersense, and the four operations of addition, subtraction, multiplication, and division on whole numbers), on one hand, and proportional reasoning, on the other.

The success in the concepts to be learnt during the upper elementary and middle school mathematics is dependent on a few important concepts: mastery of ten counting numbers (particularly decomposition/composition and sight facts); numbersense (arithmetic facts, particularly multiplication facts and place value); numeracy; divisibility rules; prime factorization; short-division; and concept and operations on fractions.

Middle School Years
The core of elementary and middle school mathematics features understanding of and operations of addition, subtraction, multiplication, and division on whole numbers, fractions, integers, rational numbers, and real numbers. With this understanding and competence, children can understand and master other related content easily and effectively.

The goal of mathematics education during the middle school years is to arrive at generalizations, extrapolations, and abstractions from arithmetic problems and express them through algebraic symbolism and then manipulate these abstractions/symbols. At the elementary school level, the focus and competence are on solving problems arising from specific contexts whereas, in algebra, the algebraic models are applied to solving classes and systems of problems rather than specific problems in particular contexts.

This generalization process begins with students realizing that algebra, initially, is generalized arithmetic and then extends to arithmetic of algebraic expressions, relations and functions.

In the middle school years, the objective is for students to make the transition from arithmetic to algebra. In other words, during the middle school years, the focus is on developing algebraic thinking and applying it. However, it should be remembered that algebra is not a collection of procedures or manipulation of symbols; it is making connections.

School mathematics and algebra have always had the goal of training students to manipulate numerical and algebraic symbols. The purpose of this manipulation is to solve problems not only by arithmetical models but also through algebraic models and systems of equations, inequalities, and representations—both algebraical and geometrical.

In CCSS-M, the framers insist that algebra is not to be introduced just as a collection of isolated procedures, and they assume that students have a reasonably well developed understanding of arithmetic principles and procedures (especially fluency in the execution of the four operations on whole numbers and integers, and the understanding that every arithmetic operation has its inverse operation, etc.).

The framers of CCSS-M advocate that algebraic concepts and operations should be introduced by developing arithmetic patterns and then converting them to the algebraic generalizations to ensure that students recognize why algebra is an important area of mathematics to learn. If the students do not see this early on, it is much harder to get them interested later. It means students clearly understand the concept of rational numbers and, in the seventh grade, extend the arithmetic operations on integers and fractions to operations on the rational number system.

In eighth grade, students should extend their understanding of the rational number system to the real number system and understand and master the arithmetic operations on the real number system. They should understand the nature of the “real number” line. They should see the applications of algebraic models in a variety of quantitative and spatial situations using real numbers.

They should generalize quantitative reasoning to algebraic reasoning—understand how to operate on algebraic expressions, equations, and inequalities; express relationships through functions, which provides the ability and tools to understand relationships between systems; know operations on functions; expand the intuitive and concrete spatial relationships (such as transformations) to formal geometrical truths and relationships; integrate quantitative and spatial reasoning and relationships in problem situations; and solve problems through arithmetic, algebraic, geometrical, statistical, and probabilistic models. More specifically, the content in these grades are:

Seventh Grade
The focus of seventh grade is to extend the number system to include rational numbers. Students should understand and master the concepts of integers and rational numbers, their relationship with other numbers and mastery of operations on them. It means:
(a) understanding the difference between numeral (representation of quantity, e.g., I took three steps is represented as the numeral 3) and number (a directed numeral, e.g., I took three steps forward is represented as +3 and I took three steps backward is represented by –3) and operations on them;

(b) understanding the definition—a number that can be written as the ratio Capture blog 6.02, where a and b are integers, b ≠ 0, and a and b are relatively prime—with 1 as the greatest common factor; representing them as decimals—a rational number can be represented by (i) terminating decimal (.459), or (ii) repeating, non-terminating decimal Capture blog 6.03; and using rational numbers in multiple contexts—quantitative and spatial reasoning;
(c) understanding and mastery of operations on rational numbers, for example, it means that: operations (such as multiplication) are extended from fractions to rational numbers by requiring that they continue to satisfy their properties, particularly the distributive property, leading to products such as (−1)(−1) = +1;
(d) understanding that the rules for operations on signed numbers are well-founded; for example, by emphasizing the properties of operations, students are able to extend the situations such as: (−m)(−n) = mn for any integer m and n to the product of rational numbers. They see them as a general case rather than a special one—indicating entry into algebraic thinking in the formal sense;
(e) interpreting the operations (e.g., products of) of rational numbers by describing real world contexts.

The second focus of seventh grade mathematics is acquiring facility in working with and applying rational numbers in multiple contexts such as extending the concept of proportional reasoning and mastering it. It should include concepts such as: proportion, unit price, scale factor (stretching and shrinking), slope, conversions, etc.
(a) from extending the understanding of and applying proportional reasoning and relationships to learning new mathematical concepts and solving real world problems
(b) deeper understanding of and operations on rational numbers involving expressions, linear equations and inequalities;
(c) solving problems involving similarity, scale drawings, rate of change, slope, and informal geometric constructions;
(d) working with two- and three-dimensional shapes to solve problems involving area, surface area, and volume, first treating them intuitively and concretely—conducting experiments such as three cones with the same base and height can fill the cylinder with the same height and base, showing that the volume of the cone is one-third the volume of the cylinder—and similarly considering generalized situations of particular situations (e.g., the surface area of a prism is equal to the product of the perimeter of the base and height plus the sum of the area of the two bases; the volume of the prism is equal to the product of the area of the base and the height) of simple and compound shapes, figures, and diagrams; and
(e) drawing inferences about populations based on samples.

Eighth Grade
In eighth grade, the focus is on understanding and operating on real numbers—each real number has a unique place on the real number line (e.g., a real number can be located on the number line, and each point on the number line represents a real number). The set of real numbers is the union of the sets of rational and irrational numbers and a real number can be represented as (a) a terminating decimal, (b) repeated non-terminating decimal, or, (c) non-repeating, non terminating decimal (e.g., .o1oo1oo1ooo1oooo1….; or .12123123412345123456…, etc.). The mastery of numeracy skills and operations on integers and rational numbers is good preparation for understanding and operating on real numbers.

Students should be fluent in using symbols so as to generalize arithmetical procedures to algebraic operations, abstracting from quantitative and spatial situations, and developing algebraical and geometrical reasoning to understand the algebra of functions, linear equations and inequalities. In particular:
(a) Formulating, reasoning, and performing operations on algebraic expressions and equations, including modeling an association in bivariate data with a linear equation, and solving linear equations and systems of linear equations and inequalities using multiple methods and knowing the appropriateness, efficiency, and limitations of a particular method;
(b) Grasping the concept of a function and using functions to describe quantitative, spatial (geometric), and probabilistic relationships;
(c) Analyzing two- and three-dimensional space, shapes, figures, and diagrams using distance, angle, similarity, and congruence; relating 2- and 3-dimensional shapes; understanding and applying quantitative and spatial relationships and results such as the Pythagorean theorem; the trigonometric ratios in a right triangle; making conjectures based on observed patterns in quantitative and spatial relationships and then supplying reasons to prove or disprove those conjectures.

In eighth grade, students should use their understanding and mastery of operations on the real numbers to solve arithmetic, algebraic, geometric, and probabilistic problems.

They should understand patterns in quantities and graphs relating two variables. The conceptual heart of the matter is understanding relations among several quantities whose values change. Eighth grade mathematics must also include

  • thinking about variables as measurable quantities that change as the situations in which they occur change;
  • understanding that variables are not usually significant by themselves but only in relation to other variables and context.

And finally, apart from considering algebra as generalized arithmetic, students should see algebra as a system with a basis in the concept of relations and functions and understand that the most useful algebraic idea for thinking about relations is the concept of function. This helps students to relate one set of representations, ideas, and properties to another and their relationships (linguistic expression, iconic, tabular, graphical/spatial, equation and systems of equations, quantitative/abstract). For example, the typical relation among two or more varying quantities may look like:

  • As time passes, the depth of water in a tidal pool increases and decreases in a periodic pattern.
  • As bank savings rates increase, the interest earned on a fixed monthly deposit also increases, but when the interest earned is compounded, the new amount increases exponentially.
  • In a sequence of squares having sides 1, 2, 3, 4, 5, …,n, …, the areas of those squares are 1, 4, 9, 16, 25, …,n2, …and the perimeters are 4, 8, 12, 16, 20, …., 4n,….
  • For any rectangle of base b and height h, the perimeter p is 2b + 2h.        

Students should know the difference between a conjecture, definition, and a theorem. They should know the difference between proof, example, and counter example; between direct and indirect proof; between justifying and providing a counter example, etc.

With strong numeracy skills and access to these tools, our students—the future mathematicians, can search for patterns in much the same way that scientists explore results from experiments by systematically manipulating variables. The experimental data of mathematics—calculations are made using appropriate algorithms and tools, and then data and calculations are displayed graphically to reveal patterns, regularities and variations. This data can be sorted and analyzed; and then patterns are observed and inferences are made. Further calculations are made to prove or refute these inferences. They should understand and appreciate that ultimate standard for verification remains a formal proof by reasoning from axiomatic foundations.

Further, students should be familiar with and able to use, when necessary and appropriate, computational capabilities of machines—both existing and envisioned. These tools suggest some exciting curricular possibilities. Calculators and computers have a profound effect on students’ understanding of the nature of mathematics. Thus, calculators and computers can be efficient means to generate understanding and interest in algorithms, in particular, and mathematics concepts in general. In this way, the role of tools (calculators, computers, sketchpad, geogebra, software apps, etc.) could be to enhance mathematical thinking rather than detracting from understanding and mastery of arithmetical and algebraic algorithms or just mindlessly learning or applying procedures.

Finally, CCCSS-M has a provision for meeting the needs of talented students in middle school. Students with high aptitude and ability in mathematics are offered pre-algebra in the seventh grade as an accelerated course that provides a transition from arithmetic to algebra and a challenging algebra course (see CCSS-M Algebra One course) in eighth grade. CCSS-M recognizes that these students are our future as they are going to invent more mathematics to provide the language for science, technology, and engineering fields. The framers understand all students should be challenged to realize their potential, but these students in particular should be provided the challenging mathematics they are capable of handling.

 

 

 

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CCSS-M: Non-negotiable Skills at the Middle School: Grades Seven and Eight

Sight Facts vs. Sight Words   

In the previous blog on CCSS-M and also in the blog on number concept, I introduced the concept of sight facts because in many of my lectures, teachers ask about sight facts. The concept of sight facts is an exercise in addition, but it is not to be seen as the formal teaching of arithmetic facts. It is students’ first attempt to understand number relationships. These number relationships are learned by observations rather than any strategies; they are not learned by counting. Learning sight facts is like learning sight words. They are learned by constant exposures. The idea of sight facts is fundamental, so it is important to know how to help children acquire sight facts.

In reading, it is important to acquire a large number of sight and high frequency words. Knowing the alphabet, a large collection of sight words, and phonemic awareness are prerequisite skills for acquiring reading skills. Similarly, it is important to acquire sight facts or high frequency arithmetic facts.

Number concept, decomposition/recomposition, and the mastery of sight facts are prerequisite skills for acquiring arithmetic facts with mastery (understanding, fluency, and applicability). The teaching of sight facts, therefore, is important for learning addition and subtraction facts efficiently.

High frequency words are best taught and acquired in context and with constant exposure. As sight words are not learned by focusing on each letter in the word, sight facts are not learned by counting discrete objects. Many average and above children can learn sight facts in spite of using fingers, TenFrames, and other counting materials. However, the most effective method is decomposition/recomposition using Visual Cluster Cards and Cuisenaire rods. They are effective for all children—with or without learning difficulties in mathematics.

Many teachers use a variety of activities for children to acquire sight words. They use a multi-sensory approach in which students say the word, say the names of the letters, trace the letters on a screen, sky write them, and then repeat the name of the word. The key is multiple and constant exposures. The same approach works for sight facts. Using Visual Cluster cards or Cuisenaire rods, children say the number represented by the Visual Cluster card (say 7), picture (visualize) the cluster on the card (3 pips in the first column, 1 in the middle, and 3 in the third column), visualize two sub-clusters on that card (6 and 1; 5 and 2; 4 and 3; 3 and 4; 2 and 5; and 1 and 6) by drawing a ring around the two sub-clusters by finger in the air or on an imaginary whiteboard (sky writing), say the corresponding sight facts as equations (6 + 1 = 7; 5 + 2 = 7; 4 + 3 = 7; 3 + 4 = 7; 2 + 5 = 7; 1 + 6 = 7; 7 = 1 + 6 = 6 + 1 = 2 + 5 = 5 + 2 = 3 + 4 = 4 + 3), then write these equations on paper. Similar exercises can be done for other numbers from 2 to 10.

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There are more than two hundred sight words (number depends upon the program being used) that children must acquire. With the help of sight words and phonemic awareness, they begin to “chunk” and “blend” and with practice they learn to read. Sight facts play the same role in acquiring arithmetic facts. The following is the list of sight facts.

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The key element is the acquisition and application of the decomposition/re-composition process. Without this process, children do not acquire fluency in addition and subtraction facts. Once children have the concept of number, the decomposition/recomposition process can also be accomplished and reinforced with Cuisenaire rods[1]. For example, the number 10 can be shown as the combination of two numbers as follows (the same process can be used for all other numbers 2, 3, 4, 5, 6, 7, 8, and 9):

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Once children have formed these combinations (all the possible sight facts for 10), the teacher helps them to make these combinations fluently and provides opportunities for applying these sight addition facts. When children have learned the sight facts of a number—recognized the combinations of sub-clusters on Visual Cluster Cards, formed them using Cuisenaire rods or InVicta Balance and can recite them, then they should be asked to record them with the help of these materials. When children can supply the decomposition/recomposition of a number in several forms, they are ready to write the sight fact equations as pointed above in the sight fact equations for number 10.

Repeated exposures to making combinations of numbers (sight facts), using Visual Cluster Cards and Cuisenaire rods, are important as a starting point for learning other facts. Children should move from oral to written form with specific, positive corrective feedback both for making combinations and acquiring fluency. The practice should involve only strategies using decomposition/recomposition[2].

When students arrive at arithmetic facts and procedures with the help of strategies, they develop mathematics conceptual understanding with robust structures rather than learning isolated facts and routine procedures. Knowledge structures here refer to conceptual schemas that students use to organize and relate language, concepts, and facts. Experts have developed complex knowledge structures with multiple and flexible interconnections based on fundamental concepts while novices and poor students have inefficient, simpler, disjointed knowledge structures with fewer connections that make it difficult to assimilate new concepts. For example, they may think addition is just “counting up” and subtraction is “counting down.” These children end up working harder with little or no pay off.

When a child does not have efficient strategies, it is important that we help develop these strategies. This requires timely and effective interventions. As soon as a teacher observes that a child is having struggle in numberness, she must arrange for interventions. All children benefit from effective math intervention. However, poor interventions where the emphasis is only on counting do not help.

The quality of instruction and intervention is dependent on the competence of individual teachers. Teacher certification for pre-K through 3rd-grade should emphasize both knowledge of the subject (specifically, a deeper knowledge of the mathematics taught in early and elementary years) and strengths in the mathematics content related pedagogy. What we now know is that mathematics instruction—initial and intervention—is far more effective when delivered by a teacher who understands both the subject matter and the most effective ways in which young children learn math. Because the conceptual complexity of elementary mathematics is underrated, a successful program will ensure that early math instructors specialize in these areas. One solution may be for a school to designate a teacher in each grade who is responsible for teaching only math to all students or at least able to provide quality interventions.

Early instruction with quality activities that develop a comprehensive numbersense can minimize and prevent failure in numeracy and even in later mathematics. For example, teaching the integration of numbersense activities with an increased focus on “sight number facts” automaticity will better prepare children for numeracy activities. Teaching these skills in isolation and without effective strategies has minimal effect both in the reduction of difficulties in mathematics for the general and LD population. Quality instruction works for students with and without learning disabilities.

[1] For derails of how to use Cuisenaire rods see Cuisenaire Rods and Mathematics Learning (Sharma, 2013).

[2] Strategies for Teaching Addition Facts in How to Teach Arithmetic Facts Effectively and Easily (Sharma, 2008)

Sight Facts vs. Sight Words   

CCSS-M: Non-Negotiable Skills in Elementary Grades

Primary Grades: Kindergarten through Second Grade
During the first three years—from Kindergarten through second grade, the goal of CCSS-M is that children understand, master and apply additive reasoning. This means children acquire true understanding and mastery of counting numbers (number concept), addition and subtraction concepts, facts, procedures and see addition and subtraction as inverse relationships—given a subtraction problem, they can solve it by addition and vice-versa. They should also understand the concept of place value (as a pattern of representing numbers in the base-ten system.

At the same time, children are able to name, recognize, and represent (draw) the commonly found objects in their environment and are able to perform simple operations such as finding their perimeters.

CCSS-M recommends three years to achieve this fundamental goal, which is the foundation of all future mathematics. Moreover, mastery should reflect understanding, fluency, and applicability of this important concept of additive reasoning.

Kindergarten
The focus of mathematics teaching and learning for children in Kindergarten is to acquire the number concept and relationships between numbers. Number concept[1] means:

  • Representing, comparing, and recognizing whole numbers in different forms and modes: discrete (sets of random objects—counting objects), continuous (visual/spatial—comparing visually the length and area of objects, such as Cuisenaire rods to determine larger, smaller, etc.), pictures (like marks on a paper, clusters, number line, etc.), abstract (forming and recognizing numbers);
  • Fluency in number relationships up to ten—mastery with understanding (decomposition/recomposition of a number, e.g., the number 7 can be seen as made of 7 and 0 (0 and 7), 6 and 1 (1 and 6), 5 and 2 (2 and 5), 4 and 3 (3 and 4);
  • Fluency of 45 sight facts[2] (can recognize the facts by sight just like children can recognize certain words by sight);
  • Expressing numbers (two-digit) through place value representation (e.g., 56 = 50 + 6 = 40 + 16 = 30 + 26 = 20 + 36 = 10 + 46 = etc.);
  • Fluently able to count numbers and understanding role of number words (difference between a quantitative and non-quantitative words, difference between cardinal and ordinal numbers); and
  • Recognizing, identifying, and naming commonly found objects and shapes in their environment and spatial relationships and corresponding vocabulary.

In order to achieve the appropriate level of competence in these concepts, a great deal of teaching and learning time (almost 70% of allotted instructional time) in Kindergarten should be devoted to the development of number concept and its mastery. In addition, at the end of Kindergarten, children should be able to represent the first 30 numbers on the number line and fluently count forward and backward by 1, 2, and 10 from any number up to 100 and beyond.

First Grade
The focus of first grade work is to build on the number concept. Children learn number relationships and place value, specifically

  • Understanding addition and subtraction concepts and learning   efficient strategies (using decomposition and recomposition of numbers and sight facts) for and fluency in (10 by10) addition facts and constructing and arriving at subtraction facts within 20 (without counting);
  • Developing an understanding of whole number relationships and place value, including groupings in tens and ones to understand two and three digit numbers fluently (e.g., 346 = 300 + 40 + 6 = 300 + 46 = 340 + 6 = 306 + 40 = etc.);
  • Developing an understanding of linear measurement and measuring lengths as iterating length units (moving from egocentric measurement to using “go between” units, e.g., the room is 35 foot lengths to it is 40 book lengths); and
  • Reasoning about, attributes of, and composing and decomposing geometric shapes commonly found in the child’s environment.

A great deal of time, in first grade, should be devoted to understanding the concept of addition, achieving mastery of addition facts[3] (adding up to 20) and understanding the concept of subtraction.

Fluency (automatization with understanding) is achieved by using efficient strategies based on decomposition/recomposition of number and sight facts (e.g., 9 + 7 is achieved by seeing that by decomposing 7 as 1 and 6 and then adding 1 to 9 transforms the sum as 10 + 6, or since 10 + 7 = 17, thus, 9 + 7 is 16; 9 + 7 = (6 + 3) + 7 = 6 + (3 + 7) = 6 + 10 = 16, or, 9 + 7 = (2 + 7) + 7 = 2 + (7 + 7) = 2 + 14 = 16, or, 9 + 7 = 9 + (9 – 2) = (9 + 9) – 2 = 18 – 2 = 16, etc. Deriving the arithmetic facts using multiple strategies provides children flexibility of thought. Similarly, in place value, they should know not only that 124 = 100 + 20 + 4 (expanded form with canonical decomposition) but also that 124 = 100 + 24 = 12 tens + 4 = 120 + 4 = 114 + 10 = 110 + 14 = etc. (expansion with non-canonical decompositions).

In first grade and then in later grades, children realize that concepts such as finding the perimeter, working with money and time are applications of numbersense (number concept, number relationships, and place value). If students have “good” numbersense, teachers can teach and students can easily learn these topics. Without numbersense, these topics are difficult for many children to learn. Teaching them in isolation, even with concrete instructional materials, limits a child’s learning and a teacher’s instructional time.

Second Grade
The focus of second grade is to master subtraction and integrate addition and subtraction into additive reasoning. This means:

  • Mastery of (fluency with strategic understanding—using decomposition and recomposition of numbers and sight facts) addition and subtraction facts;
  • Properties of numbers: even and odd;
  • Fluency in executing standard and alternative procedures of addition and subtraction with understanding that these procedures are based on place-value;
  • Mastering place value beyond hundreds; expressing the number as:

5,467 = 5,000 + 400 + 60 + 7 (expanded form–canonical decomposition)

= 5400 + 67 = 5000 + 467 = etc. (expanded from–non-canonical form);

  • Using standard units of measure (understanding that the smaller the unit of measurement the larger the iterated quantity and vice-versa); and
  • Describing and analyzing shapes.

In the second grade, a great deal of time should be devoted to achieving mastery of addition and subtraction facts (for numbers up to 20).

Fluency in addition and subtraction should be achieved using strategies and understanding (decomposition/recomposition of number and sight facts) and the addition strategies should be used and extended to deriving the subtraction facts. For example, students should able to think the subtraction problem 16 – 9 = (6 + 10) – 9 = 6 + (10 – 9) = 6 + 1= 7, or, to find 16 – 9, I know 9 is 6 + 3, so I take 6 from 6 and 3 from 10, so the answer is 7, or 10 + 6 is 16, 9 is 1 less than 10 so 9 + 7 is 16, so the answer is 7.

By the end of second grade, students should have mastered additive reasoning and its applications. Additive reasoning means: students understand the inverse relationship of these two operations and given a subtraction problem, it can be treated as an addition problem and addition problem as a subtraction problem.

[1] For details of how to teacher number concept, please see the blog on Number concept and Numbersense.

[2] Each counting number up to 10 has sight facts associated to it, e.g., 7 = 1 + 6 = 2 + 5 = 3 + 4 = 4 + 3 = 5 + 2 = 6 + 1. The total number sight facts, thus is 45. For the list see: Visual Cluster Cards and Number Concept by Sharma, 2016.

[3] For strategies of teaching addition facts see: How To Teach Arithmetic Facts Easily and Effectively by Sharma, 2008.

CCSS-M: Non-Negotiable Skills in Elementary Grades

CCSS-M Focus: Non-Negotiable Skills

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.

CCSS-M Focus: Non-Negotiable Skills

Intent of the CCSS-M

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

Capture01.1

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.

Capture02

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:

Simplify Capture03.3.PNG

Students just apply the procedure to find the prime factorization using the “Factor Tree” method and their solution is:

Capture new.PNG

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.

Intent of the CCSS-M