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.