In the quantitative domain, the focus of three-years of mathematics instruction, from Kindergarten to second grade, is that by the end of second grade, children master the concept of additive reasoning. Additive Reasoning means mastery of: (a) concepts of addition and subtraction, using multiple conceptual and instructional models, settings, diverse vocabulary and phrases that translate into addition and or subtraction concepts and operations, (b) related procedures (in standard and non-standard forms, using place-value and decomposition/ recomposition at one-digit and multi-digit levels), (c) relevant applications to solving problems in learning other mathematics concepts (e.g., multiplication), solving problems in other subject areas (e.g., time line), and relevant real-life problems (e.g., money, time, measurment), and, (d) the understanding that adddition and subtraction are inverse operations (e.g., given an addition equation, one can express it in subtraction form and a subtraction equation into an addition form and can use this knowledge in solving problem in various situations).
To achieve this goal of quantitative domain, at the end of Kindergarten, a child should have mastered: (a) Counting forward and backward by 1, 2, and 10 starting from any number up to at least 100; (b) Number vocabulary (lexical entries for number) of at least up to 100; (c) Number concept: visual clustering (generalizing subitizing)–recognizing, by observation (without counting), a cluster of objects up to 10, numberness–integrating the size of a visual cluster, its orthographic (shape of the number–“5”), and audatory (saying: f-i-v-e) representations of a number, the skill of decomposition/ recomposition: visualy and mentally breaking a cluster of obejects into two sub-clusters and, then, a number into two smaller numbers and joining two clusters into one larger number (e.g., a cluster of 7 objects is made up of a cluster of 5 and a cluster of 2, therefore, 7 = 5 + 2 and 5 + 2 = 7; (d) the 45 sight facts (using decomposition/ recomposition, by sight, one finds that a number, up to 10, is the combination of two numbers (e.g., sight facts of 5 are: 1 + 4, 2 + 3, 3 + 2, and 4 + 1)); (e) Commutative and Associative properties of addition (e.g., on a Visual Cluster card of 9, one can see that 4 + 5 = 5 + 4 and (3 + 2) + 4 = 3 + (2 + 4); (f) of Making ten (what two numbers make 10); (g) Knowing teens’ numbers (combination of 10 and a number, i.e., 10 + 5 = 15, 10 + 7 = 17, 15 = 5 + 10); (h) Concept and role of zero in forming larger numbers (10, 20, 30, etc.) and adding to and subtracted from a number; and, (i) Place-value of 2-digit numbers: what two digits make a number? and what two numbers make a number? (e.g., digits 1 and 5 make 15 and numbers 10 and 5 make 15).
Mastery of number concept is the foundation of arithmetic. The ten numbers/digits: 0, 1, 2, 3, 4, 5, 6, 7, 8, and 9 are the alphabets of the quantitative language and numeracy. Sight facts are the sight words of this language. Decomposition/recomposition is the arithmetic analog of phonemic awareness. By the help of numberness, decomposition/ recomposition, sight facts, making 10, and the knowledge of teens’ numbers, one acquires the “number-attack skills” — mastery of arithmetic facts, beyond the 45 sight facts. For example, 8 + 6 = 8 + 2 + 4 (decomposition of 6 into 2 + 2) = 10 + 4 (knowledge of the sight facts of 10, making 10, and recomposition) = 14 (knowledge of making teens’ number). The child further extends to problems such as: 68 + 6 = 60 + 8 + 6 = 60 + 8 + 2 + 4 = 60 + 10 + 4 = 70 + 4 = 74, 68 + 6 = 68 + 2 + 4 = 70 + 4 = 74.
Similarly, in the quantitative domain, at the end of second grade, a child should have mastered: (a) Concepts of addition and subtraction and extending decomposition/recomposition to numbers greater than 10 (as mentioned in the previous paragraph); (b) Addition facts (sums of two single-digit numbers up to 10 by the end of first-grade and corresponding subtraction facts (by the end of second grade); (c) Place-value of three-digit numbers, both in canonical (e.g., 59 = 50 + 9) and non-canonical forms (e.g., 59 = 50 + 9 = 40 + 19 = 30 + 29, by the end of first grade) and four-digit numbers (both in canonical and non-canonical forms, by the end of second grade); (d) Addition procedures (standard and non-standard using place-value and decompistion/recomposition at one- and two-digit level), by the end of first grade and subtraction procedures (standard and non-standard using place-value and decompistion/ recomposition at one- and two-digit level), by the end of second grade.
In the quantitative domain, the focus of three-years of mathematics instruction, from Kindergarten to second grade, is that by the end of second grade, children have mastered the concept of Additive Reasoning. Aquiring additive reasoning means mastery of: (a) the concepts of addition and subtraction, with multiple conceptual and instructional models, settings, and diverse vocabulary and phrases that translate into addition and or subtraction (b) related procedures (standard and non-standard forms, using place-value and decomposition/recomposition at one-digit and multi-digit levels), (c) Application of additive reasoning to solve problems (learning other mathematics concepts, solving problems in other subject areas, and relevant real-life problems); (d) Understanding that adddition and subtraction are inverse operaions; (e) Understanding that the operation of Addition is commutative and associative, but the operation of Subtraction is not.
The concept of mastery of mathematics concept/skill/procedure means: (a) the child possesses the appropriate numerical language (vocabulary and phrases, syntax, and ability to translate from native language to mathematics language and from mathematics language to native language) for understanding and applying; (b) appropriate strategies (effective, efficient, and elegant) for deriving an arithmetic fact, skill, or procedure accurately in standard and non-standard forms; (c) appropriate level of proficiency and fluency in producing an arithmetic fact (e.g., 2 seconds or less for an oral arithmetic fact, 3 seconds for writing a fact); (d) appropriate level of numeracy: can execute an arithmetic procedure correctly, accurately, fluently, in non-standard and standard forms (algorithm) with understanding; (e) can estimate the answer/outcome to an addition and subtraction problem in acceptable range (without counting/writing or applying a procedure), and, (f) can apply the skill, concept, and/or procedure in–learning a new mathematics concept/skill/procedure, solving a problem in another subject/discipline/area, or a real-life problem.
A strategy is appropriate if it effective, efficient, and elegant if yields result with less effort and energy. It uses the principles of decomposition and/or recomposition, place-value, or peoperty of numbers/operations. It is transparent. It does not tax the working memory and processing ability too much, i.e., it is accessible, but moderately challenging. It is applicable not just to a specific or particular problem, but is generalizable, can be extrapolated and abstracted into a principle/concept/proecedure. The learner experienced being in the “zone of proximal development.” It results in a definite expereince in metacognition for the learner. Strategies based on counting experiences (e.g., addition and subtraction facts derived by coutning forward or backward, making change by counting, finding perimeter by counting units, etc.). A strategy can be at concrete, pictorial, visualization, or abstract level. However, if it is only at the concrete or pictorial level, it should be advanced to the abstract mathematical level also.
Learning and mastering arithmetic facts is dependent on three kinds of pre-requisite skills: (a) Mathematical: number concept (numberness, 45 sight facts, making ten, knowing teens numbers, properties of operation, and the most important skill decomposition/ recomposition), (b) Executive function skills: working memory, inhibition control, organization, and flexibility of thought, and (c) cogntiive skills: ability to follow sequential directions, discerning and extending patterns, spatial orientaiton/space organizaiton, visualization, estimation, deductive and inductive reasoning. Since these skill categories have operlaps, it is important that instructional activities embed as many of these skills as possible. Integration of the use of concrete instructional models, playing games, and interactive activities is the most pedagogically sound approach to mathematics instruction, whether it is regular (intial), intervention, or remedial instruction. Moreover, these skills, when practiced in isolation do not have lasting effect, learners do not see relationships between concepts, and do not last long. As a part of regular instruction, intervention, and then in reinforcement activities, to get maximum benefit, I plan lessons that follow the principle of six levels of knowing: intuitive, concrete, pictorial, abstrct, applications, and communications. I take a child from intuitive to communicaitons. In addition, I have found that students, at all grade level, from pre-Kindergarten to Algebra, find the Number War Games exteremly engaging and productive. They incorporate many of the principles included earleir.
Before and during the game, the emphasis should be on the use of sight facts, decompositon/recomposition, making ten, and making ten. During first few games, a teacher/parent should participate in the game with the children. During the game she should develop script fot finding the facts. For example, if during the Addition War Game game a child gets two cards: 6 of diamonds and 8 of spades, he needs to find the sum 6 + 8. If the child does not know the answer readily, then the teacher/parent should practice fidning the sum of 8 + 6 by asking questions that help the child practice the necessary pre-requisite skills and develop a script that will help the child to find the answers independently. For example, (a) Look at your cards: (points the cards: 8 and 6). Which is the larger number? “8” (The numbers in quotation marks are child’s answers.) (b) Remember the strategy of adding numbers? “Make ten, first” (c) Good! How do you make 8 as 10? “by adding 2 to 8” (c) Good! You moved the 2 pips to 8 card and it became 10. Where did 2 come from? “from 6” (d) What is left in 6? “4” (e) What do you add now? “10 and 4” (f)) What is 10 plus 4? “14” (g) So, what is 8 + 6? “8 + 6 = 14.” the teacher could also use Cuisenaire rods or the Empty Number Line (ENL), to support the child in developing the script. When children have developed, visualized, and mastered such scripts, they become independent learners.
All of these questions, with the help of visual cluster cards (Cuisenaire and Empty Number Line), should be answered and practiced orally. This process develops many of these pre-requisite skills individually and then helps integrate them. For example, working with the patterns on the Visual Cluster cards and then visualization of the cards aids in the development of the working memory. The organized sequential script helps them focus, organize and develop deductive reasoning. The reorganizing the pattern on the first card into sub-patterns and then integrating them with the patterns of the second card helps with the acqusition of decomposition and recomposition skills. The game setting: playing the game involves practicing these skills again and again and soldiifes these skills. For example, in playing the Number Addition War involves making, hearning, and practiicing more than 500 addition facts. Neurologically, questions instigate neural firing and making connections, that in turn invites oligodendrocytes–(oligo) to instigate the production of myelin–creating covering around the nerve fibers, that in turn controls and improves the impulse, and the impulse speed is skill. Each time a child practicies the script, the nerve fibres get stronger and wrapping wider and wider making the pathway of the nerve impulses into a major “highway.” The integration of (a) practicing the script, (b) visualizing the action guided by the script, (c) acceleration of the neural firing (better myelination), (d) and reducing the refractory time (the wait required between one signal and the other) makes learning optimal. The increased speed abd decreased refractory time No child will practices the number examples in a formal setting as he practices in one game. With the Number Addition War Game, children master their addition facts in a very short time. And that too with great deal of pleasure.
To make the learning robust and making children super-confident, we should practice finding the answers, even to one simple fact, in multiple ways. In the script developed and used above, the practice strengthened certain nerual pathways and it opened certain “files” (e.g., sight facts, making ten, and making teen’s number files) in the long-term memory (the practice was being performed in the working memory and it was transferrdd to long-term memory), but the retrieval is easier and more useful, when the infromation is transferrd to long-term memory in more than one way (different instructional materials, stategies, models, scripts, order, modality of learning, levels, occasions, times, groupings, and settings). For example, the fact 8 + 6 can be derived using counting objects (e.g., objects, fingers, on number line, etc.), Ten Frames, Rek-n-Rek, Visual Cluster cards, Cuisenaire rods, Invicta Balance, decompositin/ recomposition, Empty Number line, orally, visualization, and abstractly (notice the order–from less efficient to more efficient, from concrete to abstract, from lower level to higher level, from less understnading to more understanding, etc.). To provide the flexibility of thought, let us consider the following. In the following discussion, child’s answers to a fact problem are dispalyed in quotations.
Display two Visual Cluster cards: 8 of dimonds and 6 of clubs. Do you know what addition problem can you make form these numbers? “8 + 6 or 6 + 8.” Good! What is 8 + 6? “14.” How did you find the answer? “8 + 2 is 10 and then 4 more is 14. So, 8 + 6 is 14.” What about 6 + 8? “14.” How did you know that quickly? “Because 8 + 6 = 6 + 8.” What property is that? “Turn-around-fact.” What is another name for that property? “Commutative Property of Addition.” Is there any way you can find 6 + 8? “I do not know.”
Display two Visual Cluster cards: 6 of dimonds and 6 of clubs. Do you know what addition problem can you make form these numbers? “6 + 6.” Good! What is 6 + 6? “16.” How did you find the answer? “6 + 4 is 10 and then 2 more is 12. So, 6 + 6 is 12.” What about 6 + 8? “14.” How did you know that quickly? What peoperty is that? Doubles property.” Good!
Display two Visual Cluster cards: 8 of dimonds and 8 of clubs. Do you know what addition problem can you make form these numbers? “8 + 8.” Good! What is 8 + 8? “16.” Can you find 8 + 6 using the fact that 8 + 8 = 16? “I do not know.” Is 8 + 6 is less than 8 + 8 or more than 8 + 8? “It is less.” If, the child begins to count. The teacher/parent should intervene. Look at the second 8-card. If you cover the 2 from the card, what do you see on the card. “a 6.” What addition problem do you have now? “8 + 6.” Can you figure out the answer for 8 + 6? “Yes, it is 14.” How do you know? “I know 8 + 6 = 14.” So, 8 + 6 is how much les than 8 + 8? “2 and 8 + 6 = 8 + 8 – 2.” Good!
Display two Visual Cluster cards: 8 of dimonds and 6 of clubs. Do you know what addition problem can you make form these numbers? “8 + 6 or 6 + 8.” Good! What is 8 + 6? “14.” How did you find the answer? “8 + 2 is 10 and then 4 more is 14. So, 8 + 6 is 14. Or, 6 + 6 + 2 = 14. Or, 8 + 8 – 2 = 14.” Do you know any other way? “I do not think so!” What if you took the one pip from the 8-card an put it on the 6-card, what problem would you have? “7 + 7.” What is 7 + 7? “14.” How do you know? Doubles property. Great! Can you apply making 10 strategy to this problem? “Yes! 7 + 3 is 10 and 10 + 4 = 14.” Great! Now, you know several ways of finding 8 + 6 or 6 + 8. How far apart are 8 and 6? “2 apart.” What number is between 6 and 8? “7.” So, 6 + 8 is same as 7 + 7. When two numbers are 2 apart, their sum is double of the middle property.
Practicing multiple strategies for finding the answer improves a child’s cognitive potential. They begin to see more realtionships, patterns, and concepts. They do not get helpless when they do not have the answer. They take action. This is an anti-dote to math anxiety.
Game Three: Number Addition War
Objective: To master addition facts
Materials: Same as above
How to Play:
- The whole deck is divided into two equal piles of cards.
- Each child gets a pile of cards. The cards are kept face down.
- Each person displays two cards face up. Each one finds the sum of the number represented by these cards. The bigger sum wins. For example, one has the three of hearts (value 3) and a 10 or a king of hearts (value 10). The sum of the numbers is 13. The other person has the seven of diamonds (7) and the seven of hearts (7). The sum is 14. The person with the sum of fourteen wins. The winner collects all the displayed cards and puts them underneath his/her pile.
- The face card and the wild card can be assigned any number value up to ten.
- If both players have the same sum, there is war. For example, one has the five of hearts (value 5) and the seven of clubs (value 7), and then the sum is 12. The other person has the six of spades (value 6) and the six of clubs (value 6). They declare war.
- Each one puts three cards face down. Then each one displays another two cards face up. The bigger sum of the last two cards wins.
- The winner collects all the cards and places them underneath his/her pile.
- The first person with an empty hand loses.
This game is appropriate for children who have not mastered/automatized addition facts.
Initially, children can count the objects on the cards. However, fairly soon they begin to rely on visual clusters on the cards to recognize and find the sums. In one game, children will encounter more than five hundred sums. Within a few weeks, they can master all the addition facts. Initially, if the child does not know his sight facts, the game can be played with dominos or with a deck of Visual Cluster cards with numbers only up to five. Then, include other cards.
I sometimes allow children to use the calculator to check their sums. The only condition I place on calculator use is that they have to give the sum before they find it using the calculator. Over a period of time, calculator use declines and after a short while, they are able to automatize the arithmetic facts. After they have learned the 10 ×10 arithmetic facts (sums up to 20), you can assign values to the face cards: Jack = 11, Queen = 12, and King = 13. The joker has a value assigned by the player. Its value can be changed from hand to hand. The joker is introduced with a variable value so that children can get the concept of variable very early on.
Variation 1: After a while, you might make a change in the rules of the game.
Each child displays three cards, discards a card of choice, and finds the sum of the remaining two cards.
Variation 2: Each child displays three or four cards, finds the sum of the three or four cards, and the bigger sum wins.
Game Three: Subtraction War
Objective: To master subtraction facts
Materials: Same as above
How to Play:
- The whole deck is divided into two equal piles of cards.
- Each child gets a pile of cards. The cards are kept face down.
- Each person displays two cards face up. Each one finds the difference of the two cards. The bigger difference wins. For example, one has the three of hearts and a king of hearts (value 10), and then the difference is 7. The other has the seven of diamonds and the seven of hearts, and then the difference is 0. The first player wins. The winner collects all cards.
- If both players have the same difference, they declare war. Each one puts down three cards face down. Then each one turns two cards face up. The bigger difference of the two displayed cards wins. The winner collects all cards.
- The first person with an empty hand loses.
As in addition, children can initially count the objects on the cards. However, fairly soon they begin to rely on visual clusters to recognize and find the difference. In one game, children will use more than five hundred subtraction facts. Within a few weeks, they can master subtraction facts. Initially, the game can be played with dominos.
I allow children to use the calculator to check their answers. As mentioned above, the only condition I place on calculator use is to give the difference before they find it using the calculator. Over a period of time, calculator use declines and after a short while, they are able to automatize the arithmetic facts. This game is appropriate for children of all ages to reinforce subtraction facts.
Variation 1: After a while, you might make a change in the rules of the game. Each child displays three cards, discards a card of choice, and finds the difference using the remaining two cards.
Variation 2: Each child displays three cards, finds the sum of any two cards, and subtracts the value of the third card. The bigger outcome of addition and difference wins.
Variation 3 :Each child displays three or four cards, an objective number is decided and finds the result by adding or subtracting of any combination of cards gets the declared number as the result. The bigger outcome of addition and difference wins. No number can be used more than once.