Question: Everyone is focusing on children knowing mathematics ideas. What does it mean to know arithmetic facts? Is mastery different than knowing? Is it important to memorize arithmetic facts?
Answer: In the context of mathematics curriculum, the term mastery is associated with several words: know, understand, master, being fluent, proficient, etc. Although it has been clearly defined in the Common Core State Standard in Mathematics (CCSS-M) documents, in our schools, the definition of mastery of mathematics curricular components still varies with individuals, individual school systems, and textbook series.
Students need to satisfy certain characteristics to have acquired mastery of mathematics curricular elements. Mastery of different curricular elements is complex; it is multi-dimensional.
Components of Mastery
A student has mastered an arithmetic fact when three conditions are satisfied: The child
- demonstrates the understanding of the concept—can provide an efficient strategy (based on decomposition/recomposition and select the most efficient strategy out of several strategies), without counting, for arriving at the arithmetic fact;
- demonstrates fluency/automatization (able to produce a fact orally in less than two seconds and in writing in less than three seconds), and
- can flexibly apply the knowledge of the fact in different situations in solving problems without counting and using concrete or pictorial models.
Learning with rigor (understanding, fluency, and applicability) is at the center of the Common Core State Standards for mathematics (CCSSM). Teaching with rigor produces flexible and long lasting mastery.
Understanding, here, means a student is able to derive a fact or procedure with effective, efficient strategies and able to explain his approach, for example, an addition fact is derived by using decomposition/recomposition of numbers, not just by counting up. This also includes flexible use of these strategies. It is widely acknowledged that practice, drill and memorization are essential if students are to become mathematically fluent. However, practice without understanding is of very little value as the facts and skills are not integrated and applications under such limited level of mastery become difficult.
The concept of “fluency” refers to knowing key mathematical facts and methods and recalling them efficiently. Fluency is automatized mastery of a fact, skill, or procedure.
Understanding and fluency facilitate the application of facts and procedures in problem solving. Applications may be learning other mathematics concepts using those facts. It may be problem solving in other disciplines or extra curricular situations.
Without this level of automatization, children become dependent on concrete or pictorial materials used in instruction such as: counting objects—number line, hash-marks, fingers, TenFrames, discrete objects, etc. Strategies based on counting materials do not help children to achieve either the understanding and fluency or the ability to apply such knowledge.
The elements of rigor (mastery) are inseparable. The absence of any one of them is problematic. For example, when children do not have understanding of the strategy and have not automatized facts, they are not able to apply their knowledge to newer situations effectively. They digress from the main problem to generate the facts needed to be used in solving a problem. Their working memory space is filled mostly in constructing facts. Then it is not available to pay attention to the instruction, to observe patterns, focus on concepts, nuances, relationships, and other subtleties involved in the concepts, procedures, and applications.
When children possess number combination mastery, their achievement increases at a steady rate in arithmetic, whereas children with low mastery in arithmetic facts make little to almost no progress in later grades. When students, without mastery of arithmetic facts, are provided classroom instruction in procedures, although they make good strides in terms of facility with these algorithms and procedures and even solving simple word problems, deficits in the retrieval of basic combinations remain. These deficits inhibit their ability to understand and participate fully in the mathematical discourse and to grasp the more complex multi-step and algebraic concepts later. Failure to instantly retrieve a basic combination, such as 8 + 7, often makes discussions of the mathematical concepts involved in algebraic equations more challenging.
Before we can have effective mathematics teaching and children can achieve higher in mathematics, everyone involved needs to have a well-defined and commonly-agreed upon definition of knowing/mastery for a concept, procedure, or skill—with clear markers for mastery. Then and only then can it be taught well, retained by students, and assessed and monitored effectively.
Essentials for Higher Achievement
To assure the conditions for higher achievement, we first need to
(a) identify non-negotiable concepts, skills, and procedures to be achieved by each child at each grade level,
(b) develop common definitions and criteria for “knowing” concepts, “mastering” skills, and achieving “proficiency” in executing procedures, across the school system, and
(c) identify and discuss the most “effective,” “efficient,” and “elegant” ways of teaching the key developmental mile-stone concepts and procedures at each grade levels. These include: number concept, arithmetic facts, place value, fractions, integers, and algebraic thinking.
When we have identified these, training should be provided for all teachers (all classroom teachers and interventionists of various kinds) and administrators by using content embedded pedagogy. The training should also include how to observe children’s work and learn from the error analysis about their level of mastery.
Observing Children’s Work
In a second grade classroom I observed a teacher assessing children’s mastery of addition and subtraction facts. The purpose of the test was to assess understanding, fluency, and applicability of addition and subtraction. The test had 25 problems on addition and subtraction.
One of the problems on the test was 17 − 9 = ☐. I identified five children from the class to observe work on the problem. Problems were written both in horizontal and vertical forms:
Here are my observations.
One child read the problem and solved it in about 20 seconds by sequentially counting on his fingers: 10, 11, 12, 13, 14, 15, 16, and 17. Then he recounted the fingers used: 1, 2, 3, 4, 5, 6, 7, 8 to find the answer and wrote the correct answer 8.
The second child answered the problem in about 50 seconds. She drew seventeen tally marks in front of 17 and nine tally marks in front of 9. She crossed one tally mark in front of 17 from the top and one from the 9. After all the nine tally marks were exhausted, she counted the remaining tally marks in front of 17 and correctly wrote the answer as 8.
The third child answered the problem in about 36 seconds. The child counted: 17, 16, 15, 14, 13, 12, 11, and 10 on his fingers and counted the fingers used. And then he wrote the correct answer in the right place.
The fourth child used the number line pasted on his desk. The child located numbers 9 and 17 on the number line and then counted the numbers from 9 to 17 and wrote down the correct answer in about 27 seconds.
The fifth child read the problem and thought for a moment and wrote down the answer (8) in 2 seconds in the correct place.
The teacher collected the papers of all the children from the class. During the debriefing, I asked her to check the problem: 17 − 9 = ☐ on these five children’s papers. She did. She put a check mark (✓) in front of the problem on their papers.
I asked her whether these five children “knew” the answer to the problem 17 − 9 = ☐ and did she have enough information to judge the responses to satisfy the criteria for knowing?
She said: “Of course. They have the correct answer on their papers.” I asked other participants to also look at the response of these five children. They agreed with the teacher.
It is true that the children had the correct answers, but they knew the fact 17 − 9 = 8 at different levels of knowing.
The first and the third child knew the fact: 17 − 9 = 8 at the concrete level and used counting as a strategy. The second and the fourth child knew the fact at the pictorial level and also used counting strategy. The second child used a simple one-to-one correspondence counting. Unlike the others, the fifth child answered the problem without any overt strategy and very quickly.
In the case above, only the fifth child, who answered the problem in 2 seconds (within the expected time for response), had automatized this subtraction fact. However, we do not know whether she just memorized the fact in a rote manner or with some strategies. Therefore, her example does not satisfy the definition of knowing.
The four children arrived at the correct answer but used inefficient strategies. They also have not achieved fluency in arriving at the fact. Very few children arrive at fluency using inefficient strategies such as counting.
The best method of automatization of arithmetic facts is to practice the facts first orally. When children have shown the mastery orally using efficient strategies, then the children should practice the facts in writing. A fact is mastered orally when the child can answer the fact in 2 seconds or less. A fact is mastered in written form when the child can answer a fact in 3 seconds or less and able to furnish a strategy when asked.
Assessment of arithmetic facts should be oral first with immediate feedback about the strategy and its efficiency. When children can answer facts orally in the prescribed time, the teacher should ask for the strategy used. Then there should be intentional effort to make children flexible by providing alternate strategies. There should be a class discussion on which strategy is most exact and of all the exact strategies which ones are efficient and then which ones are elegant.
Without focusing on children using efficient strategies and correcting papers only for correctness, we send a message to the children that they can keep using inefficient strategies. When children become fluent in using inefficient strategies, they may become fluent counters.
Several studies found that a significant area of difference between students with number combination mastery and those without was the sophistication of their strategies. The poor combination mastery group in second and third grades continued to use fingers to count when solving problems. In contrast, their peers increasingly used verbal counting or decomposing/recomposing numbers without fingers, which led much more easily to the types of mental manipulations that constitute mathematical proficiency.
In the development of counting knowledge in young children, one can observe that children use an array of strategies when solving simple computational problems. For example, when figuring out the answer to 6 + 8, a child using an unsophisticated, inefficient strategy would depend on concrete objects by picking out first 6 and then 8 objects and then counting how many objects there are all together.
A slightly mature but still inefficient counting strategy is to begin at 6 and “count up” 8. Still more mature would be to begin with the larger addend, 8, and count up 6, an approach that requires less counting. However, all of these are based on counting, and many of these children will not reach fluency in addition and subtraction. Effective teachers promote efficient use of number relationships and ultimately help children transcend counting. They help children acquire efficient number combination strategies based on decomposition/recomposition (e.g., sight facts, making ten, teens numbers, doubles, near doubles, the missing double, etc.).
For example, using decomposition/recomposition, a child might say that 6 + 8 = 6 + 4 + 4 = 10 + 4 = 14 (making ten and teens numbers), or 6 + 8 = 4 + 2 + 8 = 4 + 10 = 14 (making ten and teens numbers). Some children might give this as: 6 + 8 = 8 + 6 = 8 + 8 – 2 = 16 – 2 = 14, 6 + 8 = 6 + 6 + 2 = 12 + 2 = 14, or 6 + 8 = 6 + 1 + 7 = 7 + 7 = 14 (decomposition/recomposition and doubles). When children can furnish more than one strategy (other than counting) in arriving at a fact, they show flexibility of thought. Children with flexibility of thought are able to apply their fluency in novel situations and go higher in mathematics.
Some children will simply have this combination stored in memory and remember that 8 + 6 is 14. Ultimately, that is what we want children to reach—a level of automatization. But this should be reached with understanding and with the use of efficient strategies. Memorization of a fact with repeated use remains an isolated fact. Its application also remains isolated. However, it does have some advantages if that fact mastery is used to gain other fact mastery by using decomposition/recomposition. For example, when a child can retrieve some basic combinations (sight facts of a number, say 10 = 1 + 9 = 2 + 8 = 3 + 7 = 4 + 6 = 5 + 5 = 10), then he or she can use this information to quickly solve other problems (e.g., 6 + 5) by using decomposition (e.g., 6 + 4 + 1 = 11).
The ability to derive, store, and easily retrieving information in memory helps students to build both procedural and conceptual knowledge of abstract mathematical principles, such as commutative and the associative laws of addition and mental fluency and numbersense (e.g., 9 + 7 = 9 + 1 & can be extended to 59 + 7 = 59 + 1 + 6; 149 + 7 = 149 + 1 + 6). Immature finger or object counting creates few situations for learning these principles. Research also suggests that maturity and efficiency of strategies are valid predictors of students’ ability to profit from later mathematics instruction.
In order to achieve fluency, teachers, parents, and students should understand what fluency looks like. To achieve fluency there is a definite progression. For example, in the case of addition, it involves:
- mastering number concept—subitizing and visual clusters, decomposition/recomposition, 45 sight facts (what are sight facts, see previous blog on sight facts);
- mastering addition strategies—commutative property of addition, N + 1, making ten, teens numbers, N + 9, double numbers, near doubles, missing double, N + 2, near tens (9 and 11), and the remaining four facts (8 + 4, 4 + 8, 8 + 5, 5 + 8) using decomposition/recomposition;
- “working out” using efficient strategies with understanding to efficiently generating an answer (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, 8 + 6 = 6 + 8 = 6 + 1 + 7 = 7 + 7 = 14);
- practicing “rapid recall” and finally,
- the ultimate goal of “instant recall.”
The time taken and the expectations of efficient recall vary per topic. In mastering addition facts, the role of sight facts, decomposition/ recomposition, making ten and teens numbers is crucial. For example, a child may be able to recall the sight facts of 9 and then practice adding other numbers to it using decomposition/recomposition (9 + 2 = 9 + 1 + 1; 9 + 3 = 9 + 1 + 2, etc., or 9 + 7 = 10 + 7 – 1, etc.). Students need to keep working on decomposition/recomposition strategies till they are able to generate the facts and then keep on practicing automatization.
When a child can read fluently, we are not able to detect the strategies used for acquiring the reading skill. Fluent reading means the child has transcended the strategies and skills used for arriving at that fluency. Similarly, when we ask a child to find the sum of numbers 8 and 7 and the child counts 9-10-11-12-13-14-15 and says: 8 + 7 = 15, we should not be content with that. Instead, we need to give the child better strategies and work with the child until he or she arrives at fluency. Sequential counting is decoding of numbers; it is not mastery of a fact. By giving the feedback—“good job” to such a strategy, we are sending the wrong message that decoding of numbers is adequate. Knowing addition as “counting up” and subtraction as “counting down” are not strategies.
Mistakes are an integral part of learning, but it is equally important for teachers and children to be aware of the need for accuracy. When a child gives a wrong answer (for example, 8 + 6 = 15), the teacher needs to redirect the child:
- 8 + 7 = 15, our problem is 8 + 6, can you use that fact to find 8 + 6?
- what other strategy would you use for finding the answer? How would you make 8 as 10? Etc.
- can you use another strategy? Etc.
Through these activities—by constantly bringing their attention to the appropriate strategy, the child should derive the fact. Once the child derives the correct answer using an efficient strategy, direct the child to another equally efficient strategy.
After that, observational assessments should be used to ensure that all children are being accurate. Accuracy should be achieved first at oral level with immediate feedback when an incorrect answer or strategy is produced. This involves listening to children’s verbal responses, targeting specific children with differentiated questions and checking responses on whiteboards. Then, children should also be given responsibility for self-assessing their own work for accuracy.
Automatization and fluency at appropriate level (speed) is achieved by practice. Practice sessions for achieving speed should be brief, paced, and create a buzz of excitement in which children’s recalling and using their knowledge efficiently gives the feeling of achievement (immediate feedback).
It is important, though, to recognize that fluency is not solely about memorizing and recalling facts; it also means being able to work flexibly and choose the most appropriate method for the problem at hand. Children do need drilling in the basics, but this can be delivered in open-ended, rich and engaging ways. The key is to balance the three components outlined above. As is often the case in teaching, getting the balance right is crucial. Fluency of facts is essential, and if we teach in a style and order that suits the development of this fluency, we do not risk sacrificing creativity and contextual richness in mathematics tasks.
A short modeling demonstration by the teacher should be followed by game-like activities lasting between three and four minutes. The fast pace of these activities, combined with the emphasis on aiming to beat the child’s own personal best, makes the session exciting and engaging for the child. Each child should be working on identified facts. For example, The Addition Fact Ladders and Fact War Games are good for achieving accuracy and speed.
Thus, for automatization, first the child must have the understanding, and then accuracy, followed by speed. For that, a great deal of practice is essential.
Difficulty of the tasks (facts to be mastered) should be gradually increased while practicing and testing. The first practice and test should not be difficult, and oral practice with immediate practice is required. Teachers should point out not only the strategy but also the name of the strategy. For example, when a child is practicing a fact, say 8 + 6, the child says: I take 2 from 6 and give it to 8 and then I add 10 and 4. The answer is 14. The teacher should ask: Which strategy did you use? The child should say: “Making ten and then teens’ facts.” The strategies can be shared with parents, and the Addition Ladders can be given to them so they can practice them with their children at home.
In my experience with thousands of children, I have found that children overwhelmingly want more challenge in mathematics; they want mastery, but they also want efficient methods. What is important is giving every child a challenge that is personal to them and is attainable with the right amount of effort and practice. Children like to set their own mathematics targets in consultation with and with scaffolding from their teacher or tutor. These targets should be specific and achievable within a period of a few weeks. This motivates students to work hard and keep pushing themselves towards new goals.
During the “Tool Building Time” of daily lessons, the teacher should identify an ‘arithmetic fact of the day’ or adding a particular number to other 10 numbers (using the Addition Ladder). She should ask that fact orally and in multiple forms from all children. This process should be repeated until the fact(s) has been mastered orally. Then give them 20 facts on a sheet of paper (randomly organized and including the ten facts just practiced interspersed with previously mastered facts). The goal is one minute. At the end of one minute, children put a marker and continue till finished. The teacher reads the answers and children correct their papers. Children, in turn, give a correct strategy for a wrong answer.
One of the important elements in helping a student to memorize arithmetic facts is the questioning process and the type of questions the teacher asks. It works in the following way: Once the teacher has chosen an arithmetic fact, then she asks children the various forms of that fact.
The nature of questions to be asked should be at the level and competence of the individual student. For example, if you ask a student the question: what is 8 + 7? and the next student has difficulty with most addition facts, then the teacher should just ask: “what did the previous student give as the answer for the questions: 8 + 7? “What question did I ask and what was his/her answer?” “You know 7 + 7 is 14, then what is 7 + 8?” “What is 8 + 7?”
If the student is able to repeat it, then ask: if 8 + 7 = 15, then what is 7 + 8 = ? A child who already knows 8 + 7, then the teacher, in the next turn, should ask the child: 18 + 7 = ?, or 48 + 7 = ?, etc. She can then ask that student questions like this, out of turn also. This will help him to participate in the lesson and increase his confidence and deepen his understanding. This is an example of differentiation.
To develop flexibility in arithmetic facts, I suggest the following types of questions to be asked for each arithmetic fact.
7 + 4 = ______; 7 + ____ = 11; ____ + 4 = 11; ____ + ____ = 11; ____ = 7 + 4; 11 = 7 + ____; 11 = ____ + 4; 11 = ____ + ____
Once a student has mastered the family of facts related to a given arithmetic fact orally, then the teacher could give an Addition Fact Ladder to a pair of students and ask them to practice it orally by quizzing and helping each other.
The speed is also achieved by using the Visual Cluster cards. The teacher displays one card after another and asks questions using different strategies (see the list of Addition Strategies below):
- What is this card?
- What is one more than this number?
- What number will make it 10?
- What is the double of this number?
- What is one more than the double of the number? Etc.
This is repeated for the whole deck of Visual Cluster Cards. The same questions are repeated for making ten, doubles, double and one more, two more, making 9, making 11, etc.. After the practice has been done for each strategy individually, the teacher displays two cards and asks:
- What are these two numbers?
- What is the sum of these two numbers? If the student gives the correct answer, then ask:
- What strategy did you use?
- Can you use another strategy?
- Any other strategy? Etc. If the child doe not give the correct answer, the teacher helps him to arrive at the appropriate strategy to be used.
When To Automatize Arithmetic Facts?
To achieve mastery of numeracy by the end of fourth grade, mastering of fractions in fifth and sixth grade, and integers by the end of sixth grade, children should have mastered 45 sight facts by the end of Kindergarten; 100 addition facts by the end of first grade, 100 subtraction facts by the end of second grade, 100 multiplication facts by the end of third grade, 100 subtraction facts by the end of fourth grade. This should be with understanding, fluency, and applicability.
 Available electronically free from the Center for Teaching/Learning of Mathematics
 See Games and Their Uses by Sharma (2008).