What Does it Mean to Master Arithmetic Facts?

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:

17

 –9

Here are my observations.

Child One
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.

Child Two
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.

Child Three
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.

Child Four
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.

Child Five
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.

Fluency
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.

Accuracy
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.

Speed
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[1] and Fact War Games[2] 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.

[1] Available electronically free from the Center for Teaching/Learning of Mathematics

[2] See Games and Their Uses by Sharma (2008).

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What Does it Mean to Master Arithmetic Facts?

CCSS-M: Arithmetic and Algebraic Thinking

When we arrived as freshmen at my high school, our headmaster—a very popular, caring, and tough mathematics teacher (yes, he still taught), greeted us warmly and during his welcoming speech remarked: “Those of you who are fluent in fractions will end up in calculus and you know that fractions are dependent on multiplication. And those who do not have the mastery of multiplication tables and fractions will not enter into fields such as science, technology, engineering, mathematics, physics, and even economics. I do not want mathematics to be a gatekeeper for your aspirations. You should have all the skills that give you freedom of choice of options.” Most of my friends laughed at the remark.

After fifty years of teaching, I am convinced, more than ever, about the validity of that statement. All students should have the option to pursue any field, and a lack of proper preparation in mathematics should not close doors too early. Mathematics has become an entry to exciting and rewarding fields. Today, it is not just the STEM fields that require higher mathematics; even in the social sciences success and competence are dependent on skills in mathematics.

Throughout the twentieth century, most problems in natural and physical sciences, engineering, technology, and even social sciences could be modeled by functions – mostly by continuous and differentiable functions, but sometimes other functions such as: piece-wise, step, etc. Due to the advent of computers and related new technologies, today we can model many of the problems with only a few data points. Therefore, discrete mathematics (e.g., probability, statistics, linear programming, numerical analysis, etc.) and computer science play an important role in modeling problems in social sciences, physical and natural sciences. Similarly, the role of compu-graphics and graphing utilities in gaming systems and simulations is important in problem solving, therefore, in mathematics. This requires quantitatively and qualitatively a different kind of preparation in mathematics.

A different preparation in mathematics means that students, from the beginning of middle and high school, need to be made aware of the cumulative nature of mathematics: e.g., that mastery of multiplicative reasoning facilitates the understanding and the mastery of proportional reasoning (e.g., fractions/decimals/percent, rate, unit change, scale factor, and slope (ultimately, differential coefficient—rate of infinitesimal change in one variable with respect to the corresponding change in the other variable). The understanding and mastery of fractions are essential for success in algebra. And success in calculus is greatly dependent on facility in algebraic concepts, manipulations, and operations.

In high school, from ninth to 11th grade, students should formalize and extend the relationships between and integration of logical, arithmetic, algebraic, functional, geometric, probabilistic, linguistic, and statistical reasoning, their models and applications. Students should experience mathematics as a coherent, useful, and logical subject that makes use of their ability to make sense of problem situations. With this aim in mind, the framers of CCSS-M recommend three courses in high school: Algebra I, Geometry, and Algebra II.

The fundamental purpose of CCSS-M Algebra I is to formalize and extend the mathematics that students learned in the middle grades. CCSS-M’s Algebra I is built on the middle grades standards and is a more ambitious version of algebra traditionally taught in grades eight or nine. The topics deepen and extend understanding of linear and exponential relationships by contrasting them with each other and by applying linear models to data that exhibit a linear trend, and students engage in methods for analyzing, solving, and using quadratic functions. Solving algebraic problems procedurally, without the appropriate development of algebraic thinking, does not take students far in mathematics and the rigor of Algebra I escapes them. Such procedural work does not prepare them for higher mathematics and meaningful problem solving.

It is evident from children’s classroom work and performance on mathematics tests and examinations that most of their errors in solving problems in algebra I, such as algebraic equations, functions, and quadratic equations, are related to lack of mastery of facts and errors in operations on integers, fractions and lack of algebraic thinking. Because of these errors, even on easy word problems, the overall success rate of students using algebraic methods to solve problems is low and reflects a great deal of variation among students. In solving algebraic problems, a large number of students use few algebraic models, instead relying on arithmetic reasoning or guess and check, minimally useful but ineffective methods. The table below contrasts arithmetic and algebraic thinking.

Arithmetic Thinking Algebraic Thinking
Work from knowns to unknowns Work on unknowns to knowns
Thinking in natural language Thinking in symbolic terms and mathematics language
Unknowns transient Unknowns defined by the conditions of the problems and fixed for the particular problem

 

Equation as a formula to produce answers Equations and inequalities as descriptions of the relationships, parameters, and situations
Chains of successive calculations Chains of logically linked equalities or inequalities to transform them into simpler forms

 

Solution found to a specific problem Method and solution found to a category and classes of problems

The transition from arithmetic to algebra takes time and experience. It is the focus of middle school mathematics. However, till the rigor of CCSS-M holds in our schools, it should also get teachers’ attention during the high school years. It is important that the intervention work during the high school years focus on this aspect of transition from arithmetic to algebra.

During middle school and early part of high school, students need to see and understand the difference between arithmetic and algebraic reasoning. They need to see that for some simple problems, arithmetic methods are adequate although they do not lead to generalizations. Unless students experience several problems that are amenable to arithmetic and algebraic methods and see the inadequacy of arithmetic methods, they will have difficulty appreciating the importance of algebraic thinking. Let us consider a simple problem:

Two players, David and Mark, scored a total of 37 points in a game. David scored 5 points more than Mark. How many points did each score?

This problem can be solved by several methods.

Solution 1: Guess and check

14 + 23 = 37 but the difference is not 5

15 + 22 = 37 but the difference is not 5

16 + 21 = 37 the difference is 5

21 + 16 = 37 is also an answer as the difference is 5.

Since David scored 5 points more than Mark. David: 21 points; Mark: 16 points.

Solution 2: Arithmetic reasoning

If they both score equally, then I divide 37 by 2 = 18.5. But they did not score equally. One scored 5 more than the other. To have such a score, one is higher than 18.5 and the other is lower than 18.5 with a difference between the two scores of 5 points. Since 18.5 is the average, one is 18.5 + 2.5 = 21 and the other is 18.5 − 2.5 =16. David is 21 and Mark is 16.

Solution 3: Logical arithmetic reasoning

David’s score 5 points more than Mark’s score. I find 37 − 5 = 32. Since, 32 is the average of the two, now. I divide 32 equally. (37 − 5) ÷ 2 = 16.

David’s score = (37 – 5)/2 + 5 = 32/2 + 5 = 16 + 5

Mark’s score = (37 – 5)/2 = 32/2 = 16

Solution 4: Geometrical/pictorial

Mark’s score: ____________ (the length of the line segment represents Mark’s score)

David’s score: ____________ _____ (the line segment of the same length plus 5 points)

+ 5

Mark’s score + David’s score = 37

____________ ____________ _____ = 37 (the total = two line segments of same length + 5)

____________ ____________ = 37 – 5

____________ ____________ = 32

____________ = 32 ÷ 2 = 16

Mark’s score: 16 points

David’s score: 16 + 5 points = 21 points.

(In many countries in place of this line segment, representing an unknown, a bar is used.)

Solution 5: Algebraic approach to solution

Let us say Mark scored x points. Then David scored x + 5 points. Together they scored 37 points. So, x + x + 5 = 37.

x + (x + 5) = 37 <-> 2x + 5 = 37 <-> 2x = 32  <->   x = 16.

Mark scored 16 points; David scored 21 points.

The five approaches to solutions used by students from middle to high school are progressively algebraic in reasoning beginning with guess and check to arithmetic reasoning to more generalized algebraic thinking. These solution approaches indicate some of the ways in which arithmetic leads to algebra. For developing algebraic reasoning and the flexibility of thought and problem solving facility, students should be helped to realize the strengths, weaknesses, and limitations of each approach.

A guess and check solution is quite easy, accessible even to upper elementary school children, especially as the answer does not involve non-integral numbers and the numbers involved and the conditions of the problem are quite simple. It, thus, has limitations.

The next two solutions, the second using arithmetic reasoning and third using logical arithmetic reasoning are also simple. These methods are helpful in visualizing the problem and provide entry into the problem solving process, but they also have limitations if the numbers are not easy and if the parameters of the problem are complex. The fourth method begins to introduce the concept of unknown and the relationships between unknowns and knowns. It becomes the basis of the symbolic representation. The spatial representation (whether line segment representation or the bar method), even when the numbers in the problem are fractions, decimals, or percents, provides easy access to algebra early and effectively. Most high performing countries on mathematics use spatial representation of problems before symbolic representations.

The fifth method involves algebraic thinking and can be suggested by the spatial reasoning method, the arithmetic reasoning, and the translation from natural language to symbolic, mathematical language. Effective teachers always begin with discussing some kind of visual or spatial representation of the problem and then using logical reasoning lead to an algebraic representation.

In the first category of approaches—three methods based on arithmetic reasoning—students’ solution approaches began with the ‘knowns’ and moved to ‘unknowns.’ On the other hand, in the second category of approaches—the last two methods—the spatial reasoning and formal algebraic method, one begins from unknowns and establishes relationships between unknowns and knowns in the form of an equation. Then one applies a procedure, based on logical reasoning and properties of numbers, operations, and equality, to solve the equation.

There are fundamental differences in these two categories of solutions approaches. The three methods based on arithmetic reasoning are less applicable as they cannot be generalized. In the case of more complicated problems, the ‘guess and check’ is less straightforward and difficult to generalize, particularly when the answer is not an integer. Generalizing logical arithmetic reasoning method of Solution 2 is also hard, and even generalizing the method of Solution 3 is challenging for many students. As a result, the power of algebra in the form of integration of quantitative and spatial reasoning is needed. Although many students are unable to change or extend their approach, it can be accomplished if we begin with

  • translation from natural language to symbolic language
  • construct diagrams (Bar Graph, tables, charts, or Empty Number line), and
  • then translate into algebraic equation (integrate spatial, quantitative, and symbolic representations).

To truly understand algebraic problem solving, students should be taken through the first four methods before leading to algebraic methods. And they should arrive at the realization that when the problem is a little more difficult and involves non-integer numbers and solutions, the first four non-algebraic methods become much more difficult.

Solutions 2 and 3 of the problem above illustrate how arithmetic solutions to a problem work from ‘knowns’ (actual numbers) to unknowns (the answer) by a process of calculation. For example, from the known numbers 37 and 5, one finds the new quantity 32 (in Solution 2, the number of points to be shared equally after the extra 5 have been awarded to David), and then from this new known, one finds 16 (the points that they both score). The unknowns are transient: first the aim is to find one, by finding the first unknown, then one finds the next one, and so on in a chain of successive computations.

In contrast, the algebraic method is quite different. Instead of immediately progressing towards a solution, one understands and describes the situation in natural language: ‘One of them scores 5 more points than the other’ and then translates it into spatial and/or symbolic algebraic language. So if one scores say: x points (unknown) then, the other scores x + 5 (unknown). Then a relationship is found between unknowns and the known in the form of equation. So we have: x + (x + 5) = 37. Now from unknowns one moves to knowns by working towards a solution by calculation using the properties of numbers, operations, and equality; one then works on the whole equality, producing more (equivalent) equations until eventually the solution appears (here the transitive property of equality is hidden: first equation => second equivalent equation => etc.; the solution to the last equation, therefore, should be the solution to the first equation).

Statement                     Reasoning

x + (x + 5) = 37       Translating natural language into symbolic and relational language

(x + x) + 5 = 37       (Applying associative property of addition)

2x + 5 = 37            (Combining like terms—definition of addition)

2x + 5 –5 = 37 –5 (Addition/subtraction property of equality)

2x + 0 = 32             (Property of additive inverse and subtraction operation)

2x = 32                  (Property of additive identity), and finally, the equation:

x = 16.                 (Division property of the equality)

Solution: Mark’s score (x points) = 16 points; David’s score (x + 5 points) = 16 + 5 = 21 points.

The kind of thinking where one begins with identifying unknowns, establishes relationships between unknowns (an equation or expression), and finally moves to the known (solution process using rules, relationships, theorems, definitions, properties, etc.) is not natural. It is contrary to intuitive and arithmetical thinking.

In arithmetic, we begin with knowns and lead to unknown(s) using the logic of natural language. The translation from natural language and arithmetical thinking to algebraic language and relational thinking is a major transition for students. It requires experience and effective teaching—the development of mathematical (spatial and algebraic) language—vocabulary, syntax, and two-way translation from natural language to math and vice-versa), forming conceptual schemas (wherever possible concrete and visual and relevant models must be used), and arriving at efficient procedures.

Certain arithmetic and pre-algebraic skills need to be mastered in order to be successful in algebra. I find that along with the basic arithmetic facts and fractions[1], a few key mathematics skills are also needed for a student to be successful in algebra. The most important of these is how to operate on integers. Many of students’ problems in algebra can be traced to their limited numbersense and misconceptions about integers and lack or mastery on their operations. For example, most students conceptualize subtraction of whole numbers as ‘take away’ and multiplying and dividing integers only as repeated addition and repeated subtraction respectively. These schemas are sufficient for conceptualizing and operating on whole numbers, but they are limiting and inadequate for operating on fractions, decimals, integers, rational numbers, and algebraic expressions. Therefore, a strong understanding of numeracy (number, number relations, number operations, patterns in numbers, and properties of numbers) and proportional reasoning is essential for learning algebra, but there are also marked transformations (mathematical and cognitive) that have to occur in students’ thinking to become comfortable with problem solving using algebra rather than arithmetic. With this change, a student can access the power of tools from algebra and other higher mathematics. These tools are the language of STEM fields and social sciences such as: economics, business, psychology, geography, etc.

When students experience and engage in discussions about the efficiency and efficacy of different methods early on, they begin to think flexibly and acquire the ability to augment, extend, and adapt their methods of problem solving. When all (or almost all) of the students have solved the problem or made attempts in solving the problem, the teacher should solicit (orally or written form) all the approaches (successful or not successful). Place them on the board or show them under the document camera or a picture. She should then need to engage in the discussion that help students to see each other’s approach and thinking behind it. Then discuss the approaches using the criteria: whether the approach

  • furthers our thinking and understanding of the problem (that is the first criteria for the acceptance of a solution approach/method),
  • provides ‘exact/correct’ solution,
  • gives the correct solution efficiently (out of all the correct/exact solution approaches, we need to ask which one is more ‘efficient’. In other words, which method gives us the solution easier and with less consumption of resources and time?), and
  • is elegant

In the final stage of discussion, the whole class should focus on which method is ‘elegant’. A method is called elegant when it can be generalized, abstracted and works for many situations (class of problems rather than individual or specific problem)—a method that also leads to the standard procedure.

Arithmetic is generalizing concrete experiences to concepts and procedures. Algebra is generalizing arithmetic to relationships between numbers and concepts and then developing the concept of relationships between mathematical entities (e.g., numbers, etc.) and extending them to functions appropriate to model problems. For example, in the elementary and middle school, the formula for the area of a circle was intuitively understood or just accepted, whereas, in high school starting with a concrete model and by the use of the concept of limits, it is derived into A = πr2. The idea of deriving the formulas using fundamental principles, logic, and methods such as limits (a sum of infinite terms) is what differentiates earlier mathematics and the high school mathematics courses. Such thinking plays an important role in almost all topics of mathematics, for example, showing that a regular polygon approaches a circle when the number of sides approaches infinity or the price of a car decreases toward 0 as the number of years approaches infinity. Similarly, a diagram of a cone sectioned into cylindrical slabs gives a reasonable estimate for volume of the entire cone. The volume could be determined if these slabs were very thin, their volumes calculated and then summed. This leads to the basic idea behind integration—an important concept in calculus.

The key concepts: quantitative and spatial reasoning reach fruition by high school. For example, the concept of spatial reasoning that began in easier grades reaches its formal form in high school.

Spatial reasoning is observing objects and simple relationships between them—from spatial organization to formal concepts in geometry, trigonometry, and visualization and representation of transformations of geometrical objects. With the help of formal logic, it develops into rigorous treatment of understanding formal geometry: deriving formal definitions and formulas; making connections and inferences; proving and justifying claims using formal logic and language; describing relationships; integrating quantitative and spatial ideas to model problems into systems of equations and figures.

Quantitative reasoning ranges from conceptualizing number relationships to algebraic principles–generalizing, abstracting, extrapolatingalgebra is generalized arithmetic; reversibility of thought; pattern analysis—recognizing, extending, creating and applying patterns; propositional reasoning; analogies; moving from knowns to unknowns; from facts and procedures to relationships; expanding the set of integers to include irrationals, imaginary numbers. Once algebraic reasoning is achieved, with the integration of the spatial tools of geometry, all algebraic, geometric, trigonometric, probabilistic and even calculus tools are within the reach of a student. Of course, this is subject to the availability of effective and efficient methods teaching, support, and resources.

The focus in high school is to prepare students to see the role of mathematics in the world of natural, physical, and social sciences and to prepare them for higher education and work. At the end of high school, they should be able to apply the tools of arithmetic, algebra, geometry, trigonometry, and probability to diverse situations, such as (a) intra-mathematical (higher concepts in mathematics, e.g., calculus), (b) interdisciplinary (e.g., STEM fields, social sciences—economics, psychology, etc.), (c) extra-curricular (real life applications).

To be prepared to solve problems through mathematical modeling, at the end of high school, students should be sensitive to the presence of numerous patterns in the relationships between a variety of variables from diverse situations:

  • Direct and inverse variation—as one variable increases, another also increases (or decreases) at a similar rate.
  • Accelerated variation—as one variable increases uniformly, a second increases at an increasing rate.
  • Converging variation—as one variable increases without limit, another approaches some limiting value.
  • Cyclic variation—as one variable increases uniformly, the other increases and decreases in some repeating cycle.
  • Stepped variation—as one variable increases, another changes in jumps.

Nature of the Mastery of Curricular Elements
To make sure that students have learned the material we teach, we need to pay attention to the mastery of individual curricular elements:

  • Mastery of mathematics language: Possesses adequate vocabulary, the syntax, and can translate math expressions and equations into English and vice versa; can explain his or her thinking using mathematics language, symbols, and processes
  • Constructs, develops, and understands concepts: can demonstrate the related multiple mathematical models of a concept (e.g., for systems of equations, they know the rationale and reasons for different—graphical, substitution, elimination, matrix, and determinant—methods of   solving them)
  • Develops and executes procedures: Can execute standard procedures (including the development of them) accurately, consistently, fluently, efficiently with understanding (e.g., factoring trinomials, polynomial division, or synthetic division)
  • Automatizes skills: Produces the result in acceptable time, fluently, and consistently (e.g., can give facts about integers or laws of exponents orally in 2 seconds or less and written 3 seconds or less)

Problem solving and communication: Integrates language, concepts, procedures and skills in problem posing, solving and interpreting and communicating of results and the solution process.
As proposed by the framers of the CCSS-M, the central idea of a beginning algebra course is to become fluent in using and interpreting symbols so as to generalize the concepts from arithmetic and to see algebra as generalized arithmetic and to explore and study relationships and functions and their multiple representations. This means:

  • Understanding, mastering, and applying the operations on and properties of real and complex number systems and their applications;
  • Justifying operations on real numbers with rigor (e.g., the set of real number is complete—between any two real numbers there exists a real number);
  • Understanding the arithmetic of algebraic expressions—extending the understanding and mastery of arithmetic operations (e.g., extending long division process to division of a polynomial by a monomial and binomial) and understanding the geometric and algebraic behaviors of polynomials and rational fractions;
  • Extending the concept and operations of factorization and prime factorization to factorization of polynomials, particularly with integer coefficients;
  • Understanding—defining, mastering operations on functions—addition, subtraction, multiplication, division, composition (including trigonometric functions), and inverse of functions (including exponential and logarithmic); understanding their behavior and applications; modeling applications using functions; impact of transformations (rigid and dynamic) on functions;
  • Understanding and mastering the representation of and operations on information (data) linguistically (expressing ideas in words), arithmetically (number relationships including determinants and matrices), algebraically (expressions, systems of equations and inequalities), geometrically (pictorially, tabular, curves, figures), functionally (operations and compositions), discrete methods (determinants, matrices, flow charts), and probabilistically;
  • Intra- and interrelationships between algebra and geometry (understanding the relationship between two and three dimensional objects—generating 3-dimensiional objects from 2-dimensional objects and vice-versa), coordinate geometry—an equation represents a geometrical entity and most geometrical objects can be expressed as a system of algebraical equations; polar vs. Cartesian, etc.); e.g., a circle—a collection of points equidistant (r units) from a given point (h, k) drawn on a paper (geometrical representation) can be expressed by the Cartesian equation: (x – h)2 + (y –k)2 = r2 (derived using the distance formula—algebraic); can be converted into polar equation by sing the transformation: x = h + r cos (t) and y = k + r sin (t).
  • Understanding transformations (rigid and dynamic) and their representations (functional, matrices, etc.) and their impact on geometrical, discrete, and, algebraic systems (transformation, congruence, similarity, composition); studying systems through transformations;
  • Applications of tools—arithmetic, algebraic, geometrical, probabilistic, trigonometric, functional and technological, in learning concepts, acquiring skills, and solving problems.

Students should realize that the idea behind learning properties of whole families of relations is typical of all mathematics: recognition of structure and similarities in apparently different situations allows applications of successful reasoning methods to new problems.

 

 

 

[1] For fractions see How to Teach Fractions Effectively by Mahesh Sharma, 2008.

CCSS-M: Arithmetic and Algebraic Thinking

Order of Mathematics Operations

Question:  I read your blog on non-negotiable skills at the elementary and middle school grades. The order of operations is so important during the upper elementary and middle school level and students have so much difficulty. Do you think the order of operations is important to emphasize in the curriculum? I have always introduced it as a mnemonic device (PEMDAS). Could you suggest other ways to introduce it to children?

Answer: The order of operations is an important skill. I did not mention it among the non-negotiable skills because it is part of the mastery of numeracy. It represents the integration of arithmetic operations. It is an example of understanding the four arithmetic operations and should be introduced properly and with sound reasoning.

I was also planning to discuss this along with the Standards of Mathematics Practice (SMP) in CCSS-M, particularly under SMP Three: Construct viable arguments and critique the reasoning of others. In other words, as teachers, we should avoid as much as possible, giving children rules that we cannot support with mathematical reasoning. To teach mathematics ideas, we should present (a) concrete models, (b) set up patterns, (c) arrive at mathematics conjectures using concrete models and patterns, (d) use analogies, (e) use deductive and inductive reasoning, and (f) use formal proofs.

Before posting my response here, I questioned others who all said “I have not used PEMDAS for a long time. I think I know why we use it, but I am not sure how to explain it.” “I don’t know why we use it this way. I always put parentheses.” or “I don’t remember why, but yes it’s the way I always do.”

In the fifth or sixth grades, most teachers introduce the order of operations as a mnemonic device (e.g., PEMDAS—Parentheses, Exponents, Multiplication/Division, Addition/Subtraction; GEMDAS—Grouping symbols, Exponents, Multiplication/Division, Addition/ Subtraction; or BODMAS—Brackets, Operation—Exponents, Powers, Roots, Multiplication/Division, Addition/Subtraction).

Too many teachers simply give the order of operations as PEMDAS without explaining the reasons behind it and how different elements are related to each other. That creates problems for children. PEMDAS would be the same if it were written PEDMSA. The order of operations is based on sound mathematics reasoning and a history that students need to know.

History
The rules for the order of operations have grown gradually over several centuries and are still evolving. However, I would say that the rules actually fall into two categories: the natural rules (such as precedence of exponential over multiplicative over additive operations, and the meaning of parentheses) and the artificial rules (left-to-right evaluation, equal precedence for multiplication and division, and so on).

The former were present from the beginning of notation as the concepts were formalized (e.g., definitions of addition, multiplication, exponents, etc.). They probably existed already, though in a somewhat different form, in the geometric and verbal modes of expression that preceded algebraic symbolism. The latter, not having any absolute reason for their acceptance, were gradually agreed upon through usage and continue to evolve.

Before the invention of algebraic notation, people performed these operations in the order they were described in words using the grammar of the language. Today, however, with the definitions of operations and algebraic symbols being standardized, they have become universal, and we can give reasons behind this evolution and acceptance of a particular order of operations.

To decide the order of operations, originally a vinculum (an over-line or underline) was used, e.g., Capture1or 2 + 7 = 14. Here the expression under the vinculum is performed first. Since the introduction of modern algebraic notation and because of its definition (geometrically, it is two-dimensional), multiplication has taken precedence over addition. Thus 3 + 4 × 5 = 4 × 5 + 3 = 23. When exponents were first introduced in the 16th and 17th centuries, exponents took precedence over both addition and multiplication and could be placed only as a superscript to the right of their base. Thus 3 + 52 = 3 + 25 = 28 and 3 × 52 = 3 × 25 =75.

Today, parentheses or brackets are used to explicitly denote precedence by grouping parts of an expression that should be evaluated first. Thus resulting in 2 × (3 + 4) = 14 to force addition to precede multiplication or (3 + 5)2 = 64 to force addition to precede exponentiation (because exponentiation is multiplication repeated several times; geometrically, it is multi-dimensional). Parentheses (including braces/brackets, i.e. groupings—both apparent and hidden) are done first (following PEMDAS within the groupings) as they are multi-operational activities, then we do the unary operations like exponents and functions.

The basic rule (that multiplication has precedence over addition) appears to have arisen naturally and without much disagreement as algebraic notation was developed in the 1600s and the need for such conventions arose. That must have happened as multiplication was moved from repeated addition to groups of or the area of a rectangle (a two-dimensional concept). Before these agreed conventions, each author needed to state his conventions at the start of a book, creating a great deal of confusion. The need for the emergence of conventions was natural. The conventions also made the writing of expressions easier. An example is the emergence of different forms of multiplication symbols, e.g., ×, •Ÿ, ( ), and finally algebraic, where the product of a and b can be written as ab. For example, without these agreed conventions, we will have to write

4,567 = 4×(1000) + 5×(100) + 6×(10) + 7×(1) in place of
4,567 = 4×1000 + 5×100 + 6×10 + 7×1
and,
4, 567 = 4((10)3) + 5((10)2) + 6((10)1) + 7((10)0)
in place of
4, 567 = 4 × 102 + 5 ×102 + 6 ×101 + 7×100.

Similarly, without our order of operations, in algebra, in place of a very concise expression for the polynomial:
ax2 + bx + c
we would have to write
(a((x)2)) + (b)(x) + c
Look at the cumbersomeness of the expression.

The term “order of operations” and the mnemonics for its applications “PEMDAS/BEDMAS” mnemonics, were formalized only in this century, or at least in the late 1800s, with the growth of the textbook industry and teacher training institutions. It became more important to text authors than to mathematicians, who just informally agreed (through research journals).

Order of Operations
In mathematics, and to some extent in computer programming, the order of operations (or operator precedence) has become a convention—a collection of rules—that tells us and defines which procedures to perform first in order to evaluate a given mathematical (arithmetical or algebraic) expression (a finite combination of constants, variables, symbols and sub-expressions that is formed according to mathematics rules that depend on the context). These mathematical symbols can designate numbers (constants), variable, arithmetic operations (addition, subtraction, multiplication, division, exponents, powers, roots), functions, grouping, and other mathematical entities.

Though the order of operations has become, to a great extent, formalized in textbooks and classroom instruction, many students have difficulty applying it and many teachers have difficulty explaining it. To facilitate students’ understanding, it should be taught properly and with rigor.

One hallmark of mathematical understanding is the ability to justify, in ways appropriate to the student’s mathematical maturity, why a particular mathematical statement is true, where a mathematical rule comes from, and how and when that can be applied. There is a world of difference between a student who can summon the mnemonic PEMDAS (= Please Excuse My Dear Aunt Sally to implement the order of operations: Parentheses, Exponents, Multiplication, Division, Addition, and Subtraction) with an understanding for the underlying reasons and one who recites purely procedurally and lacks understanding. It is important to know the reasons behind this order of operations.

  1. Addition and subtraction are one-dimensional operations (linear—for example it is evident when we join two Cuisenaire rods or skip count to get the sum).
    They are at the same level and the first level of operations. If the two operations appear in the same expression, they are executed in order of appearance, first come first serve. Therefore, the order of operations at the end of second grade is in the order of their appearance. For example, 7 – 5 + 4 = 2 + 4 and 7 + 2 – 6 = 9 – 6 = 3.Capture2.PNG
    They are at the same level and the first level of operations. If the two operations appear in the same expression, they are executed in order of appearance, first come first serve Capture3. Therefore, the order of operations at the end of second grade is Capture3in the order of their appearance. For example, 7 – 5 + 4 = 2 + 4 and 7 + 2 – 6 = 9 – 6 = 3.
  2. Multiplication and division are two-dimensional operations (as represented by an array or the area of a rectangle);
    Capture4
    therefore, they are at a higher level than addition and subtraction and must be performed before addition and subtraction.And, if they both appear in an expression, they should be treated as first come first serve Capture5. Therefore, by the end of fourth grade, the order of operations should be:Capture7 . Thus, once multiplication (and division being the inverse of multiplication) has been introduced, multiplicative reasoning takes precedent over additive reasoning and performs them from left to right. The instruction in these problems should be: “simplify,” “compute,” “evaluate,” or “calculate” not “solve the problem” or “apply GEMDAS.” Even “simplify the expression by using the order of operations” is too procedural. Students should learn to make decisions in mathematics early on. That is what improves their metacognition and develops mathematical ways of thinking.
    Simplify
    : 3 + 6 × 2
    Multiplication before Addition: 3 + 6 × 2 = 3 + 12 = 15
    Simplify: 9 – 12 ÷ 3
    Division before Subtraction: 9 – 4 = 5
    Simplify:  9 – 12 ÷ 3 + 3 + 6 × 2Division and Subtraction first in order of appearance and then addition and subtraction in order of appearance: 9 – 4 + 3 + 6 × 2 = 9 – 4 + 3 + 12 = 5 + 3 + 12 = 8 + 12 = 20.If all of the four operations: addition, subtraction, multiplication, and division appear in a mathematical expression, the order should be:Capture7, higher order (2-dimensional) multiplication and division (in order of appearance) are performed first and then 1-dimensional addition and subtraction (in order of appearance) are performed next.

    The ability to apply Capture7 correctly, consistently, and fluently with understanding indicates the mastery of numeracy skills. For example, simplify the expression:

    7 + 3 × 8 ÷ 2 – 4 + 2×5 ÷2 + 5

    In the later grades (fifth grade and above), it is important to treat division as multiplication by the reciprocal (multiplicative inverse) and subtraction as addition of the opposite (additive inverse). Thus ¾ = 3 ÷ 4 = 3 • ¼; in other words the quotient of 3 and 4 equals the product of 3 and ¼. Also 3 − 4 = 3 + (−4); in other words, sometimes, the difference of 3 and 4 should be seen as the sum of positive three and negative four. Thus, 1 − 3 + 7 can be thought of as the sum of 1, negative 3, and 7, and add in any order: (1 − 3) + 7 = −2 + 7 = 5 and in reverse order (7 − 3) + 1 = 4 + 1 = 5, always keeping the negative sign with the 3.

  3. Exponential expressions are multi-dimensional (depending on the size of the exponent, e.g., a 10-cube is 3-dimensional and the exponent is 3 with a base of 10).
    Capture8
    An exponent is defined as the multi-use of multiplication; therefore, exponentiation operation is more important and higher order than multiplication (and division can be written as multiplication) and definitely higher than addition and subtraction; therefore, it must be performed before all of these four operations. Therefore, the order of operation so far is: Capture9. Start simplifying exponents (powers, roots, indices) first, then multiplication and division (in order of appearance) and then addition and subtraction (in order of appearance). Example:72 + 3 × 8 ÷ 22 – 4 + 2×5 ÷2 + 52Stacked exponents are applied from the top down, i.e., from right to left. Because exponentiation is right-associative in mathematics, we have:
    Capture10
  4. Sometimes, the intended order of computation is indicated by grouping certain operations or expressions in a given expression. For ease in reading, other grouping symbols such as braces, sometimes called curly braces { }, or brackets, sometimes called square brackets [ ], are often used along with parentheses ( ). For example:Capture12Absolute value symbol |   | is also a grouping symbol.
    |−(7 + 2) – 3| = | −9 − 3| = |−12| = 12Grouping operations such as brackets, braces, parentheses (either transparent or hidden, and absolute value; function and radical operations are also hidden operations), etc. may involve all of the above operations in multiple forms, therefore, are higher than all of the above. Therefore, the operations in the grouped expressions should be performed before. When grouping, exponents and all four operations are involved, then the order of operations should be:Capture13.Here, G represents grouping operations—both transparent and hidden. In the transparent grouping operations, the order of simplifying an expression is parentheses, braces, and brackets—from the innermost to the outermost group. They are organized in expressions in the same order—the brackets being the outermost. Brackets: (parentheses), {braces or curly braces}, or [brackets] are examples of mathematical grouping symbols and they have their own rules and impose their own order. For example, the previous expression: 2 + 3 × 4, can be reorganized into (2 + 3) × 4 giving us the simplified form as: 20. Grouping symbols are the only way to change this order and on occasion the division symbol presents itself as a grouping symbol (called hidden grouping as the Capture15read as 8 plus 2 divided by 4 + 1. Here 8 + 2 is being actually being read as (8 + 2)—as a group divided by (4 + 1) as a group and the line when used horizontally is indicating division. We should avoid writing the line as a slash to avoid misconception on the part of children.Symbols of grouping can be used to override the usual order of operations. Grouped symbols can be treated as a single expression. Symbols of grouping can be removed using the associative and distributive laws, and they can be removed if the expression inside the symbol of grouping is sufficiently simplified, so no ambiguity results from their removal.The hidden grouping operations are performed in the context. The context defines whether there is a hidden grouping or not. In a fraction expression, the numerator and denominator, because of the way we express them, define hidden grouping operations, even though there are no transparent grouping operations involved. For example, in the case of the fractionCapture16, the fraction is read as: sum of 3 and 5 divided by the difference of 3 and 1 (3 plus 5 then divided by 3 minus 1). As a result, the hidden grouping becomes transparent as: (3 + 5) ÷ (3-1). Therefore, before we simplify the fraction (performing the division operation), we first simplify the numerator and denominator—the hidden groupings. A horizontal fractional line, in an expression, acts as a symbol of grouping:
    Capture17

Many students and even some teachers are confused by texts that either teach or suggest that implicit multiplication (2x) takes precedence over explicit multiplication and division (2•Ÿx, 2/x) in expressions such as a/2b, which they would take as a/(2b), contrary to the generally accepted rules. The idea of adding new rules like this implies that the conventions are not yet completely stable; in other words, the situation is not all that different from the 1600s.

The slash sign (“/”), as indicated above, for a fraction creates a great deal of difficulty for students, even high school students. Either it should be avoided and proper fraction expression should be used, or enough time and explanations should be used to remove the ambiguity. For example, there can be ambiguity in the use of the slash (‘/’) symbol in expressions such as 1/2x. If one rewrites this expression as  1 ÷ 2 × x and then interprets the division symbol as indicating multiplication by the reciprocal, this becomes:
Capture18
With this interpretation 1/2x is equal to (1/2)x. However, in some of the academic literature, implied multiplication is interpreted as having higher precedence than division, so that 1/2x equals 1/(2x), not (1/2)x.

It is important to emphasize to the students the importance of hidden grouping. In the following expressions, there are hidden groupings:
Capture19

There are many places and concepts in middle and high school mathematics where hidden grouping appears and is important to discern, particularly in trigonometry and algebra. Students show difficulty with hidden groupings, so we need to help them discern them and practice writing expressions with hidden grouping.

The root symbol, √, requires a symbol of grouping around the radicand. The usual symbol of grouping is a bar (called vinuculum) over the radicand. Other functions, such as trigonometric functions, use parentheses around the input to avoid ambiguity. The parentheses are sometimes omitted if the input is a monomial. Thus, sin x = sin(x), but in when x and y are involved, in order to reduce the ambiguity and depending on the nature of the x and y, sin x + y = sin(x) + y, or sin (x + y) because x + y is not a monomial. Some calculators and programming languages require parentheses around function inputs, some do not.

Simplify:  7 + (6 × 52 + 3)

7 + (6 × 52 + 3) Given
7 + (6 × 25 + 3) Start inside the Grouped expression, then use Exponents First
7 + (150 + 3) We still have grouping, Multiply inside the group
7 + 153 Grouping completed, last operation is add
7 + 153 Then Add
160 DONE !

Inside a grouping expression, the same convention of order of operations is applied. Let us take an example (we are going to simplify later in detail):

72 + 3[4 × 5 − 3{21+2 + 3(28 ÷ 7 + 3)}] + 9 + 23

In this case, all the previous operations are involved. This expression has all the operations: grouping operations (both transparent and hidden), exponents, multiplication, division, addition, and subtraction. It is quite a complex expression, and these kinds of expressions appear only in higher grades. When we introduce this procedure, we should use simpler examples. However, here it will demonstrate the whole procedure.

In simplifying this expression, first we look at the grouping operations, in order. Therefore, the grouping operations (parentheses, braces, and brackets in this order) are performed first. The hidden grouping is contextual. Then exponential operations need to be performed and, after that, multiplication and division in order of their appearance. The last operations to be performed are addition and subtraction in order of their appearance.

This means that if a mathematical expression is preceded by one binary operator and followed by another, the operator higher on the list should be applied first. The properties of operations such as commutative, associative, and distributive laws of addition and multiplication allow adding terms in any order, and multiplying factors in any order—but mixed operations must obey the standard order of operations as defined above. Let us illustrate them in one problem. In the beginning, students should show their work as follows:
Capture21

When we want students to practice a concept, rule, or procedure, we should give them examples (at least four—one example becomes an exemplar of the concept, the second one begins to see the parameters of the concept, the third begins to set the pattern, and the fourth verifies the understanding of the pattern) to illustrate the idea and practice it correctly. We should also give examples where they do not work to highlight the nuances, subtleties, parameters, and conditions where the rule, concept, or procedure does not apply. The role of counter examples is very important in learning a mathematics idea. When a student applies a procedure incorrectly, we need to point out the part of the procedure, the definition/concept, or rule that was violated.

Examples:
Simplify:
8 × (5 + 3)  = 8 × 8 = 64.  (Correct)
8 × (5 + 3)  = 40 + 3 = 43.  (Incorrect)
5 × 42+ 3 = 5 × 16 +3 = 80 + 3 = 83 (Correct)
5 × 42+ 3 = 20 × 4 +3 = 80 + 3 = 83 (Incorrect procedure; Correct answer)
5 × 42+ 3 = 202 +3 = 400 + 3 = 403 (Incorrect)

Finally, w should ask students to construct an example of the concept, rule, definition, or procedure under discussion. When a student can construct an example, he or she can proceed in applying the procedure.

Mnemonics
Mnemonics are often used to help students remember the rules, but the rules taught by the use of acronyms only can be misleading. A mnemonic or memory device is only a technique that aids information retention in human memory. Mnemonics translate information into a form that the brain can retain better than its original form. Even the process of merely learning this conversion might already aid in the transfer of information to long-term memory. However, they should be used only after the concept has been understood.

When students have practiced a procedure really well and only a few are still having trouble with it, you can provide a graphic organizer or mnemonic device. Only when they have explained orally, partly or fully, what they are going to do, let them consult the graphic organizer or the mnemonic device to reinforce. You never want students to depend on organizers or mnemonic devices, and you should always ask for the reasons behind their use.

G Groupings (transparent and hidden)—first
E Exponents (i.e. Powers and Square Roots, orders, indices, etc.)
MD Multiplication and Division (left-to-right)
AS Addition and Subtraction (left-to-right)

In the United States the acronym PEMDAS is common. It stands for Parentheses, Exponents, Multiplication, Division, Addition, and Subtraction. PEMDAS is often expanded to “Please Excuse My Dear Aunt Sally”, with the first letter of each word creating the acronym PEMDAS.

Canada uses BEDMAS, standing for Brackets, Exponents, Division, Multiplication, Addition, and Subtraction. Most common in the UK and Australia are BODMAS and BIDMAS.

These mnemonics may be misleading if the user is not aware that multiplication and division are of equal precedence, as are addition and subtraction. Using any of the above rules in the order “addition first, subtraction afterward” would also give the wrong answer to the problem:Capture22
The correct answer is 9 (and not 5, as if the addition would be carried out first and the result used with the subtraction afterwards). The best way to understand a combination of addition and subtraction is to think of the subtraction as addition of a negative number. In this case, the problem can be seen as the sum of positive ten, negative three, and positive two:Capture25
All of these acronyms conflate two different ideas, operations on the one hand and symbols of grouping on the other. If not properly taught and practiced, these acronyms lead to children’s misconceptions. Even after children have mastered the order of operations, every time teachers use the acronyms, they need to point out or ask children the reason for the precedence of a certain operation in this convention. The order of operations is the foundation of all computations in mathematics; therefore, its mastery is essential for success in learning and using mathematics.
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Order of Mathematics Operations