# Attend to Precision: The Foundation of Mathematical Thinking

The sixth of the Standard of Mathematics Practice (SMP) in Common Core State Standards (CCSS-M) is: Attend to Precision. The key word in this standard is the verb “attend.” The primary focus is attention to precision of communication of mathematics—in thinking, in speech, in written symbols, in usage of reasoning, in applying it in problem solving, and in specifying the nature and units of quantities in numerical answers and in graphs and diagrams. With experience, the concepts should become more precise, and the vocabulary with which students name the concepts, accordingly, should carry more precise meanings.

The word “precision” calls to mind accuracy and correctness—accuracy of thought, speech and action. While accuracy in calculation is a part, clarity in communication is the main intent of this standard. The habit of striving for clarity, simplicity, and precision in both speech and writing is of great value in any discipline and field of study. In casual communication, we use context and people’s reasonable expectations to derive and clarify meanings so that we don’t burden our communication with too many details that the reader/listener can surmise anyway. But in mathematics (thinking, communicating, and writing), we base each new idea/concept logically on earlier ones; to do so “safely,” we must not leave room for ambiguity and misconceptions.

Students can start work with mathematics ideas without a precise definition. With experience, the concepts should become more precise, and the vocabulary with which we name the concepts can, accordingly, should carry more precise meanings. But we should strive for clarity and precision constantly. Striving for precision is also a way to refine understanding. By forcing an insight into precise language (natural language or mathematical symbols), we come to understand it better and then communicate it effectively. For example, new learners often trip over the order relationships of negative numbers until they find a way to reconcile their new learning (–12 is less than –6) with prior knowledge: 12 is bigger than 6, and –12 is twice –6, both of which pull for a intuitive feeling that –12 is the “bigger” number. Having ways to express the two kinds of “bigness” and the sign defining the direction helps distinguish them. Learners could acquire technical vocabulary, like magnitude or absolute value, or could just refer to the greater distance from 0, but being precise about what is “bigger” about –12 helps clarify thinking about what is not bigger. With such a vocabulary, one can express the relationship between the two numbers more precisely.

The standard applies equally to teachers and students and by extension to textbooks, modes and purpose of assessments, and expectations of performance. To achieve this, teachers need to be attentive to precision in their teaching and insist on its presence in students’ work. They should demonstrate, demand and expect precision in all aspects of students’ interactions relating to mathematics with them and with other students. Teachers must attend to what students pay attention to and demonstrate precision in their work, during the learning process and problem solving. This is not possible unless teachers also attend to the same standards of precision in their teaching.

Teachers, while developing students’ capacity to “attend to precision,” should focus on clarity and accuracy of process and outcomes of mathematics learning and in problem solving from the beginning of schooling and each academic year. For example, teachers can engage their students in a “mathematics language talk” to describe their mathematics activity. The emphasis on precision can begin in Kindergarten where they talk about number and number relationships and continues all the way to high school where they furnish mathematics reasoning for their selection and use of formulas and results.

Attention to precision is an overarching way of thinking mathematically and is essential to teaching, learning, and communicating in all areas of mathematical content across the grades.

For the development of precision, teachers should probe students to defend whether their requirements for a definition are adequate as an application to the problem in question, or whether there are some flaws in their group’s thinking that they need to modify, refine and correct. Just like in the writing process, one goes through the editing process, students should come to realize that in mathematics also one requires editing of expressions to make them appealing, understandable and precise.

However, communication is hard; precise and clear communication takes years to develop and often eludes even highly educated adults. With elementary school children, it is generally less reasonable to expect them to “state the meaning of the symbols they choose” in any formal way than to expect them to demonstrate their understanding of appropriate terms through unambiguous and correct use.

The expectations according to the standard are that mathematically proficient students

• communicate their understanding precisely to others using proper mathematical terms and language: “A whole number is called prime when it has exactly two factors, namely 1 and itself” rather than “A number is called prime if it can be divided by 1 and itself.
• use clear and precise definitions in discussion with others and in their own reasoning: e.g. “A rectangle is a four straight-sided closed figure with right angles only” rather than “A four-sided figure with two long sides and two short sides.”
• state the meaning of the symbols they choose, use the comparison signs ( =, >, etc.) consistently and appropriately, for example, the names of > and < are not greater and smaller than respectively, but depend on how we read them: x > 7 is read as: x is greater than 7 or 7 is less than x; 2x + 7 = -5 + 3x is bidirectional (2x + 7 => -5 + 3x and 2x + 7 <= -5 + 3x).
• are careful about the meaning of the units (e.g., “measure of an angle is the amount of rotation from the initial side to the terminal side” rather than “measure of an angle is the area inside the angle or the distance from one side to the other”), identifying and specifying the appropriate units of measure in computations, and clearly labeling diagrams (e.g., identify axes to clarify the correspondence with quantities and variables in the problem, vertices in a geometrical figure are upper case letters and lengths are lower case letters, and the side opposite to the <A in ΔABC is denoted by “a”, etc.).
• calculate accurately and efficiently, express numerical answers with a degree of precision appropriate for the problem context (e.g., the answer for the problem: “Calculate the area A of a circle with radius 2 cm” is A = 4π sq cm not A = 12.56 sq cm; if x2 = 16, then x = ± 4, not x = 4, whereas √16 = 4, etc.).
• know and state the conditions under which a particular expression, formula, or procedure works or does not work.

Beginning with the elementary grades, this means that students learn and give carefully formulated explanations to each other and to the teacher (at Kindergarten level it may mean that the child explains her answer for 8 + 1 = 9 as “I know adding by 1 means it is the next number. I know 9 is next number after 8” or can show it concretely as “Look here is the 8-rod add the 1-rod and I get the 9-rod.” By the time they reach high school, they have learned to examine claims—their own and others’ in mathematical conversations, make explicit use of definitions, formulas, and results, and proper and adequate reasoning. At the high school level the explanations are rooted in any or more of these:

• demonstrating it concretely,
• showing by creating and extending a pattern,
• application of analogous situation, or
• logical reasoning—proving it using either deductive or inductive reasoning or using an already proved result.

What Does the “Attention to Precision” Look Like?
Effective mathematics teachers who use precision and efficiency in their teaching and encourage precision in their classrooms produce mathematically proficient students. Mathematically proficient students understand the role of precision in mathematics discourse and learning. They understand that mathematics is a precise, efficient, and universal language and activity. Precision in mathematics refers to:

Language

• Appropriate vocabulary (proper terms, expressions, definitions), syntax (proper use of order of words), and accurate translation from words to mathematical symbols and from mathematical symbols to words.
• Knowledge of the difference between a pattern, definition, proof, example, counter example, non-example, lemma, analogy, etc. at the appropriate grade level.
• Reading and knowing the meaning of instructions: compute or calculate (4 × 5, √16, etc., not solve), simplify (an expression, not solve), evaluate (find the value, not solve), prove (logically, not an example), solve (an equation, problem, etc.),
• Know the difference between actions such as: sketch, draw, construct, display, etc.
• Precise language (clear definitions, appropriate mathematical vocabulary, specified units of measure, etc.).

Teacher instruction about vocabulary must be clear and correct and must help children to understand the role of vocabulary in clear communication: sometimes formal terms and words distinguish meanings that common vocabulary does not, and in those cases, they aid precision; but there are also times when formal terms/words camouflage the meaning. Therefore, while teachers and curriculum should never be sloppy in communication, we should choose our level of precision appropriately. The goal of precision in communication is clarity of communication and achieving understanding.

A teacher can use familiar vocabulary to help specify which object(s) are being discussed—which number or symbol, which feature of a geometric object—using specific attributes, if necessary, to clarify meaning. Actions such as teaching writing numerals to Kindergarten by “song and dance” is a good starting point, but ultimately the teacher should use the proper directional symbols, e.g.,

• To write number “4” the teacher first should point out the difference between the written four (4) and printed four (4). Then she needs to show the direction of writing (start from the top come down and then go to the right and then pick up the pencil and start at the same level to the right of the first starting point and come down crossing the line).
• When discussing a diagram, pointing at a rectangle from far away and saying, “No, no, that line, the long one, there,” is less clear than saying “The vertical line on the right side of the rectangle.”
• Compare “If you add three numbers and you get even, then all the numbers are even or one of them is even” with “If you add exactly three whole numbers and the sum is even, then either all three of the numbers must be even or exactly one of them must be even.”
• Compare giving an instruction or reading a problem as “when multiply 3 over 4 by 2 over 3, we multiply the two top numbers over multiply two bottom numbers” to “find the product of or multiply three-fourth by two-third, the product of numerators is divided by the product of denominators.”

Elementary school children (and, to a lesser extent, even adults) almost never learn new words effectively from definitions. Virtually all of their vocabulary is acquired from use in context. Children build their own “working definitions” based on their initial experiences. With experience and guidance, the concepts should become more precise, and the vocabulary with which children name the concepts will carry more precise meanings. Formal definitions generally come last. Children’s use of language varies with development but typically does not adhere to “clear definition” as much as to holistic images. If the teacher and curriculum serve as the “native speakers” of clear Mathematics, young students, who are the best language learners around, can learn the language from them.

Quantities
Accuracy (know the difference between exact, estimate, approximation and their appropriateness in context) and appropriate level of precision in use of numbers (level and degree of estimation, significant digits, significant powers, units of measurement), correct classification and location of number on the number line (e.g., to locate ⅞, one divides the unit segment into halves and then each half into fourths, and then each fourth into eighths and then locates ⅞ rather than arbitrarily divide the unit segment into eight parts), correct relationships between numbers (e.g., √(140) is between 11 = √(121) and 12 = √(144), because, we have 121 < 140 < 144, therefore, √(121) < √(140) < √(144), but √(140) much closer to 12 as 140 is much closer to 144 than 121), selection of appropriate range and window on graphing calculator, tool selection (when to use what tools–paper-pencil, concrete models, diagrams, abstract, or calculator), and appropriate meaning of numbers in the outcome of operations (what role do the quotient and remainder play in the outcome from the long division algorithm, etc.). Precise numbers (calculate accurately and efficiently; given a context, round to an appropriate degree of precision)

Teachers should use written symbols correctly. In particular, the equal sign (=) is used only between complete expressions and signals the equality of those two expressions. To explain one way to add 42 + 36, we sometimes see it written (incorrectly) this way: 40 + 30 = 70 + 2 = 72 + 6 = 78. This is a correct sequence of calculator buttons for this process but not a correct written mathematics expression: 40 + 30 is not equal to 70 + 4; only the last equals sign is correctly used. We need the = sign to have a single, specific meaning. Also, the equal sign should not be misused to mean “corresponds to”: writing “4 boys = 8 legs” is incorrect.

Models
Appropriate choice of concepts and models in the problem solving approach: choice of strategy in addition/subtraction (8 + 6 = 8 + 2 + 4 = 10 + 4 = 14, 8 + 6 = 4 + 4 + 6 = 10 + 4 = 14, 8 + 6 = 2 + 6 + 6 = 2 + 12, 8 + 6 = 8 + 8 – 2 = 16 – 2 = 14, or 8 + 6 = 7 + 1 + 6 = 7 + 7 = 14 rather than “counting up” 6 from 8 or 8 from 6), appropriate multiplication/division model (the only models of multiplication work for fraction multiplication are “groups of” or “area of a rectangle” not “repeated addition” and the “array” models), which exponential rule, which rule of factoring, which rule for differentiation, what parent function to relate to, what formula to use, etc.

Reasoning, Symbols, and Writing Mathematics
Appropriate and efficient use of definitions, reasons, methods of proof, and order of reasoning in solving problems and explanations. For example, children should know the reasons for using the “order of operations” or that the solutions of equations have domains and range. Precise usage of symbols and writing:

• Choose correct symbols and operators to represent a problem (knowns and unknowns; constants and variables),
• State the meaning of the symbols and operations chosen appropriate to the grade level (multiplication: 4×5, 45, 4(5), (4)5,(4)(5), a(b), (a)b, (a)(b), ab),
• Label axes, shapes, figures, diagrams, to clarify the correspondence with quantities in a problem, location of numbers,
• Show enough appropriate steps to communicate how the answer was derived,
• Organize the work so that a reader can follow the steps (know how to use paper in an organized and systematic form—left to right, top to bottom),
• Clearly explain, in writing, how to solve a specific problem,
• Use clear definitions in discussion with others and in reasoning
• Specify units of measure and dimensions,
• Calculate accurately and efficiently.

At the elementary level, even the simplest of things such as: the proper way of forming numbers and mathematical symbols, writing the problems solving steps in a sequence: ([3(4 + 8) – (4 ÷ 2)] = [3(12) – (2)] =[36 − 2]= 34 rather than 4 + 8 = 12 × 3 = 36 −2 = 34). Similarly, clarity in reading numbers and mathematical symbols needs to be  emphasized from the beginning (e.g., ¾ is read as “3 parts out of 4 equal parts” rather than “3 out of 4,” “3 divided by 4” rather than “3 over 4.”

It is difficult to change inappropriate and incorrect habits later on. For example, when elementary grade teachers do not emphasize the importance of aligning multi-digit numbers in their appropriate place values, this creates problems for children later. New symbols and operations are introduced at each grade level, so it is important for the teacher to introduce them correctly and then expect precision in their execution.

Similarly, when middle and high school students are not instructed to write fractions properly, it creates problems. The following high school lesson illustrates the point. The problem on the board was:

To solve the equation, in order to eliminate fractions in the equation, the student suggested we multiply the whole equation by the common denominator of all the fractions in the equation (a correct and efficient method). When I asked for the common denominator, the student said: 9x because the denominators are 3x, 9 and 3. The error is purely because of lack of precision in writing fractions in the equations.

Precision often means including units when specifying numerical quantities. But not always. The purpose of precision is never to create work, only to create clarity. Sometimes a number is clear by itself, other times a unit is needed, sometimes a whole sentence is required: the situation determines the need. For the same reason, label graphs and diagrams sufficiently to make their meaning and the meanings of their parts clear.

Exposure and consistent questions from the teacher such as the following help students to be accurate, precise and efficient:

• Is this the right way of writing the expression (number, symbol, etc.)?
• Does the diagram you have drawn show the elements asked for or given in the problem?
• Is this the right unit for the quantities/numbers given in the problem?
• What mathematical terms apply in this situation?
• Is the term you used the right one in this situation?
• How do you know your solution is reasonable and accurate?
• Explain how you might show that your solution answers the problem?
• How are you showing the meaning of the quantities given in the problem (e.g., problem says: “the length of the rectangle is 3 more than twice the width)? Does your rectangle demonstrate the right dimensions? Your rectangle looks like a square.
• What symbols or mathematical notations are important in this problem?
• What mathematical language, definitions, known results, properties, can you use to explain ….?
• Can you read this number (symbol, expression, formula, etc.) more efficiently?
• Is ___ reading (saying, writing, drawing, etc.) correctly? If not, can you state it correctly and more efficiently?
• How could you test your solution to see if it answers the problem?
• Of all the solutions and strategies presented in the classroom, which ones are exact/correct?
• Which one of the strategies is efficient (can achieve the goal more effectively)?
• What would be a more efficient strategy?
• Which one is the most elegant (can be generalized and applied to more complex problems) strategy? Etc.

The number and quality of questions in a classroom bring the attention of students to appropriate and precise conversation. In a fourth grade geometry lesson, I had the following exchange with the students:

Sharma: Look at this rectangle (I was holding one of the 10 by 10 by 1 rectangular solids in my hand) while touching the 10 by 10 face, I asked: What are the dimensions of this rectangle?

A student raised his hand and said: “That is not a rectangle. It is a square.”

I said: “yes, it is a square. Can you also call it a rectangle? Is it also rectangle?”

“No!” He declared emphatically.

I asked the class: “How many of you believe that it is not a rectangle?” Almost every hand went up.

When I asked them what the definition of a rectangle was, almost all of them said: “A rectangle has two long sides and two shorter sides.” I drew a quadrilateral with 2 long sides and 2 short sides that did not like a rectangle.

Another student said: “The sides are parallel.” I drew a parallelogram.

The student said: “No! That is not what I mean. Let me show you what I mean.” He drew a correct rectangle.

One student said: “A rectangle has four right angles and 2 longer sides and 2 shorter sides. Like this.” He drew a correct rectangle.

We had a nice discussion and came to the conclusion that a rectangle is: A straight-sided closed figure with four right angles. I also emphasized the meaning of the word “rectangle.”  It is made up of two words “recta” and “angle.” The word “recta” means right.  Therefore, a rectangle has only right angles. With this discussion and the precise definition, they were able to accept and see the face of the object I was showing as a rectangle.

This episode, in one form or the other, is repeated in many classes, from urban to rural classrooms, in many elementary schools. The same misconception is present even in many classrooms in many middle and high schools students. This is an example of lack of precision in teaching and, therefore, lack of precision in student understanding and expression.

There are many examples of such misconceptions. For instance, children often misunderstand the meaning of the equal sign. The equal sign means is “the same as,” “equal in value” “equal in some specified characteristic—length, area, quantity, volume, or weight,” but most primary students believe the equal sign tells you that the answer is coming up to the right of the equal sign. When children only see examples of number sentences with an operation to the left side of the equal sign and the answer on the right, this misconception is formed and generalized. Teachers should, therefore, emphasize the true meaning of the equal sign. From the very beginning—Kindergarten children should be shown that the equal sign “=” is a two-way implication. For example, Kindergarteners should be shown and know the simple facts as: 2 + 8 = 10, 8 + 2 = 10 & 10 = 2 + 8, 8 + 2 = 10 and first graders need to see equations written in multiple ways, for example 5 + 7 = 12, 7 + 5 = 12, 12 = 5 + 7, 12 = 7 + 5, and 5 + 7 = 2 + 10, 5 + 7 = ☐+10, ☐ + 2 = 9 + ☐. Although most above average and many average children are able to realize this level of understanding of the concept of equal or equal sign, there are many average and children with learning disabilities who have difficulty in reaching that level of understanding. This level of precision in understanding can be achieved by using Cuisenaire rods, the Invicta math balance for teaching arithmetic facts, and proper and appropriate language usage and questioning by teachers.

If students are taught using imprecise language, they will necessarily learn imprecise language and concepts, because language is the basis of mathematics learning. Later, they will not only resist when asked to use precise language in mathematics, but they will also have difficulty applying the concepts. A sequence of ideas begins to take place in students’ mind when we ask questions and emphasize language.
Questions instigate language.
Language instigates models.
Models instigate thinking.
Thinking instigates understanding.
Understanding produces conceptual schemas.
Conceptual schemas produce competent performance.
Competent performance produces long lasting self-esteem.
Self-esteem produces willingness to inquire and learn.

With proper language and conceptual models a great deal can be achieved. It is not too late to instill precision even at the high school level; however, if it is not emphasized at the elementary and middle school levels, it is much more difficult to do so. This does not mean we give up; it only means we redouble our effort and find better ways of doing it, such as using concrete models, patterns, and analogies when we are introducing new mathematics concepts and procedures.

As students progress into the higher grades, their ability to attend to precision will expand to be more explicit and complex if we constantly use proper language and symbols.

As students develop mathematical language, they learn to use algebraic notation to express what they already know and to translate among words, symbols, and diagrams. Possibly the most profound idea is giving names to objects. When we give numbers names, not just values, then we can talk about general cases and not just specific ones.

Correct use of mathematical terms, symbols, and conventions can always achieve mathematical precision but can also produce speech and writing that is opaque, especially to learners, often to teachers, and sometimes even to mathematicians. Good mathematical thinking, therefore, requires being correct, but with the right simplicity of language and lack of ambiguity to maintain both correctness and clarity for the intended audience. If we are particular about this in the first few grades, it becomes much easier to attend to precision in later grades.

# Reason Quantitatively and Abstractly: Specific vs. General

Common Core State Standards-Mathematics (CCSS-M) define what students should understand and be able to do in their study of mathematics. But asking a student to understand and do something also means asking a teacher to first help the student to learn it and then assess whether the student has understood it. So how do teachers gauge mathematical understanding? One way is to ask the student to justify, in a way that is appropriate to the student’s mathematical maturity and to the context of the problem and concept, why a particular mathematical statement is true or where a mathematical rule comes from. Mathematical understanding results from the practice of these justifications and, in the process, procedural skills are strengthened, particularly when mathematical tasks experienced by students are of sufficient richness.

Reason Abstractly and Quantitatively
Mathematics learning is the continuous movement between the particular and universal. Resolving the tension in mathematics between understanding at an abstract, context-free level and providing some kind of context for the problem at hand is at the heart of teaching and learning of mathematics. For children, mathematics begins with specific and concrete tasks, and they ultimately reach the most important and high-level thought process in mathematics—the abstraction process. It means to know the abstract and general, on the one hand, and the particular and specific, on the other. Taking the child from understanding a concept at the specific, concrete level to generalizing and extrapolating it to the abstract, symbolic level is the mark of a good teacher.

Abstraction is to capture essential properties common to a set of objects, problems, or processes while hiding irrelevant distinctions and uniqueness among them. Abstraction gives the power to deal with a class of problems that are diverse and complex. For example, children encounter specific shapes, figures, and diagrams in geometry in different contexts. At the same time, all geometrical shapes are abstractions, that is representations of concrete objects from multiple settings and contexts, e.g., a circle drawn on a paper represents a family of circular objects. Similarly, students encounter different kinds of numbers and diverse relationships between them. On the other hand, definitions, theorems, and standard procedures are abstractions, that are general cases derived from specific contexts and relationships—properties such as: associative and commutative property of addition, distributive property of multiplication over addition/subtraction, long division procedure, prime factorization, divisibility rules, solving equations. Abstract thinking enables a learner to bend computation to the needs of the problem.

A mathematically proficient student makes sense of quantities and their relationships in a given problem situation, looks for principle(s) applicable to that problem, and takes the problem situation to a general situation. The specific case is dependent on the context, but generalization happens only when we decontextualize the relationship(s). For example, the expression 12 ÷ 3 in a specific case represents: if 12 children are divided into 3 teams of equal number of students, how many are in each team? However, it is an abstraction of several situations, the numbers 12 and 3 can represent a variety of objects—concrete and abstract and from several settings and forms:

1. How many groups of 3 are there in 12? (repeated subtraction)
2. If we divide 12 into 3 equal parts/shares/sets/groups, what is the size of each part? (groups of/partitioning model)
3. If we organize 12 chairs in 3 rows with equal number, how many will be in each row? (array model)
4. If we organize 12 unit square tiles into a rectangle with a vertical height of 3 units, what will be the size of the horizontal side? (area model).

Thus, the expression 12 ÷ 3 no longer represents a contextual, concrete problem; it has been decontextualized; it is context free. To abstract a given situation and represent it symbolically and manipulate the representing symbols as if they have a life of their own is the key to true mathematical thinking.

Decontextualizing, thus, means abstracting, going from specific situations to general and representing them abstractly, symbolically and then to manipulate these symbols without necessarily attending to their referents and contexts. However, once the solution is found, it needs to be interpreted from the context of the original problem.

Quantitative reasoning entails habits of creating a coherent representation of the problem at hand; considering the units involved; attending to the meaning of quantities, not just how to compute them; and knowing and flexibly using different properties of operations and objects.

For effective learning of mathematics and solving problems, students need two complementary abilities—understanding quantitative and abstract relationships—how to contextualize and decontextualize. Many students, even when they may show skills at each of these levels separately, show gaps in reasoning at these two levels simultaneously or making connections between them. Proficient students reason at both levels—to reason quantitatively and abstractly, to understand the context of the problem and then to decontextualize it.

For most students, to understand a problem and apply mathematical reasoning, the context of the problem matters. However, the ultimate goal is a context-independent understanding of problem solving. Everyday examples, models, context, analogies, and metaphors are critical in linking the problem to students’ prior knowledge and to illustrate different aspects of the subject matter and facilitate students’ transition from specific to general and vice versa. Mathematically proficient students make sense of quantities and their relationships in problem situations. At the same time, they are able to generalize and abstract from these specific situations.

As an example of transition from specific to abstract, consider this problem:

Children collected 45 bottle caps each school day for a week. How many bottle caps did they collect?

Initially, children see this as a series of additions (45 + 45 + 45 + 45 + 45)—a context specific approach to the problem, but then they abstract it into a multiplication concept connecting with the schemas of multiplication as repeated addition or (5 ×45) or “5 groups of 45”—a one-dimensional concept. When several such problems are handled successfully, they begin to see the general situations that are translated to a × b, where a and b are numbers representing a variety of settings and later the multiplication is extended to the array and area model—two-dimensional models and application to a diversity of numbers (multi-digit, fractions, decimals, algebraic expressions) and mathematical entities, such as: functions, determinants, matrices, etc.

Similarly, at the high school level, students know the role of numbers in a situation represented by algebraic relationships. For example, in the linear equation p = 25n + 45, they understand that p describes the cost in \$ of n items where the cost of manufacturing per item is \$25, and \$45 represents the start up costs. This is the context—this is a specific case. Representing these in a table and developing a pattern helps students to reach the general case.

 Case/State # of items p = Total Cost in \$ Start 0 45 1st 1 25×1+ 45 2nd 2 25×2+ 45 3rd 3 25×3+ 45 — — — 100th 100 25×100+ 45 — — — nth n p=25×n+ 45=25n+ 45

Decontextualizing here means that if the cost per item or the start up costs are changed, then we will have different numbers in place of 25 and 45; we will have a new equation. In the most general case, the equation will be p = an +b, where p is the cost of n items, a is the cost of manufacturing one item and b is the start up costs. This is a complex idea and many students have difficulty arriving at this point. Only with a great deal of scaffolded questioning and examples can a teacher achieve this with all students.

Contextualizing is also the movement from general to specific or seeing the role of context on quantities and probing into the referents for the symbols and numbers in the problem. It is to take an abstract symbol or an equation and to look for its context—its special case. In the manufacturing equation, it means that if we want to find the cost of manufacturing 1 item, we will change n to 1 and if we want to know how many items we can manufacture for \$245, we will change p to 245. Here we are going from general to specific. And we understand the specific case that even if no item has been manufactured, there is a cost of \$45.00 incurred. Or, when the variables in the equation are changed, the student still understands the roles of the variables.

For example, in a right triangle ABC, with the right angle at vertex C, when the 2 legs and hypotenuse are given, in several settings, one observes and then derives: the sum of the squares of the legs is equal to the square of the hypotenuse. Then, generalizes this result into, form specific right triangle to any right triangle, a2 + b2 = c2, the decontextualized form as Pythagoras Theorem. Further, one applies this universal result into specific contexts (special cases) in solving problems. Every middle and high school student understands and masters the specific and general result about right triangles. However, when the name of the triangle is changed to ABC with the right angle at vertex B (e.g., a2 + c2 = b2), or with the triangle PQS, with the right angle at Q, (p2 + r2 = q2), they have difficulty relating to the Pythagorean result. In other words, for them the result is contextual to a particular right triangle. Thus, mathematics learning is closing the loop:

In meaningful problem solving, the decontextualizing and contextualizing processes are intertwined. The process starts when students first read the problem and understand the context of the quantities. They

• understand and convert what they have read into mathematical equivalents—numbers, symbols, operators (contextualize),
• use knowledge of arithmetic, algebra, geometry, calculus, etc., to write expressions, equations/inequalities, functions, systems (de-contextualize),
• compute, evaluate, solve equation(s) and systems, simplify expressions, etc., to generate answers to the questions posed in the problem (context to general and back to context),
• refer the solution/answer back to the original context of the problem, interpret and understand the meaning of the answer to realize a solution (contextualize and decontextualize), and
• extend the solution approach to other similar problems to generalize the approach (contextualize and decontextualize).

Decontextualizing and contextualizing also mean thinking about a problem at multiple levels—going beneath the surface and making connections. It goes beyond the ability to merely find the value of the unknown (say, x) in the equation. It is also to find the meaning about the solution and the uniqueness and efficiency of the solution process. For example,

Find the distance between a submarine, 250 ft below the surface, and a satellite tracer orbiting 23,000 ft directly above the submarine at a particular time.

The following steps describe the contextualizing to decontextualizing process that provide entry to the solution process.

•  As a start, student represents this information on a vertical line (contextualizing) locating the zero as the sea level and the locations of these two objects as points on the vertical line with relative positions and distances (submarine = −250, satellite = +23,000 (de-contextualization);
• The student tries to remember how to find the distance between two points (e.g., y1 and y2) on a number line (in this case, y-axis) as distance = |y2−y1| (decontextualizing); and
• Relate the formula to the objects = |23000 −250| (contextualizing).
• Finally, they simplify the expression and respond to the question in the problem and express the result contextually: The distance between the satellite and the submarine is 23,250 ft.

Let us take a similar problem and use another approach for solving it and make connections to make generalizations to prior knowledge.

The temperature in the morning was 450F and in the evening it went down to -120F. How much colder was in the evening? How much warmer was in the morning? What was the difference in temperature in the morning and evening? The temperature from morning to evening went down by how many degrees?

In a seventh grade classroom, when students initially saw the problem, quite a few of them answered it quickly as 330F. These students did not contextualize it. Others wrote: 45 – 12 = 330F. These students started with quantities without contextualizing the problem.

However, if they had represented the problem (contextualized), they would have been able to solve this problem, answer all the questions raised in the problem, and even others of the same type (decontextualized).

By the help of this diagram, they compute the distance between the points to 45 –(-12) = 45 + 12 = 57 and infer that it is 570F cooler in the evening. Therefore, it is 570 F warmer in the morning than evening. And, the difference between the temperature in the morning and evening is 570F.

This problem can also be solved by starting from 450F and getting to -120F by moving left rather than right adding a level of generalization (decontextualize).

Quantitative reasoning is important in its own right; however, the goal is to learn, apply, generalize, and reason with numbers and use them to make meaningful inferences, create conjectures to arrive at generalizations.

For successful execution of the solution process with understanding, quantitative reasoning should be comprehensive—contextualized, decontextualized, and contextualized; it must go beyond mere computational proficiency.

Comprehensive quantitative reasoning entails the habits of creating a coherent representation of the problem; considering and understanding the units involved; attending to the meaning of quantities and efficiently computing with them; and knowing and flexibly using different properties of operations and objects. Thinking quantitatively and abstractly also means that students know the proper use of mathematical symbols, terms and expressions.

Comprehensive reasoning—to think abstractly and quantitatively separately and then together, develops when teachers employ a range of questions to help students focus on understanding quantities (e.g., type and nature of numbers), language (vocabulary, syntax, sentence structure, and translation), concepts and the associated schemas, and operations involved in the problem. We need to help students focus on the specific as well as the general and abstract, particular and the universal. It means:

1. Making sense of quantities in the problem (units, size, meaning, and context) and their relationships:

• What do the numbers/quantities in the problem represent?
• What is the relationship between these quantities?
• How is _____ related to ______?
• What is the significance of units associated with these quantities?
• Are all the units of measurement uniform?
• What are the relationships _____ units and _____ units?

2. Creating multiple representations of quantities and relationships in the problem (concrete, iconic and pictorial representations, symbolic expressions—equations, inequalities, diagrams, etc.).

These representations should be appropriate to the grade level (for example, thinking of division “as groups of” and performing it by sequential counting is appropriate at the third grade level, but it is not appropriate at the sixth or seventh grade levels. At that time, we should be thinking of the area model of division).

The teacher should provide a range of representations of mathematical ideas and problem situations and encourage varied solution paths.

• What are some of the ways to represent the quantities and their   relationships?
• Is there another form that the numbers can be represented by (table, chart, graph, bars, model, etc.)?
• What is an equation(s) or expression(s) that matches the pattern, diagram, number line, chart, table, graph, …?
• What formula(s) might apply in this situation? Why?

As an illustration let us consider the problem: 91− 59.

At the concrete level the solution can be derived by using BaseTen blocks or Cuisenaire rods. But Cuisenaire rods are more efficient as there is no counting involved. Then we can use Empty Number Line in multiple ways (ENL) to find the difference. The ENL helps develop numbersense and mental arithmetic. Once students have facility with ENL, they should explore this problem using compatible numbers and decomposition/re-composition. For example,

All of these problems are equivalent and develop a deeper understanding of numbersense, quantitative reasoning and mental arithmetic.

3. Forming and manipulating equations (attending to the meaning of the quantities, not just computing them):

• Is it the most efficient relationship or equation representing the quantities in the problem?
• Which property or rule can make this equation simpler?
• What property of the equation (equality, procedure, number, operation, etc.) did you apply in solving the equation?
• Could you use another operation or property to solve this task? Why or why not?

4. Making sense of the given problem and applying that understanding to consider if the answer makes sense.

• How does this solution relate to the problem?
• Can you relate the solution of the problem to a real life situation?
• What does this answer mean? For example, what does the slope of this line mean in the context of the problem?
• Can this solution approach be generalized to other number systems, operations, ……, ……?

Levels of Knowing Mathematics
For any concept or procedure to be mastered by a child, it has to go through several levels of knowing: intuitive, concrete, pictorial/representational, abstract/symbolic, applications, and communication.

Intuitive level of knowing means the student is trying to connect the new concept with the schemas of prior knowledge—language, concepts, skills, and procedures. It is like relating subtraction to addition, division to multiplication, laws of exponents with base 10 to other bases, or laws of exponents in the case of whole numbers to integers, rational, or real numbers. In the process, previous schemas get transformed—extended, amalgamated, reorganized, even destroyed and replaced by new schemas. This is how a person enters into the new mathematics concept, learning, or problem.

Concrete level of knowing means the student represents the concept, procedure, problem through concrete models based on the intuitive level understanding. The concrete model should be efficient and transparent in representing the concept or problem. Of all the efficient models, one should look for elegant models. A model is efficient and elegant when it takes the student to representation level easily.

Pictorial and representational level of knowing means seeing the concept using pictures (iconic or representational), diagrams, or graphic organizers. There is a difference between an iconic representation and pictorial representation. For example, representing a problem with pictures of Cuisenaire rods or Base Ten blocks is iconic, whereas Empty Number Lines or Bar diagrams are pictorial. Iconic representation is the true copy of the concrete model and keeps the learner longer on a concrete and contextual level. As a result, many children do not become proficient in abstract or de-contextualization. On the other hand, an efficient pictorial representation leads the student to generalization and abstract representation of the concept. Efficient and elegant models facilitate such decontextualization.

When a concept is learned at the abstract level, it is easier for a student to apply it to general problems (applications level of knowing) and the exposure from intuitive to concrete to pictorial to abstract helps the student to become fluent in communicating understanding and mastery (communications level of knowing).

Let us consider an example of writing an addition equation to describe a situation (first grade level) that illustrates the transition from contextualizing to decontextualizing:

The team scored 33 and 25 points in two games, respectively. How many points in all did the team score in the two games?

First step, using Cuisenaire rods, 33 can be represented by 3 tens (3 orange rods) and 3 ones (1 light green rod) and 25 can be represented by 2 tens (2 orange rods) and 5 ones (one yellow rod), then the sum is 5 tens (5 orange rods) and the 3-rod and the 5-rod gives the 8-rod (brown) equals 58 (concrete).

Second step: the sum can be represented by an empty number line (pictorial level). Several ENLs can be created for this computation. Finally, the total score in two games can be expressed as a sum of 33 and 25. Total = 33 + 25. (abstract)

Only after students understand the concept should a teacher move to abstract.

After the understanding is gained from this decomposition/recomposition, we should move to the standard addition procedure.

After this, one can use the procedure to solve problems or extended to multi-digit additions with regrouping.

Let us consider another example to examine how to go from specific to general.

The length of a rectangle is 3 more than two times the width. The perimeter is 78 in. What is the width of the rectangle?

Solution One: (Contextualizing: Quantitative reasoning)
Each expression from the problem is translated into mathematical expressions:

Solution Two: (De-contextualizing: Generalizing)
We express the length in terms of the width: length in inches = 2x + 3, where x = width in inches.

To have proficiency in mathematics, to decontextualize and to represent abstractly, students need to learn to use symbols correctly. This begins with number concept and the fundamental concepts such as equality. Many students misunderstand the concepts of equation and equality. Their misconceptions originate from not knowing the concept of “=” in its proper form.

It is difficult to understand the concept of and working with equations, without understanding the concept of equality. Understanding and using the concept of equality is a good example of going from a particular situation to a general situation. Though the concept of equality is so germane to mathematics, most children have difficulty in answering problems such as (a question that has appeared on several national standardized tests):

What should be placed in the place of in the equation?  9 + 5 = + 7.

Many students from second to eighth grade would place 14 in place of . These students have no idea what the symbol “=” means. For them it is an operation and is used when two numbers are added. They see it as one-way implication (). They do not have the idea that the two expressions on either side of the equal symbol need to be compared to see if they are equal. They need to see it as a two way implication ( equivalent to =).

When students, in the early grades, have not experimented with materials such as a mathematics balance or Cuisenaire rods to see the equivalence of two expressions, they have difficulty understanding the concept of equality or equation.

The diagram suggests that 9 + 5 = 7 + 2 + 5; therefore, there should be 7 in the box. The use of concrete models is a good starting point for proper understanding of these fundamental concepts. Practice without conceptual understanding does not lead to generalizations and abstractions.

A group of teachers was asked how they or their students would respond to 4 = 6? Almost everyone replied: “Well, we just know it is not true.” When asked how they would prove their statements, one of them said: “If you compare 6 items and 4 items by one-to-one correspondence, you find that six has two more items, so 6 does not equal 4.” This shows that they have the reasoning for the concept of inequality.

When they were asked: “How they or their students would explain 2 + 3 = 5,” one of them answered: “My students would get 3 things and then 2 things and put them together and you would know you have five things.” That is finding a total of 2 objects and 3 objects. That is not a proof for equality. That is right, but that is not the question.

To prove the equation “2 + 3 = 5” concretely, we put on one side of a balance two Unifix cubes and three more with the two already in the rocker balance. Now we place 5 Unifix (of the same size and weight) cubes on the other side of the balance, and the balance balances. Now we see that 2 and 3 are 5. Now if we take 5 cubes on one side, once again, we find that 5 is not equal to zero. However, if we put 3 cubes and then 2 more, we find that the two sides balance. We have shown that 2 + 3 equals 5 and 5 equals 2 + 3. It shows it as a two-way implication.

Similarly, if we take the red Cuisenaire rod (representing 2, if the white represents 1) and place the light green rod (representing 3) next to the red rod making a train, we find that the yellow rod (representing 5) is equal in length to the two rods. Now we can read (in color): red + light green = yellow and yellow = red + light green. Therefore (in numbers), 2 + 3 = 5 and 5 = 2 + 3. In both cases, we have shown that the equation is true using concrete materials. We can do the same in later grades using abstract formal arguments using the properties of numbers and axioms.

When a group of middle and high school students were asked: “What is the definition of an equation?” Answers varied:
“When two sides are equal.”
“When we are solving something.”
“When there is variable in it.”
Although there was a lot of discussion, none of them could clearly define an equation.

We have an equation when two mathematics statements/expressions are equated. Examples: (a) 2 + 3 and 4 + 1 are two mathematical expressions. When they are equated we have an equation: 2 + 3 = 4 + 1.   (3x + 5) + 9 and x2 + 2(3x + 7) are two mathematical expressions, when we equate them, we get an equation: (3x + 5) + 9 = x2 + 2(3x + 7).

In the early grades, we need to ask students to use quantities and units as descriptions whenever possible. We should inundate them with questions that ask how many, how many more, how many less, what is the total, why can you do this, what is the reason, what do you infer from this, what conclusion can be drawn from this, can we form a conjecture from this, can you give another example for this procedure, concept or word, etc. The role of examples, counter examples, non-examples, specific cases of a definitions, and theorems are effective means of relating to the specific and general. Unless students regularly connect different concepts, procedures, and language, they will have difficulty in focusing on the specific and general and the quantitative and abstract.

# Make Sense of Problems and Persevere in Solving Them: Engagement with Mathematics

On encountering a new problem that they cannot solve, many students immediately give up. It doesn’t have to be this way.

There is a difference between students who welcome and remain engaged in the problem and those who give up easily. The difference is not due to innate factors, but it is mostly the outcome of teaching. With effective teaching all children can acquire attitudes and strategies to become proficient in problem solving—understand the problem, approach the solution process, and stay engaged in the problem using different perspectives.

According to the framers of the Common Core State Standards-Mathematics (CCSS-M) and Standards of Mathematics Practices (SMP), helping students to understand a problem, initiate a solution process, remain with the problem by exploring it from multiple perspectives are important characteristics of teaching. This helps students acquire the ability to enter the solution process and develop mathematical stamina.

Making Sense of Problems
Making sense of the problem means understanding the language, the concept, and the conditions and parameters involved in the problem. Students identify the objectives of the problem. They may not engage in the problem and remain engaged in the problem if they do not understand the problem. To initiate a solution process and to pursue it, students should associate appropriate schemas and procedures with the language, symbols, and concepts involved in the problem.

Mathematically proficient students read a problem carefully, understand the meaning and context of the problem, and explain to themselves the role of particular numbers, expressions, and actions in the problem. They analyze the givens, study the constraints on the quantities in the problem, understand, identify, or determine the unknowns in the problem. They understand intra- and interrelationships amongst knowns and unknowns. They understand the nature of these relationships. They seek entry points to the solution process keeping focus on the goal/s of the problem.

Mathematically proficient students analyze the problem, consider analogous situations, try special cases, and simpler forms of the problem (changing numbers, e.g., changing fractions into whole numbers, relaxing constraints in the problem, or reducing the number of variables) to gain insight into the problem and solution process.

They classify and organize the information into tables, charts, or groups. They search for regularity, patterns, or trends. They make conjectures about these patterns. They observe and explain correspondences between variables (knowns and unknowns) by forming equations, verbal descriptions, inequalities or diagrams of important features, relationships, and representations. Through these conjectures about the form and meaning of the data, they plan solution pathways and enter the solution process, rather than simply jumping into a solution attempt by choosing a formula or procedure.

Students who have acquired a concept, skill, or procedure using diverse language and a multiplicity of strategies have flexibility of thought to explore multiple ways of entering the problem.

Building Mathematics Stamina: Perseverance
Perseverance means having the self-discipline to continue a task in spite of difficulties and dead ends. It is a function of skills and attitudes. Albert Einstein said, “It’s not that I’m so smart, it’s just that I stay with problems longer.” Perseverance is a necessary ingredient for student achievement. One of the reasons students do not persevere in solving problems is lack of flexibility of thought. When they exhaust their ability and options to think about the problem, they do not have stamina for solving problems. Students develop perseverance when they are taught with rigor.

The requirements of rigor—understanding, fluency, and ability to apply, means a student demonstrates intra- and inter-conceptual understanding, fluency in performing computational procedures and their interrelationships, knowledge of the appropriateness of a particular mathematical conceptual and procedural tool, and ability to apply mathematics concepts and procedures in solving meaningful, mathematics and real-life problems. Finally, it is demonstrated in the ability to communicate this understanding. To achieve a level of mastery/rigor among students, mathematics educators need to balance expectations, instruction, and assessments.

We can help students continue thinking about a problem by modeling the many different questions they can ask about a difficult problem.

Effective teachers use a variety of language, questions, and methods to derive a concept or procedure. For example, let us consider a problem:

A science book has 251 pages and a mathematics book has 197 pages.

Teacher: What question can we ask so we have a subtraction problem from this information? Students formulate questions. If they do not, she articulates several questions:

• How many more/extra pages are in the science book than the math book?
• How many less/fewer pages are in the math book than the science book?
• What is the difference in the number of pages in the science and math books?
• How many pages should be added to the math book so that it will have the same number of pages as the science book?
• How many fewer pages should be in the science book to have the same number of pages in the math book?
• How many pages are left in the science book if we took away as many pages as the math book?

Through this process of generating questions, over a period of time, students develop flexibility of thought about additive reasoning, in general, and subtraction, in particular. These children, in future, will find several ways to enter the solution process of any subtraction problem involving numbers other than whole (e.g., fractions, decimals, integers, algebraic expressions, etc.). Effective questions build student stamina for problem solving. This should be a regular process in a mathematics class.

Using Effective Concrete Materials
To build stamina, younger students should be exposed to a multiplicity of concrete objects (e.g., Visual Cluster cards, TenFrames, Cuisenaire rods, fraction strips, Base-Ten blocks, Unifix cubes, pattern blocks, Invicta Balance, etc.) and diagrams and pictures (number line, Venn diagram, empty number line, bar model, graphic organizers, tape diagrams, tables, charts, graph paper, etc.) to understand strategies based on decomposition/ recomposition of numbers and facts and solve problems. Counting materials and strategies based on them build neither the flexibility nor the stamina for problem solving.

Middle and high school students may, depending on the context of the problem, transform numbers (fractions, decimals, and percents, algebraic expressions) using concrete models and the properties of numbers, operations (associative, commutative, or distributive properties to simplify numbers and expressions), change the viewing window on their graphing calculator to get the information they need (e.g., to observe the behavior of a polynomial, trigonometric, or rational function near the origin or at a specific point; compare it with the “parent function,” etc.), or use Algebra tiles, Geoboard, geogebra, Invicta Balance, etc., to arrive at relationships and equations involving variables.

Monitoring Progress and Evaluating Success
Mathematically proficient students monitor and evaluate their progress and change course if necessary. They check their answers to problems using a different method, and they continually ask themselves: “Does this make sense?” “Does this answer the questions in the problem?” (e.g., analyze partial and final answers). They can explain their solution approach and try to understand others’ approaches to solving problems, and they identify correspondences between different solution approaches. All of these activities, habits, and attitudes help them to be engaged in the problem resulting in perseverance. Students develop and improve perseverance when they realize that mathematics is thinking and making mistakes. It is also a process, not just finding the answer. It happens when we ask:

• Step One: What is it that we are trying to find out here? This is the question we ask in the real world. And this is the most important part of doing mathematics. People, including our students, need practice and opportunities in asking the right questions. This should be a group activity as group work as a strategy is critical to good mathematics work and student engagement. Group work generates better understanding of problems and then multiple entry points. It is also critical in countering inequities in mathematics achievement by different groups of students in the classroom.
• Step Two: Next is to take that problem and turn it from a real world problem into a mathematics problem—express it as a relationship between the elements (variables and quantities) that define or have created the problem. This translation from real word situation expressed in the native language to mathematics language is an important step in doing mathematics.
• Step Three: Once we have defined a relationship (an expression, an equation/inequality, or a system of equations/inequalities, etc.), we manipulate these relationship(s) and that involves formal mathematics—this is the computation step. Through computation, we transform the relationships into an answer in a mathematical form. This is an important step, but for developing interest in mathematics, we should not begin with this step.
• Step Four: When we have dealt with the computation part of mathematics, we need to then turn it back to the real world. We ask the question: Did it answer the question? And we also verify it—a crucial step.
• Step Five: To create interest and involvement, we need to now engage students in collective reflections by sharing different strategies and their relative efficiencies and elegance.

Completing this loop keeps our students grounded in the reality and power of mathematics. The majority of students will repeat these steps in their real life. And a small percent of students will have the satisfaction of repeating the steps in the context of mathematics and sciences only.

Teacher Attitudes
Teachers need certain attitudes, skills, and habits of mind for developing children into effective problem solvers with stamina. They need to practice the following:

• Believe in each child’s ability to improve and achieve higher in mathematics.
• Expect and help them to finish what they start and when they are stuck, providing scaffolding with enabling questions to continue in the task.
• Avoid accepting excuses for unfinished work.
• Give positive feedback when a child puts forth extra effort or takes initiative.
• Help students realize that everyone makes mistakes, but what is important is to keep trying.
• Demonstrate and motivate them to try new things.
• Encourage children to take responsibility for their work and make constructive choices.

Students become mathematically proficient and persevere in solving problems when teachers model these skills and choose meaningful problems to solve. They create conditions for students’ engagement in problems; that in turn develops perseverance. Students are engaged when problems are contextual, moderately challenging yet accessible, have multiple entry points, and are amenable to various solution approaches (intuitive, concrete, pictorial, abstract, on the one hand, and arithmetical, geometrical, and algebraical, on the other). It develops a variety of tools.

For example, using the Empty Number Line (ENL) approach to solving addition and subtraction problems rather than jumping into applying the standard procedure has many more entry points to the solution and can be solved using multiple ENLs (e.g., the problem: the difference 231 – 197 can be arrived at by at least different ENLs with a deeper understanding of numbersense (number concept, arithmetic facts, and place value) and problems solving. Arriving at the answer this way will keep them engaged.

Similarly, the Bar Model (BM) is an effective problem solving tool involving fractions, decimals, percents, and deriving algebraic equations easily. The area model of multiplication and division is effective for whole numbers, fractions, decimals, and algebraic numbers and for deriving properties of operations (e.g., commutative, associative, and distributive properties of multiplication and subtraction, etc.).

Tools are not enough, however, unless teachers scaffold student work. Questioning, based on formative assessment, is the key to the scaffolding process. Scaffolding is a function of a teacher’s ability in

• doing task analysis—know and establish the trajectory of the development of a concept, skill or procedure, and help students to know the goal of the task,
• being aware of the student’s capabilities, as well as their limits;
• doing continuous formative assessments of students’ assets—cognitive and content (conceptual and skill sets),
• asking enabling questions to move students toward the goal, and gradually fade and remove the support structures, and
• knowing models and approaches best suited for connecting concepts with students.

The response to “good” questions develops conceptual understanding, stick-to-it-ness, and helps them refine the tools—make them effective, efficient, and elegant. The better a teacher gets at asking “why” questions, the better her students are at understanding concepts, staying on, applying tools, and solving problems.

Effective questioning is more than giving students a solution approach, steps for solving a problem, or identifying the typology of the problem. Effective questions invite students to enter the solution process and stay with it. They may include:

• What question(s) are you trying to answer in the problem? What are you trying to find? Can you state that in your own words?
• What information do you have that can help you answer the question in the problem? Do you have enough information to answer the questions raised in the problem?
• Do you know any relationships among the information you have and what you do not have?
• Can you write this information using mathematical symbols?
• Can you write a fact, equation, inequality, formula or a relationship between symbols in the problem?
• How would you show the information in the problem in a different way?
• What other information do you need to answer the question?
• Where might you get that information?
• What other questions do you need to answer before you can answer the question in the problem?
• Have you solved another problem like this before?
• Could you solve the problem if the numbers were simpler?

When students have solved the problem, the teacher reengages them by asking:

• Do you have the answer to the problem/question?
• Have you answered the question raised in the problem?
• Which question in the problem does this answer?
• Does this answer make sense?
• Have you expressed the answer in the appropriate units of measurement or order of magnitude?
• What did you learn from this problem?
• Is there any information in the problem that was not necessary for answering the problem?
• Can we relax the conditions of the problem and still answer the problem?
• Can you write another problem similar to the given problem?
• Can you formulate a more difficult problem?

To demonstrate some of the questions, let us consider a problem:

In a village, 20% of voting age people did not vote during the last election. If only 4,280 people voted, what was the voting age population of the village?

Teacher: What are we looking for?
Students: The total voting age population of the village.
T: What information do we have?
S: The number of people voted? 4,280
S: The percentage of people did not vote? 20%
T: What else do we have? What information can we derive from the given information?
S: The percentage of people who did vote: 80%
S: The percentage of voting age population: 100%
T: What are we trying to find?
S: The voting age population of the village.
T: Can you represent the information by diagram, table, equation, or relationship? Make a start and try to solve it. We will discuss all of the methods used by the class. I will visit all of you and keep an eye on your progress. You can ask me questions when you need help.

At the end she asks children to share all of their methods and their relative merits are discussed. The approaches are shown here.

Method One: Visual Representation Method (line segment, Bar Model or Pie Chart)
The following bar represents the total voting age population.

# of people of voting age = ______________________________ =100% = ?

# of people who voted     = _______________________    = 80% = 4,280

# of people did not vote = ______                                  = 20% =

As we do not know the total population, we represent it by a “?” mark, which is made up of those who voted (longer line or a bar) and those who did not vote (shorter line or shorter bar) (see the bar model below).

Because of 80% and 20% distribution, the line/bar is divided in two sections: the larger section is 4 equal parts and the smaller section is 1 part. The number 4,280 is equal to 4 equal parts and the missing part is one part. Therefore, one part is equal to 4,280 ÷ 4 = 1,070. Then the total number of people of voting age is 5 parts (4 parts + 1 part): 1,070 × 5 = 5,350.

Method Two: Applications of Fractions
The fraction of people who did not vote = 20% (= ⅕) of total number of people of voting age. The fraction of people who voted = 80% (= ⅘) of total number of people of voting age = ⅘ of total = 4280 (the 4 parts out of the 5 equal parts). So 1 part is 4,280 ÷ 4 = 1,070. Therefore, the total = 5 parts =1,070 × 5 = 5,350.

Method Three: Ratio and Proportion Method

Here part = number of people voted, whole = number of people of voting age, percent of people voted is 80% as percent of people did not vote is 20%. We can compare the number of people who voted in two forms: 80 percent vs actual number (4,380) and similarly compare the total # of people of voting age as 100 percent vs. actual number that we do not know and we consider as “?”. We have

4×? = 5(4280) (multiply both sides by 5 and ?; or cross-multiply);

? = 5(4280) ÷ 4 (isolate the “?,” divide both sides by 4),

? = 5(1070)

? = 5,350 (# of people of voting age).

Or, the total number of people of voting age = Number of people who voted + number of people who did not vote = 4280 + 1070 = 5350.

Method Four: Algebraic Method
Let us assume the number of people of voting age is x.
The number of people who did not vote is 20% of x. The number of people who did vote is 80% of x.

Thus,       80% of x = 4,280
.80 × x = 4,280 or of x = 4,280 or x = 4,280÷
x = 4280 ÷ .8 = 5,350.
Therefore, the total number of people of voting age = 5,350.

Method Five: Shortcut
To solve the problem, many teachers will just give the formula:
They will say to solve this problem is easy:
If 80% of a number is 4,280, then what is that number?
First, underline is and of in the problem. Then, the number just before is is the number to be placed in place of is and the number in place of of is to be placed in place of of, in the formula. Therefore, we have . Then, they will ask students to solve next ten to twenty problems on a sheet of paper.

This is purely a procedural method and does not emphasize much mathematics. The consequence is that students are unable to apply it if the problem is slightly different or the numbers are placed in a different form or different language. In this method, there is no involvement with language or concepts of mathematics. There are no connections made with other procedures or concepts. Students get the impression that mathematics is just a collection of procedures, and if they can recall the formula but can’t apply it, they give up.

Shortcut methods do not develop perseverance. Perseverance is reached when teachers apply methods that have mathematics and thinking behind them rather than methods that appear like tricks. Students who are familiar with the above four methods will be able to see where this formula comes from and then use it effectively.

Exposure to multiple approaches helps students understand concepts and acquire “stamina” for problem solving. As in any exercise, the stamina is a function of optimal (conceptually efficient) methods, regular and intentional practice, guided reinforcement (coaching and well-designed exercises and homework) and discussions of mathematics processes.

When teachers encourage students to share with the class, their

• understanding of the problem—language and concepts involved in the problem,
• entry points to the problem,
• approaches and strategies to and nature of the solution, and
• the mathematics concepts and procedures involved

students work hard and their mathematics stamina is strengthened. The crucial point is that students need to understand, know and experience that mathematics is not equal to computation. That is what develops perseverance.

Well-crafted mathematics classroom tasks, exercises, and assignments (including homework) hold the potential to make learning and teaching of mathematics focused and relevant and making all students achieve. In planning lessons, effective teachers make decisions about context, mathematics language, content, and rigor. Since homework is generally for reinforcement and practice, they assign homework that achieves those goals and needs to ensure that large chunks of class time are devoted to “why” and “how” questions to develop and reinforce mathematics concepts.

If mathematics is taught using deep learning—emphasis on concepts, language, and multiple models, instead of a performance subject—applying just “closed end” standard procedures, students will see it as important knowledge. Mathematics will become a collection of powerful tools that empower them to think quantitatively to solve problems in their work and lives.

We need to give all students the opportunity to taste real mathematics. Once students have acquired and mastered numeracy and algebraic skills with understanding, fluency and have the ability to apply, then we should use more efficient methods for computations. For example, computers and calculators can do a better job than any human as long as we know what we are doing and when such tools should be used. When relevant and efficient, we ought to use calculators and computers to do computation and engage students to spend more effort on conceptualizing and solving problems.

# Effective Teaching of Mathematics

The framers of CCSS-M have identified Standards of Mathematics Practice (SMP) based on practices and research on teaching and learning that consistently produce mathematics learners with high achievements. To be an effective teacher, one needs to know the content, pedagogy and models to deliver that content, and understand how students learn.

CCSS-M gives us what is important to teach and learn at what level. It describes the content and levels and nature of content mastery. However, teachers decide how to teach and assess. The goal of the first four to five years of students’ mathematics experience is to become proficient and comfortable in number concept, numbersense, and numeracy. After that, students use their understanding and fluency in numeracy skills to learn mathematics—algebraic and geometric models, and their integration.

The Standards for Mathematical Practice (SMP) describe ways in which students should be engaged in increasingly demanding subject matter as they grow in mathematics expertise and content throughout the elementary, middle and high school years.

To support students’ growth in mathematical maturity, designers of curricula, assessments, and professional development should connect mathematical instructional practices to the mathematics content espoused in the CCSS-M. Without connecting content standards with instruction practice standards, we cannot achieve the goals of CCSS-M. Assessment, on the other hand, is to see whether that connection has been made or not.

The Mathematical objects (e.g., numbers of various kinds—natural to complex; geometric entities—shapes, figures, diagrams, functions; operations and procedures of different types—decomposition/ recomposition, manipulations of numbers of different kinds and forms, e.g., long-division; transformations and functions—static and dynamic, congruence and similarity, matrices and determinants, etc. are examples of components and manifestation of content standards.

Mathematics practice standards, on the other hand, describe what actions teachers are to take so that students make the mathematics content—language, concepts, procedures, and skills their own. Thus while we usually pay attention to nouns in content standards, for practice standards we must pay attention to verbs. The Standards of Mathematics Practice are action steps to make the content of the CCSS-M possible to take hold in the classroom and make students learners of mathematics.

Why SMP?
The National Mathematics Advisory Panel concluded that our students have reasonable, though incomplete, factual and procedural knowledge, but poor conceptual knowledge (e.g., many do not fully understand the base-10 number system, concepts of fractions, how decimals and fractions are related, the differences in arithmetic and algebraic reasoning). This is because of our over emphasis on procedural knowledge throughout children’s mathematics experience.

At the same time, the last 20 years of mathematics education reform indicate that American students, even many college students, neither have automatized fact retrieval nor have achieved fluency with procedures. For example, many students can perform routine procedures (e.g., procedures on fractions) but cannot justify the reasons for the steps involved or provide estimated answers before they execute the procedures.

Our students’ lack of conceptual understanding is a major cause of concern and requires investment after they leave school. For example, many corporations spend large sums of money in training high school graduates in their use of simple arithmetic. The problem has also reached college and university levels. Half of the students at community colleges, and 1 in 5 students at four-year institutions, require remedial courses in writing and mathematics, with community colleges spending more than 2 billion dollars on remediation and four-year colleges \$500-million. As one university leader pointed out,

Many of us in higher education have observed an increasing number of students arriving at our doorstep not fully prepared to pursue a college degree. This is our collective problem as a nation. (Chronicle of Higher Education, 2014, June 14)

Framers and supporters of CCSS-M and SMP believe that this latest educational reform, if properly implemented, can alleviate some of these problems and narrow the college and career preparation gap. Their aim is to improve instruction so that students acquire mathematics ideas with conceptual understanding and procedural fluency so they can apply mathematical tools effectively and provide reasons for what they do.

Let us consider an analogy to illustrate this point: You can watch two people swim a length of a pool. They take nearly the same time to swim the same distance, but one of them churns the water more and takes more strokes. When this swimmer gets out s/he is breathing a bit heavier but is in great shape, so it is not too noticeable. The other swimmer took fewer strokes and seemed to glide through the water.

If we assess the performance only by a stopwatch, we will conclude that they are swimmers of the same competence. In actuality, they are not.  As the lengths pile up and the task gets harder, the second swimmer will do much better. No matter what the stopwatch said, s/he is a much better swimmer than the other. And if we ask the first swimmer to swim a long distance over deep water, s/he may very well drown.

Students who only memorize facts, formulas and procedures, without understanding, are like the first swimmer. They churn and work hard, and if they are gifted with an outstanding memory, they can pull it off for a while. These students can – and often do – get by in the early grades when they can rely on their strong counting skills and contextual clues (e.g., concrete materials and pictures) to “find” a fact but they typically hit a wall sometime around 4-5th grade when they have to deal with fractions and decimal numbers with problems involving unfamiliar content and complex mathematics vocabulary words. On the other hand, children who have the rich language, robust conceptual schemas, and clear understanding of and fluency in executing procedures develop into graceful, effortless swimmers in the waters of mathematics.

In cultivating greater conceptual knowledge, effective teachers do not sacrifice procedural or factual knowledge. Procedural or factual knowledge without conceptual knowledge is shallow and unlikely to transfer to new contexts. At the same time, conceptual knowledge without procedural or factual knowledge is ineffectual and inefficient in execution. It needs to be connected to procedures so that students learn that the “how” has a meaningful “why” associated with it. It is more effective to move from conceptual knowledge to procedural knowledge. Increased conceptual knowledge helps students move from competence with facts and procedures to the automaticity needed to be good problem solvers.

When teaching conceptual, procedural and factual knowledge, effective teachers ensure that students gain automaticity. Their students know that automaticity and understanding of procedures and facts is important because it frees their minds to think about concepts and making connections. This requires some memorization and ample practice and the ability to communicate with the support of reason.

What the Research Says
The data from the 13 million students who took the Program for International Student Assessment (PISA) tests shows that the lowest achieving students worldwide are those who use memorization strategies – those who think of mathematics as methods to remember and who approach mathematics by trying to memorize facts and procedural steps. The implication is that we should not encourage rote memorization without understanding (i.e., in dividing fractions, teachers should not continue to use statements such as: “just invert and multiply”) but, instead, we should present appropriate models to create conceptual schemas and arrive at and master the procedure with proper understanding and mastery that is rooted in reasoning.

Because of the emphasis on procedural teaching, the U.S. has more memorizers than most other comparable countries. Perhaps not surprisingly mathematics teachers, driven by narrow and subjective tests (e.g., end of section tests in textbooks that value only the methods considered in the textbook), have valued those students over all others, communicating to other students that they do not belong in the mathematics class. Current research about how we learn and how our brains receive and process information shows that the students who are better memorizers may not have more ability or potential, but, unfortunately, we continue to value the faster memorizers over those who think slowly, deeply and creatively. We need students with mastery or conceptual, procedural, and factual knowledge for our scientific and technological future. Certain things should be memorized to relieve our work memory from mundane fact work to focus on creativity and applications.

Poor conceptual understanding comes at a cost. For example, if a student thinks that an equal sign means “put the answer here in the box,” she will be confused the first time she sees an equation with terms involving variables and multiple operations on both sides of the equal sign. Similarly, when a student first encounters factoring (whether in the case of whole numbers or polynomials), she ought to see its relationship to division and multiplication. But she may not be able to do so unless she has a deep conceptual understanding of multiplicative reasoning—that division and multiplication are inverse operations. She also will be slowed in factoring if she hasn’t memorized the multiplication tables, divisibility rules, prime factorization, and short division.

Factual and procedural knowledge are acquired by practice and stick-to-ness to tasks and practice. But, to be engaged in the task deeply, the task should be meaningful and the student should have understanding of the concepts and skills involved in it.

Of the three varieties of knowledge that students need, conceptual knowledge is difficult to acquire. It is difficult because knowledge is never transferred from one person to another directly nor is it developed without explorations, concrete manipulations, effective reasoning, and questioning. Rather, new concepts must build upon something that students already know when they explore the new concepts and ideas. Examples that are familiar to students and analogous to the current concept are useful to understand the concept.

SMP Principles
The writers of the CCSS-M were careful to balance the development of conceptual understanding, procedural skill and fluency, and application at each grade level. The standards are based on the idea that procedural skill and fluency expectations hinge on conceptual understanding. Fact fluency and procedural fluency help students to develop concepts, make connections, observe patterns, and form relationships between ideas, concepts, skills, and procedures thereby facilitating mathematical thinking. With mathematical thinking, students take interest in mathematics and develop mathematics stamina.

To achieve the different kinds of knowledge, we need to adopt pedagogical principles in every lesson that are informed by the Standards of Mathematics Practice:

• Make sense of problems and persevere in solving them.
• Reason abstractly and quantitatively.
• Construct viable arguments and critique the reasoning of others.
• Model with mathematics.
• Use appropriate tools strategically.
• Attend to precision.
• Look for and make use of structure.
• Look for and express regularity in repeated reasoning.

Each of these standards adds to a teacher’s ability to develop the different components of knowledge, helping children acquire mathematical ways of thinking, creating interest for mathematics, and recognizing the power of mathematics. We need to incorporate these practices in our lessons if we want to have students who enjoy doing mathematics and achieve higher.

Examples of Standards
In future posts, I want to comprehensively develop how to implement each of these standards. At present, I want to consider a few illustrative examples.

Standard number one, for example, deals with developing understanding and engagement with a problem and creating mathematics stamina in solving it.

Solving a problem almost always depends on what tools (linguistic, conceptual, and procedural) one knows and how to connect those tools with the current problem. As students advance and encounter new problems, new concepts will increasingly depend on old conceptual knowledge. For example, understanding and solving algebraic equations depend on the understanding of the concepts of equality, variable, arithmetic operations, operations on fractions and integers, ability to generalize, etc. In solving problems factual, procedural, and conceptual knowledge all go together. And to stay engaged with the problem requires a student to have mastery of these prerequisite tools.

Familiarity is not the only ingredient necessary for successful problem solving. Students are more likely to understand abstract ideas when they see many diverse examples in the classroom that depict the conceptual components, schemas, and constraints of the ideas. In such instances, they can learn the essential properties in the concept of the problem (e.g., in fractions, the division of the object into equal parts and what is equal there) and which properties are incidental (e.g., in fractions, that the resulting parts need to be whole numbers).

Standard number seven: to look for structure in mathematics concepts and procedures is at the heart of understanding mathematics. Students need to realize that mathematics is the study of patterns—the underlying structures. Students frequently fail to understand the concept if they are not helped to discern patterns—to look for the structure—commonalities among examples and what is different in these examples.

Indeed, when the teacher introduces a concept through an abstract definition alone, e.g., the standard deviation is a measure of the dispersion of a distribution, students miss the conceptual understanding of the concept of standard deviation. The standard deviation is dependent on the spread; therefore, our examples should show the impact of that spread on standard deviation: Two groups of people have the same average height, but one group has many tall and many short people, and thus has a larger distribution and standard deviation, whereas the other group mostly has people’s heights right around the average, and thus has a small standard deviation.

If we introduce students to the formal procedure of finding the standard deviation too quickly, they won’t realize the relationship of spread on standard deviation. To realize the importance of such a component in the concept, we should also consider special cases and non-examples: e.g., to find the standard deviation when every piece of the data is the same, the spread has a particular meaning.

The third standard is the key to developing conceptual understanding in students. No conceptual understanding can be developed without emphasizing reasoning. To emphasize the importance of reasoning— concrete and abstract, examples, non examples, and counter examples play a crucial role.

Let us consider the definition of prime number: A number is called prime if it has exactly two distinct factors, namely, 1 and itself. The definition of prime number has two key features: it has factors and there are exactly two of them. 2, 3, 5, etc. are prime numbers as they have two factors only. It is important to give examples of prime numbers and examples of numbers that are not prime and the reasoning behind the choice. Examples emphasize the components of the concept, and “non examples” help students see the subtleties and nuances of the concept. For example, why is 1 not a prime number? (It has only one factor.) Why is 0 not a prime number? (0 is divisible by any non-zero number; therefore, it has more than two factors.)

The discussion to discern similarities and differences—comparing and contrasting examples, helps students to acquire the language and conceptual knowledge with rigor. With the help of appropriate language and transparent models, conceptual knowledge is converted into procedural knowledge. For this to happen, the conceptual (representational models—concrete and pictorial) should be congruent with abstract, procedural models. And with practice, procedural knowledge is then converted into factual knowledge. This practice should take place in a variety of problems and problem situations that are related to the procedure to arrive at the appropriate level of fluency.

If students fail to gain conceptual understanding, it will become harder to catch up, as new conceptual knowledge depends on the old. Students will also become more likely to simply memorize algorithms and apply them without understanding.

Helping Students Learn Concepts and Procedures
In our schools, much is made of the use of manipulatives to help children understand abstract concepts in mathematics, but many manipulatives and models themselves are abstract (students treat them as a symbol for something else), and not all manipulatives help learning—they sometimes impede it. This is most likely when manipulatives are so visually interesting that they distract from their purpose, when their relationship to the concept to be represented is obscure, or when they are used for rote counting. Manipulatives seem helpful because they are concrete; to be helpful, they should satisfy certain properties.

To illustrate the idea of a fraction, one might divide a cookie in two for the purpose of sharing it with a student. The concreteness of this example is likely less important than its familiarity. In contrast, suppose I cut a hexagon into two pieces and said, “See? Now there are two equal pieces. Each one is half a hexagon.” That example is concrete but less effective because it is unfamiliar; the student has no experience with divided hexagons, and the purpose of sharing is also missing.

Concreteness, in itself, is not a magical property that allows teachers to pour content into students’ minds. It is the familiarity that helps because it allows the teacher to prompt students to think in new ways about things they already know. However, familiarity also may create some misconceptions, half a pizza, half a cookie, half a glass are not precise as key characteristics of fractions may be missed. Students know a fraction when they focus on: (a) What is my whole here? (b) How many parts are there in this whole? (c) Are the parts equal? (d) Do all the parts together make the whole? (e) What is the name of each part? (f) How many of these parts will make the whole? And (g) What is the new name of the whole in the light of these parts? The teacher’s language, questions, and sequence of activities with materials transform the concrete models into representations—pictorial and abstract.

A teacher must move from familiar materials and models to the form that shows all the attributes of the concept and then can lead to abstract representations that are congruent to the abstract procedure. As concepts become more complex, it becomes harder to generate familiar examples from students’ lives to generate mathematics conceptual schemas, and teachers may have to use analogies more often. In such cases, a familiar situation is offered as analogous to the concept under discussion, not as an example of the concept.

An Example of Implementing SMP
In order to focus instruction responding to CCSS-M and applying SMP, teachers need to identify the essential language, concepts, procedure and skills in each major standard. SMP does not advocate a “one size fits all” model or a boxed curriculum. Essential Elements of each lesson informed by SMP include: Integration of language, concepts, and procedures, Multisensory (appropriate and efficient models), decomposition/ recomposition of problem components, Synthetic-Analytic (seeing patterns and analyzing the problem), Structure (logical language categories), Sequential (simple to complex), Cumulative (continually making connections), Repetitive, cognitive (meta), Diagnostic and Prescriptive (design lesson to assure progress and plan next lesson around noted errors that need additional reinforcement).

Let us illustrate this process in one specific mathematics standard. Mathematics standard 4.OA.A.3 (CCSS-M, 4th grade) says:

Solve multistep word problems posed with whole numbers and having whole-number answers using the four operations, including problems in which remainders must be interpreted. Represent these problems using equations with a letter standing for the unknown quantity. Assess the reasonableness of answers using mental computation and estimation strategies including rounding.

This standard includes only a few sentences, but involves several different terms, concepts, procedures, and skills. To make sense of and understand them, students need to know the meaning of these terms and concepts and execute the procedures.

There is a need to delineate these elements, focus on them, teach them, help students master them, connect them with other concepts and procedures, and then assess all of these elements.

There should be a clear understanding of what and how to represent each concept, procedure and the skill involved in this standard. Every concept and procedure involved in this standard should be transformed into a set of concepts and skills to be learned, mastered, and applied by the students. In the context of CCSS-M, teaching should be to acquire understanding; students should arrive at fluency and should be able to apply concepts and skills contextually.

Step # 1
Language and Concepts

• Know the meaning of each word and term in order to translate from English to mathematical equations
• Identify the unknowns and understand the role of these unknowns; know the relationship(s) between knowns and unknowns

Step # 2
Language and Concepts

• Represent terms and words into appropriate mathematics symbols; translate multi-step word problems into/by equation(s)

Skills and Facts

• Identify the units and the domain and the range of the variable(s) involved in the problem

Step # 3
Procedures

• Solve multistep word problems by establishing the sequence of arithmetic operations

Skills and Facts

• Know and apply the properties of equality; mastery of arithmetic facts; execute procedures for whole numbers efficiently; know the order of operations

Step # 4
Concepts and Procedure

• Assess the reasonableness of the answer

Skills and Facts

• Numbersense: Use mental computations such as rounding to estimate the outcome of an operation

Step # 5
Language and Concepts

• Interpret the answer including the remainder if involved; express the division problems in multiple ways

Skills and Facts

• Add, subtract, multiply and divide whole numbers fluently with understanding; know the role of numbers in each operation, e.g., know the role of remainder in practical situations

Learning with rigor using SMP means that the students not only understand the concept and procedures but also see that a particular method(s) may have limitations and that the context of the problem defines the applicability and efficiency of the method.