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:
- How many groups of 3 are there in 12? (repeated subtraction)
- If we divide 12 into 3 equal parts/shares/sets/groups, what is the size of each part? (groups of/partitioning model)
- If we organize 12 chairs in 3 rows with equal number, how many will be in each row? (array model)
- 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 $|
|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 have an equal sign in it.”
“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.