Reason quantitatively and abstractly
March 11, 2021 20210401 13:51Reason quantitatively and abstractly
Reason Quantitatively and Abstractly: Specific vs. General
Common Core State StandardsMathematics (CCSSM) 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, contextfree 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 highlevel 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:
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, a^{2} + b^{2} = c^{2}, 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., a^{2} + c^{2} = b^{2}), or with the triangle PQS, with the right angle at Q, (p^{2} + r^{2} = q^{2}), 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

 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).
Case/State  # of items  p = Total Cost in $ 
Start  0  45 
1^{st}  1  25×1+ 45 
2^{nd}  2  25×2+ 45 
3^{rd}  3  25×3+ 45 
—  —  — 
100^{th}  100  25×100+ 45 
—  —  — 
n^{th}  n  p=25×n+ 45=25n+ 45 
 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 (decontextualize),
 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).
 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 (decontextualization);
 The student tries to remember how to find the distance between two points (e.g., y_{1} and y_{2}) on a number line (in this case, yaxis) as distance = y_{2}−y_{1} (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.
 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?
 Creating multiple representations of quantities and relationships in the problem (concrete, iconic and pictorial representations, symbolic expressions—equations, inequalities, diagrams, etc.).
 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?
 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?
 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, ……, ……?
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