When we arrived as freshmen at my high school, our headmaster—a very popular, caring, and tough mathematics teacher (yes, he still taught), greeted us warmly and during his welcoming speech remarked: “Those of you who are fluent in fractions will end up in calculus and you know that fractions are dependent on multiplication. And those who do not have the mastery of multiplication tables and fractions will not enter into fields such as science, technology, engineering, mathematics, physics, and even economics. I do not want mathematics to be a gatekeeper for your aspirations. You should have all the skills that give you freedom of choice of options.” Most of my friends laughed at the remark.
After fifty years of teaching, I am convinced, more than ever, about the validity of that statement. All students should have the option to pursue any field, and a lack of proper preparation in mathematics should not close doors too early. Mathematics has become an entry to exciting and rewarding fields. Today, it is not just the STEM fields that require higher mathematics; even in the social sciences success and competence are dependent on skills in mathematics.
Throughout the twentieth century, most problems in natural and physical sciences, engineering, technology, and even social sciences could be modeled by functions – mostly by continuous and differentiable functions, but sometimes other functions such as: piece-wise, step, etc. Due to the advent of computers and related new technologies, today we can model many of the problems with only a few data points. Therefore, discrete mathematics (e.g., probability, statistics, linear programming, numerical analysis, etc.) and computer science play an important role in modeling problems in social sciences, physical and natural sciences. Similarly, the role of compu-graphics and graphing utilities in gaming systems and simulations is important in problem solving, therefore, in mathematics. This requires quantitatively and qualitatively a different kind of preparation in mathematics.
A different preparation in mathematics means that students, from the beginning of middle and high school, need to be made aware of the cumulative nature of mathematics: e.g., that mastery of multiplicative reasoning facilitates the understanding and the mastery of proportional reasoning (e.g., fractions/decimals/percent, rate, unit change, scale factor, and slope (ultimately, differential coefficient—rate of infinitesimal change in one variable with respect to the corresponding change in the other variable). The understanding and mastery of fractions are essential for success in algebra. And success in calculus is greatly dependent on facility in algebraic concepts, manipulations, and operations.
In high school, from ninth to 11th grade, students should formalize and extend the relationships between and integration of logical, arithmetic, algebraic, functional, geometric, probabilistic, linguistic, and statistical reasoning, their models and applications. Students should experience mathematics as a coherent, useful, and logical subject that makes use of their ability to make sense of problem situations. With this aim in mind, the framers of CCSS-M recommend three courses in high school: Algebra I, Geometry, and Algebra II.
The fundamental purpose of CCSS-M Algebra I is to formalize and extend the mathematics that students learned in the middle grades. CCSS-M’s Algebra I is built on the middle grades standards and is a more ambitious version of algebra traditionally taught in grades eight or nine. The topics deepen and extend understanding of linear and exponential relationships by contrasting them with each other and by applying linear models to data that exhibit a linear trend, and students engage in methods for analyzing, solving, and using quadratic functions. Solving algebraic problems procedurally, without the appropriate development of algebraic thinking, does not take students far in mathematics and the rigor of Algebra I escapes them. Such procedural work does not prepare them for higher mathematics and meaningful problem solving.
It is evident from children’s classroom work and performance on mathematics tests and examinations that most of their errors in solving problems in algebra I, such as algebraic equations, functions, and quadratic equations, are related to lack of mastery of facts and errors in operations on integers, fractions and lack of algebraic thinking. Because of these errors, even on easy word problems, the overall success rate of students using algebraic methods to solve problems is low and reflects a great deal of variation among students. In solving algebraic problems, a large number of students use few algebraic models, instead relying on arithmetic reasoning or guess and check, minimally useful but ineffective methods. The table below contrasts arithmetic and algebraic thinking.
|Arithmetic Thinking||Algebraic Thinking|
|Work from knowns to unknowns||Work on unknowns to knowns|
|Thinking in natural language||Thinking in symbolic terms and mathematics language|
|Unknowns transient||Unknowns defined by the conditions of the problems and fixed for the particular problem
|Equation as a formula to produce answers||Equations and inequalities as descriptions of the relationships, parameters, and situations|
|Chains of successive calculations||Chains of logically linked equalities or inequalities to transform them into simpler forms
|Solution found to a specific problem||Method and solution found to a category and classes of problems|
The transition from arithmetic to algebra takes time and experience. It is the focus of middle school mathematics. However, till the rigor of CCSS-M holds in our schools, it should also get teachers’ attention during the high school years. It is important that the intervention work during the high school years focus on this aspect of transition from arithmetic to algebra.
During middle school and early part of high school, students need to see and understand the difference between arithmetic and algebraic reasoning. They need to see that for some simple problems, arithmetic methods are adequate although they do not lead to generalizations. Unless students experience several problems that are amenable to arithmetic and algebraic methods and see the inadequacy of arithmetic methods, they will have difficulty appreciating the importance of algebraic thinking. Let us consider a simple problem:
Two players, David and Mark, scored a total of 37 points in a game. David scored 5 points more than Mark. How many points did each score?
This problem can be solved by several methods.
Solution 1: Guess and check
14 + 23 = 37 but the difference is not 5
15 + 22 = 37 but the difference is not 5
16 + 21 = 37 the difference is 5
21 + 16 = 37 is also an answer as the difference is 5.
Since David scored 5 points more than Mark. David: 21 points; Mark: 16 points.
Solution 2: Arithmetic reasoning
If they both score equally, then I divide 37 by 2 = 18.5. But they did not score equally. One scored 5 more than the other. To have such a score, one is higher than 18.5 and the other is lower than 18.5 with a difference between the two scores of 5 points. Since 18.5 is the average, one is 18.5 + 2.5 = 21 and the other is 18.5 − 2.5 =16. David is 21 and Mark is 16.
Solution 3: Logical arithmetic reasoning
David’s score 5 points more than Mark’s score. I find 37 − 5 = 32. Since, 32 is the average of the two, now. I divide 32 equally. (37 − 5) ÷ 2 = 16.
David’s score = (37 – 5)/2 + 5 = 32/2 + 5 = 16 + 5
Mark’s score = (37 – 5)/2 = 32/2 = 16
Solution 4: Geometrical/pictorial
Mark’s score: ____________ (the length of the line segment represents Mark’s score)
David’s score: ____________ _____ (the line segment of the same length plus 5 points)
Mark’s score + David’s score = 37
____________ ____________ _____ = 37 (the total = two line segments of same length + 5)
____________ ____________ = 37 – 5
____________ ____________ = 32
____________ = 32 ÷ 2 = 16
Mark’s score: 16 points
David’s score: 16 + 5 points = 21 points.
(In many countries in place of this line segment, representing an unknown, a bar is used.)
Solution 5: Algebraic approach to solution
Let us say Mark scored x points. Then David scored x + 5 points. Together they scored 37 points. So, x + x + 5 = 37.
x + (x + 5) = 37 <-> 2x + 5 = 37 <-> 2x = 32 <-> x = 16.
Mark scored 16 points; David scored 21 points.
The five approaches to solutions used by students from middle to high school are progressively algebraic in reasoning beginning with guess and check to arithmetic reasoning to more generalized algebraic thinking. These solution approaches indicate some of the ways in which arithmetic leads to algebra. For developing algebraic reasoning and the flexibility of thought and problem solving facility, students should be helped to realize the strengths, weaknesses, and limitations of each approach.
A guess and check solution is quite easy, accessible even to upper elementary school children, especially as the answer does not involve non-integral numbers and the numbers involved and the conditions of the problem are quite simple. It, thus, has limitations.
The next two solutions, the second using arithmetic reasoning and third using logical arithmetic reasoning are also simple. These methods are helpful in visualizing the problem and provide entry into the problem solving process, but they also have limitations if the numbers are not easy and if the parameters of the problem are complex. The fourth method begins to introduce the concept of unknown and the relationships between unknowns and knowns. It becomes the basis of the symbolic representation. The spatial representation (whether line segment representation or the bar method), even when the numbers in the problem are fractions, decimals, or percents, provides easy access to algebra early and effectively. Most high performing countries on mathematics use spatial representation of problems before symbolic representations.
The fifth method involves algebraic thinking and can be suggested by the spatial reasoning method, the arithmetic reasoning, and the translation from natural language to symbolic, mathematical language. Effective teachers always begin with discussing some kind of visual or spatial representation of the problem and then using logical reasoning lead to an algebraic representation.
In the first category of approaches—three methods based on arithmetic reasoning—students’ solution approaches began with the ‘knowns’ and moved to ‘unknowns.’ On the other hand, in the second category of approaches—the last two methods—the spatial reasoning and formal algebraic method, one begins from unknowns and establishes relationships between unknowns and knowns in the form of an equation. Then one applies a procedure, based on logical reasoning and properties of numbers, operations, and equality, to solve the equation.
There are fundamental differences in these two categories of solutions approaches. The three methods based on arithmetic reasoning are less applicable as they cannot be generalized. In the case of more complicated problems, the ‘guess and check’ is less straightforward and difficult to generalize, particularly when the answer is not an integer. Generalizing logical arithmetic reasoning method of Solution 2 is also hard, and even generalizing the method of Solution 3 is challenging for many students. As a result, the power of algebra in the form of integration of quantitative and spatial reasoning is needed. Although many students are unable to change or extend their approach, it can be accomplished if we begin with
- translation from natural language to symbolic language
- construct diagrams (Bar Graph, tables, charts, or Empty Number line), and
- then translate into algebraic equation (integrate spatial, quantitative, and symbolic representations).
To truly understand algebraic problem solving, students should be taken through the first four methods before leading to algebraic methods. And they should arrive at the realization that when the problem is a little more difficult and involves non-integer numbers and solutions, the first four non-algebraic methods become much more difficult.
Solutions 2 and 3 of the problem above illustrate how arithmetic solutions to a problem work from ‘knowns’ (actual numbers) to unknowns (the answer) by a process of calculation. For example, from the known numbers 37 and 5, one finds the new quantity 32 (in Solution 2, the number of points to be shared equally after the extra 5 have been awarded to David), and then from this new known, one finds 16 (the points that they both score). The unknowns are transient: first the aim is to find one, by finding the first unknown, then one finds the next one, and so on in a chain of successive computations.
In contrast, the algebraic method is quite different. Instead of immediately progressing towards a solution, one understands and describes the situation in natural language: ‘One of them scores 5 more points than the other’ and then translates it into spatial and/or symbolic algebraic language. So if one scores say: x points (unknown) then, the other scores x + 5 (unknown). Then a relationship is found between unknowns and the known in the form of equation. So we have: x + (x + 5) = 37. Now from unknowns one moves to knowns by working towards a solution by calculation using the properties of numbers, operations, and equality; one then works on the whole equality, producing more (equivalent) equations until eventually the solution appears (here the transitive property of equality is hidden: first equation => second equivalent equation => etc.; the solution to the last equation, therefore, should be the solution to the first equation).
x + (x + 5) = 37 Translating natural language into symbolic and relational language
(x + x) + 5 = 37 (Applying associative property of addition)
2x + 5 = 37 (Combining like terms—definition of addition)
2x + 5 –5 = 37 –5 (Addition/subtraction property of equality)
2x + 0 = 32 (Property of additive inverse and subtraction operation)
2x = 32 (Property of additive identity), and finally, the equation:
x = 16. (Division property of the equality)
Solution: Mark’s score (x points) = 16 points; David’s score (x + 5 points) = 16 + 5 = 21 points.
The kind of thinking where one begins with identifying unknowns, establishes relationships between unknowns (an equation or expression), and finally moves to the known (solution process using rules, relationships, theorems, definitions, properties, etc.) is not natural. It is contrary to intuitive and arithmetical thinking.
In arithmetic, we begin with knowns and lead to unknown(s) using the logic of natural language. The translation from natural language and arithmetical thinking to algebraic language and relational thinking is a major transition for students. It requires experience and effective teaching—the development of mathematical (spatial and algebraic) language—vocabulary, syntax, and two-way translation from natural language to math and vice-versa), forming conceptual schemas (wherever possible concrete and visual and relevant models must be used), and arriving at efficient procedures.
Certain arithmetic and pre-algebraic skills need to be mastered in order to be successful in algebra. I find that along with the basic arithmetic facts and fractions, a few key mathematics skills are also needed for a student to be successful in algebra. The most important of these is how to operate on integers. Many of students’ problems in algebra can be traced to their limited numbersense and misconceptions about integers and lack or mastery on their operations. For example, most students conceptualize subtraction of whole numbers as ‘take away’ and multiplying and dividing integers only as repeated addition and repeated subtraction respectively. These schemas are sufficient for conceptualizing and operating on whole numbers, but they are limiting and inadequate for operating on fractions, decimals, integers, rational numbers, and algebraic expressions. Therefore, a strong understanding of numeracy (number, number relations, number operations, patterns in numbers, and properties of numbers) and proportional reasoning is essential for learning algebra, but there are also marked transformations (mathematical and cognitive) that have to occur in students’ thinking to become comfortable with problem solving using algebra rather than arithmetic. With this change, a student can access the power of tools from algebra and other higher mathematics. These tools are the language of STEM fields and social sciences such as: economics, business, psychology, geography, etc.
When students experience and engage in discussions about the efficiency and efficacy of different methods early on, they begin to think flexibly and acquire the ability to augment, extend, and adapt their methods of problem solving. When all (or almost all) of the students have solved the problem or made attempts in solving the problem, the teacher should solicit (orally or written form) all the approaches (successful or not successful). Place them on the board or show them under the document camera or a picture. She should then need to engage in the discussion that help students to see each other’s approach and thinking behind it. Then discuss the approaches using the criteria: whether the approach
- furthers our thinking and understanding of the problem (that is the first criteria for the acceptance of a solution approach/method),
- provides ‘exact/correct’ solution,
- gives the correct solution efficiently (out of all the correct/exact solution approaches, we need to ask which one is more ‘efficient’. In other words, which method gives us the solution easier and with less consumption of resources and time?), and
- is elegant
In the final stage of discussion, the whole class should focus on which method is ‘elegant’. A method is called elegant when it can be generalized, abstracted and works for many situations (class of problems rather than individual or specific problem)—a method that also leads to the standard procedure.
Arithmetic is generalizing concrete experiences to concepts and procedures. Algebra is generalizing arithmetic to relationships between numbers and concepts and then developing the concept of relationships between mathematical entities (e.g., numbers, etc.) and extending them to functions appropriate to model problems. For example, in the elementary and middle school, the formula for the area of a circle was intuitively understood or just accepted, whereas, in high school starting with a concrete model and by the use of the concept of limits, it is derived into A = πr2. The idea of deriving the formulas using fundamental principles, logic, and methods such as limits (a sum of infinite terms) is what differentiates earlier mathematics and the high school mathematics courses. Such thinking plays an important role in almost all topics of mathematics, for example, showing that a regular polygon approaches a circle when the number of sides approaches infinity or the price of a car decreases toward 0 as the number of years approaches infinity. Similarly, a diagram of a cone sectioned into cylindrical slabs gives a reasonable estimate for volume of the entire cone. The volume could be determined if these slabs were very thin, their volumes calculated and then summed. This leads to the basic idea behind integration—an important concept in calculus.
The key concepts: quantitative and spatial reasoning reach fruition by high school. For example, the concept of spatial reasoning that began in easier grades reaches its formal form in high school.
Spatial reasoning is observing objects and simple relationships between them—from spatial organization to formal concepts in geometry, trigonometry, and visualization and representation of transformations of geometrical objects. With the help of formal logic, it develops into rigorous treatment of understanding formal geometry: deriving formal definitions and formulas; making connections and inferences; proving and justifying claims using formal logic and language; describing relationships; integrating quantitative and spatial ideas to model problems into systems of equations and figures.
Quantitative reasoning ranges from conceptualizing number relationships to algebraic principles–generalizing, abstracting, extrapolating—algebra is generalized arithmetic; reversibility of thought; pattern analysis—recognizing, extending, creating and applying patterns; propositional reasoning; analogies; moving from knowns to unknowns; from facts and procedures to relationships; expanding the set of integers to include irrationals, imaginary numbers. Once algebraic reasoning is achieved, with the integration of the spatial tools of geometry, all algebraic, geometric, trigonometric, probabilistic and even calculus tools are within the reach of a student. Of course, this is subject to the availability of effective and efficient methods teaching, support, and resources.
The focus in high school is to prepare students to see the role of mathematics in the world of natural, physical, and social sciences and to prepare them for higher education and work. At the end of high school, they should be able to apply the tools of arithmetic, algebra, geometry, trigonometry, and probability to diverse situations, such as (a) intra-mathematical (higher concepts in mathematics, e.g., calculus), (b) interdisciplinary (e.g., STEM fields, social sciences—economics, psychology, etc.), (c) extra-curricular (real life applications).
To be prepared to solve problems through mathematical modeling, at the end of high school, students should be sensitive to the presence of numerous patterns in the relationships between a variety of variables from diverse situations:
- Direct and inverse variation—as one variable increases, another also increases (or decreases) at a similar rate.
- Accelerated variation—as one variable increases uniformly, a second increases at an increasing rate.
- Converging variation—as one variable increases without limit, another approaches some limiting value.
- Cyclic variation—as one variable increases uniformly, the other increases and decreases in some repeating cycle.
- Stepped variation—as one variable increases, another changes in jumps.
Nature of the Mastery of Curricular Elements
To make sure that students have learned the material we teach, we need to pay attention to the mastery of individual curricular elements:
- Mastery of mathematics language: Possesses adequate vocabulary, the syntax, and can translate math expressions and equations into English and vice versa; can explain his or her thinking using mathematics language, symbols, and processes
- Constructs, develops, and understands concepts: can demonstrate the related multiple mathematical models of a concept (e.g., for systems of equations, they know the rationale and reasons for different—graphical, substitution, elimination, matrix, and determinant—methods of solving them)
- Develops and executes procedures: Can execute standard procedures (including the development of them) accurately, consistently, fluently, efficiently with understanding (e.g., factoring trinomials, polynomial division, or synthetic division)
- Automatizes skills: Produces the result in acceptable time, fluently, and consistently (e.g., can give facts about integers or laws of exponents orally in 2 seconds or less and written 3 seconds or less)
Problem solving and communication: Integrates language, concepts, procedures and skills in problem posing, solving and interpreting and communicating of results and the solution process.
As proposed by the framers of the CCSS-M, the central idea of a beginning algebra course is to become fluent in using and interpreting symbols so as to generalize the concepts from arithmetic and to see algebra as generalized arithmetic and to explore and study relationships and functions and their multiple representations. This means:
- Understanding, mastering, and applying the operations on and properties of real and complex number systems and their applications;
- Justifying operations on real numbers with rigor (e.g., the set of real number is complete—between any two real numbers there exists a real number);
- Understanding the arithmetic of algebraic expressions—extending the understanding and mastery of arithmetic operations (e.g., extending long division process to division of a polynomial by a monomial and binomial) and understanding the geometric and algebraic behaviors of polynomials and rational fractions;
- Extending the concept and operations of factorization and prime factorization to factorization of polynomials, particularly with integer coefficients;
- Understanding—defining, mastering operations on functions—addition, subtraction, multiplication, division, composition (including trigonometric functions), and inverse of functions (including exponential and logarithmic); understanding their behavior and applications; modeling applications using functions; impact of transformations (rigid and dynamic) on functions;
- Understanding and mastering the representation of and operations on information (data) linguistically (expressing ideas in words), arithmetically (number relationships including determinants and matrices), algebraically (expressions, systems of equations and inequalities), geometrically (pictorially, tabular, curves, figures), functionally (operations and compositions), discrete methods (determinants, matrices, flow charts), and probabilistically;
- Intra- and interrelationships between algebra and geometry (understanding the relationship between two and three dimensional objects—generating 3-dimensiional objects from 2-dimensional objects and vice-versa), coordinate geometry—an equation represents a geometrical entity and most geometrical objects can be expressed as a system of algebraical equations; polar vs. Cartesian, etc.); e.g., a circle—a collection of points equidistant (r units) from a given point (h, k) drawn on a paper (geometrical representation) can be expressed by the Cartesian equation: (x – h)2 + (y –k)2 = r2 (derived using the distance formula—algebraic); can be converted into polar equation by sing the transformation: x = h + r cos (t) and y = k + r sin (t).
- Understanding transformations (rigid and dynamic) and their representations (functional, matrices, etc.) and their impact on geometrical, discrete, and, algebraic systems (transformation, congruence, similarity, composition); studying systems through transformations;
- Applications of tools—arithmetic, algebraic, geometrical, probabilistic, trigonometric, functional and technological, in learning concepts, acquiring skills, and solving problems.
Students should realize that the idea behind learning properties of whole families of relations is typical of all mathematics: recognition of structure and similarities in apparently different situations allows applications of successful reasoning methods to new problems.
 For fractions see How to Teach Fractions Effectively by Mahesh Sharma, 2008.