CCSS3

 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
• 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).
• furthers our thinking and understanding of the problem (that is the first criteria for the acceptance of a solution approach/method),
• provides ‘exact/correct’ solution,
• gives the correct solution efficiently (out of all the correct/exact solution approaches, we need to ask which one is more ‘efficient’. In other words, which method gives us the solution easier and with less consumption of resources and time?), and
• is elegant
In the final stage of discussion, the whole class should focus on which method is ‘elegant’. A method is called elegant when it can be generalized, abstracted and works for many situations (class of problems rather than individual or specific problem)—a method that also leads to the standard procedure. Arithmetic is generalizing concrete experiences to concepts and procedures. Algebra is generalizing arithmetic to relationships between numbers and concepts and then developing the concept of relationships between mathematical entities (e.g., numbers, etc.) and extending them to functions appropriate to model problems. For example, in the elementary and middle school, the formula for the area of a circle was intuitively understood or just accepted, whereas, in high school starting with a concrete model and by the use of the concept of limits, it is derived into A = πr2. The idea of deriving the formulas using fundamental principles, logic, and methods such as limits (a sum of infinite terms) is what differentiates earlier mathematics and the high school mathematics courses. Such thinking plays an important role in almost all topics of mathematics, for example, showing that a regular polygon approaches a circle when the number of sides approaches infinity or the price of a car decreases toward 0 as the number of years approaches infinity. Similarly, a diagram of a cone sectioned into cylindrical slabs gives a reasonable estimate for volume of the entire cone. The volume could be determined if these slabs were very thin, their volumes calculated and then summed. This leads to the basic idea behind integration—an important concept in calculus. The key concepts: quantitative and spatial reasoning reach fruition by high school. For example, the concept of spatial reasoning that began in easier grades reaches its formal form in high school. Spatial reasoning is observing objects and simple relationships between them—from spatial organization to formal concepts in geometry, trigonometry, and visualization and representation of transformations of geometrical objects. With the help of formal logic, it develops into rigorous treatment of understanding formal geometry: deriving formal definitions and formulas; making connections and inferences; proving and justifying claims using formal logic and language; describing relationships; integrating quantitative and spatial ideas to model problems into systems of equations and figures. Quantitative reasoning ranges from conceptualizing number relationships to algebraic principles–generalizing, abstracting, extrapolatingalgebra is generalized arithmetic; reversibility of thought; pattern analysis—recognizing, extending, creating and applying patterns; propositional reasoning; analogies; moving from knowns to unknowns; from facts and procedures to relationships; expanding the set of integers to include irrationals, imaginary numbers. Once algebraic reasoning is achieved, with the integration of the spatial tools of geometry, all algebraic, geometric, trigonometric, probabilistic and even calculus tools are within the reach of a student. Of course, this is subject to the availability of effective and efficient methods teaching, support, and resources. The focus in high school is to prepare students to see the role of mathematics in the world of natural, physical, and social sciences and to prepare them for higher education and work. At the end of high school, they should be able to apply the tools of arithmetic, algebra, geometry, trigonometry, and probability to diverse situations, such as (a) intra-mathematical (higher concepts in mathematics, e.g., calculus), (b) interdisciplinary (e.g., STEM fields, social sciences—economics, psychology, etc.), (c) extra-curricular (real life applications). To be prepared to solve problems through mathematical modeling, at the end of high school, students should be sensitive to the presence of numerous patterns in the relationships between a variety of variables from diverse situations:
• Direct and inverse variation—as one variable increases, another also increases (or decreases) at a similar rate.
• Accelerated variation—as one variable increases uniformly, a second increases at an increasing rate.
• Converging variation—as one variable increases without limit, another approaches some limiting value.
• Cyclic variation—as one variable increases uniformly, the other increases and decreases in some repeating cycle.
• Stepped variation—as one variable increases, another changes in jumps.
Nature of the Mastery of Curricular Elements To make sure that students have learned the material we teach, we need to pay attention to the mastery of individual curricular elements:
• Mastery of mathematics language: Possesses adequate vocabulary, the syntax, and can translate math expressions and equations into English and vice versa; can explain his or her thinking using mathematics language, symbols, and processes
• Constructs, develops, and understands concepts: can demonstrate the related multiple mathematical models of a concept (e.g., for systems of equations, they know the rationale and reasons for different—graphical, substitution, elimination, matrix, and determinant—methods of   solving them)
• Develops and executes procedures: Can execute standard procedures (including the development of them) accurately, consistently, fluently, efficiently with understanding (e.g., factoring trinomials, polynomial division, or synthetic division)
• Automatizes skills: Produces the result in acceptable time, fluently, and consistently (e.g., can give facts about integers or laws of exponents orally in 2 seconds or less and written 3 seconds or less)
Problem solving and communication: Integrates language, concepts, procedures and skills in problem posing, solving and interpreting and communicating of results and the solution process. As proposed by the framers of the CCSS-M, the central idea of a beginning algebra course is to become fluent in using and interpreting symbols so as to generalize the concepts from arithmetic and to see algebra as generalized arithmetic and to explore and study relationships and functions and their multiple representations. This means:
• Understanding, mastering, and applying the operations on and properties of real and complex number systems and their applications;
• Justifying operations on real numbers with rigor (e.g., the set of real number is complete—between any two real numbers there exists a real number);
• Understanding the arithmetic of algebraic expressions—extending the understanding and mastery of arithmetic operations (e.g., extending long division process to division of a polynomial by a monomial and binomial) and understanding the geometric and algebraic behaviors of polynomials and rational fractions;
• Extending the concept and operations of factorization and prime factorization to factorization of polynomials, particularly with integer coefficients;
• Understanding—defining, mastering operations on functions—addition, subtraction, multiplication, division, composition (including trigonometric functions), and inverse of functions (including exponential and logarithmic); understanding their behavior and applications; modeling applications using functions; impact of transformations (rigid and dynamic) on functions;
• Understanding and mastering the representation of and operations on information (data) linguistically (expressing ideas in words), arithmetically (number relationships including determinants and matrices), algebraically (expressions, systems of equations and inequalities), geometrically (pictorially, tabular, curves, figures), functionally (operations and compositions), discrete methods (determinants, matrices, flow charts), and probabilistically;
• Intra- and interrelationships between algebra and geometry (understanding the relationship between two and three dimensional objects—generating 3-dimensiional objects from 2-dimensional objects and vice-versa), coordinate geometry—an equation represents a geometrical entity and most geometrical objects can be expressed as a system of algebraical equations; polar vs. Cartesian, etc.); e.g., a circle—a collection of points equidistant (r units) from a given point (h, k) drawn on a paper (geometrical representation) can be expressed by the Cartesian equation: (x – h)2 + (y –k)2 = r2 (derived using the distance formula—algebraic); can be converted into polar equation by sing the transformation: x = h + r cos (t) and y = k + r sin (t).
• Understanding transformations (rigid and dynamic) and their representations (functional, matrices, etc.) and their impact on geometrical, discrete, and, algebraic systems (transformation, congruence, similarity, composition); studying systems through transformations;
• Applications of tools—arithmetic, algebraic, geometrical, probabilistic, trigonometric, functional and technological, in learning concepts, acquiring skills, and solving problems.
Students should realize that the idea behind learning properties of whole families of relations is typical of all mathematics: recognition of structure and similarities in apparently different situations allows applications of successful reasoning methods to new problems.        For fractions see How to Teach Fractions Effectively by Mahesh Sharma, 2008.

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Webinars

Tuesday Mathematics Education Webinars (Free)
For teachers, parents, and curriculum coordinators.

By Professor Mahesh Sharma
Assisted by: Sanjay Raghav January 18 8:00 AM US EST

Topic:Trajectory of Multiplication across the grades: Its language, conceptual models, and procedures.

Zoom ID: 5084944608
PC: mathforall