Standards of Program Quality and Effectiveness for Elementary Subject Matter Programs
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Content Specifications in Mathematics
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Part I: Content Domains for Subject Matter Understanding and Skill in Mathematics
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Domain 1:Number Sense
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1.1 Numbers, Relationships Among Numbers, and Number Systems. Candidates for Multiple Subject Teaching Credentials understand base ten place value, number theory concepts (e.g., greatest common factor), and the structure of the whole, integer, rational, and real number systems. They order real numbers, including integers, mixed numbers, rational numbers (e.g., fractions, decimals, percents) and irrational numbers on a number line. They represent and perform operations on numbers in exponential and scientific notation. They describe the relationships between the algorithms for addition, subtraction, multiplication, and division. They understand properties of number systems and their relationship to the algorithms, [e.g., 1 is the multiplicative identity; 27 + 34 = 2 × 10 + 7 + 3 × 10 + 4 = (2 + 3) × 10 + (7 + 4)]. Candidates perform operations with positive, negative, and fractional exponents, as they apply to whole numbers and fractions.
1.2 Computational Tools, Procedures, and Strategies. Candidates demonstrate fluency in standard algorithms for computation and evaluate the correctness of nonstandard algorithms. They demonstrate an understanding of the order of operations. They round numbers, estimate the results of calculations, and place numbers accurately on a number line. They demonstrate the ability to use technology, such as calculators or software, for complex calculations.
Domain 2: Algebra and Functions
2.1 Patterns and Functional Relationships. Candidates represent patterns, including relations and functions, through tables, graphs, verbal rules, or symbolic rules. They use proportional reasoning such as ratios, equivalent fractions, and similar triangles, to solve numerical, algebraic, and geometric problems. They use mathematics to represent and analyze quantitative relationships between dependent and independent variables in real- world problems.
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2.2 Linear and Quadratic Equations and Inequalities. Candidates are able to find equivalent expressions for equalities and inequalities, explain the meaning of symbolic expressions (e.g., relating an expression to a situation and vice versa), find the solutions, and represent them on graphs. They recognize and create equivalent algebraic expressions (e.g., 2(a+3) = 2a + 6), and represent geometric problems algebraically (e.g., the area of a triangle). They use mathematics to solve real-world problems using numerical and algebraic expressions and equations. Candidates have a basic understanding of linear equations and their properties (e.g., slope, perpendicularity); the multiplication, division, and factoring of polynomials; and graphing and solving quadratic equations through factoring and completing the square. They interpret graphs of linear and quadratic equations and inequalities, including solutions to systems of equations.
Domain 3: Measurement and Geometry
3.1 Two- and Three-dimensional Geometric Objects. Candidates for Multiple Subject Teaching Credentials understand characteristics of common two- and three-dimensional figures, such as triangles (e.g., isosceles and right triangles), quadrilaterals, and spheres. They are able to draw conclusions based on the congruence, similarity, or lack thereof, of two figures. They identify different forms of symmetry, translations, rotations, and reflections. They understand the Pythagorean theorem and its converse. They are able to work with properties of parallel lines.
3.2 Representational Systems, Including Concrete Models, Drawings, and Coordinate Geometry. Candidates use concrete representations, such as manipulatives, drawings, and coordinate geometry to represent geometric objects. They construct basic geometric figures using a compass and straightedge, and represent three-dimensional objects through two-dimensional drawings. They combine and dissect two- and three- dimensional figures into familiar shapes, such as dissecting a parallelogram and rearranging the pieces to form a rectangle of equal area.
3.3 Techniques, Tools, and Formulas for Determining Measurements. Candidates estimate and measure time, length, angles, perimeter, area, surface area, volume, weight/mass, and temperature through appropriate units and scales. They identify relationships between different measures within the metric or customary systems of measurements and estimate an equivalent measurement across the two systems. They calculate perimeters and areas of two-dimensional objects and surface areas and volumes of three- dimensional objects, and use mathematics to solve real-world problems involving the volume of cones, cylinders, and spheres. They relate proportional reasoning to the construction of scale drawings or models. They use measures such as miles per hour to analyze and solve problems.
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Domain 4: Statistics, Data Analysis, and Probability
4.1 Collection, Organization, and Representation of Data. Candidates represent a collection of data through graphs, tables, or charts, incorporating technology as appropriate. They understand the mean, median, mode, and range of a collection of data. They have a basic understanding of the design of surveys, such as the role of a random sample.
4.2 Inferences, Predictions, and Arguments Based on Data. Candidates interpret a graph, table, or chart representing a data set. They investigate patterns of association in bivariate data (e.g., linear associations, goodness of fit) in scatter plots and frequency tables. They draw conclusions about a population from a random sample, and identify potential sources and effects of bias.
4.3 Basic Notions of Chance and Probability. Candidates can define the concept of probability in terms of a sample space of equally likely outcomes. They use their understanding of complementary, mutually exclusive, dependent, and independent events to calculate probabilities of simple events. They can express probabilities in a variety of ways, including ratios, proportions, decimals, and percents. They find probabilities of compound events using various representations (e.g., organized lists, tables, tree diagrams, simulations).
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Part II: Subject Matter Skills and Abilities Applicable to the Content Domains in Mathematics
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Candidates for Multiple Subject Teaching Credentials identify and prioritize relevant and missing information in mathematical problems. They make sense of problems and persevere in solving them. They look for and make use of structure, analyzing complex problems to identify similar simple problems that might suggest solution strategies. They model with mathematics, representing a problem in alternate ways, such as with words, symbols, concrete models, diagrams, and technology in order to gain greater insight. They consider examples and patterns as means to formulating a conjecture.
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Candidates reason abstractly and quantitatively, and apply logical reasoning and techniques from arithmetic, algebra, geometry, and probability/statistics to solve mathematical problems. They look for and express regularity in repeated reasoning, use appropriate tools strategically, and analyze problems to identify alternative solution strategies. They evaluate the truth of mathematical statements (i.e., whether a given statement is always, sometimes, or never true). They apply different solution strategies (e.g., estimation) to check the reasonableness of a solution. They demonstrate whether or not a solution is correct.
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Candidates explain their mathematical reasoning through a variety of methods, such as words, numbers, symbols, charts, graphs, tables, diagrams, and concrete models. They use academic language to construct viable arguments and critique the reasoning of others. They use appropriate mathematical notation with clear and accurate language, and they attend to precision. They explain how to derive a result based on previously developed ideas, and explain how a result is related to other ideas.