Mathematics (2013)

Demonstrate ability to embed CCSS-M mathematical practices in the instructional process to deepen conceptual understanding.

• 1.A – Make sense of problems and persevere in solving them.
• 1.B – Reason abstractly and quantitatively.
• 1.C – Construct viable arguments and critique the reasoning of others.
• 1.D – Model with mathematics.
• 1.E – Use appropriate tools strategically.
• 1.F – Attend to precision.
• 1.G – Look for and make use of structure.
• 1.H – Look for and express regularity in repeated reasoning.

Candidates demonstrate a conceptual understanding of and procedural facility with operations and number systems.

• 2.A – Understand the structure, properties, characteristics of, and relationships between number systems including whole numbers, integers, rational, real, and complex numbers.
• 2.B Understand arithmetic operations of different number systems and their properties.
• 2.C Understand vectors and matrices.
  • 2.C.1 – Represent and model with vector quantities and matrices.
  • 2.C.2 – Perform operations on vectors and matrices.
  • 2.C.3 – Understand the properties of operations using vectors and matrices.

Candidates demonstrate a conceptual understanding of and procedural facility with algebra concepts emphasizing functions.

• 3.A – Use functional notation and interpret functions as they arise in different mathematical contexts.
• 3.B – Understand operations on algebraic expressions and functions (e.g., polynomials, rationals, and roots).
• 3.C – Understand and prove properties of algebraic systems.
• 3.D – Create mathematical models using algebraic expressions.
• 3.E – Solve algebraic equations and inequalities with one or more variables.
• 3.F – Write equations and inequalities in equivalent forms (e.g., graphs, tables, and algebraic expressions) to gain a more complete understanding to solve problems.
• 3.G – Explain the connection among equivalent forms and their algebraic purposes.
• 3.H – Conceptually understand functions and build models of relationships between two quantities.
• 3.I – Analyze and model functions (e.g., linear quadratic, exponential, and trigonometric).

Candidates prove and understand geometric theorems and transformations as they apply to congruence, similarity, lines, triangles, trigonometry, circles, and measurements. In addition, the following content should be included; the relationship between geometric figures and the Cartesian coordinate system.

• 4.A – Understand congruence in terms of rigid motions and prove geometric theorems.
• 4.B – Apply transformations and use similarity and congruence in mathematical situations.
• 4.C – Perform geometric constructions physically and/or with technology.
• 4.D – Define trigonometric ratios and solve problems involving right triangles and general triangles.
• 4.E – Derive the Pythagorean theorem and apply it to problem solving situations.
• 4.F – Identify and describe relationships among angles, radii, and chords.
• 4.G – Derive formulas for arc lengths and areas of sectors of circles.
• 4.H – Translate between the geometric description and the equation for a conic section.
• 4.I – Use coordinates to prove geometric theorems algebraically.
• 4.J – Derive area, surface area, and volume formulas and use them to solve problems.
• 4.K – Solve real life and mathematical problems involving angle measures and/or polygons.
• 4.L – Visualize and describe two-dimensional figures and three-dimensional objects as well as the relationships among them.
• 4.M – Apply geometric concepts to model real world situations.

Candidates demonstrate a conceptual understanding of and procedural facility with statistics and probability.

• 5.A – Summarize, represent, and interpret single variable and multivariate data that can be categorical or continuous.
• 5.B – Represent and interpret linear regression models.
• 5.C – Apply statistical concepts and representations to model real world situations.
• 5.D – Use appropriate technology to collect, represent, and analyze data.
• 5.E – Make inferences, collect data from random experiments, and justify conclusions.
  • 5.E.1 – Conduct random sampling.
  • 5.E.2 – Evaluate a statistical experiment with respect to the assumption of randomization.
  • 5.E.3 – Make and justify conclusions about inferences from appropriate statistical analysis of data.
• 5.F – Understand and apply the principles of probability (complementary events, mutually exclusive events, independent, and dependent events) to compute probabilities of compound events.
• 5.G – Apply probability concepts to model real world situations.
• 5.H – Use probability to make and inform decisions.
  • 5.H.1 – Calculate expected values.
  • 5.H.2 – Make inferences from binomial probability distributions (e.g., sampling populations from a binary sample space).

Candidates recognize, analyze, and represent equivalent ratios, rates, and proportional relationships and use them to solve problems.

Candidates will be able to connect mathematics with real life problems through the use of mathematical modeling and technology.

• 7.A – Construct mathematical models in the content strands (e.g., look at a real life situation and transpose it into a mathematical problem, solve the problem, and interpret the solution in real life.)
• 7.B – Use the appropriate technology available.
  • 7.B.1 – Explore conjectures, visualize, and analyze the mathematics.
  • 7.B.2 – Develop concepts and apply them to a context.

Candidates demonstrate a conceptual understanding of and procedural facility with calculus concepts.

• 8.A – Demonstrate a conceptual understanding of limit, continuity, differentiation, and integration and have a thorough background in the techniques and application of the calculus.
• 8.B – Apply concepts of function, geometry, and trigonometry in solving problems involving calculus.
• 8.C – Use the concepts of calculus and mathematical modeling to represent and solve problems taken from real world contexts.

Candidates apply the fundamental ideas of discrete mathematics in the formulation and solution of problems.

• 9.A – Develop the general techniques of mathematical proof including direct proofs, proof by contradiction, contrapositive proof, and proof by induction.
• 9.B – Demonstrate knowledge of basic elements of discrete mathematics such as graph theory, recurrence relations, finite difference approaches, and combinatorics.
• 9.C – Apply the fundamental ideas of discrete mathematics in the formulation and solution of problems arising from real world situations.

Candidates possess a deep understanding of how students learn mathematics and of the pedagogical knowledge specific to mathematics teaching and learning.

• 10.A – Select, use, and determine suitability of the available mathematics curricula, teaching materials, and other resources including manipulatives for the learning of mathematics for all students.
• 10.B – Demonstrate ability to present mathematical concepts using multiple representations (e.g., numerical, graphical, analytical, and contextual).
• 10.C – Demonstrate the ability to guide student discourse in mathematical problem solving, argumentation (creation and critiquing), literacy, and in-depth conceptual understanding.
• 10.D – Demonstrate knowledge of learning progressions, including conceptual and procedural milestones and common misconceptions, within each content domain and connections to instruction.
  • 10.D.1 – Demonstrate knowledge of major, supporting, and additional clusters for each grade level.
  • 10.D.2 – Demonstrate an understanding of the concept of mathematical rigor including conceptual understanding, procedural skill and fluency, and application.
  • 10.D.3 – Demonstrate an understanding of coherent connections within clusters at a grade level and the progression from grade level to grade level that builds on previous learning.
• 10.E – Engage in developmentally and culturally responsive teaching of mathematics that minimizes power and status issues, nurtures a positive mathematics disposition, and utilizes students’ cultural funds of knowledge and experiences as resources for lessons.