Middle level 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 (integers, rational, and irrational numbers).
• 2.C – Understand the progression of learning that begins with the base-ten number system and operations thereof, builds into understanding of and operations with fractions and rational numbers, and extends to understanding of and operations with real numbers.

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

• 3.A – Solve and graphically represent real life and mathematical problems using numerical and algebraic expressions, equations, inequalities, and systems of equations and inequalities.
• 3.B – Understand the connections between proportional relationships, lines, and linear equations and use them to solve real world and mathematical problems.
• 3.C – Use functional notation and interpret expressions for functions as they arise in terms of the situation they model (e.g., linear, quadratic, simple rational, and exponential).
• 3.D – Understand operations on algebraic expressions and functions (e.g., polynomials, rationals, and roots).
• 3.E – Apply arithmetic properties to algebraic expressions and equations.
• 3.F – Write equations and inequalities in equivalent forms.
• 3.G – Analyze and model functions.
• 3.H – Explain the interrelationship between the various representations of a function (e.g., graphs, tables, algebraic expressions, concrete models, and contexts).

Candidates demonstrate a conceptual understanding of geometric properties and relationships as they apply to congruence, similarity, geometric figures, and the Cartesian coordinate system.

• 4.A – Understand congruence in terms of rigid motion.
• 4.B – Prove theorems involving triangle congruency and similarity.
• 4.C – Apply transformations and use similarity and congruence in mathematical situations.
• 4.D – Understand and perform geometric constructions physically and/or with technology.
• 4.E – Understand the Pythagorean theorem and apply it to problem solving situations.
• 4.F – Solve real life and mathematical problems involving lines, angle measure, area, surface area, and volume.
• 4.G – Classify, visualize, and describe two-dimensional figures and three-dimensional objects as well as the relationship among them.
• 4.H – Apply geometric concepts to model real world situations.

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

• 5.A – Use appropriate measures of central tendency and distributions to summarize, represent, and interpret categorical and quantitative data.
• 5.B – Understand and evaluate random processes underlying statistical experiments and use random sampling to make inferences about whole populations.
• 5.C – Understand and use the rules of probability to make predictions, evaluate decisions, and solve problems.
• 5.D – Apply probability concepts to model real world situations.

Candidates demonstrate conceptual understanding and procedural fluency in analyzing proportional relationships and solving real world mathematical problems.

• 6.A – Describe and determine additive versus multiplicative perspectives.
• 6.B – Reason and compute with ratios and the constant of proportionality (unit rate) to solve real world and mathematical problems.
• 6.C – Recognize, describe, and represent equivalent ratios, rates, and proportional relationships.
• 6.D – Represent and analyze proportional relationships using tables, graphs, equations, diagrams, concrete and mathematical models, and verbal descriptions of proportional relationships.
• 6.E – Compute the constant of proportionality (unit rate) associated with rational numbers.
• 6.F – Recognize and connect proportional relationships to geometry, measurement, statistics, probability, and function.
• 6.G – Use ratio reasoning to convert measurement units.
• 6.H – Apply ratio and proportion concepts to model real world situations.

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

• 8.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.
• 8.B – Demonstrate ability to present mathematical concepts using multiple representations (e.g., numerical, graphical, analytical, and contextual).
• 8.C – Demonstrate the ability to guide student discourse in mathematical problem solving, argumentation (creation and critiquing), literacy, and in-depth conceptual understanding.
• 8.D – Demonstrate knowledge of learning progressions, including conceptual and procedural milestones and common misconceptions, within each content domain and connections to instruction.
  • 8.D.1 – Demonstrate knowledge of major, supporting, and additional clusters for each grade level.
  • 8.D.2 – Demonstrate an understanding of the concept of mathematical rigor including conceptual understanding, procedural skill and fluency, and application.
  • 8.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.
• 8.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.