Elementary mathematics specialist – specialty endorsement (2007)

Candidates possess a deep understanding of the development of mathematical and spatial learning from early childhood through adolescence. This includes mastery of mathematics content assessed by the middle level mathematics teacher candidate assessment (WEST-E / NES or NBCT in early adolescence mathematics). Additionally, the following competencies will be assessed by the preparation program.

• 1.A – Standards for mathematical practices: demonstrate ability to embed common core state standards for mathematical practices in the instructional process to deepen conceptual understanding of mathematics content.
  • 1.A.1 – Make sense of problems and persevere in solving them.
  • 1.A.2 – Reason abstractly and quantitatively.
  • 1.A.3 – Construct viable arguments and critique the reasoning of others.
  • 1.A.4 – Model with mathematics.
  • 1.A.5 – Use appropriate tools strategically.
  • 1.A.6 – Attend to precision.
  • 1.A.7 – Look for and make use of structure.
  • 1.A.8 – Look for and express regularity in repeated reasoning.
• 1.B – Number and operations: demonstrate a conceptual understanding of, procedural fluency with, and application of operations, number systems, and properties.
  • 1.B.1 – Demonstrate an understanding of pre-number concepts: non-quantified comparisons (less than, more than, the same), containment (e.g., 5 contains 3), 1-to-1 correspondence, cardinality, and ordinality.
  • 1.B.2 – Have a comprehensive repertoire of interpretations of the four operations of arithmetic and of the common ways they can be applied in real world situations.
  • 1.B.3 – Demonstrate an understanding of place value: the structure of place-value notation in general and base-10 notation in particular; how place-value notations efficiently represent very large and very small numbers; the use of place value notations to order numbers, estimate, and represent order of magnitude (e.g., using scientific notation).
  • 1.B.4 – Demonstrate an understanding of multi-digit calculations, including standard algorithms, alternative algorithms, mental math, and informal reasoning.
  • 1.B.5 – Demonstrate an understanding of basic number systems (whole numbers, integers, rational numbers, and real numbers) and operations, relationships among them, locations of numbers in each system on the number line, and extension of operations from whole numbers to other systems.
  • 1.B.6 – Demonstrate an understanding and apply the fundamental ideas of number theory; odd and even numbers, factors, multiples, primes, least common multiple, and greatest common factor.
• 1.C – Algebra and functions: demonstrate a conceptual understanding of, procedural fluency with, and application of algebra concepts emphasizing functions.
  • 1.C.1 – Solve and graphically represent real world and mathematical problems using numerical and algebraic expressions, equations, inequalities, and systems of equations and inequalities.
  • 1.C.2 – Understand the connections and distinctions between proportional relationships, lines, and linear equations and use them to solve real world and mathematical problems.
  • 1.C.3 – 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).
  • 1.C.4 – Understand operations on algebraic expressions and functions (e.g., polynomials, rationals, and roots).
  • 1.C.5 – Apply arithmetic properties to algebraic expressions and equations.
  • 1.C.6 – Write equations and inequalities in equivalent forms.
  • 1.C.7 – Analyze and model functions; find functions to model various kinds of growth, both numerical and geometric.
  • 1.C.8 – Explain the relationships between various representations of a function (e.g., graphs, tables, algebraic expressions and equations, concrete models, and contexts).
• 1.D – Geometry and measurement: demonstrate a conceptual understanding of, procedural fluency with, and application of geometric figures and their properties and relationships as they apply to congruence, similarity, and the Cartesian Coordinate System.
  • 1.D.1 – Classify, visualize, and describe two- and three-dimensional figures based on their properties and the relationships among them.
  • 1.D.2 – Compose and decompose geometric shapes.
  • 1.D.3 – Understand and apply transformations and use similarity and congruence in mathematical situations.
  • 1.D.4 – Make and prove conjectures about geometric shapes and their relationships; prove theorems involving triangle congruence and similarity.
  • 1.D.5 – Understand and perform geometric constructions physically and/or with technology.
  • 1.D.6 – Apply geometric concepts to model real world situations.
  • 1.D.7 – Understand and use the cartesian coordinate system to support connections between algebraic expressions and geometric objects.
  • 1.D.8 – Understand and apply the pythagorean theorem to problem solving situations.
  • 1.D.9 – Understand and apply the concept of geometric measurement as a way of attaching a numerical quantity to a geometric figure by using a standard or non-standard unit.
  • 1.D.10 – Apply measurement concepts to solve real world and mathematical problems.
  • 1.D.11 – Understand and apply non-standard units and U.S. Customary and metric systems of measurement, including relative sizes of units, conversion of units, and estimation.
• 1.E – Statistics and probability: demonstrate a conceptual understanding of, procedural fluency with, and application of statistics and probability.
  • 1.E.1 – Collect, summarize, represent, interpret, and make inferences from categorical and quantitative data. This includes the use of appropriate measures of central tendency, variability, and distribution.
  • 1.E.2 – Understand and evaluate random processes underlying statistical experiments and use random sampling to make inferences about whole populations.
  • 1.E.3 – Understand concepts of likelihood, uncertainty, and randomness and use the rules of probability to make predictions, evaluate decisions, and solve problems.
  • 1.E.4 – Apply probability concepts to model real world situations.
• 1.F – Ratios and proportional relationships: demonstrate conceptual understanding of, procedural fluency with, and application of proportional relationships.
  • 1.F.1 – Identify and describe additive versus multiplicative comparisons and relationships.
  • 1.F.2 – Recognize, describe, and represent equivalent ratios, equivalent rates, and proportional relationships.
  • 1.F.3 – Represent and analyze proportional relationships using concrete models, diagrams, tables, coordinate graphs, equations, and verbal descriptions.
  • 1.F.4 – Reason and compute with ratios and the constant of proportionality (unit rate) to solve real world and mathematical problems.
  • 1.F.5 – Recognize and connect proportional relationships to statistics, probability, functions, geometry, and measurement, including using ratio reasoning to convert units.
  • 1.F.6 – Apply ratio and proportion concepts to model real world situations.

Candidates connect mathematics with real world problems through the use of mathematical modeling and technology.

• 2.A – Construct mathematical models in the content domains (e.g., look at a real world situation and transpose it into a mathematical problem, solve the problem, and interpret the solution).
  2.B – Use appropriate technology to make conjectures, explore, visualize, and analyze mathematics and to develop concepts and apply them in meaningful contexts.

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

• 3.A – Select, use, adapt, and determine suitability of the available mathematics curricula, teaching materials, and other resources that align to state standards.
• 3.B – Present mathematical concepts using multiple representations.
• 3.C – Guide student discourse in mathematical problem solving, including the formulation and critique of arguments to develop mathematical literacy and in-depth conceptual understanding.
• 3.D – Use questions to effectively probe mathematical understanding and make productive use of responses to inform ongoing instruction.
• 3.E – Demonstrate an understanding of the Common Core state standards for mathematics (CCSS-M) as they affect instructional methodology.
  • 3.E.1 – Demonstrate knowledge of learning progressions and their connections to instruction, including conceptual and procedural milestones and common misconceptions, within each content domain.
  • 3.E.2 – Demonstrate an understanding of major, supporting, and additional clusters for each grade level within the CCSS-M and use that knowledge to focus instruction.
  • 3.E.3 – Demonstrate an understanding of coherent connections within CCSS-M clusters at a grade level and the progression from grade level to grade level.
  • 3.E.4 – Demonstrate an understanding of the concept of mathematical rigor including conceptual understanding, procedural skill and fluency, and application.
• 3.F – Emphasize and use accurate and precise language in mathematics, including appropriate mathematical vocabulary, precise definitions, and use mathematical language to communicate relationships and concepts.
• 3.G – Engage in developmentally appropriate, language sensitive, and culturally responsive teaching of mathematics that nurtures a positive mathematics disposition and utilizes students’ cultural funds of knowledge and experiences as resources for lessons.
• 3.H – Develop skillful and flexible use of different instructional formats—explicit and implicit instruction, whole group, small group, partner, and individual—to support differentiated learning goals.
• 3.I – Plan, develop, implement, and evaluate formative and summative mathematics assessments.
• 3.J – Use the results of formative assessments to probe mathematical understanding and inform ongoing instruction.

Candidates are prepared to assume collegial non-evaluative leadership roles within their schools and districts for the purpose of supporting and facilitating professional growth with a focus on the improvement of teacher content knowledge and skill, use of effective pedagogy, and student learning.

• 4.A – Demonstrate knowledge of current research, local, state, and national resources, professional development opportunities, and critical issues related to mathematics teaching and learning.
• 4.B – Communicate professionally about students, curriculum, instruction, and assessment to educational constituents, including parents and other caregivers, teachers, school administrators, and school boards.
• 4.C – Plan, develop, implement, and evaluate professional development programs that align with school, district, and state goals.
• 4.D – Foster a safe, collaborative environment among teachers.
  • 4.D.1 – Support teachers through use of reflective dialogue about student work, data, and research-based instructional practices.
  • 4.D.2 – Promote and support collegial interactions and dialogue, including the use of professional learning communities.
  • 4.D.3 – Provide a framework for evidence-based data collection and interventions.
• 4.E – Evaluate educational structures and policies that affect students’ equitable access to high quality mathematics instruction, and help to assure that all students have appropriate opportunities to learn mathematics at or above grade-level.
• 4.F – Use leadership skills to support improvement of mathematics programs at the school and district levels, e.g., develop appropriate classroom- or school-level learning environments; build relationships among teachers, administrators and the community; collaborate to create a shared vision and develop an action plan for school improvement; and mentor new and experienced teachers.