Essential Competencies for Middle Grades Mathematics Teachers

MATHEMATICS CONTENT COMPETENCIES

1. Understand and apply concepts of number, number theory and number systems
2. Understand and apply numerical computation and estimation techniques and extend them to algebraic expressions
3. Use algebra to describe patterns, relations and functions and to model and solve problems
4. Understand and represent functions algebraically and graphically
5. Understand and apply calculus as modeling dynamic change, including an intuitive understanding of differentiation and integration and apply calculus concepts to real-world settings
6. Explore and use geometric concepts and relationships to describe and model mathematical ideas and real-world situations
7. Understand and apply the major concepts of Euclidean geometry from a variety of perspectives including coordinate and transformational
8. Understand and apply the process of measurement to two- and three-dimensional objects using customary and metric units
9. Design investigations and formulate questions that can be answered through experiments
10. Use both descriptive and inferential statistics to analyze data, make predictions and make decisions
11. Interpret probability in real-world situations, construct sample spaces, model and compare experimental probabilities with mathematical expectations; use probability to make predictions
12. Use graph theory and networks to solve problems

MATHEMATICS PROCESS COMPETENCIES

13. Problem Solving
14. Communicating
15. Logical Reasoning and Proof
16. Connections
17. Using Technology

INSTRUCTIONAL (PEDAGOGY) COMPETENCIES

18. Select instructional strategies and activities appropriate to the content being studied
19. Plan instruction that addresses students' needs
20. Use technology in planning for instruction
21. Plan for assessment
22. Reflect on the effectiveness of lessons

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(1) Understand and apply concepts of number, number theory and number systems

• Understand and use field axioms on the set of real numbers
• Understand decimal representation and other base representations
• Use prime factorization and relate it to algebra
• Make conjectures about prime and composite numbers and provide justifications to prove or disprove them

(2) Understand and apply numerical computation and estimation techniques and extend them to algebraic expressions

• Understand the mathematics behind algorithms, including alternative algorithms and samples of student-generated algorithms
• Represent arithmetic operations using multiple models and manipulatives
• Read, write, compare, order, represent, estimate and compute with numbers in a variety of forms: integers, rational numbers, decimals, percents, square roots, perfect squares, irrational numbers, complex numbers, and numbers written in scientific and exponential notation
• Compare, estimate, order, model, and convert all forms of numbers, including multiple estimation techniques
• Use order of operations, properties of operations, inverse operations and ratios to solve problems
• Determine when to use operations
• Use mental mathematics, paper and pencil, and calculator methods in working with numbers

(3) Use algebra to describe patterns, relations and functions to model and solve problems

• Understand algebra as a language to represent or describe scientific or mathematical situations
• Represent, analyze, extend, generalize, conjecture about and verbalize a variety of patterns
  • Understand the concept of variable
  • Analyze, extend and find missing values in complex patterns involving multiple operations and powers using a variety of strategies (e.g., using a rule, writing a rule or drawing a picture)
  • Analyze and use patterns in other contexts (e.g., Pascal's triangle)
• Use matrices for solving systems of equations
• Graph functions and their inverses and understand physical situations calling for each
• Work with symbols fluently
• Write, simplify and solve algebraic equations and inequalities using substitution, the order of operations, the properties of operations and the properties of equalities and inequalities
  • Graph and solve linear and quadratic equations and inequalities using pencil and paper and graphing calculators
  • Solve multi-step one-variable linear and quadratic equations and inequalities and word problems
  • Graph linear equations using the slope-intercept method with graphing calculators and relate proportional reasoning to linear functions.

(4) Understand and represent functions algebraically and graphically

• Distinguish between relations and functions
• Graph, create, and understand functions and relations (including definitions and vocabulary of functions)
• Understand that functions can be represented using multiple representations and notations
• Connect symbolic, graphical, tabular and numerical representations for linear and nonlinear functions
• Identify, contrast, and graph linear and non-linear functions
• Use the vertical line test to determine if a relation is a function
• Recognize change patterns associated with linear and non-linear functions and their inverses

(5) Understand and apply calculus as modeling dynamic change, including an intuitive understanding of differentiation and integration and apply calculus concepts to real-world settings

• Demonstrate an intuitive understanding of limit (e.g., in context of repeating decimals, circumference, fractals)
• Connect concepts (such as optimization problems) to rate of change for students in middle grades
• Understand differentiation as a limit that gives a rate of change and integration as a limit that gives an accumulated quantity

(6) Explore and use geometric concepts and relationships to describe and model mathematical ideas and real-world situations

• Develop spatial reasoning, including the use of symmetry, rotations and dilations
• Use technology tools to explore geometric ideas (e.g., dynamic drawing tools to investigate pi)
• Identify two- and three-dimensional figures using their properties
• Recognize and write valid statements using "if-then," "all," "some," and "none" about geometric figures
• Construct and explain basic inductive and deductive arguments concerning geometric ideas and relationships

(7) Understand and apply the major concepts of Euclidean geometry from a variety of perspectives including coordinate and transformational

• Make connections to rudimentary non-Euclidean geometry
• Use matrix representations for transformations
• Describe and perform transformations, including those in the coordinate plane
• Understand and apply properties of simple and composite shapes

(8) Understand and apply the process of measurement to two- and three-dimensional objects using customary and metric units

• Understand how a measurement instrument influences accuracy
• Understand the concept of measurement through the use of non-standard units
• Understand that the concept of measurement is the assignment of a number to an attribute of a figure or object
• Compute perimeter, area, surface area, and volume using appropriate units (customary and metric), techniques, formulas, and levels of accuracy using significant digits
• Understand and apply the Pythagorean Theorem to solve real-world problems, even when the existence of right triangles is not apparent
• Use dimensional analysis to convert between measures
• Write and solve proportions and word problems involving similar figures, indirect measurement and scale drawings
• Use right triangle ratios for sine, cosine and tangent

(9) Design investigations and formulate questions that can be answered through experiments

• Make decisions on how to collect data
• Understand survey design and bias
• Understand statistical design, including the misuse of statistics, margin of error and confidence intervals

(10) Use descriptive and inferential statistics to analyze data, make predictions and make decisions

• Understand margin of error, standard deviation, confidence intervals, sampling techniques, correlation versus cause-and-effect and normal distributions
• Gather, organize, display and interpret data
  • Understand the difference between qualitative and quantitative data
  • Distinguish between sampling, random sampling, whole population and target populations
  • Read and make a variety of graphical data displays, including bar graphs, line graphs, pictographs, circle graphs, line plots, stem-and-leaf plots, histograms, scatterplots, box-and-whisker graphs and lines of best fit
  • Choose and use the appropriate measures of central tendency and range to analyze data and solve problems
  • Determine and explain situations of skewed and misleading data

(11) Interpret probability in real-world situations, construct sample spaces, model and compare experimental probabilities with mathematical expectations; use probability to make predictions

• Identify the sample space and compute mathematical probabilities for compound events
• Use combinations and permutations to solve problems
• Use modeling to solve probability problems (e.g., tree diagrams and areas models)
• Design and solve experimental and theoretical probability problems
• Identify and distinguish between independent and dependent events and between complementary and mutually exclusive events
• Calculate odds for and odds against

(12) Use graph theory and networks to solve problems

• Use counting to enumerate and order
• Use matrices, finite graphs, trees and diagrams to model situations
• Describe basic algorithms for accomplishing tasks

(13) Problem Solving

• Use mathematical modeling to solve real-world problems and to investigate mathematical concepts
  • Use multiple models for the same problem
  • Use models that result from natural sciences, social sciences, business and engineering
• Analyze others' work and approaches to solving problems (both for student and teacher work)
• Use a variety of strategies to solve non-routine problems
  • Solve multi-step problems with and without multiple solutions
  • Solve problems with extraneous information, insufficient information or no solution
  • Solve problems that require choosing and using a formula
• Determine reasonableness of answers

(14) Communicating

• Know and use the language of mathematics (students and teachers)
• Write explanations for solving problems
• Write hypotheses and conclusions that include the reasons for making them
• Ask appropriate questions regarding others' solutions
• Use cause and effect words (e.g., "if-then" and "because") in describing reasoning

(15) Logical Reasoning and Proof

• Understand the difference between reasoning and proof
  • Understand the role of axiomatic systems and proofs in different branches of mathematics, such as algebra and geometry
  • Use conjectures, reasoning and proof with such topics as the Euclidean algorithm and odd/even numbers
  • Compare and contrast inducting reasoning, deductive reasoning and proof
• Complete simple truth tables
• Make and evaluate basic logical arguments containing conditional statements, conjunctions, disjunctions and negations
• Evaluate students' solutions, solution strategies and reasoning
• Make general predictions (e.g., the outcomes of events or the resulting areas or volumes from changes in dimensions)

(16) Connections

• Understand the historical development in mathematics that includes the contributions of under-represented groups and diverse cultures
• Recognize and use connections among mathematical ideas
• Understand how mathematical ideas interconnect and build on one another to produce a coherent whole
• Recognize and apply mathematics in contexts outside of mathematics

(17) Using Technology

• Know when it is appropriate to use technology
• Use a calculator to perform arithmetic operations, graph simple equations, find the mean and median of data, explore large numbers and number patterns, and explore trigonometric ratios.
• Use software to create spreadsheets that include totals and mean values; create bar, line and circle graphs; and graph functions and explore other types of equations.
• Use dynamic drawing tools to explore geometric concepts

(18) Select instructional strategies and activities appropriate to the content being studied

• Connect lessons to state and national standards
• Select strategies common to mathematics and other content areas (e.g., cooperative learning, literacy development)
• Select strategies specific to mathematics (e.g., manipulatives)

(19) Plan instruction that addresses students' needs

• Look at student records and classroom progress to provide extra help and support, to address achievement gaps and to provide differentiation of instruction and assessment (including special populations)
• Interpret and use mathematics test data in planning instruction
• Identify prerequisite skills needed to learn a particular topic
• Help students make connections between strategies and examine information in multiple ways
• Know and anticipate common misconceptions that students develop when exploring mathematical ideas
• Analyze student work that may result from common misconceptions

(20) Use technology in planning for instruction

• Use the Internet to find resources for mathematics instruction; including the use of effective searching techniques and lesson plan repositories.
• Find and evaluate the quality of resources
• Use technology to manage the best practices of instruction -communications, assessments, record keeping, etc.
• Utilize mathematics-specific resources and supplies (e.g., manipulatives, measuring devices, geometric tools)

(21) Plan for assessment

• Design assessments that produce student work that provides convincing evidence of what students know and can do
• Use assessment results to modify instruction beyond simply re-teaching

(22) Reflect on the effectiveness of lessons

• Discuss lesson plans, rubrics and samples of student work in the context of the effectiveness of instruction
• Analyze the impact of lessons on horizontal and vertical articulation within the school and with feeder schools.