PROFESSIONAL STANDARDS

Below is a summary of principles of mathematics teaching as developed by AMATYC.
See the AMATYC website for more details.

• Intellectual Development
• Problem Solving
• Modeling
• Reasoning
• Connecting with Other Disciplines
• Communicating
• Using Technology
• Developing Mathematical Power
• Content
• Number Sense
• Symbolism & Algebra
• Geometry
• Function
• Discrete Mathematics
• Probability & Statistics
• Deductive Proof
• Pedagogy
• Teaching with Technology
• Interactive & Collaborative Learning
• Connecting with Other Experiences
• Multiple Approaches
• Experiencing Mathematics

Intellectual Development (AMATYC Standards)

• Problem Solving:

  Students will engage in substantial mathematical problem solving.
  
  Problems should be relevant to the needs and interests of students in order to provide a context and purpose for learning new skills, concepts and theories.


• Students will use problem-solving strategies that require
  • persistence
  • recognizing inappropriate assumptions
  • intellectual risk-taking, rather than simple procedural approaches
  • posing questions
  • organizing information
  • drawing diagrams
  • analyzing situations through trial and error, graphing and modeling
  • drawing conclusions by translating, illustrating and verifying results
  
  
• Students will communicate and interpret results.

• Modeling:

  Students will learn mathematics through modeling real-world situations.


• Students will create mathematical models.
• Students will make informed predictions and decisions based on their models.

• Reasoning:

  Students will expand their mathematical reasoning skills as they develop convincing mathematical arguments.


• Students will regularly apply inductive and deductive reasoning techniques.
• Students will develop conjectures based on experience and intuition.
• Students will learn to test conjectures through proof, framing examples and statistical reasoning.
• Students will explore the meaning and role of identities and verify them.
• Student will judge the validity of mathematical arguments and draw appropriate conclusions.

• Connecting with other disciplines:

  Students will develop the view that mathematics is a growing discipline, interrelated with human culture, and understand its connections to other disciplines.


• Students will encounter modern topics.
• Students will research sources beyond textbooks to determine how mathematics provides a language for science, art, music, literature, economics, business and history.

• Communicating:

  Students will acquire the ability to read, write, listen to, and speak mathematics.


• Students will use appropriate mathematical vocabulary and notation.
• Students will read, listen to and understand mathematical presentations and arguments

• Using Technology:

  Students will use appropriate technology to enhance their mathematical thinking and understanding and to solve mathematical problems and judge the reasonableness of their results.


• Technology will be used to enhance understanding of mathematical principles through concrete imagery and computation.
• Technology will be used routinely to aid in the solution of realistic mathematical problems.

• Developing Mathematical Power:

  Students will engage rich experiences that encourage independent, nontrivial exploration in mathematics, develop and reinforce tenacity and confidence in their abilities to use mathematics, and inspire them to pursue the study of mathematics and related disciplines.


• Students will develop self-confidence and persistence.
• Students will encounter problems that do not have unique answers and provide experiences that develop the ability to conduct independent explorations.
• Students will learn and be able to transfer problem-solving strategies.
• Students will appreciate mathematics as a discipline.
• Students will develop an awareness of careers in mathematics and related fields.

Content (AMATYC Standards)

• Number Sense:

  Students will perform arithmetic operations, as well as reason and draw conclusions from numberical information.


• Students will develop the ability to:.
• perform arithmetic operations
• estimate reliably
• judge the reasonableness of numerical results
• understand orders of magnitude
• think proportionally


• Topics include:
• pattern recognition
• data representation and interpretation
• estimation
• proportionality
• comparison

• Symbolism and Algebra:

  Students will translate problem situations into their symbolic representations and use those representations to solve problems.


• Students will develop the ability to:.
• represent mathematical situations symbolically
• use algebraic, graphical and numerical methods


• Topics include:
• derivation of formulas
• translation of realistic problems into mathematical statements
• solution of equations by graphical, numerical and algebraic methods

• Geometry:

  Students will develop a spatial and measurement sense.


• Students will develop the ability to:.
• visualize, compare and transform objects
• draw one-, two- and three-dimensional objects and extend the concept to higher dimensions
• determine area, perimeter and volume of plane and solid figures


• Topics include:
• comparison of geometric objects, including congruence and similarity
• graphing
• prediction from graphs
• measurement
• vectors

• Function:

  Students will demonstrate understanding of the concept of function by several means (verbally, numerically, graphically, and symbolically) and incorporate it as a central theme into their use of mathematics.


• Students will develop the ability to:.
• interpret functional relationships between two or more variables
• formulate functional relationships when presented with data sets
• transform functional information from one representation to another


• Topics include:
• generalizations about families of function s
• use of functions to model realistic problems
• behavior of functions

• Discrete Mathematics:

  Students will use discrete mathematical algorithms and develop combinatorical abilities in order to solve problems of finite character and enumerate sets without direct counting.


• Students will develop the ability to apply non-continuous numerical models throughout the curriculum, not just in specialized courses such as discrete and finite math.


• Topics include:
• sequences & series
• recursion
• permutations & combinations
• difference equations
• linear programming
• finite graphs
• voting systems
• matrices

• Probability and Statistics:

  Students will analyze data and use probability and statistical models to make inferences about real-world situations.


• The basic concepts of probability and statisitcs should be integrated throughout the introductory college mathematics curriculum at an intuitive level.


• Students will:
• gather, organize, display and summarize data
• draw conclusions or make predictions from data and assess the chances of the occurance of certain events


• Topics include:
• basic sampling techniques
• tabulation techniques
• creating and interpreting charts and graphs
• data transformation
• curve fitting
• measures of center and dispersion
• simulations
• probability laws
• sampling distributions

• Deductive Proof:

  Students will appreciate the deductive nature of mathematics as an identifying characteristic of the discipline, recognize the roles of definitions, axioms and theorems, and identify and construct valid deductive arguments.


• Students will engage in activities that will lead them to form statements of conjecture, test them by seeking counterexamples and identify and construct arguments verifying or disproving the statements.



GUIDELINES FOR DEVELOPING CONTENT

INCREASED ATTENTION	DECREASED ATTENTION
+ pattern recognition, drawing inferences	- rote application of formulas
+ number sense, mental arithmetic, estimation	- arithmetic drill exercises, routine operations with real numbers
+ connections between mathematics and other disciplines	- presentation of mathematics as an abstract entity
+ integration of topics throughout the curriculum	- algebra, trigonometry, analytic geometry and so forth as separate courses
+ discovery of geometric relationships through the use of models, technology and manipulatives	- establishing geometric relationships solely through formal proofs
+ visual presentation of concepts	- rote memorization of formulas
+ integration of the concept of function across topics within and among courses	- separate and unconnected units on linear, quadratic, polynomial, radical, exponential, and logarithmic functions
+ analysis of the general behavior of a variety of functions in order to check the reasonableness of graphs produced by graphing utilities	- paper-and-penci evaluation of functions and hand-drawn graphs based on plotting points
+ connection of functional behavior to the situation modeled by the function	- emphasis on manipulation of complicated radical expressions, factoring, rational expressions, logarithms and exponents
+ connections among a problem situation, its model as a function in symbolic form, and the graph of that function	- cookbook problem-solving without connections
+ modeling problems of chance by constructing probability distributions or by actual experiment	- theoretical development of probability theorems
+ collection of real data for analysis of both descriptive and inferential statistical techniques	- analysis of contrived data
+ exploratory graphical analysis as part of inferential procedures	- cookbook approaches to applying statistical computations and tests which fail to focus on the logic behind the processes
+ use of curve-fitting to model real data, including transformation of data when needed	- reliance on out-of-context functions that are overly simplistic
+ discussion of the meaning of nonzero correlation and the independence of correlations from any implications of cause and effect	- blind acceptance of r
+ use of statistical software and graphing calculators	- paper-and-pencil calculations and four-function calculators
+ problems related to the ordinary lives of students	- problems unrelated to the daily lives of most students
+ matrices to organize and analyze information from a wide variety of settings	- requiring a system of equations to be solved by three methods
+ graph theory and algorithms as a means of solving problems	- algebraically derived exact answers




Pedagogy (AMATYC Standards)

• Teaching with Technology:

  Mathematics faculty will model the use of appropriate technology in the teaching of mathematics so that students can benefit from the opportunities it presents as a medium of instruction.

• Interactive and Collaborative Learning:

  Mathematics faculty will foster interactive learning through student writing, reading, speaking, and collaborative activities so that students can learn to work effectively in groups and communicate about mathematics both orally and in writing.


• Students will experience cooperative learning, oral and written reports, writing in journals, open-ended projects.


• Alternative assessments will be used such as essay questions and portfolios.

• Connecting with Other Experiences:

  Mathematics faculty will actively involve students in meaningful mathematics problems that build upon their experiences, focus on broad mathematical themes, and build connections within branches of mathematics and between mathematics and other disciplines so that students will view mathematics as a connected whole relevant to their lives.

• Multiple Approaches:

  Mathematics faculty will model the use of multiple approaches (numerical, graphical, symbolic and verbal) to help students learn a variety of techniques for solving problems.

• Experiencing Mathematics:

  Mathematics faculty will provide learning activities, including projects and apprenticeship, that promote independent thinking and require sustained effort and time so that students will have the confidence to access and use needed mathematics and other technical information independently, to form conjectures from an array of specific examples, and to draw conclusions from general principles.


• Open-ended classroom and lab projects will be assigned.


• Partnerships with area business and industry will be formed.

GUIDELINES FOR DEVELOPING PEDAGOGY

INCREASED ATTENTION	DECREASED ATTENTION
+ active involvement of students	- passive listening
+ technology to aid concept development	- paper-and-pencil drill
+ problem-solving and multi-step problems	- one-step single-answer problems
+ mathematical reasoning	- memorization of facts and procedures
+ conceptual understanding	- rote manipulation
+ realistic problems encountered by adults	- contrived exercises
+ an integrated curriculum with ideas developed in context	- isolated topic approach
+ multiple approaches to problem-solving	- requiring a particular method for solving a problem
+ diverse and frequent assessment both in class and outside of class	- tests and a final exam as the sole assessment
+ open-ended problems	- problems with only one possible answer
+ oral and written communication to explain solutions	- requiring only short, numerical answers, or multiple-choice responses
+ variety of teaching strategies	- lecturing

• Number and Operations:
  • Understand numbers, ways of representing numbers, relationships among numbers and number systems.
  • Understand meanings of operations and how they relate to one another.
  • Compute fluently and make reasonable estimates.
• Algebra:
  • Understand patterns, relations and functions.
  • Represent and analyze mathematical situations and structures using algebraic symbols.
  • Use mathematical models to represent and understand quantitative relationships.
  • Analyze change in various contexts.
• Geometry:
  • Analyze characteristics and properties of two- and three-dimensional geometric shapes and develop mathematical arguments about geometric relationships.
  • Specify locations and describe spatial relationships using coordinate geometry and other representational systems.
  • Apply transformations and use symmetry to analyze mathematical situations.
  • Use visualization, spatial reasoning, and geometric modeling to solve problems.
• Measurement:
  • Understand measurable attributes of objects and the units, systems and processes of measuerment.
  • Apply appropriate techniques, tools and formulas to determine measurements.
• Data Analysis and Probability:
  • Formulate questions that can be addressed with data and collect, organize, and display relevant data to answer them..
  • Select and use appropriate statistical methods to analyze data.
  • Develop and evaluate inferences and predictions that are based on data.
  • Understand and apply basic concepts of probability.
• Problem Solving:
  • Build new mathematical knowledge through problem solving.
  • Solve problems that arise in mathematics and in other contexts.
  • Apply and adapt a variety of appropriate strategies to solved problems.
  • Monitor and reflect on the process of mathematical problem solving.
• Reasoning and Proof:
  • Recognize reasoning and proof as fundamental aspects of mathematics.
  • Make and investigate mathematical conjectures.
  • Develop and evaluate mathematical arguments and proofs.
  • Select and use various types of reasoning and methods of proof.
• Communication:
  • Organize and consolidate mathematical thinking through communication.
  • Communicate mathematical thinking coherently and clearly to peers, instructors and others.
  • Analyze and evaluate the mathematical thinking and strategies of others.
  • Use the language of mathematics to express mathematical ideas precisely.
• Connections:
  • 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.
• Representation:
  • Create and use representations to organize, record, and communicate mathematical ideas.
  • Select, apply and translate among mathematical representations to solve problems.
  • Use representations to model and interpret physical, social and mathematical phenomena.

QUANTITATIVE REASONING ACROSS THE CURRICULUM

Quantitative literacy is an institutional learning outcome that many colleges have adopted. The Math Department needs to have a discussion on this topic at some point. Should Mission College ever move in that direction, with or without the momentum coming from the Math Department, MAA has done a thorough job of outlining the issues, creating definitions and extending the NCTM standards. This is worth a look. You can find the outline and details of these standards here.