# Mathematics Fundamental Process

Doing and understanding mathematics fosters growth in the following fundamental processes:

Big Ideas Enduring Understandings Essential Questions
Problem Solving
• Tools and strategies are strategically selected and used to solve particular applications.
• Reflection on the process and reasonableness of the solution moves students from the symbolic to the practical.
• How do I know where to begin when solving a problem?
• What are the characteristics of a successful problem solver?
• What is the best method or technique for working toward a solution?
• What is the meaning of the solution and does it make sense?
Connections
• Connections exist within mathematical concepts and can broaden understanding of the world.
• Mathematical ideas interconnect and build on one another to produce a coherent whole.
• Where do mathematical ideas surface in the world?
• How can I use mathematical knowledge in different contexts?
• How is mathematics a coherent discipline that contains internal structure and interconnectedness?
Reasoning
• Abstract and quantitative reasoning are required to process mathematical information and solve problems.
• Mathematical conjectures are developed and investigated through observing patterns.
• Sound reasoning requires the ability to distinguish between valid and invalid arguments and to critique the reasoning of others.
• How do I select appropriate reasoning tools to progress mathematically?
• What is a possible approach? What happens if I follow that approach?
• Is my / their reasoning sound? What does a well-reasoned argument entail?
• What constitutes proof?
Communication
• Mathematical ideas must be communicated clearly in written, visual, or oral form.
• Communication of mathematical thinking should demonstrate clear and concise organization.
• Mathematical language can be used to express ideas symbolically, numerically, and graphically.
• How do I communicate mathematical ideas clearly?
• How can I organize and consolidate my thinking through communication?
• How can mathematics be perceived as a language? What terms, notations, and representations most accurately and succinctly convey my ideas?
Representation
• Symbols, graphs, pictures, and tables can be used to represent real situations.
• Flexibility in one’s ability to read and interpret various forms is important in understanding problems and solutions.
• Various mathematical representations are useful for problem solving and communicating a solution.
• What representation best illuminates this relationship?
• How do I move between mathematical abstraction and physical reality?
Number and Operations
• Understanding numbers, their representations, properties, and relationships assist in higher level thinking.
• How does the context of a problem dictate the number system that should be used?
• What operation is most appropriate given the context of the problem?
• When is a rough estimate, an approximation, or an exact answer suitable for a solution to a problem?
Algebraic Understanding
• Patterns, relations, and functions are mathematical ways to describe connectedness and dependence.
• Mathematical situations and structures can be represented and analyzed using symbols to advance algebraic thinking.
• Mathematical models can be used to represent and understand quantitative relationships.
• Change can be modeled in a variety of mathematically ways.
• How can I generalize patterns, describe relationships, and analyze functions?
• How do I use the tools of symbolic algebra to judge the reasonableness of mathematical representations?
• How is the idea of change analyzed?
Geometric
Understanding and
Measurement
• Two and three dimensional shapes have properties and relationships similar to each other.
• Coordinate geometry can be used to describe spatial relationships and location.
• The study of transformations and symmetry provides a deeper understanding of physical change.
• Visualization, spatial reasoning, and geometric modeling are strategies to enhance problem solving.
• In order to assign numerical values to spatial and physical attributes, objects can be measured using appropriate systems, units, and processes.
• How do the tools of geometry such as definitions, theorems, and properties foster an increasing ability to spatially visualize and logically deduce conclusions?
• What is the relationship between units and physical quantity?
Data Analysis and Probability
• Data collection and its organization helps formulate relevant questions that can be answered using mathematical tools.
• Selection of the appropriate statistical method to
• analyze data will progress students toward solutions and subsequent inferences.
• A study of probability helps illuminate the randomness of our everyday world.
• What types of questions can be answered by analyzing data?
• When is data analysis valid and what is its purpose?
• How can I become a critical interpreter of data?
• How does probability relate to the real world?