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Mathematics

Process Standard

 

 

Problem Solving Standard

 

Instructional programs should enable all students to -

Build new mathematical knowledge through problem solving

Classroom examples

·         Take risks and have confidence in confronting unknown problems

·         Discover new mathematical ideas

·         Develop proficiency with previously learned skills by applying them to a new situation

 

Solve problems that arise in mathematics and in other contexts

Classroom examples

·         Design conflict-free schedule for high school course

·         Use mathematics to solve problems from a science class

·         Formulate and solve interesting problems based on a wide variety of real-world situations

 

Apply and adapt a variety of appropriate strategies to solve problems

Classroom examples

·         Model the problem with manipulative

·         Make a picture or diagram

·         Use or make a table

·         Make an organized list

·         Estimate and check

·         Use or look for a pattern

·         Solve a simpler problem

·         Work a problem backward

·         Break a problem into smaller parts

·         Predict what a solution will look like

·         Relate the problem to one you already know how to solve

 

Monitor and reflect on the process of mathematical problem solving

Classroom examples

·         Reflect on a problem in a math journal

·         Reflect orally on the process used to solve a problem

·         Recognize situations when more than one strategy might be used to solve a problem

 

 

Reasoning and Proof Standard

 

Instructional programs should enable all students to -

Recognize reasoning and proof as fundamental aspects of mathematics

Classroom examples

·         Recognize that explaining and justifying the process used to solve a problem is important

·         Appreciate mathematics as a logically consistent and sequential system

·         Understand the difference between proving that something is true and inferring that it is true

·         Distinguish between how something is done and why it works

 

Make and investigate mathematical conjectures

Classroom examples

·         Make, investigate, and test predictions based on patterns observed

·         Disprove a faulty conjecture using a counterexample

·         Explain whether a conjecture is sometimes true, always true, or never true

 

Develop and evaluate mathematical arguments and proofs

Classroom examples

·         Estimate an answer and defend its reasonableness

·         Create a story problem

·         Create lists of examples and non-examples when making definitions

 

Select and use various types of reasoning and methods of proof

Classroom examples

·         Use various methods to justify a solution (diagrams, graphs, written argument, oral explanation, manipulatives, etc.)

·         Base arguments on a logical analysis of a situation

·         Reason inductively from study of patterns and specific cases

 

 

Communication Standard

 

Instructional programs should enable all students to -

Organize and consolidate their mathematical thinking through communication

Classroom examples

·         Read and understand a mathematical problem

·         Describe and reflect on a problem solving process in writing

·         Listen to other students’ ideas and explanations

·         Restate a mathematical idea in their own words

·         Ask productive and specific questions of peers and teachers

 

Communicate their mathematical thinking coherently and clearly to peers, teachers, and others

Classroom examples

·         Explain a mathematical idea to another student

·         Clarify a mathematical idea by asking questions

·         Write a narrative explaining the solution steps taken to solve a complex problem

 

Analyze and evaluate the mathematical thinking and strategies of others

Classroom examples

·         Compare and analyze different solution methods to a problem

·         Identify errors in the worked solution to a problem (own errors and errors of others)

·         Follow and understand a logical argument (textbook explanation or proof)

 

Use the language of mathematics to express mathematical ideas precisely

Classroom examples

·         Use vocabulary to explain ideas and ask questions

·         Use vocabulary to describe geometric objects (diagonal, parallel, right angle, etc.)

·         Use vocabulary to describe relationships (exponential growth, linear growth, etc.)

 

 

Connections Standard

 

Instructional programs should enable all students to -

Recognize and use connections among mathematical ideas

Classroom examples

·         Understand multiplication is repeated addition

·         Understand addition and subtraction as opposites

·         Understand that fractions are a kind of division

·         Understand that fractions, decimals, and percents are different ways of representing parts of wholes

·         Draw a geometric figure to illustrate an algebraic idea (algebra tiles, distributive property, etc.)

 

Understand how mathematical ideas interconnect and build on one another to produce a coherent whole

Classroom examples

·         Explain how an idea being studied in class is related to an idea studied in previous classes

·         Use different approaches to solve the same problem and understand the connections between the two

·         Understand how mathematical ideas being studied now are related to ideas that will be studied in the future

 

Recognize and apply mathematics in contexts outside of mathematics

Classroom examples

·         Use maps to measure and understand scale, position, and distance

·         Use mathematical ideas to make designs and artwork

·         Connect geometric to architecture

·         Use statistical techniques to gather, understand, and represent scientific data

·         Use networks to model social relationships

·         Find and recognize mathematical ideas in literature and music

 

 

Representation Standard

 

Instructional programs should enable all students to -

Create and use representations to organize, record, and communicate mathematical ideas

Classroom examples

·         Organize data in a table of numbers

·         Write or verbally explain a mathematical idea

·         Write a symbolic representation (equation or a rule) for a situation

·         Use manipulatives to illustrate a mathematical idea

·         Make a graph or a chart from data

·         Draw a picture of a mathematical idea

·         Use Venn diagrams to represent the relationships between things

 

Select, apply, and translate among mathematical representations to solve problems

Classroom examples

·         Plot a graph from data in a table

·         Make an equation given a graph or a table

·         Write an equation based on a written explanation of a situation

·         Create an equation from a story problem

·         Write a story problem based on an equation

·         Create a specific example given a general rule and vice versa

 

Use representations to model and interpret physical, social, and mathematical relationships

Classroom examples

·         Interpret information from a graph or table

·         Make predictions based on a graph or table (extrapolation and interpolation)

·         Use manipulatives to model and understand mathematical relationships

·         Draw pictures to model and understand mathematical relationships

·         Collect data and create a graph or equation modeling a real-life situation