Mathematics

 

STANDARD: Upon completion of the Associate of Arts in Teaching Degree with an emphasis in Elementary Education, the candidate will "know, understand, and use the major concepts, procedures, and reasoning processes of mathematics that define number systems and number sense, geometry, measurement, statistics and probability, and algebra in order to foster student understanding and use of patterns, quantities, and spatial relationships that can represent phenomena, solve problems, and manage data" (NCATE Program Standards for Elementary Teacher Preparation, February 2000).

The ten areas described in this document are taken from the National Council for the Teachers of Mathematics Principles and Standards for School Mathematics (2000). The first five areas address mathematics content and the last five address mathematical processes. An effective elementary school teacher of mathematics requires a solid conceptual base in these ten areas. Emphasized throughout these recommendations are the development of deep understanding, active engagement in mathematics, flexibility of thinking, use of technology to enhance learning, effective communication, and reflective work. Instruction will model a variety of student-centered pedagogical techniques including but not limited to discovery, hands-on learning, and cooperative learning.

The indicators for each outcome within the ten content and process areas rely heavily on performance based, alternative assessments. Candidates need to do math in meaningful settings, not simply parrot procedures and algorithms. The outcomes for the five process areas (problem-solving, reasoning, communication, connections, and representation) must be considered in conjunction with the outcomes for the five content areas; they cannot be separated. In addition, the content areas are often interwoven, with outcomes for one area connected to outcomes in other areas.

All assessments, with the exception of selected response items, will be evaluated for evidence of mathematical understanding, use of examples and diagrams when appropriate, depth of thought, clarity and organization, conciseness, and grammatical and mechanical integrity.

Standard 1: Number and Operation Sense

Outcomes

Indicators

Assessment Type

Sample Assessment Tasks

1. Teacher candidates will understand numbers, ways of representing numbers, relationships among numbers, and number systems

 

 

 

 

 

 

 

 

 

2. Teacher candidates will understand the meaning of operations and how they relate to each other

 

 

 

3. Teacher candidates will compute fluently and make reasonable estimations

 

 

 

 

 

 

a. Represent numbers in a variety of ways using physical materials, drawings, and symbols.

b. Recognize and generate equivalent representations for whole numbers, fractions, decimals and percents.

c. Use ratios and proportions to represent quantitative relationships.

d. Describe, explain and model place value in multiple number systems

e. Identify multiple interpretations of operations.

f. Describe and demonstrate the relationships between operations.

g. Describe and use the properties of operations.

h. Select and apply appropriate methods and tools for computing with whole numbers, fractions, decimals (e.g., mental computation, estimation, calculators, computers, paper & pencil).

i. Develop and use strategies to estimate the results of whole number, fraction and decimal computations and judge the reasonableness of the results.

Develop and analyze range of reliable algorithms and useful, interesting strategies for computation.

!Selected response

!Extended constructed response

!Learning log with diagrams and written explanations

!Research paper

!Given a number, represent it using two different manipulatives. Draw the representations and provide a written explanation of each.

!Describe the conventional algorithm and one alternative algorithm for multiplying simple fractions. Provide a written analysis of the two methods, describing the underlying mathematical concepts.

!Research three numeration systems and compare notation, place value, use of zero, base, etc.

Standard 2: Algebra

Outcomes

Indicators

Assessment Type

Sample Assessment Tasks

1. Teacher candidates will understand patterns, relations and functions

 

 

 

 

 

2. Teacher candidates will represent and analyze mathematical situations and structures using algebraic symbols

3. Teacher candidates will use mathematical models and analyze change in both real and abstract contexts

 

4. Teacher candidates will analyze change in various contexts

a. Represent, analyze and generalize a variety of patterns with physical materials, tables, graphs, words, and symbolic rules.

b. Compare different forms of representation for a relationship.

 

c. Use numeric, graphic, verbal and symbolic representations of functions

 

 

 

d. Model and solve problem situations using various representations such as graphs, table, and equations.

 

 

e. Describe how a change in one variable relates to a change in a second variable.

Identify and describe situations with constant or varying rates of change and compare them.

!Investigative project

!Extended constructed response

!Learning log with diagrams and written explanation

!Presentation

!Given a series of operations to perform in which the solution is the same regardless of the initial number used, explain the result in algebraic terms. Use physical models to demonstrate the explanation.

!Select a real life example that involves rate. Show changes in rate using a line graph. Provide written and/or oral explanation.

!From a set of patterns select the ones that are functions.

 

Standard 3: Geometry

Outcomes

Indicators

Assessment Type

Sample Assessment Tasks

1. Teacher candidates will analyze characteristics and properties of two- and three-dimensional geometric shapes and develop mathematical arguments about geometric relationships

 

 

2. Teacher candidates will specify locations and describe spatial relationships using coordinate geometry and other representational systems

 

3. Teacher candidates will apply transformations and use symmetry to analyze mathematical situations

 

4. Teacher candidates will use visualization and spatial reasoning to solve problems

a. Using mathematical vocabulary, describe and classify a variety of 2-D and 3-D shapes, noting their properties and relationships.

b. Make and test conjectures about geometric properties and relationships and develops logical arguments to justify conclusions.

c. Describe location and movement using common language and geometric vocabulary.

d. Use coordinate geometry to represent and examine the properties of geometric shapes.

e. Describe sizes, positions, and orientations of shapes under informal transformations such as flips, turns, slides and scaling.

f. Describe congruence, similarity, and line and rotational symmetry.

g. Identify and build a 3-D object from a 2-D representation (and vice versa).

h. Use geometric models to solve problems in number, area, volume, and algebra.

!Extended constructed response

!Model

!Demonstrate using models and oral explanation

!Investigative project

!Build a model of a 3-D shape and describe its properties. In a group, use pantomime to describe the shape.

!Use algebra tiles to give a geometric demonstration of algebraic statements, such as (x+2)(x-4).

!Use Geometer’s Sketch pand or Cabri Software to illustrate the relationships of lines, angles, parallels triangles, etc.

!Describe how geometric models help solve word problems.

Standard 4: Measurement

Outcomes

Indicators

Assessment Type

Sample Assessment Tasks

1. Teacher candidates will understand measurable attributes of objects and the units, systems and processes of measurement

2. Teacher candidates will apply appropriate techniques, tools, and formulas to determine measurement

 

 

a. Use both standard and nonstandard units for measuring length, area, volume, weight, time and temperature

b. Describe metric and customary systems of measurement

c. Convert between units within the metric system.

d. Make reasonable estimates of measurement

e. Select and apply appropriate units and tools to accurately measure length, area, volume, weight, time, temperature, angles to appropriate levels of accuracy

f. Develop and use formulas for determining circumference, perimeter, area, and volume of geometric figures

g. Solve problems involving scale factors and rates 

!Extended constructed response

!Demonstration

!Research paper and presentation

!Build a three-dimensional construction from 2 multi-link cubes. Then build a similar construction in which the length, width and height of the original are doubled. Compare the changes in length, area and volume.

!Research the development of metric and customary systems of measurement and share findings in a multimedia presentation.

!Create non-standard units of measurement for use in measuring length, area, volume, weight, time, and temperature.

 

Standard 5: Data Analysis and Probability

Outcomes

Indicators

Assessment Type

Sample Assessment Tasks

1. Teacher candidates will formulate questions that can be addressed with data and collect, organize, and display relevant data to answer them

2. Teacher candidates will select and use appropriate statistical methods to analyze data

3. Teacher candidates will develop and evaluate inferences and predictions that are based on data 

4. Teacher candidates will understand and apply basic concepts of probability

a. Design investigations to address a questions, select an appropriate data collection method, and collect data using observations, surveys, and experiments

 

b. Select, create and use appropriate graphical representations of data, including line plots, bar graphs, line graphs, histograms, box and whisker plots, stem and leaf plots, and scatterplots.

c. Compare and evaluate different representations of the same data

d. Find, use and interpret measures of center and spread

e. Propose and justify conclusions and predictions that are based on data

f. Design further investigations based on conclusions and predictions

g. Describe degrees of likelihood.

Predict the probability of outcomes and test them.

!Investigative project

Spreadsheet, graph, etc.

!Extended constructed response

!Learning log with diagrams and explanation

!Demonstration

!Pose a question about an environmental issue; collect data (using CBLs if appropriate); represent the data in at least two forms; analyze the effectiveness of each method; and describe the significance of the data.

!Predict and illustrate with a tree diagram the outcome of flipping multiple coins ten times.

 

Standard 6: Problem Solving

Outcomes

Indicators

Assessment Type

Sample Assessment Tasks

1. Teacher candidates will build new mathematical knowledge through problem solving 

2. Teacher candidates will solve problems that arise in mathematics and in other contexts

3. Teacher candidates will apply and adapt a wide variety of appropriate strategies to solve problems

 

 

 

 

4. Monitor and reflect on processes of mathematical problem solving

 

a. Analyze situations carefully in mathematical terms

 

 

b. Pose mathematical problems based on experiences and seek solutions

 

c. Identify multiple problem solving strategies such as making diagrams, looking for patterns, working backward, creating a simpler but similar problem, etc.

Select and use most appropriate strategy for given problem

d. Be persistent when solving challenging problems

e. Assess effectiveness of problem solving strategies and adjust as necessary

!Investigative project

!Learning log with diagrams and written explanation

!Reflective journal

!Extended response

!Solve a mathematical situation in multiple ways; evaluate the effectiveness of each strategy used.

!Select a problem of personal interest; pose mathematical questions and seek solutions (e.g., buying a car – how much can you spend, do you buy a new or used car, how much is insurance and upkeep, etc.)

!In a reflective journal, describe the difficulties and successes encountered when solving the above problem.

 

Standard 7: Reasoning and Proof

Outcomes

Indicators

Assessment Type

Sample Assessment Tasks

1. Teacher candidates will recognize reasoning and proof as fundamental aspects of mathematics

2. Teacher candidates will make and investigate mathematical conjectures

 

 

 

 

 

3. Teacher candidates will develop and evaluate mathematical arguments and proofs

 

4. Teacher candidates will select and use various types of reasoning and methods of proof

a. Note patterns, structure, or regularities in both real-world situations and symbolic objects

 

b. Pose mathematical questions and speculate results

c. Seek solutions to questions posed through the use of concrete materials, calculators and other tools, diagrams, and mathematical symbols

d. Provide evidence for mathematical judgments based on specific assumptions and rules

 

e. Demonstrate a repertoire of reasoning types, such as algebraic, geometric, proportional, probabilistic, statistical, inductive, deductive, etc.

!Group project and presentation

!Learning log with diagrams and written explanation

!Reflective journal

!Extended responses

!Given a series of triangular numbers (e.g., 3, 6, 10, 15), describe the pattern, predict other numbers in the sequence and derive a formula for determining the value of any triangular number. Demonstrate the process with the aid of calculators and manipulatives.

 

Standard 8: Communication

Outcomes

Indicators

Assessment Type

Sample Assessment Tasks

1. Teacher candidates will organize and consolidate mathematical thinking through communication

 

 

 

 

2. Teacher candidates will express mathematical ideas coherently and clearly to peers, teachers, and others

 

 

 

3. Teacher candidates will analyze and evaluate the mathematical thinking and strategies of others

 

 

 

4. Teacher candidates will use the language of mathematics to express mathematical ideas precisely

a. Draw, diagram, act, manipulate objects, verbally explain, write and use mathematical symbols to present methods for solving problems, justify mathematical reasoning to others, or formulate questions

 

b. Through discussion and writing, present clear and complete arguments for mathematical reasoning

Justify mathematical procedures and reasoning in a variety of ways

c. Listen to, paraphrase, question and interpret others’ mathematical ideas

Examine others’ mathematical methods and strategies and determine their strengths and weaknesses

d. Write well-constructed mathematical arguments using the special meanings of mathematical language and the representations and standards of explanation and proof. 

!Presentation/ demonstration

!Written paper

!Reflective journal

!Group discussion

!Diagrams, spreadsheets, etc.

!In a learning log, explain the process used to solve a problem; include diagrams, drawings of manipulatives used, written explanations, mathematical symbols and equations.

!Verbally justify mathematical thinking to a group of peers and provide constructive feedback to group members.

 

Standard 9: Corrections

Outcomes

Indicators

Assessment Type

Sample Assessment Tasks

1. Teacher candidates will recognize and use connections among different mathematical ideas 

2. Teacher candidates will understand how mathematical ideas interconnect and build on one another to produce a coherent whole

3. Teacher candidates will recognize and apply mathematics in contexts outside of mathematics

a. Describe how mathematical concepts and skills are related, not isolated

 

b. Use links among mathematical concepts for solving problems 

 

 

 

c. Use mathematics in science, and social sciences. 

!Investigative project

!Simulation

!Research paper

!Make a presentation on a topic such as bridges, trees, cities, textiles, etc., demonstrating how mathematics, science, social studies, the arts, etc. are interrelated and contribute to understanding of the topic.

 

Standard 10: Representation

Outcomes

Indicators

Assessment Type

Sample Assessment Tasks

1. Teacher candidates will create and use represent-ations to organize, record, and communicate mathematical ideas

2. Teacher candidates will select, apply and translate among mathematical representations to solve problems

3. Teacher candidates will use representations to model and interpret physical, social and mathematical phenomena

a. Examine and compare the efficiency and effectiveness of historical, current and technological forms of representation of mathematical ideas

b. Demonstrate appropriate use of conventional forms of representation

 

 

c. Create and interpret models, such as manipulative models, simulations, tables, graphs, pictures, equations, technology software models, etc. to represent real-world phenomena.

!Demonstration of models

!Simulation

!Extended response

!Write an explanatory essay describing how various models (e.g., manipulatives, equations, graphs, diagrams, drawings, pictures, simulations, etc.) serve as tools for thinking about and solving problems.