Equipped for the Future

Performance Continuum

for Use Math to Solve

Problems and Communicate

In order to fulfill responsibilities

as parents/family members, citizens/community members,

and workers, adults must be able to:

COMMON ACTIVITIES

· Gather, Analyze, and Use Information

· Manage Resources

· Work Within the Big Picture

· Work Together

· Provide Leadership

· Guide and Support Others

· Seek Guidance and Support From Others

· Develop and Express Sense of Self

· Respect Others and Value Diversity

· Exercise Rights and Responsibilities

· Create and Pursue Vision and Goals

· Use Technology and Other Tools to Accomplish Goals

· Keep Pace With Change

Equipped for the Future

Use Math to Solve Problems and Communicate Performance Continuum

PERFORMANCE LEVEL 1

Use Math to Solve Problems and Communicate

How adults at Level 1 Use Math to Solve Problems and Communicate:

· Understand, interpret, and work with pictures, numbers, and symbolic information.

· Apply knowledge of mathematical concepts and procedures to figure out how to answer a question, solve a problem, make a prediction, or carry out a task that has a mathematical dimension.

· Define and select data to be used in solving the problem.

· Determine the degree of precision required by the situation.

· Solve problem using appropriate quantitative procedures and verify that the results are reasonable.

Communicate results using a variety of mathematical representations, including graphs, charts, tables and algebraic models.

Level 1 Indicators

Use Key Knowledge, Skills, and Strategies

Adults performing at Level 1 can:

Read, write, and interpret very simple types of mathematical information such as

Numbers and number sense: whole numbers (up to three digits), common monetary values, and benchmark fractions (1/2, 1/4) and percents (50%);

Patterns/Functions/Relationships: very simple patterns, commonly-used denominations/groupings (5s, 10s, 25s), and very simple proportions (2:1, 1:2);

Space/Shape/Measurement: high frequency standard units of measurement (pounds, feet, quarts, gallons), geometric shapes, and concepts of length and width; and

Data/Statistics: very simple ways to interpret and represent data (checksheets, picture graphs, unambiguous bar graphs, line plots) emphasizing frequency of occurrence.

Recall and use a few simple mathematical procedures such as very basic estimating, counting, sorting, ordering, grouping, adding on (using counting or a calculator), orally counting by 2s, 5s and 10s, and measuring length and weight using tools calibrated with whole numbers (rulers, manipulatives).
Evaluate the degree of precision needed for the solution.

Extract discrete information from simple and concrete data and graphs, and measure with appropriate tools, describe patterns, and/or use computational procedures effectively to solve a problem and to verify that the solution is reasonable.

Communicate the solution to the problem orally, in role plays, with pictures, or by entries on a simple chart.

Show Fluency, Independence, and Ability to Perform in a Range of Settings

Adults performing at Level 1 can easily select and apply the knowledge, skills, and strategies at this level to independently accomplish simple, well-defined, and highly structured math tasks in one or more comfortable and familiar settings.

Level 1 Examples of Proficient Performance

Adults performing at Level 1 can count, order, and group to accomplish a variety of goals, such as:

· Make change using the “building” method

· Determine the amount of flooring needed in a room by counting square units of material within the shape of the room

· Estimate daily/weekly wages by adding on (counting) per-hour amounts

· Fit furniture into a space by counting square units within the shape of the space or by estimating using informal measurement units

· Interpret an unambiguous horizontal or vertical bar graph in a brochure from a local clinic in order to understand information about children’s health concerns

· Measure ingredients for simple recipes using benchmark fractions

· Use tallies to determine number of useable items from total number of items produced

· Sort coins into like piles, and then determine the value of each pile

· Use manipulatives, mental math, a calculator, or paper and pencil to calculate how much it will cost for 2 people to go to the movies

· Develop a schedule for how and when to take medication according to a doctor’s order

· Double a recipe for chocolate chip cookies for a children’s party

Equipped for the Future

Use Math to Solve Problems and Communicate Performance Continuum

PERFORMANCE LEVEL 2

Use Math to Solve Problems and Communicate

How adults at Level 2 Use Math to Solve Problems and Communicate:

· Understand, interpret, and work with pictures, numbers, and symbolic information.

· Apply knowledge of mathematical concepts and procedures to figure out how to answer a question, solve a problem, make a prediction, or carry out a task that has a mathematical dimension.

· Define and select data to be used in solving the problem.

· Determine the degree of precision required by the situation.

· Solve problem using appropriate quantitative procedures and verify that the results are reasonable.

Communicate results using a variety of mathematical representations, including graphs, charts, tables, and algebraic models.

Level 2 Indicators

Use Key Knowledge, Skills, and Strategies

Adults performing at Level 2 can:

Read, write, and interpret some common types of mathematical information such as

Numbers and number sense: whole numbers, monetary values and prices, benchmark fractions (3/4, 1/10), decimals (.25, .50, .75, .10) and percents (25%, 75%, 10%, 100%); Patterns/Functions/Relationships: simple patterns, probability and proportions (1:4, 4:1);

Space/Shape/Measurement: commonly used standard units of measurement, common geometric shapes, and the concept of “area”; and

Data/Statistics: simple ways to interpret and represent data (tables, bar graphs with and without gridlines, line graphs and pie graphs).

Recall and use mathematic procedures such as addition and subtraction (with or without use of calculator), grouping, comparing 2 numbers, and basic estimating; and measure length, weight, and areas of standard and non-standard shapes using tools calibrated with whole numbers and benchmark equivalents (rulers, manipulatives).
Evaluate the degree of precision needed for the solution.

· Define, select, and organize simple data, and measure with appropriate tools, describe patterns, and/or use computational procedures effectively to solve a problem and to verify that the solution is reasonable.

Communicate the solution to the problem orally, in pictures, or in writing.

Show Fluency, Independence, and Ability to Perform in a Range of Settings

Adults performing at Level 2 can easily select and apply the knowledge, skills, and strategies at this level to independently accomplish simple, well-defined, and structured math tasks in a range of comfortable and familiar, or highly structured, settings.

Level 2 Examples of Proficient Performance

Adults performing at Level 2 can select, record and work with simple mathematical information to accomplish a variety of goals, such as:

· Stay within a set budget while shopping for work clothes

· Determine the area of a room by grouping the square units within the shape and performing repeated addition

· Describe a simple equation for determining weekly pay based on a consistent, predictable pattern, such as $5 per hour

· Review a restaurant check for a group of 5 people to see if there are errors; estimate the sales tax on the meal

· Measure the approximate number of feet of baseboard that will be needed for a room

· Design a garden with at least 2 rectangular flowerbeds; cost out fencing and plants using a catalogue

· Survey a group on the topic of your choice and create a bar graph to display data; explain findings

· Listen to 2 weather forecasters and keep track (by graphing) of the actual high and low temperatures each day for a week to see which forecaster is more accurate

· Use a calculator to add total receipts from 6 cash registers

· Quadruple a recipe for chocolate chip cookies for a community bake sale

Equipped for the Future

Use Math to Solve Problems and Communicate Performance Continuum

PERFORMANCE LEVEL 3

Use Math to Solve Problems and Communicate

How adults at Level 3 Use Math to Solve Problems and Communicate:

· Understand, interpret, and work with pictures, numbers, and symbolic information.

· Apply knowledge of mathematical concepts and procedures to figure out how to answer a question, solve a problem, make a prediction, or carry out a task that has a mathematical dimension.

· Define and select data to be used in solving the problem.

· Determine the degree of precision required by the situation.

· Solve problem using appropriate quantitative procedures and verify that the results are reasonable.

Communicate results using a variety of mathematical representations, including graphs, charts, tables, and algebraic models.

Level 3 Indicators

Use Key Knowledge, Skills, and Strategies

Adults performing at Level 3 can:

· Read, write, and interpret a variety of common mathematical information such as

Numbers and number sense: monetary values, extensions of benchmark fractions (1/8, 1/3, 1/5, etc), decimals, and percents (15%, 30%, etc.);

Patterns/Functions/Relationships: patterns and simple formulas (such as d=rt, a=lw); Space/Shape/Measurement: standard units of measurement including fractional units and benchmark angle measurements (90 degrees, 360 degrees, etc), geometric shapes including shapes containing a combination of common shapes, concept of pi, and concept of converting between units of measurement; and

Data/Statistics: ways to interpret and represent data (tables and graphs with scaling, basic statistical concepts such as range, mode, mean, and median).

Recall and use a good store of mathematical procedures such as estimation, rounding, multiplication and division (with and without use of a calculator), adding and subtracting common fractional amounts, measure length, weight, area and circumference using tools calibrated to varying degrees of precision and converting units of measurement as appropriate.
Evaluate the degree of precision needed for the solution.
Define, select and organize a variety of common mathematical data and measure with appropriate tools, describe patterns, and/or use appropriate procedures effectively to solve a problem and verify that the solution is reasonable.
Communicate the solution to the problem orally, with visual representations, in writing, by entries in a table or appropriate graph, or with basic statistics (range, mode, mean, median).

Show Fluency, Independence, and Ability to Perform in a Range of Settings

Adults performing at Level 3 can easily select and apply the knowledge, skills, and strategies at this level to independently accomplish well-defined and structured math tasks in a range of comfortable and familiar settings.

Level 3 Examples of Proficient Performance

Adults performing at Level 3 can select, record, and work with mathematical information to accomplish a variety of goals, such as:

· Figure a tip on a restaurant bill

· Balance a checking account

· Write a children’s book on multiplication facts

· Determine how much flooring is needed for a room by multiplying the length times the width of the room

· Design a survey regarding a community issue, and collect and organize the results

· Develop a monthly budget

· Design a garden to get the most space with the least amount of fencing needed

· Decide which product to buy based on a comparison of nutritional information

· Figure estimated taxes

Equipped for the Future

Use Math to Solve Problems and Communicate Performance Continuum

PERFORMANCE LEVEL 4

Use Math to Solve Problems and Communicate

How adults at Level 4 Use Math to Solve Problems and Communicate:

· Understand, interpret, and work with pictures, numbers, and symbolic information.

· Apply knowledge of mathematical concepts and procedures to figure out how to answer a question, solve a problem, make a prediction, or carry out a task that has a mathematical dimension.

· Define and select data to be used in solving the problem.

· Determine the degree of precision required by the situation.

· Solve problem using appropriate quantitative procedures and verify that the results are reasonable.

Communicate results using a variety of mathematical representations, including graphs, charts, tables, and algebraic models.

Level 4 Indicators

Use Key Knowledge, Skills, and Strategies

Adults performing at Level 4 can:

Read, write, and interpret a wide variety of mathematical information such as

Numbers and number sense: money/expenses/prices, percentages, decimals and fractions;

Patterns/Functions/Relationships: patterns and formulas (such as a=pr2);

Space/Shape/Measurement: units of measurement including fractional units, geometrical shapes including shapes containing a combination of common shapes, and concept of volume; and

Data/Statistics: ways to interpret, represent and draw implications from data (graphs, tables, and simple forms of statistical analysis).

· Recall and use multi-step mathematical procedures (such as keeping accounts) that involve whole numbers as well as fractions, decimals, and/or percents, and measure volume using tools with different calibrations.

Evaluate the degree of precision needed for the solution.
Define, select, organize, and integrate mathematical information of different types in carrying out procedures, describing patterns, and/or measuring with appropriate tools to solve the problem and to verify that the solution is reasonable.

· Create appropriate visual or graphic representations such as charts, tables, graphs, etc. and clearly communicate the solution process and results orally or in writing to a variety of audiences.

Show Fluency, Independence, and Ability to Perform in a Range of Settings

Adults performing at Level 4 can easily select and apply the knowledge, skills, and strategies at this level to independently accomplish structured math tasks in a variety of comfortable and familiar settings.

Level 4 Examples of Proficient Performance

Adults performing at Level 4 can select, analyze (by categorizing and comparing), and work with mathematical information to accomplish a variety of goals, such as:

· Design and measure shelves for a closet

· Keep track of monthly income and expenses

· Design a garden and determine the amount of fertilizer, fencing, and plants needed

· Develop a yearly budget and illustrate it by creating a graph

· Create a presentation using assorted charts and graphs to influence a committee

· Choose which car to buy based on information such as down payment required, monthly installments, and insurance costs

· Record product measurements on a chart to monitor whether the process is in control

· Estimate the gallons of water in an irregularly shaped garden pond in order to purchase the right amount of pond clarifier

Equipped for the Future

Use Math to Solve Problems and Communicate Performance Continuum

PERFORMANCE LEVEL 5

Use Math to Solve Problems and Communicate

How adults at Level 5 Use Math to Solve Problems and Communicate:

· Understand, interpret, and work with pictures, numbers, and symbolic information.

· Apply knowledge of mathematical concepts and procedures to figure out how to answer a question, solve a problem, make a prediction, or carry out a task that has a mathematical dimension.

· Define and select data to be used in solving the problem.

· Determine the degree of precision required by the situation.

· Solve problem using appropriate quantitative procedures and verify that the results are reasonable.

Communicate results using a variety of mathematical representations, including graphs, charts, tables and algebraic models.

Level 5 Indicators

Use Key Knowledge, Skills, and Strategies

Adults performing at Level 5 can:

Read, write, and interpret a wide variety of (often) complex mathematical information such as

Numbers and number sense: money/expenses/pricing;

Patterns/Functions/Relationships: formulas for a variety of calculations;

Space/Shape/Measurement: architectural symbols/ models and scale modeling; and

Data/Statistics: ways to interpret, represent, identify trends in or draw inferences from data (complex tables and graphs; advanced forms of statistical analysis; graphing equations and generating equations from data and/or line graphs; using concept of slope).

Research, select and apply sophisticated, multi-step mathematical concepts and procedures (such as scale modeling, cost analysis, earnings/deductions analysis)

· Evaluate the degree of precision needed for the solution.

Independently research, select, organize and integrate mathematical information of different types in carrying out procedures, describing patterns, and/or measuring with appropriate tools, to solve the problem and to verify that the solution in reasonable.
Create appropriate visual or graphic representations such as charts, tables, graphs, etc. and clearly communicate the solution process and results orally or in writing to a variety of audiences.

Show Fluency, Independence and Ability to Perform in a Range of Settings

Adults performing at Level 5 can easily select and apply the knowledge, skills, and strategies at this level to independently accomplish minimally structured, complex math tasks in a variety of comfortable and familiar settings.

Level 5 Examples of Proficient Performance

Adults performing at Level 5 can select, analyze, integrate, and use mathematical information to accomplish a variety of goals, such as:

· Analyze effects of deductions on earnings and projecting annual income

· Design a “dream house”

· Use an amortization table to decide whether to refinance the mortgage on a house

· Determine a budget for a grant proposal

· Design an archway or bridge to scale

· Make a decision about how to consolidate bills and credit card payments

Equipped for the Future

Use Math to Solve Problems and Communicate Performance Continuum

Performance levels for the EFF Content Standard
Use Math to Solve Problems and Communicate
mapped to the NRS Educational Functioning Levels

EFF Use Math to

Solve Problems and

Communicate

Performance Levels

Can be used to define an exit point for the

NRS ABE Educational Functioning Level …

Use Math

Performance Level 1

Beginning Basic Education

Use Math

Performance Level 2

Low Intermediate Basic Education

Use Math

Performance Level 3

High Intermediate Basic Education

Use Math

Performance Level 4

Low Adult Secondary Education

Use Math

Performance Level 5

High Adult Secondary Education

Equipped for the Future

Use Math to Solve Problems and Communicate Performance Continuum

How to read the EFF Use Math to Solve Problems and Communicate Performance Continuum

The EFF Use Math to Solve Problems and Communicate Performance Continuum is a developmental description of performance on the Use Math to Solve Problems and Communicate Standard. The continuum portrays development along four dimensions: the structure and depth of knowledge, and the fluency, independence, and range of performance. The five EFF Levels described here are points along the continuum that serve as benchmarks for key stages in development and increasing ability to accomplish important activities in daily life that require adults to use math to solve problems and communicate.

The five levels of performance described in this document cover only a portion of the performance levels possible. There are aspects of math development and performance that fall below the performance described in Level 1 and there are many levels of proficiency leading toward higher levels of expertise beyond the knowledge, skills, strategies, and performance descriptions at Level 5 on the performance continuum.

The descriptions of performance at each level of the performance continuum are anchored in analysis of data on adult learner performance. They were developed by analyzing data on use of the Use Math to Solve Problems and Communicate Standard by adult learners in adult basic education programs (including adult literacy, adult ESOL, family literacy, and adult secondary education). This empirical evidence of performance on the standard went through extensive analysis by research staff and was reviewed and amended by a panel of content experts. At each step in this process, cognitive science and math theory and research was used to guide and refine the definition of performance criteria. The number of levels defined for the EFF Use Math to Solve Problems and Communicate Performance Continuum (five) was determined through analysis and review of data. Each level describes a qualitatively distinct stage in the development of proficiency on the standard. Each level builds on the previous levels. Thus, an adult who is able to perform at Level 3 also has mastered the performance on the Standard described at Levels 1 and 2.

Each performance level on the Use Math to Solve Problems and Communicate Performance Continuum is described on a single page in this document. Each page is divided into four sections.

Section 1 is the definition of the Standard. The performance-level description starts with the components of performance of the standard. These components define the content standard and they remain the same at each level of performance. This repetition serves as a reminder that the integrated skill process defined by the components of performance for each standard is constant across all levels, from novice to expert levels of performance. What changes from level to level is the growth and complexity of the underlying knowledge base and the resulting increases in fluency and independence in using the standard to accomplish an increasing range and variety of tasks. Equipped for the Future

Use Math to Solve Problems and Communicate Performance Continuum

This definition of the standard is a useful tool for communicating to adults and their teachers the essential features of the construct, or set of targeted abilities, for each standard. By making it clear how the skill process is defined (or “unmasking the construct,” as described by Gitomer & Bennett, 2002), adult learners are better able to articulate their own learning goals for improving proficiency and teachers are better able to focus learning and instructional activities that build toward the goal of increasing ability to use the standard to accomplish everyday activities. Here is how the standard is defined for all performance levels of Use Math to Solve Problems and Communicate:

· Understand, interpret, and work with pictures, numbers, and symbolic information;

· Apply knowledge of mathematical concepts and procedures to figure out how to

answer a question, solve a problem, make a prediction, or carry out a task that has a

mathematical dimension;

· Define and select data to be used in solving the problem;

· Determine the degree of precision required by the situation;

· Solve problem using appropriate quantitative procedures and verify that the results are reasonable; and

· Communicate results using a variety of mathematical representations, including graphs, charts, tables, and algebraic models.

Section 2 is a list of key knowledge, skills and strategies that can be observed in proficient performance at that level. These are the primary indicators (or benchmarks) of the growth and organization of the knowledge base needed for proficient performance on the standard at each level. This list is thus of central importance for designing assessments to measure performance on the standard. Because the performance levels are designed primarily as guides for assessment and not as a curriculum framework, the list does not specify details of knowledge, skills, and strategies that might be studied and taught. Nonetheless, the list can serve as a way of identifying instructional objectives for each level and provides a guide for developing criteria for placement of learners in instructional levels.

Because this list of key knowledge, skills, and strategies focuses only on those features of performance that indicate qualitative changes in what a learner knows and can do, it can be used by curriculum developers and instructors to set instructional objectives for each level and to develop more detailed curricula or learning plans that will prepare learners to develop these abilities, meet the criteria, and move on to the next level in their development of expertise.

Section 3 defines the fluency, independence and ability to perform in a range of settings expected for proficient performance on the standard at each level. Together with the descriptions of key knowledge, skills, and strategies, these descriptions serve as the primary behavioral indicators (benchmarks) of proficient performance at each level. As such, section 3 descriptions also provide a basis for designing learning, instruction and assessment that is appropriate to that level.

Section 4 of the performance level descriptions provides a short list of examples of the purposes for using math (math activities) that can be accomplished by an adult who is performing at each level. Like Sections 2 and 3, the descriptions of math activities in Section 4 are specific to each performance level. These examples of things that adults can accomplish in the real world at each level of performance are useful to adult learners and to their teachers as ways of making concrete the purpose and need for attaining increasing knowledge and proficiency in math. By making it clear what can be accomplished at each level, the descriptions of math activities in Section 3 also provide motivation for higher levels of learning. The listing of real-world accomplishments also provides guidance for selecting and designing the content for instructional materials and assessments.

Equipped for the Future

Use Math to Solve Problems and Communicate Performance Continuum

What are the Guides to Using the EFF Performance Continua?

The EFF Assessment Resource Collection contains guides to ten of the EFF performance continua (http://eff.cls.utk.edu/assessment/guides.htm). The purpose of each guide is to introduce you to one of the EFF performance continua and show you how to use it to plan for instruction and for classroom-based assessment. Since accountability assessments on the standard will be based on the same performance continuum, the guide may also help you to better understand what to expect once a performance assessment process based on the standard is in place.

The EFF Continua of Performance are multidimensional, developmental descriptions of performance on the EFF Content Standards. They allow for descriptions of performance ranging from the novice level to the expert level. Currently the descriptions of performance describe from three to six levels each beginning with a novice level and extending to levels that correlate with exit points for adult basic education. Each continuum is built around the four EFF Dimensions of Performance, and performance levels are defined by identifying key features of performance at various points along the continuum. The performance continua make up one part of the EFF Assessment Framework.

How Can I Use the Guides?

Currently guides are available for 10 of the 16 EFF standards. Each guide contains a description of the standard and a two-page chart showing the performance continuum itself. The guide also includes: 1) information to help you understand how to read the continuum; 2) tools for lesson planning and assessment; 3) a scenario describing how one teacher used these tools; 4) information on the research basis for the standard; and 5) tips for where to go for more information. Some of the tools are available as Word documents to allow you to adapt them to your needs. You can get to each guide by clicking on the following link: http://eff.cls.utk.edu/assessment/guides.htm

Where Can I Find Examples of Completed Planning Guides?

The guides for each of the standards also contain scenarios describing how a teacher used the performance continuum for the standard along with lesson planning and assessment tools to prepare lesson plans. Embedded within each of the guides are excerpts from completed 1 page planning guides based on the scenarios. You will find the full 1 page versions of the completed planning guides for 10 standards here: http://eff.cls.utk.edu/assessment/planguides.htm.

Equipped for the Future

Use Math to Solve Problems and Communicate Performance Continuum

Notes on the Research Base for the Use Math to Solve Problems and Communicate Standard and Performance Continuum

In recent years, researchers and educators in the United States and abroad have been moving towards a common identification of the mathematics that adults need to function well in the world, to prepare for further education or training programs, and to prepare for and function well within the workforce. While the empirical research in adults’ mathematics learning still is limited, the development of the Use Math to Solve Problems and Communicate Standard and Performance Continuum has been informed by and builds upon the extensive cognitive research on children’s learning of mathematics, parallel standards development for the K–12 system, and the input of the business community.

It is clear that people learn mathematics that is useful to them in many settings outside of the classroom. Frequently the ways adults perform mathematical tasks are predicated on strategies that make sense in the environment in which they were developed and demonstrate a flexibility and conceptual basis that is closely tied to the context in which they are used. However, these procedures and strategies are often quite different from the decontextualized procedures and strategies that are taught in formal schooling. For some, the informal strategies and procedures were developed and are used alongside existing school-based strategies and procedures (Lave, Murtaugh, & de la Rocha, 1984). For others who did not attend school, informal strategies and procedures were developed to meet real-world needs (Nunes, Schliemann, Carraher, 1993; Millroy, 1992).

Adult learners in the United States generally have experienced some schooling and show evidence of both school-based and informally developed procedures and strategies. However, the knowledge and skills they have typically are fragmented and patchy and may be accompanied by “buggy” algorithms, misunderstandings and negative attitudes (Ginsburg, Gal & Schuh, 1996; Ginsburg & Gal, 1997; similar findings in the United Kingdom and the Netherlands: Evans, 2000; van Groenestijn, 2001). To fill the gaps in adults’ conceptual understanding and to address the limitations of their informal and formal mathematics knowledge, the Use Math to Solve Problems and Communicate Standard and Performance Continuum emphasize the following:

· the development of facility with multiple representations of mathematical

concepts,

· movement from the familiar and meaningful to the less familiar,

· connections within and across mathematical procedures, and

• the reality of multiple effective strategies to achieve the same ends.

By embedding the mathematics in realistic contexts, the boundaries and limitations between formal and informal mathematics are lessened, reducing the difficulty of transferring knowledge from decontextualized instruction to application within real world contexts.