Introduction

The Development of the Massachusetts ABE Curriculum Framework

for Mathematics and Numeracy

Over the past number of years, several initiatives have set the stage for writing the Massachusetts ABE Curriculum Frameworks for Mathematics and Numeracy.

The First Version: Changing the Way We Teach Math

In 1989, the National Council of Teachers of Mathematics (NCTM) published the Curriculum and Evaluation Standards for School Mathematics, a document that served as a template for reforming and improving K-12 mathematics education across the nation. In 1994, sixteen Massachusetts ABE/GED teachers formed a team and studied the Massachusetts K-12 standards to see how some of the ideas might play out in their adult education classrooms. After a year of action research in their classes, these teachers published two documents: a set of adult education math standards and stories of what changes looked like in their classrooms. Their adult math standards were incorporated into the Massachusetts ABE Math Standards (1995) and were the first set of ABE frameworks to hit the press. As such, they served as an early template for the Massachusetts ABE Curriculum Frameworks in other subjects that were subsequently developed.

In 1996, in the wake of education reform and a national science and math initiative in the state (which included Adult Basic Education), the Massachusetts ABE Math Standards were subsumed into the document, Massachusetts Curriculum Frameworks: Achieving Mathematical Power (1996). This state curriculum framework was to be used for both grades K-12 and for Adult Basic Education. In 2000, when the Massachusetts K-12 frameworks were revised, it was decided that the adult education math framework should be rewritten and revised, and developed as a separate document. This current version of the Massachusetts ABE Mathematics Curriculum Frameworks is a second revision of that first framework, but it is heavily influenced by developments in the adult education field since then, both nationally and internationally.

National Influences: The Adult Numeracy Frameworks and Equipped for the Future

In March 1994, the first national Conference on Adult Mathematical Numeracy, co-sponsored by the National Council of Teachers, the National Center on Adult Literacy (NCAL), and the U.S. Department of Education/Office of Vocation and Adult Education, brought policy makers, researchers, publishers, and practitioners together to discuss the issues of adult numeracy needs and mathematical education. Out of this conference came at least two significant events: the formation of the Adult Numeracy Network (ANN), a national network of practitioners, and the development of the “honest list: what math we should be teaching adults.”

In October 1995, the ANN was granted one of eight planning grants for system reform and improvement, funded by the National Institute for Literacy as part of the Equipped for the Future (EFF) project. Over the course of a year, through teacher-led focus groups of learners, business, and other state policy stakeholders in five states (including Massachusetts), and an on-line virtual study group, the ANN expanded upon the “honest list” developed from the conference. The teacher teams studied, among other documents, the teacher-developed Massachusetts ABE math standards, the report of the Secretary’s Commission on Achieving Necessary Skills (SCANS, 1991), and Equipped for the Future. Out of their research and focus groups, the teams developed seven themes which serve as the foundation for adult numeracy standards: Relevance/Connections, Problem-Solving/Reasoning/Decision-Making, Communication, Number and Number Sense, Data, Geometry: Spatial Sense and Measurement, Algebra: Patterns and Functions. In 1996, they published A Framework for Adult Numeracy Standards: The Mathematical Skills and Abilities Adults Need to be Equipped for the Future (1996).

As a result of this work, mathematics was included in the Equipped for the Future Content Standards: What Adults Need to Know for the 21^st Century (Stein, 2000), a framework for adult instruction that is grounded in data gathered from adults on their roles as workers, parents, and community members. Of the sixteen EFF standards, one specifically addresses numeracy or mathematics: listed under Decision-Making Skills, it is Use Math to Solve Problems and Communicate.

International Influences: Looking at Adult Numeracy

In addition to studying state and national mathematics curriculum frameworks, the ABE Math Frameworks 2001 Development Team considered several numeracy frameworks from other countries, including Australia, the United Kingdom, and the Netherlands, as well as the numeracy framework developed for the Adult Literacy and Lifeskills Survey (ALL), an international, large-scale comparative survey of basic skills in the adult populations of participating countries.

The term numeracy is a word that was first used in 1959 in Great Britain and is used more often internationally than in this country. Numeracy has been described as the mirror image of literacy (Crowther Report, 1959) and is often thought to deal just with “numbers.” But since the 1980’s, work by adult educators in Australia, the UK, and other countries, has expanded the notion that numeracy refers just to the ability to perform basic calculations. For example, in the Australian curriculum frameworks, numeracy denotes the ability to perform a wider range of math skills, such as measuring and designing, interpreting statistical information, and giving and following directions, as well as using formulas and other advanced topics to pursue further knowledge. Moreover, numeracy and literacy are presented as interconnected and on an equal footing. The frameworks are written so as to address the purposes for learning mathematics and do not proceed from a school-based mathematics curriculum model so much as looking at the mathematics that is used in the context of adult lives. The Massachusetts ABE Curriculum Frameworks for Mathematics and Numeracy incorporate some of these ideas in the current revision.

What is Numeracy? A Definition of Numerate Behavior

For purposes of this framework, the following definition is incorporated for describing numeracy and what it means to be a numerate adult:

Numerate behavior involves:

Managing a situation or solving a problem in a real context

everyday life

work

societal

further learning

by responding

identifying or locating

acting upon

interpreting

communicating about

to information about mathematical ideas

quantity and number

dimension and shape

pattern and relationships

data and chance

change

that is represented in a range of ways

objects and pictures

numbers and symbols

formulae

diagrams and maps

graphs

tables

texts

and requires activation of a range of

enabling knowledge, behaviors, and processes.

mathematical knowledge and understanding

mathematical problem-solving skills

literacy skills

beliefs and attitudes.

Source: Gal, I., van Groenestijn, M., Manly, M., Schmitt, M.J., and Tout, D. (1999). Adult Literacy and Lifeskills Survey Numeracy Framework Working Draft. Ottawa: Statistics Canada.

How to use This Document (Teacher's Guide)

The Mathematics Frameworks presents four learning strands: Number Sense; Patterns, Functions, and Algebra; Statistics and Probability; Geometry and Measurement which are described beginning on page 16 (in the Section on Content Strands and Learning Standards.) In order to present a document that makes sense practically, as well as theoretically, the Outline of Learning Levels on page 21 presents each of the strands and their standards at six performance levels:

§ Level 1: Beginning Adult Numeracy

§ Level 2: Beginning ABE Mathematics

§ Level 3: Intermediate ABE Mathematics

§ Level 4: Pre-GED/ABE Mathematics

§ Level 5: ASE/GED Mathematics

§ Level 6: ASE/Bridge to College Mathematics

At each level the strands are given in a chart, as shown below.

Level ÞLevel 1: Beginning Adult Numeracy

Strand Þ Number Sense

Learners engage in problem solving within adult contextual situations by communicating, reasoning, and connecting to:

Standard Þ	Standard 2P-3. Recognize and use algebraic symbols to model mathematical and contextual situations
	Benchmark: At this level an adult will be expected to:	Enabling Knowledge and Skills	Examples of Where Adults Use It
Benchmark Þ Assessment (See page 10) Þ	2P-3.4 Read and understand positive and negative numbers as showing direction and change. Assessed by 3P-3.7	2P-3.4.1 Know that positive refers to values greater than zero 2P-3.4.2 Know that negative refers to values less than zero 2P-3.4.3 Use a horizontal or vertical number line to show positive and negative values	Reading thermometers Riding an elevator below ground level Staying "in the black" or going "into the red" on bill paying
	2P-3.5 Use a number line to represent the counting numbers.	2P-3.5.1 Demonstrate an understanding that a horizontal number line moves from left to right using lesser to greater values	Reading and interpreting scales
		Ý Enabling skill	Ý Application

Benchmark Column (e.g. At this level an adult will be expected to:)

Benchmarks describe the set of skills learners need to develop and achieve in order to meet the more broadly stated standards. By providing more detailed information on the specific skills and contexts for learners to meet the standard, benchmarks show teachers and learners what a standard “looks like” at each of the six levels.

The strands and standards are arranged by performance levels so that each level can build on the previous ones. At each level, the four strands and their standards are outlined with the skills appropriate for that level. The skills defined at each level are ones to be achieved while working through the level. The teacher can use the frameworks as a curriculum guide. Each level builds on the previous levels, so it is recommended that teachers familiarize themselves not only with the level of their own class, but with the preceding levels as well.

Enabling Knowledge and Skills Column

The study of mathematics is developmental, but many adult learners have gaps in their learning of math. At times a learner may struggle with a skill because he or she has not grasped an enabling skill on which it is based. To present problems and practice with a skill, we must first lay the proper groundwork. Since not all adult education teachers have experience teaching math at an elementary level, the skills needed for the development of each performance skill are outlined.

Examples of Where Adults Use It Column

Teaching mathematics to adults is different than teaching it to children. As stated in the Common Chapters for the Massachusetts Adult Basic Education Curriculum Frameworks, “Adult learners value education and the power it has, but they rarely see it as an end in and of itself. Rather, education is seen as a means to other kinds of opportunities and achievements.”[1] Adult learners need to know that what they are learning in the classroom is relevant to the lives and goals outside of the classroom. For this reason, we have included an application for each skill by giving an example of using the skill in an adult context.

It is our expectation that this format will be a useful tool for:

§ Lesson planning

§ Curriculum development

§ Presenting practical applications for adult use of the math skills

§ Assessing student math levels for placement, informal classroom instruction, and for pre- and post-test assessment

§ Connecting pre- and post-test assessment to curriculum and instruction

The standards and benchmarks for each level are ambitious. They set the bar to be reached by learners, not the expectation of what is covered in a given class in a given year. However, the Framework does assume that the teaching of numeracy and mathematics be given a significant amount of time and attention in a program’s class offerings and curriculum.

Mathematical understanding progresses from the concrete (counting two groups of blocks) to the representative (adding numbers presented in pictorial or verbal problems) to the abstract (using symbols and graphs). Presenting adults with problems or situations that allow them to develop their own approach to an inquiry model gives learners opportunities to talk about, write about, and represent math situations. During such inquiry, a learner can experience this progression in his or her own thinking. This affords an opportunity to see interconnections within math and between math and other disciplines.

The numbering system used with the Standards and benchmarks was developed so the specific benchmarks or enabling skills can be referred to (e.g. in a lesson plan, curriculum, or scope and sequence). In the number 2P-3.4.1, for example, the system is as follows:

2 refers to the Proficiency Level 2
P- refers to the Strand, Patterns, Functions and Algebra (N for Number Sense, and so on)
3 refers to the Standard (Recognize and use algebraic symbols to model mathematical and contextual situations)
4 refers to the Benchmark (Read and understand positive and negative numbers as showing direction and change)
1 refers to the Enabling Knowledge and Skills (Know that positive refers to values greater than zero)

How to use This Document in

Connecting Curriculum, Instruction, and Assessment

The University of Massachusetts Center for Educational Assessment, working with the Adult and Community Learning Services of the Massachusetts Department of Education, has developed an assessment to measure adult learners’ skills as outlined in the Massachusetts ABE Curriculum Framework for Math and Numeracy.

The ABE Curriculum Framework for Math and Numeracy is not an end in itself but a part of the broader goal of aligning curriculum, instruction and assessment. To this end, Adult and Community Learning Services and ABE practitioners have worked closely with the University of Massachusetts’ Center of Educational Assessment to develop a math and numeracy assessment that is designed to measure the skills outlined in the Framework. This assessment will be capable of measuring more accurately and capturing more comprehensively, the skills that adult learners have acquired or need to acquire through the instruction provided in adult basic education classes. Both the ABE Curriculum Framework for Math and Numeracy and the results of the new math assessment are valuable tools that should be used to inform classroom instruction.

The Frameworks provide teachers with Standards, Benchmarks and Examples that describe what it is adult learners need to know and be able to do, while the new math assessment will help identify how well students are acquiring the skills and knowledge as well as their ability to apply the skills and knowledge outlined in the Frameworks. By using the Frameworks and assessment results to inform instruction, programs and teachers can achieve the goal of aligning curriculum, instruction and assessment.

The skill numbers in the frameworks directly correspond with the skill numbers on the math test. The skills within each level are assessed at that level unless otherwise noted as shown in the example on page 8, and below.

At this level an adult will be expected to:

Enabling Knowledge and Skills

Examples of Where Adults Use It

Skill Þ

Assessment Þ

(See page 11)

2P-3.4 Read and understand positive and negative numbers as showing direction and change

Assessed by 3P-3.7

2P-3.4.1 Know that positive refers to values greater than zero

2P-3.4.2 Know that negative refers to values less than zero

2P-3.4.3 Use a horizontal or vertical number line to show positive and negative values.

Reading thermometers

Riding an elevator below ground level

Staying "in the black" or going "into the red" on bill paying

The math frameworks endeavor to expose students at all levels to the four strands: N-Number Sense; P-Patterns, Functions, and Algebra; S-Statistics and Probability; and G-Geometry and Measurement with the realization that some material introduced at one level might need to be expanded on in a later level. For this reason, there is overlap between the levels. Positive and negative numbers, for example, may be discussed with basic applications at Level 2, but the learner will not be expected to demonstrate knowledge and skill with the topic until Level 3 as shown above with the reference to 3P-3.7

Adult learners come to our classes with a wide range of prior learning, but often they have gaps in their knowledge. A student who is well-read may be familiar with interpreting graphs and tables, but struggle to understand the principles of area and volume relating to home decor. Some adults who are very capable with computation may have developed a mental block against algebraic notation. The Frameworks, therefore; encourages multi-level exploration within the classroom while more clearly defining skills to be demonstrated at each assessment level.

Core Concepts

Adults develop numeracy skills and mathematical fluency through actions involving problem solving, reasoning, decision-making, communicating and connecting in curriculums that link to their own mathematics knowledge, experiences, strategies and goals. Fluency is enhanced by instruction that requires learners to strive for a constant interplay of accuracy, efficiency and flexibility in their work.

Problem solving is an important key to independence for adults. Problem solving enables learners to:

§ reach their own solutions,

§ generalize problem solving strategies to a wide range of significant and relevant problems,

§ use appropriate problem solving tools including real objects, calculators, computers, and measurement instruments.

Mathematical reasoning provides adults with access to information and the ability to orient themselves to the world. It enables learners to:

§ validate their own thinking and intuition,

§ pose their own mathematical questions,

§ evaluate their own arguments, and

§ feel confident as math problem solvers.

Success as an adult involves decision-making as a parent, citizen and worker. Mathematical decision-making enables learners to:

§ determine the degree of precision required by a situation,

§ define and select data to be used in solving a problem, and

§ 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.

The ability to communicate mathematically means having an expanded voice and being heard in a wider audience. Mathematical communication enables learners to:

§ interact with others,

§ define everyday, work-related or test-related mathematical situations using concrete, pictorial, graphical or algebraic methods,

§ reflect and clarify their own thinking about mathematical outcomes, and

§ make convincing arguments and decisions based on discussion and reflection.

Connecting everyday life with mathematics helps adults access essential information and make informed decisions. Mathematical connections enable the learner to:

§ view mathematics as an integrated whole that is connected to past learning, the real world, adult life skills, and work-related settings, and

§ apply mathematical thinking and modeling to solve problems that arise in other disciplines, as well as in the real world and work-related settings.

The thinking skills of accuracy, efficiency and flexibility are essential tools for success in a rapidly changing world. In mathematics, such fluency enables the learner to:

§ develop a sense of the appropriate ballpark for a solution,

§ be able to keep track of how a solution is reached,

§ develop the practice of double-checking results,

§ use robust strategies that work efficiently for solving different kinds of problems, and

§ take more than one approach to solving a class of problems.

Habits of Mind

Habits of Mind are practices that strengthen learning. In numeracy instruction, habits of mind involve reflection, inquiry and action. They are developed by teachers and programs that offer challenging mathematical tasks in settings that support learners’ curiosity, respect for evidence, persistence, ownership, and reflection about what is learned and how it is learned. These habits flourish in instructional environments that favor uncovering mathematical concepts and connections rather than mimicking algorithms.

The following chart defines the habits of mind crucial to adults’ numeracy development. It also lists questions students and teachers may share to assess their own mathematical habits.

Habits of Mind

Habit

Learner Question

Curiosity

A curious and open attitude towards the presentation of new ideas or ways of approaching problems, even when confusion arises, facilitates learning.

Do I ask “Why,” “How,” or “What If” questions?

Respect for Evidence

To evaluate reasoning, it is essential to see evidence. Reasoning is demonstrated by the appropriate use of verbal and visual mathematical evidence to support solutions and ideas.

Do I listen carefully for others’ use of

evidence, and do I include evidence to support my solutions and ideas?

Persistence

Solutions in mathematics are not always apparent at first glance. Persistence is necessary to work through challenging problems that stretch our understanding.

Do I keep going when I feel lost or discouraged while solving problems?

Ownership

What we own has meaning for us, and taking ownership of our work encourages us to do our best. Although someone else might assign a mathematical task to us, we must treat the problem as important to us, as though it was our own, if we are to produce high quality work and learn from experience.

In what ways do I show that my work is purposeful and important to me?

Reflection

To become an autonomous learner, it is necessary to think about how our learning happens. We need to consider how we learn from mathematical experiences.

Do I notice and analyze how and what I learn?

Content Strands and Learning Standards

Following is a chart that outlines the content strands and learning standards for the Mathematics and Numeracy curriculum framework. After this chart, you will find a more detailed explanation of each content strand and the learning standards that go along with it.

Strands

Standards

Learners will demonstrate the ability to…

Number Sense

Massachusetts Department of Education

TABLE OF CONTENTS

Acknowledgments

Introduction

The Development of the Massachusetts ABE Curriculum Framework

The First Version: Changing the Way We Teach Math

National Influences: The Adult Numeracy Frameworks and Equipped for the Future

International Influences: Looking at Adult Numeracy

What is Numeracy? A Definition of Numerate Behavior

How to use This Document (Teacher's Guide)

How to use This Document in

Connecting Curriculum, Instruction, and Assessment

Core Concepts

Guiding Principles

Habits of Mind

Habits of Mind

Habit

Learner Question

Curiosity

Respect for Evidence

Persistence

Ownership

Reflection

Content Strands and Learning Standards

Standards