A FRAMEWORK FOR ADULT NUMERACY STANDARDS:
The Mathematical Skills and Abilities Adults Need
To Be Equipped for the Future
By
Donna Curry
Mary Jane Schmitt
Sally Waldron
The Adult Numeracy Practitioners Network
System Reform Planning Project
July, 1996
Funded by the
National Institute for Literacy
A Framework for Adult Numeracy Standards:
The Mathematical Skills and Abilities Adults Need
To Be Equipped for the Future
The terms "mathematical literacy" and "numeracy" are used interchangeably in this
document. Both terms should be viewed as loosely referring to the aggregate of skills,
knowledge, beliefs, patterns of thinking, and related communicative and problem-solving processes individuals need to effectively interpret and handle real-world quantitative
situations, problems, and tasks. (Proceedings of the Conference on Adult Mathematical
Literacy, March, 1994)
INTRODUCTION
In October 1995, the National Institute for Literacy (NIFL) funded eight planning
grants for system reform and improvement as part of the Equipped for the Future (EFF)
project. World Education, Inc., in cooperation with five state literacy resource
centers, accepted the grant on behalf of the Adult Numeracy Practitioners Network (ANPN).
The purpose of the ANPN Planning Grant is to begin the work of developing Adult
Numeracy Standards for adult basic education. We are augmenting previous work done
in this area (e.g., NCTM, SCANS, Massachusetts ABE Math Standards) by interviewing adult learners,
teachers and other stakeholders.
This project, while furthering the work of other projects, was exciting in that the
voices of the adult learner as well as stakeholders were added to the mix. Based
on all the voices along with the work done previously in the area of adult numeracy,
the following seven themes emerged and 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
Along with the seven themes noted above, adult learner and stakeholder voices also
gave us greater insight into affective issues. A section on Competence and Self-confidence
was added to ensure that adults' voices were heard and their feelings considered as this document is read.
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We not only asked adults what they need to know and be able to do to be productive
citizens, workers, and parents, but also we encouraged the adults we spoke with to
share their opinions on how math instruction should be changed. Adult learners shared
how math instruction should be changed in the classroom while stakeholders tended to
look at system reform issues. From the uniformity of voices of adults across the
country, Recommendations for System Reform have been drafted and are reflected in
the final chapter of this document.
What We Need Is an Honest List!
In March, 1994, over 100 adult educators, mathematics educators and other stakeholders
in the field of adult education and training came together for three days to discuss
the topic of adult numeracy. One of the major suggestions of the Conference on Adult Mathematical Literacy was that an important next step would be to develop an honest
list of the skills and knowledge that adults really need to be mathematically literate.
The participants called for a serious rethinking of the content and relevance of the adult basic education mathematics classes as they are currently taught. Through
analysis of the mathematical demands on adults in today s society, educators can
refocus the adult numeracy curriculum in a meaningful way.
A Massachusetts cohort of adult education teachers, inspired by the National Council
of Teachers of Mathematics Curriculum and Evaluation Standards for School Mathematics,
had already begun this task in earnest. However, because the cohort reflected the
input of teachers only, many participants at the conference felt that more research
was needed. That research should include consideration of data from the Secretary's
Commission on Achieving Necessary Skills (SCANS) Report, other recent documents,
employer and community needs assessment, and especially the opinions of adult learners.
In this report, our grassroots organization formed at the conference, the Adult Numeracy
Practitioners Network (ANPN) will bring together essential documents and the many
voices of adult learners, teachers, and employer and community stakeholders. Through the Equipped for the Future Initiative, an ANPN working group was encouraged by the
National Institute for Literacy to carefully listen to the data by analyzing the
transcribed tapes of twenty-one learner focus groups, five stakeholder focus groups,
and five teacher study groups. In addition, the ANPN working group examined pertinent
documents such as Equipped for the Future, the SCANS Report, the 1994 Conference
Proceedings, the NCTM Standards, and The Massachusetts ABE Math Standards (See Table 1).
The ANPN working group was struck by the fact that all the voices -- from SCANS to
The Massachusetts ABE Math Standards to the focus group partici-
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pants -- resonate,
all be it with diverse perspectives. It is this resonance that must and will guide
our honest list .
Responding to SCANS Research
The SCANS Report for America 2000, What Work Requires of Schools lists the following
foundation skills :
- Basic Skills: reading, writing, mathematics (arithmetical computation and mathematical reasoning), listening, and speaking;
- Thinking Skills: creative thinking, making decisions, solving problems, seeing
things in the mind s eye, knowing how to learn, and reasoning; and
- Personal Qualities: individual responsibility, self-esteem, sociability, self-
management, and integrity.
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The basic skills are the irreducible minimum for anyone who wants to get even a low-skill
job . . . the thinking skills, by contrast, permit workers to analyze, synthesize
and evaluate complexity. (SCANS, p. 17)
SCANS holds that even these foundation skills are not enough. They must be integrated
with other kinds of competencies to make them fully operational. Competencies such
as managing or using resources, interpersonal skills, information, systems, and technology are needed by everyone from the entry level or unskilled worker to managers
and executives.
The seven adult numeracy themes in our Framework reflect this "more than the basics"
slant. Relevance/Connections, Problem-Solving/Reasoning/ Decision-Making, and Communication
combine with the four content areas of Number and Number Sense, Data, Geometry: Spatial Sense and Measurement, and Algebra: Patterns and Functions to deliver an
up-to-date, SCANS-friendly definition of mathematical literacy.
Building upon The Massachusetts ABE Math Standards
The Massachusetts ABE Math Standards posited Problem-Solving, Communication, Reasoning,
and Connections as the four over-arching standards for mathematical literacy. Through
ANPN s further research, these four standards, often referred to as process standards, were consolidated into three adult numeracy themes: Relevance/Connections,
Problem-Solving/ Reasoning/Decision-Making, and Communication. Responses showed
that it was difficult for individuals to differentiate between problem-solving and
reasoning, both key skills in decision-making. Our data also revealed that the issue of
relevance frequently occurred.
The remaining seven MA ABE Standards have been integrated into four adult numeracy
content themes. Number and Number Sense includes two MA ABE standards, Estimation
and Number, Operations, and Computation. Data is similar to the MA ABE standard
called Statistics and Probability. Geometry: Spatial Sense and Measurement incorporates two
MA ABE standards, Geometry and Spatial Sense and Measurement. Two MA ABE standards
-- Patterns, Relationships, and Functions and Algebra -- correspond to the adult
numeracy theme, Algebra: Patterns and Functions. This reorganized structure is a reflection
of the words of adult learners, teachers and stakeholders as they told us about the
math that they need and use.
The final standard of The Massachusetts ABE Math Standards is Evaluation and Assessment.
The Framework for Adult Numeracy Standards, while not choosing to include a separate
theme to address these topics, addresses assessment under System Reform. Many focus group participants saw evaluation and assessment
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as issues that need to be approached
through system reform efforts.
Connecting to Equipped for the Future s Four Key Purposes
The Equipped for the Future (EFF) project focused on goal 6 of the National Education
Goals which stated: By the year 2000, every adult American will be literate and
will possess the knowledge and skills necessary to compete in a global economy and
exercise the right and responsibilities of citizenship. When asked what they needed to
compete in a global economy and exercise the rights and responsibilities of citizenship,
adult learners responded with four key purposes:
- to have access to information and orient themselves in the world;
- to give voice to their ideas and opinions and to have the confidence that their voice will be heard and taken into account;
- to solve problems and make decisions on their own, acting independently as a parent, citizen and worker, for the good of their families, their communities, and their nation; and
- to be able to keep on learning in order to keep up with a rapidly changing world.
The Framework for Adult Numeracy Standards supports these four key purposes through the seven adult numeracy themes. Literacy for Access and Orientation includes access
to the broader world of ideas and opportunities that surround them and they know literacy -- including the ability to work with numbers as well as read and write for themselves
-- is the price of the ticket. (Equipped for the Future, p.11) The four content
themes: Number and Number Sense; Data; Geometry: Spatial Sense and Measurement;
and Algebra: Patterns and Functions provide access to the world of mathematical thought.
Literacy as Voice refers to adults ability to communicate to others what they
think and feel. The Communication theme addresses this issue and is considered a
key process that is integrated in all other math areas. Literacy for Independent Action
reflects adults desires to be able to act independently and make informed decisions.
All focus group participants in the Adult Numeracy project could clearly describe
how decisions were made involving math. The process theme Problem-Solving/Reasoning/
Decision-Making is an outcome of their comments. The fourth key purpose -- Literacy
as a Bridge to the Future -- explains why adults participate in adult education programs. They realize that education is a key to future success, not only for themselves
but for their children. Over and over again, the adult learners in our project shared
that a key reason for wanting to learn math was to help their children be successful.
In essence, they saw the importance of our third process theme, Relevance/Connections.
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The ANPN Planning Project for System Reform sought to discover what math adults needed
to accomplish the EFF key purposes in the roles as parent, worker, and citizen.
Focus group participants were specifically asked what math they need to know and
be able to do in order to be successful in their roles as parent, worker, and citizen. The
feedback from these questions is integrated throughout the Framework for Adult Numeracy
Standards. Table 2, on the following page, compares ANPN s Numeracy Themes with
The Massachusetts ABE Math Standards, Equipped for the Future, and SCANS.
How Much Closer to the Honest List Are We?
This document is not a set of standards, but a framework for developing standards.
Sometimes one has to step back before really going forward. The ANPN is being true
to the data collected. Through this project, ANPN has spent an enormous amount of
time listening to learners, teachers, employers and other stakeholders in a systematic,
structured manner. This has given us a rich base from which to derive the honest
list , the next step.
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METHODOLOGY
The ultimate goal of the Adult Numeracy Practitioners Network Planning Project was
to change adult numeracy education within larger reform of the ABE system. To accomplish
this, two objectives had to be met. First, the project would create opportunities
for teachers, stakeholders, and specifically, adult learners, to participate in the
discussion of math knowledge and skills needed by adults in their roles as parents,
workers, and citizens. The second objective was to develop a framework for math
content standards and an initial plan for system reform around adult numeracy. This project
was designed and executed with those two objectives in mind.
DEMOGRAPHICS
ANPN has involved hundreds of learners, teachers, and other stakeholders during this
project. We gathered data from seven states across the nation, for a total of nearly
300 individuals participating in some phase of data collection. (See appendix.)
Adult Learners
Twenty-one Learner Focus Groups from seven states participated in this project, including
six each from Illinois and Virginia, two from Ohio, one from each of the New England
states of New Hampshire, Vermont, and Rhode Island, and four from Oregon. There
were 171 adult learners, all of whom were enrolled in adult education mathematics
classes, participating in the Learner Focus Groups. Over half of the learners were
female (59% females). Almost three-quarters (71%) of the learners came from urban
areas rather than rural. Since a key role for adults is that of parent, it was interesting
to note that 69% of adult learner focus group participants were parents. There were
more unemployed than employed adult learners (60% unemployed).
Efforts were made to gather information from regions across the country as well as
diverse groups of learners. Of those adult learner focus group members, half were
white. About a quarter were African American (26%). Hispanics represented 12% of
the adult learners. Seven percent were Asian and 3% were Native American. The ethnic background
of two percent of the adult learners was unreported.
Adult learner focus groups came from a variety of adult education classes. Almost
half (49%) were participating in GED classes, while just over a quarter (26%) were
Adult Basic Education class participants. The other 25% of the adult learners were
involved in other adult education programs such as English for Speakers of Other Languages,
workplace, and developmental college courses.
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And, although data was not collected
on the number of adult learners who came from correctional institutions, three adult
learner focus groups were held in correctional facilities.
Stakeholders
In January and February, five Stakeholder Focus Groups were held in various parts
of the country. The states of Illinois, Massachusetts, Ohio, Oregon, and Virginia
were represented. Most of the 61 stakeholders were in roles involved directly or
indirectly with adult education, training, or employment. Included were state and municipal
administrators/policy makers, college and university personnel, staff developers,
and publishers. Those employers who were involved in the focus groups represented
food services, and the automobile and printing industries.
Data was also collected over a period of time from a "Virtual" Study Group. This
group consisted of a mixture of teachers and stakeholders directly involved with
adult education. There were a total of 21 "Virtual" study group participants, including
researchers, graduate students, adult and mathematics educators from across the world,
who communicated via a closed electronic discussion network.
Teachers
Data for this project was collected from five Study Groups. Forty-one teachers came
from the four states of Illinois, Ohio, Oregon, and Virginia as well as a New England
Regional Math Group with representatives of all the New England states. These adult
education teachers came from a variety of settings: community colleges, correctional
facilities, school districts (Local Education Agencies), and community-based organizations.
Although it appears that adult education teachers represent the smallest group in
the data collection, The Massachusetts ABE Math Standards project was also considered
as a data point. This project involved 22 adult educators from Massachusetts. Through
the NUMERACY Electronic list and The Math Practitioner newsletter, the general membership
of ANPN were also invited to respond to the Study Group Questions.
PROJECT DESIGN
There were many layers of participation from the initial design of the project to
the final draft. The design included a Working Group, a Teacher Study Group for each
region, a Stakeholder Focus Group for each region, a Virtual Study Group, and at
least two Learner Focus Groups per region. Each of these groups pro-
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vided data for the project
with different groups having different levels of responsibility to the project.
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The Working Group consisted of representatives from each of the regions chosen to
participate in the project. The five regions/states -- New England, Illinois, Oregon,
Virginia, and Ohio -- were selected because they already had active Math Teams connected to ANPN. Members of these Math Teams were to be used as a basis for choosing the
Study Groups. Early in October 1995, the Working Group was asked to begin to think
about the make-up of their own Study Group members. Membership would include interested Math Team members and also a key regional stakeholder.
In late October the Working Group held its first of two meetings to begin to map out
the work ahead of them. The Group left the first meeting with a sense of the protocol
for collecting data. The Working Group, with feedback from their Study Group members, actively participated in the formulation of the focus group questions.
Focus Group Questions
Focus Group questions were developed using input from the Working Group and each Study
Group. The Focus Group Questions for the learners and stakeholders were very similar.
For stakeholders, we wanted to know how they themselves viewed and applied math
as well as what math skills they felt were needed by employees.
Each Learner Focus Group began with an ice breaker math autobiography activity where
each learner was asked to respond to the following questions: "Where did you learn
math?" "What was the BEST learning situation for you?" "Who was involved?" "What
was a frustrating situation?" "Who was involved?"
The Stakeholder Focus Groups were asked to think about a time when they were learning
math: "What were you doing?" "Where were you?" "How did you feel?" "What skills
were you using?"
After the icebreaker, Focus Group participants were asked to respond to the following
four questions:
- Please describe a time that you made a decision using amounts, money, measurement,
graphs, or another kind of math. What was the decision? How did you make it? What
did you do? What skills did you use?
- (Learner Groups) What math skills to you need to be successful as a:
parent or family member?
worker?
community member?
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(Stakeholder Groups) What math skills do you (and/or your employees)
need to be successful as a:
parent or family member?
worker?
community member?
- Working in groups of three, please look at the following math topics, then pick the four most important and order them from most important to least important. You can add any other topics that you think are important that are missing. You can also t
ake out any topics that you don t think are important. You should agree on the top four in your group. Be ready to explain why you picked these four topics and why you ordered them the way you did. The topics were: Problem-Solving, Communicati
on,
Reasoning, Estimation, Decimals, Fractions, Percent, Algebra, Measurement, Whole Number Computation, Patterns and Relationships, Statistics and Probability, and Geometry and Spatial
Sense.
- What recommendations do you have for improving basic skills/workforce development/family literacy/workplace education (etc.) math instruction and programs?
In January 1996, the seemingly overwhelming task of convening focus groups began.
Focus groups were run by Study Group members. The protocol required that two individuals
share the task of running each focus group -- one individual to ask the questions
and guide the discussion and the other to take notes. The protocol also specified
that each focus group be tape recorded. Later, all the discussions were transcribed
and forwarded to the product coordinator. All quotes by focus group members were
to be verbatim. (See appendix.)
Analyzing the Results
In each region, as learner focus groups were completed, the Study Group met to conduct
an initial analysis of their data. Topics were noted and discussed by the group,
then coded after consensus was reached.
By early March all learner focus groups had met, the data initially analyzed by the
Study Groups and forwarded to the product coordinator. Also, each region had conducted
one Stakeholder Focus Group and forwarded the data.
The two project co-directors and the product coordinator then met to compile all five
regions' data. Thematic topics sometimes varied from region to region, especially
those in more rural areas as compared to more urban areas. The data was combined
and recoded in order to look at the data as a whole rather than regionally. Using the
coding from the Study Groups as well as our own, we
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found several overall thematic
areas. The five areas were Empowerment, Math Content and Skills, Using Math, Ways
of Learning and Teaching, and System Reform. Each of these five areas were then further coded.
(See appendix.)
The data, after being initially coded, was then entered into one large data pool.
Because we wanted to ensure that whole thoughts were captured, we used a simple
coding method to identify region and focus group number at the end of each data item.
At the beginning of each data item, we coded for major themes. Because a quote often included
themes from more than one area, all data bits were cross-referenced. For example,
a comment made by a learner may have included bits of information focusing on Empowerment, Ways of Learning/Teaching, as well as on the Content area of Algebra. The
quote, therefore, would be found under three areas: Empowerment, Ways of Learning/Teaching,
and Content.
In late April 1996, the Working Group again met. At this meeting they were divided
into groups and given all the data for a particular theme. Since there were five
major themes, but many sub-topics under each theme, we decided to use the four key
math topics that were chosen by focus groups as being most important. (See focus group question
3). These four topics were Communication, Problem-Solving, Whole Numbers and Estimation.
(For complete results of the prioritization, see the Appendix.) Each group chose one topic to further analyze the data from learners, stakeholders, and teachers.
After looking at the focus group data, they were then tasked to review other sources
-- The Massachusetts ABE Math Standards, SCANS, Equipped for the Future -- and come to consensus about what was being said about the topic. Their task was to arrive
at five key points that reflected all the data sources. Each of the five key points
had to be substantiated by specific documentation from the data. (See appendix for
direction sheets.)
Later, each of the regional Study Groups had an opportunity to participate in the
same process but with the five content math skills groupings: Algebra/Patterns and
Relationships, Fractions/Decimals/Percents, Measurement/Time, Probability/Statistics/
Graphing, and Geometry/Spatial Sense.
The feedback from the Working Group and the Study Group was then collapsed into three
process themes and four content themes which serve as a framework for standards.
The three process themes are Communication, Connections/Relevance, and Problem-Solving/Reasoning/Decision-Making. The four content themes are Number and Number Sense, Algebra:
Patterns and Functions, Geometry: Spatial Sense and Measurement, and Data. Number
includes whole number operations, estimation, money, and fractions/decimals/percents. Algebra: Patterns and Functions is a combination of algebra and patterns and
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relationships.
Geometry: Spatial Sense and Measurement includes measurement, geometry, and spatial
sense. Data involves probability, statistics, and graphing.
In addition to the three process and four content themes, a good deal of the data
categorized as mathematical empowerment provided us with a set of affective issues
which continually emerged. These issues include learner self-confidence, attitudes
about mathematics, and beliefs about what one can or cannot accomplish in mathematics.
After much discussion about the title of this section, we finally came to a consensus
that when adults talk about the affective aspect of math, they are referring to their
self-confidence in doing math and their sense of competency around tasks involving math.
Therefore, the section on adults' feelings and attitudes about math has been titled
"Competence and Self-confidence."
Copies of all the primary documents (verbatim transcriptions of the focus groups,
notes of study group meetings, and coding) are on file at World Education.
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THE SEVEN NUMERACY THEMES
RELEVANCE/CONNECTIONS
I remember teachers asking me if a drop of water is falling, at what point will it
pick up the highest speed, the beginning or the end? . . . Like who cares?! This
is the stuff they put in there just to mess you up.
. . . they just started this when I was in 8th grade, but we had a business class
where you'd go in there, and they'd give you a stock book and, you know, they'd give
you so much money and you got to invest in these companies and the guy that came
in there actually used the real paper and he'd tell you how much money you made, how much
money you lost. You were your own broker and that was real neat. You know you invested,
you put so much money in like Nike, for instance, and then McDonalds. You know,
you go across the board and he tells you what you made and what you want to invest it
in and what you lost. Whoever made the most money, they get something like a free
pizza, whatever. And it was using real numbers . . .
Overview
In school, when math was made relevant, the concepts were better remembered. All
too often, though, the school experiences involving math have not been positive or
interesting. They tended to be like those described by the first adult learner.
On the other hand, when adults talk about math in every day life, they tend to perk
up. When adults use math at work, at home, or in the community, they often present
a brighter picture about using math. This is because the math has relevance to them.
They are able to apply the math and see connections. Adults need to see connections
in math -- connections within the domain of math itself, connections to other disciplines,
and connections to real life and work situations.
Key Findings
Math takes on greater meaning and understanding when it is directly applied in the
workplace or in real-life situations. Adult learners provided specific examples
of how they learned math best. "You know when I was young I used to empty out coke
bottles at home and take them in to get candy bars. That's how we learned to make change."
"The best learning was when I am at work using my tape measure." "I worked in
a Chevrolet parts department and learned more math on my job than in school."
Many of the adult learners participating in the focus group discussions felt that
their best math situation was when they learned math at work. This suggests that
the math they learned on the job was directly applicable for them. "The best situation
I've been
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in was when I got a construction job about a year and a half ago -- being good
with numbers and stuff, it made it that much more easier on me doing the job as far
as measuring, weighing. Everything just seemed to flow naturally. I felt comfortable."
Adults see little relevance or connections between math and their everyday living
and working conditions. Adults often ask, What is it used for? about math topics
that they, thus far, have seen little relevance or connections to in their everyday
living and working situations. ". . .And I started off using basically pennies for math,
adding and subtracting which was real helpful. I liked math all the way up till
about high school and then algebra and geometry. I kind of lost it right there.
The way I seen it responding to the same question, What is it used for? You don't use it
unless you're teaching it or unless you're going into some kind of manufacturing
type deal where you're actually making diagrams and stuff like that, but otherwise
it is of no use. . . I use math every day, fractions and so on and so forth, but just don't
use algebra or geometry." The adult learner quoted above shares the same sense of
frustration as this stakeholder: "I remember my father standing over me at the dining
room table attempting to drill into my head the algebra x, y and x + y. I couldn't understand
how anyone could understand it and why anyone would want to."
Adults feel they are more successful when they are able to link any new learning to
something they already know. This adult learner has a clear sense of how to make
math learning more meaningful: "I learn better if I start off with something I already
know, if you go back to the basic formula and link it to an easier way. Because the
more I learn the easier it gets. I can go all the way through from basic multiplication
and division all the way to algebra; if you would just refer back to the other form
of math. Link it to something you already know and you'll get it; you'll remember
it. I mean, you can sit down and read a book. Within 15 minutes, you've lost 80%
of what you've just read, but if you link it to something else, I mean, it is that
much easier to remember."
Textbook math, and particularly word problems, seem to have little relevance to what
adults perceive as math in everyday life. The phrase "who cares" often seems to
be used by adult learners when asked about word problems. "When it comes to math,
you've got to remember the word frustrating. . . It gets so frustrating and it is not that
I don't like it, it is like I don't care how many cookies Sally made. And I don't
care how many were oatmeal and I don't care how many were chocolate chip and I could
care less who ate them. You know, I'll never in my life forget that problem as long as
I live. Who cares?! You cook 'em, you eat 'em." ". . . Start with the bad: the
most frustrating part in math are the word problems in my class because you do them
endlessly. They are senseless. You do not use them later on in life. They confuse you
for days.
Adults' real math skills often don't show when they do meaningless word problems.
Adults often actually use math successfully in their daily lives, yet fail to see
any con
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nections to word problems presented in the class. An adult learner explained
this lack of connection: ". . . Math was good for me. I always liked it. English was hard.
I wanted to be a draftsman. I went to school at PCC. I chose building things.
I succeeded, I got into that work, hurt my back. I used math for building. After
my back I lost it [the math]. I feel good now -- I think some of what I knew are really
coming back. Reading is hard for me -- word problems. Working with math is one
thing. Reading with math is a whole other."
Implications for Teaching/Learning
Teach math in the context of real-life and workplace situations. "For all adults,
mathematics learning should be connected to real life situations. (The Massachusetts
ABE Math Standards, p. 32) When math is taught in context, adults understand that
there is a practical application for that skill. "The thing that I regret when I didn't
learn math was how to use a calculator. You know, problems with subtraction; how
to use it in life. I can add but when it comes to things like when I want to cash
my check or write checks, I've got to be able to subtract, etc.", explained an adult learner.
Several stakeholders echoed the same sentiment about relevance and connections.
"Whatever skills are needed, there needs to be relevance to life and application
across activities." ". . . One of our hiring practices is to run through a simulated
production. You need to develop interesting programs, have a cash register in the
classroom, do medical calculations, simulate real life in the classroom. Textbooks
get boring. Come visit our plant. Make it as real as possible." "Mathematics should
be taught as an experience in context, not as a lecture. To this end mathematics
needs to be taught using REAL problems, not textbook reality."
"Many adults already do complex math on their jobs and in their everyday life. it
is important for math teachers to use this as a basis for developing new ideas or
extending old ones to different places." Carrying this teacher suggestion one step
further, teachers need to become more knowledgeable about the world of work in order to offer
relevant math curricula.
Use learner-centered approaches to teaching to ensure that learners see the relevance
of what they are learning. Math learning for adults should be relevant to their
own personal goals, whether it be to attain a GED certificate, a job, or whatever.
"I think a lot of times people use the excuse 'Well, I don't need to know this.' But
maybe just pointing out to them why they need to know it, then it becomes more valuable
to them to know it." (learner)
Adult learners need to have a voice in what is taught in the classroom. A teacher
suggested, "The student has to 'buy into' the math item/concepts in order for her/him
to internalize them." The teacher may think something is important for the adults
to learn, but unless the learner sees the relevance to his own life, s/he finds little
value in the
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topic. This stakeholder further explained: ". . . To get a hook in
them [the student] and bring them along. Because I think once they see the relevancy
and understand what they can do, they will pursue it on their own. It becomes a mind
set for them to proceed to the higher levels of mathematics if they desire to. But
right now it is difficult. They have such low skills and they have been functioning,
at least in their estimation, fine in society. You know, if they don't have a checkbook,
'Why do I need to do it? I'll never have one.' They pay cash for their needs, everything
is done on a cash basis. So, in most cases, until they have a need to proceed to the next level or see that there is something beyond the level where they are already
at, they come with limited experiences in many cases." Whether teachers "hook" the
students or get them to "buy into" the math, adults will find the relevance when
the material is relevant to their needs and goals.
Use an interdisciplinary approach to teaching. Math should be an integral part of
other content areas. "Integrate math instruction with other literacy development
-- use reading and writing in teaching math and show that math content/skill (e.g.
reasoning, problem-solving) are vital to making sense of the world -- in other disciplines
and in the workplace and as citizens and parents." (teacher)
Link new math learning to previous learning. Linkages should be made with other math
concepts and skills as well as with other prior knowledge. A stakeholder suggested,
"We need to connect prior knowledge of the learner with formal instruction in math."
Not only should the new learning be connected to prior learning, but there should
also be a connection between knowing how to perform a skill and being interested
in performing that skill. "Shouldn't the whole thing, if you say you're not interested
in something, wouldn't it be easier to try a different style to get you interested in
that kind of stuff, from the teacher's perspective, to find out why you're bored
with it. You know, then go about a different teacher style that would make it easier
for you to learn, because once you start learning stuff, you get interested in it because,
I mean, the only reason you're not interested in it is because you don't know how
to do it. I mean, if you don't know how to do it, speak up and then there will be
someone there to help you. I mean, find a different way to make some difference and then
you won't have that problem, math especially, you've got to link it to something
else." (learner)
Teach concepts before rules. For example, teach the concept of area of a rectangle
as a counting of square units. The grouping of those units into rows and multiplying
units per row times number of rows is a shortcut that can be summarized in a formula, but formulas are easily forgotten with no connection to models, examples, experience,
etc., offered one teacher.
Help adults see the relevance of learning by seeing the "big picture". In SCANS'
terms, this would be considered the "Systems" competency, the ability to understand
how all the small pieces work in relation to the total system. When adults are shown
17
how math skills are interconnected with one another, they begin to see relationships and
the relevance of what they are learning. This stakeholder explained how she learned
quicker by having the big picture: ". . . When we ran a business, I did most of
the accounting. I had no accounting background whatsoever, but when it is your own money,
you learn real fast. We were running a business and I was the office manager. But
he [husband] taught me how to do that by giving me the big picture. In a very short
time, I learned the concepts. You didn't have to go back and take an accounting class
because when you are dealing with it in real time with real things that have real
consequences, you learn pretty fast." And this stakeholder expressed his concern
about seeing the whole puzzle: "One of the concerns I've noticed . . .[is the] need to learn,
of seeing the whole puzzle before we put it together. [There] will always be a dilemma
taking a person from the known to the unknown. How can you show them the unknown in its entirety before you get there? What we are doing with people now is we are
telling designers before you design something, come and meet with the binders, meet
with the press person, meet with the pre-press person, as a team, so that you understand what the limitations and possibilities are. As far as math class, this is the dilemma
that we see in our industry -- taking people from the known to the unknown and giving
them a picture of what it's going to be when they get there."
Support teachers in making their classrooms more relevant and connected. Before the
curriculum can be changed and any of the above strategies implemented, teachers need
to be retrained. A stakeholder offered this recommendation: "We need to teach teachers to teach math concepts and connections rather than rules and to convince their
students of the importance of these. Some of the rules are not that important in
life, but their development is and transfers to solving problems on the job and in
life."
Connecting to the Four Purposes
Learner 1, "You use math almost every day of your life, everything you do practically."
Learner 2, "You don't think about it that often; it's just there.
Learner 1, "You just do it."
The conversation between two adult learners above illustrates how math skills play
a critical role in literacy. Whether it be to access information, to have a voice,
to act independently, or to prepare for the future, adults connect math to their
every day lives -- whether it be at work, at home, or in the community.
When adults see the relevance of math, they are able to use it to their benefit.
They are able to understand the wealth of information surrounding them and know what
pieces of information they need to access in order to solve problems. When adults
are able to make connections to everyday life and work situations, they are better equipped
to express their opinions and make informed decisions as this adult learner explained:
18
" As a business man in a community, I use math everyday to run my business. Without it, my business would come to a standstill. I use it from controlling my inventory,
receivables, payables, and accounting. It is probably the most important aspect
to understand to run my business. Without having any math skills, it would probably
be impossible to run my business."
19
PROBLEM-SOLVING/REASONING/DECISION-MAKING
My dad owned a bakery for twenty years about the late 70s, early 80s. Starting in
the mid 70s he started saying that his employees, the young kids that he was hiring
as helpers, baker apprentices -- wasn't much of an apprenticeship program. He said,
'They can't think anymore. Nobody knows what to do when something goes wrong. They just
do whatever and go, 'I just followed the instructions.' What he was saying was they
needed to predict. If it was really humid, they needed to know that the bread needed to spend less time in the steam box and they needed to know they needed to change
ingredients by adding things just slightly. My dad did not know how to articulate
it, but I was among those who could think. What I learned was that certain adjustments,
certain ingredients needed to be added, not all of them. My dad didn't know which
ones to tell somebody, but he could tell that someone was not taking all the raw
data in and making a judgment on all the raw data: increasing time, predicting
what was going to happen because the temperature was 75 degrees and it was 80% humidity and the
bread was going to have to spend five extra minutes in the steam box. You have to
change ingredients, less salt. He couldn't articulate it, but nobody who was working
for him during that time period could interpret those changes critically. You talk about
critical thinking. But it is basically day to day understanding of adjustments that
is as important as knowing how and when to use it.
Overview
To the stakeholder above, it is clear that problem-solving, reasoning, and decision-making
are three very interconnected processes adults engage in continuously, whether they
are using numbers or words. SCANS classified problem-solving, reasoning, and decision-making under foundation "Thinking Skills": "Creative thinking, making decisions,
solving problems, seeing things in the mind's eye, knowing how to learn, and reasoning".
The SCANS Report further defined the higher order thinking skills of problem-solving,
reasoning, and decision-making: "Problem Solving. Recognizes that a problem exists
(i.e., there is a discrepancy between what is and what should or could be), identifies
possible reasons for the discrepancy, and devises and implements a plan of action
to resolve it. Evaluates and monitors progress, and revises plan as indicated by
findings. Decision Making. Specifies goals and constraints, generates alternatives,
considers risks, and evaluates and chooses best alternatives. Reasoning. Discovers a rule
or principle underlying the relationship between two or more objects and applies
it in solving a problem. For example, uses logic to draw conclusions from available
information, extracts rules or principles from a set of objects or written text, applies rules
and principles
20
to a new situation, or determines which conclusions are correct when
given a set of facts and a set of conclusions."
Reasoning is a key step in problem-solving and decision-making. Adults use reasoning
to analyze information in order to solve problems which, in turn, allows them to
make reasonable decisions. One stakeholder defined reasoning this way: "Reasoning.
We thought in daily situations you probably see math statistics and math numbers. You're
seeing different information that you need to reason and draw conclusions based on
this: Is this a good sale, not a good sale? and so forth depends on the whole reasoning process. Looking at graphs and charts, looking at your paycheck and whatever --
just being presented with information and attempting to draw conclusions."
Key Findings
Math skills are integrated in the problem-solving and decision-making processes.
Although it is clear that math skills are integrated in the problem-solving and decision-making
process, the skills needed vary from problem to problem. One instructor stated: On one hand we all agree that people should learn to problem solve, reason, communicate,
etc. These might be called process skills and all adults do these all the time;
we all strive to improve in these areas, whether or not we are in adult ed. classes. Indeed, these skills are not particularly mathematical skills, but rather skills
that cover all domains. On the other hand, a person needs meaningful information
and knowledge to be able to solve problems, reason about, and have something to communicate. Therefore, I guess I see these process skills as the goals of all education
and learning, no matter what the domain. To me, our challenge here is to prioritize
the specific mathematical content that is necessary and useful to support the kinds
of reasoning, problem-solving and communicating that people need to do at the end of
the 20th century.
The math skills needed to solve problems and make decisions are integrated throughout
the process, with more than one math operation generally being required to come to
final decisions. The following adult learner provided an example of how integrated
math skills are in the problem-solving process in his role as citizen: Problem-solving,
when working with the school department for our children, doing fund raisers for
the sports programs our children were involved in. Being able to use fractions,
multiplying, adding, subtracting, knowing your math formulas to help build playgrounds, churches,
and homes.
According to SCANS (p. xvi), "Virtually all employees will be required to maintain
records, estimate results, use spreadsheets, or apply statistical process controls
as they negotiate, identify trends, or suggest new courses of action. Most of us
will not leave our mathematics behind us in school. Instead, we will find ourselves using
it on the job, for example, to reconcile differences between inventory and financial
records, estimate
21
discounts on the spot while negotiating sales, use spreadsheet
programs to monitor expenditures, employ statistical process control procedures to check
quality, and project resource needs over the next planning period."
Problem-solving is a process that includes seeking to understand the problem, and
figuring out what information and math skills are important to use to solve the problem.
From comments made by learners, it is clear that the process of solving problems
requires an understanding of the situation. "Communication and problem-solving seemed
more relevant because, obviously, you have to communicate to understand the problem
. You have to know how to do problem-solving before you solve a problem. You
have to understand the problem in order to solve it.
Adult learners also know that without this understanding, the problem cannot be solved.
If you don t know what you re doing, you can t solve it. You can know a formula
or how to add, subtract, multiply, or divide, but don t understand and then you can
t solve it."
Adult learners shared specific examples of how they defined problems and determined
how to go about figuring out how to solve them: "I had bought an old truck. I was
restoring it. I had to average out how much I make a month with how much I could
put into it. I didn't anticipate going to another job and getting paid less. All my numbers
was from when I was making more money. This was when I was living at home when I
bought this truck. Then a couple months later, I got my own apartment. I didn't
average it all out right, so I am still working on that truck. I used a lot of estimation
about what I think I'm gonna need a month versus what I got to spend on that truck.
I know I need to do the rest first, the basic necessities, and then how much I can
play with, what I got left to entertain myself."
One employer summed up the problem-solving process this way: Our philosophy in the
workplace is whatever it takes . We will use whatever it takes to make it work,
we will try it. And to be open to try.
It is important for adults to have a repertoire of strategies and tools to solve problems.
And then with our problem solving techniques that we use, either textbook, brain
power, calculator, or whatever, then we can go through all the different equations, all the different geometry, algebra, whatever else you have, and come right down
to the decimal point or what you need. Maybe you won t have to figure it out.
Maybe you could use a calculator or computer, use some other source to try to help
you achieve your goal as far a solving your problem. You could use another kind of tool or
an unconventional method. That s when [you] sit down and you think about how you
re gonna do this. Are you gonna add it up? If it s too big to add up by your hands
or pen, you gotta get out a computer or calculator or whatever. "I use measurements
in our cooking classes every day. You have to know how many tablespoons are in a
gallon, how many teaspoons are in a tablespoon. If he give you six gallons, you
gotta know
22
how many quarts. You can use a calculator if you want, but you have to know the
formula. If the teacher gives you a recipe, you have to make a decision if you are
gonna convert it down to a smaller amount. If you have a cake mix, you have to decide
if it's cheaper to make it from scratch." The learners' comments above, along with those
of other learners, instructors, and stakeholders, point out that the process of problem-solving
involved a variety of strategies.
The use of calculators came up fairly often in discussions around problem-solving.
One stakeholder offered his insight into the use of calculators:"Just to be able
to use the calculator, to me you're doing a form of problem-solving to know how to
manipulate the numbers."
Learners often had strong opinions about the use of calculators as a tool for solving
problems. Some felt calculators were a useful tool and could be used to access information
in order to solve problems and make decisions. "I just want to make a comment about calculators: math is my hardest subject. I don't trust my own adding and
subtracting math. With a calculator you press certain buttons and you know you're
going to get the correct answer." "Let us bring in calculators to help us know how
to use the calculator with the problem. We should all learn how to write down the problem
first." "You need calculators to keep up with the pace of life."
Other learners, however, felt that calculators should not be used."Take away computers.
Everybody relies on the calculator and computers and they can't figure out anything
without them. So I think you ought to get back to basics to make sure people do
understand instead of 'push this button'. " [Response by another learner] "I agree.
I've seen too many times when people just type in a bunch of numbers, hit "enter"
and whatever the computer prints out, this must be the truth because the printer
just printed it out. They may have typed in a wrong number or one of the formulas may have
been programmed wrong. You have a wrong number and people don't take the time to
work through by hand to make sure it's right." "I think a lot of people have gotten
lazy. They don't do it in their heads; [they] use a calculator." "I got four kids. If
they can't do it [math] on paper, I'm sure not buying them a calculator."
One of the five competencies spelled out in the SCANS Report is "Technology". Employees,
in order to be successful on the job, need to be able to select the appropriate technology
and apply technology to different tasks.
Problem-solving and decision-making often involve teamwork. On the job and in daily
situations at home, problems are solved and decisions made with the advice and input
of others. While in school situations teamwork is not often encouraged, at work,
at home, and in the community, individuals must work together to solve problems and move
forward. According to SCANS [pp. xviii-xix], "More and more, work involves listening
carefully to clients and co-workers and clearly articulating one's own point of
23
view. Today's worker has to listen and speak well enough to explain schedules and procedures,
communicate with customers, work in teams, understand customer concerns, describe
complex systems and procedures, probe for hidden meanings, teach others, and solve
problems. "
Parents, workers, and community members use problem-solving and reasoning to reach
decisions. "Problem-solving for me would be something on the floor that we make,
you know by using one of my ... like a caliper to measure it and if it is over by
so much, you know you fix it. I send my part somewhere and they have problems with it, saying
your gauges, you know this part won t screw into this part. You know it s got to
be fixed so that would be problem-solving."
Being able to problem-solve successfully in the workplace gives workers more confidence,
which, in turn, gives them more of a voice. In the manufacturing area, I was 20
years old and three top managers couldn t figure out an algebraic formula and one
of them very jokingly said, Here, see if you can solve this, as they all laughed at
the thought. However, in one minute, I solved the problem to determine their daily
production which they couldn t do.
Adults use problem-solving strategies as parents to survive . They need to maintain
budgets and comparison shop. "When I was married, my wife was smart, so she took
care of the bills. When I got divorced I took my boys. I was off work for a year
and a half, plus I had bad debt for credit cards. Made $6 an hour with two kids. Had to
feed them and pay some on my bills. So basically, I'd figure out how much food we'd
need for the week and how much gas it took to get to work and pay for that first.
Then I'd pay some on the bills. When the kids needed clothes that came first with the
food and gas. I didn't have a checking, savings account. I can't spell or do math.
I'm really crappy with numbers, I always get them turned around so I have almost
always just paid cash for everything." "I had $45. The decision I had to make was to buy
shoes or buy meat and put the shoes on hold. I came to the decision by telling myself
that my feet weren't dragging on the ground but with no food, my children and I would've had growling stomachs. I used addition to decide how many packs of meat I could
get with my $45."
Adults also use problem-solving strategies to better understand how their money is
manipulated. They feel that understanding what is REALLY happening with their out-go
will help them better budget in order to create the best environment for their families. I bought a house last year. The price of the house sounds pretty inexpensive,
but when you add up the interest on it . .. the points they charge you, the closing
fees, the maintenance . . . it s like on a 30-year loan, you end up paying three
times as much as the house is worth . . . You gotta compute simple interest, compounded interest,
all that sort of stuff . . . First I took what I made a month. I took an average,
then I deducted all my expenses, then I had a budget saying what I could afford to pay a month . . . simple math . . . only you divide that if you have a roommate or
whatever . .. just basic plan-
24
ning and basic math skills . . . averages . . .When
they first tell you, just put down 5% or 10%, then pay this much a month, you take
it like that and you don t know what it really costs you. You gotta figure everything else.
That s what math does, it makes you organize, makes you think in a certain manner.
Adults, in their role as citizens, have to solve problems and make decisions using
numbers. Even when you are dealing with your bills, when you are looking in your
checkbook, if you have two bills then one of them is going to be late. You have to
decide which one you want to pay first. Like for instance, we had a phone bill and a credit
card bill and one was going on late and you don t know which one. You got to make
a decision between the two of them based on how much money each one is, how much
you got in the bank and how much money it gives you to spend for the rest of the week.
(Response by another adult learner) You can always let your phone bill go because
they don t charge you nothing. Sometimes you have to do the math when you go to
buy something. Like you have to ask yourself, like how much you gonna pay for this, if you
can afford it or not. If you say like when you buy the furniture, or maybe car,
or maybe even TV or something like that, you have to pay like a month. Especially
like with the car, you have to know how much you gonna pay every month, how much interest is
gonna be on it, how much tax you gonna pay for that car, and you have to add everything
on it. If you say how much you gonna pay on insurance . . . say like $2000 for the
car, like $100, if you are a young male, so it's gonna be $300. How much you gonna
spend for gas, if any problem happens, say like anything broken, if there is no guarantee
on it, how much you gonna pay for that. How you gonna afford that car or not. The numbers are very important, if you can afford the car but you cannot afford the
repair, that means you don't have the car. If you cannot afford the insurance, that
means you gonna have a problem one day with that. Like everything, like furniture,
you have to remember if any problem happens, say like if you ... the job, you gonna afford
these things after that? The owner is gonna ask you to bring it back, he needs his
money in about a month; he gonna ask you in a specific time, so you have to prepare
for any problems that happen in the future."
Another example of using math as a community member is this learner's experience working
with children at a town gathering: "Well, you use it [math] when you to Town Meeting
with different groups to raise funds. I've gone there to help 6th graders raise
funds to go on their Great Adventure. They sell coffee, hot dogs, that type of thing
right during Town Meeting. They buy a cup of coffee off a kid, a hot dog, a lot
of different things like that. The kids each brought in so much of everything.
They had to figure out so many people in town, and how many hot dogs they were going to need
for that and how much bread. Then they had to decide who was going to work what
shift. But, of course, you would have to figure out to make sure they figured it
out right. And you had to make sure there was enough people on each shift because Town Meeting
ran all day. We started out with a certain amount of money. The two kids, when
they finished a shift, had to check and then report to the next shift exactly how
much they had."
25
Implications for Learning and Teaching
Embed math content skills in processes like problem solving, reasoning, and decision
making. Processes such as problem-solving are viewed as more than just a topic to
be covered in an adult education classroom. According to one instructor, How can
anyone function if they can not solve problems? This is more than just the word problems
in a book. This is the real understanding of a work problem or a community problem
that needs to be solved. Making mathematics real, not pseudo-real, is important.
Other instructors echoed the same philosophy: "Students should be able to apply problem-solving
skills including mathematical modeling to solve problems found in life situations.
"Good to do 'projects' like floor plans that have interest, life value, and problem solving all rolled into one." "Teach math in a meaningful context of relevant
problems, and recognize basic skills and problem-solving skills as mutually reinforcing,
and therefore, encourage students to identify or frame and solve problems themselves."
A manager shared what one company is doing to integrate math and thinking skills:
"When I look at Motorola most of the training that is going on now is around problem-solving
and something like that and the math is embedded in the material. The people don't even know they are teaching math. You know the anxiety that comes with a math
class is less and they come through with the actual application."
Integrate reasoning and problem-solving in all teaching. Even when teaching basic
skills, such as reading, writing, and math, higher level thinking skills such as
reasoning, problem-solving, and decision-making should be incorporated into the lessons.
According to SCANS (p. 27): "Proposing an effective menu requires creativity and mental
visualization. Learning how to use a spreadsheet program -- by definition -- cannot
be accomplished without knowing how to learn. Recommending equipment requires decision making. Developing a training plan that does not upset production schedules requires
problem-solving and reasoning skills."
One stakeholder offered her philosophy of the higher level skills: "On one hand we
all agree that people should learn to problem solve, reason, communicate, etc. These
might be called process skills and all adults (kids, too) do these all the time;
we all strive to improve in these areas, whether or not we are in adult ed classes. Indeed,
these skills are not particularly mathematical skills, rather skills that cover all
domains. On the other hand, a person needs meaningful information and knowledge
to be able to solve problems, reason about, and have something to communicate. Therefore,
I guess I see these process skills as the goals of all education and learning no
matter what the domain. To me, our challenge here is to prioritize the 'specific
mathematical content' that is necessary and useful to support the kinds of reasoning, problem
solving and communicating that people need to do at the end of the 20th century."
26
SCANS suggests, "Reading and mathematics become less abstract and more concrete when
they are embedded in one or more of the competencies; that is, when the learning
is 'situated' in a systems or a technological problem. When skills are taught in
the context of the competencies, students will learn the skill more rapidly and will be more
likely to apply it in real situations." "Choosing between teaching the foundation
and the competencies is false; students usually become more proficient faster if
they learn both simultaneously. In sum, learning in order 'to know' must never be separated
from learning in order 'to do'. Knowledge and its uses belong together." (SCANS,
p. 20)
Provide opportunities for learners to work in groups. Learners do learn from one
another, as they have readily testified: "Group work is helpful. A lot of times
you get the way other people think." "I really like the math class. All of you are
so helpful. We all learn so well together." "If you get into a class where you have more people
in the same situations as you are, you seem to learn a little bit more. You sort
of feed off each other. Somebody else comes up with an idea that you're having trouble with. You can sort of learn from that. I find that really troubled groups together
or really quick groups together works a little bit easier for me."
Working together in groups also gives learners opportunities to hone their personal
qualities, an important part of the Foundation Skills (SCANS. page xviii): "Personal
qualities: Displays responsibility, self-esteem, sociability, self-management, and
integrity and honesty." Not only is interaction with others a key foundation skill according
to SCANS, but it is also one of the five major competencies needed in today's workplace.
The SCANS Report considers Interpersonal Skills to be key for employees to succeed. "Interpersonal skills. Competent employees are skilled team members and teachers
of new workers; they serve clients directly and persuade co-workers either individually
or in groups; they negotiate with others to solve problems or reach decisions; they work comfortably with colleagues from diverse backgrounds; and they responsibly
challenge existing procedures and policies."
Skills such as those listed above are not developed overnight, nor are they simply
"picked up". Learners need to interact with their peers in problem-solving teams
within the classroom environment. Adult learners need to work in group situations
in order to learn to check reasoning and take advice and suggestions from others. According
to The Massachusetts ABE Math Standards, "Genuine respect and support of each other's
ideas is essential for learners to be able to explain and justify their thinking
and to be able to understand that how the problem is solved is as important as its answer.
In all adult basic education math settings, the development of critical thinking
skills is crucial. Statements should be open to question, reaction and elaboration
from others."
(p. 30)
27
Connecting to the Four Purposes
Problem-solving and reasoning in math are vehicles for independent action for adults.
Adults want to be able to find information, analyze it themselves, and then make
reasonable decisions based on the data. Adult learners and stakeholders alike were
able to provide specific examples of how they use problem-solving and reasoning to make
decisions in their roles as parents, workers, and citizens. This adult learner's
problem-solving enabled him to make a realistic decision about whether to rent an
apartment or not: "A couple weeks ago, I was looking at an apartment. It was one right near
my house. I compared how much he wanted for a month, plus all the utilities and
add in the security deposit. I compared it to my income and found that I would have
about 50 cents at the end of each month to live on. So I just decided not to rent it.
I used addition and subtraction and all that good simple stuff."
Problem-solving and reasoning in math are needed for access and orientation. Without
these critical thinking skills, it is difficult for adults to sort through the myriad
of information surrounding them and make sense of which pieces of information they
need to make decisions. For example, listen to this learner talk about all the information
that he has to work through in order to determine where he is credit-wise: "If you
get a certain percentage of interest from the bank, which is 3% or 4%, you could
put it in a long-term mutual fund or CD, you get this much. If you have bills, you
pay 20% interest. Is it better to pay off the bills and not save anything, or is
it better to save the money? You think, what's the prime rate or the interest rate.
You need math every day to survive. I mean when they say they're gonna hike up your phone
bill so much percentage points or you get a refund from the water company for this
much; I think it's all based on numbers." And this learner explained how he gathered his information before making decisions about a family vacation: "I'm leaving Saturday,
going to Florida, so I called every airlines to get the cheapest rates to Florida
and then I did the same thing with the hotels. Then I'm taking my two grandkids
with me so I had to multiply by three."
28
COMMUNICATION
Math for me is the same as for W. I grew up with math and numbers. Raised in
the South, being poor, we had a love for money, a need for money, so it was kind
of natural. When I became older, math became like a second language. I relate to
it like a second language. It's automatic. I've never had difficulty in math or anything
pertaining to numbers. I've always loved it. I've always found great success and
accomplishments dealing with math and numbers.
Overview
As the learner quoted above affirms, math is a language. Mathematical communication
is an overarching process which includes understanding, expressing and conveying
ideas mathematically in order to reflect on and clarify one's thinking, to make convincing arguments, and to reach decisions. As noted in What Work Requires of Schools, a
SCANS Report for America 2000, arithmetic and mathematics are essential basic skills
and part of the foundation each worker needs to be successful. Effective workers
must be able to "interpret and communicate information" and "communicate ideas to justify
positions". In the workplace, much of this information and many of these ideas are
mathematical.
Mathematical communication can occur in any relationship and context. In the ABE
setting, communication happens among learners and between learners and their teachers;
at work among workers and between workers and their supervisors; at home among family
members and between children and their parents; and in the community among individuals
and between community members and public officials. Good mathematical communication
is like all other effective communication requiring listening, speaking, reading
and writing skills along with interpersonal skills.
The adult learners who participated in the ANPN Focus Groups were asked to prioritize
different mathematical topics; they rated communication as the most important area.
Stakeholders representing public and private employers, k-12 and higher education
practitioners and administrators, professional development providers, ABE math teachers,
workforce development officials, educational publishers, and researchers and other
academicians also included communication among their top four priorities. (Please
see the appendices for further information.)
Key Findings
First and foremost, communication is essential for understanding. As stated by learners,
"We need communication so that we can understand and be understood;" "You have to
communicate in order to listen and understand with somebody else;" "We
29
looked at
communication as being the very basic, the very foundation; and "Communication because
it's the first thing you have to understand. If you don't understand the problem,
if you don't understand the words ... then you really can't solve anything."
Communication provides the foundation for learning in school and in life. "If you can't communicate, you can't learn." "So we thought communication was very important
in being able to understand and making progress in learning basic skills, being able
to understand the other things on the list (other math topics)." "Without the communication
of math--no teaching or learning happens."
Communication includes knowing when and being able to ask for help in the ABE math
classroom and in life in general. As stated by learners, "First is communication.
Let's say you're going into math ignorant. You gotta be able to communicate or
to get you a grasp on the other stuff. If you don't know how to talk and ask for help, you'll
never get nowhere." "People can go through life not asking for help and they never
get nowhere. You can't just expect everybody goes into a situation knowing something
about it. They're gonna have to communicate. They have to get some help from others."
Communication, in math as in other aspects of life, is the bridge to finding and exchanging
ideas, to identifying problems, and to seeking and finding solutions to these problems.
"Communication--we felt it necessary in being able to do reasoning, to discuss with others, to help identify problems ..." "We didn't think there could be any
exchange of ideas if we couldn't communicate." "To me this is the bridge--communication.
If you can't articulate how you got to where you're at or what it means ..." "Basically you need communication to come up with new and better ideas."
Communication is essential to working collaboratively at home, in school, at work
and in the community. "You need communication to get along with one another." While
some learners and stakeholders focused on the relationship between learners and teachers, others talked about communication across all aspects of their lives. "We picked
communication first because you have to communicate with the teacher and, like I
said before, I tell the teacher what I don't know so she can work with me helping
me understand what I don't know." And "The teacher has got to get through to you." But also:
"Without communication, you can't do anything." "You have to communicate with one
another to solve problems." "We based ours on everyday life. Communication is the
key." "...being able to talk openly about and convince somebody else ..." "Communication--a
chance to come together, present opinions, organize ideas.
Communication is the link that makes other math skills effective. As one stakeholder
notes, "The other big piece that we do is that it's fine and dandy to do all this
math but then you had better be able to communicate what you've just done to some-
30
one
else. It's one thing to do it on a piece of paper and come up with a number but part
of that becomes the ability to communicate what you've just done."
When asked to give examples of decisions involving math and to think about the math
skills they use as family members, workers and citizens, both learners and other
stakeholders articulated examples of work, family and consumer-related situations
in which mathematical communication skills are important. For example, talking about buying/selling
a house, one learner states, "... communication is very important. You know, say
for instance, you are working on a mortgage company, and then you're communicating with other salespersons which is selling the house and then they told you, well it
is this much rate and you look in the papers and it is a different percentage rate
on that. And so you communicated with the real estate person. Sometimes they want
you to pay all the points for the house and you say, 'Why don't we pay half and half?
Half to the seller and half to the buyer.' And that is communication." And another
learner responds, "Because if you don't communicate with the people wherever you
go, whatever you buy, you don't do it. You have to communicate with everybody.
Mathematical communication is also important within the family. "You know, the parent
who can explain to the child why there isn't the extra money for the pair of Nike
tennis shoes that cost $150 a pair. They need to learn a little bit about budgeting.
And so putting that in communication, putting some of that together with the other
basic skills." (stakeholder)
Talking about work, another stakeholder states, "Consider, for example, that people's
ability to understand and communicate about (rather than compute) what an average
or percent is, understand notions of sampling and representation, make sense of and
make choices about probability and risk (e.g., likelihood of accidents, errors out of
xxx products), and anything that is decreasing, increasing, or changing its magnitude
..."
Mathematical communication--the representation of a problem in mathematical language--also
happens in the "other" direction, especially as individuals interact with technology.
As explained by a teacher who participated in the Virtual Study Group, "But beyond this, as technology becomes more pervasive, it is necessary for one to be
able to distill the elements of a real situation into a mathematical expression-the
universal language, if you will. In order to communicate the problem to any one
of our technological aids, it first must be translated to symbols and then the results from
the machine must be interpreted in light of the situation. (Reminds me of the fact
that although the new dishwashers can scrub the pots and leave the glassware gleaming,
they cannot load or unload themselves.) I believe it was M. who made the suggestion
that from the beginning of the discussion of percent, symbols representing the words,
concepts, and operations could be included."
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Implications for Teaching and Learning
Increase the focus on mathematical communication. Teaching mathematical communication
is integral to the success of math reform efforts, and both learners and stakeholders
recognize the need to increase the focus on mathematical communication within the
ABE setting. For example, one learner states, "More materials and better communication.
More work and more materials one-on-one, verbal explanation, groups, slides, visual
effects. You know, different ways that everybody can understand, different ways.
If you don't understand the problem, you need to go to someone else to communicate
the problem in other ways. A lot of people, they don't know how to read, so maybe
pictures. Yeah, that makes it interesting."
Another learner, recognizing that talking about math has increased her comfort with
math states, "I would like the opportunity to do like we're doing today, sitting
here in a circle and discussing math. The more you can talk about something in a
group, the more comfortable you become. Then it becomes more like other classes I have that
I like. I never thought math had anything to offer me. It just seemed like a teacher
up there at the board, with lots of subjects that weren't relevant to me, and there's no interaction. To be able to relax and learn that math is just like any other class."
Stakeholders also recognize that communication is key. "Most ABE/GED classes, because
of their rolling enrollment, are taught on an individualized basis. In order to
function in the 21st century, our students need to communicate mathematical ideas
and use them to solve non-traditional problems. To accomplish that we need classrooms where
discussion is a key part of instruction.
And, "I agree with much of what P. said. Open-entry, open-exit individualized instruction
may not be the culprit, however. Students always working by themselves with no requirements
that they communicate their problem solutions to anyone else might be. Pairings and small group work seem viable alternatives to whole-class instruction
when it is difficult to find many students attending regularly and learning at the
same rate." And "Use a variety of approaches, models and manipulatives and have
the students involved in talking about their work with each other on a frequent and regular
basis."
Encourage good mathematical communication for work, home and community situations
through group discussions. As noted by one teacher, "Good communication. Math
should be taught using a well-defined vocabulary of math terms so that what the teacher
believes is being taught is what is being received by the student. This should involve
verbal and written feedback from the students to confirm that they understand and
can express to others what they know. As a skill necessary for future employees,
students should be able to express mathematical ideas and concepts orally and in writing.
Also very few employees will work totally by themselves. More and
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more, work will
involve listening carefully to clients and co-workers and clearly articulating one's
point of view."
Connecting to the Four Purposes
Mathematical communication mirrors the four purposes of literacy identified by learners
in Equipped for the Future. Mathematical literacy, or numeracy, being able to understand,
interpret and express ideas mathematically, is important for access and orientation. Being able to communicate mathematically to others what one thinks and feels
is math as voice and numeracy is a vehicle for independent and collective action.
And, finally, the ability to communicate mathematically is one, primary bridge to
the future.
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NUMBER AND NUMBER SENSE
"Workers can't be afraid of numbers!"
GM Parts Plant Manager
Overview
Being able to handle numbers comfortably and competently is important to adults as
parents, workers and community members. This competence relies upon having developed
a kind of "number sense" about whole numbers, money, fractions, decimals, and percentages. Number sense includes calculation skills with numbers as well as a sense of number
and operation and an ability to appropriately use estimation, mental math, computation,
calculators or other tools. The learners, teachers and employers that were in the ANPN focus groups had lots of opinions about NUMBER. Learners ranked whole numbers,
estimation, and fractions/decimals high as important math topics (3rd, 4th, and 6th);
stakeholders concurred, ranking whole numbers fifth and estimation fourth. (Please
see the appendices for more information.)
Key Findings
Whole number computational skills are necessary but not sufficient. "If you don't
know whole numbers, that is the basis that everything else is built upon... I mean
how can you do any other type of thing if you can't do the simple whole number computations? ... Right there you're starting with one foot in the hole... " "I think whole
number computation is most important... it's a basic fundamental thing. For example,
there is a tree. If the root is weak, the tree's life will not be long. So we have
to know whole number computation first, because it's basic, the root of the tree."
The adult learners quoted above share a belief held by learners, educators and employers
alike about how important it is for adults to be solidly grounded in whole numbers.
What should be included in the root of that tree? What's the nature of a good solid understanding of number?
The Massachusetts ABE Math Standards states: "To be efficient workers or consumers
in today's world, adults must have a strongly developed conceptual understanding
of arithmetic operations as well as procedural knowledge of computation and number
facts. They must be able to perceive the idea of place value and be able to read, write and
represent whole numbers and numerical relationships in a wide variety of ways. Simple
paper and pencil computation skills are not enough. Adults must be able to make decisions regarding the best method of computation (mental math, paper-and-pencil, calculator/computer)
to use for a particular situation." (p. 38)
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But even deeper in the root of the tree, are some very basic understandings such as
sorting and classifying, comparing, ordering, counting and pattern recognition and
development. These "pre-number and pre-operational" understandings apply to all numbers.
One adult education math teacher reflected on how important it is to be able to do
those "pre-number things with fractions before one can make sense of the operations."
Other math educators included among the number/number sense basics:
Operation sense or understanding how the four operations (addition, subtraction, multiplication
and division) work means "recognizing conditions in real-world situations that indicate
that the operation would be useful in those situations." (NCTM, 1989)
"Multiplicative reasoning is basic as it leads to the understanding of multiplication,
division, and proportional reasoning. The notion of unitizing or forming units
of units where a person begins to group objects together and consider them as sets
or wholes, e.g., five candies per bag, six bags, gives 30 candies. The five candies
are considered a unit." (teacher)
Proportional reasoning was mentioned as critical to "people's ability to understand
and communicate about (rather than compute) what an average or percent is... Anything
that is decreasing/increasing or changing magnitude relies heavily on deep understanding of proportions, rates, ratios, relations and relative comparisons." (stakeholder)
All these elements add to a dynamic definition of what learners, employers and teachers
mean by being able "to do whole numbers."
Estimation and mental math are essential to sense making with numbers. ANPN focus
group members emphasized how critical the skill of estimation was. One
employer said: "I realize in the last few years in my career path, I use estimation
so much more now than I ever knew (I would).. you discuss something on the phone
and How much will I get paid for this and that? I can do a quick estimation, you
know. It is going to be about a thousand dollars commission here. Just being able to do
that -- I probably do estimation 80% of the time and there is 20% of the time when
I actually need to figure out a shipping charge or whatever. But most of the time
in dealing with all my sales reps and customer in general, most of the time, it is like estimating
90% of the time -- giving them a ballpark figure and they are comfortable with that.
I've actually had an awakening with estimation in the last couple of years. I didn't realize I was using it all the time. Yet someone right next to me says: How
did you know that, how did you figure that out? Obviously, I've learned the process
because I practice the process."
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The SCANS Report suggests that work competencies and skills require estimation. Workers
need to be able to "summarize information, set upper and lower limits and estimate
if it falls within acceptable range, understand precision both as consumer and producer, estimate time and costs, troubleshoot and anticipate consequences." (p. )
And the teachers who wrote The Massachusetts ABE Math Standards concurred. "Estimation
is probably the most used and useful skill for adults and continually plays an important
role in the adult learner's life. Adults use informal measurements in activities such as cooking, shopping, buying clothes or estimating the time required for daily
tasks. Good estimators use a variety of strategies and techniques for computational
estimation..." (p. 35)
Adults use estimation everyday and all the time; it's woven into the fabric of daily
decision making. "Having $50 to use at the store and seeing how far I can stretch
it. I kind of round the amounts off and keep a running total in my head til I think
I'm out of money." Adults use estimation to predict and to plan. "If you are going to
the store you usually estimate whether you can afford that coat or not and how much
it would be and how much off and then reason, making conclusions based on that estimation." And they use it to check outcomes. "Does this make sense?" Approximations guide
thinking all along the way. It is a kind of sense making activity. In the final analysis,
people need to be able to decide how precise they need to be in a particular situation.
Fractions, decimals, percentages and ratios are necessary and challenging. Learners,
educators and employers are clear about the need to understand and use decimals,
fractions and percentages. "It's never just whole numbers... it's always fractional
amounts and decimals." "Nurses using measurements and fractions to give the right amount
of medicine." Many also thought that adults should be comfortable handling numbers
in a variety of forms. One teacher cited "understanding number relationships, about
how percents and decimals and fractions are related" as essential.
This is supported by The Massachusetts ABE Math Standards which state that adults
should "understand, represent and use numbers in a variety of equivalent forms (integers,
fractions, decimal, percent, exponential, and scientific notation) in real-world,
work-related and mathematical problem situations." (p. 41)
Fractions were frequently mentioned as a hard topic in school math, both in childhood
and in adult education classes. "Fractions, decimals and percents have always been
really hard to do." "Math was OK until seventh grade when we started fractions."
"When I was in school and we started on decimals and fractions, I could not catch on
and my teachers wouldn't help so I got behind in class and couldn't keep up with
everyone else so I just gave up completely on all of school so that no one knew that
I couldn't do it and quit school..." "I never understood how to work problems like $1.23
x 33 1/3. I
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missed school when this was being explained.. I was never able to learn
this throughout school."
Knowledge of numbers is useful to adults in making decisions about issues that relate
to their families, communities, and workplaces. No matter what the occupation, employers
and employees furnished many examples of how critical number is in their lines of work:
An automotive parts plant manager from New England ticked off aspects of number proficiency
needed in his workplace. The workers are usually on a forklift making some quick
calculations such as knowing how many boxes of filters to load if a dealer orders 100 and there are a dozen per box. They also need to be able to read and retain
an identification (SKU) number ... to be able to break up a number, repeat it, recognize
it and locate it. Finally, he said how important it is for workers to be able to
use logic in solving problems in the workplace.
A banker noted the importance of calculators in banking, but was concerned that employees
were now doing things that people used to do in their heads, (like adding and doubling),
but not using the calculators for more complicated problems. He would like to see more of that.
A restaurant owner, stressing the need for mental math ability, says she noticed over
the past ten years, employees' skills have gotten worse... I've had to change the
equipment because people didn't know how to do math. I have to put calculators around
the restaurant and change registers. If some item costs $5.25, and you give them a
ten and then you say you have a quarter, they're lost and have to start over!"
A learner states, "Everyday at work I use math. I'm a cashier and gas station attendant
without a cash register. Therefore I have to figure out change on my own and if
people get the wrong change back, they become highly upset and critical. In order
to make change at work I usually use (mental) addition, subtraction, and multiplication."
Parents, family members and caregivers had no trouble citing instances where they
use number in the daily care and survival of their families. "Just the overall running
of the household...checkbooks...there's bills, rent..." They mentioned such things
as mixing formulas, dividing candy or toys, how much fun money you get after you estimate
the bills, cooking, house expenses, buying cars and houses, grocery shopping, comparing
prices, and doing taxes. "When my husband is short on money, he'll claim more deductions on his W4 for a few weeks and then change it back. He has to be really
careful - last year he got a little carried away, and we ended up having to pay at
the end of the year."
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Parents see helping their children "do math" as a key responsibility. Sometimes that
means being available to help with homework. "My kids ask me all the time - like
a division question or something". Other times it's informal teaching. "Asking the
child to spend her own money for some groceries or practical things, so she learns how much
things cost." "I use math with snacks with my children. How many crackers the want.
How many they have."
Citizens make personal decisions as consumers. "My biggest (math) decision based on
money is probably where I'm sitting right here and now. I decided to come back to
school... I figured out how much it would cost per quarter, I figured out how much
income was already coming in and allotted for other expenditures. I came up with my numbers
to see if I could afford to come to school and still be able to maintain the same
life style. Or at least maintain a lifestyle! You know to be able to come back to
school and cut my hours at work and comparing the numbers.. like the percentage on my student
loans... how much money I would get from the government, different funding.. trying
to work at that... plus, still go on vacation. It was everything from simple addition, just adding up what I make each month... figuring out different percentages,
what's my loan going to cost each month. It ran the gamut, even figuring out ratio
and proportion."
Implications for Teaching and Learning
Teach and learn about numbers in context. The teaching and learning about number
(whole numbers, fractions, decimals and percents) must be done in context right from
the beginning, because as an adult learner said, "Although whole numbers are nice,
they are not the numbers of real life." Neatly controlled pages of decontextualized computation
are not the way adults learn best. "My best learning situation is probably work...
I got the basics in school, the really simple stuff, that wasn't so great. But I've been working at the same company for 11 years. We're really a big textile distributor,
and they run quality control, so it's constantly figuring out, we're running hundreds
of pieces. It's different numbers, it's never just whole numbers, it's always fractional amounts and decimals... We get stuff from different people. We need
to get so many small pieces running from a linear yard. We have to figure out cuts,
how much they can get out of what, so it's just a constant use of it, the sheer volume
of doing math. It's just constantly going over stuff that's made me have the decent
knowledge that I have."
So many others concurred. "We expect lifelong learning so we should use real-life
problems." "Learning how to compute percentages in the context of a real life budget
problem will be much more profitable than if taught in the abstract or with artificial
word problems."
The SCANS Report insists that "the most effective way of teaching skills is in context.
Placing learning objectives within real environments is better than insisting that
students first learn in the abstract what they will be expected to apply... Students do not need to
38
learn basic skills before they learn problem-solving skills. The
two go together. They are not sequential but mutually reinforcing." "Real know-how
-- foundation and competencies -- cannot be taught in isolation; students need practice
in the application of these skills." (p. 19)
The Massachusetts ABE Math Standards holds that "computation skills should be practiced
in the context of problem solving and not as a set of isolated skills. Adults should
be encouraged to develop and share their own tricks and ways of computing percentages; for example, sharing short-cuts to determining the tip on a meal tab or finding
a discount." (p. 40)
Build upon adult's personal number sense. Traditional "school math" calculation methods
are not always useful. One of the teacher/authors of The Massachusetts ABE Math
Standards related this happening:
"I asked a group of my GED math students to tell me how much it would cost if you
bought four shirts for $7.98 each. They were told they could figure it out any way
they wanted, except they could not use paper and pencil. I watched as they used
their fingers in the air or "wrote " on the desk. Most we're able to multiply and get the right
answer. When I asked HOW they got their answer, all agreed they needed to multiply
$7.98 by four.
"I then asked if they were in a store and had to figure out the same problem would
they have done it the same way. All agreed they probably would NOT solve it the
same way in real life. Some said they would have multiplied four by seven plus
four by one and then subtracted eight cents from that total. Others said they would have rounded
$7.98 to $8.00, multiplied that by four and then subtracted $.08 for the product.
I then asked why no one admitted to solving the problem like that in class. The
response was this is math class so they needed to do it out." (vol. 2, p. 60-61)
This notion that adults should do it the "right way" or the teacher's way robs adults
of their mathematical power. Good numeracy instruction must build upon an adult's
personal number sense and help further develop that sense so that he or she can handle
real life situations.
Adult educators must question the teaching of "school math" especially when those
strategies or techniques are rarely utilized by other competent adults. The way estimation,
for example, is taught has nothing to do with the way people really use it in the
workplace. The choice of teaching complicated fraction computation which will never
be used in real life must be weighed against more important and realistic skills.
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Connecting to the Four Purposes
It is easy to see how number sense is connected to each of the four purposes for literacy.
Number sense enables adults to be able to interpret (access) and represent (give
voice to) the world in which they work and live. Good number sense supports the
judgements and decisions that lead to independent action. Number sense is the cornerstone
of mathematics ... It is exemplified every day, whether we consider notions as complex
as the consumer price index, as pivotal as the impact of the Great Depression on United States history, or as personal as a blood pressure reading. (The Massachusetts
Mathematics Curriculum Framework, p. 32)
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DATA:
DATA ANALYSIS, PROBABILITY AND STATISTICS, AND GRAPHING
In the workplace the common thing would be -- we measure everything. If someone
is measuring something and they are taking samples and they see that is it going
out of the acceptable margin. I mean, they need to be able to stop everything and
get someone to find out what is happening. Upper and lower limits, check that against a
blueprint and see that the upper and lower limits have been put in properly, then
compare that with how it was running earlier that day. Someone has got to understand;
it could be a supervisory level. Someone's got to be putting in the data properly. It
is computerized, of course, but it is only as good as the information that someone
is putting in. We try to tell people, numbers have got to make sense. They have
got to make sense. You have to be able to see if for twenty-five days it has been operating
like this and suddenly it looks like this. What does your gut tell you? There is
something wrong here. Something is going out, that kind of thing. Look at the chart.
People should have enough understanding and knowledge to read all the results of all
our quality measure that we post on a regular basis. And if they see something that
they don't understand, they need to question that. Say, 'what is that?' How does
that impact that? Where does that come from? Everyone of those things impacts something
else. In a workplace setting, there is this domino effect. Everything builds on
the next thing. From the time that raw material comes in the door and one process
after another through departments takes place. You are adding value, but you are also
adding costs. We need to catch those numbers if they start to go out of alignment.
Again going back to -- people need to see the big picture first. The numbers stand
out, how things operate. So they see the big math picture. You get to the big math picture
through all these tiny calculations.>blockquote>
Overview
Adults make decisions based on data in their daily lives and in the workplace. According
to Equipped for the Future, "Adults are also interested in learning and strengthening
the skills associated with using information to have an impact on the world. They identify the need to develop the problem solving and critical thinking skills that
have to do with analyzing and reflecting on information in order to make good decisions.
. ." (p. 24) Reading charts and graphs, interpreting the data, and making decisions based on the information are key skills to being a successful worker and an informed
citizen. Being an informed citizen includes understanding statistics and probability
as well. Adults
41
cannot make reasonable decisions unless they understand from where the statistics come.
Charts and graphs are essential in the workplace. According to SCANS documentation,
tomorrow's workers must have reading skills that enable employees "to read well enough
to understand and interpret diagrams, directories, correspondence, manuals, records, charts, graphs, tables, and specifications. Without the ability to read a diverse
set of materials, workers cannot locate the descriptive and quantitative information
needed to make decisions or to recommend courses of action." (p. xvi)
Data from charts and graphs are used to make decisions. Graphs are useful tools in
that they organize data so that information becomes clearer. This organized information
can then be used to draw conclusions, to make decisions, or to influence others.
Data is organized in a variety of fashions, from charts and graphs, to computer-generated
spreadsheets.
In comparing the comments made by stakeholders --often individuals in managerial positions
-- with those made by adult learners, several interesting distinctions were noted.
Stakeholders tended to use the data in charts and graphs to make inferences and
decisions. Adult learners, on the other hand, were more inclined to use charts and
graphs in a more literal way - simply to gather information. Adult learners who created
charts and graphs used them to help themselves while stakeholders (management) used
charts and graphs to influence others. In addition, adult learners claimed either to
not have a use for charts and graphs or felt they used them when they needed information.
Stakeholders shared a concern for the lack of "chart literacy".
Key Findings
Data collection, analysis, and graphing are essential in the workplace. SCANS proposes
five competencies needed by employees for success in the workplace. The competency
Information clearly suggests that data analysis and graphing are necessary skills
for tomorrow's employees. An example of the level of proficiency for the competency
Information includes the ability to ". . . analyze statistical control charts to
monitor error rate. Develop, with other team members, a way to bring performance
in production line up to that of best practice in competing plants." (p. xx)
Many industries, manufacturing in particular, now use statistical control processes
(SPC) to monitor their processes in order to ensure quality products. Often the
front-line employee is required to collect the data used for charting the manufacturing
process; therefore, employees at all levels should be knowledgeable about and comfortable
with using a variety of charts. As more and more quality teams -- consisting of
a variety of employees -- are charged with the task of ensuring quality products,
employees will need to have an understanding of probability and sampling. During a focus
42
group session, when asked how math is used in the workplace, an adult learner responded,
"Sometimes I have SPC graphs. It kind of determines if something is wrong with the machine and pretty simple things, nothing really major. One out of 6 is bad, shut
the machine down. A lot of times we don't count the parts. They'll just be in a
can and we write down on an overlay. You know, if you red tag some parts that are
no good, you tag -- nothing really major."
It is interesting to note that only stakeholders seem to be acutely aware of the need
to have the ability to read and interpret statistical process control charts. "There
are two other areas and we mentioned this before about statistical process control.
Our industry is moving into really using numbers to determine whether the production
process is functioning or not. And they are using the concept of time as well as
numbers. How fast it takes to do a particular process. Budgetary hourly rates is
another phrase that is kicked around. . . " Yet another stakeholder commented, "In the
workplace today, employers want everyone to understand quality. Any chart or graph
that shows production uses statistics."
Other forms of charting are also used in the workplace to make decisions as well as
gauge accuracy. One learner shared how he uses blueprints to determine whether a
part is within tolerance: "Basically at work every day, you know, just looking at
parts, I use a blueprint. That gives you a tolerance, a couple thousandths here, couple
thousandths there, sometimes 5, -2, and if you've got a part that's right on the
borderline of tolerance you want to decide if you want to just keep on running it
or fix it or basically see what kind of problem it is going to cause." And another adult learner
has an idea of what will be required of her in the workplace: For myself, I m hoping
to get into social work. So I need to be able to read and understand graphs (for
statistics). That way, I ll be able to compare past trends with current trends and
hopefully predict results. Also making my own graphs. At least I think that s what
statistics will be like.
Statistical knowledge is important in problem-solving and decision-making. Adults,
often without even realizing it, make decisions based on statistical information.
It may be via the television, radio, or it may be through print materials. The
following adult learner made her decision based on what she had seen in a magazine. "That
reminds me of a fee that I thought was too high. There was a newsletter I wanted
to subscribe to, but it charged $35 a year, and I couldn't understand why a little
paper would cost so much. But then on the inside it showed a circle graph, with sections like
a pie, and it showed what the money was spent for. Then I could see that it really
was a reasonable amount to pay." It is important that adults know that they are
using statistical information in their reasoning. A stakeholder weighed statistical information
to decide whether it was relevant to him: "I was reading an article about what to
look for if a kid is using drugs. And I thought about my 14-year old grandson and how different it is now that he is a teenager. I had to take my random knowledge
of a person and decide whether it was statistically relevant to drug use." Another
stakeholder agreed that sta
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tistical information does influence decision-making.
"You're seeing different information that you need to reason and draw conclusions based on
this. Is this a good sale, not a good sale, and so forth depends on the whole reasoning
process -- looking at graphs and charts -- looking at your paycheck and whatever
-- just being presented with information and attempting to draw conclusions."
Statistical information is used to communicate information and sometimes influence
others. Understanding the flood of statistical information allows adults to make
more informed decisions. A teacher said it very well, "[When] we understand math,
we can use it to take control of our lives. Do our own figures so as not to be the victim
of scams." A stakeholder explained how she thought a nice graph could be an influencing
agent: ". . . Not only did he do nice histograms and circle graphs but inside he
did it by ethnic, by cities that the students are from, and so forth, and then he has
a final chart on the money that he's asking on the back. I think it is a nice piece
of work. On the back is a financial chart which could hopefully affect the budget."
Graphs, tables, and statistics make data easier to understand. Adults create graphs
for clarity and understanding, for themselves as well as for others. Sometimes seeing
the data in chart form makes the decision making process easier since the information is clearer. The following adult learner provides an example of how charts helped
them see the issues more clearly. "When I bought my car, I put a $1000 down payment
on it. I owed them like $8000 on it for five years which I was gonna end up paying
a lot of interest . . . so I made a plan to pay as soon as possible so I save the interest.
So I figured out how much I could spare each month. I did a budget so I would send
them like 2 payments, 3 payments a month. I did a graph to see how far it would
go. So like in a year and two months I called the company and I asked them how much
money I save from all the interest from me paying early and I end up with almost
$2000 and I ask them if I can take this $2000 I saved and add it to what I pay and
they said I could."
Even when the charts and graphs are not initiated by adults, they do tend to make
the information easier to digest. From an adult learner, "I watch this thing on PBS.
It's like this physics show -- and it's actually enjoyable. They give images and
graphs; they give you things about different theories with physics. It is hard to understand,
but the way they put it on TV, it's very simplified. You can actually see it with
your eyes." There are those, however, who don't agree that charts make understanding information easier. A stakeholder commented, "Right now all the changes that
are going on. There's ATM. You can't go to a bank anymore necessarily and talk
to a person. Statements are getting more, well, they are supposed to be easier to
read . . . things are constantly changing on them, you know."
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Charts and graphs are also used for record keeping such as spreadsheets and data bases.
According to the SCANS report (p. xviii), employees of the near future will need
to be able to use spreadsheet programs for tasks such as monitoring expenditures
There is a concern for the lack of understanding and ability to read and interpret
statistical information, including charts and graphs. There is also worry about the
use and misuse of statistical information. While adult learners did not have this
concern, stakeholders and instructors agree that adults do tend to have difficulty deciphering
what the numbers and charts mean. Stakeholders shared their concerns: "I think
transferability is really hard for adults. To know a concept is one thing, but to
be able to look at a table and say, 'I understand this table, or I can read this table,
or I can interpret what this means' is hard to do." "I tried something in the workforce
here about a month ago based on quality where we measure quality based on the number of errors per 1000 lines shipped. We benchmark ourselves against Toyota, Ford,
Chrysler, and all the other automotive people. I had a lot of blank stares going
back from the audience. They could not associate what I was trying to talk about
. . . statistics to the competition. They simply had a block saying, 'But we're different.'
Math is math and I was having a hard time with that comparison. They were not able
to connect. I kind of lost them on that one. So we are going to try again going
back to more graphs I guess is probably the best way to try to communicate this. You know,
'A picture is worth a thousand words.' I'll try that out next time."
Yet another stakeholder believed that understanding statistical data involved much
more than just looking at numbers in a literal sense. "I'd like to also suggest
that, underlying understanding of statistics and some key aspects of both measurement
and number sense lies the fuzzy yet critical domain of proportional reasoning. Consider,
for example, that people's ability to understand and communicate about (rather than
compute) what an average or percents, understand notions of sampling and representation, make sense of and make choices about probability and risk (e.g., likelihood of
accidents, errors out of XXX products), and anything that is decreasing/ increasing/changes
its magnitude, relies heavily (though of course not exclusively) on deep understanding of proportions, rates, ratios, relations, and relative comparisons, which are
all parts of the same conceptual system that mathematicians (and I'm not one of them)
call Rational number concepts. "
Adults use charts, graphs, and statistical information in their roles as workers,
parents, and citizens. As workers, adults used data to monitor the quality of the
products being made. They also make decisions based on the data. A stakeholder uses
blueprints and statistical process control charts: "People that I work with have to know
at least the basic math skills in order to perform the SPC and we do a lot of blueprint
reading -- a lot of math involved in reading a blueprint." A learner explained how
he uses a variety of skills to do his job. "My daily job was office manager. My responsibilities
were to do a daily cash reconciliation, post accounts receivable, accounts payable,
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keep up with hourly employee time cards and keep track of everyone's vacation and sick time. I always made decisions using amounts, money, graphs, and basic
addition and subtraction skills. I used all these skills on a daily basis to reconcile
and solve any problems regarding my specific job requirements."
As citizens, adults need to understand the data that they are continually being bombarded
with -- through all forms of media. This stakeholder clarified the importance of
understanding data as it relates to elections. "I want to switch from workplace to
community and society and all that data that we get inundated with -- try to make,
you know, what's going on in the world -- what does it mean to win a primary and
say that is 14 electoral votes and, all of a sudden, you're supposed to be the front
runner and how do you gauge the real significance of that. Then the next week you're blown
out of the water supposedly because something else happens. I remember the election
last year when the polls and the data, they became the driving force themselves as
you watched one go up and one go down. How do people really assess that because it is
such a big part -- you're talking about one arena, but locally, we're having a school
tax referendum. People are being surrounded by numbers in which they've got to make
decisions. This was cut in half. Well, what does it mean that this was cut in half?
Half of what? Is it really that big of a difference . . or whatever. So I think
something in adult ed. I think we're tending to look at the work stuff a lot and
a lot of the sort of consumer needs, but I think some of the data in terms of broader societal
issues. We're tying to get people more engaged." And another stakeholder, when
asked what were the three most important math concepts that should be taught, included
statistics: " . . . As members of a community we use statistics to understand our
community better and to help create a better environment for ourselves."
Implications for Teaching and Learning
Introduce more work-related charts and graphs and other statistical information to
better prepare adult learners for the world of work. According to the Massachusetts
ABE Math Standards [pg. 50], to become successful employees, adult learners need
to have the opportunity to "systematically collect, organize and describe data; and construct,
read and interpret tables, charts and graphs". Adult learners need much more than
simple activities where they are asked to find literal bits of information in charts
and graphs. They need opportunities to collect their own data, then create their
own charts and graphs. In designing their own charts, adult learners begin to understand
how data can be represented. Employees at all levels are being required to read
and interpret charts and graphs, so adult learners need to be prepared. As one stakeholder
put it, ". . . Being able to be chart literate and being able to read those charts
and graphs that we produce and we put up in our plant everywhere; all our quality
charts -- the lowest level, entry-level employee should be able to read those."
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Provide hands-on experience collecting, organizing, and interpreting data. It is
not enough that adult education classes give learners practice in simply reading
and finding literal information based on charts and graphs. Providing adult learners
with the actual experience of gathering data, deciding on how to represent the data, and
interpreting the results will give them a deeper understanding of statistical information.
According to The Massachusetts ABE Math Standards (p. 50), adult learners should
be able to "make inferences and convincing arguments that are based on data analysis;
and evaluate arguments that are based on data analysis." Adult learners need opportunities
to interpret charts and graphs and discuss their findings and implications with others. A stakeholder added, ". . .Need basic level of mathematics to survive,
for public discourse -- the use, abuse, and misuse of statistics today -- how and
why -- more observation -- reading charts and polls. Why they're done and how they're
used."
Connecting to the Four Purposes
Stakeholders interviewed for this project were concerned about many adults' inability
to read and interpret statistical information. This suggests that many adults, at
least when it comes to statistical information including charts and graphs, need
to become more literate for access and information. The National Adult Literacy Survey
includes the literacy tasks of reading and interpreting statistical information under
the heading of quantitative literacy. In fact, while most adult learners viewed
charts and graphs as a medium for accessing information, there were a few exceptions as illustrated
by this interaction between three adult learners and the focus group facilitator.
BB [facilitator]: "How about if you're reading the paper and you see a graph comparing the number of high school dropouts in 1965 and 1995. Can you read and understand
information presented that way?" M [first learner]: "You need to know the number
of students in '65 compared to the number of students in '95." C [second learner]:
I can't read graphs, no." BB: "Would it be important to you to be able to?" C:
"No." M: "Yes, it would be important. S [third learner]: "It is if you're doing
a test." BB: "Any other reason?" S: "No, not really." C: "Yes, actually it would
be important to know." S: "My brother-in-law uses his computer to graph his income, you
know?" M: "The light bill is a graph."
The creation of charts and graphs based on data collection is one method of giving
voice to the data. Literacy as voice requires that adults be able to communicate
to others; charts, graphs, statistics are each a means of communicating what the
data is suggesting.
As seen in earlier examples, adults use statistical information to guide their decision-making.
They often create charts and graphs to clarify the problem, then make decisions
based on the interpretation they give. The following adult learner provides an example of how adults use charts and graphs to take action and make decisions: "We
had to reduce our hours at work. We made a big chart on the chalkboard. We compared
four-hour shifts, eight-and ten-hour ..."
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GEOMETRY: SPATIAL SENSE AND MEASUREMENT
I need math when I redid my house, measuring dry wall. That was a problem but we
did it. How much did I need? How much to go buy? ... I cut dry wall to remodel
my kitchen. When we put it up it didn't fit. It's uneven but most people can't
tell.
Overview
As told by the learner quoted above and noted in The Massachusetts ABE Math Standards,
"adult learners who attend basic mathematics classes at any level share a wealth
of pragmatic experience surrounding geometric and spatial concepts. They've probably
built a bookcase, laid out a garden, applied wallpaper or tiled a floor, all the while
discovering informally the rules which formally govern the study of geometry itself.
For many adult students, geometry is one math topic that immediately makes sense
to them and gives them confidence in their ability to learn." (p. 51) It is also true,
however, that many adults associate geometry, like algebra, with failure. "In seventh
grade I started to have trouble with geometry. I still have trouble with the GED
geometry. I don't know why we have to learn it. It's so confusing." And "the hardest
part for me is geometry."
Measurement, a foundation skill for geometry, is also an essential life skill, one
that adults use in many different but familiar contexts: "on-the-job, for home improvement
projects, in the daily task of food preparation." ( Massachusetts ABE Math Standards, p. 53) Or as one learner states, "Measuring. You can put it under workplace,
family. You're always measuring something. You can be at home or stuff where you're
measuring out ingredients, whatever, like cooking.
Key Findings
Measurement is not an end in itself. It is a tool used in many contexts: home, work
and community. We measure many different attributes of physical objects and time
in many different ways in many different situations and contexts. As learners state,
Measuring, well, cough medicine, anything like that. Temperature, yeah. You re not
using it, but it s on the thermometer, so that s a form of math. 98.6 is normal,
right? So that s math. And When I worked in a factory, we made fan belts ... We
had to measure them if they re too long or too short. We had to use a cutting machine to
adjust, if too long or too short. What I had to do if they were too long, I had
to cut em or sew em together ... measured by two sticks to check if they were right.
Measured in meters I think it was. And During the windstorm our fence got blown down.
We had to go back out, measure everything, and, you know, put it up. How far apart
everything would be and then figure out how much fencing we needed.
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Measurement is essential to our sense of ourselves and our orientation to the world.
For example, as one teacher