Publications:
A Framework
for Adult Numeracy Standards:
The Mathematical Skills and Abilities Adults Need
To Be Equipped for the Future
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
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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-
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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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