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