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