Publications:
A Framework
for Adult Numeracy Standards:
The Mathematical Skills and Abilities Adults Need
To Be Equipped for the Future
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 states, I work with 80% welfare mothers, 20% low income
... They work with people in nursing homes ... Some don t have that concept, measurement. They have to do measurement. How tall someone is. How much they weigh. 5'10"
can be like 4'10" to them. Kind of things we take for granted about how we see our
world. We try to bring them to something that s real in their lives so that they
will tie into it and try to generate some trust in that.
Because measurement is used so often and in so many contexts, many learners have great
confidence in their measurement skills. For example, We sell fry food and chicken
and fish. The fish, we sell a lot of fish, fried fish. For example, the fish is
$3.99 a pound. And the people say, I want $15. I don t have to go to machine and
check how many pounds is $15. We think and we fry a lot of fish and we separate
and exactly for the customer $15. I do that every day. And, I am night manager
in a restaurant, so I have to order every night. I have to do a balance for everything. Everything
by the pounds, like sugar. See how many pounds I have and how many pounds I need
to order.
For ESOL learners, teaching measurement is very important as a cross-cultural component
of mathematics and second language learning, since many of these learners have used
the metric measurement system much more than the U.S. system. For example, as one
learner states, In addition, it s very difficult for us to use the American system.
I think about mathematic, and American system about labor, pounds, miles, yards,
because in European you have metric system. I heard United States try to change
to this system, 2000. You change now?
Learners and stakeholders recognize that measurement skills can be critically important.
As one learner states, I worked for Nabisco as a mixer. You had to know the correct
scale and formulas. I kept messing up. I lost my job. It doesn t look too good on the record. If you don t know math, you can t succeed. Or, I just remember,
in my job, I use in my country ... nurse. When the doctor tell me you have to give
25 milligrams, I have to. This is very important with medicine. You have insulin,
you have to know. And, carpentry work. Making sure the measurements are correct so
that you don t waste too much and so that you don t have to do twice the work, by
having to redo the work over. And as one employer states, In the workplace, the
common thing would be--we measure everything. If someone is measuring something and they
are taking samples and then see that it is 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.
Time management is another critical measurement skill. As one employer states, Time
management, that is math. People have no sense. It is a work ethic, or break down
of. We used to have a schedule ... Some people don t get what being to work on time
means ... being late and putting stress on all that must be completed by the time
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the
door opens ... forgetting how long it would require to do all the food prep. Or
as one learner states, In my work I must count the time I work. The time I begin
work and the time I stop working. I write it or not. Good to check. That week if the
boss count it exactly, I can check it. And as another learner says, You need to
figure out how much time to plan so you can get to places on time. You need to figure
out when the bus comes, how long it takes to walk to the doctor s appointment, and everything.
And from another, Just time basically, time in a day.
Some adult learners identify geometry (along with algebra) with failure. " The hardest
part for me is geometry. Learning geometry was really hard."
Other learners recognize their excellent everyday skills in geometry, although they
may or may not use the term geometry in relation to these skills."Three or four
years ago, I took up quilting and realized how much geometry you need when you go
to modify patterns or create patterns. Estimating only goes so far when you re dealing
with little, tiny pieces. When they re fairly big, you can estimate and you re okay.
I did addition, subtraction, estimating, did a lot of work with angles. Who knows
what else. I don t even remember my geometry well enough to remember the terms but some
of it took going back to books and some of it was pretty straightforward." And,"
I use math to figure how to shoot pool shots. I know where you need to hit the
ball so it will go in the pocket. "
Some adult learners don't see geometry as useful. For example, as one learner states,"
Like geometry is so esoteric, with the angles and things. You think to yourself,
what am I gonna use it for. You think, what's the point? ... This stuff they put
in there just to mess you up."
However, geometry is and can be related to all aspects of life: home, school, work
and community. Geometry and spatial sense can be used to describe the physical world.
For, as another learner sees, "Geometry is everywhere. Not everything is square."
And, "We have four horses. Once we had to figure how much hay we d need for a year.
Then we had to figure if the hay would fit in the barn. So one horse eats about
a bale a week, so multiply by four and so on. Then we had to know how big the bale
was. And how big the loft was." Or as another learner states, "I remember six months ago
we had moved from a house in Annandale to Alexandria and the house we had moved in,
it was no carpet on the floor. So we had to use our old carpet. It was good, not
that bad. We had that carpet. The house we used to be before, it was bigger. This one
was smaller. The carpet was bigger than we need ... We had to measure the room and
the hallway exactly what it was. We had to cut the pieces of the carpet that we
had to fit exactly the room. So we had to make a map of the paper and how we gonna cut
it. How many feet. How many centimeters. Exactly how it s gonna be and it was
good using the math. We did some mistake. We had some left over. The next time,
maybe we do fine."
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Implications for Teaching and Learning
Use exact and estimated measurements to describe and compare phenomena to increase
the understanding of the structure, concepts and process of measurement. " Despite the fact that competency in measurement is vital, some adult basic education learners have difficulty selecting and determining appropriate units of measure as well as using the appropriate tools of measurement ... Teachers should use concrete activities
(with non-standard and standard units) to help ABE learners develop an understanding
of the many measurable attributes of physical objects (length, time, temperature, capacity,
weight, mass, area, volume, and angle). This is the natural way of building a vocabulary
for measurement, and for comprehension of what it means to measure. " (Massachusetts ABE Math Standards, p. 53)
Address the impact of measurement skills on self-efficacy and self-reliance. As one
learner points out, "Recently, I transformed my storage room into a walk-in closet.
I needed to decide how large I wanted the clothing racks to be which depended on
the size of the closet, the amount of space I needed left over for walking space, a bureau,
and other items that are stored inside. Basically, math is everywhere, and to be
independent and survive on a limited budget you need to be able to do things yourself
and find the best values along the way."
Extend measurement skills to concept areas such as volume, proportion, and problem
solving. As one learner points out, "I bought a couch last fall and miscalculated
the footage on it. Then I had to rearrange my whole living room. It needed to go
where the entertainment center was, but it was also big. It really got complicated, and
I really was surprised at how challenging it was to get everything to fit." Or,
after Oregon flooded, "We had a situation where we had to move some stuff out because
of the flooding. We had to rent a truck. We needed to figure out how big a truck to rent.
Was it really going to be more cost effective to rent one big truck or two little
trucks? We had to figure out how may boxes we would need, and how many boxes would
fit in a trailer."
Increase the awareness of acceptable tolerances (margins and upper and lower limits)
and the consequences of being within and outside of these tolerances. To return
to the workplace, where "we measure everything ... They are taking samples and then
see that it is 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 ... 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."
Start from the learner 's strengths and make the instruction practical and useful for
learners to overcome their fears regarding geometry. Provide opportunities
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for learners
to make connections between instruction and real-life situations common to their
lives. As one workplace learner states, "I think like in these examples you have in
here where we used real workplace kind of examples rather than some kind of theoretical
examples that didn t apply. How many rolls of paper fit in the truck? What s the
area of the truck, the volume of the truck. Real examples like that--that means something
to your everyday job." Or as another learner says, "Learning volume here is easy
because I can see it as something in front of me. It's easier for me to figure out
if it's hands-on." Or as another learner states, "I like learning volume and shapes
because in landscaping you can visualize in your head the shape to determine how
much fill or sod. You need math to compute the job." Or, "the best learning was
when I am at work using my tape measure."
Focus on hands-on problem-solving and give special attention to developing spatial
sense in order for learners to develop an understanding of geometric principles.
As one participant in the Virtual Study Group states, Spatial reasoning which
in my mind includes not only geometry, but measurement and the ability to visualize. It is
often the visual and concrete models that can help people understand and learn what
we want to teach about number and statistics. In addition, being able to realize
that this kind of reasoning, this part of mathematics, often helps students who have talents
in this direction realize and accept that they do have mathematical potential.
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