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Markus , Nancy

Geometry in the Adult Education Classroom.

Kent State Univ. , OH. Ohio Literacy Resource Center.

Ohio State Dept, of Education, Columbus. Div. of
Career-Technical and Adult Education.

2001-03-00

5p.

Ohio Literacy Resource Center, Research 1--1100 Summit
Street, Kent State University, P.O. Box 5190, Kent, Ohio
44242-0001; Tel: 800-765-2897 (Toll Free); Tel:
330-672-2007; Fax: 330-672-4841.

Collected Works - Serials (022) -- Guides - Classroom -

Teacher (052)

Math Literacy News; nlO Mar 2001
MF01/PC01 Plus Postage.

Adult Basic Education; *Adult Literacy; Classroom
Techniques; Education Work Relationship; Estimation
(Mathematics) ; Geometric Concepts; ^Geometry; High School
Equivalency Programs; ^Learning Activities; ^Literacy
Education; ^Mathematics Instruction; Mathematics Skills;
♦Measurement; Numeracy; Student Attitudes; Teaching Methods

ABSTRACT

For many adults, geometry is a mathematics topic that
immediately makes sense to them and gives them confidence in their ability to
learn, while other adult learners identify geometry with failure. Most
adults, however, do recognize the need for measurement, and many have a basic
understanding of measurement concepts, although they may need to learn
English measurements if they already know metric measurement. Implications
for teaching and learning are the following: (1) teachers must use exact and

estimated measurements to describe and compare phenomena to increase the
understanding of the structure, concepts, and process of measurement; (2)
teachers must address the impact of measurement skills on self-efficacy and
self-reliance; (3) measurement skills should be extended to concept areas
such as volume, proportion, and problem solving; (4) teachers must increase
the awareness of acceptable tolerances and the consequences of being within
and outside these tolerances; (5) teachers should start from the learners'
strengths; and (6) hands-on problem solving and attention to development of
spatial sense is necessary for learners to develop an understanding of
geometric principles. This brief includes suggestions for classroom
activities in measurement, perimeter and area, angles, and circles. (KC)

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Math Literacy News

March 2001, Issue #10

Geometry in the Adult Education Classroom

Nancy Markus

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March 2001— Issue #10

GEOMETRY IN THE ADULT EDUCATION CLASSROOM

Nancy Markus

For many adults, geometry is a math topic that immediately
makes sense to them and gives them confidence in their
ability to learn. It is also true, however, that many adults
associate geometry, like algebra, with failure. Measure-
ment, a foundation skill for geometry, is also an essential
life skill, one that adults use in many different but familiar
contexts.

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 contexts. Measure-
ment is essential to our sense of ourselves and our orienta-
tion to the world. Because measurement is used so often
and in so many contexts, many learners have great confi-
dence in their measurement skills. For ESL learners, teach-
ing measurement is very important as a cross-cultural com-
ponent of mathematics and second language learning, since
many of these learners have used the metric system much
more than the U.S. (standard) system. Measurement skills
can be critically important.

Some adult learners identify geometry with failure. Other
learners recognize their excellent everyday skills in geome-
try, although they may or may not use the term “geometry”
in relation to these skills. Some adult learners don’t see
geometry as useful, however, geometry is and can be re-
lated to all aspects of life: home, school, work, and com-
munity. Geometry and spatial sense describe the physical
world.

Implications for Teaching and Learning:

♦ We must 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 com-

petence in measurement is vital, some adult learners
have difficulty selecting and determining appropriate
units of measure, as well as appropriate tools of meas-
urement. Concrete activities with non-standard and
standard units help learners develop an understanding of
the many measurable attributes of physical objects
(length, time, temperature, capacity, weight, mass, area,
volume, and angle). This is a natural way of building a
vocabulary for measurement and for comprehension of
what it means to measure.

♦ We must address the impact of measurement skills on
self-efficacy and self-reliance. Math is everywhere and
to be independent and survive on a limited budget, one
needs to be able to do things oneself and find the best
values along the way.

♦ Measurement skills should be extended to concept
areas such as volume, proportion, and problem solv-
ing.

♦ We must increase the awareness of acceptable toler-
ances (margins, upper and lower limits) and the con-
sequences of being within and outside of these toler-
ances. In the workplace, everything is measured. Some-
one has to understand what upper and lower limits are,
and how to input data. Much is computerized but the
results are only as good as the information inputted.

♦ The place to start is the learner’s strengths. Instruc-
tion must be practical and useful for learners to over-
come their fears regarding geometry. Opportunities
must be provided for learners to make connections be-
tween instruction and real-life situations common to their
lives.

♦ Finally, it is necessary to focus on hands-on problem
solving and to give special attention to developing
spatial sense in order for learners to develop an un-
derstanding of geometric principles. Spatial reasoning
includes not only geometry, but also measurement and

Ohio Literacy Resource Center

Enhancing Adult Literacy in Ohio

O

ERIC

Math Literacy News is a publication of the Ohio Literacy Resource Center
Research 1—1100 Summit St., Kent State University
PO Box 5190, Kent, OH 44242-0001
Phone: (800) 765-2897 or (330) 672-2007
FAX: (330)672-4841 Email address: olrc@literacy.kent.edu
Website: http://literacv.kent.edu

3

Math Literacy News

•

Page 2

realize that this kind of reasoning, this part of mathemat-
ics, often helps students who have talents in this direc-
tion realize and accept that they do have mathematical
potential.

CLASSROOM ACTIVITIES

MEASUREMENT

There is a tendency to teach students how to measure rather
than teaching what it means to measure. Here are some
ideas that help foster understanding.

Introduction: Have students measure a piece of paper us-

ing one inch square tiles. How long is the paper? How
wide? What would be easier than using the square tile over
and over again? (Students can put marks on the paper for
each inch.) When using a standard sheet of typing paper \
what about the shorter side? (This paper is 8 1/2 by 11
inches.) What about the extra distance at the end of the
short side? (Students will put numbers for each inch within
the inch area. They are often surprised that a standard ruler
puts the number at the END of each inch.)

Enlarged Inch : Students often have trouble understanding
what the lines mean on a ruler. In this activity, the student
will pretend that the length of one inch has been magnified
to the length of a larger piece of paper. Start by folding the
paper in half end to end. How many sections does your pa-
per have? Draw a line along the fold and label
it l A to indicated that it is l A way along the paper. The stu-
dent can measure something (his/her desk?) using this
enlarged half inch!

The paper is then folded again in half. Shorter lines are
drawn on each of those folds and labeled % and 3 /». The pa-
per can be folded again and again in order to put eighths and
sixteenths on the enlarged inch. (Most rulers don’t go be-
yond sixteenths.) With each step, compare the enlarged inch
with a standard ruler.

PERIMETER AND AREA

Introduction: Often teachers assume that students have the
background needed to study a topic formally, when students
have gaps in knowledge and vocabulary. Often area and
perimeter are confusing to students. Put the words on a
board or large sheet of paper and discuss their meanings.
Questions to ask might include:

0 What do you know about these two con-
cepts?

0 For what things in our lives do we need
perimeter measurements? (fences, wallpa-
per boarders, molding, fringe)

0 For what things in our lives do we need
area measurements ? (pastures, sod,
paint, carpeting, tiles, material for sew-
ing)

0 How do we measure area? (Area is spa-
tial. We use square inches (draw one),
square feet (draw one), square yards
(draw one), square millimeters, square
meters, etc.)

0 How do we measure perimeter ?
(Perimeter is a distance measure, a linear
measurement. This is the measurement
we are most used to. Perimeter is as if
we are taking steps around something.
How many steps do we need to walk
around it?)

0 How can we remember which is which ?
{peRIMeter: what is the rim of a glass?)

This discussion might take ten or fifteen minutes. It will
eliminate much confusion.

Fences and Pastures : In a discussion, relate perimeter to
fences in that they both surround a space on all sides, and
have an area (or pasture) within it. Students are given
pieces of paper cut into shapes and must determine how
many one inch rods (can be made out of toothpicks or pa-
per) would go around each shape. Then students are given
larger rods to measure the same figure. Will it take the
same number of the larger rods to measure the perimeter
of the figures? Why does it take fewer of the larger rods?

Students are then asked to use the one inch rods to build a
rectangle, and then square tiles are fit inside. What is the
number of square tiles, the area , that is needed to fill each
pasture? Why do different people have different number of
square tiles for the same number of fence (perimeter)
pieces?

Once students can count the sides of the square tiles, the
one inch rods are not necessary. Take 24 square tiles and
see how many different rectangles can be made. Make a
chart of the area (all will be the same), the dimensions, and
the perimeters. Discuss why perimeter and area may use
the same numbers but are different. An example such as
the perimeter of a comb, that an ant is walking around,
gives a good visual that a perimeter may be large, but the
area small! Teachers note: a square has the smallest pe-
rimeter for the same amount of area.

Students can put on graph paper the rectangles formed.
The dimensions of the rectangles are also the factors of 24!

Floor Plans : Floor plans can be obtained from local news-
papers or home magazines. They can be copied, enlarged

4

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Page 3

and have dimensions added if needed. Looking at these house
plans, teachers can review or reinforce the concepts of area
and perimeter. Students will practice finding the perimeters
for baseboards and wallpaper borders, and area for carpeting
and concrete. Square feet and square yards can be discussed
and explored.

Square Inch : Visualization is important in geometry and ma-
nipulatives can be simple yet extremely helpful. Have each
student cut a square inch out of paper. Have the students look
at the square inch and identify it. It is written inch. Discuss
what a square foot would be. How many square inches are in
one square foot? Many students will think there are 12 square
inches in a square foot! Hand out square foot pieces of paper.
(These are not as easy to make; most paper is 8 Vi by 11
inches.) How many of your square inches would fit on this
square foot? Encourage the students to put the square inch on
the square foot to measure "it. Most students will see that it
will take many more than 12 square inches to fill the square
foot! A large square yard can be drawn on the black board or
a chart. How many square feet will fit into the square yard?
How many square inches will fit into the square yard? Stu-
dents who have experienced these activities seem to have
fewer problems with the geometry problems dealing with car-
peting, tiling, etc. Give the student the tiny square inch as a
bookmark so he/she can see the figure!

ANGLES

Adult education students often come to class with some
knowledge of angles. Most students know what a “360” or
“180” is when talking about cars! Students know or learn that
triangles have 180 degrees, however they don’t always really
believe it! Students can cut out triangles from paper, any size,
any shape. Have the students draw the sides with a ruler in
order to make sure the angles are accurate. Color the tips of
each angle and rip them from the triangle. No matter what
size or shape triangle, putting the three angles together will
make a straight 180 degree line.

CIRCLES

Students can carefully measure the circumference and diame-
ter of many different circles (lids, cans, etc.) Using a tape
measure is easiest, but using taut string and a ruler is also pos-
sible. Students wrap the string around the jars lids, tubes,
cans, etc. and mark the circumference and then measure using
a ruler. Yam, or other stretchy material, doesn’t do an accu-
rate job! After making a chart, the students can use a calcula-
tor to determine and record the ratio of circumference to di-
ameter. With careful measurement, the results should be
close to 3.1 or 3.2. Pi is an irrational number (it never ends)
that is close to 3.1415926. What is most important about this
activity is that students will develop a clear understanding that
pi is the ratio in any circle, no matter what the size!

In conclusion, there are many other hands-on activities that
will help adult students master geometry and spatial sense.
Other ideas can be found in the bibliographical materials
listed. Another resource is the National Institute for Liter-
acy’s Science and Numeracy Special Collection of websites.
This can be found at http://literacynet.org/sciencelincs/ Re-
member, geometry is an area that can be actively, concretely
explored. It can and should be an area of success.

Bibliography

Adult Numeracy Themes - Ohio Mathematical Planning Committee
Coordinated by Nancy L. Markus
Funded through 353 funds from the Ohio Dept, of Ed
August 1996

Available online http://literacy.kent.edu

Elementary School Mathematics: Teaching Developmentally
John A. Van De Walle
Longman, New York
1994

ISBN #0-8013-1184-5
Family Math

Joan Kerr Stenmark, Virginia Thompson, & Ruth Cossey

Lawrence Hall of Science

University of California, Berkeley, CA

1986

ISBN #0-91251 1-06-0

5

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