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A Phi Delta Kappa Professional Development Institute
MAKING ALGEBRA CHILD'S PLAY! ®
The One-Day Hands-On Equations® Workshop
(for teachers of grades 3 to adult)
~
This one-day workshop empowers teachers to understand and use
the visuaJ and kinesthetic HANDS-ON EQUATIONS patented teaching
methodology for successfully presenting essential algebraic concepts
to students in grades 3 to adult. Through this methodology,
practically all upper elementary and middle school students can
experience success with basic algebraic concepts, thus raising
student mathematical aspirations.
Hands-On Equations Inventor
Dr. Henry Borenson
Participants in this workshop will:
• Observe a live demonstration lesson with young children
• Experience the "teacher-as-the-coach" mode of instruction
• Understand the rationale for introducing algebraic concepts
early and concretely
• Understand effective pedagogy for use in the HANDS-ON
EQUATIONS learning environment
• Use the manipulatives to understand and solve such
algebraic equations as: 4x + 2 =2x + 1o,
2(x+ 4)+ x
=2x + 10, and 2x + (-x) + 3 =2(-x)+l5
• Use the manipulatives to set up and solve verbal problems
• Receive an individual set of HANDS-ON EQUATIONS
(Complete program for use with one student)
This workshop is recommended for upper elementary teachers,
middle and junior high math teachers, math coordinators,
school principals, and curriculum directors. The workshop
will also be of interest to teachers of the gifted (grades 2-5)
and teachers of LD students (grades 4- 12). All workshops
are conducted by certified instructional staff.
For additional infonnation contact· Sorenson and Associates. (800) 993·6284,
http:// www.Borenson.com, or Center for Professional Development and Services.
Pbi Delta Kappa, (800) 766-1156.
r---------------------,
I
I
I
I
I
I
A Phi Delta Kappa Professional Development Institute
MAKJNG AlGEBRA
(lcx:ation)
CHILD'S
PLAY! ®: A Hands-On
Equations® Workshop
(dale)
Distri ct/Organization - - - -- - - -- - - --
Workshop lime:
9:00a.m. to 4:00 p.m.
Regismnion Fee: $150
(Includes an individual set of
1 Name _ _ _ _ _ __ _ __ _ _ _ __ _ ___ HANDS-ON EQUATIONS
and lunch)
I Address - - - - - - -- - - - - -- - - - -- - - - - -- I City - - -- - - - - - - -- - - -- - - - -- 1
Make check(s) payable lo:
Phi Della Kappa
1 State/Zip - - - - - - - - - - - - - - - - Mail to:
Phillip Harris
1 Work Telephone ( _ ) --- - - -- - -- - - Phi Della Kappa
P.O. Box 789
1
Bloomington, IN 47402.(}789
I
Home Telephone ( _ ) - - -- - -- - - - -
TCM-10/98
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1998 FALL SCHEDULE
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matics. The comments of these professionals will
help in the continual improvement of the joumal.
A new twist on nines
Katie Morrison, a student in Joanna Pose's thirdgrade class at Park Avenue School in Des Moines,
Iowa. recently discovered the following twist on
the strategy for learning the multiplication facts for
nines. She wrote, " I have been multiplying by 9s
and I found an easier way. For the multiplication
sentence 9 x 7, l think of 9 + 7 = 16. The 'six' in
sixteen tells me that a 6 will be in the tens place
when I multiply 9 x 7. We tried this with all the
nine , My teachers had never thought of that
before. r made a chart:
9X I
9X2
9X3
9X 8
=Q_; think 9 + l = l Q
=l_: think 9 + 2 = I 1
=2 _; think 9 + 3 = I 2
=1_; think 9 + 8 = I 1"
Joan Barle
Anna Corbett
Beverly Faust
Linda C. Fowler
Clark E. Gardener
Leah Gamer
Cathy T. Ha le
Susan Heidemann
Kathy M. Higgins
Carolyn Hoyt
Marsha Ingrao
Garry Katz
Sally Livingston
Shauna Lund
Judy McLeod
Denise P. Masullo
Tawana Miller
Susan Nowosad
Linda Pazos
Yvonne Rothe
Cherie Schafer
Kathy Schanbacher
Martha Poole Simmons
Helen Ca erez Smith
Tamsy Sneed
Janet Steele
Mary Jane Stewart
Peggy Taylor
Theodora A. Wieland
Judith Beebe Zoeller
Congratulations to Cherie Schafer. who won the
$25 gift certificate from NCTM. Thank you to all
again for your participation.
In NCTM journals
Katie has not studied algebra yet, but the reason
her method works can be shown algebraically.
When we multiply any one-digit variable by 9, we
get the following:
Readers of Teachi11g Children Mathematics might
enjoy the following articles and deprutments in the
October 1998 issue of Mathematics Teaching in the
Middle School:
9n=( l0-l)n= lOn - 11
= I On - 11 + J0 - I 0
• "Roll the Dice-an Introduction to Probability,"
Andrew Freda
• "Cartoon Comer: Working on Study Habits; And
the Date Is Approximately ... ,.. Julie A. Fisher
• " Reflections on Practice: Exploring How One
Problem Contributes to Student Learning,"
Susan N. Friel
=(l On -
l 0) + I 0 - n
II)
= 10(11 -1) + ( 10 -
When we add this same var iable to 9, we get
9 + n = ( I 0 - l ) + n = I 0 + (11 - I ).
The variable expression in boldface type is the
same in both ca es, which shows that her method
will work for any one-digit number.
I try to nurture this type of exploration among
my students, but am especially pleased by Katie's
ability to communicate her conjecture so c learly.
Cheryl L. Arevalo
Des Moines Independent
Community School District
Des Moines. lA 50309-3382
Thanks!
The following NCTM members, who were selected
randomly from the readers of the journal, were generous enough to complete an extensive evaluation of
the March 1998 issue of Teachi11g Children MatheOCTOBER 1998
For a complete listing of the contents of this and
other NCTM journal , ee the NCTM Web site at
www.nctm. org. .A.
The Editorial Pmwl appreciates the imerest and I'OI11es
the views of those who take the time to send 11s tlleirt·omme/1/s. Readers wlto are commellling on articles are
encouraged to send copies of their correspondence to the
cwthors. Because of space limiwtions, lellet:\ and rejoinders from tmtlwrs an• limited to 250 wonts each. Leiters
are also edited for style and comem. Please doublespace all leiters that are to be consideredfor p11blicllli<1n.
SEEKING K-12 AUTHORS
We publish workbooks that help teachers teach. Write
for our free manuscript guidelines and complete catalog.
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Authors, Fducators Publishing Service, Inc.
31 Smith Place, Cambridge, MA 02138
800-435-7728, x 252 • www.epsbooks.com
TEACHING CHILDREN
VOLUME
ON
5 ,
NUMBER
THE
2
COVER
OCTOBER
c
0
T
N
E
N
1998
T
s
Readers' Exchange
Standards 2000:
Refining Our Efforts
Glenda Lappan
A Problem Worth Revisiting
Linda Schulman and Rebeka Eston
In "A Problem Worth Revisiting," Linda Schulman and
Rebeka Eston explore how to present a classic problem to
students, add new ingredients along the way, and return to
it without students' remarking, "Leftovers again?" Photograph by Sue Collum; all rights reserved. Readers are
encouraged to submit color photographs or color slides of
children actively involved in exploring mathematical ideas
for possible use on the cover, accompanied by a brief
explanation of the activity. Please send such submissions
to the "Readers' Exchange" department, Teaching Children
Mathematics, 1906 Association Drive, Reston, VA 201911593. Include a statement from the parents of the children
pictured, indicating their permission for NCTM to publish
the photograph, and include an appropriate credit line for
the photographer.
The mission of the National Council ofTcochcl"\ of Malhcmaucs is 10 provide
'·~•on and leadership '" impro' mg lhe 1cachmg and learnmg of mathematiC\ \0
1ha1 C\'Cry Student i~ ensured an equitable S1111ttfards-bascd malhemai iC\ education and every lcncher of mathematics
~~
Exploring Interplanetary
Algebra to Understand Earthly
Mathematics
Robert M. Berkman
Early Childhood
Corner
Kindergarten Is More Than
Counting
Kate Kline
Feisty Females: Using
Children's Literature with
Strong Female Characters
Karen Karp, Candy Allen, Linda G.
Allen, and Elizabeth Todd Brown
cn;ured the opponunity 10 grow profe>sionally.
Teaching Children Mathematics •s an ofTic13l JOUrnal of the Natrona! Council of Teachers
of Malhematic>. It 1s a forum for the c'changc of ideas and a ~ourcc of ac1iv1tics and pedagogical sLmtegiC\ for malhemniiCS education prc-K-6. II prc>Cnl~ new developments in curnculum. m~lrucuon. learnmg. and teacher cducat1on; interpretS the re>uh' of research: and
m general prov1de' infonna1ion on any aspt.'CI of lhe broad 'pectrum of malhcmauc' education appropriate for prcservice and in-service lc;&chcrs. The publ ication~ of lhe Council pre>ent a ,·ariel) of newpoml\. The vie";, e~prc~ or unphcd in lh1<, publication. unlc'~
other\\~
66
noted. ~hould not be mterprelcd 3'> oOicial po"uons of the Counc1l.
TEACHING CHILDREN MATHEMATICS
Journal Staff
Math by the Month
Using Mathematics to
Tell Stories
Stuart J. Murphy
Math Storybooks
Virginia Vogel Zanger
Problem Solvers
Guess the Weight!
Solution to the "How Much
Film?" Problem
Judith Olson
Kids + Conjecture =
Mathematics Power
Danise Cantlon
News from the Net
Cut-the-Knot
Beth Lazerick
Guide to Advertisers
Reviewing and
Viewing
Computer Materials, Paul G.
Becher and Douglas H. Clements: New
Books, David J. Whitin: Etcetera
Authors Needed for the
2001 NCTM Yearbook
JOHN A. THORPE, £tecu/llt Dm·ctor
HARRY B. TUNIS, Otrt.'clor of Pu/JiicnlltJIIf
ANDY REEVES, Dtrecwr of Eduarwl Sen·ice<
JOAN ARMISTEAD, Semnr Jaunwl Editor
DANIEL H. BREIDENBACH, Jtmnwl &litor
KATHLEEN CHAPMAN, Jounwll:.'tlimr
ANN M. BUTTERFIELD, Joumal Pmtluctimt Manager
NANCY K. GREEN, PAMELA A. HALONEN, Editors
BETH HAHN, Keybomrling SpecialiM
LYNN S. GATES, Supervt.wr of Rc•view St!n•ices
SHEILA J . BARKER, Revu•w Srrvirr.1 A w.<ta/11
ANN E. JENKINS, Re1•iew Srn'k<'< Ani11a111
ROSEMARIE ROLLO, Revie11 Sen ICI'\ A Hi.f talll
Marketing Staff
CYNTHIA C. ROSSO, Dtrutor of Marketing ServiC'es
TOM PEARSON, Ad1·emsmgl£fltilmr Mmwger
PATTY MARKUSSON, Adw•nlllnJI/&Itibits Manager
SANDRA S . BELSLEY, Adl·emlln!i/Exhtbits Assisltlnt
ROSA Q, SNIECHOSKI, Markrtmg Anistnnr
NCTM Board of Directors
GLENDA LAPPAN, Michig:m SlniC Universily; Presitlem. GAIL BURRILL,
Univcrsi1y of Wisconsin-M:~dhon; PttSI President. JOHN A. THORPE, NCTM;
&emrive Director. PATRICIA F. CAMPBELL, Univcrsily of Maryland 31 College
Park. ANN CARLYLE, Ellwood Elcmcnlary School. California. LORINO (TERRY)
COES Ill, Rocky Hill School. Rhode hland. DWIGHT A. COOLEY, Fore>l Oak Middle School, Texas. LINDA M. GOJAK, Ha" l..cn School Ohio. RITA C . JANES,
Newfound Educational As\OCtale\, Nc" foundland. RICHARD KOPAN, Calgary.
Albena. STEVEN J. LEINWAND, ConnccltCUI Depan:mcm of EducaltOn. TOM
LEWlS, Jane Addams Elemenlllr) School. lllinoi<. KA.R EN A. LONGHART, Flal·
head High School. Momana. JOHNNY W. LOTT, Universi1y of Momana. JOHN
VAN DE WALLE, Virginia Common"callh Umvcrsily.
Editorial Panel
DEANN HUINKER, Univcrsily of Wiscon<in-Milw:tukee: Chair
ANGELA G IGLIO ANDREWS, Scoll Elcmenlary School, lllinoi>
JAMES BARTA, U1ah S1a1e Univcr,ily
LILLIAN (NORDIE) DEAL, Chrislma School Districl. Delaware
DAVID FUYS, Brooklyn College, New York
CHARLES P. GEER, Texa> Tech Unhcr-;tty
RICHARD KOPAN, Calgary. Albena: Boord of Directors Unison
DANIEL H. BREIDENBACH, NCTM: Staff liaison
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Inc. Prinled in the U.S.A.
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67
Two students' recordings with invented code
(a)
There are 6 sea creatures in the tide pool.
Some are stars and some are hermit crabs.
How many sea stars and hermit crabs
are in the tide pool?
(b)
I see 8 pieces of fruit in a basket.
I see some pineapples and some pears.
How many pineapples and pears
do I see in the basket?
One student's use of invented code and conventional symbols
1.\ 7-;.8 2 f6.:: .r
lZ-t.i ~G> 6-% J-;.
y~L\ ~8
~~5
;:;3 '5-t ~--;. g
\~tO==- 8
quickly. The recordings shown in figure 4 were
completed by the same child on the same day. The
recording io figure 4(a ) shows the child's invented
system. Eston asked the child to describe the system. The child responded, "The number tells how
many. These [pointing to a number in a circle] are
the seeds that have not germinated, and these
[pointing to a numbe r within arcs] have. There are
lots of opposites. When you put them together [tracing the horizontal lines within the symbol between
the numbers], you get a total of eight seeds." Eston
asked if he could see another way to record this
information. The child's eyes grew wide. "I've got
to do thi over," he exclaimed. The recording shown
in figure 4(b) is the result. It is interesting to note
that even this child, when asked if he had found all
the ways, responded, "I think so."
76
,
~
..........
~
o~g;;.g
(a)
We planted 8 sunflower seeds in a cup.
Some have germinated. Some have not.
What might the cups look like now?
8
1~7-:-9
_.;;..
7·d... ::- 5I
(b)
We planted 8 sunflower seeds in a cup.
Some have germinated. Some have not.
What might the cups look like now?
Conclusion
Young children can sustain interest in significant
mathematical questions when those questions
relate to their personal and classroom experiences
and allow a variety of entry poiJ1ts. Presenting children with opportunities to share and record findings encourages the invention of terms and symbols that have meaning. Such inventions can lead
to the use of the conventional forms .
Revisiting a problem throughout the school year
affords opportunities for children to build important
generalizations. The revisits allow mathematical
ideas. strategies, and skills to unfold for both teachers
and children. Through recording sheets and annotated
observations, these problems yield important assessment data as changes are noted easily across revisits.
For most children, this type of problem may
TEACHING CHILDREN MATHEMATICS
continue to be explored during the first and second
grades as well. Older children may work with larger numbers and record their thinking in more
sophisticated ways. They may also explain how
they know that they have named all solutions. O ver
the course of three years, only one child at the
kindergarten level ended the year by finding all the
possible combinations of eight in a systematic
sequence and announced proudly that sbe knew
that she had found all the ways.
References
Atkinson. Sue. "A New Approach to Maths." In Mathematics
with Reason, edited by Sue Atkinson. Portsmouth, N.H.:
Heinemann Educational Books, 1992.
Brooks. Jacq ueline G.. and Martin G. Brooks. The Case for
Constructivist Classrooms. Alexandria. Va.: Association for
Supervision and Curriculum Developmem, 1993.
Ginsberg, Herbert P. Children's Arithmetic: How They Learn It
and How You Teach lt. Austin: ProEd, 1989.
Hughes. Martin. Children and Number: Difficulties in Leanring
Mathematics. Oxford: Basil Blackwell, 1986.
Mills. Heidi, Timothy O'Keefe. and David Whirin. Mathematics in the Making. Portsmouth. N. H.: Heinemann Educational Books, 1996.
Nelson, Doyal. and Joan Worth. How to Choose and Create
Good Problems for Prima1y Children. Reston, Va.: National Council of Teachers of Mathematics, 1983.
Polacco, Patricia. Jusr Plain Fancy. New York: Bantam Books, 1990.
Ward, Leila. I Am Eyes: Ni macho. New York: Greenwi1Jow
Books. 1978. A
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77
Robert M . Berkman
Ro/Jerr Berkman, [email protected]. teaches mathematics and science at the Salk School
for Sciences. New York, NY 10003, a public school. His imeresrs include concept developme/If. reclmology in education, and school-based managemem.
78
TEACHING CHILDREN MATHEMATICS
t the beginning of class with my
What the space visitor saw
fifth graders one day, l draw a dia(fig. I ) on the chalkboard. I
tell a story about a visitor from another galaxy who
spots a group of Earthlings sitting in an unusual
formation in a field somewhere on Earth.
Excited by her discovery, I continue, the space
alien takes a picture and re ports the following to
her cohorts waiting on the mother ship, which is
parked at a meter in the nearby Andromeda
galaxy: "1Uxo*!3... )()(4 JU*x ¢oo¢oo¢!" Anybody _w ho has studied foreign languages in high
school knows that this exclamation roughly translates into "Yo, dudes, I just caught an excellent
look at these Earth folks sitting in a field, and boy,
are they funny looking!" The official intergalactic
census taker looks at the picture and writes down
a number. Did he write "13," "7," or "23"?
Earthlings look at this picture and scratch their
heads. If they counted using their fingers, they
would find 13; and if they used the formation , 7.
But why should "23" be considered a possibility?
When prompted for reasons that the answer might
be 23, children can be very creative. One child
explained that the space alien added 13 to the
number 7 to make 20, then added another 3
because it was part of the 13, which became 23.
Perhaps a closer look at the visitor who made
this observation would help, I suggest. As luck
would have it, I happen to have a photograph that
she left behind (fi g. 2).
Still confused? A hint is that like us Earthlings,
our space travelers first organized their number
systems by using their fingers. This hint immediately leads students to jump out of their seats.
''She counts '23' because there is a two on one
hand and a three on the other !'' one student
explains. Everybody nods their heads in agreement. l counter with a question: "So, if I took one
of these people out of the field, how many would
the alien count now?" The students scratch their
heads. I fo llow up with another question: "Why
do we count the way we do?" One child suggests
grouping the people in twos and threes but cannot
explain how doing so would result in the number
"23 ." 1 suggest that they look at how their own
fingers are organized. The students hold their fingers in front of them, wiggling them back and
forth. A minute or two later, the lights go on.
One of my students explains that since this
alien has three fingers on one hand and two on the
OCTOBER 1998
The five-fingered space traveler
Earthly and extraterrestrial counting systems
How the Earthling
counts:
1 group of ten and
31eft over= 13
How the extraterrestrial
counts:
2 groups of five and
3 left over = "23"
79
other, they probably count in groups of five.
instead of ten like us. This extraterre trial groups
her people differently (fig. 3).
The student's explanation provokes a big
"aha;· and when I suggest that they can try this
system on their parents, and perhaps win some
money in the process, pencils and notebooks suddenly appear out of nowhere. The students are
hooked, and J am ready to dig deeper and stretch
their thinking further.
Suppose that the traveler shown in fi gure 4
gazed at the . arne field ? How would he group the
people he sees? How many ''leftovers .. would
there be?
"If you said '15,' " 1 explain to my students,
"you arc well on your way to a lucrative career
with the Galaxywide Census Service. Of course,
since you have lO report your findings to hundreds
of different civilizations, you should be prepared
to be able to supervise your census takers. For
example, suppo e that the previously mentioned
space traveler with eight fingers was employed by
you and . aid that he had counted '26' aliens. How
would you explain to another Earthling how many
aliens there are?" My students write out their
explanations in their notebooks and share their
ideas with one another.
Russel writes. "The alien in the problem saw
'26' aliens. If you want to find out what the real
an wer is. I have to translate it. 2 is the first digit,
so there are 2 groups of something. 2 groups of
eight is ixteen, and ixteen plus six i twentytwo." As I look over his explanation, l a k him to
tell us about the 6 that was added to the 16. He
adds to his explanation, "l got the '6' from the
second digit of '26.' These are the people who
couldn't make up another group, so they're left
over."
A different extraterrestrial counting system
80
When I am sati lied that the group understands
these types of problems, I give them an assignment. "Make up your own alien braintea er, and
put the answer to it on the back of your paper." We
talk about things that they should consider when
creating their teaser. For example, does the alien
always have to count people? Does the alien have
to have five or eight tingers? What number of lingers would be very difficult for counting large
groups? The students start their problem!. in class,
doing a rough draft of what the alien will look
like, the number of fingers on its hands, and the
answers. T look over their teasers before the s tudents go home to finish them for homework.
The following day. I have the children switch
problems with one another. T hey work on the
problems and discuss them. I then pop the question, Is there some way of solving this problem so
that it works with all kinds of aliens? Some students are confused until I clarify the question, Is
there some sort of rule or recipe for changing
alien number to Earth numbers? After working
on several of these problems, the students recognize a method for solving them; the goal then i to
have them put it into words. This act of generalization is the foundation for algebraic thinking.
The children work in groups to discuss their
ideas, each member having a job to perform. l
walk around and monitor the groups' progress.
checking to see that they have explained their
ideas clearly. A typical explanation by a tudent
reporter ounds like this: "You would count the
number of fingers and multiply it by the first
digit, and then add the remainder which didn't
make a group." At the same time, I ask students
whether they can abbreviate their explanations. I
"seed'' each group with examples of other abbreviations that we use, like "x" for "multiplied by"
and '·+." whic h can mean "plus" or "added to.''
The formula "Groups Digit x Fingers+ Remainder= Human Number" is a typical response. As
the group arrive at thi level of symbolis m. l
wonder aloud whether it i possible to abbreviate
the words. I give examples like ..Mr." for Mister,
and "em" for "centimeters." The groups come up
with formulas like "GD x F + R =£#,"where GD
is Groups Digit, F is the number of fingers on an
alien's hands, R is the "remainder" of people not
put in a group. and £#is the "Earth number." As I
monitor each group's progress, I ask whether
other people will know what these symbols mean,
which inspires students to create a key to the different letters. As the class concludes. we look at
the explanation. created by each group and compare the notations they used, noting which ones
are clear and which ones may confuse the reader.
The next day, a new problem (fi g. 5) appears.
TEACHING CHILDREN MATHEMATICS
one that is related to the previous situation but
requires children to manipulate their thinking into
a new context. This problem was easily understood by the children: they recognized that for
this group to be called "30," the fifteen people
must be divided into three equal groups, with
nothing left over. Therefore, the alien must have
five fingers. When I told them that another alien
had reported " 17 ," they solved the problem by
subtracting the seven "remainders" and saw that
the eight left over was the "group" of 1. This variation set them up for a more difficult problem that
combines the first two problems: Suppose the
alien reported "23" people? How many fingers
did she have on her hands?
I bad the children work together in groups on
this problem, emphasizing that I would expect a
clear _explanation of how they figured it out during our wrapping-up session at the conclusion of
the lesson. One group stated that "we took away
the '3,' because that was the leftover; and we
knew that the ones that were left over were in two
groups, so we divided it by '2' and we knew the
alien had six fingers. We knew we were right
because if we had two groups of six, it would be
twelve, plus the three would be fifteen, and that's
how many people there were." Other groups
employed different strategies. One group divided
the number of people by two and got seven with a
remainder of one, which they knew was inconect.
They knew that the answer could not be five fingers because that amount would make three
groups of five with no remainder, so they tried six
and checked to see if it would yield the correct
answer. Theirs was a guess-and-check strategy in
which incorrect answers provided clues about the
correct one.
But Is It Algebra?
I designed this activity several years ago to introduce fifth graders to the concept of base systems,
but while analyzing ways of solving the problems
with my students, I realized that a lot of informal
algebraic thinking is taking place. When I pointed
out this notion to my students, they were quite
horrified: "How can this be algebra?" one young
lady challenged. "Isn ' t algebra supposed to use a
lot of letters and be really hard? That's what my
brother told me."
Introductory concepts in algebra include the
study of unknowns, and in the context of taking a
census of space aliens, these unknowns, or variables, are very "friendly." When children look for
generalizable rules and express these relationships using symbols to represent operations and
variables, they are creating their own kind of algeOCTOBER 1998
A variation of the problem
Draw a picture of the space alien who reported that the
group of people shown here was "30:'
bra. However, just as children are encow·aged to
invent their own spellings in the "whole language" approach to writing, I encourage them to
invent their own symbols. My goal is not to have
children learn the formal notation of algebra but
rather to understand that their equations are a system of communication that others can use for a
practical purpose. My goal is for these children to
make the connections when they see the same
relationship expressed in a more conventional
way further down the road.
This informal algebra is not limited to creating
formulas for changing alien numbers to Earth
numbers; it can also be applied to setting up and
solving equations. For example. when the children are told that the alien numeral "23" is equal
to Earth number fifteen, they can reason that two
"full groups" plus three " leftovers" equals fifteen,
or, in the more conventional algebraic notation, 2x
+ 3 = 15. Many students realize that if two groups
plus three is fifteen, then two groups alone is
twelve, and one group is six; therefore, the alien
had six fingers. The connection to solving algebraic equations should be clear. Students are
incorporating "inverse operations" to solve this
problem; that is, when the equation says to add
three, they need to subtract three to "undo" it;
when they divide the remaining pieces by two,
they are ·'undoing" the multiplication. Again,
algebra is not being taught directly, but the
instruction incorporates algebraic thinking as a
vehicle for problem solving.
Starting Out Simple
Figure 6 gives examples of three kinds of
"alien" problems. The example that opens this
article is a complex problem that is difficult for
children to solve on their own, but it is a good
motivational tool because it presents children
with a mystery to be solved. Once they "see" the
way space visitors group things, an "aha" expe81
Examples of alien problems
Type A: The Earth number is known, and the
alien's hand is known. Find the alien number.
. •. • ••
Type B: The alien number is known, and the
number of fingers on the alien's hand is
known. Find out the Earth number.
~
Suppose a four-fingered alien saw this
illustration of these planets; what number would
she write down? ("22")
A space traveler with the hands shown here
wrote down "42:' How many objects did he
actually see? (26)
Type C: The Earth number is known, and the alien number
is known. Find how many fingers are on the alien's hand.
A space alien saw this group of stars and wrote
down "20." How many fingers did this alien have
on his hand? (8)
rience usually takes place. Many of my students
take the problem home and present it to their
parents as a brain teaser. From there, we learn
how to count in a variety of " planetary" number
systems. One example involves making a set o f
" handprints" of three different aliens and matching the number of items each would count given
the same number of objects (see fig. 7). The
hands shou ld be drawn large enough for children
to cou nt the fingers. Student should al o be
given fifteen ·'asteroid counters" that can be
!-.pread out and matched up with the hands. Some
Mudents may want to place the counters on top
of the hand to find the solution . whereas other
may want to u e loops of trings to show the
groupings. Through thi activity the children
develop an informal, intuitive approach to solving algebraic equations, which can be written
out in sentences using unknowns to represent the
missi ng information. Thus, to match ·'23," they
need to interpret it to mean "2 sets of alien hands
+ 3
fifteen,.. whereas "17'' means '·I set of
alien hands plus 7 more= fifteen."
A variation of this problem involves figuring
=
82
out how many objects were counted by comparing
the alien handprint with the alien number. The
same materials shown in figure 7 can be used, but
the correct numbers are matched with the handprints and the students are asked how many asteroids or spaceships each alien counted. Suppose
that an eight-fi ngered space creature wrote down
''24." What number would she be writing in earth
numbers?
The most complex problem involves comparing alien numbers with Earth numbers and figuring out the appearance of the alien hands. The
opening problem in this article is an example of
thi kind of problem. and children will struggle to
olve problems like it. It is important. therefore.
that the students have many opportunities to work
with the first two type of problems before moving on to the third type. Again, using the concrete
materials is important for students to develop and
master the concepts involved. lt is important to
support the children as they develop and master
their strategies and share them with others. Algebra is more than solving equations correctly; it is
understanding the various " attack'' methods.
TEACHING CHILDREN MATHEMATICS
An activity that combines all three types of problems
Handprint "A"
Handprint "8"
Handprint "C"
Fifteen asteroids are circling a planet.
Match the numerals each alien wrote out below with their handprints above.
·~· ·c· ·;·
•••••
G 0 EJ
(8)
I give the children ample time to master the
three basic types of problems: converting from
Earth numbers to alien numbers, converting from
alien numbers to Earth numbers, and figuring out
the appearance of a space alien after comparing
an Earth number with an alien number. After that,
I throw it open to them. For example, l have students make up their own "alien puzzlers" that use
these types of problems. I n one instance, a student
wrote out the question, showed the numbers,
illustrated the problem, and hid the alien hands
under a flap of paper. Another child made an overJay from a piece of transparent film, which could
be placed over the picture to show the groupings
that the alien would make as she counted the
number of relatives in a photograph. This activity
taps into children's creativity and Jove of the
fantastic.
As stated previously, my stude nts never believe
me when I tell them that what they are doing
involves algebra. They are so wrapped up in envisioning space creatures with different types of
hands zooming around the universe counting up
objects that it never occurs to them that what they
are doing involves such serious mathematical
concepts as base systems, inverse operations, and
working with variables. Perhaps the moral of this
story is that by Laking our students beyond the
stars, we can help them better understand mathematics here on Earth. A
OCTOBER 1998
(C)
(A)
The A IMS Education
Foundati on is now
accepti ng appl ications
to host A Week with AIMS
workshops f or 1999.
83
Kindergarten Is More
Than Counting
T
eachers of primary-grade children realize
the importance of helping their students
develop an understanding of number relationships. It is important to encourage the kind of
thinking that allows children to readily decompose
numbers into parts and know how to put parts
together to make a whole. This thinking sets the
foundation for working with larger numbers, using
reasoning to approach computation, and devel.oping sophisticated mental strategies. Parker (1998)
describes the importance of building what she calls
"fluency with small numbers." She defines fluency
as being able to take apart and put back together
numbers without even thinking, or with automaticity, and believes that before the third grade, children should be fluent with numbers to I 0.
So how does fluency develop? The NCTM's
curriculum standards for grades K-4 describe the
need to develop number relationships by composing and decomposing sets of objects (1989, 39).
Mathematics Their Way (Baratta-Lorton 1976), one
of the most popular K- 2 mathematics programs,
conta.ins a multitude of activities focusing on parts
and wholes with manipulatives. T hree activities
among the many with which readers may be familiar- The Hand Game, Lift the Bowl, and
Peek through the Wall-encourage children
to think about number combinations and
how smaller parts can be put together to
make the whole. In Peek through the Wall ,
Kare Kline. [email protected], teaches mathematics education courses at Westem Michigan Universiry, Kalamazoo, MI 49008. She is imeresred in investigating ways to develop
young children's understanding of number.
Tlris departmem addresses the early childhood reacher's need to support young children's
emerging mathematics understandings and skills in pre-K through second-grade classrooms in
a com ext rhar conforms with currem knowledge about tire way that young children/eam mathematics. Readers are encouraged to submit manuscripts ro editors Kate Kline, Departmem of
Mathematics. Western Michigan Universi~y. Kalamazoo. M/49008. or Sally Rober ts, College
of Education, Wayne State Universiry. Detroil. M/48202.
for example, children use a see-through wall to
divide a set of objects into two parts, such as three
and one for a set of four, and respond out loud,
"three and one." This task is repeated for the
remaining combinations for four (two and two,
four and zero).
The philosophy underlying these activities is
that with repeated exposure, children will eventually begin to remember these combinations. However, in my experiences teaching primary-grade
children, I found that many students did not
remember these combinations. As long as the
man.ipulatives were available, they were more likely simply to count by ones to figure out the proper
response. Even in the Peek through the Wall activity, many children continued to count those three
objects on the other side of the wall at the end of
the year! Many children continued to rely on
counting by ones as their primary strategy in ftrst
or second grade and even beyond.
I began, then, to ask myself some serious questions. Why do some children seem to get stuck in
counting by ones and never develop fluency with
numbers? Were all the concrete objects I was using
to promote fluency actually encouraging children
to count by ones? How could I help all my students
become more fluent?
As I began talking to colleagues and looking for
information to answer these questions, 1 encountered many stimulating ideas. [ found a wealth of
literature on developing young children's mental
imagery with numbers (Baroody and Standifer
1993; Payne and Huinker 1993; Van de Walle
1990; Wirtz 1980). The essential element of these
recommendations was to use specially patterned
arrangements of numbers, such as dot patterns on
dice and dominoes and ten-frames, to encourage
children to develop mental imagery of those numbers (see fig. 1).
The recommendation was made that these models be used in a quick-image format. To do so, flash
TEACHING CHILDREN MATHEMATICS
Examples of dot patterns and ten-frame models
•
•
•
•
•
•
•
One of the dot patterns for 7
one particular image on the overhead projector
screen for three seconds and then hide it from view.
Ask s.tudents to tell how many dots were shown
and to describe what they saw. This tactic will
encourage them to think about the parts of the
images. You may want to flash the image a second
time for three seconds, then hide it again, to give
students a chance to adapt their visual images. As
students become better at recognizing the patterns
instantaneousl y, second looks are rarely required.
It is important to flash the image f or only three seconds. lf you show it for too l ong, students w ill
work from the pi cture rather than w ith a mental
image, and i f you show it too bri efly, students w ill
not have time to form a mental image.
Using Quick Images
These activi ties can serve well as five-to-ten-minute
warm-up activities throughout the year. ln addition,
a variety of ways can be used to al ter the quickimage activities, maki ng i t possible to present new
ideas throughout the year as well. For example, cllildren may draw picwres of the quick images rather
than describe them orally, hold up a numeral card to
represent the number they saw, or show on their fingers the number they saw. Showing responses on
fingers leads to interesting discussions when you ask
two or tJu·ee chil dren to compare their representations. For a quick i mage for 6, some children will
hold up three fingers on each hand; some, four on
one hand and two on the other; and sri ll others, all
fi ve fingers on one hand and one on the other.
Fluency with 8
Encouraging children to fin d different ways to see
and describe the quick images is ex tremely valuable. I t creates an environment that promotes flexibl e thinking and sharing of ideas. Figure 2 shows
a sampl.e conversation that I had w ith students
about the ten-frame model for 8. I often asked
questions such as " How many would there be i f l
OCTOBER 1998
Ten-frame for 6
Ten-frames are filled from left to right, and
the top row must be filled before filling the
bottom row.
Classroom dialogue about the ten-frame model for 8
Students discuss how many dots they saw altogether and how they saw
them, after the following was displayed on the overhead projector for three
seconds and then hidden from view.
•
•
•
•
•
•
•
•
Teacher: How many dots did you see altogether?
Ping: Eight.
Teacher: What helped you remember that there were eight dots?
Ping: Well, I saw that the whole top row was filled, and that makes five.
And there were three on the bottom row, and five and three make eight.
Teacher: Did anyone see it in a different way?
Mohammed: I saw two empty boxes in the bottom row, and ten take away
two is eight.
Teacher: How else did others know that there were eight?
Aeysha: I saw a [group of] six. I thought of the three dots on top and the
three dots on the bottom row as six. And then there were two more on the
top, and six and two is eight.
(Display the ten-frame again, ask the following question, wait three seconds, and hide from view.]
Teacher: If I take away two dots from the ten-frame, how many will be left?
Celia: Six.
Teacher. Did anyone get a different number? [Pause] All right, so, Celia,
how did you think about this problem?
Celia: Well, if you take two away from the top row, that would leave three
on the top and three on the bottom. And three and three is six.
Teacher: Who else would like to explain their thinking?
Cherise: I knew it was six left, because three take away two is one.
Teacher: And how did you use that information to find the answer?
Cherise: Well, if you have one left, just add it to the five on top, and five
and one makes six!
85
----------
took two away (or added two)?" to encourage
thinking based on the identified parts of a number.
Cherise's response was typical of students who had
developed tluency with a particular number.
As l attempted to explain Cherise's reasoning in
numerical form to another teacher, I wrote the
following:
8=5+3
8 - 2=5+3-2
8 - 2 = 5 + (3 - 2)
8 - 2=5+1
8 - 2=6
---------
We both realized the complexity of the reasoning
that Cherise displayed and the flexibil ity with
which this would allow her to work with numbers.
I do not claim that Cherise was thinking in this
numerical way. However, she was able to decompose the 8 into two pruts-5 and 3-and operate on
only one of those parts. This flex ibility in working
with numbers will provide a strong foundation for
mental mathematics with larger numbers and for
developing computation strategies.
Numbers between 10 and 20
Although I would not expect kindergarten students
to be fluent with numbers in the teens, it is important to help them develop visual imagery for these
numbers using the ten-frame. I found that the most
effective model was two ten-frames in a horizontal
orientation with the ones in the top frame and the
ten in the bottom frame (see fig. 3). The reader may
wonder why it would not be more helpful to use a
vertical orientation for the ten-frames, with a full
ten-frame on the left and the ones on the right,
since this configuration matches the way we write
numbers in the teens. For example, we would write
a I below the full ten-frame to represent one 10 and
then write a 6 below the other ten-frame to repreHorizontal ten-frame model for 16; 6 and 10 make 16.
•
•
•
•
86
•
•
•
•
•
•
•
•
•
•
•
•
sent six 1's. Although this approach may make
sense to adults who already know how to write the
symbols to represent the numbers in the teens, my
kindergarten students did not find it very useful.
The names for the numbers in the teens fo!Jow a
pattern in that we say the ones fu·st and then we say
"teen" to represent 10 (except for the number
names for II and 12). J found that it was much
more helpful to work on the names of the numbers
first before working on their symbols. And showing the ones in the first ten-frame followed by the
ten, as in figure 3, matches more closely with our
names for those numbers.
Once my students knew the names for the teen
numbers and easily recognized them on the tenframe model, they were much more flexible in
using different orientations for the ten-frames.
They were able to recognize that 10 and 7 make 17
and that viewing the frames in a ve11ical orientation
does not change the number.
Assessment Ideas
ln addition to observing students during quickimage sessions, 1 also use two tasks to assess their
fluency with numbers. One is to connt out objects
into your hand, hide some, and leave the rest showing. When working on fluency for 5, for example,
count out five objects into your hand. Ask the child,
"How many do I have?" lf the child is able to
respond, "Five," then you may proceed. Take two
away and leave three visible in your hand. Ask, "I
took some away. If I have three showing, how
many did I take away?" Repeat for other combinations for five. If a child is able to respond correctly
and with ease for all combinations for a pa1ticular
number, then the child is fluent with that number.
A second assessment task is to use story problems
to identify those students who are transferring what
they know about numbers to real situations. One
example is to provide a list of toys at various prices
(see fig. 4). Several story problems can be created
from this list. such as "Imagine you have ten cents to
shop in this toy store. What would you buy?" Alex's
exceptional response to this problem is shown in figure 5. I knew from his response that he had a good
understanding of how to combine numbers to make
10. He ftrst drew pictures of all toys possible then
checked those that he had decided to buy. Interestingly, he chose to buy two bowling balls, a car, and a
truck and added 1 to the cost of the car (I + 3) and to
the cost of the truck ( l + 5) to get 4 and 6. I asked
him if he had spent aU of his money, and he replied,
"Yes, because 1 know six and four make ten."
In the years I taught kindergartners and worked
with other teachers using quick-image activities, I
saw dramatic increases in students' fluency with
TEACHING CHILDREN MATHEMATICS
numbers to J0. Previously, very few students
would have been successful on the tasks desctibed
here, especially the hand-hiding task. But after
repeated experiences with quick-image activities
throughout the year, the majority of students were
confident in their ability to think about numbers
and could put them together and take them apart
with flexibility and ease.
List of toys and prices for story problems
Concluding Comments
ft is important to understand that quick-image
activities are nor a replacement for concrete activities. Opportunities to work with concrete manipulatives are essential as children develop fluency
with numbers. However, equally important are
quick-image activities to encourage children to
move beyond a counting-by-ones strategy and
think about number combinations.
Fortunately, some new curricula, including
reform curricula based on the NCTM's Standards,
use dot-pattern and ten-frame images to develop
number sense. However, the only program that
uses these models in a quick-image format on a
continual basis is !nvestigations in Numbet; Data,
and Space (Kliman and Russell 1998), which uses
them throughout first and second grade as warm-up
routines to extended investigations. I invite readers
to use some of the quick-image activities described
in that program or in this article and to share with
colleagues the increased fluency that students will
undoubtedly display.
Alex's solution to toy-store word problem
What would you buy with ten cents?
References
Baratta-Lonon. Mary. Mathemmics Their Way. New York:
Addison-Wesley Publishing Co.. 1976.
Baroody. Arthur, and Dorothy Standifer. "Addition and Subtraction in the Primary Grades." In Resea rch Ideas for the
Classroom: Early Childhood Mathematics, edited by Robert
Jensen. New York: Macmillan, 1993.
Kliman. Marlene. and Susan Jo Russell. ·'Building Number
Sense." Ln Investigations in Numbet: Data, and Space, Grade
/.White Plains, N.Y.: Dale Seymour Publications, 1998.
Parker, Ruth. "Building Pacility wi th Numbers: Grade Level
Appropriate Expectmions for Mental Computation." Handout at community session, Watertown, Mass., 1998.
Payne, Joseph, and DeA nn Hui nker. "Early Numbers and
Numeration." In Research Ideas for the Classroom: Early
Childhood Mathematics, edited by Robert Jensen. New
York, Macmillan, 1993.
Van de Walle, John. "Concepts of Number." In Mathematics for
the Young Child. edited by Joseph Payne. Reston, Va.:
National Council of Teachers of Mathematics. 1990.
Van de Walle, John, and Karen Bowman Watkins. "Early Development of Number Sense." ln Research Ideas fo r the Classroom: Early Childhood Mathematics. edited by Robert
Jensen. New York: Macmillan, 1993.
Wirtz. Robert. New Beginnings. A Guide to the Think, Talk,
Read Math Cemer for Beginners. Monterey. Calif.: Curriculum Development Associates. 1980. A
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87
I
Using Children's
G
iris enter school more mathematics
ready tha n boys. By the time they
graduate from high school, however,
females have been outdistanced by males in the
numbe r of higher-level mathematics courses taken
and in the results of c rucial tests, such as the mathe matics portion of the Scholastic Achievement
Test (American Association of University Women
I 99 1). They are also much less likely to pursue
majors and careers that relate to mathematics.
Why? What happe ns to girls as they age from
eight to sixteen that puts the m at a disadvantage in
mathe matics?
I
The change in girls' interest and confidence in
mathematics during this developmental period is
part of a larger phenomenon. As girls progress
through the late el.eme ntary school years and e nter
middle school, they are frequ ently seen as losing
their daring nature (Brown and Gilligan
1992). They change from outspoken tenKaren Karp, Candy Allen,
Linda G. Allen, and
year-aids into young adolescents who
respond, " I don' t know," to most quesElizabeth Todd Brown
tions. S pecifically, researche rs Ouellette
Karen Karp. [email protected], reaches marhemarics educarion ar rhe Universiry of Louisville. Louisville, KY 40292. Her professional imeresrs focus 011 equiry issues in
marhemarics educario11. Ca11dy Alle11. cca33 / @aol.com. and Linda Alle11 teach eight- and
nine-year-old childre11 ar Goshen Elememary School, Goshen. KY 40026. They share a11 imer·
est ;, connecti11g children's lirera111re to mtllhematics and scie11ce. Todd Brown.
[email protected]. reaches fo urrh- a11d fifth-grade s111de111s ar Wheeler Elementary School.
Louis1•ille, KY 40291. Her pro[essio11al imeresrs are ;, children's literature, particularly
books wirh strong fe male proragonists.
88
terature witl
and Pacelli ( 1983) report that adolescent females
often lac k a " hardy personality," which they define
as the propensity to look forward to changes and
challe nges, feel in control of one's life, be responsible for one's owD actions, aDd survive unfavorable conditions.
Interestingly, Ouelle tte and Pacelli also connect
the characteristics of a hardy personality to successful problem-solving skills. Although the proble m-solving skills that
they discuss are not exclusively mathematical, their findings could connect
with young fe males' loss of interest
in mathe ma tics at this same age.
Whe n female students lack confidence in their ability to prevail in
novel or chall.enging situations, the n
approaching mathematics problems
with risk-takj ng behaviors seems
unLikely. Mathematics educators
suggest that the kinds of
mathematical problem-solving
strategies that we teach
should directly relate to
solving the problems
that students face in the
real world. The reverse
seems true, as well. Girls
who do not develop hardy inner
personalities, or what we
describe as the characteristics
of a "feisty fe male," may
not be prepared to tackle
proble m-solving situations. How can we build
this stre ngth a nd nurture these stude nts to mature
mathematically?
One strategy is to present girls with
models of hardy female personalities
through examples of young proble m
solvers found in childre n's lite rature. These
characters can act as springboards to mathematics lessons as teachers link these
TEACHING CHILDREN MATHEMATICS
•
•
5trong Female Characters
"feisty females'' to mathematical actiVIIIC. . Children's literature is a way to supply a powerful context in which to build mathematical tasks and is a
strong inlluence in the development of children's
perceptions about their world. Through books we
can find young and feisty female who face adventure bravely; make hard decisions; solve problems
OCTOBER 1998
of their own and of others; and u e their connection.
to. and relationships with, other people as ways to
both develop a sense of spirit and face new challenges. Walkerdi nc states that stories are "one of the
powerful ways in which constructions of gender are
authorized and regulated" ( 1994. 128). Gilbert notes
that "through constant repetition and layering, story
89