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Learning Math With Manipulatives -- The Abacus

The abacus has been around in various formsquite a simple process. Begin by representing
for over 2300 years. It was used for variousthe first number. Add the value of each place
counting and operational tasks. One mightvalue in the second and subsequent numbers
even call it the original math manipulativeone at a time beginning with the lowest place
(unless you count fingers and stones). In myvalue  and  regroup  as  necessary.
younger years, abaci were relegated to the
bottom shelf or used as a toy for theConsider this simple example, 178 + 255. The
kinesthetic kids. These days, abaci can meetstudent would represent 178 on the abacus to
the same fate that the abaci of my youth did.begin. She would then add five to the ones
The first known abacus, the Salamis tablet,row. Since there aren't five more beads to
collected dust for over 2100 years. For alladd, this first move would also involve
those lonely and banished abaci on dustyregrouping. The student would move the two
shelves everywhere, I dedicate this articleremaining ones, then regroup by sliding all
on how to represent, add and subtract wholeten ones back and replacing them with a ten.
and  decimal  numbers.She would then move three more beads since
she already moved two of them for a total of
As most teachers know, the use offive. Since there was some regrouping, there
manipulatives by younger elementary studentswould now be eight tens. The students needs
helps them to understand the concepts ofto add five more, so there would be another
place value and operations later on. In myregrouping, this time of ten tens to make a
search for a variety of manipulatives tohundred. Finally, the student moves two
teach number sense, addition and subtraction,additional hundred beads; this time
I came across a convenient tool in theregrouping isn't necessary. If everything was
abacus. I'm sure it was no coincidence thatdone correctly, the student would end up with
each row on the abacus included exactly tenfour hundreds beads, three tens beads and
beads, but there was no operators manual withthree  ones  beads.
the abacus I found. When I found an
instruction manual several years later, IA variation on addition is to add the second
found that the manufacturer of the abacus sawand subsequent numbers from the highest place
it as no more than a counting device and hadvalue  to  the  lowest  place  value.
no idea of the place value power inherent in
the  design.Subtracting is much the same as addition, but
it involves "removing" beads. The procedure
Representing  Numbers  With  a  Dusty  Abacusfor subtracting is to represent the first
number then to subtract the value of each
When I first started using an abacus as aplace value in the second and subsequent
manipulative in math class, I was teachingnumbers beginning with the highest place
grade six. In the grade six curriculum,value.
students were supposed to represent whole
numbers greater that one million and decimalConsider this example, 3.252 - 1.986. The
numbers to thousandths. If you count thestudent would first represent 3.252 using the
number of places from one million down toabacus. He would begin by subtracting one
thousandths, you get ten places.one. This is fairly straight forward because
Coincidentally, the abacus had ten rods ofthere are enough ones available. In the next
ten beads each. I'm sure what I discoveredstep, though, the student has to subtract
was discovered long ago, and somenine tenths from two tenths. He begins by
manufacturers probably even send out bettersubtracting two of the nine tenths, but he
instruction manuals that make note of this,then has to regroup one of the remaining ones
but at the time, it was a completely newinto ten tenths. Once he has ten more tenths,
discovery.he can subtract the remaining seven tenths.
He continues by subtracting eight hundredths
To make a long story short, I assigned eachfrom five hundredths, and again, he has to
row a specific place value starting withregroup, this time, one of the tenths into
millions at the top, and thousandths at theten hundredths. The final step also involves
bottom. One could use a strip of tape or anregrouping since six thousandths must be
indelible marker to label the rows. Tosubtracted from two thousandths. In the end,
represent a number, a student would simplythe student hopefully ends up with one one,
move the number of beads for the value oftwo tenths, six hundredths, and six
each place in the number they were given. Forthousandths  (1.266).
example, the number 325,729 was represented
by moving three of the hundred thousandsSubtraction could also be accomplished by
beads, two of the ten thousands beads, fivesubtracting the lowest place value first, but
of the thousands beads, seven of the hundredsthis sometimes means more manipulations of
beads, two of the tens beads and nine of thethe  beads which means more chance for error.
ones  beads.
Conclusion
I didn't have a class set of abaci, so I made
up little sketches of an abacus (six or soThe use of the abacus takes a little bit of
per page) and students showed representationstime to master. It is important that the
of  numbers  using  these.teacher and the students use the correct
place value terminology (e.g. "regroup ten
Adding and Subtracting Numbers With ahundreds to make one thousand" instead of
Polished  Abacus"turn ten green beads into one blue bead"),
so the concepts of place value, addition, and
Once students are familiar with representingsubtraction can be transfered to mental
numbers using an abacus, they can move ontostrategies and paper/pencil algorithms.
adding and subtracting numbers. The idea ofRemember, the best way to dust and polish an
adding using an abacus and place value isabacus is with little fingers!



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