Understanding Abacus Math

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In our concern about the memorization of math facts or solvingproblems, we must not forget that the root of mathematical study isthe creation of mental pictures in the imagination and manipulatingthose images and relationships using the power of reason and logic.–Mindy HolteMathematics is changing.

• Our world is becoming ever more mathematized.• Math itself is expanding, e.g., fractals, statistics, encryption.• Computers and calculators both change and are changed by

new advances in math.• Math itself is becoming more visual, e.g., graphing calcula-

tors, fractals, encryption. More geometry is being used. Mathematics education is changing.

• More known about brain and learning, e.g., child understress stops learning.

• Real learning means making connections.• Learning styles: majority of children do not learn best by lis-

tening. A study showed teachers spend > 80% time talking. • Visual thinkers, the gifted and many with LD, find rote

memorizing difficult. They need to see whole picture, notsmall steps.

• Standards suggest what, when, and how math is taught.• More than arithmetic must be taught: Geometry, algebra,

probability, and statistics are all on state exams from K on.Economics of mathematics education.

• In international studies, such as TIMSS and PISA, the U.S.scores low compared to other countries.

• In 2004 of the 1.2 million students who took the ACT test,only 40% were deemed ready to study college algebra.

• 25% of college students take remedial math, 37% in CA.• 50% of engineers & computer scientists receiving Ph.Ds are

foreign born.• Only 51% of patents going to U.S. citizens, down from 90%.

Why understanding is necessary.• Understanding aids memory: 93 min to learn earn 200 non-

sense syllables, 24 min to learn 200 words of prose, and 10min to learn 200 words of poetry.

• Better learning.• Less memorization and review needed.• Essential for applying to real problems.• Impossible to memorize advanced math.

Counting, a rote activity, does not help a child master mathconcepts.

• The alphabet example of how we teach children simpleadding shows some of the difficulties. (F + E = K)

• Ignores place value.

AL Abacus + Card Games = Understanding Math

Yellow is the SunYellow is the sun.Six is five and one.Why is the sky so blue?Seven is five and two.Salty is the sea.Eight is five and three.Hear the thunder roar.Nine is five and four.Ducks will swim and dive.Ten is five and five.

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An even number.

An odd number.

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3 2Part-part-whole. Helps

children solve problemsand write equations.


• Young children don’t realize counting represents quantity.• Very error prone. Children under 6 are not good counters.• No motivation to learn facts.

• Inefficient and time-consuming.Visualizing quantities.

• Babies, at 5 months, can add and subtract up to 3.• Group by 5s. Impossible to imagine 8 objects without

grouping.Place value is the most important concept of arithmetic.

• Teach math way of counting: after 10, say ten 1 (11), ten 2(12), ten 3 (13), . . , 2-ten (20), 2-ten 1 (21), . . . , 9-ten 9 (99).

• All Asian children learn math with math way of namingnumbers they understand place value in first grade. Aver-age U.S. child understands it at the end of fourth grade.

• Essential for understanding algorithms.• Place-value cards: encourage reading in normal order; start-

ing with ones column and then tens columns is backwards.• Essential to use 4-digit numbers to understand trading

(carrying).What makes a good manipulative (according to Japanese).

• Easily visualized.• Representative of the structure of mathematics.• Easily managed.

The AL Abacus.• Grouped in fives and tens.• Used for operations, strategies, money.• Evens and odds; also needed for side 2 of abacus.

Some addition strategies.• What makes 10: seen on abacus, Go to the Dump game.• Adding 9: complete the 10.• Two 5s: two fives = 10; then add “leftovers.” For 8 + 7, the

leftovers are 3 + 2; so the sum is 15. See figure at left.Learning the facts.

• Strategies first: It takes time for new strategy to becomeautomatic.

• Games far superior to flash cards. • Timed tests and graphs.Importance of mental computation.

• Understanding more important than procedures.• Develops number sense (common sense with numbers).• Necessary for estimating.• Easier to start at the left: e.g. 34 + 48 = 34 + 40 [74] + 8 [82].

Adding 4-digit numbers on the abacus.• Important for understand trading: that 10 ones = 1 ten, 10

tens = 1 hundred, 10 hundreds = 1 thousand.• Children need to write down on paper what happens after

number is added on the abacus.

AL abacus (side 1) with37 entered.

AL abacus (side 2) with6438 entered.

Place value cards.

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Transforming 9 + 4 into 10 + 3.

Seeing the sum of 8 and 7 as 10(2 fives, the black beads) and5, the number of white beads.


Why thousands so early.• To appreciate a pattern, we need at least three samples.

• To understand trading, the child must trade 10 ones for 1ten, 10 tens for 1 hundred, and 10 hundreds for 1 thousand.

Introducing subtraction.• Only after addition is mastered. It is psychologically

negative.• Going up is easier for some facts; e.g. 11 – 9. Used for mak-

ing change. Also, avoid “take away.”• 15 – 9: subtract 9 from 10.

Skip counting (multiples).• Needed for multiplication facts, fractions, and algebra.• Start as soon as 1-100 is understood; use patterns.

Skip counting pattern explanations.• Twos. The second row is 10 plus the first row. They are the

even numbers.• Threes. Consider the ones: they increase starting at the low-

er left with 0 (30) and continue up the first column andover to bottom of the second column and to the thirdcolumn. Next consider the tens: 0, 1, 2 in each column.Sum of the digits: 3 in the first column (1 + 2, 2 + 1, 3 +0), 6 in the second column, and 9 in the third.

• Fours. The second row is 20 more than the first row, everyother even number.

• Fives. They have an obvious singsong pattern.• Sixes. The first row is the even 3s. Second row is 30 more

than the first row.• Sevens. Within each row the tens increase by 1. The ones in-

crease by 1 starting at the upper right (21) and continu-ing down the column and over to the next column.

• Eights. In each row the ones are the decreasing even num-bers. The second row is 40 more than the first row, alsoevery other 4.

• Nines. The sum of the digits in all cases is 9. The ones de-crease while the tens increase. The second row has thedigits of the first row reversed, as shown by the arrow.

• Skip Counting Memory game.

Multiplication seen visually on abacus. (See figures at left.)Goal of math—not to turn students into $7 calculators— butthinking persons who can apply math to new situations.Galileo: “The great book of nature can be read only by thosewho know the language in which it was written. And this lan-guage is mathematics.

Ref: Cotter, Joan. Using Language and Visualization to TeachPlace Value. (for NCTM members:http://my.nctm.org/eresourcesview_article.asp?article_id=2261)

A downloadable PowerPoint presentation on this topic is availableat http://www.alabacus.com.

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Skip counting patterns.

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6 x 3 (6 taken 3 times).

Subtracting 15 – 9 by subtract-ing the 9 from 10, then 1 + 5 = 6.

7 x 5 (7 taken 5 times).

888-272-3291 • [email protected] • www.RightStartMath rev Oct. 2004

Fractions in general. Most problems with fractions occur because thechild doesn’t really understand what a fraction is.Children need a visual representation, not a “fish tank.”Start early. Linear model first; pies later. (Pies can’t show fractions > 1.)Informal fraction work first. What is 3⁄7 divided by 1⁄7?Word fraction means to break, derived from Latin “frangere.”

Fraction chart. See figure.Young children like to construct the chart as a puzzle.Start with strips of paper same length as the “1.” Then divide intohalves and fourths.Reading fractions: use ordinal names except one-half.

Fraction stairs. Made with unit fractions (fractions with a 1 in the numera-tor). See figure. How does it differ from ordinary stairs?Which is smaller (less), 1⁄4 or 1⁄5?Which is larger (greater), 7⁄8 or 9⁄10?Meaning of 2⁄3: either two 1⁄3s (easier to understand) or 2 divided by 3.What is missing in each row to make 1? [1⁄2, 2⁄3, 3⁄4, 4⁄5 and so on]

Game: Concentrating on One. A memory game whose pairs equal 1.

Comparing fractions. Lay out halves, fourths, and eighths. Ask the fol-lowing questions and ask for explanations:1. How many halves do you need to make a whole? [2]2. How many fourths do you need to make a whole? [4]3. How many eighths do you need to make a whole? [8]4. What is the same as two 1⁄4s? [1⁄2]5. Which is more, 1⁄4 or 1⁄8? [1⁄4]6. What is the same as two 1⁄8s? [1⁄4]7. What is a half of one-half? [1⁄4]8. What is a half of one-fourth? [1⁄8]9. How many eighths do you need to make a half? [4]

Game: Fraction War. Make the “ruler” chart as shown in the figure. Usecards with 1s, 1⁄2s, 1⁄4s, and 1⁄8s; card with higher fraction takes both.

Relating halves and fourths (quarters). The word quarter means one-fourth; e.g. of a dollar, of an hour, of a whole note.1. How many quarter notes equal a whole note? [4]2. What is one fourth of a dollar? [25¢]3. What is one half of an hour? [30 min]

Finding fractions of objects and figures. See the figure. Extra lines are“perceptual distracters.”

Given a fraction, draw the whole. See the figure.

Mixed fractions. Fractions such as 11⁄4 are difficult for children. Demonstrate with fraction pieces.Use examples such as 11⁄2 triangles.

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The fraction chart.

The fraction stairs.

The “ruler” chart.

Crosshatch (shade)1⁄⁄⁄⁄2 of each figure.

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Draw the whole.

© Joan A. Cotter 2007 • www.ALabacus.com

GO TO THE DUMP(From Math Card Games: 300 Games for Learning and Enjoying Math. Fourth edition byJoan A. Cotter (2005); published by Activities for Learning: Hutchinson, MN.)

Objective To learn the combinations that total 10

Number of players 2 to 4

Cards 4 or 6 of each basic number card 1 to 9

Deal Each player takes 5 cards; the remaining cards face down form the dump, or stack.

Object of the game To collect the most pairs that equal 10

Preparation Before starting, the players check over their hands for pairs that total 10.To do this, they look at each card in turn, determine what is needed tomake 10 and look for that number among their other cards. (Some chil-dren may need to spread the cards out on the playing surface.) Encour-age the children to use their abacuses.Store paired cards face up on two piles. (This allows verification andkeeps the cards shuffled for the next game.)

Play When all are ready, the first player asks the player on the left for a num-ber needed to complete a pair. If the second player has it, it must given tothe first player, whereupon the first player receives another turn. If theplayer asked does not have it, the player says, “Go to the Dump,” whichis also the signal to begin a turn. The second player takes a turn by ask-ing the player on the left and so forth.Meanwhile, the first player concludes the turn by picking up the topcard from the dump. Additional turns are not given even if a neededcard is picked up. A player running out of cards takes five more cards, but the turn is end-ed. When the dump is exhausted, players may ask any player (not onlythe players on their left) for a card.At the end of the game, players combine their two stacks and comparethe heights. (Counting the cards is too time consuming.) No shuffling isnecessary for subsequent games.

Note One way to monitor two games simultaneously is to sit in the middle ofa figure 8. See the figure below.

g1

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Monitoring two games: g1 isone game, g2 is the othergame, and T is the teacher.

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6 is neededwith 4 tomake 10.

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(The PowerPoint presentation and handout available at www.ALabacus.com)

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ROWS AND COLUMNS(From Math Card Games: 300 Games for Learning and Enjoying Math. Fourth edition byJoan A. Cotter (2005); published by Activities for Learning: Hutchinson, MN.)

Objective To practice adding three numbers

Number of players 2-3. (With three, the game may become too long.)

Cards 12 of each basic number card 1 to 9

Layout Sixteen cards are laid face up in a 4 x 4array. The remaining cards form the stock.

Object of the game To collect the most cards

Play During a turn, the player checks each rowand column for two or more cards that total15. The same card cannot be used for both arow and a column.In the figure, the 7 and 8 can be collectedfrom the first row. Also the 5, 1, and 9 fromthe first column and the 6, 7, and 2 from thelast column can be collected. Alternately, the5, 3, and 7 from the second row and the 9and 6 from the last row could be picked up.After a turn, fill in the array for the next player. If a player cannot play,she skips her turn and replaces the four corners.

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ROWS AND COLUMNS SOLITAIRE(From Math Card Games: 300 Games for Learning and Enjoying Math. Fourth edition byJoan A. Cotter (2005); published by Activities for Learning: Hutchinson, MN.)

Objective To practice adding three numbers

Number of players 1 or more. (Explain that in a “solitaire,” they want to beat the cards.)

Cards 6 of each basic number card 1 to 9

Layout Sixteen cards are laid face up in a 4 x 4 array. The remaining cards formthe stock.

Object of the game To collect all the cards

Play From each row or column, collect the facts of cards that total 15. Then fillin the gaps from the stock and again collect the facts. Continue until thestock is exhausted, at which time the cards may be combined regardlessof rows or columns. If no mistakes have been made, the cards will comeout in even groups.


CORNERS GAMEObjectives To learn the sums that total 5, 10, and 15

To mentally add 5, 10, and 15To enjoy using mathematics

Number of players 2 to 6

Cards The set of Corners cards; each of the 50 cards has 4 colored numbers. Notwo cards are alike.

Deal The stack of cards is placed face down on the table. Each player draws 4cards initially and draws another card every time after playing a card.Players’ cards are laid out face up in full view of all players.

Object of the game To join the cards to make the highest possible score.It is suggested that before playing this game, the player first play theCorners Exercise (Lesson 59). Emphasize that in this game, in contrast tothe Corners Exercise, play is to the last card played (or to a Corner).

Play The player with the card having the lowest green number starts by plac-ing that card in the center of the table. In the event there is more than 1card with the lowest green number, of those cards with the lowest greennumber, the card with the lowest blue number starts.

The player to the left of the one who started takes a turn, playing 1 cardwhile observing the following rules:

1.The colors of the numbers where 2 cards join must be the same. (Onenumber will be upside down.)

2. The sum of 2 adjoining numbers must total 5, 10, 15, or 20, or the 2numbers may be the same. However, only those sums totaling 5, 10, 15,or 20 will add to the score. That is, joining 1 with 1, 2 with 2, 3 with 3, 4with 4, 6 with 6, 7 with 7, 8 with 8, or 9 with 9 is legal, but does not givepoints for scoring. See Fig. 2 for an example.

3. Play is to the last card played or to a Corner. A Corner is a space wherea card could fit by joining edges with 2 or more cards. See Fig. 3 on thenext page. All edges must match according to the rules above. Beginnersusually ignore the Corners.

Players continue by taking turns playing 1 card. A player must play ifpossible. The game is over when the all the cards are played or no onecan play. For beginners the winner can be the first person reaching 100.

Scoring Players do their own scoring, which is to done mentally with only the fi-nal result of each turn written down. See scoring in the figures below. (Aperson playing to a Corner may write down the results for each sidejoined.)

It may be helpful to practice with the players the following mental exer-cises before scoring on paper:

What is 20 + 10? [30] What is 40 + 10? [50]What is 10 + 5? [15] What is 30+ 5? [35]

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What is 75+ 10? [85] What is 65 + 10? [75]What is 25 + 5? [30] What is 55 + 5? [60]

To add 15, first add the 10 and then add the 5.What is 30+ 15? [40, 45] What is 50 + 15? [60, 65]What is 65 + 15? [80] What is 35+ 15? [50]

Sample game Shown below is a progression of a sample game between two players.Four type styles are used to designate the four colors.

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SKIP COUNTING MEMORYObjective To learn the skip counting patterns on page 7.

Preparation To prepare the envelopes, see page 9. The players use the envelopes forreference during the game to memorize the patterns.

Number of players 2 or 2 teamsCards Each player chooses an envelope and removes the cards. Mix the cards to-

gether and shuffle lightly. Lay the cards out face down in a 5 by 4 array.Object of the game To be the first player to collect in order the complete set of cards

Play The first player turns over one card so both players can see it. If it is theneeded card, the player collects the card and receives another turn. If it isnot the needed card, the card is returned. Next the second player takes aturn. Turns alternate until one player has picked up all ten cards.Stress the importance of returning the cards to the correct envelopes following a game.

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A game in progress: Theplayer on the left collectsthe 2s while the player onthe right collects the 5s.

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MULTIPLICATION MEMORYObjective To help the players master the multiplication facts.

Cards 20 cards: use 10 basic number cards, 1 to 10, and one set of product cards.Number of players Two. Beginners should sit on the same side of the cards.Object of the game To collect the most cards by matching the multiplier with the product.

Layout Lay the basic number cards face down in two rows. To the right in separaterows lay the product cards.

Play The first player turns over a basic number card and states the fact. For ex-ample, if the card is 4, the player states, “Three taken four times is 12.” Hethen decides where it could be among the product cards. If he is correct, hecollects both cards and takes another turn. If he is not, both cards are re-turned face down in their original place, and the other player takes a turn.

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[email protected] • www.ALabacus.com8-05

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Cut out and tape each of the 10 skipcounting layouts onto 10 envelopes. In-sert into each envelope 10 cards with thesame numbers as are on the envelope.

Skip Counting Patterns

888-272-3291 • [email protected] • www.RightStartMath.com rev Oct. 2004

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Education

Understanding Abacus Math