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- 1. 1 3 Enriching Montessori National Math Crisis Mathematics with Visualization • 25% of college freshmen take remedial math. • In 2009, of the 1.5 million students who took theHandout and by Joan A. Cotter, Ph.D. ACT test, only 42% are ready for college algebra.Presentation: JoanCotter@ALabacus.com • A generation ago, the US produced 30% of theALabacus.com world’s college grads; today it’s 14%. (CSM 2006) AMS Fall Conference October 22, 2010 San Diego, California • Two-thirds of 4-year degrees in Japan and China are in science and engineering; one-third in the U.S. 7 • U.S. students, compared to the world, score high at 4th grade, average at 8th, and near bottom at 12th. 5 2 • Ready, Willing, and Unable to Serve says that 75% of 7x7 VII 17 to 24 year-olds are unfit for military service. (2010) © Joan A. Cotter, 2010 © Joan A. Cotter, 2010 2 4 Key Decisions of a First-year Math Education is Changing ‘Turnaround’ Principal • The field of mathematics is doubling every 7 years. D. Duke and M. Salmonowicz • Math is used in many new ways. The workplace needs analytical thinkers and problem solvers. Educational Administration Management & Leadership, 2010 • State exams require more than arithmetic: including 1) Elimination of an ineffective instructional program. geometry, algebra, probability, and statistics. 2) Creation of a culture of teacher accountability. • Brain research is providing clues on how to better facilitate learning, including math. 3) Development of an effective reading program. • Increased emphasis on mathematical reasoning, less emphasis on rules and procedures. © Joan A. Cotter, 2010 © Joan A. Cotter, 2010
- 2. 5 7 Calendar Math Drawbacks Yellow is the Sun• The calendar is not a number line. Yellow is the sun. Six is five and one. • No quantity is involved. Why is the sky so blue? • Numbers are in spaces, not at lines like a ruler. Seven is five and two.• Children need to see the whole month, not just part. Salty is the sea. • Purpose of calendar is to plan ahead. Eight is five and three. • Many ways to show the current date. Hear the thunder roar. Nine is five and four.• Calendars give a narrow view of patterning. Ducks will swim and dive. • Patterns do not necessarily involve numbers. Ten is five and five. • Patterns rarely proceed row by row. –Joan A. Cotter • Patterns go on forever; they don’t stop at 31. © Joan A. Cotter, 2010 © Joan A. Cotter, 2010 6 8 Memorizing Math Counting Model Drawbacks Counting: Percentage Recall • Is not natural. Immediately After 1 day After 4 wks • Provides poor concept of quantity.Rote 32 23 8Concept 69 69 58 • Ignores place value. • Is very error prone. Math needs to be taught so 95% is • Is inefficient and time-consuming. understood and only 5% memorized. • Is a hard habit to break for mastering Richard Skemp the facts. © Joan A. Cotter, 2010 © Joan A. Cotter, 2010
- 3. 9 11 Recognizing 5 Materials for Visualizing • Representative of structure of numbers. • Easily manipulated by children. • Imaginable mentally. Japanese Council of Mathematics Education 5 has a middle; 4 does not. © Joan A. Cotter, 2010 © Joan A. Cotter, 2010 10 12 Materials for Visualizing Materials for Visualizing“In our concern about the memorization of math “Mathematics is the activity offacts or solving problems, we must not forget creating relationships, many of whichthat the root of mathematical study is thecreation of mental pictures in the imagination are based in visual imagery.”and manipulating those images and relationships Wheatley and Cobbusing the power of reason and logic.” Mindy Holte (E I) © Joan A. Cotter, 2010 © Joan A. Cotter, 2010
- 4. 13 15 Materials for Visualizing Spindle BoxThe role of physical manipulativeswas to help the child form those 0 1 2 3 4visual images and thus to eliminatethe need for the physicalmanipulatives. Ginsberg and others © Joan A. Cotter, 2010 © Joan A. Cotter, 2010 14 16 Number Rods Spindle Box 5 6 7 8 9 © Joan A. Cotter, 2010 © Joan A. Cotter, 2010
- 5. 17 19 Bead Frame Challenges Adding 4+3= 7• Distracting: Room is visible through the frame.• Not visual: Beads need to be grouped in fives.• Inconsistent with equation order when beads aremoved right: Beads need to be moved left.• Hierarchies represented sideways: They need to bein vertical columns.• Trading done before second number is completelyadded: Addends need to combined before trading.• Answer is read going up: We read top to bottom. © Joan A. Cotter, 2010 © Joan A. Cotter, 2010 18 20 AL Abacus Sums Adding to Ten 1000 100 10 1 © Joan A. Cotter, 2010 © Joan A. Cotter, 2010
- 6. 21 23 Math Way of Naming Numbers Part-Whole Circles • Only 11 words are needed to count to 100 the math way, 28 in English. (All Indo-European languages are non-standard in number naming.) 10 • Asian children learn mathematics using the math way of counting. 4 6 • They understand place value in first grade; only half of U.S. children understand place value at the end of fourth grade. What is the other part? • Mathematics is the science of patterns. The patterned math way of counting greatly helps children learn number sense. © Joan A. Cotter, 2010 © Joan A. Cotter, 2010 22 24Language Effect on Counting Math Way of Counting Compared to Reading 100 Chinese U.S. Average Highest Number Counted 90 Korean formal [math way] Korean informal [not explicit] 80 70 60 • Just as reciting the alphabet doesn’t teach reading, 50 counting doesn’t teach arithmetic. 40 30 • Just as we first teach the sound of the letters, we 20 first teach the name of the quantity (math way). 10 0 4 5 6 Ages (yrs.) Song, M., & Ginsburg, H. (1988). p. 326. The effect of the Korean number system on young childrens counting: A natural experiment in numerical bilingualism. International Journal of Psychology, 23, 319-332. © Joan A. Cotter, 2010 © Joan A. Cotter, 2010
- 7. 25 27 Adding 7 1000 100 10 13-ten 7 30 7 8 +6 © Joan A. Cotter, 2010 © Joan A. Cotter, 2010 26 28Strategy: Two Fives Adding 1000 100 10 18 + 7 = 10 + 5 = 15 8 +6 14 © Joan A. Cotter, 2010 © Joan A. Cotter, 2010
- 8. 29 31 The Multiplication Board “Pie” Model Difficulties 7x7 • Perpetuates cultural myth that fractions < 1. • It does not give child the “big picture.” • A fraction is much more than “a part of a set of part of a whole.” • Difficult for the child to see how fractions relate to each other. • Is the user comparing angles, arcs, or area? © Joan A. Cotter, 2010 © Joan A. Cotter, 2010 30 32 Fraction Chart Simplifying Fractions 1 1 2 3 4 5 6 7 8 9 10 1 1 2 2 2 4 6 8 10 12 14 16 18 20 1 1 1 3 3 3 3 6 9 12 15 18 21 24 27 30 21 1 1 1 1 4 4 4 4 4 8 12 16 20 24 28 32 36 40 28 1 1 1 1 1 5 5 5 5 5 5 10 15 20 25 30 35 40 45 50 1 1 1 1 1 1 6 12 18 24 30 36 42 48 54 60 45 1 6 1 6 1 6 1 6 1 6 1 6 1 72 7 7 7 7 7 7 7 7 14 21 28 35 42 49 56 63 70 1 1 1 1 1 1 1 1 8 8 8 8 8 8 8 8 8 16 24 32 40 48 56 64 72 80 1 1 1 1 1 1 1 1 1 9 9 9 9 9 9 9 9 9 9 18 27 36 45 54 63 72 81 90 1 1 1 1 1 1 1 1 1 110 10 10 10 10 10 10 10 10 10 10 20 30 40 50 60 70 80 90 100How many fourths make a whole? How many sixths? © Joan A. Cotter, 2010 © Joan A. Cotter, 2010
- 9. !inger (ar*s APPENDI 1© Activities for Learning, Inc. 2010 This page may be duplicated for a single teacher or a single family’s use.
- 10. 5 GO TO THE DUMP (From Math Card Games: 300 Games for Learning and Enjoying Math. Fifth edition by Joan A. Cotter (2010); published by Activities for Learning, Inc.: Hazelton, ND.) Objective To learn the combinations that total 10Number of players 2 to 4 Cards 4 or 6 of each basic number card 1 to 9 Deal Each player takes five cards; the remaining cards face down form the dump, or stack.Object of the game To collect the most pairs that equal 10 Materials Beginners need an abacus or at least a list of the facts. 1+9 2+8 3+7 4+6 6 is needed with 4 to make 10. 5+5 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 to make 10 and look for that number among their other cards. (Some children may need to spread the cards out on the playing surface.) Store paired cards face up on two piles. (This allows verification and keeps the cards shuffled for the next game.) 4 6 8 2 4 6 8 2 Player 1. Player 2. Play When all are ready, the first player asks the player on her left for a number needed to complete a pair. If he has it, he must give it to her, whereupon she receives another turn. If he does not have it, he says, “Go to the Dump,” which is also the signal for him to begin his turn. He takes a turn by asking the player on his left and so forth.Meanwhile, the first player concludes her turn by picking up the top card from the dump. She does not receive an additional turn even if she picks up a needed card. However, she may put a new pair on top of her other pairs. A player running out of cards takes five more cards, but the turn is ended. When the dump is exhausted, players may ask any player (not only the players on their left) for a card. At the end of the game, players combine their two stacks and compare the heights. (Counting the cards is too time consuming.) No shuffling is necessary for subsequent games. © 2010 Joan A. Cotter, Ph.D. • JoanCotter@ALabacus.com • alabacus.com
- 11. SKIP COUNTING MEMORY Objective To learn the skip counting patterns on previous page. Preparation To prepare the envelopes, see page 13. The players use the envelopes for reference during the game to memorize the patterns.Number of players 2 or 2 teams Cards Each player or team chooses an envelope and removes the cards. Mix the cards together 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 the needed card, the player collects the card and receives another turn. If it is not the needed card, the card is returned. Next the second player takes a turn. Turns alternate until one player has picked up all ten cards. Stress the importance of returning the cards to the correct 5 10 envelopes following a game. 15 20 25 30 2 4 6 8 10 35 40 12 14 16 18 20 45 50 2 4 6 5 10 2 4 6 5 10 A game in progress: The player on the left collects the 2s while the player on 12 the right collects the 5s. 12 MULTIPLICATION MEMORY Objective To help the players master the multiplication facts. Cards 10 basic number cards with numbers 1 to 10 and one set of product cards. Also a sticky note with the set number and “×” and another note with “=.”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 separate rows lay the product cards. Play The first player turns over a basic number card and states the fact. For example, if the card is 4, the player says, “Three taken four times is 12.” He then decides where it could be among the product cards. If he is correct, he collects both cards and takes another turn. If it is not a match, both cards are returned face down in their original places, and the other player takes a turn. 4 4 3× = 12 12 © 2010 Joan Cotter • JoanCotter@ALabacus.com • More Games at: alabacus.com > Resources > Presentations
- 12. CONCENTRATING ON ONE (From Math Card Games: 300 Games for Learning and Enjoying Math. Fifth edition by Joan A. Cotter (2010); published by Activities for Learning, Inc.: Hazelton, ND.) Objective To help the children realize that two halves, three thirds, and so forth, equal one. Being told this fact does not necessarily mean understanding it. 2 1 Background Explain that – means two –s. Then lay down various fraction cards and ask 3 3 the children to find the equivalent fraction pieces. 3 Now, ask a child to lay the fraction pieces for – under the 1. Then ask her 5 1 how many more fifths are needed to make 1. [Two 5 Repeat this for other –s] 1 7 1 fractions, such as 6 and —. Children often have a problem with 2 – 10 –. Some children find the fraction chart to be very 1 helpful. With it they can see what they have and 1 1 2 2 count how many more are needed. With the left 1 3 1 3 1 3 index finger, the child counts what she has. With 1 4 1 4 1 4 1 4 the left finger still in place, she counts with her 1 5 1 5 1 5 1 5 1 5 right index finger how many more she needs. 1 6 1 6 1 6 1 6 1 6 1 6 Explain that these are the pairs for this game. 1 7 1 7 1 7 1 7 1 7 1 7 1 7 1 1 1 1 1 1 1 1 Cards Twenty fraction cards are needed: two 1⁄2s and 8 8 8 1 1 1 1 1 1 1 1 1 8 8 8 8 8 one of each of the following: 1⁄3, 2⁄3, 1⁄4, 3⁄4, 1⁄5, 9 9 9 9 9 9 9 9 9 1 1 1 1 1 1 1 1 1 1 2⁄5, 3⁄5, 4⁄5, 1⁄6, 5⁄6, 1⁄8, 3⁄8, 5⁄8, 7⁄8, 1⁄10, 3⁄10, 7⁄10, 10 10 10 10 10 10 10 10 10 10 and 9⁄10. The fraction chart.Number of players Two or two teams. Layout Lay the fraction cards out on the table face down in rows as shown.Object of the game To collect the most pairs of fractions totaling one. Play The first player turns over a card and decides how many more are needed to make 1. She then chooses a probable card. If she is correct, she collects both cards and takes another turn. If they do not match, both cards are returned face down. The second player then takes his turn. Turns continue until all the cards are collected. 1 Showing that five 1 equal 1. –s 5 1 1 1 1 1 5 5 5 5 5 5 8 A beginning game showing 3 two fractions that equal 1. 8 © 2010 Joan Cotter • JoanCotter@ALabacus.com • More Games at: alabacus.com > Resources > Presentations
- 13. FRACTION WAR (From Math Card Games: 300 Games for Learning and Enjoying Math. Fifth edition by Joan A. Cotter (2010); published by Activities for Learning, Inc.: Hazelton, ND.) Objective To provide practice in comparing two fractions between the 1s, halves, fourths, and eighths, the fractions needed for reading a ruler. Materials The 1, halves, fourths, and eighths of the fraction pieces, arranged as shown below. 1 1 1 2 2 1 1 1 1 4 4 4 4 1 1 1 1 1 1 1 1 8 8 8 8 8 8 8 8 The fraction pieces forming a “ruler.” Cards The fraction cards with 1s, halves, fourths, and eighths.Number of players Two only. Deal With the cards face down, divide the stack in half by comparing heights.Object of the game To capture all the cards. Play Each player takes the top card from his stack and lays it down in the middle of the table face up. The player whose card is greater takes both cards. Players should alternate deciding whose card is higher. Players continue comparing cards until they put down cards of equal value, which constitutes a “war.” To resolve a war, both players play two cards face down and then play a third face up to be compared. The player who has the high card in the last comparison takes all eight cards. © 2010 Joan A. Cotter, Ph.D. • JoanCotter@ALabacus.com • alabacus.com

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