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Linda Stulz

- The document discusses issues with traditional counting models used to teach mathematics. It notes that counting is not natural, provides a poor concept of quantity, ignores place value, and is error-prone. - An alternative model is presented using letters to represent numbers, with examples of how addition, subtraction, and other math concepts can be demonstrated in a more intuitive way. Hands-on Montessori materials are also noted as being incorporated into teaching counting.

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© Joan A. Cotter, Ph.D., 2012 New Discoveries ! Montessori Academy! August 31, 2012
 Hutchinson, Minnesota by Joan A. Cotter, Ph.D.
 JoanCotter@RightStartMath.com" RightStart™ Mathematics in a
 Montessori Environment Other presentations available: rightstartmath.com 7 x 7 1000 10 1 100 5 3 5 2
© Joan A. Cotter, Ph.D., 2012 National Math Crisis
© Joan A. Cotter, Ph.D., 2012 National Math Crisis • 25% of college freshmen take remedial math.
© Joan A. Cotter, Ph.D., 2012 National Math Crisis • 25% of college freshmen take remedial math. • In 2009, of the 1.5 million students who took the ACT test, only 42% are ready for college algebra.
© Joan A. Cotter, Ph.D., 2012 National Math Crisis • 25% of college freshmen take remedial math. • In 2009, of the 1.5 million students who took the ACT test, only 42% are ready for college algebra. • A generation ago, the US produced 30% of the world’s college grads; today it’s 14%. (CSM 2006)
© Joan A. Cotter, Ph.D., 2012 National Math Crisis • 25% of college freshmen take remedial math. • In 2009, of the 1.5 million students who took the ACT test, only 42% are ready for college algebra. • A generation ago, the US produced 30% of the world’s college grads; today it’s 14%. (CSM 2006) • Two-thirds of 4-year degrees in Japan and China are in science and engineering; one-third in the U.S.
© Joan A. Cotter, Ph.D., 2012 National Math Crisis • 25% of college freshmen take remedial math. • In 2009, of the 1.5 million students who took the ACT test, only 42% are ready for college algebra. • A generation ago, the US produced 30% of the world’s college grads; today it’s 14%. (CSM 2006) • Two-thirds of 4-year degrees in Japan and China are in science and engineering; one-third in the U.S. • U.S. students, compared to the world, score high at 4th grade, average at 8th, and near bottom at 12th.
© Joan A. Cotter, Ph.D., 2012 National Math Crisis • Ready, Willing, and Unable to Serve says that 75% of 17 to 24 year-olds are unfit for military service. (2010) • 25% of college freshmen take remedial math. • In 2009, of the 1.5 million students who took the ACT test, only 42% are ready for college algebra. • A generation ago, the US produced 30% of the world’s college grads; today it’s 14%. (CSM 2006) • Two-thirds of 4-year degrees in Japan and China are in science and engineering; one-third in the U.S. • U.S. students, compared to the world, score high at 4th grade, average at 8th, and near bottom at 12th.
© Joan A. Cotter, Ph.D., 2012 Math Education is Changing
© Joan A. Cotter, Ph.D., 2012 Math Education is Changing • The field of mathematics is doubling every 7 years.
© Joan A. Cotter, Ph.D., 2012 Math Education is Changing • The field of mathematics is doubling every 7 years. • Math is used in many new ways. The workplace needs analytical thinkers and problem solvers.
© Joan A. Cotter, Ph.D., 2012 Math Education is Changing • The field of mathematics is doubling every 7 years. • Math is used in many new ways. The workplace needs analytical thinkers and problem solvers. • State exams require more than arithmetic: including geometry, algebra, probability, and statistics.
© Joan A. Cotter, Ph.D., 2012 Math Education is Changing • The field of mathematics is doubling every 7 years. • Math is used in many new ways. The workplace needs analytical thinkers and problem solvers. • State exams require more than arithmetic: including geometry, algebra, probability, and statistics. • Brain research is providing clues on how to better facilitate learning, including math.
© Joan A. Cotter, Ph.D., 2012 Math Education is Changing • The field of mathematics is doubling every 7 years. • Math is used in many new ways. The workplace needs analytical thinkers and problem solvers. • State exams require more than arithmetic: including geometry, algebra, probability, and statistics. • Brain research is providing clues on how to better facilitate learning, including math. • Calculators and computers have made computation with many digits an unneeded skill.
© Joan A. Cotter, Ph.D., 2012 Math Education is Changing • The field of mathematics is doubling every 7 years. • Math is used in many new ways. The workplace needs analytical thinkers and problem solvers. • State exams require more than arithmetic: including geometry, algebra, probability, and statistics. • Brain research is providing clues on how to better facilitate learning, including math. • Calculators and computers have made computation with many digits an unneeded skill. • There is a greater emphasis on STEM subjects.
© Joan A. Cotter, Ph.D., 2012 Counting Model
© Joan A. Cotter, Ph.D., 2012 Counting Model From a child's perspective Because we’re so familiar with 1, 2, 3, we’ll use letters. A = 1 B = 2 C = 3 D = 4 E = 5, and so forth
© Joan A. Cotter, Ph.D., 2012 Counting Model From a child's perspective F + E!
© Joan A. Cotter, Ph.D., 2012 Counting Model From a child's perspective A F + E!
© Joan A. Cotter, Ph.D., 2012 Counting Model From a child's perspective A B F + E!
© Joan A. Cotter, Ph.D., 2012 Counting Model From a child's perspective A CB F + E!
© Joan A. Cotter, Ph.D., 2012 Counting Model From a child's perspective A FC D EB F + E!
© Joan A. Cotter, Ph.D., 2012 Counting Model From a child's perspective AA FC D EB F + E!
© Joan A. Cotter, Ph.D., 2012 Counting Model From a child's perspective A BA FC D EB F + E!
© Joan A. Cotter, Ph.D., 2012 Counting Model From a child's perspective A C D EBA FC D EB F + E!
© Joan A. Cotter, Ph.D., 2012 Counting Model From a child's perspective A C D EBA FC D EB F + E! What is the sum?! (It must be a letter.)!
© Joan A. Cotter, Ph.D., 2012 Counting Model From a child's perspective K G I J KHA FC D EB F + E
© Joan A. Cotter, Ph.D., 2012 Counting Model From a child's perspective Now memorize the facts!! G! + D!
© Joan A. Cotter, Ph.D., 2012 Counting Model From a child's perspective Now memorize the facts!! G! + D!
© Joan A. Cotter, Ph.D., 2012 Counting Model From a child's perspective Now memorize the facts!! G! + D! D! + C!
© Joan A. Cotter, Ph.D., 2012 Counting Model From a child's perspective Now memorize the facts!! G! + D! C! + G! D! + C!
© Joan A. Cotter, Ph.D., 2012 Counting Model From a child's perspective Now memorize the facts!! G! + D! C! + G! D! + C!
© Joan A. Cotter, Ph.D., 2012 Counting Model From a child's perspective Try subtracting by “taking away” H – E
© Joan A. Cotter, Ph.D., 2012 Counting Model From a child's perspective Try skip counting by B’s to T: B, D, . . . T.
© Joan A. Cotter, Ph.D., 2012 Counting Model From a child's perspective Try skip counting by B’s to T: B, D, . . . T. What is D  E?
© Joan A. Cotter, Ph.D., 2012 Counting Model From a child's perspective L is written AB! because it is A J ! and B A’s !
© Joan A. Cotter, Ph.D., 2012 Counting Model From a child's perspective L is written AB! because it is A J ! and B A’s ! huh?
© Joan A. Cotter, Ph.D., 2012 Counting Model From a child's perspective L is written AB! because it is A J ! and B A’s ! (twelve)
© Joan A. Cotter, Ph.D., 2012 Counting Model From a child's perspective L is written AB! because it is A J ! and B A’s ! (12) (twelve)
© Joan A. Cotter, Ph.D., 2012 Counting Model From a child's perspective L is written AB! because it is A J ! and B A’s ! (12) (one 10) (twelve)
© Joan A. Cotter, Ph.D., 2012 Counting Model From a child's perspective L is written AB! because it is A J ! and B A’s ! (12) (one 10) (two 1s). (twelve)
© Joan A. Cotter, Ph.D., 2012 Counting Model • Number Rods • Spindle Boxes • Decimal materials • Snake Game • Dot Game • Stamp Game • Multiplication Board • Bead Frame In Montessori, counting is pervasive:
© Joan A. Cotter, Ph.D., 2012 Counting Model Summary
© Joan A. Cotter, Ph.D., 2012 Counting Model • Is not natural; it takes years of practice. Summary
© Joan A. Cotter, Ph.D., 2012 Counting Model • Is not natural; it takes years of practice. • Provides poor concept of quantity. Summary
© Joan A. Cotter, Ph.D., 2012 Counting Model • Is not natural; it takes years of practice. • Provides poor concept of quantity. • Ignores place value. Summary
© Joan A. Cotter, Ph.D., 2012 Counting Model • Is not natural; it takes years of practice. • Provides poor concept of quantity. • Ignores place value. • Is very error prone. Summary
© Joan A. Cotter, Ph.D., 2012 Counting Model • Is not natural; it takes years of practice. • Provides poor concept of quantity. • Ignores place value. • Is very error prone. • Is tedious and time-consuming. Summary
© Joan A. Cotter, Ph.D., 2012 Counting Model • Is not natural; it takes years of practice. • Provides poor concept of quantity. • Ignores place value. • Is very error prone. • Is tedious and time-consuming. Summary • Does not provide an efficient way to master the facts.
© Joan A. Cotter, Ph.D., 2012 Calendar Math August 29! 22! 15! 8! 1! 30! 23! 16! 9! 2! 24! 17! 10! 3! 25! 18! 11! 4! 26! 19! 12! 5! 27! 20! 13! 6! 28! 21! 14! 7! 31! Sometimes calendars are used for counting.!
© Joan A. Cotter, Ph.D., 2012 Calendar Math August 29! 22! 15! 8! 1! 30! 23! 16! 9! 2! 24! 17! 10! 3! 25! 18! 11! 4! 26! 19! 12! 5! 27! 20! 13! 6! 28! 21! 14! 7! 31! Sometimes calendars are used for counting.!
© Joan A. Cotter, Ph.D., 2012 Calendar Math August 29! 22! 15! 8! 1! 30! 23! 16! 9! 2! 24! 17! 10! 3! 25! 18! 11! 4! 26! 19! 12! 5! 27! 20! 13! 6! 28! 21! 14! 7! 31!
© Joan A. Cotter, Ph.D., 2012 Calendar Math August 29! 22! 15! 8! 1! 30! 23! 16! 9! 2! 24! 17! 10! 3! 25! 18! 11! 4! 26! 19! 12! 5! 27! 20! 13! 6! 28! 21! 14! 7! 31! This is ordinal, not cardinal counting. The 3 doesn’t include the 1 and the 2.!
© Joan A. Cotter, Ph.D., 2012 Calendar Math August 29! 22! 15! 8! 1! 30! 23! 16! 9! 2! 24! 17! 10! 3! 25! 18! 11! 4! 26! 19! 12! 5! 27! 20! 13! 6! 28! 21! 14! 7! 31! This is ordinal, not cardinal counting. The 4 doesn’t include 1, 2 and 3.!
© Joan A. Cotter, Ph.D., 2012 Calendar Math August 29! 22! 15! 8! 1! 30! 23! 16! 9! 2! 24! 17! 10! 3! 25! 18! 11! 4! 26! 19! 12! 5! 27! 20! 13! 6! 28! 21! 14! 7! 31! 1 2 3 4 5 6 A calendar is NOT a ruler. On a ruler the numbers are not in the spaces.!
© Joan A. Cotter, Ph.D., 2012 Calendar Math August 8! 1! 9! 2! 10! 3! 4! 5! 6! 7! Always show the whole calendar. A child needs to see the whole before the parts. Children also need to learn to plan ahead.!
© Joan A. Cotter, Ph.D., 2012 Calendar Math The calendar is not a number line. • No quantity is involved. • Numbers are in spaces, not at lines like a ruler.
© Joan A. Cotter, Ph.D., 2012 Calendar Math The calendar is not a number line. • No quantity is involved. • Numbers are in spaces, not at lines like a ruler. Children need to see the whole month, not just part. • Purpose of calendar is to plan ahead. • Many ways to show the current date.
© Joan A. Cotter, Ph.D., 2012 Calendar Math The calendar is not a number line. • No quantity is involved. • Numbers are in spaces, not at lines like a ruler. Children need to see the whole month, not just part. • Purpose of calendar is to plan ahead. • Many ways to show the current date. Calendars give a narrow view of patterning. • Patterns do not necessarily involve numbers. • Patterns rarely proceed row by row. • Patterns go on forever; they don’t stop at 31.
© Joan A. Cotter, Ph.D., 2012 Memorizing Math Percentage Recall Immediately After 1 day After 4 wks Rote 32 23 8 Concept 69 69 58
© Joan A. Cotter, Ph.D., 2012 Memorizing Math Percentage Recall Immediately After 1 day After 4 wks Rote 32 23 8 Concept 69 69 58
© Joan A. Cotter, Ph.D., 2012 Memorizing Math Percentage Recall Immediately After 1 day After 4 wks Rote 32 23 8 Concept 69 69 58
© Joan A. Cotter, Ph.D., 2012 Memorizing Math Percentage Recall Immediately After 1 day After 4 wks Rote 32 23 8 Concept 69 69 58
© Joan A. Cotter, Ph.D., 2012 Memorizing Math Percentage Recall Immediately After 1 day After 4 wks Rote 32 23 8 Concept 69 69 58
© Joan A. Cotter, Ph.D., 2012 Memorizing Math Percentage Recall Immediately After 1 day After 4 wks Rote 32 23 8 Concept 69 69 58
© Joan A. Cotter, Ph.D., 2012 Memorizing Math Math needs to be taught so 95% is understood and only 5% memorized. Richard Skemp Percentage Recall Immediately After 1 day After 4 wks Rote 32 23 8 Concept 69 69 58
© Joan A. Cotter, Ph.D., 2012 Memorizing Math Flash cards: 9! + 7!
© Joan A. Cotter, Ph.D., 2012 • Are often used to teach rote. Memorizing Math Flash cards: 9! + 7!
© Joan A. Cotter, Ph.D., 2012 • Are often used to teach rote. • Are liked only by those who don’t need them. Memorizing Math Flash cards: 9! + 7!
© Joan A. Cotter, Ph.D., 2012 • Are often used to teach rote. • Are liked only by those who don’t need them. • Don’t work for those with learning disabilities. Memorizing Math Flash cards: 9! + 7!
© Joan A. Cotter, Ph.D., 2012 • Are often used to teach rote. • Are liked only by those who don’t need them. • Don’t work for those with learning disabilities. • Give the false impression that math isn’t about thinking. Memorizing Math Flash cards: 9! + 7!
© Joan A. Cotter, Ph.D., 2012 • Are often used to teach rote. • Are liked only by those who don’t need them. • Don’t work for those with learning disabilities. • Give the false impression that math isn’t about thinking. • Often produce stress – children under stress stop learning. Memorizing Math 9! + 7!Flash cards:
© Joan A. Cotter, Ph.D., 2012 • Are often used to teach rote. • Are liked only by those who don’t need them. • Don’t work for those with learning disabilities. • Give the false impression that math isn’t about thinking. • Often produce stress – children under stress stop learning. • Are not concrete – use abstract symbols. Memorizing Math 9! + 7!Flash cards:
© Joan A. Cotter, Ph.D., 2012 Research on Counting Karen Wynn’s research Show the baby two teddy bears.!
© Joan A. Cotter, Ph.D., 2012 Research on Counting Karen Wynn’s research Then hide them with a screen.!
© Joan A. Cotter, Ph.D., 2012 Research on Counting Karen Wynn’s research Show the baby a third teddy bear and put it behind the screen.!
© Joan A. Cotter, Ph.D., 2012 Research on Counting Karen Wynn’s research Show the baby a third teddy bear and put it behind the screen.!
© Joan A. Cotter, Ph.D., 2012 Research on Counting Karen Wynn’s research Raise screen. Baby seeing 3 won’t look long because it is expected.!
© Joan A. Cotter, Ph.D., 2012 Research on Counting Karen Wynn’s research Researcher can change the number of teddy bears behind the screen.!
© Joan A. Cotter, Ph.D., 2012 Research on Counting Karen Wynn’s research A baby seeing 1 teddy bear will look much longer, because it’s unexpected.!
© Joan A. Cotter, Ph.D., 2012 Research on Counting Other research
© Joan A. Cotter, Ph.D., 2012 Research on Counting •  Australian Aboriginal children from two tribes. Brian Butterworth, University College London, 2008. Other research These groups matched quantities without using counting words.!
© Joan A. Cotter, Ph.D., 2012 Research on Counting •  Australian Aboriginal children from two tribes. Brian Butterworth, University College London, 2008. •  Adult Pirahã from Amazon region. Edward Gibson and Michael Frank, MIT, 2008. Other research These groups matched quantities without using counting words.!
© Joan A. Cotter, Ph.D., 2012 Research on Counting •  Australian Aboriginal children from two tribes. Brian Butterworth, University College London, 2008. •  Adult Pirahã from Amazon region. Edward Gibson and Michael Frank, MIT, 2008. •  Adults, ages 18-50, from Boston. Edward Gibson and Michael Frank, MIT, 2008. Other research These groups matched quantities without using counting words.!
© Joan A. Cotter, Ph.D., 2012 Research on Counting •  Australian Aboriginal children from two tribes. Brian Butterworth, University College London, 2008. •  Adult Pirahã from Amazon region. Edward Gibson and Michael Frank, MIT, 2008. •  Adults, ages 18-50, from Boston. Edward Gibson and Michael Frank, MIT, 2008. •  Baby chicks from Italy. Lucia Regolin, University of Padova, 2009. Other research These groups matched quantities without using counting words.!
© Joan A. Cotter, Ph.D., 2012 Research on Counting In Japanese schools: • Children are discouraged from using counting for adding.
© Joan A. Cotter, Ph.D., 2012 Research on Counting In Japanese schools: • Children are discouraged from using counting for adding. • They consistently group in 5s.
© Joan A. Cotter, Ph.D., 2012 Research on Counting Subitizing • Subitizing is quick recognition of quantity without counting.
© Joan A. Cotter, Ph.D., 2012 Research on Counting Subitizing • Subitizing is quick recognition of quantity without counting. • Human babies and some animals can subitize small quantities at birth.
© Joan A. Cotter, Ph.D., 2012 Research on Counting Subitizing • Subitizing is quick recognition of quantity without counting. • Human babies and some animals can subitize small quantities at birth. • Children who can subitize perform better in mathematics.—Butterworth
© Joan A. Cotter, Ph.D., 2012 Research on Counting Subitizing • Subitizing is quick recognition of quantity without counting. • Human babies and some animals can subitize small quantities at birth. • Children who can subitize perform better in mathematics.—Butterworth • Subitizing “allows the child to grasp the whole and the elements at the same time.”—Benoit
© Joan A. Cotter, Ph.D., 2012 Research on Counting Subitizing • Subitizing is quick recognition of quantity without counting. • Human babies and some animals can subitize small quantities at birth. • Children who can subitize perform better in mathematics.—Butterworth • Subitizing “allows the child to grasp the whole and the elements at the same time.”—Benoit • Subitizing seems to be a necessary skill for understanding what the counting process means.— Glasersfeld
© Joan A. Cotter, Ph.D., 2012 Research on Counting Finger gnosia • Finger gnosia is the ability to know which fingers can been lightly touched without looking.
© Joan A. Cotter, Ph.D., 2012 Research on Counting Finger gnosia • Finger gnosia is the ability to know which fingers can been lightly touched without looking. • Part of the brain controlling fingers is adjacent to math part of the brain.
© Joan A. Cotter, Ph.D., 2012 Research on Counting Finger gnosia • Finger gnosia is the ability to know which fingers can been lightly touched without looking. • Part of the brain controlling fingers is adjacent to math part of the brain. • Children who use their fingers as representational tools perform better in mathematics—Butterworth
© Joan A. Cotter, Ph.D., 2012 Visualizing Mathematics
© Joan A. Cotter, Ph.D., 2012 Visualizing Mathematics “In our concern about the memorization of math facts or solving problems, we must not forget that the root of mathematical study is the creation of mental pictures in the imagination and manipulating those images and relationships using the power of reason and logic.” Mindy Holte (E1)
© Joan A. Cotter, Ph.D., 2012 Visualizing Mathematics “Think in pictures, because the brain remembers images better than it does anything else.”   Ben Pridmore, World Memory Champion, 2009
© Joan A. Cotter, Ph.D., 2012 Visualizing Mathematics “Mathematics is the activity of creating relationships, many of which are based in visual imagery.” Wheatley and Cobb
© Joan A. Cotter, Ph.D., 2012 Visualizing Mathematics “The process of connecting symbols to imagery is at the heart of mathematics learning.” Dienes
© Joan A. Cotter, Ph.D., 2012 Visualizing Mathematics “The role of physical manipulatives was to help the child form those visual images and thus to eliminate the need for the physical manipulatives.” Ginsberg and others
© Joan A. Cotter, Ph.D., 2012 • Representative of structure of numbers. • Easily manipulated by children. • Imaginable mentally. Visualizing Mathematics Japanese criteria for manipulatives Japanese Council of Mathematics Education
© Joan A. Cotter, Ph.D., 2012 Visualizing Mathematics • Reading • Sports • Creativity • Geography • Engineering • Construction Visualizing also needed in:
© Joan A. Cotter, Ph.D., 2012 Visualizing Mathematics • Reading • Sports • Creativity • Geography • Engineering • Construction • Architecture • Astronomy • Archeology • Chemistry • Physics • Surgery Visualizing also needed in:
© Joan A. Cotter, Ph.D., 2012 Visualizing Mathematics Ready: How many?
© Joan A. Cotter, Ph.D., 2012 Visualizing Mathematics Ready: How many?
© Joan A. Cotter, Ph.D., 2012 Visualizing Mathematics Try again: How many?
© Joan A. Cotter, Ph.D., 2012 Visualizing Mathematics Try again: How many?
© Joan A. Cotter, Ph.D., 2012 Visualizing Mathematics Try again: How many?
© Joan A. Cotter, Ph.D., 2012 Visualizing Mathematics Ready: How many?
© Joan A. Cotter, Ph.D., 2012 Visualizing Mathematics Try again: How many?
© Joan A. Cotter, Ph.D., 2012 Visualizing Mathematics Try to visualize 8 identical apples without grouping.
© Joan A. Cotter, Ph.D., 2012 Visualizing Mathematics Try to visualize 8 identical apples without grouping.
© Joan A. Cotter, Ph.D., 2012 Visualizing Mathematics Now try to visualize 5 as red and 3 as green.
© Joan A. Cotter, Ph.D., 2012 Visualizing Mathematics Now try to visualize 5 as red and 3 as green.
© Joan A. Cotter, Ph.D., 2012 Visualizing Mathematics I II III IIII V VIII 1 2 3 4 5 8 Early Roman numerals Romans grouped in fives. Notice 8 is 5 and 3.!
© Joan A. Cotter, Ph.D., 2012 Visualizing Mathematics Who could read the music? : Music needs 10 lines, two groups of five.!
© Joan A. Cotter, Ph.D., 2012 Research on Counting Teach Counting • Finger gnosia is the ability to know which fingers can been lightly touched without looking. • Part of the brain controlling fingers is adjacent to math part of the brain. • Children who use their fingers as representational tools perform better in mathematics—Butterworth
© Joan A. Cotter, Ph.D., 2012 Very Early Computation Numerals In English there are two ways of writing numbers: Three thousand five hundred seventy eight 3578
© Joan A. Cotter, Ph.D., 2012 Very Early Computation Numerals Three thousand five hundred seventy eight 3578 In English there are two ways of writing numbers: In Chinese there is only one way of writing numbers: 3 Th 5 H 7 T 8 U (8 characters)
© Joan A. Cotter, Ph.D., 2012 Very Early Computation Calculating rods Because their characters are cumbersome to use for computing, the Chinese used calculating rods, beginning in the 4th century BC.
© Joan A. Cotter, Ph.D., 2012 Very Early Computation Calculating rods
© Joan A. Cotter, Ph.D., 2012 Very Early Computation Calculating rods Numerals for Ones and Hundreds (Even Powers of Ten)
© Joan A. Cotter, Ph.D., 2012 Very Early Computation Calculating rods Numerals for Ones and Hundreds (Even Powers of Ten)
© Joan A. Cotter, Ph.D., 2012 Very Early Computation Calculating rods Numerals for Tens and Thousands (Odd Powers of Ten) Numerals for Ones and Hundreds (Odd Powers of Ten)
© Joan A. Cotter, Ph.D., 2012 Very Early Computation Calculating rods 3578
© Joan A. Cotter, Ph.D., 2012 Very Early Computation Calculating rods 3578 3578,3578 They grouped, not in thousands, but ten-thousands!
© Joan A. Cotter, Ph.D., 2012 Naming Quantities Using fingers
© Joan A. Cotter, Ph.D., 2012 Naming Quantities Using fingers Naming quantities is a three-period lesson.
© Joan A. Cotter, Ph.D., 2012 Naming Quantities Using fingers Use left hand for 1-5 because we read from left to right.!
© Joan A. Cotter, Ph.D., 2012 Naming Quantities Using fingers
© Joan A. Cotter, Ph.D., 2012 Naming Quantities Using fingers
© Joan A. Cotter, Ph.D., 2012 Naming Quantities Using fingers Always show 7 as 5 and 2, not for example, as 4 and 3.!
© Joan A. Cotter, Ph.D., 2012 Naming Quantities Using fingers
© Joan A. Cotter, Ph.D., 2012 Naming Quantities Yellow 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. –Joan A. Cotter Yellow is the Sun Also set to music. Listen and download sheet music from Web site.!
© Joan A. Cotter, Ph.D., 2012 Naming Quantities Recognizing 5
© Joan A. Cotter, Ph.D., 2012 Naming Quantities Recognizing 5
© Joan A. Cotter, Ph.D., 2012 Naming Quantities 5 has a middle; 4 does not. Recognizing 5 Look at your hand; your middle finger is longer to remind you 5 has a middle.!
© Joan A. Cotter, Ph.D., 2012 Naming Quantities Tally sticks Lay the sticks flat on a surface, about 1 inch (2.5 cm) apart.!
© Joan A. Cotter, Ph.D., 2012 Naming Quantities Tally sticks
© Joan A. Cotter, Ph.D., 2012 Naming Quantities Tally sticks
© Joan A. Cotter, Ph.D., 2012 Naming Quantities Tally sticks Stick is horizontal, because it won’t fit diagonally and young children have problems with diagonals.!
© Joan A. Cotter, Ph.D., 2012 Naming Quantities Tally sticks
© Joan A. Cotter, Ph.D., 2012 Naming Quantities Tally sticks Start a new row for every ten.!
© Joan A. Cotter, Ph.D., 2012 Naming Quantities What is 4 apples plus 3 more apples? Solving a problem without counting How would you find the answer without counting?!
© Joan A. Cotter, Ph.D., 2012 Naming Quantities What is 4 apples plus 3 more apples? Solving a problem without counting To remember 4 + 3, the Japanese child is taught to visualize 4 and 3. Then take 1 from the 3 and give it to the 4 to make 5 and 2.!
© Joan A. Cotter, Ph.D., 2012 Naming Quantities 1" 2" 3" 4" 5! Number Chart
© Joan A. Cotter, Ph.D., 2012 Naming Quantities 1" 2" 3" 4" 5! Number Chart To help the child learn the symbols
© Joan A. Cotter, Ph.D., 2012 Naming Quantities 6!1" 7!2" 8!3" 9!4" 10!5! Number Chart To help the child learn the symbols
© Joan A. Cotter, Ph.D., 2012 Naming Quantities Pairing Finger Cards Use two sets of finger cards and match them.!
© Joan A. Cotter, Ph.D., 2012 Naming Quantities Ordering Finger Cards Putting the finger cards in order.!
© Joan A. Cotter, Ph.D., 2012 Naming Quantities 10! 5! 1! Matching Numbers to Finger Cards Match the number to the finger card.!
© Joan A. Cotter, Ph.D., 2012 Naming Quantities 9! 4! Matching Fingers to Number Cards 1! 6!10! 2! 8!3! 5!7! Match the finger card to the number.!
© Joan A. Cotter, Ph.D., 2012 Naming Quantities Finger Card Memory game Use two sets of finger cards and play Memory.!
© Joan A. Cotter, Ph.D., 2012 Naming Quantities Number Rods
© Joan A. Cotter, Ph.D., 2012 Naming Quantities Number Rods
© Joan A. Cotter, Ph.D., 2012 Naming Quantities Number Rods Using different colors.!
© Joan A. Cotter, Ph.D., 2012 Naming Quantities Spindle Box 45 dark-colored and 10 light-colored spindles. Could be in separate containers.!
© Joan A. Cotter, Ph.D., 2012 Naming Quantities Spindle Box 45 dark-colored and 10 light-colored spindles in two containers.!
© Joan A. Cotter, Ph.D., 2012 Naming Quantities Spindle Box 1 2 30 4 The child takes blue spindles with left hand and yellow with right.!
© Joan A. Cotter, Ph.D., 2012 Naming Quantities Spindle Box 6 7 85 9 The child takes blue spindles with left hand and yellow with right.!
© Joan A. Cotter, Ph.D., 2012 Naming Quantities Spindle Box 6 7 85 9 The child takes blue spindles with left hand and yellow with right.!
© Joan A. Cotter, Ph.D., 2012 Naming Quantities Spindle Box 6 7 85 9 The child takes blue spindles with left hand and yellow with right.!
© Joan A. Cotter, Ph.D., 2012 Naming Quantities Spindle Box 6 7 85 9 The child takes blue spindles with left hand and yellow with right.!
© Joan A. Cotter, Ph.D., 2012 Naming Quantities Spindle Box 6 7 85 9 The child takes blue spindles with left hand and yellow with right.!
© Joan A. Cotter, Ph.D., 2012 6 7 85 9 Naming Quantities Spindle Box The child takes blue spindles with left hand and yellow with right.!
© Joan A. Cotter, Ph.D., 2012 Naming Quantities “Grouped in fives so the child does not need to count.” Black and White Bead Stairs A. M. Joosten This was the inspiration to group in 5s.!
© Joan A. Cotter, Ph.D., 2012 AL Abacus 1000 10 1100 Double-sided AL abacus. Side 1 is grouped in 5s.! Trading Side introduces algorithms with trading. !
© Joan A. Cotter, Ph.D., 2012 AL Abacus Cleared
© Joan A. Cotter, Ph.D., 2012 3 AL Abacus Entering quantities Quantities are entered all at once, not counted.!
© Joan A. Cotter, Ph.D., 2012 5 AL Abacus Entering quantities Relate quantities to hands.!
© Joan A. Cotter, Ph.D., 2012 7 AL Abacus Entering quantities
© Joan A. Cotter, Ph.D., 2012 AL Abacus 10 Entering quantities
© Joan A. Cotter, Ph.D., 2012 AL Abacus The stairs Can use to “count” 1 to 10. Also read quantities on the right side.!
© Joan A. Cotter, Ph.D., 2012 AL Abacus Adding
© Joan A. Cotter, Ph.D., 2012 AL Abacus Adding 4 + 3 =
© Joan A. Cotter, Ph.D., 2012 AL Abacus Adding 4 + 3 =
© Joan A. Cotter, Ph.D., 2012 AL Abacus Adding 4 + 3 =
© Joan A. Cotter, Ph.D., 2012 AL Abacus Adding 4 + 3 =
© Joan A. Cotter, Ph.D., 2012 AL Abacus Adding 4 + 3 = 7 Answer is seen immediately, no counting needed.!
© Joan A. Cotter, Ph.D., 2012 Go to the Dump Game Aim: To learn the facts that total 10: 1 + 9! 2 + 8! 3 + 7! 4 + 6! 5 + 5! Children use the abacus while playing this “Go Fish” type game.!
© Joan A. Cotter, Ph.D., 2012 Go to the Dump Game Aim: To learn the facts that total 10: 1 + 9! 2 + 8! 3 + 7! 4 + 6! 5 + 5! Object of the game: To collect the most pairs that equal ten. Children use the abacus while playing this “Go Fish” type game.!
© Joan A. Cotter, Ph.D., 2012 Go to the Dump Game The ways to partition 10.!
© Joan A. Cotter, Ph.D., 2012 Go to the Dump Game Starting A game viewed from above.!
© Joan A. Cotter, Ph.D., 2012 7 2 7 9 5 7 42 61 3 8 3 4 9 Go to the Dump Game Starting Each player takes 5 cards.!
© Joan A. Cotter, Ph.D., 2012 Go to the Dump Game Finding pairs 7 2 7 9 5 7 42 61 3 8 3 4 9 Does YellowCap have any pairs? [no]!
© Joan A. Cotter, Ph.D., 2012 Go to the Dump Game Finding pairs 7 2 7 9 5 7 42 61 3 8 3 4 9 Does BlueCap have any pairs? [yes, 1]!
© Joan A. Cotter, Ph.D., 2012 Go to the Dump Game Finding pairs 7 2 7 9 5 7 42 61 3 8 3 4 9 Does BlueCap have any pairs? [yes, 1]!
© Joan A. Cotter, Ph.D., 2012 Go to the Dump Game Finding pairs 7 2 7 9 5 7 2 1 3 8 3 4 9 4 6 Does BlueCap have any pairs? [yes, 1]!
© Joan A. Cotter, Ph.D., 2012 Go to the Dump Game Finding pairs 7 2 7 9 5 7 2 1 3 8 3 4 9 4 6 Does PinkCap have any pairs? [yes, 2]!
© Joan A. Cotter, Ph.D., 2012 Go to the Dump Game Finding pairs 7 2 7 9 5 7 2 1 3 8 3 4 9 4 6 Does PinkCap have any pairs? [yes, 2]!
© Joan A. Cotter, Ph.D., 2012 Go to the Dump Game Finding pairs 7 2 7 9 5 2 1 8 3 4 9 4 67 3 Does PinkCap have any pairs? [yes, 2]!
© Joan A. Cotter, Ph.D., 2012 Go to the Dump Game Finding pairs 7 2 7 9 5 1 3 4 9 4 62 82 8 Does PinkCap have any pairs? [yes, 2]!
© Joan A. Cotter, Ph.D., 2012 Go to the Dump Game Playing 7 2 7 9 5 1 3 4 9 4 62 82 8 The player asks the player on her left.!
© Joan A. Cotter, Ph.D., 2012 Go to the Dump Game BlueCap, do you have a 3? BlueCap, do you have an 3? Playing 7 2 7 9 5 1 3 4 9 4 62 82 8 The player asks the player on her left.!
© Joan A. Cotter, Ph.D., 2012 Go to the Dump Game BlueCap, do you have a 3? BlueCap, do you have an 3? Playing 7 2 7 9 5 1 3 4 9 4 62 82 8 The player asks the player on her left.!
© Joan A. Cotter, Ph.D., 2012 Go to the Dump Game BlueCap, do you have a 3? BlueCap, do you have an 3? Playing 2 7 9 5 1 4 9 4 62 82 8 7 3
© Joan A. Cotter, Ph.D., 2012 Go to the Dump Game BlueCap, do you have a 3? BlueCap, do you have an 8? Playing 2 7 9 5 1 4 9 4 62 82 8 7 3 YellowCap gets another turn.!
© Joan A. Cotter, Ph.D., 2012 Go to the Dump Game BlueCap, do you have a 3? BlueCap, do you have an 8? Go to the dump. Playing 2 7 9 5 1 4 9 4 62 82 8 7 3 YellowCap gets another turn.!
© Joan A. Cotter, Ph.D., 2012 2 Go to the Dump Game BlueCap, do you have a 3? BlueCap, do you have an 8? Go to the dump. Playing 2 7 9 5 1 4 9 4 62 82 8 7 3
© Joan A. Cotter, Ph.D., 2012 Go to the Dump Game Playing 2 2 7 9 5 1 4 9 4 62 82 8 7 3
© Joan A. Cotter, Ph.D., 2012 Go to the Dump Game PinkCap, do you have a 6? Playing 2 2 7 9 5 1 4 9 4 62 82 8 7 3
© Joan A. Cotter, Ph.D., 2012 Go to the Dump Game PinkCap, do you have a 6? Playing Go to the dump. 2 2 7 9 5 1 4 9 4 62 82 8 7 3
© Joan A. Cotter, Ph.D., 2012 5 Go to the Dump Game Playing 2 2 7 9 5 1 4 9 4 62 82 8 7 3
© Joan A. Cotter, Ph.D., 2012 Go to the Dump Game Playing 5 2 2 7 9 5 1 4 9 4 62 82 8 7 3
© Joan A. Cotter, Ph.D., 2012 Go to the Dump Game YellowCap, do you have a 9? Playing 5 2 2 7 9 5 1 4 9 4 62 82 8 7 3
© Joan A. Cotter, Ph.D., 2012 Go to the Dump Game YellowCap, do you have a 9? Playing 5 2 2 7 5 1 4 9 4 62 82 8 7 3
© Joan A. Cotter, Ph.D., 2012 Go to the Dump Game YellowCap, do you have a 9? Playing 5 2 2 7 5 1 4 9 4 62 82 8 7 3 9
© Joan A. Cotter, Ph.D., 2012 Go to the Dump Game Playing 5 2 2 7 5 4 9 4 62 81 9 7 3
© Joan A. Cotter, Ph.D., 2012 2 9 1 7 7 Go to the Dump Game Playing 5 2 2 7 5 4 9 4 62 81 9 7 3 PinkCap is not out of the game. Her turn ends, but she takes 5 more cards.!
© Joan A. Cotter, Ph.D., 2012 Go to the Dump Game Winner? 5 54 6 9 1
© Joan A. Cotter, Ph.D., 2012 Go to the Dump Game Winner? 5546 91 No counting. Combine both stacks.!
© Joan A. Cotter, Ph.D., 2012 Go to the Dump Game Winner? 5546 91 Whose stack is the highest?!
© Joan A. Cotter, Ph.D., 2012 Go to the Dump Game Next game No shuffling needed for next game.!
© Joan A. Cotter, Ph.D., 2012 “Math” Way of Naming Numbers
© Joan A. Cotter, Ph.D., 2012 “Math” Way of Naming Numbers 11 = ten 1
© Joan A. Cotter, Ph.D., 2012 “Math” Way of Naming Numbers 11 = ten 1 12 = ten 2
© Joan A. Cotter, Ph.D., 2012 “Math” Way of Naming Numbers 11 = ten 1 12 = ten 2 13 = ten 3
© Joan A. Cotter, Ph.D., 2012 “Math” Way of Naming Numbers 11 = ten 1 12 = ten 2 13 = ten 3 14 = ten 4
© Joan A. Cotter, Ph.D., 2012 “Math” Way of Naming Numbers 11 = ten 1 12 = ten 2 13 = ten 3 14 = ten 4 . . . . 19 = ten 9
© Joan A. Cotter, Ph.D., 2012 “Math” Way of Naming Numbers 11 = ten 1 12 = ten 2 13 = ten 3 14 = ten 4 . . . . 19 = ten 9 20 = 2-ten Don’t say “2-tens.” We don’t say 3 hundreds eleven for 311.!
© Joan A. Cotter, Ph.D., 2012 “Math” Way of Naming Numbers 11 = ten 1 12 = ten 2 13 = ten 3 14 = ten 4 . . . . 19 = ten 9 20 = 2-ten 21 = 2-ten 1 Don’t say “2-tens.” We don’t say 3 hundreds eleven for 311.!
© Joan A. Cotter, Ph.D., 2012 “Math” Way of Naming Numbers 11 = ten 1 12 = ten 2 13 = ten 3 14 = ten 4 . . . . 19 = ten 9 20 = 2-ten 21 = 2-ten 1 22 = 2-ten 2 Don’t say “2-tens.” We don’t say 3 hundreds eleven for 311.!
© Joan A. Cotter, Ph.D., 2012 “Math” Way of Naming Numbers 11 = ten 1 12 = ten 2 13 = ten 3 14 = ten 4 . . . . 19 = ten 9 20 = 2-ten 21 = 2-ten 1 22 = 2-ten 2 23 = 2-ten 3 Don’t say “2-tens.” We don’t say 3 hundreds eleven for 311.!
© Joan A. Cotter, Ph.D., 2012 “Math” Way of Naming Numbers 11 = ten 1 12 = ten 2 13 = ten 3 14 = ten 4 . . . . 19 = ten 9 20 = 2-ten 21 = 2-ten 1 22 = 2-ten 2 23 = 2-ten 3 . . . . . . . . 99 = 9-ten 9
© Joan A. Cotter, Ph.D., 2012 “Math” Way of Naming Numbers 137 = 1 hundred 3-ten 7 Only numbers under 100 need to be said the “math” way.!
© Joan A. Cotter, Ph.D., 2012 “Math” Way of Naming Numbers 137 = 1 hundred 3-ten 7 or 137 = 1 hundred and 3-ten 7 Only numbers under 100 need to be said the “math” way.!
© Joan A. Cotter, Ph.D., 2012 “Math” Way of Naming Numbers 0 10 20 30 40 50 60 70 80 90 100 4 5 6 Ages (yrs.) Song, M., & Ginsburg, H. (1988). p. 326. The effect of the Korean number system on young children's counting: A natural experiment in numerical bilingualism. International Journal of Psychology, 23, 319-332. ! Korean formal [math way] Korean informal [not explicit] Chinese U.S. Shows how far children from 3 countries can count at ages 4, 5, and 6.! AverageHighestNumberCounted
© Joan A. Cotter, Ph.D., 2012 “Math” Way of Naming Numbers 0 10 20 30 40 50 60 70 80 90 100 4 5 6 Ages (yrs.) Song, M., & Ginsburg, H. (1988). p. 326. The effect of the Korean number system on young children's counting: A natural experiment in numerical bilingualism. International Journal of Psychology, 23, 319-332. ! Korean formal [math way] Korean informal [not explicit] Chinese U.S. Purple is Chinese. Note jump between ages 5 and 6.! AverageHighestNumberCounted
© Joan A. Cotter, Ph.D., 2012 “Math” Way of Naming Numbers 0 10 20 30 40 50 60 70 80 90 100 4 5 6 Ages (yrs.) Song, M., & Ginsburg, H. (1988). p. 326. The effect of the Korean number system on young children's counting: A natural experiment in numerical bilingualism. International Journal of Psychology, 23, 319-332. ! Korean formal [math way] Korean informal [not explicit] Chinese U.S. Dark green is Korean “math” way.! AverageHighestNumberCounted
© Joan A. Cotter, Ph.D., 2012 “Math” Way of Naming Numbers 0 10 20 30 40 50 60 70 80 90 100 4 5 6 Ages (yrs.) Song, M., & Ginsburg, H. (1988). p. 326. The effect of the Korean number system on young children's counting: A natural experiment in numerical bilingualism. International Journal of Psychology, 23, 319-332. ! Korean formal [math way] Korean informal [not explicit] Chinese U.S. Dotted green is everyday Korean; notice smaller jump between ages 5 and 6.! AverageHighestNumberCounted
© Joan A. Cotter, Ph.D., 2012 “Math” Way of Naming Numbers 0 10 20 30 40 50 60 70 80 90 100 4 5 6 Ages (yrs.) Song, M., & Ginsburg, H. (1988). p. 326. The effect of the Korean number system on young children's counting: A natural experiment in numerical bilingualism. International Journal of Psychology, 23, 319-332. ! Korean formal [math way] Korean informal [not explicit] Chinese U.S. Red is English speakers. They learn same amount between ages 4-5 and 5-6.! AverageHighestNumberCounted
© Joan A. Cotter, Ph.D., 2012 Math Way of Naming Numbers • 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.)
© Joan A. Cotter, Ph.D., 2012 Math Way of Naming Numbers • 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.) • Asian children learn mathematics using the math way of counting.
© Joan A. Cotter, Ph.D., 2012 Math Way of Naming Numbers • 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.) • Asian children learn mathematics using the math way of counting. • They understand place value in first grade; only half of U.S. children understand place value at the end of fourth grade.
© Joan A. Cotter, Ph.D., 2012 Math Way of Naming Numbers • 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.) • Asian children learn mathematics using the math way of counting. • They understand place value in first grade; only half of U.S. children understand place value at the end of fourth grade. • Mathematics is the science of patterns. The patterned math way of counting greatly helps children learn number sense.
© Joan A. Cotter, Ph.D., 2012 Math Way of Naming Numbers Compared to reading:
© Joan A. Cotter, Ph.D., 2012 Math Way of Naming Numbers • Just as reciting the alphabet doesn’t teach reading, counting doesn’t teach arithmetic. Compared to reading:
© Joan A. Cotter, Ph.D., 2012 Math Way of Naming Numbers • Just as reciting the alphabet doesn’t teach reading, counting doesn’t teach arithmetic. • Just as we first teach the sound of the letters, we must first teach the name of the quantity (math way). Compared to reading:
© Joan A. Cotter, Ph.D., 2012 Math Way of Naming Numbers • Just as reciting the alphabet doesn’t teach reading, counting doesn’t teach arithmetic. • Just as we first teach the sound of the letters, we must first teach the name of the quantity (math way). • Montessorians do use the math way of naming numbers but are too quick to switch to traditional names. Use the math way for a longer period of time. Compared to reading:
© Joan A. Cotter, Ph.D., 2012 Math Way of Naming Numbers “Rather, the increased gap between Chinese and U.S. students and that of Chinese Americans and Caucasian Americans may be due primarily to the nature of their initial gap prior to formal schooling, such as counting efficiency and base-ten number sense.” Jian Wang and Emily Lin, 2005 Researchers
© Joan A. Cotter, Ph.D., 2012 Math Way of Naming Numbers Using 10s and 1s, ask the child to construct 48. Research task:
© Joan A. Cotter, Ph.D., 2012 Math Way of Naming Numbers Using 10s and 1s, ask the child to construct 48. Research task: Then ask the child to subtract 14.
© Joan A. Cotter, Ph.D., 2012 Math Way of Naming Numbers Using 10s and 1s, ask the child to construct 48. Research task: Then ask the child to subtract 14. Children thinking of 14 as 14 ones count 14.
© Joan A. Cotter, Ph.D., 2012 Math Way of Naming Numbers Using 10s and 1s, ask the child to construct 48. Research task: Then ask the child to subtract 14. Children thinking of 14 as 14 ones counted 14.
© Joan A. Cotter, Ph.D., 2012 Math Way of Naming Numbers Using 10s and 1s, ask the child to construct 48. Research task: Then ask the child to subtract 14. Children thinking of 14 as 14 ones counted 14.
© Joan A. Cotter, Ph.D., 2012 Math Way of Naming Numbers Using 10s and 1s, ask the child to construct 48. Research task: Then ask the child to subtract 14. Children thinking of 14 as 14 ones counted 14.
© Joan A. Cotter, Ph.D., 2012 Math Way of Naming Numbers Using 10s and 1s, ask the child to construct 48. Research task: Then ask the child to subtract 14. Children thinking of 14 as 14 ones counted 14.
© Joan A. Cotter, Ph.D., 2012 Math Way of Naming Numbers Using 10s and 1s, ask the child to construct 48. Research task: Then ask the child to subtract 14. Children thinking of 14 as 14 ones counted 14.
© Joan A. Cotter, Ph.D., 2012 Math Way of Naming Numbers Using 10s and 1s, ask the child to construct 48. Research task: Then ask the child to subtract 14. Children thinking of 14 as 14 ones counted 14.
© Joan A. Cotter, Ph.D., 2012 Math Way of Naming Numbers Using 10s and 1s, ask the child to construct 48. Research task: Then ask the child to subtract 14. Children thinking of 14 as 14 ones counted 14.
© Joan A. Cotter, Ph.D., 2012 Math Way of Naming Numbers Using 10s and 1s, ask the child to construct 48. Research task: Then ask the child to subtract 14. Children who understand tens remove a ten and 4 ones.
© Joan A. Cotter, Ph.D., 2012 Math Way of Naming Numbers Using 10s and 1s, ask the child to construct 48. Research task: Then ask the child to subtract 14. Children who understand tens remove a ten and 4 ones.
© Joan A. Cotter, Ph.D., 2012 Math Way of Naming Numbers Using 10s and 1s, ask the child to construct 48. Research task: Then ask the child to subtract 14. Children who understand tens remove a ten and 4 ones.
© Joan A. Cotter, Ph.D., 2012 Math Way of Naming Numbers Traditional names 4-ten = forty The “ty” means tens.
© Joan A. Cotter, Ph.D., 2012 Math Way of Naming Numbers Traditional names 4-ten = forty The “ty” means tens. The traditional names for 40, 60, 70, 80, and 90 follow a pattern.!
© Joan A. Cotter, Ph.D., 2012 Math Way of Naming Numbers Traditional names 6-ten = sixty The “ty” means tens.
© Joan A. Cotter, Ph.D., 2012 Math Way of Naming Numbers Traditional names 3-ten = thirty “Thir” also used in 1/3, 13 and 30.
© Joan A. Cotter, Ph.D., 2012 Math Way of Naming Numbers Traditional names 5-ten = fifty “Fif” also used in 1/5, 15 and 50.
© Joan A. Cotter, Ph.D., 2012 Math Way of Naming Numbers Traditional names 2-ten = twenty Two used to be pronounced “twoo.”
© Joan A. Cotter, Ph.D., 2012 Math Way of Naming Numbers Traditional names A word game fireplace place-fire Say the syllables backward. This is how we say the teen numbers.!
© Joan A. Cotter, Ph.D., 2012 Math Way of Naming Numbers Traditional names A word game fireplace place-fire paper-news newspaper Say the syllables backward. This is how we say the teen numbers.!
© Joan A. Cotter, Ph.D., 2012 Math Way of Naming Numbers Traditional names A word game fireplace place-fire paper-news box-mail mailbox newspaper Say the syllables backward. This is how we say the teen numbers.!
© Joan A. Cotter, Ph.D., 2012 Math Way of Naming Numbers Traditional names ten 4 “Teen” also means ten.
© Joan A. Cotter, Ph.D., 2012 Math Way of Naming Numbers Traditional names ten 4 teen 4 “Teen” also means ten.
© Joan A. Cotter, Ph.D., 2012 Math Way of Naming Numbers Traditional names ten 4 teen 4 fourteen “Teen” also means ten.
© Joan A. Cotter, Ph.D., 2012 Math Way of Naming Numbers Traditional names a one left
© Joan A. Cotter, Ph.D., 2012 Math Way of Naming Numbers Traditional names a one left a left-one
© Joan A. Cotter, Ph.D., 2012 Math Way of Naming Numbers Traditional names a one left a left-one eleven
© Joan A. Cotter, Ph.D., 2012 Math Way of Naming Numbers Traditional names two left Two pronounced “twoo.”
© Joan A. Cotter, Ph.D., 2012 Math Way of Naming Numbers Traditional names two left twelve Two pronounced “twoo.”
© Joan A. Cotter, Ph.D., 2012 Composing Numbers 3-ten
© Joan A. Cotter, Ph.D., 2012 Composing Numbers 3-ten
© Joan A. Cotter, Ph.D., 2012 Composing Numbers 3-ten 3 0
© Joan A. Cotter, Ph.D., 2012 Composing Numbers 3-ten 3 0 Point to the 3 and say 3.!
© Joan A. Cotter, Ph.D., 2012 Composing Numbers 3-ten 3 0 Point to 0 and say 10. The 0 makes 3 a ten.!
© Joan A. Cotter, Ph.D., 2012 Composing Numbers 3-ten 7 3 0
© Joan A. Cotter, Ph.D., 2012 Composing Numbers 3-ten 7 3 0
© Joan A. Cotter, Ph.D., 2012 Composing Numbers 3-ten 7 3 0 7
© Joan A. Cotter, Ph.D., 2012 3 0 Composing Numbers 3-ten 7 7 Place the 7 on top of the 0 of the 30.!
© Joan A. Cotter, Ph.D., 2012 Composing Numbers 3-ten 7 Notice the way we say the number, represent the number, and write the number all correspond. 3 07
© Joan A. Cotter, Ph.D., 2012 Composing Numbers 7-ten 8 7 88 Another example.!
© Joan A. Cotter, Ph.D., 2012 Composing Numbers 10-ten
© Joan A. Cotter, Ph.D., 2012 Composing Numbers 10-ten 1 0 0
© Joan A. Cotter, Ph.D., 2012 Composing Numbers 10-ten 1 0 0
© Joan A. Cotter, Ph.D., 2012 Composing Numbers 10-ten 1 0 0
© Joan A. Cotter, Ph.D., 2012 Composing Numbers 1 hundred
© Joan A. Cotter, Ph.D., 2012 Composing Numbers 1 hundred 1 0 0
© Joan A. Cotter, Ph.D., 2012 Composing Numbers 1 hundred 1 0 0 Of course, we can also read it as one-hun-dred.!
© Joan A. Cotter, Ph.D., 2012 Composing Numbers 1 hundred 1 01 01 0 0 Of course, we can also read it as one-hun-dred.!
© Joan A. Cotter, Ph.D., 2012 Composing Numbers 1 hundred 1 0 0 Of course, we can also read it as one-hun-dred.!
© Joan A. Cotter, Ph.D., 2012 2584 8 Composing Numbers To read a number, students are often instructed to start at the right (ones column), contrary to normal reading of numbers and text: Reading numbers backward
© Joan A. Cotter, Ph.D., 2012 2584 58 Composing Numbers To read a number, students are often instructed to start at the right (ones column), contrary to normal reading of numbers and text: Reading numbers backward
© Joan A. Cotter, Ph.D., 2012 2584258 Composing Numbers To read a number, students are often instructed to start at the right (ones column), contrary to normal reading of numbers and text: Reading numbers backward
© Joan A. Cotter, Ph.D., 2012 2584258 Composing Numbers 4 To read a number, students are often instructed to start at the right (ones column), contrary to normal reading of numbers and text: Reading numbers backward
© Joan A. Cotter, Ph.D., 2012 2584258 Composing Numbers 4 To read a number, students are often instructed to start at the right (ones column), contrary to normal reading of numbers and text: Reading numbers backward The Decimal Cards encourage reading numbers in the normal order.
© Joan A. Cotter, Ph.D., 2012 Composing Numbers In scientific notation, we “stand” on the left digit and note the number of digits to the right. (That’s why we shouldn’t refer to the 4 as the 4th column.) Scientific Notation 4000 = 4 x 10 3!
© Joan A. Cotter, Ph.D., 2012 Fact Strategies
© Joan A. Cotter, Ph.D., 2012 Fact Strategies • A strategy is a way to learn a new fact or recall a forgotten fact.
© Joan A. Cotter, Ph.D., 2012 Fact Strategies • A strategy is a way to learn a new fact or recall a forgotten fact. • A visualizable representation is part of a powerful strategy.
© Joan A. Cotter, Ph.D., 2012 Fact Strategies Complete the Ten 9 + 5 =
© Joan A. Cotter, Ph.D., 2012 Fact Strategies Complete the Ten 9 + 5 =
© Joan A. Cotter, Ph.D., 2012 Fact Strategies Complete the Ten 9 + 5 =
© Joan A. Cotter, Ph.D., 2012 Fact Strategies Complete the Ten 9 + 5 = Take 1 from the 5 and give it to the 9.
© Joan A. Cotter, Ph.D., 2012 Fact Strategies Complete the Ten 9 + 5 = Take 1 from the 5 and give it to the 9. Use two hands and move the beads simultaneously.!
© Joan A. Cotter, Ph.D., 2012 Fact Strategies Complete the Ten 9 + 5 = Take 1 from the 5 and give it to the 9.
© Joan A. Cotter, Ph.D., 2012 Fact Strategies Complete the Ten 9 + 5 = 14 Take 1 from the 5 and give it to the 9.
© Joan A. Cotter, Ph.D., 2012 Fact Strategies Two Fives 8 + 6 =
© Joan A. Cotter, Ph.D., 2012 Fact Strategies Two Fives 8 + 6 =
© Joan A. Cotter, Ph.D., 2012 Fact Strategies Two Fives 8 + 6 = Two fives make 10.!
© Joan A. Cotter, Ph.D., 2012 Fact Strategies Two Fives 8 + 6 = Just add the “leftovers.”!
© Joan A. Cotter, Ph.D., 2012 Fact Strategies Two Fives 8 + 6 = 10 + 4 = 14 Just add the “leftovers.”!
© Joan A. Cotter, Ph.D., 2012 Fact Strategies Two Fives 7 + 5 = Another example.!
© Joan A. Cotter, Ph.D., 2012 Fact Strategies Two Fives 7 + 5 =
© Joan A. Cotter, Ph.D., 2012 Fact Strategies Two Fives 7 + 5 = 12
© Joan A. Cotter, Ph.D., 2012 Fact Strategies Going Down 15 – 9 =
© Joan A. Cotter, Ph.D., 2012 Fact Strategies Difference 7 – 4 = Subtract 4 from 5; then add 2.
© Joan A. Cotter, Ph.D., 2012 Fact Strategies Going Down 15 – 9 =
© Joan A. Cotter, Ph.D., 2012 Fact Strategies Going Down 15 – 9 = Subtract 5; then 4.
© Joan A. Cotter, Ph.D., 2012 Fact Strategies Going Down 15 – 9 = Subtract 5; then 4.
© Joan A. Cotter, Ph.D., 2012 Fact Strategies Going Down 15 – 9 = Subtract 5; then 4.
© Joan A. Cotter, Ph.D., 2012 Fact Strategies Going Down 15 – 9 = 6 Subtract 5; then 4.
© Joan A. Cotter, Ph.D., 2012 Fact Strategies Subtract from 10 15 – 9 =
© Joan A. Cotter, Ph.D., 2012 Fact Strategies Subtract from 10 15 – 9 = Subtract 9 from 10.
© Joan A. Cotter, Ph.D., 2012 Fact Strategies Subtract from 10 15 – 9 = Subtract 9 from 10.
© Joan A. Cotter, Ph.D., 2012 Fact Strategies Subtract from 10 15 – 9 = Subtract 9 from 10.
© Joan A. Cotter, Ph.D., 2012 Fact Strategies Subtract from 10 15 – 9 = 6 Subtract 9 from 10.
© Joan A. Cotter, Ph.D., 2012 Fact Strategies Going Up 13 – 9 =
© Joan A. Cotter, Ph.D., 2012 Fact Strategies Going Up 13 – 9 = Start with 9; go up to 13.
© Joan A. Cotter, Ph.D., 2012 Fact Strategies Going Up 13 – 9 = Start with 9; go up to 13.
© Joan A. Cotter, Ph.D., 2012 Fact Strategies Going Up 13 – 9 = Start with 9; go up to 13.
© Joan A. Cotter, Ph.D., 2012 Fact Strategies Going Up 13 – 9 = Start with 9; go up to 13.
© Joan A. Cotter, Ph.D., 2012 Fact Strategies Going Up 13 – 9 = 1 + 3 = 4 Start with 9; go up to 13.
© Joan A. Cotter, Ph.D., 2012 Money Penny
© Joan A. Cotter, Ph.D., 2012 Money Nickel
© Joan A. Cotter, Ph.D., 2012 Money Dime
© Joan A. Cotter, Ph.D., 2012 Money Quarter
© Joan A. Cotter, Ph.D., 2012 Money Quarter
© Joan A. Cotter, Ph.D., 2012 Money Quarter
© Joan A. Cotter, Ph.D., 2012 Money Quarter
© Joan A. Cotter, Ph.D., 2012 Base-10 Picture Cards One
© Joan A. Cotter, Ph.D., 2012 Base-10 Picture Cards Ten One
© Joan A. Cotter, Ph.D., 2012 Base-10 Picture Cards Hundred Ten One
© Joan A. Cotter, Ph.D., 2012 Base-10 Picture Cards Thousand Hundred Ten One
© Joan A. Cotter, Ph.D., 2012 Base-10 Picture Cards 3658! +2724! Add using the base-10 picture cards.
© Joan A. Cotter, Ph.D., 2012 Base-10 Picture Cards 3 0 0 06 0 05 08
© Joan A. Cotter, Ph.D., 2012 Base-10 Picture Cards 3 0 0 06 0 05 08
© Joan A. Cotter, Ph.D., 2012 Base-10 Picture Cards 3 0 0 06 0 05 08
© Joan A. Cotter, Ph.D., 2012 Base-10 Picture Cards 3 0 0 06 0 05 08
© Joan A. Cotter, Ph.D., 2012 Base-10 Picture Cards 2 0 0 07 0 02 04
© Joan A. Cotter, Ph.D., 2012 Base-10 Picture Cards 2 0 0 07 0 02 04
© Joan A. Cotter, Ph.D., 2012 Base-10 Picture Cards 3 0 0 06 0 05 08 2 0 0 07 0 02 04 Add them together.
© Joan A. Cotter, Ph.D., 2012 Base-10 Picture Cards 3 0 0 06 0 05 08 2 0 0 07 0 02 04
© Joan A. Cotter, Ph.D., 2012 Base-10 Picture Cards 3 0 0 06 0 05 08 2 0 0 07 0 02 04 Trade 10 ones for 1 ten.
© Joan A. Cotter, Ph.D., 2012 Base-10 Picture Cards Trade 10 ones for 1 ten. 3 0 0 06 0 05 08 2 0 0 07 0 02 04
© Joan A. Cotter, Ph.D., 2012 Base-10 Picture Cards Trade 10 ones for 1 ten. 3 0 0 06 0 05 08 2 0 0 07 0 02 04
© Joan A. Cotter, Ph.D., 2012 Base-10 Picture Cards 3 0 0 06 0 05 08 2 0 0 07 0 02 04
© Joan A. Cotter, Ph.D., 2012 Base-10 Picture Cards 3 0 0 06 0 05 08 2 0 0 07 0 02 04 Trade 10 hundreds for 1 thousand.
© Joan A. Cotter, Ph.D., 2012 Base-10 Picture Cards Trade 10 hundreds for 1 thousand. 3 0 0 06 0 05 08 2 0 0 07 0 02 04
© Joan A. Cotter, Ph.D., 2012 Base-10 Picture Cards Trade 10 hundreds for 1 thousand. 3 0 0 06 0 05 08 2 0 0 07 0 02 04
© Joan A. Cotter, Ph.D., 2012 Base-10 Picture Cards Trade 10 hundreds for 1 thousand. 3 0 0 06 0 05 08 2 0 0 07 0 02 04
© Joan A. Cotter, Ph.D., 2012 Base-10 Picture Cards 6 0 0 03 0 08 02 3 0 0 06 0 05 08 2 0 0 07 0 02 04
© Joan A. Cotter, Ph.D., 2012 Bead Frame 1 10 100 1000
© Joan A. Cotter, Ph.D., 2012 Bead Frame 8 + 6 1 10 100 1000
© Joan A. Cotter, Ph.D., 2012 Bead Frame 8 + 6 1 10 100 1000
© Joan A. Cotter, Ph.D., 2012 Bead Frame 8 + 6 1 10 100 1000
© Joan A. Cotter, Ph.D., 2012 Bead Frame 8 + 6 1 10 100 1000
© Joan A. Cotter, Ph.D., 2012 Bead Frame 8 + 6 1 10 100 1000
© Joan A. Cotter, Ph.D., 2012 Bead Frame 8 + 6 1 10 100 1000
© Joan A. Cotter, Ph.D., 2012 Bead Frame 8 + 6 1 10 100 1000
© Joan A. Cotter, Ph.D., 2012 Bead Frame 8 + 6 1 10 100 1000
© Joan A. Cotter, Ph.D., 2012 Bead Frame 8 + 6 1 10 100 1000
© Joan A. Cotter, Ph.D., 2012 8 + 6 14 1 10 100 1000 Bead Frame
© Joan A. Cotter, Ph.D., 2012 Bead Frame Difficulties for the child 1 10 100 1000
© Joan A. Cotter, Ph.D., 2012 • Distracting: Room is visible through the frame. Bead Frame Difficulties for the child 1 10 100 1000
© Joan A. Cotter, Ph.D., 2012 • Distracting: Room is visible through the frame. • Not visualizable: Beads need to be grouped in fives. Bead Frame Difficulties for the child 1 10 100 1000
© Joan A. Cotter, Ph.D., 2012 • Distracting: Room is visible through the frame. • Not visualizable: Beads need to be grouped in fives. • When beads are moved right, inconsistent with equation order: Beads need to be moved left. Bead Frame Difficulties for the child 1 10 100 1000
© Joan A. Cotter, Ph.D., 2012 • Distracting: Room is visible through the frame. • Not visualizable: Beads need to be grouped in fives. • When beads are moved right, inconsistent with equation order: Beads need to be moved left. • Hierarchies of numbers represented sideways: They need to be in vertical columns. Bead Frame Difficulties for the child 1 10 100 1000
© Joan A. Cotter, Ph.D., 2012 • Distracting: Room is visible through the frame. • Not visualizable: Beads need to be grouped in fives. • When beads are moved right, inconsistent with equation order: Beads need to be moved left. • Hierarchies of numbers represented sideways: They need to be in vertical columns. • Trading done before second number is completely added: Addends need to be combined before trading. Bead Frame Difficulties for the child 1 10 100 1000
© Joan A. Cotter, Ph.D., 2012 • Distracting: Room is visible through the frame. • Not visualizable: Beads need to be grouped in fives. • When beads are moved right, inconsistent with equation order: Beads need to be moved left. • Hierarchies of numbers represented sideways: They need to be in vertical columns. • Trading done before second number is completely added: Addends need to be combined before trading. • Answer is read going up: We read top to bottom. Bead Frame Difficulties for the child 1 10 100 1000
© Joan A. Cotter, Ph.D., 2012 Trading Side Cleared 1000 10 1100
© Joan A. Cotter, Ph.D., 2012 1000 10 1100 Trading Side Thousands
© Joan A. Cotter, Ph.D., 2012 1000 10 1100 Trading Side Hundreds The third wire from each end is not used.!
© Joan A. Cotter, Ph.D., 2012 1000 10 1100 Trading Side Tens The third wire from each end is not used.!
© Joan A. Cotter, Ph.D., 2012 1000 10 1100 Trading Side Ones The third wire from each end is not used.!
© Joan A. Cotter, Ph.D., 2012 1000 10 1100 Trading Side Adding 8 + 6
© Joan A. Cotter, Ph.D., 2012 1000 10 1100 Trading Side Adding 8 + 6
© Joan A. Cotter, Ph.D., 2012 1000 10 1100 Trading Side Adding 8 + 6
© Joan A. Cotter, Ph.D., 2012 1000 10 1100 Trading Side Adding 8 + 6
© Joan A. Cotter, Ph.D., 2012 1000 10 1100 Trading Side Adding 8 + 6 14
© Joan A. Cotter, Ph.D., 2012 1000 10 1100 Trading Side Adding 8 + 6 14 Too many ones; trade 10 ones for 1 ten. You can see the 10 ones (yellow).!
© Joan A. Cotter, Ph.D., 2012 1000 10 1100 Trading Side Adding 8 + 6 14 Too many ones; trade 10 ones for 1 ten.
© Joan A. Cotter, Ph.D., 2012 1000 10 1100 Trading Side Adding 8 + 6 14 Too many ones; trade 10 ones for 1 ten.
© Joan A. Cotter, Ph.D., 2012 1000 10 1100 Trading Side Adding 8 + 6 14 Same answer before and after trading.
© Joan A. Cotter, Ph.D., 2012 1000 10 1100 Trading Side Cleared
© Joan A. Cotter, Ph.D., 2012 1000 10 1100 Trading Side Bead Trading game Object: To get a high score by adding numbers on the green cards.
© Joan A. Cotter, Ph.D., 2012 1000 10 1100 Trading Side Bead Trading game Object: To get a high score by adding numbers on the green cards. 7
© Joan A. Cotter, Ph.D., 2012 1000 10 1100 Trading Side Bead Trading game Object: To get a high score by adding numbers on the green cards. 7
© Joan A. Cotter, Ph.D., 2012 1000 10 1100 Trading Side Bead Trading game Turn over another card. Enter 6 beads. Do we need to trade?! 6
© Joan A. Cotter, Ph.D., 2012 1000 10 1100 Trading Side Bead Trading game Turn over another card. Enter 6 beads. Do we need to trade?! 6
© Joan A. Cotter, Ph.D., 2012 1000 10 1100 Trading Side Bead Trading game Turn over another card. Enter 6 beads. Do we need to trade?! 6
© Joan A. Cotter, Ph.D., 2012 1000 10 1100 Trading Side Bead Trading game Trade 10 ones for 1 ten. 6
© Joan A. Cotter, Ph.D., 2012 1000 10 1100 Trading Side Bead Trading game 6
© Joan A. Cotter, Ph.D., 2012 1000 10 1100 Trading Side Bead Trading game 6
© Joan A. Cotter, Ph.D., 2012 1000 10 1100 Trading Side Bead Trading game 9
© Joan A. Cotter, Ph.D., 2012 1000 10 1100 Trading Side Bead Trading game 9
© Joan A. Cotter, Ph.D., 2012 1000 10 1100 Trading Side Bead Trading game Another trade. 9
© Joan A. Cotter, Ph.D., 2012 1000 10 1100 Trading Side Bead Trading game Another trade. 9
© Joan A. Cotter, Ph.D., 2012 1000 10 1100 Trading Side Bead Trading game 3
© Joan A. Cotter, Ph.D., 2012 1000 10 1100 Trading Side Bead Trading game 3
© Joan A. Cotter, Ph.D., 2012 Trading Side Bead Trading game • In the Bead Trading game 10 ones for 1 ten occurs frequently;
© Joan A. Cotter, Ph.D., 2012 Trading Side Bead Trading game • In the Bead Trading game 10 ones for 1 ten occurs frequently; 10 tens for 1 hundred, less often;
© Joan A. Cotter, Ph.D., 2012 Trading Side Bead Trading game • In the Bead Trading game 10 ones for 1 ten occurs frequently; 10 tens for 1 hundred, less often; 10 hundreds for 1 thousand, rarely.
© Joan A. Cotter, Ph.D., 2012 Trading Side Bead Trading game • In the Bead Trading game 10 ones for 1 ten occurs frequently; 10 tens for 1 hundred, less often; 10 hundreds for 1 thousand, rarely. •  Bead trading helps the child experience the greater value of each column from left to right.
© Joan A. Cotter, Ph.D., 2012 Trading Side Bead Trading game • In the Bead Trading game 10 ones for 1 ten occurs frequently; 10 tens for 1 hundred, less often; 10 hundreds for 1 thousand, rarely. •  Bead trading helps the child experience the greater value of each column from left to right. • To detect a pattern, there must be at least three examples in the sequence. (Place value is a pattern.)
© Joan A. Cotter, Ph.D., 2012 1000 10 1100 Trading Side Adding 4-digit numbers 3658 + 2738
© Joan A. Cotter, Ph.D., 2012 1000 10 1100 Trading Side Adding 4-digit numbers 3658 + 2738 Enter the first number from left to right.
© Joan A. Cotter, Ph.D., 2012 1000 10 1100 Trading Side Adding 4-digit numbers 3658 + 2738 Enter the first number from left to right.
© Joan A. Cotter, Ph.D., 2012 1000 10 1100 Trading Side Adding 4-digit numbers 3658 + 2738 Enter the first number from left to right.
© Joan A. Cotter, Ph.D., 2012 1000 10 1100 Trading Side Adding 4-digit numbers 3658 + 2738 Enter the first number from left to right.
© Joan A. Cotter, Ph.D., 2012 1000 10 1100 Trading Side Adding 4-digit numbers 3658 + 2738 Enter the first number from left to right.
© Joan A. Cotter, Ph.D., 2012 1000 10 1100 Trading Side Adding 4-digit numbers 3658 + 2738 Enter the first number from left to right.
© Joan A. Cotter, Ph.D., 2012 1000 10 1100 Trading Side Adding 4-digit numbers 3658 + 2738 Add starting at the right. Write results after each step.
© Joan A. Cotter, Ph.D., 2012 1000 10 1100 Trading Side Adding 4-digit numbers 3658 + 2738 Add starting at the right. Write results after each step.
© Joan A. Cotter, Ph.D., 2012 1000 10 1100 Trading Side Adding 4-digit numbers 3658 + 2738 Add starting at the right. Write results after each step.
© Joan A. Cotter, Ph.D., 2012 1000 10 1100 Trading Side Adding 4-digit numbers 3658 + 2738 Add starting at the right. Write results after each step.
© Joan A. Cotter, Ph.D., 2012 1000 10 1100 Trading Side Adding 4-digit numbers 3658 + 2738 6 Add starting at the right. Write results after each step. . . . 6 ones. Did anything else happen?!
© Joan A. Cotter, Ph.D., 2012 1000 10 1100 Trading Side Adding 4-digit numbers 3658 + 2738 6 Add starting at the right. Write results after each step. 1 Is it okay to show the extra ten by writing a 1 above the tens column?!
© Joan A. Cotter, Ph.D., 2012 1000 10 1100 Trading Side Adding 4-digit numbers 3658 + 2738 6 Add starting at the right. Write results after each step. 1
© Joan A. Cotter, Ph.D., 2012 1000 10 1100 Trading Side Adding 4-digit numbers 3658 + 2738 6 Add starting at the right. Write results after each step. 1 Do we need to trade? [no]!
© Joan A. Cotter, Ph.D., 2012 1000 10 1100 Trading Side Adding 4-digit numbers 3658 + 2738 96 Add starting at the right. Write results after each step. 1
© Joan A. Cotter, Ph.D., 2012 1000 10 1100 Trading Side Adding 4-digit numbers 3658 + 2738 96 Add starting at the right. Write results after each step. 1
© Joan A. Cotter, Ph.D., 2012 1000 10 1100 Trading Side Adding 4-digit numbers 3658 + 2738 96 Add starting at the right. Write results after each step. 1 Do we need to trade? [yes]!
© Joan A. Cotter, Ph.D., 2012 1000 10 1100 Trading Side Adding 4-digit numbers 3658 + 2738 96 Add starting at the right. Write results after each step. 1 Notice the number of yellow beads. [3] Notice the number of blue beads left. [3] Coincidence? No, because 13 – 10 = 3.!
© Joan A. Cotter, Ph.D., 2012 1000 10 1100 Trading Side Adding 4-digit numbers 3658 + 2738 96 Add starting at the right. Write results after each step. 1
© Joan A. Cotter, Ph.D., 2012 1000 10 1100 Trading Side Adding 4-digit numbers 3658 + 2738 396 Add starting at the right. Write results after each step. 1
© Joan A. Cotter, Ph.D., 2012 1000 10 1100 Trading Side Adding 4-digit numbers 3658 + 2738 396 Add starting at the right. Write results after each step. 11
© Joan A. Cotter, Ph.D., 2012 1000 10 1100 Trading Side Adding 4-digit numbers 3658 + 2738 396 Add starting at the right. Write results after each step. 11
© Joan A. Cotter, Ph.D., 2012 1000 10 1100 Trading Side Adding 4-digit numbers 3658 + 2738 396 Add starting at the right. Write results after each step. 11
© Joan A. Cotter, Ph.D., 2012 1000 10 1100 Trading Side Adding 4-digit numbers 3658 + 2738 6396 Add starting at the right. Write results after each step. 11
© Joan A. Cotter, Ph.D., 2012 1000 10 1100 Trading Side Adding 4-digit numbers 3658 + 2738 6396 Add starting at the right. Write results after each step. 11
© Joan A. Cotter, Ph.D., 2012 Role of the AL Abacus “Neither is it strange to us, looking back, that there should have come a result quite unforeseen by the educators of that time, namely, a loss of the power of real insight into number [by not using abacuses].” David Eugene Smith, 1903
© Joan A. Cotter, Ph.D., 2012 Role of the AL Abacus • Provides a visual organization of quantity. • Allows child to handle quantities in 5s and 10s. • Models trading tens needed for algorithms on the trading side. • Along with the math way of number naming, makes place value transparent. • Physical abacus leads to developing mental abacus. • Shows strategies concretely for learning facts. Its functions
© Joan A. Cotter, Ph.D., 2012 Role of the AL Abacus 1. The abacus is needed for all number activities. 2. It’s used selectively for new concepts or unsure facts. 4. The abacus becomes completely internalized. Since this occurs at different times for different children, they must be encouraged to use the abacus whenever they need it. 3. Beads are moved on an imaginery abacus. Stages
© Joan A. Cotter, Ph.D., 2012 Mental Addition You need to find twenty-four plus thirty-eight. How do you do it? You are sitting at your desk with a calculator, paper and pencil, and a box of teddy bears. Research shows a majority of people do it mentally. “How would you do it mentally?” Discuss methods.!
© Joan A. Cotter, Ph.D., 2012 Mental Addition 24 + 38 = A very efficient way, taught to Dutch children, especially oral.! Dutch method
© Joan A. Cotter, Ph.D., 2012 Mental Addition 24 + 38 = + 24 A very efficient way, taught to Dutch children, especially oral.! Dutch method
© Joan A. Cotter, Ph.D., 2012 Mental Addition 24 + 38 = + 30 24 + A very efficient way, taught to Dutch children, especially oral.! Dutch method
© Joan A. Cotter, Ph.D., 2012 Mental Addition 24 + 38 = + 30 24 + 8 = A very efficient way, taught to Dutch children, especially oral.! Dutch method
© Joan A. Cotter, Ph.D., 2012 Multiplication on the AL Abacus Basic facts 6  4 = (6 taken 4 times)
© Joan A. Cotter, Ph.D., 2012 Multiplication on the AL Abacus Basic facts 6  4 = (6 taken 4 times)
© Joan A. Cotter, Ph.D., 2012 Multiplication on the AL Abacus Basic facts 6  4 = (6 taken 4 times)
© Joan A. Cotter, Ph.D., 2012 Multiplication on the AL Abacus Basic facts 6  4 = (6 taken 4 times)
© Joan A. Cotter, Ph.D., 2012 Multiplication on the AL Abacus Basic facts 6  4 = (6 taken 4 times)
© Joan A. Cotter, Ph.D., 2012 Multiplication on the AL Abacus Basic facts 9  3 =
© Joan A. Cotter, Ph.D., 2012 Multiplication on the AL Abacus Basic facts 9  3 =
© Joan A. Cotter, Ph.D., 2012 Multiplication on the AL Abacus Basic facts 9  3 = 30
© Joan A. Cotter, Ph.D., 2012 Multiplication on the AL Abacus Basic facts 9  3 = 30 – 3 = 27
© Joan A. Cotter, Ph.D., 2012 Multiplication on the AL Abacus Basic facts 4  8 =
© Joan A. Cotter, Ph.D., 2012 Multiplication on the AL Abacus Basic facts 4  8 =
© Joan A. Cotter, Ph.D., 2012 Multiplication on the AL Abacus Basic facts 4  8 =
© Joan A. Cotter, Ph.D., 2012 Multiplication on the AL Abacus Basic facts 4  8 = 20 + 12 = 32
© Joan A. Cotter, Ph.D., 2012 Multiplication on the AL Abacus Basic facts 7  7 =
© Joan A. Cotter, Ph.D., 2012 Multiplication on the AL Abacus Basic facts 7  7 =
© Joan A. Cotter, Ph.D., 2012 Multiplication on the AL Abacus Basic facts 7  7 = 25 + 10 + 10 + 4 = 49
© Joan A. Cotter, Ph.D., 2012 Multiplication on the AL Abacus Commutative property 5  6 =
© Joan A. Cotter, Ph.D., 2012 Multiplication on the AL Abacus Commutative property 5  6 =
© Joan A. Cotter, Ph.D., 2012 Multiplication on the AL Abacus Commutative property 5  6 =
© Joan A. Cotter, Ph.D., 2012 Multiplication on the AL Abacus Commutative property 5  6 = 6  5
© Joan A. Cotter, Ph.D., 2012 The Multiplication Board 1! 2! 3! 4! 5! 6! 7! 8! 9! 10! 6! 6  4 6 x 4 on original multiplication board.!
© Joan A. Cotter, Ph.D., 2012 1! 2! 3! 4! 5! 6! 7! 8! 9! 10! 6! The Multiplication Board 6  4 Using two colors.!
© Joan A. Cotter, Ph.D., 2012 The Multiplication Board 1! 2! 3! 4! 5! 6! 7! 8! 9! 10! 7! 7  7 7 x 7 on original multiplication board.!
© Joan A. Cotter, Ph.D., 2012 1! 2! 3! 4! 5! 6! 7! 8! 9! 10! 7! The Multiplication Board 7  7 Upper left square is 25, yellow rectangles are 10. So, 25, 35, 45, 49.!
© Joan A. Cotter, Ph.D., 2012 The Multiplication Board 7  7 Less clutter.!
© Joan A. Cotter, Ph.D., 2012 Multiples Patterns Twos 2 4 6 8 10 12 14 16 18 20 Recognizing multiples needed for fractions and algebra.
© Joan A. Cotter, Ph.D., 2012 Multiples Patterns Twos 2 4 6 8 10 12 14 16 18 20 The ones repeat in the second row. Recognizing multiples needed for fractions and algebra.
© Joan A. Cotter, Ph.D., 2012 Multiples Patterns Twos 2 4 6 8 10 12 14 16 18 20 The ones repeat in the second row. Recognizing multiples needed for fractions and algebra.
© Joan A. Cotter, Ph.D., 2012 Multiples Patterns Twos 2 4 6 8 10 12 14 16 18 20 The ones repeat in the second row. Recognizing multiples needed for fractions and algebra.
© Joan A. Cotter, Ph.D., 2012 Multiples Patterns Twos 2 4 6 8 10 12 14 16 18 20 The ones repeat in the second row. Recognizing multiples needed for fractions and algebra.
© Joan A. Cotter, Ph.D., 2012 Multiples Patterns Twos 2 4 6 8 10 12 14 16 18 20 The ones repeat in the second row. Recognizing multiples needed for fractions and algebra.
© Joan A. Cotter, Ph.D., 2012 Multiples Patterns Fours 4 8 12 16 20 24 28 32 36 40
© Joan A. Cotter, Ph.D., 2012 Multiples Patterns Fours 4 8 12 16 20 24 28 32 36 40 The ones repeat in the second row.
© Joan A. Cotter, Ph.D., 2012 Multiples Patterns Sixes and Eights 6 12 18 24 30 36 42 48 54 60 8 16 24 32 40 48 56 64 72 80
© Joan A. Cotter, Ph.D., 2012 Multiples Patterns Sixes and Eights 6 12 18 24 30 36 42 48 54 60 8 16 24 32 40 48 56 64 72 80
© Joan A. Cotter, Ph.D., 2012 Multiples Patterns Sixes and Eights 6 12 18 24 30 36 42 48 54 60 8 16 24 32 40 48 56 64 72 80 Again the ones repeat in the second row.
© Joan A. Cotter, Ph.D., 2012 Multiples Patterns Sixes and Eights 6 12 18 24 30 36 42 48 54 60 8 16 24 32 40 48 56 64 72 80 The ones in the 8s show the multiples of 2.
© Joan A. Cotter, Ph.D., 2012 Multiples Patterns Sixes and Eights 6 12 18 24 30 36 42 48 54 60 8 16 24 32 40 48 56 64 72 80 The ones in the 8s show the multiples of 2.
© Joan A. Cotter, Ph.D., 2012 Multiples Patterns Sixes and Eights 6 12 18 24 30 36 42 48 54 60 8 16 24 32 40 48 56 64 72 80 The ones in the 8s show the multiples of 2.
© Joan A. Cotter, Ph.D., 2012 Multiples Patterns Sixes and Eights 6 12 18 24 30 36 42 48 54 60 8 16 24 32 40 48 56 64 72 80 The ones in the 8s show the multiples of 2.
© Joan A. Cotter, Ph.D., 2012 Multiples Patterns Sixes and Eights 6 12 18 24 30 36 42 48 54 60 8 16 24 32 40 48 56 64 72 80 The ones in the 8s show the multiples of 2.
© Joan A. Cotter, Ph.D., 2012 Multiples Patterns Sixes and Eights 6 12 18 24 30 36 42 48 54 60 8 16 24 32 40 48 56 64 72 80 6  4 6  4 is the fourth number (multiple).
© Joan A. Cotter, Ph.D., 2012 Multiples Patterns Sixes and Eights 6 12 18 24 30 36 42 48 54 60 8 16 24 32 40 48 56 64 72 80 8  7 8  7 is the seventh number (multiple).
© Joan A. Cotter, Ph.D., 2012 Multiples Patterns Nines 9 18 27 36 45 90 81 72 63 54 The second row is written in reverse order. Also the digits in each number add to 9.
© Joan A. Cotter, Ph.D., 2012 Multiples Patterns Threes 3 6 9 12  15 18 21 24 27 30 The 3s have several patterns: Observe the ones.
© Joan A. Cotter, Ph.D., 2012 Multiples Patterns Threes 3 6 9 12  15 18 21 24 27 30 The 3s have several patterns: Observe the ones.
© Joan A. Cotter, Ph.D., 2012 Multiples Patterns Threes 3 6 9 12  15 18 21 24 27 30 The 3s have several patterns: Observe the ones.
© Joan A. Cotter, Ph.D., 2012 Multiples Patterns Threes 3 6 9 12 15 18 21 24 27 30 The 3s have several patterns: Observe the ones.
© Joan A. Cotter, Ph.D., 2012 Multiples Patterns Threes 3 6 9 12 15 18 21 24 27 30 The 3s have several patterns: Observe the ones.
© Joan A. Cotter, Ph.D., 2012 Multiples Patterns Threes 3 6 9 12 15 18 21 24 27 30 The 3s have several patterns: Observe the ones.
© Joan A. Cotter, Ph.D., 2012 Multiples Patterns Threes 3 6 9 12 15 18 21 24 27 30 The 3s have several patterns: Observe the ones.
© Joan A. Cotter, Ph.D., 2012 Multiples Patterns Threes 3 6 9 12 15 18 21 24 27 30 The 3s have several patterns: Observe the ones.
© Joan A. Cotter, Ph.D., 2012 Multiples Patterns Threes 3 6 9 12 15 18 21 24 27 30 The 3s have several patterns: Observe the ones.
© Joan A. Cotter, Ph.D., 2012 Multiples Patterns Threes 3 6 9 12 15 18 21 24 27 30 The 3s have several patterns: Observe the ones.
© Joan A. Cotter, Ph.D., 2012 Multiples Patterns Threes 3 6 9 12 15 18 21 24 27 30 The 3s have several patterns: Observe the ones.
© Joan A. Cotter, Ph.D., 2012 Multiples Patterns Threes 3 6 9 12 15 18 21 24 27 30 The 3s have several patterns: The tens are the same in each row.
© Joan A. Cotter, Ph.D., 2012 Multiples Patterns Threes 3 6 9 12 15 18 21 24 27 30 The 3s have several patterns: Add the digits in the columns.
© Joan A. Cotter, Ph.D., 2012 Multiples Patterns Threes 3 6 9 12 15 18 21 24 27 30 The 3s have several patterns: Add the digits in the columns.
© Joan A. Cotter, Ph.D., 2012 Multiples Patterns Threes 3 6 9 12 15 18 21 24 27 30 The 3s have several patterns: Add the digits in the columns.
© Joan A. Cotter, Ph.D., 2012 Multiples Patterns Threes 3 6 9 12 15 18 21 24 27 30 The 3s have several patterns: Add the “opposites.”
© Joan A. Cotter, Ph.D., 2012 Multiples Patterns Threes 3 6 9 12 15 18 21 24 27 30 The 3s have several patterns: Add the “opposites.”
© Joan A. Cotter, Ph.D., 2012 Multiples Patterns Threes 3 6 9 12 15 18 21 24 27 30 The 3s have several patterns: Add the “opposites.”
© Joan A. Cotter, Ph.D., 2012 Multiples Patterns Threes 3 6 9 12 15 18 21 24 27 30 The 3s have several patterns: Add the “opposites.”
© Joan A. Cotter, Ph.D., 2012 Multiples Patterns Sevens 7 14 21 28 35 42 49 56 63 70 The 7s have the 1, 2, 3… pattern.
© Joan A. Cotter, Ph.D., 2012 Multiples Patterns Sevens 7 14 21 28 35 42 49 56 63 70 The 7s have the 1, 2, 3… pattern.
© Joan A. Cotter, Ph.D., 2012 Multiples Patterns Sevens 7 14 21 28 35 42 49 56 63 70 The 7s have the 1, 2, 3… pattern.
© Joan A. Cotter, Ph.D., 2012 Multiples Patterns Sevens 7 14 21 28 35 42 49 56 63 70 The 7s have the 1, 2, 3… pattern.
© Joan A. Cotter, Ph.D., 2012 Multiples Patterns Sevens 7 14 21 28 35 42 49 56 63 70 Look at the tens.
© Joan A. Cotter, Ph.D., 2012 Multiples Patterns Sevens 7 14 21 28 35 42 49 56 63 70 Look at the tens.
© Joan A. Cotter, Ph.D., 2012 Multiples Patterns Sevens 7 14 21 28 35 42 49 56 63 70 Look at the tens.
© Joan A. Cotter, Ph.D., 2012 Multiples Memory “Multiples” are sometimes referred to as “skip counting.”!
© Joan A. Cotter, Ph.D., 2012 Multiples Memory Aim: To help the players learn the multiples patterns. “Multiples” are sometimes referred to as “skip counting.”!
© Joan A. Cotter, Ph.D., 2012 Multiples Memory Object of the game: To be the first player to collect all ten cards of a multiple in order. Aim: To help the players learn the multiples patterns.
© Joan A. Cotter, Ph.D., 2012 Multiples Memory The 7s envelope contains 10 cards, each with one of the numbers listed. 7 14 21 28 35 42 49 56 63 70
© Joan A. Cotter, Ph.D., 2012 Multiples Memory The 8s envelope contains 10 cards, each with one of the numbers listed. 8 16 24 32 40 48 56 64 72 80
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Right startmath phpapp02

  • 1. © Joan A. Cotter, Ph.D., 2012 New Discoveries ! Montessori Academy! August 31, 2012
 Hutchinson, Minnesota by Joan A. Cotter, Ph.D.
 JoanCotter@RightStartMath.com" RightStart™ Mathematics in a
 Montessori Environment Other presentations available: rightstartmath.com 7 x 7 1000 10 1 100 5 3 5 2
  • 2. © Joan A. Cotter, Ph.D., 2012 National Math Crisis
  • 3. © Joan A. Cotter, Ph.D., 2012 National Math Crisis • 25% of college freshmen take remedial math.
  • 4. © Joan A. Cotter, Ph.D., 2012 National Math Crisis • 25% of college freshmen take remedial math. • In 2009, of the 1.5 million students who took the ACT test, only 42% are ready for college algebra.
  • 5. © Joan A. Cotter, Ph.D., 2012 National Math Crisis • 25% of college freshmen take remedial math. • In 2009, of the 1.5 million students who took the ACT test, only 42% are ready for college algebra. • A generation ago, the US produced 30% of the world’s college grads; today it’s 14%. (CSM 2006)
  • 6. © Joan A. Cotter, Ph.D., 2012 National Math Crisis • 25% of college freshmen take remedial math. • In 2009, of the 1.5 million students who took the ACT test, only 42% are ready for college algebra. • A generation ago, the US produced 30% of the world’s college grads; today it’s 14%. (CSM 2006) • Two-thirds of 4-year degrees in Japan and China are in science and engineering; one-third in the U.S.
  • 7. © Joan A. Cotter, Ph.D., 2012 National Math Crisis • 25% of college freshmen take remedial math. • In 2009, of the 1.5 million students who took the ACT test, only 42% are ready for college algebra. • A generation ago, the US produced 30% of the world’s college grads; today it’s 14%. (CSM 2006) • Two-thirds of 4-year degrees in Japan and China are in science and engineering; one-third in the U.S. • U.S. students, compared to the world, score high at 4th grade, average at 8th, and near bottom at 12th.
  • 8. © Joan A. Cotter, Ph.D., 2012 National Math Crisis • Ready, Willing, and Unable to Serve says that 75% of 17 to 24 year-olds are unfit for military service. (2010) • 25% of college freshmen take remedial math. • In 2009, of the 1.5 million students who took the ACT test, only 42% are ready for college algebra. • A generation ago, the US produced 30% of the world’s college grads; today it’s 14%. (CSM 2006) • Two-thirds of 4-year degrees in Japan and China are in science and engineering; one-third in the U.S. • U.S. students, compared to the world, score high at 4th grade, average at 8th, and near bottom at 12th.
  • 9. © Joan A. Cotter, Ph.D., 2012 Math Education is Changing
  • 10. © Joan A. Cotter, Ph.D., 2012 Math Education is Changing • The field of mathematics is doubling every 7 years.
  • 11. © Joan A. Cotter, Ph.D., 2012 Math Education is Changing • The field of mathematics is doubling every 7 years. • Math is used in many new ways. The workplace needs analytical thinkers and problem solvers.
  • 12. © Joan A. Cotter, Ph.D., 2012 Math Education is Changing • The field of mathematics is doubling every 7 years. • Math is used in many new ways. The workplace needs analytical thinkers and problem solvers. • State exams require more than arithmetic: including geometry, algebra, probability, and statistics.
  • 13. © Joan A. Cotter, Ph.D., 2012 Math Education is Changing • The field of mathematics is doubling every 7 years. • Math is used in many new ways. The workplace needs analytical thinkers and problem solvers. • State exams require more than arithmetic: including geometry, algebra, probability, and statistics. • Brain research is providing clues on how to better facilitate learning, including math.
  • 14. © Joan A. Cotter, Ph.D., 2012 Math Education is Changing • The field of mathematics is doubling every 7 years. • Math is used in many new ways. The workplace needs analytical thinkers and problem solvers. • State exams require more than arithmetic: including geometry, algebra, probability, and statistics. • Brain research is providing clues on how to better facilitate learning, including math. • Calculators and computers have made computation with many digits an unneeded skill.
  • 15. © Joan A. Cotter, Ph.D., 2012 Math Education is Changing • The field of mathematics is doubling every 7 years. • Math is used in many new ways. The workplace needs analytical thinkers and problem solvers. • State exams require more than arithmetic: including geometry, algebra, probability, and statistics. • Brain research is providing clues on how to better facilitate learning, including math. • Calculators and computers have made computation with many digits an unneeded skill. • There is a greater emphasis on STEM subjects.
  • 16. © Joan A. Cotter, Ph.D., 2012 Counting Model
  • 17. © Joan A. Cotter, Ph.D., 2012 Counting Model From a child's perspective Because we’re so familiar with 1, 2, 3, we’ll use letters. A = 1 B = 2 C = 3 D = 4 E = 5, and so forth
  • 18. © Joan A. Cotter, Ph.D., 2012 Counting Model From a child's perspective F + E!
  • 19. © Joan A. Cotter, Ph.D., 2012 Counting Model From a child's perspective A F + E!
  • 20. © Joan A. Cotter, Ph.D., 2012 Counting Model From a child's perspective A B F + E!
  • 21. © Joan A. Cotter, Ph.D., 2012 Counting Model From a child's perspective A CB F + E!
  • 22. © Joan A. Cotter, Ph.D., 2012 Counting Model From a child's perspective A FC D EB F + E!
  • 23. © Joan A. Cotter, Ph.D., 2012 Counting Model From a child's perspective AA FC D EB F + E!
  • 24. © Joan A. Cotter, Ph.D., 2012 Counting Model From a child's perspective A BA FC D EB F + E!
  • 25. © Joan A. Cotter, Ph.D., 2012 Counting Model From a child's perspective A C D EBA FC D EB F + E!
  • 26. © Joan A. Cotter, Ph.D., 2012 Counting Model From a child's perspective A C D EBA FC D EB F + E! What is the sum?! (It must be a letter.)!
  • 27. © Joan A. Cotter, Ph.D., 2012 Counting Model From a child's perspective K G I J KHA FC D EB F + E
  • 28. © Joan A. Cotter, Ph.D., 2012 Counting Model From a child's perspective Now memorize the facts!! G! + D!
  • 29. © Joan A. Cotter, Ph.D., 2012 Counting Model From a child's perspective Now memorize the facts!! G! + D!
  • 30. © Joan A. Cotter, Ph.D., 2012 Counting Model From a child's perspective Now memorize the facts!! G! + D! D! + C!
  • 31. © Joan A. Cotter, Ph.D., 2012 Counting Model From a child's perspective Now memorize the facts!! G! + D! C! + G! D! + C!
  • 32. © Joan A. Cotter, Ph.D., 2012 Counting Model From a child's perspective Now memorize the facts!! G! + D! C! + G! D! + C!
  • 33. © Joan A. Cotter, Ph.D., 2012 Counting Model From a child's perspective Try subtracting by “taking away” H – E
  • 34. © Joan A. Cotter, Ph.D., 2012 Counting Model From a child's perspective Try skip counting by B’s to T: B, D, . . . T.
  • 35. © Joan A. Cotter, Ph.D., 2012 Counting Model From a child's perspective Try skip counting by B’s to T: B, D, . . . T. What is D  E?
  • 36. © Joan A. Cotter, Ph.D., 2012 Counting Model From a child's perspective L is written AB! because it is A J ! and B A’s !
  • 37. © Joan A. Cotter, Ph.D., 2012 Counting Model From a child's perspective L is written AB! because it is A J ! and B A’s ! huh?
  • 38. © Joan A. Cotter, Ph.D., 2012 Counting Model From a child's perspective L is written AB! because it is A J ! and B A’s ! (twelve)
  • 39. © Joan A. Cotter, Ph.D., 2012 Counting Model From a child's perspective L is written AB! because it is A J ! and B A’s ! (12) (twelve)
  • 40. © Joan A. Cotter, Ph.D., 2012 Counting Model From a child's perspective L is written AB! because it is A J ! and B A’s ! (12) (one 10) (twelve)
  • 41. © Joan A. Cotter, Ph.D., 2012 Counting Model From a child's perspective L is written AB! because it is A J ! and B A’s ! (12) (one 10) (two 1s). (twelve)
  • 42. © Joan A. Cotter, Ph.D., 2012 Counting Model • Number Rods • Spindle Boxes • Decimal materials • Snake Game • Dot Game • Stamp Game • Multiplication Board • Bead Frame In Montessori, counting is pervasive:
  • 43. © Joan A. Cotter, Ph.D., 2012 Counting Model Summary
  • 44. © Joan A. Cotter, Ph.D., 2012 Counting Model • Is not natural; it takes years of practice. Summary
  • 45. © Joan A. Cotter, Ph.D., 2012 Counting Model • Is not natural; it takes years of practice. • Provides poor concept of quantity. Summary
  • 46. © Joan A. Cotter, Ph.D., 2012 Counting Model • Is not natural; it takes years of practice. • Provides poor concept of quantity. • Ignores place value. Summary
  • 47. © Joan A. Cotter, Ph.D., 2012 Counting Model • Is not natural; it takes years of practice. • Provides poor concept of quantity. • Ignores place value. • Is very error prone. Summary
  • 48. © Joan A. Cotter, Ph.D., 2012 Counting Model • Is not natural; it takes years of practice. • Provides poor concept of quantity. • Ignores place value. • Is very error prone. • Is tedious and time-consuming. Summary
  • 49. © Joan A. Cotter, Ph.D., 2012 Counting Model • Is not natural; it takes years of practice. • Provides poor concept of quantity. • Ignores place value. • Is very error prone. • Is tedious and time-consuming. Summary • Does not provide an efficient way to master the facts.
  • 50. © Joan A. Cotter, Ph.D., 2012 Calendar Math August 29! 22! 15! 8! 1! 30! 23! 16! 9! 2! 24! 17! 10! 3! 25! 18! 11! 4! 26! 19! 12! 5! 27! 20! 13! 6! 28! 21! 14! 7! 31! Sometimes calendars are used for counting.!
  • 51. © Joan A. Cotter, Ph.D., 2012 Calendar Math August 29! 22! 15! 8! 1! 30! 23! 16! 9! 2! 24! 17! 10! 3! 25! 18! 11! 4! 26! 19! 12! 5! 27! 20! 13! 6! 28! 21! 14! 7! 31! Sometimes calendars are used for counting.!
  • 52. © Joan A. Cotter, Ph.D., 2012 Calendar Math August 29! 22! 15! 8! 1! 30! 23! 16! 9! 2! 24! 17! 10! 3! 25! 18! 11! 4! 26! 19! 12! 5! 27! 20! 13! 6! 28! 21! 14! 7! 31!
  • 53. © Joan A. Cotter, Ph.D., 2012 Calendar Math August 29! 22! 15! 8! 1! 30! 23! 16! 9! 2! 24! 17! 10! 3! 25! 18! 11! 4! 26! 19! 12! 5! 27! 20! 13! 6! 28! 21! 14! 7! 31! This is ordinal, not cardinal counting. The 3 doesn’t include the 1 and the 2.!
  • 54. © Joan A. Cotter, Ph.D., 2012 Calendar Math August 29! 22! 15! 8! 1! 30! 23! 16! 9! 2! 24! 17! 10! 3! 25! 18! 11! 4! 26! 19! 12! 5! 27! 20! 13! 6! 28! 21! 14! 7! 31! This is ordinal, not cardinal counting. The 4 doesn’t include 1, 2 and 3.!
  • 55. © Joan A. Cotter, Ph.D., 2012 Calendar Math August 29! 22! 15! 8! 1! 30! 23! 16! 9! 2! 24! 17! 10! 3! 25! 18! 11! 4! 26! 19! 12! 5! 27! 20! 13! 6! 28! 21! 14! 7! 31! 1 2 3 4 5 6 A calendar is NOT a ruler. On a ruler the numbers are not in the spaces.!
  • 56. © Joan A. Cotter, Ph.D., 2012 Calendar Math August 8! 1! 9! 2! 10! 3! 4! 5! 6! 7! Always show the whole calendar. A child needs to see the whole before the parts. Children also need to learn to plan ahead.!
  • 57. © Joan A. Cotter, Ph.D., 2012 Calendar Math The calendar is not a number line. • No quantity is involved. • Numbers are in spaces, not at lines like a ruler.
  • 58. © Joan A. Cotter, Ph.D., 2012 Calendar Math The calendar is not a number line. • No quantity is involved. • Numbers are in spaces, not at lines like a ruler. Children need to see the whole month, not just part. • Purpose of calendar is to plan ahead. • Many ways to show the current date.
  • 59. © Joan A. Cotter, Ph.D., 2012 Calendar Math The calendar is not a number line. • No quantity is involved. • Numbers are in spaces, not at lines like a ruler. Children need to see the whole month, not just part. • Purpose of calendar is to plan ahead. • Many ways to show the current date. Calendars give a narrow view of patterning. • Patterns do not necessarily involve numbers. • Patterns rarely proceed row by row. • Patterns go on forever; they don’t stop at 31.
  • 60. © Joan A. Cotter, Ph.D., 2012 Memorizing Math Percentage Recall Immediately After 1 day After 4 wks Rote 32 23 8 Concept 69 69 58
  • 61. © Joan A. Cotter, Ph.D., 2012 Memorizing Math Percentage Recall Immediately After 1 day After 4 wks Rote 32 23 8 Concept 69 69 58
  • 62. © Joan A. Cotter, Ph.D., 2012 Memorizing Math Percentage Recall Immediately After 1 day After 4 wks Rote 32 23 8 Concept 69 69 58
  • 63. © Joan A. Cotter, Ph.D., 2012 Memorizing Math Percentage Recall Immediately After 1 day After 4 wks Rote 32 23 8 Concept 69 69 58
  • 64. © Joan A. Cotter, Ph.D., 2012 Memorizing Math Percentage Recall Immediately After 1 day After 4 wks Rote 32 23 8 Concept 69 69 58
  • 65. © Joan A. Cotter, Ph.D., 2012 Memorizing Math Percentage Recall Immediately After 1 day After 4 wks Rote 32 23 8 Concept 69 69 58
  • 66. © Joan A. Cotter, Ph.D., 2012 Memorizing Math Math needs to be taught so 95% is understood and only 5% memorized. Richard Skemp Percentage Recall Immediately After 1 day After 4 wks Rote 32 23 8 Concept 69 69 58
  • 67. © Joan A. Cotter, Ph.D., 2012 Memorizing Math Flash cards: 9! + 7!
  • 68. © Joan A. Cotter, Ph.D., 2012 • Are often used to teach rote. Memorizing Math Flash cards: 9! + 7!
  • 69. © Joan A. Cotter, Ph.D., 2012 • Are often used to teach rote. • Are liked only by those who don’t need them. Memorizing Math Flash cards: 9! + 7!
  • 70. © Joan A. Cotter, Ph.D., 2012 • Are often used to teach rote. • Are liked only by those who don’t need them. • Don’t work for those with learning disabilities. Memorizing Math Flash cards: 9! + 7!
  • 71. © Joan A. Cotter, Ph.D., 2012 • Are often used to teach rote. • Are liked only by those who don’t need them. • Don’t work for those with learning disabilities. • Give the false impression that math isn’t about thinking. Memorizing Math Flash cards: 9! + 7!
  • 72. © Joan A. Cotter, Ph.D., 2012 • Are often used to teach rote. • Are liked only by those who don’t need them. • Don’t work for those with learning disabilities. • Give the false impression that math isn’t about thinking. • Often produce stress – children under stress stop learning. Memorizing Math 9! + 7!Flash cards:
  • 73. © Joan A. Cotter, Ph.D., 2012 • Are often used to teach rote. • Are liked only by those who don’t need them. • Don’t work for those with learning disabilities. • Give the false impression that math isn’t about thinking. • Often produce stress – children under stress stop learning. • Are not concrete – use abstract symbols. Memorizing Math 9! + 7!Flash cards:
  • 74. © Joan A. Cotter, Ph.D., 2012 Research on Counting Karen Wynn’s research Show the baby two teddy bears.!
  • 75. © Joan A. Cotter, Ph.D., 2012 Research on Counting Karen Wynn’s research Then hide them with a screen.!
  • 76. © Joan A. Cotter, Ph.D., 2012 Research on Counting Karen Wynn’s research Show the baby a third teddy bear and put it behind the screen.!
  • 77. © Joan A. Cotter, Ph.D., 2012 Research on Counting Karen Wynn’s research Show the baby a third teddy bear and put it behind the screen.!
  • 78. © Joan A. Cotter, Ph.D., 2012 Research on Counting Karen Wynn’s research Raise screen. Baby seeing 3 won’t look long because it is expected.!
  • 79. © Joan A. Cotter, Ph.D., 2012 Research on Counting Karen Wynn’s research Researcher can change the number of teddy bears behind the screen.!
  • 80. © Joan A. Cotter, Ph.D., 2012 Research on Counting Karen Wynn’s research A baby seeing 1 teddy bear will look much longer, because it’s unexpected.!
  • 81. © Joan A. Cotter, Ph.D., 2012 Research on Counting Other research
  • 82. © Joan A. Cotter, Ph.D., 2012 Research on Counting •  Australian Aboriginal children from two tribes. Brian Butterworth, University College London, 2008. Other research These groups matched quantities without using counting words.!
  • 83. © Joan A. Cotter, Ph.D., 2012 Research on Counting •  Australian Aboriginal children from two tribes. Brian Butterworth, University College London, 2008. •  Adult Pirahã from Amazon region. Edward Gibson and Michael Frank, MIT, 2008. Other research These groups matched quantities without using counting words.!
  • 84. © Joan A. Cotter, Ph.D., 2012 Research on Counting •  Australian Aboriginal children from two tribes. Brian Butterworth, University College London, 2008. •  Adult Pirahã from Amazon region. Edward Gibson and Michael Frank, MIT, 2008. •  Adults, ages 18-50, from Boston. Edward Gibson and Michael Frank, MIT, 2008. Other research These groups matched quantities without using counting words.!
  • 85. © Joan A. Cotter, Ph.D., 2012 Research on Counting •  Australian Aboriginal children from two tribes. Brian Butterworth, University College London, 2008. •  Adult Pirahã from Amazon region. Edward Gibson and Michael Frank, MIT, 2008. •  Adults, ages 18-50, from Boston. Edward Gibson and Michael Frank, MIT, 2008. •  Baby chicks from Italy. Lucia Regolin, University of Padova, 2009. Other research These groups matched quantities without using counting words.!
  • 86. © Joan A. Cotter, Ph.D., 2012 Research on Counting In Japanese schools: • Children are discouraged from using counting for adding.
  • 87. © Joan A. Cotter, Ph.D., 2012 Research on Counting In Japanese schools: • Children are discouraged from using counting for adding. • They consistently group in 5s.
  • 88. © Joan A. Cotter, Ph.D., 2012 Research on Counting Subitizing • Subitizing is quick recognition of quantity without counting.
  • 89. © Joan A. Cotter, Ph.D., 2012 Research on Counting Subitizing • Subitizing is quick recognition of quantity without counting. • Human babies and some animals can subitize small quantities at birth.
  • 90. © Joan A. Cotter, Ph.D., 2012 Research on Counting Subitizing • Subitizing is quick recognition of quantity without counting. • Human babies and some animals can subitize small quantities at birth. • Children who can subitize perform better in mathematics.—Butterworth
  • 91. © Joan A. Cotter, Ph.D., 2012 Research on Counting Subitizing • Subitizing is quick recognition of quantity without counting. • Human babies and some animals can subitize small quantities at birth. • Children who can subitize perform better in mathematics.—Butterworth • Subitizing “allows the child to grasp the whole and the elements at the same time.”—Benoit
  • 92. © Joan A. Cotter, Ph.D., 2012 Research on Counting Subitizing • Subitizing is quick recognition of quantity without counting. • Human babies and some animals can subitize small quantities at birth. • Children who can subitize perform better in mathematics.—Butterworth • Subitizing “allows the child to grasp the whole and the elements at the same time.”—Benoit • Subitizing seems to be a necessary skill for understanding what the counting process means.— Glasersfeld
  • 93. © Joan A. Cotter, Ph.D., 2012 Research on Counting Finger gnosia • Finger gnosia is the ability to know which fingers can been lightly touched without looking.
  • 94. © Joan A. Cotter, Ph.D., 2012 Research on Counting Finger gnosia • Finger gnosia is the ability to know which fingers can been lightly touched without looking. • Part of the brain controlling fingers is adjacent to math part of the brain.
  • 95. © Joan A. Cotter, Ph.D., 2012 Research on Counting Finger gnosia • Finger gnosia is the ability to know which fingers can been lightly touched without looking. • Part of the brain controlling fingers is adjacent to math part of the brain. • Children who use their fingers as representational tools perform better in mathematics—Butterworth
  • 96. © Joan A. Cotter, Ph.D., 2012 Visualizing Mathematics
  • 97. © Joan A. Cotter, Ph.D., 2012 Visualizing Mathematics “In our concern about the memorization of math facts or solving problems, we must not forget that the root of mathematical study is the creation of mental pictures in the imagination and manipulating those images and relationships using the power of reason and logic.” Mindy Holte (E1)
  • 98. © Joan A. Cotter, Ph.D., 2012 Visualizing Mathematics “Think in pictures, because the brain remembers images better than it does anything else.”   Ben Pridmore, World Memory Champion, 2009
  • 99. © Joan A. Cotter, Ph.D., 2012 Visualizing Mathematics “Mathematics is the activity of creating relationships, many of which are based in visual imagery.” Wheatley and Cobb
  • 100. © Joan A. Cotter, Ph.D., 2012 Visualizing Mathematics “The process of connecting symbols to imagery is at the heart of mathematics learning.” Dienes
  • 101. © Joan A. Cotter, Ph.D., 2012 Visualizing Mathematics “The role of physical manipulatives was to help the child form those visual images and thus to eliminate the need for the physical manipulatives.” Ginsberg and others
  • 102. © Joan A. Cotter, Ph.D., 2012 • Representative of structure of numbers. • Easily manipulated by children. • Imaginable mentally. Visualizing Mathematics Japanese criteria for manipulatives Japanese Council of Mathematics Education
  • 103. © Joan A. Cotter, Ph.D., 2012 Visualizing Mathematics • Reading • Sports • Creativity • Geography • Engineering • Construction Visualizing also needed in:
  • 104. © Joan A. Cotter, Ph.D., 2012 Visualizing Mathematics • Reading • Sports • Creativity • Geography • Engineering • Construction • Architecture • Astronomy • Archeology • Chemistry • Physics • Surgery Visualizing also needed in:
  • 105. © Joan A. Cotter, Ph.D., 2012 Visualizing Mathematics Ready: How many?
  • 106. © Joan A. Cotter, Ph.D., 2012 Visualizing Mathematics Ready: How many?
  • 107. © Joan A. Cotter, Ph.D., 2012 Visualizing Mathematics Try again: How many?
  • 108. © Joan A. Cotter, Ph.D., 2012 Visualizing Mathematics Try again: How many?
  • 109. © Joan A. Cotter, Ph.D., 2012 Visualizing Mathematics Try again: How many?
  • 110. © Joan A. Cotter, Ph.D., 2012 Visualizing Mathematics Ready: How many?
  • 111. © Joan A. Cotter, Ph.D., 2012 Visualizing Mathematics Try again: How many?
  • 112. © Joan A. Cotter, Ph.D., 2012 Visualizing Mathematics Try to visualize 8 identical apples without grouping.
  • 113. © Joan A. Cotter, Ph.D., 2012 Visualizing Mathematics Try to visualize 8 identical apples without grouping.
  • 114. © Joan A. Cotter, Ph.D., 2012 Visualizing Mathematics Now try to visualize 5 as red and 3 as green.
  • 115. © Joan A. Cotter, Ph.D., 2012 Visualizing Mathematics Now try to visualize 5 as red and 3 as green.
  • 116. © Joan A. Cotter, Ph.D., 2012 Visualizing Mathematics I II III IIII V VIII 1 2 3 4 5 8 Early Roman numerals Romans grouped in fives. Notice 8 is 5 and 3.!
  • 117. © Joan A. Cotter, Ph.D., 2012 Visualizing Mathematics Who could read the music? : Music needs 10 lines, two groups of five.!
  • 118. © Joan A. Cotter, Ph.D., 2012 Research on Counting Teach Counting • Finger gnosia is the ability to know which fingers can been lightly touched without looking. • Part of the brain controlling fingers is adjacent to math part of the brain. • Children who use their fingers as representational tools perform better in mathematics—Butterworth
  • 119. © Joan A. Cotter, Ph.D., 2012 Very Early Computation Numerals In English there are two ways of writing numbers: Three thousand five hundred seventy eight 3578
  • 120. © Joan A. Cotter, Ph.D., 2012 Very Early Computation Numerals Three thousand five hundred seventy eight 3578 In English there are two ways of writing numbers: In Chinese there is only one way of writing numbers: 3 Th 5 H 7 T 8 U (8 characters)
  • 121. © Joan A. Cotter, Ph.D., 2012 Very Early Computation Calculating rods Because their characters are cumbersome to use for computing, the Chinese used calculating rods, beginning in the 4th century BC.
  • 122. © Joan A. Cotter, Ph.D., 2012 Very Early Computation Calculating rods
  • 123. © Joan A. Cotter, Ph.D., 2012 Very Early Computation Calculating rods Numerals for Ones and Hundreds (Even Powers of Ten)
  • 124. © Joan A. Cotter, Ph.D., 2012 Very Early Computation Calculating rods Numerals for Ones and Hundreds (Even Powers of Ten)
  • 125. © Joan A. Cotter, Ph.D., 2012 Very Early Computation Calculating rods Numerals for Tens and Thousands (Odd Powers of Ten) Numerals for Ones and Hundreds (Odd Powers of Ten)
  • 126. © Joan A. Cotter, Ph.D., 2012 Very Early Computation Calculating rods 3578
  • 127. © Joan A. Cotter, Ph.D., 2012 Very Early Computation Calculating rods 3578 3578,3578 They grouped, not in thousands, but ten-thousands!
  • 128. © Joan A. Cotter, Ph.D., 2012 Naming Quantities Using fingers
  • 129. © Joan A. Cotter, Ph.D., 2012 Naming Quantities Using fingers Naming quantities is a three-period lesson.
  • 130. © Joan A. Cotter, Ph.D., 2012 Naming Quantities Using fingers Use left hand for 1-5 because we read from left to right.!
  • 131. © Joan A. Cotter, Ph.D., 2012 Naming Quantities Using fingers
  • 132. © Joan A. Cotter, Ph.D., 2012 Naming Quantities Using fingers
  • 133. © Joan A. Cotter, Ph.D., 2012 Naming Quantities Using fingers Always show 7 as 5 and 2, not for example, as 4 and 3.!
  • 134. © Joan A. Cotter, Ph.D., 2012 Naming Quantities Using fingers
  • 135. © Joan A. Cotter, Ph.D., 2012 Naming Quantities Yellow 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. –Joan A. Cotter Yellow is the Sun Also set to music. Listen and download sheet music from Web site.!
  • 136. © Joan A. Cotter, Ph.D., 2012 Naming Quantities Recognizing 5
  • 137. © Joan A. Cotter, Ph.D., 2012 Naming Quantities Recognizing 5
  • 138. © Joan A. Cotter, Ph.D., 2012 Naming Quantities 5 has a middle; 4 does not. Recognizing 5 Look at your hand; your middle finger is longer to remind you 5 has a middle.!
  • 139. © Joan A. Cotter, Ph.D., 2012 Naming Quantities Tally sticks Lay the sticks flat on a surface, about 1 inch (2.5 cm) apart.!
  • 140. © Joan A. Cotter, Ph.D., 2012 Naming Quantities Tally sticks
  • 141. © Joan A. Cotter, Ph.D., 2012 Naming Quantities Tally sticks
  • 142. © Joan A. Cotter, Ph.D., 2012 Naming Quantities Tally sticks Stick is horizontal, because it won’t fit diagonally and young children have problems with diagonals.!
  • 143. © Joan A. Cotter, Ph.D., 2012 Naming Quantities Tally sticks
  • 144. © Joan A. Cotter, Ph.D., 2012 Naming Quantities Tally sticks Start a new row for every ten.!
  • 145. © Joan A. Cotter, Ph.D., 2012 Naming Quantities What is 4 apples plus 3 more apples? Solving a problem without counting How would you find the answer without counting?!
  • 146. © Joan A. Cotter, Ph.D., 2012 Naming Quantities What is 4 apples plus 3 more apples? Solving a problem without counting To remember 4 + 3, the Japanese child is taught to visualize 4 and 3. Then take 1 from the 3 and give it to the 4 to make 5 and 2.!
  • 147. © Joan A. Cotter, Ph.D., 2012 Naming Quantities 1" 2" 3" 4" 5! Number Chart
  • 148. © Joan A. Cotter, Ph.D., 2012 Naming Quantities 1" 2" 3" 4" 5! Number Chart To help the child learn the symbols
  • 149. © Joan A. Cotter, Ph.D., 2012 Naming Quantities 6!1" 7!2" 8!3" 9!4" 10!5! Number Chart To help the child learn the symbols
  • 150. © Joan A. Cotter, Ph.D., 2012 Naming Quantities Pairing Finger Cards Use two sets of finger cards and match them.!
  • 151. © Joan A. Cotter, Ph.D., 2012 Naming Quantities Ordering Finger Cards Putting the finger cards in order.!
  • 152. © Joan A. Cotter, Ph.D., 2012 Naming Quantities 10! 5! 1! Matching Numbers to Finger Cards Match the number to the finger card.!
  • 153. © Joan A. Cotter, Ph.D., 2012 Naming Quantities 9! 4! Matching Fingers to Number Cards 1! 6!10! 2! 8!3! 5!7! Match the finger card to the number.!
  • 154. © Joan A. Cotter, Ph.D., 2012 Naming Quantities Finger Card Memory game Use two sets of finger cards and play Memory.!
  • 155. © Joan A. Cotter, Ph.D., 2012 Naming Quantities Number Rods
  • 156. © Joan A. Cotter, Ph.D., 2012 Naming Quantities Number Rods
  • 157. © Joan A. Cotter, Ph.D., 2012 Naming Quantities Number Rods Using different colors.!
  • 158. © Joan A. Cotter, Ph.D., 2012 Naming Quantities Spindle Box 45 dark-colored and 10 light-colored spindles. Could be in separate containers.!
  • 159. © Joan A. Cotter, Ph.D., 2012 Naming Quantities Spindle Box 45 dark-colored and 10 light-colored spindles in two containers.!
  • 160. © Joan A. Cotter, Ph.D., 2012 Naming Quantities Spindle Box 1 2 30 4 The child takes blue spindles with left hand and yellow with right.!
  • 161. © Joan A. Cotter, Ph.D., 2012 Naming Quantities Spindle Box 6 7 85 9 The child takes blue spindles with left hand and yellow with right.!
  • 162. © Joan A. Cotter, Ph.D., 2012 Naming Quantities Spindle Box 6 7 85 9 The child takes blue spindles with left hand and yellow with right.!
  • 163. © Joan A. Cotter, Ph.D., 2012 Naming Quantities Spindle Box 6 7 85 9 The child takes blue spindles with left hand and yellow with right.!
  • 164. © Joan A. Cotter, Ph.D., 2012 Naming Quantities Spindle Box 6 7 85 9 The child takes blue spindles with left hand and yellow with right.!
  • 165. © Joan A. Cotter, Ph.D., 2012 Naming Quantities Spindle Box 6 7 85 9 The child takes blue spindles with left hand and yellow with right.!
  • 166. © Joan A. Cotter, Ph.D., 2012 6 7 85 9 Naming Quantities Spindle Box The child takes blue spindles with left hand and yellow with right.!
  • 167. © Joan A. Cotter, Ph.D., 2012 Naming Quantities “Grouped in fives so the child does not need to count.” Black and White Bead Stairs A. M. Joosten This was the inspiration to group in 5s.!
  • 168. © Joan A. Cotter, Ph.D., 2012 AL Abacus 1000 10 1100 Double-sided AL abacus. Side 1 is grouped in 5s.! Trading Side introduces algorithms with trading. !
  • 169. © Joan A. Cotter, Ph.D., 2012 AL Abacus Cleared
  • 170. © Joan A. Cotter, Ph.D., 2012 3 AL Abacus Entering quantities Quantities are entered all at once, not counted.!
  • 171. © Joan A. Cotter, Ph.D., 2012 5 AL Abacus Entering quantities Relate quantities to hands.!
  • 172. © Joan A. Cotter, Ph.D., 2012 7 AL Abacus Entering quantities
  • 173. © Joan A. Cotter, Ph.D., 2012 AL Abacus 10 Entering quantities
  • 174. © Joan A. Cotter, Ph.D., 2012 AL Abacus The stairs Can use to “count” 1 to 10. Also read quantities on the right side.!
  • 175. © Joan A. Cotter, Ph.D., 2012 AL Abacus Adding
  • 176. © Joan A. Cotter, Ph.D., 2012 AL Abacus Adding 4 + 3 =
  • 177. © Joan A. Cotter, Ph.D., 2012 AL Abacus Adding 4 + 3 =
  • 178. © Joan A. Cotter, Ph.D., 2012 AL Abacus Adding 4 + 3 =
  • 179. © Joan A. Cotter, Ph.D., 2012 AL Abacus Adding 4 + 3 =
  • 180. © Joan A. Cotter, Ph.D., 2012 AL Abacus Adding 4 + 3 = 7 Answer is seen immediately, no counting needed.!
  • 181. © Joan A. Cotter, Ph.D., 2012 Go to the Dump Game Aim: To learn the facts that total 10: 1 + 9! 2 + 8! 3 + 7! 4 + 6! 5 + 5! Children use the abacus while playing this “Go Fish” type game.!
  • 182. © Joan A. Cotter, Ph.D., 2012 Go to the Dump Game Aim: To learn the facts that total 10: 1 + 9! 2 + 8! 3 + 7! 4 + 6! 5 + 5! Object of the game: To collect the most pairs that equal ten. Children use the abacus while playing this “Go Fish” type game.!
  • 183. © Joan A. Cotter, Ph.D., 2012 Go to the Dump Game The ways to partition 10.!
  • 184. © Joan A. Cotter, Ph.D., 2012 Go to the Dump Game Starting A game viewed from above.!
  • 185. © Joan A. Cotter, Ph.D., 2012 7 2 7 9 5 7 42 61 3 8 3 4 9 Go to the Dump Game Starting Each player takes 5 cards.!
  • 186. © Joan A. Cotter, Ph.D., 2012 Go to the Dump Game Finding pairs 7 2 7 9 5 7 42 61 3 8 3 4 9 Does YellowCap have any pairs? [no]!
  • 187. © Joan A. Cotter, Ph.D., 2012 Go to the Dump Game Finding pairs 7 2 7 9 5 7 42 61 3 8 3 4 9 Does BlueCap have any pairs? [yes, 1]!
  • 188. © Joan A. Cotter, Ph.D., 2012 Go to the Dump Game Finding pairs 7 2 7 9 5 7 42 61 3 8 3 4 9 Does BlueCap have any pairs? [yes, 1]!
  • 189. © Joan A. Cotter, Ph.D., 2012 Go to the Dump Game Finding pairs 7 2 7 9 5 7 2 1 3 8 3 4 9 4 6 Does BlueCap have any pairs? [yes, 1]!
  • 190. © Joan A. Cotter, Ph.D., 2012 Go to the Dump Game Finding pairs 7 2 7 9 5 7 2 1 3 8 3 4 9 4 6 Does PinkCap have any pairs? [yes, 2]!
  • 191. © Joan A. Cotter, Ph.D., 2012 Go to the Dump Game Finding pairs 7 2 7 9 5 7 2 1 3 8 3 4 9 4 6 Does PinkCap have any pairs? [yes, 2]!
  • 192. © Joan A. Cotter, Ph.D., 2012 Go to the Dump Game Finding pairs 7 2 7 9 5 2 1 8 3 4 9 4 67 3 Does PinkCap have any pairs? [yes, 2]!
  • 193. © Joan A. Cotter, Ph.D., 2012 Go to the Dump Game Finding pairs 7 2 7 9 5 1 3 4 9 4 62 82 8 Does PinkCap have any pairs? [yes, 2]!
  • 194. © Joan A. Cotter, Ph.D., 2012 Go to the Dump Game Playing 7 2 7 9 5 1 3 4 9 4 62 82 8 The player asks the player on her left.!
  • 195. © Joan A. Cotter, Ph.D., 2012 Go to the Dump Game BlueCap, do you have a 3? BlueCap, do you have an 3? Playing 7 2 7 9 5 1 3 4 9 4 62 82 8 The player asks the player on her left.!
  • 196. © Joan A. Cotter, Ph.D., 2012 Go to the Dump Game BlueCap, do you have a 3? BlueCap, do you have an 3? Playing 7 2 7 9 5 1 3 4 9 4 62 82 8 The player asks the player on her left.!
  • 197. © Joan A. Cotter, Ph.D., 2012 Go to the Dump Game BlueCap, do you have a 3? BlueCap, do you have an 3? Playing 2 7 9 5 1 4 9 4 62 82 8 7 3
  • 198. © Joan A. Cotter, Ph.D., 2012 Go to the Dump Game BlueCap, do you have a 3? BlueCap, do you have an 8? Playing 2 7 9 5 1 4 9 4 62 82 8 7 3 YellowCap gets another turn.!
  • 199. © Joan A. Cotter, Ph.D., 2012 Go to the Dump Game BlueCap, do you have a 3? BlueCap, do you have an 8? Go to the dump. Playing 2 7 9 5 1 4 9 4 62 82 8 7 3 YellowCap gets another turn.!
  • 200. © Joan A. Cotter, Ph.D., 2012 2 Go to the Dump Game BlueCap, do you have a 3? BlueCap, do you have an 8? Go to the dump. Playing 2 7 9 5 1 4 9 4 62 82 8 7 3
  • 201. © Joan A. Cotter, Ph.D., 2012 Go to the Dump Game Playing 2 2 7 9 5 1 4 9 4 62 82 8 7 3
  • 202. © Joan A. Cotter, Ph.D., 2012 Go to the Dump Game PinkCap, do you have a 6? Playing 2 2 7 9 5 1 4 9 4 62 82 8 7 3
  • 203. © Joan A. Cotter, Ph.D., 2012 Go to the Dump Game PinkCap, do you have a 6? Playing Go to the dump. 2 2 7 9 5 1 4 9 4 62 82 8 7 3
  • 204. © Joan A. Cotter, Ph.D., 2012 5 Go to the Dump Game Playing 2 2 7 9 5 1 4 9 4 62 82 8 7 3
  • 205. © Joan A. Cotter, Ph.D., 2012 Go to the Dump Game Playing 5 2 2 7 9 5 1 4 9 4 62 82 8 7 3
  • 206. © Joan A. Cotter, Ph.D., 2012 Go to the Dump Game YellowCap, do you have a 9? Playing 5 2 2 7 9 5 1 4 9 4 62 82 8 7 3
  • 207. © Joan A. Cotter, Ph.D., 2012 Go to the Dump Game YellowCap, do you have a 9? Playing 5 2 2 7 5 1 4 9 4 62 82 8 7 3
  • 208. © Joan A. Cotter, Ph.D., 2012 Go to the Dump Game YellowCap, do you have a 9? Playing 5 2 2 7 5 1 4 9 4 62 82 8 7 3 9
  • 209. © Joan A. Cotter, Ph.D., 2012 Go to the Dump Game Playing 5 2 2 7 5 4 9 4 62 81 9 7 3
  • 210. © Joan A. Cotter, Ph.D., 2012 2 9 1 7 7 Go to the Dump Game Playing 5 2 2 7 5 4 9 4 62 81 9 7 3 PinkCap is not out of the game. Her turn ends, but she takes 5 more cards.!
  • 211. © Joan A. Cotter, Ph.D., 2012 Go to the Dump Game Winner? 5 54 6 9 1
  • 212. © Joan A. Cotter, Ph.D., 2012 Go to the Dump Game Winner? 5546 91 No counting. Combine both stacks.!
  • 213. © Joan A. Cotter, Ph.D., 2012 Go to the Dump Game Winner? 5546 91 Whose stack is the highest?!
  • 214. © Joan A. Cotter, Ph.D., 2012 Go to the Dump Game Next game No shuffling needed for next game.!
  • 215. © Joan A. Cotter, Ph.D., 2012 “Math” Way of Naming Numbers
  • 216. © Joan A. Cotter, Ph.D., 2012 “Math” Way of Naming Numbers 11 = ten 1
  • 217. © Joan A. Cotter, Ph.D., 2012 “Math” Way of Naming Numbers 11 = ten 1 12 = ten 2
  • 218. © Joan A. Cotter, Ph.D., 2012 “Math” Way of Naming Numbers 11 = ten 1 12 = ten 2 13 = ten 3
  • 219. © Joan A. Cotter, Ph.D., 2012 “Math” Way of Naming Numbers 11 = ten 1 12 = ten 2 13 = ten 3 14 = ten 4
  • 220. © Joan A. Cotter, Ph.D., 2012 “Math” Way of Naming Numbers 11 = ten 1 12 = ten 2 13 = ten 3 14 = ten 4 . . . . 19 = ten 9
  • 221. © Joan A. Cotter, Ph.D., 2012 “Math” Way of Naming Numbers 11 = ten 1 12 = ten 2 13 = ten 3 14 = ten 4 . . . . 19 = ten 9 20 = 2-ten Don’t say “2-tens.” We don’t say 3 hundreds eleven for 311.!
  • 222. © Joan A. Cotter, Ph.D., 2012 “Math” Way of Naming Numbers 11 = ten 1 12 = ten 2 13 = ten 3 14 = ten 4 . . . . 19 = ten 9 20 = 2-ten 21 = 2-ten 1 Don’t say “2-tens.” We don’t say 3 hundreds eleven for 311.!
  • 223. © Joan A. Cotter, Ph.D., 2012 “Math” Way of Naming Numbers 11 = ten 1 12 = ten 2 13 = ten 3 14 = ten 4 . . . . 19 = ten 9 20 = 2-ten 21 = 2-ten 1 22 = 2-ten 2 Don’t say “2-tens.” We don’t say 3 hundreds eleven for 311.!
  • 224. © Joan A. Cotter, Ph.D., 2012 “Math” Way of Naming Numbers 11 = ten 1 12 = ten 2 13 = ten 3 14 = ten 4 . . . . 19 = ten 9 20 = 2-ten 21 = 2-ten 1 22 = 2-ten 2 23 = 2-ten 3 Don’t say “2-tens.” We don’t say 3 hundreds eleven for 311.!
  • 225. © Joan A. Cotter, Ph.D., 2012 “Math” Way of Naming Numbers 11 = ten 1 12 = ten 2 13 = ten 3 14 = ten 4 . . . . 19 = ten 9 20 = 2-ten 21 = 2-ten 1 22 = 2-ten 2 23 = 2-ten 3 . . . . . . . . 99 = 9-ten 9
  • 226. © Joan A. Cotter, Ph.D., 2012 “Math” Way of Naming Numbers 137 = 1 hundred 3-ten 7 Only numbers under 100 need to be said the “math” way.!
  • 227. © Joan A. Cotter, Ph.D., 2012 “Math” Way of Naming Numbers 137 = 1 hundred 3-ten 7 or 137 = 1 hundred and 3-ten 7 Only numbers under 100 need to be said the “math” way.!
  • 228. © Joan A. Cotter, Ph.D., 2012 “Math” Way of Naming Numbers 0 10 20 30 40 50 60 70 80 90 100 4 5 6 Ages (yrs.) Song, M., & Ginsburg, H. (1988). p. 326. The effect of the Korean number system on young children's counting: A natural experiment in numerical bilingualism. International Journal of Psychology, 23, 319-332. ! Korean formal [math way] Korean informal [not explicit] Chinese U.S. Shows how far children from 3 countries can count at ages 4, 5, and 6.! AverageHighestNumberCounted
  • 229. © Joan A. Cotter, Ph.D., 2012 “Math” Way of Naming Numbers 0 10 20 30 40 50 60 70 80 90 100 4 5 6 Ages (yrs.) Song, M., & Ginsburg, H. (1988). p. 326. The effect of the Korean number system on young children's counting: A natural experiment in numerical bilingualism. International Journal of Psychology, 23, 319-332. ! Korean formal [math way] Korean informal [not explicit] Chinese U.S. Purple is Chinese. Note jump between ages 5 and 6.! AverageHighestNumberCounted
  • 230. © Joan A. Cotter, Ph.D., 2012 “Math” Way of Naming Numbers 0 10 20 30 40 50 60 70 80 90 100 4 5 6 Ages (yrs.) Song, M., & Ginsburg, H. (1988). p. 326. The effect of the Korean number system on young children's counting: A natural experiment in numerical bilingualism. International Journal of Psychology, 23, 319-332. ! Korean formal [math way] Korean informal [not explicit] Chinese U.S. Dark green is Korean “math” way.! AverageHighestNumberCounted
  • 231. © Joan A. Cotter, Ph.D., 2012 “Math” Way of Naming Numbers 0 10 20 30 40 50 60 70 80 90 100 4 5 6 Ages (yrs.) Song, M., & Ginsburg, H. (1988). p. 326. The effect of the Korean number system on young children's counting: A natural experiment in numerical bilingualism. International Journal of Psychology, 23, 319-332. ! Korean formal [math way] Korean informal [not explicit] Chinese U.S. Dotted green is everyday Korean; notice smaller jump between ages 5 and 6.! AverageHighestNumberCounted
  • 232. © Joan A. Cotter, Ph.D., 2012 “Math” Way of Naming Numbers 0 10 20 30 40 50 60 70 80 90 100 4 5 6 Ages (yrs.) Song, M., & Ginsburg, H. (1988). p. 326. The effect of the Korean number system on young children's counting: A natural experiment in numerical bilingualism. International Journal of Psychology, 23, 319-332. ! Korean formal [math way] Korean informal [not explicit] Chinese U.S. Red is English speakers. They learn same amount between ages 4-5 and 5-6.! AverageHighestNumberCounted
  • 233. © Joan A. Cotter, Ph.D., 2012 Math Way of Naming Numbers • 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.)
  • 234. © Joan A. Cotter, Ph.D., 2012 Math Way of Naming Numbers • 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.) • Asian children learn mathematics using the math way of counting.
  • 235. © Joan A. Cotter, Ph.D., 2012 Math Way of Naming Numbers • 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.) • Asian children learn mathematics using the math way of counting. • They understand place value in first grade; only half of U.S. children understand place value at the end of fourth grade.
  • 236. © Joan A. Cotter, Ph.D., 2012 Math Way of Naming Numbers • 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.) • Asian children learn mathematics using the math way of counting. • They understand place value in first grade; only half of U.S. children understand place value at the end of fourth grade. • Mathematics is the science of patterns. The patterned math way of counting greatly helps children learn number sense.
  • 237. © Joan A. Cotter, Ph.D., 2012 Math Way of Naming Numbers Compared to reading:
  • 238. © Joan A. Cotter, Ph.D., 2012 Math Way of Naming Numbers • Just as reciting the alphabet doesn’t teach reading, counting doesn’t teach arithmetic. Compared to reading:
  • 239. © Joan A. Cotter, Ph.D., 2012 Math Way of Naming Numbers • Just as reciting the alphabet doesn’t teach reading, counting doesn’t teach arithmetic. • Just as we first teach the sound of the letters, we must first teach the name of the quantity (math way). Compared to reading:
  • 240. © Joan A. Cotter, Ph.D., 2012 Math Way of Naming Numbers • Just as reciting the alphabet doesn’t teach reading, counting doesn’t teach arithmetic. • Just as we first teach the sound of the letters, we must first teach the name of the quantity (math way). • Montessorians do use the math way of naming numbers but are too quick to switch to traditional names. Use the math way for a longer period of time. Compared to reading:
  • 241. © Joan A. Cotter, Ph.D., 2012 Math Way of Naming Numbers “Rather, the increased gap between Chinese and U.S. students and that of Chinese Americans and Caucasian Americans may be due primarily to the nature of their initial gap prior to formal schooling, such as counting efficiency and base-ten number sense.” Jian Wang and Emily Lin, 2005 Researchers
  • 242. © Joan A. Cotter, Ph.D., 2012 Math Way of Naming Numbers Using 10s and 1s, ask the child to construct 48. Research task:
  • 243. © Joan A. Cotter, Ph.D., 2012 Math Way of Naming Numbers Using 10s and 1s, ask the child to construct 48. Research task: Then ask the child to subtract 14.
  • 244. © Joan A. Cotter, Ph.D., 2012 Math Way of Naming Numbers Using 10s and 1s, ask the child to construct 48. Research task: Then ask the child to subtract 14. Children thinking of 14 as 14 ones count 14.
  • 245. © Joan A. Cotter, Ph.D., 2012 Math Way of Naming Numbers Using 10s and 1s, ask the child to construct 48. Research task: Then ask the child to subtract 14. Children thinking of 14 as 14 ones counted 14.
  • 246. © Joan A. Cotter, Ph.D., 2012 Math Way of Naming Numbers Using 10s and 1s, ask the child to construct 48. Research task: Then ask the child to subtract 14. Children thinking of 14 as 14 ones counted 14.
  • 247. © Joan A. Cotter, Ph.D., 2012 Math Way of Naming Numbers Using 10s and 1s, ask the child to construct 48. Research task: Then ask the child to subtract 14. Children thinking of 14 as 14 ones counted 14.
  • 248. © Joan A. Cotter, Ph.D., 2012 Math Way of Naming Numbers Using 10s and 1s, ask the child to construct 48. Research task: Then ask the child to subtract 14. Children thinking of 14 as 14 ones counted 14.
  • 249. © Joan A. Cotter, Ph.D., 2012 Math Way of Naming Numbers Using 10s and 1s, ask the child to construct 48. Research task: Then ask the child to subtract 14. Children thinking of 14 as 14 ones counted 14.
  • 250. © Joan A. Cotter, Ph.D., 2012 Math Way of Naming Numbers Using 10s and 1s, ask the child to construct 48. Research task: Then ask the child to subtract 14. Children thinking of 14 as 14 ones counted 14.
  • 251. © Joan A. Cotter, Ph.D., 2012 Math Way of Naming Numbers Using 10s and 1s, ask the child to construct 48. Research task: Then ask the child to subtract 14. Children thinking of 14 as 14 ones counted 14.
  • 252. © Joan A. Cotter, Ph.D., 2012 Math Way of Naming Numbers Using 10s and 1s, ask the child to construct 48. Research task: Then ask the child to subtract 14. Children who understand tens remove a ten and 4 ones.
  • 253. © Joan A. Cotter, Ph.D., 2012 Math Way of Naming Numbers Using 10s and 1s, ask the child to construct 48. Research task: Then ask the child to subtract 14. Children who understand tens remove a ten and 4 ones.
  • 254. © Joan A. Cotter, Ph.D., 2012 Math Way of Naming Numbers Using 10s and 1s, ask the child to construct 48. Research task: Then ask the child to subtract 14. Children who understand tens remove a ten and 4 ones.
  • 255. © Joan A. Cotter, Ph.D., 2012 Math Way of Naming Numbers Traditional names 4-ten = forty The “ty” means tens.
  • 256. © Joan A. Cotter, Ph.D., 2012 Math Way of Naming Numbers Traditional names 4-ten = forty The “ty” means tens. The traditional names for 40, 60, 70, 80, and 90 follow a pattern.!
  • 257. © Joan A. Cotter, Ph.D., 2012 Math Way of Naming Numbers Traditional names 6-ten = sixty The “ty” means tens.
  • 258. © Joan A. Cotter, Ph.D., 2012 Math Way of Naming Numbers Traditional names 3-ten = thirty “Thir” also used in 1/3, 13 and 30.
  • 259. © Joan A. Cotter, Ph.D., 2012 Math Way of Naming Numbers Traditional names 5-ten = fifty “Fif” also used in 1/5, 15 and 50.
  • 260. © Joan A. Cotter, Ph.D., 2012 Math Way of Naming Numbers Traditional names 2-ten = twenty Two used to be pronounced “twoo.”
  • 261. © Joan A. Cotter, Ph.D., 2012 Math Way of Naming Numbers Traditional names A word game fireplace place-fire Say the syllables backward. This is how we say the teen numbers.!
  • 262. © Joan A. Cotter, Ph.D., 2012 Math Way of Naming Numbers Traditional names A word game fireplace place-fire paper-news newspaper Say the syllables backward. This is how we say the teen numbers.!
  • 263. © Joan A. Cotter, Ph.D., 2012 Math Way of Naming Numbers Traditional names A word game fireplace place-fire paper-news box-mail mailbox newspaper Say the syllables backward. This is how we say the teen numbers.!
  • 264. © Joan A. Cotter, Ph.D., 2012 Math Way of Naming Numbers Traditional names ten 4 “Teen” also means ten.
  • 265. © Joan A. Cotter, Ph.D., 2012 Math Way of Naming Numbers Traditional names ten 4 teen 4 “Teen” also means ten.
  • 266. © Joan A. Cotter, Ph.D., 2012 Math Way of Naming Numbers Traditional names ten 4 teen 4 fourteen “Teen” also means ten.
  • 267. © Joan A. Cotter, Ph.D., 2012 Math Way of Naming Numbers Traditional names a one left
  • 268. © Joan A. Cotter, Ph.D., 2012 Math Way of Naming Numbers Traditional names a one left a left-one
  • 269. © Joan A. Cotter, Ph.D., 2012 Math Way of Naming Numbers Traditional names a one left a left-one eleven
  • 270. © Joan A. Cotter, Ph.D., 2012 Math Way of Naming Numbers Traditional names two left Two pronounced “twoo.”
  • 271. © Joan A. Cotter, Ph.D., 2012 Math Way of Naming Numbers Traditional names two left twelve Two pronounced “twoo.”
  • 272. © Joan A. Cotter, Ph.D., 2012 Composing Numbers 3-ten
  • 273. © Joan A. Cotter, Ph.D., 2012 Composing Numbers 3-ten
  • 274. © Joan A. Cotter, Ph.D., 2012 Composing Numbers 3-ten 3 0
  • 275. © Joan A. Cotter, Ph.D., 2012 Composing Numbers 3-ten 3 0 Point to the 3 and say 3.!
  • 276. © Joan A. Cotter, Ph.D., 2012 Composing Numbers 3-ten 3 0 Point to 0 and say 10. The 0 makes 3 a ten.!
  • 277. © Joan A. Cotter, Ph.D., 2012 Composing Numbers 3-ten 7 3 0
  • 278. © Joan A. Cotter, Ph.D., 2012 Composing Numbers 3-ten 7 3 0
  • 279. © Joan A. Cotter, Ph.D., 2012 Composing Numbers 3-ten 7 3 0 7
  • 280. © Joan A. Cotter, Ph.D., 2012 3 0 Composing Numbers 3-ten 7 7 Place the 7 on top of the 0 of the 30.!
  • 281. © Joan A. Cotter, Ph.D., 2012 Composing Numbers 3-ten 7 Notice the way we say the number, represent the number, and write the number all correspond. 3 07
  • 282. © Joan A. Cotter, Ph.D., 2012 Composing Numbers 7-ten 8 7 88 Another example.!
  • 283. © Joan A. Cotter, Ph.D., 2012 Composing Numbers 10-ten
  • 284. © Joan A. Cotter, Ph.D., 2012 Composing Numbers 10-ten 1 0 0
  • 285. © Joan A. Cotter, Ph.D., 2012 Composing Numbers 10-ten 1 0 0
  • 286. © Joan A. Cotter, Ph.D., 2012 Composing Numbers 10-ten 1 0 0
  • 287. © Joan A. Cotter, Ph.D., 2012 Composing Numbers 1 hundred
  • 288. © Joan A. Cotter, Ph.D., 2012 Composing Numbers 1 hundred 1 0 0
  • 289. © Joan A. Cotter, Ph.D., 2012 Composing Numbers 1 hundred 1 0 0 Of course, we can also read it as one-hun-dred.!
  • 290. © Joan A. Cotter, Ph.D., 2012 Composing Numbers 1 hundred 1 01 01 0 0 Of course, we can also read it as one-hun-dred.!
  • 291. © Joan A. Cotter, Ph.D., 2012 Composing Numbers 1 hundred 1 0 0 Of course, we can also read it as one-hun-dred.!
  • 292. © Joan A. Cotter, Ph.D., 2012 2584 8 Composing Numbers To read a number, students are often instructed to start at the right (ones column), contrary to normal reading of numbers and text: Reading numbers backward
  • 293. © Joan A. Cotter, Ph.D., 2012 2584 58 Composing Numbers To read a number, students are often instructed to start at the right (ones column), contrary to normal reading of numbers and text: Reading numbers backward
  • 294. © Joan A. Cotter, Ph.D., 2012 2584258 Composing Numbers To read a number, students are often instructed to start at the right (ones column), contrary to normal reading of numbers and text: Reading numbers backward
  • 295. © Joan A. Cotter, Ph.D., 2012 2584258 Composing Numbers 4 To read a number, students are often instructed to start at the right (ones column), contrary to normal reading of numbers and text: Reading numbers backward
  • 296. © Joan A. Cotter, Ph.D., 2012 2584258 Composing Numbers 4 To read a number, students are often instructed to start at the right (ones column), contrary to normal reading of numbers and text: Reading numbers backward The Decimal Cards encourage reading numbers in the normal order.
  • 297. © Joan A. Cotter, Ph.D., 2012 Composing Numbers In scientific notation, we “stand” on the left digit and note the number of digits to the right. (That’s why we shouldn’t refer to the 4 as the 4th column.) Scientific Notation 4000 = 4 x 10 3!
  • 298. © Joan A. Cotter, Ph.D., 2012 Fact Strategies
  • 299. © Joan A. Cotter, Ph.D., 2012 Fact Strategies • A strategy is a way to learn a new fact or recall a forgotten fact.
  • 300. © Joan A. Cotter, Ph.D., 2012 Fact Strategies • A strategy is a way to learn a new fact or recall a forgotten fact. • A visualizable representation is part of a powerful strategy.
  • 301. © Joan A. Cotter, Ph.D., 2012 Fact Strategies Complete the Ten 9 + 5 =
  • 302. © Joan A. Cotter, Ph.D., 2012 Fact Strategies Complete the Ten 9 + 5 =
  • 303. © Joan A. Cotter, Ph.D., 2012 Fact Strategies Complete the Ten 9 + 5 =
  • 304. © Joan A. Cotter, Ph.D., 2012 Fact Strategies Complete the Ten 9 + 5 = Take 1 from the 5 and give it to the 9.
  • 305. © Joan A. Cotter, Ph.D., 2012 Fact Strategies Complete the Ten 9 + 5 = Take 1 from the 5 and give it to the 9. Use two hands and move the beads simultaneously.!
  • 306. © Joan A. Cotter, Ph.D., 2012 Fact Strategies Complete the Ten 9 + 5 = Take 1 from the 5 and give it to the 9.
  • 307. © Joan A. Cotter, Ph.D., 2012 Fact Strategies Complete the Ten 9 + 5 = 14 Take 1 from the 5 and give it to the 9.
  • 308. © Joan A. Cotter, Ph.D., 2012 Fact Strategies Two Fives 8 + 6 =
  • 309. © Joan A. Cotter, Ph.D., 2012 Fact Strategies Two Fives 8 + 6 =
  • 310. © Joan A. Cotter, Ph.D., 2012 Fact Strategies Two Fives 8 + 6 = Two fives make 10.!
  • 311. © Joan A. Cotter, Ph.D., 2012 Fact Strategies Two Fives 8 + 6 = Just add the “leftovers.”!
  • 312. © Joan A. Cotter, Ph.D., 2012 Fact Strategies Two Fives 8 + 6 = 10 + 4 = 14 Just add the “leftovers.”!
  • 313. © Joan A. Cotter, Ph.D., 2012 Fact Strategies Two Fives 7 + 5 = Another example.!
  • 314. © Joan A. Cotter, Ph.D., 2012 Fact Strategies Two Fives 7 + 5 =
  • 315. © Joan A. Cotter, Ph.D., 2012 Fact Strategies Two Fives 7 + 5 = 12
  • 316. © Joan A. Cotter, Ph.D., 2012 Fact Strategies Going Down 15 – 9 =
  • 317. © Joan A. Cotter, Ph.D., 2012 Fact Strategies Difference 7 – 4 = Subtract 4 from 5; then add 2.
  • 318. © Joan A. Cotter, Ph.D., 2012 Fact Strategies Going Down 15 – 9 =
  • 319. © Joan A. Cotter, Ph.D., 2012 Fact Strategies Going Down 15 – 9 = Subtract 5; then 4.
  • 320. © Joan A. Cotter, Ph.D., 2012 Fact Strategies Going Down 15 – 9 = Subtract 5; then 4.
  • 321. © Joan A. Cotter, Ph.D., 2012 Fact Strategies Going Down 15 – 9 = Subtract 5; then 4.
  • 322. © Joan A. Cotter, Ph.D., 2012 Fact Strategies Going Down 15 – 9 = 6 Subtract 5; then 4.
  • 323. © Joan A. Cotter, Ph.D., 2012 Fact Strategies Subtract from 10 15 – 9 =
  • 324. © Joan A. Cotter, Ph.D., 2012 Fact Strategies Subtract from 10 15 – 9 = Subtract 9 from 10.
  • 325. © Joan A. Cotter, Ph.D., 2012 Fact Strategies Subtract from 10 15 – 9 = Subtract 9 from 10.
  • 326. © Joan A. Cotter, Ph.D., 2012 Fact Strategies Subtract from 10 15 – 9 = Subtract 9 from 10.
  • 327. © Joan A. Cotter, Ph.D., 2012 Fact Strategies Subtract from 10 15 – 9 = 6 Subtract 9 from 10.
  • 328. © Joan A. Cotter, Ph.D., 2012 Fact Strategies Going Up 13 – 9 =
  • 329. © Joan A. Cotter, Ph.D., 2012 Fact Strategies Going Up 13 – 9 = Start with 9; go up to 13.
  • 330. © Joan A. Cotter, Ph.D., 2012 Fact Strategies Going Up 13 – 9 = Start with 9; go up to 13.
  • 331. © Joan A. Cotter, Ph.D., 2012 Fact Strategies Going Up 13 – 9 = Start with 9; go up to 13.
  • 332. © Joan A. Cotter, Ph.D., 2012 Fact Strategies Going Up 13 – 9 = Start with 9; go up to 13.
  • 333. © Joan A. Cotter, Ph.D., 2012 Fact Strategies Going Up 13 – 9 = 1 + 3 = 4 Start with 9; go up to 13.
  • 334. © Joan A. Cotter, Ph.D., 2012 Money Penny
  • 335. © Joan A. Cotter, Ph.D., 2012 Money Nickel
  • 336. © Joan A. Cotter, Ph.D., 2012 Money Dime
  • 337. © Joan A. Cotter, Ph.D., 2012 Money Quarter
  • 338. © Joan A. Cotter, Ph.D., 2012 Money Quarter
  • 339. © Joan A. Cotter, Ph.D., 2012 Money Quarter
  • 340. © Joan A. Cotter, Ph.D., 2012 Money Quarter
  • 341. © Joan A. Cotter, Ph.D., 2012 Base-10 Picture Cards One
  • 342. © Joan A. Cotter, Ph.D., 2012 Base-10 Picture Cards Ten One
  • 343. © Joan A. Cotter, Ph.D., 2012 Base-10 Picture Cards Hundred Ten One
  • 344. © Joan A. Cotter, Ph.D., 2012 Base-10 Picture Cards Thousand Hundred Ten One
  • 345. © Joan A. Cotter, Ph.D., 2012 Base-10 Picture Cards 3658! +2724! Add using the base-10 picture cards.
  • 346. © Joan A. Cotter, Ph.D., 2012 Base-10 Picture Cards 3 0 0 06 0 05 08
  • 347. © Joan A. Cotter, Ph.D., 2012 Base-10 Picture Cards 3 0 0 06 0 05 08
  • 348. © Joan A. Cotter, Ph.D., 2012 Base-10 Picture Cards 3 0 0 06 0 05 08
  • 349. © Joan A. Cotter, Ph.D., 2012 Base-10 Picture Cards 3 0 0 06 0 05 08
  • 350. © Joan A. Cotter, Ph.D., 2012 Base-10 Picture Cards 2 0 0 07 0 02 04
  • 351. © Joan A. Cotter, Ph.D., 2012 Base-10 Picture Cards 2 0 0 07 0 02 04
  • 352. © Joan A. Cotter, Ph.D., 2012 Base-10 Picture Cards 3 0 0 06 0 05 08 2 0 0 07 0 02 04 Add them together.
  • 353. © Joan A. Cotter, Ph.D., 2012 Base-10 Picture Cards 3 0 0 06 0 05 08 2 0 0 07 0 02 04
  • 354. © Joan A. Cotter, Ph.D., 2012 Base-10 Picture Cards 3 0 0 06 0 05 08 2 0 0 07 0 02 04 Trade 10 ones for 1 ten.
  • 355. © Joan A. Cotter, Ph.D., 2012 Base-10 Picture Cards Trade 10 ones for 1 ten. 3 0 0 06 0 05 08 2 0 0 07 0 02 04
  • 356. © Joan A. Cotter, Ph.D., 2012 Base-10 Picture Cards Trade 10 ones for 1 ten. 3 0 0 06 0 05 08 2 0 0 07 0 02 04
  • 357. © Joan A. Cotter, Ph.D., 2012 Base-10 Picture Cards 3 0 0 06 0 05 08 2 0 0 07 0 02 04
  • 358. © Joan A. Cotter, Ph.D., 2012 Base-10 Picture Cards 3 0 0 06 0 05 08 2 0 0 07 0 02 04 Trade 10 hundreds for 1 thousand.
  • 359. © Joan A. Cotter, Ph.D., 2012 Base-10 Picture Cards Trade 10 hundreds for 1 thousand. 3 0 0 06 0 05 08 2 0 0 07 0 02 04
  • 360. © Joan A. Cotter, Ph.D., 2012 Base-10 Picture Cards Trade 10 hundreds for 1 thousand. 3 0 0 06 0 05 08 2 0 0 07 0 02 04
  • 361. © Joan A. Cotter, Ph.D., 2012 Base-10 Picture Cards Trade 10 hundreds for 1 thousand. 3 0 0 06 0 05 08 2 0 0 07 0 02 04
  • 362. © Joan A. Cotter, Ph.D., 2012 Base-10 Picture Cards 6 0 0 03 0 08 02 3 0 0 06 0 05 08 2 0 0 07 0 02 04
  • 363. © Joan A. Cotter, Ph.D., 2012 Bead Frame 1 10 100 1000
  • 364. © Joan A. Cotter, Ph.D., 2012 Bead Frame 8 + 6 1 10 100 1000
  • 365. © Joan A. Cotter, Ph.D., 2012 Bead Frame 8 + 6 1 10 100 1000
  • 366. © Joan A. Cotter, Ph.D., 2012 Bead Frame 8 + 6 1 10 100 1000
  • 367. © Joan A. Cotter, Ph.D., 2012 Bead Frame 8 + 6 1 10 100 1000
  • 368. © Joan A. Cotter, Ph.D., 2012 Bead Frame 8 + 6 1 10 100 1000
  • 369. © Joan A. Cotter, Ph.D., 2012 Bead Frame 8 + 6 1 10 100 1000
  • 370. © Joan A. Cotter, Ph.D., 2012 Bead Frame 8 + 6 1 10 100 1000
  • 371. © Joan A. Cotter, Ph.D., 2012 Bead Frame 8 + 6 1 10 100 1000
  • 372. © Joan A. Cotter, Ph.D., 2012 Bead Frame 8 + 6 1 10 100 1000
  • 373. © Joan A. Cotter, Ph.D., 2012 8 + 6 14 1 10 100 1000 Bead Frame
  • 374. © Joan A. Cotter, Ph.D., 2012 Bead Frame Difficulties for the child 1 10 100 1000
  • 375. © Joan A. Cotter, Ph.D., 2012 • Distracting: Room is visible through the frame. Bead Frame Difficulties for the child 1 10 100 1000
  • 376. © Joan A. Cotter, Ph.D., 2012 • Distracting: Room is visible through the frame. • Not visualizable: Beads need to be grouped in fives. Bead Frame Difficulties for the child 1 10 100 1000
  • 377. © Joan A. Cotter, Ph.D., 2012 • Distracting: Room is visible through the frame. • Not visualizable: Beads need to be grouped in fives. • When beads are moved right, inconsistent with equation order: Beads need to be moved left. Bead Frame Difficulties for the child 1 10 100 1000
  • 378. © Joan A. Cotter, Ph.D., 2012 • Distracting: Room is visible through the frame. • Not visualizable: Beads need to be grouped in fives. • When beads are moved right, inconsistent with equation order: Beads need to be moved left. • Hierarchies of numbers represented sideways: They need to be in vertical columns. Bead Frame Difficulties for the child 1 10 100 1000
  • 379. © Joan A. Cotter, Ph.D., 2012 • Distracting: Room is visible through the frame. • Not visualizable: Beads need to be grouped in fives. • When beads are moved right, inconsistent with equation order: Beads need to be moved left. • Hierarchies of numbers represented sideways: They need to be in vertical columns. • Trading done before second number is completely added: Addends need to be combined before trading. Bead Frame Difficulties for the child 1 10 100 1000
  • 380. © Joan A. Cotter, Ph.D., 2012 • Distracting: Room is visible through the frame. • Not visualizable: Beads need to be grouped in fives. • When beads are moved right, inconsistent with equation order: Beads need to be moved left. • Hierarchies of numbers represented sideways: They need to be in vertical columns. • Trading done before second number is completely added: Addends need to be combined before trading. • Answer is read going up: We read top to bottom. Bead Frame Difficulties for the child 1 10 100 1000
  • 381. © Joan A. Cotter, Ph.D., 2012 Trading Side Cleared 1000 10 1100
  • 382. © Joan A. Cotter, Ph.D., 2012 1000 10 1100 Trading Side Thousands
  • 383. © Joan A. Cotter, Ph.D., 2012 1000 10 1100 Trading Side Hundreds The third wire from each end is not used.!
  • 384. © Joan A. Cotter, Ph.D., 2012 1000 10 1100 Trading Side Tens The third wire from each end is not used.!
  • 385. © Joan A. Cotter, Ph.D., 2012 1000 10 1100 Trading Side Ones The third wire from each end is not used.!
  • 386. © Joan A. Cotter, Ph.D., 2012 1000 10 1100 Trading Side Adding 8 + 6
  • 387. © Joan A. Cotter, Ph.D., 2012 1000 10 1100 Trading Side Adding 8 + 6
  • 388. © Joan A. Cotter, Ph.D., 2012 1000 10 1100 Trading Side Adding 8 + 6
  • 389. © Joan A. Cotter, Ph.D., 2012 1000 10 1100 Trading Side Adding 8 + 6
  • 390. © Joan A. Cotter, Ph.D., 2012 1000 10 1100 Trading Side Adding 8 + 6 14
  • 391. © Joan A. Cotter, Ph.D., 2012 1000 10 1100 Trading Side Adding 8 + 6 14 Too many ones; trade 10 ones for 1 ten. You can see the 10 ones (yellow).!
  • 392. © Joan A. Cotter, Ph.D., 2012 1000 10 1100 Trading Side Adding 8 + 6 14 Too many ones; trade 10 ones for 1 ten.
  • 393. © Joan A. Cotter, Ph.D., 2012 1000 10 1100 Trading Side Adding 8 + 6 14 Too many ones; trade 10 ones for 1 ten.
  • 394. © Joan A. Cotter, Ph.D., 2012 1000 10 1100 Trading Side Adding 8 + 6 14 Same answer before and after trading.
  • 395. © Joan A. Cotter, Ph.D., 2012 1000 10 1100 Trading Side Cleared
  • 396. © Joan A. Cotter, Ph.D., 2012 1000 10 1100 Trading Side Bead Trading game Object: To get a high score by adding numbers on the green cards.
  • 397. © Joan A. Cotter, Ph.D., 2012 1000 10 1100 Trading Side Bead Trading game Object: To get a high score by adding numbers on the green cards. 7
  • 398. © Joan A. Cotter, Ph.D., 2012 1000 10 1100 Trading Side Bead Trading game Object: To get a high score by adding numbers on the green cards. 7
  • 399. © Joan A. Cotter, Ph.D., 2012 1000 10 1100 Trading Side Bead Trading game Turn over another card. Enter 6 beads. Do we need to trade?! 6
  • 400. © Joan A. Cotter, Ph.D., 2012 1000 10 1100 Trading Side Bead Trading game Turn over another card. Enter 6 beads. Do we need to trade?! 6
  • 401. © Joan A. Cotter, Ph.D., 2012 1000 10 1100 Trading Side Bead Trading game Turn over another card. Enter 6 beads. Do we need to trade?! 6
  • 402. © Joan A. Cotter, Ph.D., 2012 1000 10 1100 Trading Side Bead Trading game Trade 10 ones for 1 ten. 6
  • 403. © Joan A. Cotter, Ph.D., 2012 1000 10 1100 Trading Side Bead Trading game 6
  • 404. © Joan A. Cotter, Ph.D., 2012 1000 10 1100 Trading Side Bead Trading game 6
  • 405. © Joan A. Cotter, Ph.D., 2012 1000 10 1100 Trading Side Bead Trading game 9
  • 406. © Joan A. Cotter, Ph.D., 2012 1000 10 1100 Trading Side Bead Trading game 9
  • 407. © Joan A. Cotter, Ph.D., 2012 1000 10 1100 Trading Side Bead Trading game Another trade. 9
  • 408. © Joan A. Cotter, Ph.D., 2012 1000 10 1100 Trading Side Bead Trading game Another trade. 9
  • 409. © Joan A. Cotter, Ph.D., 2012 1000 10 1100 Trading Side Bead Trading game 3
  • 410. © Joan A. Cotter, Ph.D., 2012 1000 10 1100 Trading Side Bead Trading game 3
  • 411. © Joan A. Cotter, Ph.D., 2012 Trading Side Bead Trading game • In the Bead Trading game 10 ones for 1 ten occurs frequently;
  • 412. © Joan A. Cotter, Ph.D., 2012 Trading Side Bead Trading game • In the Bead Trading game 10 ones for 1 ten occurs frequently; 10 tens for 1 hundred, less often;
  • 413. © Joan A. Cotter, Ph.D., 2012 Trading Side Bead Trading game • In the Bead Trading game 10 ones for 1 ten occurs frequently; 10 tens for 1 hundred, less often; 10 hundreds for 1 thousand, rarely.
  • 414. © Joan A. Cotter, Ph.D., 2012 Trading Side Bead Trading game • In the Bead Trading game 10 ones for 1 ten occurs frequently; 10 tens for 1 hundred, less often; 10 hundreds for 1 thousand, rarely. •  Bead trading helps the child experience the greater value of each column from left to right.
  • 415. © Joan A. Cotter, Ph.D., 2012 Trading Side Bead Trading game • In the Bead Trading game 10 ones for 1 ten occurs frequently; 10 tens for 1 hundred, less often; 10 hundreds for 1 thousand, rarely. •  Bead trading helps the child experience the greater value of each column from left to right. • To detect a pattern, there must be at least three examples in the sequence. (Place value is a pattern.)
  • 416. © Joan A. Cotter, Ph.D., 2012 1000 10 1100 Trading Side Adding 4-digit numbers 3658 + 2738
  • 417. © Joan A. Cotter, Ph.D., 2012 1000 10 1100 Trading Side Adding 4-digit numbers 3658 + 2738 Enter the first number from left to right.
  • 418. © Joan A. Cotter, Ph.D., 2012 1000 10 1100 Trading Side Adding 4-digit numbers 3658 + 2738 Enter the first number from left to right.
  • 419. © Joan A. Cotter, Ph.D., 2012 1000 10 1100 Trading Side Adding 4-digit numbers 3658 + 2738 Enter the first number from left to right.
  • 420. © Joan A. Cotter, Ph.D., 2012 1000 10 1100 Trading Side Adding 4-digit numbers 3658 + 2738 Enter the first number from left to right.
  • 421. © Joan A. Cotter, Ph.D., 2012 1000 10 1100 Trading Side Adding 4-digit numbers 3658 + 2738 Enter the first number from left to right.
  • 422. © Joan A. Cotter, Ph.D., 2012 1000 10 1100 Trading Side Adding 4-digit numbers 3658 + 2738 Enter the first number from left to right.
  • 423. © Joan A. Cotter, Ph.D., 2012 1000 10 1100 Trading Side Adding 4-digit numbers 3658 + 2738 Add starting at the right. Write results after each step.
  • 424. © Joan A. Cotter, Ph.D., 2012 1000 10 1100 Trading Side Adding 4-digit numbers 3658 + 2738 Add starting at the right. Write results after each step.
  • 425. © Joan A. Cotter, Ph.D., 2012 1000 10 1100 Trading Side Adding 4-digit numbers 3658 + 2738 Add starting at the right. Write results after each step.
  • 426. © Joan A. Cotter, Ph.D., 2012 1000 10 1100 Trading Side Adding 4-digit numbers 3658 + 2738 Add starting at the right. Write results after each step.
  • 427. © Joan A. Cotter, Ph.D., 2012 1000 10 1100 Trading Side Adding 4-digit numbers 3658 + 2738 6 Add starting at the right. Write results after each step. . . . 6 ones. Did anything else happen?!
  • 428. © Joan A. Cotter, Ph.D., 2012 1000 10 1100 Trading Side Adding 4-digit numbers 3658 + 2738 6 Add starting at the right. Write results after each step. 1 Is it okay to show the extra ten by writing a 1 above the tens column?!
  • 429. © Joan A. Cotter, Ph.D., 2012 1000 10 1100 Trading Side Adding 4-digit numbers 3658 + 2738 6 Add starting at the right. Write results after each step. 1
  • 430. © Joan A. Cotter, Ph.D., 2012 1000 10 1100 Trading Side Adding 4-digit numbers 3658 + 2738 6 Add starting at the right. Write results after each step. 1 Do we need to trade? [no]!
  • 431. © Joan A. Cotter, Ph.D., 2012 1000 10 1100 Trading Side Adding 4-digit numbers 3658 + 2738 96 Add starting at the right. Write results after each step. 1
  • 432. © Joan A. Cotter, Ph.D., 2012 1000 10 1100 Trading Side Adding 4-digit numbers 3658 + 2738 96 Add starting at the right. Write results after each step. 1
  • 433. © Joan A. Cotter, Ph.D., 2012 1000 10 1100 Trading Side Adding 4-digit numbers 3658 + 2738 96 Add starting at the right. Write results after each step. 1 Do we need to trade? [yes]!
  • 434. © Joan A. Cotter, Ph.D., 2012 1000 10 1100 Trading Side Adding 4-digit numbers 3658 + 2738 96 Add starting at the right. Write results after each step. 1 Notice the number of yellow beads. [3] Notice the number of blue beads left. [3] Coincidence? No, because 13 – 10 = 3.!
  • 435. © Joan A. Cotter, Ph.D., 2012 1000 10 1100 Trading Side Adding 4-digit numbers 3658 + 2738 96 Add starting at the right. Write results after each step. 1
  • 436. © Joan A. Cotter, Ph.D., 2012 1000 10 1100 Trading Side Adding 4-digit numbers 3658 + 2738 396 Add starting at the right. Write results after each step. 1
  • 437. © Joan A. Cotter, Ph.D., 2012 1000 10 1100 Trading Side Adding 4-digit numbers 3658 + 2738 396 Add starting at the right. Write results after each step. 11
  • 438. © Joan A. Cotter, Ph.D., 2012 1000 10 1100 Trading Side Adding 4-digit numbers 3658 + 2738 396 Add starting at the right. Write results after each step. 11
  • 439. © Joan A. Cotter, Ph.D., 2012 1000 10 1100 Trading Side Adding 4-digit numbers 3658 + 2738 396 Add starting at the right. Write results after each step. 11
  • 440. © Joan A. Cotter, Ph.D., 2012 1000 10 1100 Trading Side Adding 4-digit numbers 3658 + 2738 6396 Add starting at the right. Write results after each step. 11
  • 441. © Joan A. Cotter, Ph.D., 2012 1000 10 1100 Trading Side Adding 4-digit numbers 3658 + 2738 6396 Add starting at the right. Write results after each step. 11
  • 442. © Joan A. Cotter, Ph.D., 2012 Role of the AL Abacus “Neither is it strange to us, looking back, that there should have come a result quite unforeseen by the educators of that time, namely, a loss of the power of real insight into number [by not using abacuses].” David Eugene Smith, 1903
  • 443. © Joan A. Cotter, Ph.D., 2012 Role of the AL Abacus • Provides a visual organization of quantity. • Allows child to handle quantities in 5s and 10s. • Models trading tens needed for algorithms on the trading side. • Along with the math way of number naming, makes place value transparent. • Physical abacus leads to developing mental abacus. • Shows strategies concretely for learning facts. Its functions
  • 444. © Joan A. Cotter, Ph.D., 2012 Role of the AL Abacus 1. The abacus is needed for all number activities. 2. It’s used selectively for new concepts or unsure facts. 4. The abacus becomes completely internalized. Since this occurs at different times for different children, they must be encouraged to use the abacus whenever they need it. 3. Beads are moved on an imaginery abacus. Stages
  • 445. © Joan A. Cotter, Ph.D., 2012 Mental Addition You need to find twenty-four plus thirty-eight. How do you do it? You are sitting at your desk with a calculator, paper and pencil, and a box of teddy bears. Research shows a majority of people do it mentally. “How would you do it mentally?” Discuss methods.!
  • 446. © Joan A. Cotter, Ph.D., 2012 Mental Addition 24 + 38 = A very efficient way, taught to Dutch children, especially oral.! Dutch method
  • 447. © Joan A. Cotter, Ph.D., 2012 Mental Addition 24 + 38 = + 24 A very efficient way, taught to Dutch children, especially oral.! Dutch method
  • 448. © Joan A. Cotter, Ph.D., 2012 Mental Addition 24 + 38 = + 30 24 + A very efficient way, taught to Dutch children, especially oral.! Dutch method
  • 449. © Joan A. Cotter, Ph.D., 2012 Mental Addition 24 + 38 = + 30 24 + 8 = A very efficient way, taught to Dutch children, especially oral.! Dutch method
  • 450. © Joan A. Cotter, Ph.D., 2012 Multiplication on the AL Abacus Basic facts 6  4 = (6 taken 4 times)
  • 451. © Joan A. Cotter, Ph.D., 2012 Multiplication on the AL Abacus Basic facts 6  4 = (6 taken 4 times)
  • 452. © Joan A. Cotter, Ph.D., 2012 Multiplication on the AL Abacus Basic facts 6  4 = (6 taken 4 times)
  • 453. © Joan A. Cotter, Ph.D., 2012 Multiplication on the AL Abacus Basic facts 6  4 = (6 taken 4 times)
  • 454. © Joan A. Cotter, Ph.D., 2012 Multiplication on the AL Abacus Basic facts 6  4 = (6 taken 4 times)
  • 455. © Joan A. Cotter, Ph.D., 2012 Multiplication on the AL Abacus Basic facts 9  3 =
  • 456. © Joan A. Cotter, Ph.D., 2012 Multiplication on the AL Abacus Basic facts 9  3 =
  • 457. © Joan A. Cotter, Ph.D., 2012 Multiplication on the AL Abacus Basic facts 9  3 = 30
  • 458. © Joan A. Cotter, Ph.D., 2012 Multiplication on the AL Abacus Basic facts 9  3 = 30 – 3 = 27
  • 459. © Joan A. Cotter, Ph.D., 2012 Multiplication on the AL Abacus Basic facts 4  8 =
  • 460. © Joan A. Cotter, Ph.D., 2012 Multiplication on the AL Abacus Basic facts 4  8 =
  • 461. © Joan A. Cotter, Ph.D., 2012 Multiplication on the AL Abacus Basic facts 4  8 =
  • 462. © Joan A. Cotter, Ph.D., 2012 Multiplication on the AL Abacus Basic facts 4  8 = 20 + 12 = 32
  • 463. © Joan A. Cotter, Ph.D., 2012 Multiplication on the AL Abacus Basic facts 7  7 =
  • 464. © Joan A. Cotter, Ph.D., 2012 Multiplication on the AL Abacus Basic facts 7  7 =
  • 465. © Joan A. Cotter, Ph.D., 2012 Multiplication on the AL Abacus Basic facts 7  7 = 25 + 10 + 10 + 4 = 49
  • 466. © Joan A. Cotter, Ph.D., 2012 Multiplication on the AL Abacus Commutative property 5  6 =
  • 467. © Joan A. Cotter, Ph.D., 2012 Multiplication on the AL Abacus Commutative property 5  6 =
  • 468. © Joan A. Cotter, Ph.D., 2012 Multiplication on the AL Abacus Commutative property 5  6 =
  • 469. © Joan A. Cotter, Ph.D., 2012 Multiplication on the AL Abacus Commutative property 5  6 = 6  5
  • 470. © Joan A. Cotter, Ph.D., 2012 The Multiplication Board 1! 2! 3! 4! 5! 6! 7! 8! 9! 10! 6! 6  4 6 x 4 on original multiplication board.!
  • 471. © Joan A. Cotter, Ph.D., 2012 1! 2! 3! 4! 5! 6! 7! 8! 9! 10! 6! The Multiplication Board 6  4 Using two colors.!
  • 472. © Joan A. Cotter, Ph.D., 2012 The Multiplication Board 1! 2! 3! 4! 5! 6! 7! 8! 9! 10! 7! 7  7 7 x 7 on original multiplication board.!
  • 473. © Joan A. Cotter, Ph.D., 2012 1! 2! 3! 4! 5! 6! 7! 8! 9! 10! 7! The Multiplication Board 7  7 Upper left square is 25, yellow rectangles are 10. So, 25, 35, 45, 49.!
  • 474. © Joan A. Cotter, Ph.D., 2012 The Multiplication Board 7  7 Less clutter.!
  • 475. © Joan A. Cotter, Ph.D., 2012 Multiples Patterns Twos 2 4 6 8 10 12 14 16 18 20 Recognizing multiples needed for fractions and algebra.
  • 476. © Joan A. Cotter, Ph.D., 2012 Multiples Patterns Twos 2 4 6 8 10 12 14 16 18 20 The ones repeat in the second row. Recognizing multiples needed for fractions and algebra.
  • 477. © Joan A. Cotter, Ph.D., 2012 Multiples Patterns Twos 2 4 6 8 10 12 14 16 18 20 The ones repeat in the second row. Recognizing multiples needed for fractions and algebra.
  • 478. © Joan A. Cotter, Ph.D., 2012 Multiples Patterns Twos 2 4 6 8 10 12 14 16 18 20 The ones repeat in the second row. Recognizing multiples needed for fractions and algebra.
  • 479. © Joan A. Cotter, Ph.D., 2012 Multiples Patterns Twos 2 4 6 8 10 12 14 16 18 20 The ones repeat in the second row. Recognizing multiples needed for fractions and algebra.
  • 480. © Joan A. Cotter, Ph.D., 2012 Multiples Patterns Twos 2 4 6 8 10 12 14 16 18 20 The ones repeat in the second row. Recognizing multiples needed for fractions and algebra.
  • 481. © Joan A. Cotter, Ph.D., 2012 Multiples Patterns Fours 4 8 12 16 20 24 28 32 36 40
  • 482. © Joan A. Cotter, Ph.D., 2012 Multiples Patterns Fours 4 8 12 16 20 24 28 32 36 40 The ones repeat in the second row.
  • 483. © Joan A. Cotter, Ph.D., 2012 Multiples Patterns Sixes and Eights 6 12 18 24 30 36 42 48 54 60 8 16 24 32 40 48 56 64 72 80
  • 484. © Joan A. Cotter, Ph.D., 2012 Multiples Patterns Sixes and Eights 6 12 18 24 30 36 42 48 54 60 8 16 24 32 40 48 56 64 72 80
  • 485. © Joan A. Cotter, Ph.D., 2012 Multiples Patterns Sixes and Eights 6 12 18 24 30 36 42 48 54 60 8 16 24 32 40 48 56 64 72 80 Again the ones repeat in the second row.
  • 486. © Joan A. Cotter, Ph.D., 2012 Multiples Patterns Sixes and Eights 6 12 18 24 30 36 42 48 54 60 8 16 24 32 40 48 56 64 72 80 The ones in the 8s show the multiples of 2.
  • 487. © Joan A. Cotter, Ph.D., 2012 Multiples Patterns Sixes and Eights 6 12 18 24 30 36 42 48 54 60 8 16 24 32 40 48 56 64 72 80 The ones in the 8s show the multiples of 2.
  • 488. © Joan A. Cotter, Ph.D., 2012 Multiples Patterns Sixes and Eights 6 12 18 24 30 36 42 48 54 60 8 16 24 32 40 48 56 64 72 80 The ones in the 8s show the multiples of 2.
  • 489. © Joan A. Cotter, Ph.D., 2012 Multiples Patterns Sixes and Eights 6 12 18 24 30 36 42 48 54 60 8 16 24 32 40 48 56 64 72 80 The ones in the 8s show the multiples of 2.
  • 490. © Joan A. Cotter, Ph.D., 2012 Multiples Patterns Sixes and Eights 6 12 18 24 30 36 42 48 54 60 8 16 24 32 40 48 56 64 72 80 The ones in the 8s show the multiples of 2.
  • 491. © Joan A. Cotter, Ph.D., 2012 Multiples Patterns Sixes and Eights 6 12 18 24 30 36 42 48 54 60 8 16 24 32 40 48 56 64 72 80 6  4 6  4 is the fourth number (multiple).
  • 492. © Joan A. Cotter, Ph.D., 2012 Multiples Patterns Sixes and Eights 6 12 18 24 30 36 42 48 54 60 8 16 24 32 40 48 56 64 72 80 8  7 8  7 is the seventh number (multiple).
  • 493. © Joan A. Cotter, Ph.D., 2012 Multiples Patterns Nines 9 18 27 36 45 90 81 72 63 54 The second row is written in reverse order. Also the digits in each number add to 9.
  • 494. © Joan A. Cotter, Ph.D., 2012 Multiples Patterns Threes 3 6 9 12  15 18 21 24 27 30 The 3s have several patterns: Observe the ones.
  • 495. © Joan A. Cotter, Ph.D., 2012 Multiples Patterns Threes 3 6 9 12  15 18 21 24 27 30 The 3s have several patterns: Observe the ones.
  • 496. © Joan A. Cotter, Ph.D., 2012 Multiples Patterns Threes 3 6 9 12  15 18 21 24 27 30 The 3s have several patterns: Observe the ones.
  • 497. © Joan A. Cotter, Ph.D., 2012 Multiples Patterns Threes 3 6 9 12 15 18 21 24 27 30 The 3s have several patterns: Observe the ones.
  • 498. © Joan A. Cotter, Ph.D., 2012 Multiples Patterns Threes 3 6 9 12 15 18 21 24 27 30 The 3s have several patterns: Observe the ones.
  • 499. © Joan A. Cotter, Ph.D., 2012 Multiples Patterns Threes 3 6 9 12 15 18 21 24 27 30 The 3s have several patterns: Observe the ones.
  • 500. © Joan A. Cotter, Ph.D., 2012 Multiples Patterns Threes 3 6 9 12 15 18 21 24 27 30 The 3s have several patterns: Observe the ones.
  • 501. © Joan A. Cotter, Ph.D., 2012 Multiples Patterns Threes 3 6 9 12 15 18 21 24 27 30 The 3s have several patterns: Observe the ones.
  • 502. © Joan A. Cotter, Ph.D., 2012 Multiples Patterns Threes 3 6 9 12 15 18 21 24 27 30 The 3s have several patterns: Observe the ones.
  • 503. © Joan A. Cotter, Ph.D., 2012 Multiples Patterns Threes 3 6 9 12 15 18 21 24 27 30 The 3s have several patterns: Observe the ones.
  • 504. © Joan A. Cotter, Ph.D., 2012 Multiples Patterns Threes 3 6 9 12 15 18 21 24 27 30 The 3s have several patterns: Observe the ones.
  • 505. © Joan A. Cotter, Ph.D., 2012 Multiples Patterns Threes 3 6 9 12 15 18 21 24 27 30 The 3s have several patterns: The tens are the same in each row.
  • 506. © Joan A. Cotter, Ph.D., 2012 Multiples Patterns Threes 3 6 9 12 15 18 21 24 27 30 The 3s have several patterns: Add the digits in the columns.
  • 507. © Joan A. Cotter, Ph.D., 2012 Multiples Patterns Threes 3 6 9 12 15 18 21 24 27 30 The 3s have several patterns: Add the digits in the columns.
  • 508. © Joan A. Cotter, Ph.D., 2012 Multiples Patterns Threes 3 6 9 12 15 18 21 24 27 30 The 3s have several patterns: Add the digits in the columns.
  • 509. © Joan A. Cotter, Ph.D., 2012 Multiples Patterns Threes 3 6 9 12 15 18 21 24 27 30 The 3s have several patterns: Add the “opposites.”
  • 510. © Joan A. Cotter, Ph.D., 2012 Multiples Patterns Threes 3 6 9 12 15 18 21 24 27 30 The 3s have several patterns: Add the “opposites.”
  • 511. © Joan A. Cotter, Ph.D., 2012 Multiples Patterns Threes 3 6 9 12 15 18 21 24 27 30 The 3s have several patterns: Add the “opposites.”
  • 512. © Joan A. Cotter, Ph.D., 2012 Multiples Patterns Threes 3 6 9 12 15 18 21 24 27 30 The 3s have several patterns: Add the “opposites.”
  • 513. © Joan A. Cotter, Ph.D., 2012 Multiples Patterns Sevens 7 14 21 28 35 42 49 56 63 70 The 7s have the 1, 2, 3… pattern.
  • 514. © Joan A. Cotter, Ph.D., 2012 Multiples Patterns Sevens 7 14 21 28 35 42 49 56 63 70 The 7s have the 1, 2, 3… pattern.
  • 515. © Joan A. Cotter, Ph.D., 2012 Multiples Patterns Sevens 7 14 21 28 35 42 49 56 63 70 The 7s have the 1, 2, 3… pattern.
  • 516. © Joan A. Cotter, Ph.D., 2012 Multiples Patterns Sevens 7 14 21 28 35 42 49 56 63 70 The 7s have the 1, 2, 3… pattern.
  • 517. © Joan A. Cotter, Ph.D., 2012 Multiples Patterns Sevens 7 14 21 28 35 42 49 56 63 70 Look at the tens.
  • 518. © Joan A. Cotter, Ph.D., 2012 Multiples Patterns Sevens 7 14 21 28 35 42 49 56 63 70 Look at the tens.
  • 519. © Joan A. Cotter, Ph.D., 2012 Multiples Patterns Sevens 7 14 21 28 35 42 49 56 63 70 Look at the tens.
  • 520. © Joan A. Cotter, Ph.D., 2012 Multiples Memory “Multiples” are sometimes referred to as “skip counting.”!
  • 521. © Joan A. Cotter, Ph.D., 2012 Multiples Memory Aim: To help the players learn the multiples patterns. “Multiples” are sometimes referred to as “skip counting.”!
  • 522. © Joan A. Cotter, Ph.D., 2012 Multiples Memory Object of the game: To be the first player to collect all ten cards of a multiple in order. Aim: To help the players learn the multiples patterns.
  • 523. © Joan A. Cotter, Ph.D., 2012 Multiples Memory The 7s envelope contains 10 cards, each with one of the numbers listed. 7 14 21 28 35 42 49 56 63 70
  • 524. © Joan A. Cotter, Ph.D., 2012 Multiples Memory The 8s envelope contains 10 cards, each with one of the numbers listed. 8 16 24 32 40 48 56 64 72 80