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  • Montessori math materials ingeniously introduces children to our decimal system, but current research suggests that mathematical mastery can be better facilitated with simple enhancements in teaching techniques and material extensions. In this workshop, learn about research-based math discoveries, and explore ideas for Montessori math refinements, such as grouping the materials in fives to reduce counting and help the child in forming abstract images.
  • Show the baby 2 bears.
  • Show the baby 2 bears.
  • Show the baby 2 bears.
  • Show the baby 2 bears.
  • Show the baby 2 bears.
  • Stairs
  • Montessori math materials ingeniously introduces children to our decimal system, but current research suggests that mathematical mastery can be better facilitated with simple enhancements in teaching techniques and material extensions. In this workshop, learn about research-based math discoveries, and explore ideas for Montessori math refinements, such as grouping the materials in fives to reduce counting and help the child in forming abstract images.

NJMAC Visualization NJMAC Visualization Presentation Transcript

  • Enriching Montessori Mathematics with Visualization by Joan A. Cotter, Ph.D. JoanCotter@rightstartmath.com 1000 3 2 5 5 100 10 7 x7 1NJMAC Conference March 2, 2012 Edison, New Jersey Presentations available: rightstartmath.com © Joan A. Cotter, Ph.D., 2012
  • Verbal Counting Model2 © Joan A. Cotter, Ph.D., 2012
  • Verbal Counting Model From a childs 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 forth3 © Joan A. Cotter, Ph.D., 2012
  • Verbal Counting Model From a childs perspective F +E4 © Joan A. Cotter, Ph.D., 2012
  • Verbal Counting Model From a childs perspective F +E A5 © Joan A. Cotter, Ph.D., 2012
  • Verbal Counting Model From a childs perspective F +E A B6 © Joan A. Cotter, Ph.D., 2012
  • Verbal Counting Model From a childs perspective F +E A B C7 © Joan A. Cotter, Ph.D., 2012
  • Verbal Counting Model From a childs perspective F +E A B C D E F8 © Joan A. Cotter, Ph.D., 2012
  • Verbal Counting Model From a childs perspective F +E A B C D E F A9 © Joan A. Cotter, Ph.D., 2012
  • Verbal Counting Model From a childs perspective F +E A B C D E F A B10 © Joan A. Cotter, Ph.D., 2012
  • Verbal Counting Model From a childs perspective F +E A B C D E F A B C D E11 © Joan A. Cotter, Ph.D., 2012
  • Verbal Counting Model From a childs perspective F +E A B C D E F A B C D E What is the sum? (It must be a letter.)12 © Joan A. Cotter, Ph.D., 2012
  • Verbal Counting Model From a childs perspective F +E K A B C D E F G H I J K13 © Joan A. Cotter, Ph.D., 2012
  • Verbal Counting Model From a childs perspective Now memorize the facts!! G +D14 © Joan A. Cotter, Ph.D., 2012
  • Verbal Counting Model From a childs perspective Now memorize the facts!! H + G F +D15 © Joan A. Cotter, Ph.D., 2012
  • Verbal Counting Model From a childs perspective Now memorize the facts!! H + G F +D D +C16 © Joan A. Cotter, Ph.D., 2012
  • Verbal Counting Model From a childs perspective Now memorize the facts!! H + G F +D D C +C +G17 © Joan A. Cotter, Ph.D., 2012
  • Verbal Counting Model From a childs perspective Now memorize the facts!! H E + G I F + +D D C +C +G18 © Joan A. Cotter, Ph.D., 2012
  • Verbal Counting Model From a childs perspective Try subtracting H by “taking away” –E19 © Joan A. Cotter, Ph.D., 2012
  • Verbal Counting Model From a childs perspective Try skip counting by B’s to T: B, D, . . . T.20 © Joan A. Cotter, Ph.D., 2012
  • Verbal Counting Model From a childs perspective Try skip counting by B’s to T: B, D, . . . T. What is D × E?21 © Joan A. Cotter, Ph.D., 2012
  • Verbal Counting Model From a childs perspective L is written AB because it is A J and B A’s22 © Joan A. Cotter, Ph.D., 2012
  • Verbal Counting Model From a childs perspective L is written AB because it is A J and B A’s huh?23 © Joan A. Cotter, Ph.D., 2012
  • Verbal Counting Model From a childs perspective L (twelve) is written AB because it is A J and B A’s24 © Joan A. Cotter, Ph.D., 2012
  • Verbal Counting Model From a childs perspective L (twelve) is written AB (12) because it is A J and B A’s25 © Joan A. Cotter, Ph.D., 2012
  • Verbal Counting Model From a childs perspective L (twelve) is written AB (12) because it is A J (one 10) and B A’s26 © Joan A. Cotter, Ph.D., 2012
  • Verbal Counting Model From a childs perspective L (twelve) is written AB (12) because it is A J (one 10) and B A’s (two 1s).27 © Joan A. Cotter, Ph.D., 2012
  • Verbal Counting Model Summary28 © Joan A. Cotter, Ph.D., 2012
  • Verbal Counting Model Summary • Is not natural; it takes years of practice.29 © Joan A. Cotter, Ph.D., 2012
  • Verbal Counting Model Summary • Is not natural; it takes years of practice. • Provides poor concept of quantity.30 © Joan A. Cotter, Ph.D., 2012
  • Verbal Counting Model Summary • Is not natural; it takes years of practice. • Provides poor concept of quantity. • Ignores place value.31 © Joan A. Cotter, Ph.D., 2012
  • Verbal Counting Model Summary • Is not natural; it takes years of practice. • Provides poor concept of quantity. • Ignores place value. • Is very error prone.32 © Joan A. Cotter, Ph.D., 2012
  • Verbal Counting Model Summary • 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.33 © Joan A. Cotter, Ph.D., 2012
  • Verbal Counting Model Summary • 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. • Does not provide an efficient way to master the facts.34 © Joan A. Cotter, Ph.D., 2012
  • Calendar Math August 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31Sometimes calendars are used for counting. © Joan A. Cotter, Ph.D., 201235
  • Calendar Math August 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31Sometimes calendars are used for counting. © Joan A. Cotter, Ph.D., 201236
  • Calendar Math August 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 3137 © Joan A. Cotter, Ph.D., 2012
  • Calendar Math August 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31This is ordinal, not cardinal counting. The 3 doesn’t include the 1 and the 2.Joan A. Cotter, Ph.D., 2012 ©38
  • Calendar Math Septemb 1234567 August 89101214 1 113 11921 2 15112628 122820 8 67527 9 3 4 10 11 12 13 14 5 6 7 2234 20 15 16 17 18 19 20 21 29 3 22 23 24 25 26 27 28 29 30 31This is ordinal, not cardinal counting. The 4 doesn’t include 1, 2 and 3. © Joan A. Cotter, Ph.D., 201239
  • Calendar Math Septemb 1234567 August 89101214 1 113 11921 2 15112628 122820 8 67527 9 3 4 5 10 11 12 13 14 6 7 2234 20 15 16 17 18 19 20 21 29 3 22 23 24 25 26 27 28 29 30 31 1 2 3 4 5 6A calendar is NOT a ruler. On a ruler the numbers are not in the spaces. © Joan A. Cotter, Ph.D., 201240
  • Calendar Math August 1 2 3 4 5 6 7 8 9 10Always show the whole calendar. A child needs to see the wholebefore the parts. Children also need to learn to plan ahead. © Joan A. Cotter, Ph.D., 201241
  • Calendar Math The calendar is not a number line. • No quantity is involved. • Numbers are in spaces, not at lines like a ruler.42 © 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.43 © 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 31.44 © Joan A. Cotter, Ph.D., 2012
  • Memorizing Math 9 +7 Flash cards:• 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 aboutthinking.• Often produce stress – children under stressstop learning.• Are not concrete – use abstract symbols. © Joan A. Cotter, Ph.D., 2012
  • Learning Arithmetic Compared to reading: • A child learns to read. • Later a child uses reading to learn. • A child learns to do arithmetic. • Later a child uses arithmetic to solve problems.Show the baby two teddy bears. © Joan A. Cotter, Ph.D., 2012
  • Research on Counting Karen Wynn’s researchShow the baby two teddy bears. © Joan A. Cotter, Ph.D., 2012
  • Research on Counting Karen Wynn’s researchShow the baby two teddy bears. © Joan A. Cotter, Ph.D., 2012
  • Research on Counting Karen Wynn’s researchThen hide them with a screen. © Joan A. Cotter, Ph.D., 201249
  • Research on Counting Karen Wynn’s researchShow the baby a third teddy bear and put it behind the screen. © Joan A. Cotter, Ph.D., 201250
  • Research on Counting Karen Wynn’s researchShow the baby a third teddy bear and put it behind the screen. © Joan A. Cotter, Ph.D., 201251
  • Research on Counting Karen Wynn’s researchRaise screen. Baby seeing 3 won’t look long because it is expected. © Joan A. Cotter, Ph.D., 201252
  • Research on Counting Karen Wynn’s researchResearcher can change the number of teddy bears behind the screen. © Joan A. Cotter, Ph.D., 201253
  • Research on Counting Karen Wynn’s researchA baby seeing 1 teddy bear will look much longer, because it’s unexpected.Joan A. Cotter, Ph.D., 2012 ©54
  • Research on Counting Other research55 © Joan A. Cotter, Ph.D., 2012
  • Research on Counting Other research • Australian Aboriginal children from two tribes. Brian Butterworth, University College London, 2008.These groups matched quantities without using counting words. © Joan A. Cotter, Ph.D., 201256
  • Research on Counting Other research • Australian Aboriginal children from two tribes. Brian Butterworth, University College London, 2008. • Adult Pirahã from Amazon region. Edward Gibson and Michael Frank, MIT, 2008.These groups matched quantities without using counting words. © Joan A. Cotter, Ph.D., 201257
  • Research on Counting Other research • 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.These groups matched quantities without using counting words. © Joan A. Cotter, Ph.D., 201258
  • Research on Counting Other research • 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.These groups matched quantities without using counting words. © Joan A. Cotter, Ph.D., 201259
  • Research on Counting In Japanese schools: • Children are discouraged from using counting for adding.60 © 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.61 © Joan A. Cotter, Ph.D., 2012
  • Research on Counting Subitizing • Subitizing is quick recognition of quantity without counting.62 © 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.63 © 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 long term.—Butterworth64 © 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 long term.—Butterworth • Subitizing “allows the child to grasp the whole and the elements at the same time.”—Benoit65 © 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 long term.—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.— Glasersfeld66 © Joan A. Cotter, Ph.D., 2012
  • Visualizing Mathematics67 © 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 (E I)68 © 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, 200969 © 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 others70 © Joan A. Cotter, Ph.D., 2012
  • Visualizing Mathematics Japanese criteria for manipulatives• Representative of structure of numbers.• Easily manipulated by children.• Imaginable mentally. Japanese Council of Mathematics Education © Joan A. Cotter, Ph.D., 2012
  • Visualizing Mathematics Visualizing also needed in:• Reading• Sports• Creativity• Geography• Engineering• Construction © Joan A. Cotter, Ph.D., 2012
  • Visualizing Mathematics Visualizing also needed in:• Reading • Architecture• Sports • Astronomy• Creativity • Archeology• Geography • Chemistry• Engineering • Physics• Construction • Surgery © 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 MathematicsTry to visualize 8 identical apples without grouping. © Joan A. Cotter, Ph.D., 2012
  • Visualizing MathematicsTry to visualize 8 identical apples without grouping. © Joan A. Cotter, Ph.D., 2012
  • Visualizing MathematicsNow try to visualize 5 as red and 3 as green. © Joan A. Cotter, Ph.D., 2012
  • Visualizing MathematicsNow try to visualize 5 as red and 3 as green. © Joan A. Cotter, Ph.D., 2012
  • Visualizing Mathematics Early Roman numerals 1 I 2 II 3 III 4 IIII 5 V 8 VIIIRomans 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., 201286
  • Very Early Computation Numerals In English there are two ways of writing numbers: Numerals: 357887 © Joan A. Cotter, Ph.D., 2012
  • Very Early Computation Numerals In English there are two ways of writing numbers: Numerals: 3578 Words: Three thousand five hundred seventy-eight88 © Joan A. Cotter, Ph.D., 2012
  • Very Early Computation Numerals In English there are two ways of writing numbers: Numerals: 3578 Words: Three thousand five hundred seventy-eight In ancient Chinese there was only one way of writing numbers: 3 Th 5 H 7 T 8 U (8 characters)89 © 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.90 © Joan A. Cotter, Ph.D., 2012
  • Very Early Computation Calculating rods91 © Joan A. Cotter, Ph.D., 2012
  • Very Early Computation Calculating rods Numerals for Ones and Hundreds (Even Powers of Ten)92 © Joan A. Cotter, Ph.D., 2012
  • Very Early Computation Calculating rods Numerals for Ones and Hundreds (Even Powers of Ten)93 © Joan A. Cotter, Ph.D., 2012
  • Very Early Computation Calculating rods Numerals for Ones and Hundreds (Even Powers of Ten) Numerals for Tens and Thousands (Odd Powers of Ten)94 © Joan A. Cotter, Ph.D., 2012
  • Very Early Computation Calculating rods 357895 © Joan A. Cotter, Ph.D., 2012
  • Very Early Computation Calculating rods 3578 3578,3578 They grouped, not in thousands, but ten-thousands!96 © Joan A. Cotter, Ph.D., 2012
  • Naming Quantities Using fingers © Joan A. Cotter, Ph.D., 2012
  • Naming Quantities Using fingersNaming quantities is a three-period lesson. © Joan A. Cotter, Ph.D., 2012
  • Naming Quantities Using fingersUse left hand for 1-5 because we read from left to right. © Joan A. Cotter, Ph.D., 2012
  • Naming Quantities Using fingers100 © Joan A. Cotter, Ph.D., 2012
  • Naming Quantities Using fingers101 © Joan A. Cotter, Ph.D., 2012
  • Naming Quantities Using fingersAlways show 7 as 5 and 2, not for example, as 4 and 3. © Joan A. Cotter, Ph.D., 2012102
  • Naming Quantities Using fingers103 © Joan A. Cotter, Ph.D., 2012
  • Naming Quantities Yellow is the Sun 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. CotterAlso 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 Recognizing 5 5 has a middle; 4 does not.Look at your hand; your middle finger is longer to remind you 5 has a middle. A. Cotter, Ph.D., 2012 © Joan
  • Naming Quantities Tally sticksLay the sticks flat on a surface, about 1 inch (2.5 cm) apart. © Joan A. Cotter, Ph.D., 2012
  • Naming Quantities Tally sticks109 © Joan A. Cotter, Ph.D., 2012
  • Naming Quantities Tally sticks110 © Joan A. Cotter, Ph.D., 2012
  • Naming Quantities Tally sticksStick is horizontal, because it won’t fit diagonally and young children haveproblems with diagonals.111 © Joan A. Cotter, Ph.D., 2012
  • Naming Quantities Tally sticks112 © Joan A. Cotter, Ph.D., 2012
  • Naming Quantities Tally sticksStart a new row for every ten. © Joan A. Cotter, Ph.D., 2012113
  • Naming Quantities Solving a problem without counting What is 4 apples plus 3 more apples?How would you find the answer without counting? © Joan A. Cotter, Ph.D., 2012114
  • Naming Quantities Solving a problem without counting What is 4 apples plus 3 more apples?To remember 4 + 3, the Japanese child is taught to visualize 4 and 3. Thentake 1 from the 3 and give it to the 4 to make 5 and 2. © Joan A. Cotter, Ph.D., 2012115
  • Naming QuantitiesNumberChart 1 2 3 4 5 © Joan A. Cotter, Ph.D., 2012
  • Naming Quantities Number Chart 1 2To help the 3child learnthe symbols 4 5 © Joan A. Cotter, Ph.D., 2012
  • Naming Quantities Number Chart 1 6 2 7To help the 3 8child learnthe symbols 4 9 5 10 © Joan A. Cotter, Ph.D., 2012
  • Naming Quantities Pairing Finger Cards QuickTimeª and a QuickTimeª and a QuickTimeª and a QuickTimeª and a QuickTimeª and a TIFF (LZW) decompressor TIFF (LZW) decompressor TIFF (LZW) decompressor TIFF (LZW) decompressor TIFF (LZW) decompressor are needed to see this picture. are needed to see this picture. are needed to see this picture. are needed to see this picture. are needed to see this picture. QuickTimeª and a TIFF (LZW) decompressor are needed to see this picture. QuickTimeª and a QuickTimeª and a QuickTimeª and a QuickTimeª and a QuickTimeª and a TIFF (LZW) decompressor TIFF (LZW) decompressor TIFF (LZW) decompressor TIFF (LZW) decompressor TIFF (LZW) decompressor are needed to see this picture. are needed to see this picture. are needed to see this picture. are needed to see this picture. are needed to see this picture. QuickTimeª and a TIFF (LZW) decompressor are needed to see this picture. QuickTimeª andand a TIFFQuickTimeª and aa QuickTimeª and a QuickTimeª areTIFF (LZW) decompressor TIFF (LZW) decompressor (LZW) decompressor areTIFF (LZW)to see this picture. needed(LZW)seeathis picture. see decompressor are neededto see this picture. to to see this picture. QuickTimeª this picture. are needed to and needed decompressor TIFF are neededUse two sets of finger cards and match them. © Joan A. Cotter, Ph.D., 2012119
  • Naming Quantities Ordering Finger Cards QuickTimeª and a TIFF (LZW) decompressor are needed to see this picture. QuickTimeª and a TIFF (LZW) decompressor QuickTimeª and a are needed to see this picture. TIFF (LZW) decompressor are needed to see this picture. QuickTimeª and a TIFF (LZW) decompressor are needed to see this picture. QuickTimeª and a TIFF (LZW) decompressor are needed to see this picture. QuickTimeª and a TIFF (LZW) decompressor are needed to see this picture. QuickTimeª and a TIFF (LZW) decompressor are needed to see this picture. QuickTimeª and a TIFF (LZW) decompressor are needed to see this picture. QuickTimeª and a TIFF (LZW) decompressor are needed to see this picture. QuickTimeª and a TIFF (LZW) decompressor are needed to see this picture.Putting the finger cards in order. © Joan A. Cotter, Ph.D., 2012120
  • Naming Quantities Matching Numbers to Finger Cards QuickTimeª and a QuickTimeª and a QuickTimeª and a QuickTimeª and a QuickTimeª and a TIFF (LZW) decompressor TIFF (LZW) decompressor TIFF (LZW) decompressor TIFF (LZW) decompressor TIFF (LZW) decompressor are needed to see this picture. are needed to see this picture. are needed to see this picture. are needed to see this picture. are needed to see this picture. 5 1 QuickTimeª and a QuickTimeª and a QuickTimeª and a QuickTimeª and a QuickTimeª and a TIFF (LZW) decompressor TIFF (LZW) decompressor TIFF (LZW) decompressor TIFF (LZW) decompressor TIFF (LZW) decompressor are needed to see this picture. are needed to see this picture. are needed to see this picture. are needed to see this picture. are needed to see this picture. 10Match the number to the finger card. © Joan A. Cotter, Ph.D., 2012121
  • Naming Quantities Matching Fingers to Number Cards 9 1 10 4 6 QuickTimeª and a TIFF (LZW) decompressor are needed to see this picture. 2 3 7 8 5 QuickTimeª and a TIFF (LZW) decompressor are needed to see this picture. QuickTimeª and aa QuickTimeª and a QuickTimeª and a TIFF (LZW) decompressor QuickTimeª and a TIFF (LZW) decompressor QuickTimeª and a TIFF (LZW) decompressor are needed QuickTimeªpicture. and are neededtotoseedecompressor TIFF (LZW) this picture. see TIFF (LZW) decompressor are neededtotoseedecompressor TIFF (LZW) this picture. are needed toseethis picture. are needed toseethis picture. are needed seethis picture. thisMatch the finger card to the number. © Joan A. Cotter, Ph.D., 2012122
  • Naming Quantities Finger Card Memory game QuickTimeª and a QuickTimeª and a QuickTimeª and a QuickTimeª and a TIFF (LZW) decompressor TIFF (LZW) decompressor TIFF (LZW) decompressor are needed to see this picture. needed to see this picture. are TIFF (LZW) decompressor are needed to see this picture. are needed to see this picture. QuickTimeª and a QuickTimeª and a QuickTimeª and a QuickTimeª and a TIFF (LZW) decompressor TIFF (LZW) decompressor TIFF (LZW) decompressor TIFF (LZW) decompressor are needed to see this picture. are needed to see this picture. are needed to see this picture. are needed to see this picture. QuickTimeª and a QuickTimeª and a QuickTimeª and a QuickTimeª and a TIFF (LZW) decompressor TIFF (LZW) decompressor TIFF (LZW) decompressor TIFF (LZW) decompressor are needed to see this picture. are needed to see this picture. are needed to see this picture. needed to see this picture. are QuickTimeª and a QuickTimeª and a QuickTimeª and a QuickTimeª and a TIFF (LZW) decompressor TIFF (LZW) decompressor TIFF (LZW) decompressor TIFF (LZW) decompressor are needed to see this picture. are needed to see this picture. are needed to see this picture. needed to see this picture. are QuickTimeª and a QuickTimeª and a QuickTimeª and a QuickTimeª and a TIFF (LZW) decompressor TIFF (LZW) decompressor TIFF (LZW) decompressor TIFF (LZW) decompressor are needed to see this picture. are needed to see this picture. are needed to see this picture. needed to see this picture. areUse two sets of finger cards and play Memory. © Joan A. Cotter, Ph.D., 2012123
  • Naming Quantities Number Rods124 © Joan A. Cotter, Ph.D., 2012
  • Naming Quantities Number Rods125 © Joan A. Cotter, Ph.D., 2012
  • Naming Quantities Number RodsUsing different colors. © Joan A. Cotter, Ph.D., 2012126
  • Naming Quantities Spindle Box45 dark-colored and 10 light-colored spindles. Could be in separate containers. Cotter, Ph.D., 2012 © Joan A.127
  • Naming Quantities Spindle Box45 dark-colored and 10 light-colored spindles in two containers. © Joan A. Cotter, Ph.D., 2012128
  • Naming Quantities Spindle Box 0 1 2 3 4The child takes blue spindles with left hand and yellow with right. © Joan A. Cotter, Ph.D., 2012129
  • Naming Quantities Spindle Box 5 6 7 8 9The child takes blue spindles with left hand and yellow with right. © Joan A. Cotter, Ph.D., 2012130
  • Naming Quantities Spindle Box 5 6 7 8 9The child takes blue spindles with left hand and yellow with right. © Joan A. Cotter, Ph.D., 2012131
  • Naming Quantities Spindle Box 5 6 7 8 9The child takes blue spindles with left hand and yellow with right. © Joan A. Cotter, Ph.D., 2012132
  • Naming Quantities Spindle Box 5 6 7 8 9The child takes blue spindles with left hand and yellow with right. © Joan A. Cotter, Ph.D., 2012133
  • Naming Quantities Spindle Box 5 6 7 8 9The child takes blue spindles with left hand and yellow with right. © Joan A. Cotter, Ph.D., 2012134
  • Naming Quantities Spindle Box 5 6 7 8 9The child takes blue spindles with left hand and yellow with right. © Joan A. Cotter, Ph.D., 2012135
  • Naming Quantities Black and White Bead Stairs “Grouped in fives so the child does not need to count.” A. M. JoostenThis was the inspiration to group in 5s. © Joan A. Cotter, Ph.D., 2012136
  • AL Abacus Cleared © Joan A. Cotter, Ph.D., 2012
  • AL Abacus Entering quantities 3Quantities are entered all at once, not counted. © Joan A. Cotter, Ph.D., 2012
  • AL Abacus Entering quantities 5Relate quantities to hands. © Joan A. Cotter, Ph.D., 2012139
  • AL Abacus Entering quantities 7140 © Joan A. Cotter, Ph.D., 2012
  • AL Abacus Entering quantities 10141 © Joan A. Cotter, Ph.D., 2012
  • AL Abacus The stairsCan use to “count” 1 to 10. Also read quantities on the right side. © Joan A. Cotter, Ph.D., 2012142
  • AL Abacus Adding © Joan A. Cotter, Ph.D., 2012
  • AL Abacus Adding4+3= © Joan A. Cotter, Ph.D., 2012
  • AL Abacus Adding4+3= © Joan A. Cotter, Ph.D., 2012
  • AL Abacus Adding4+3= © Joan A. Cotter, Ph.D., 2012
  • AL Abacus Adding4+3= © Joan A. Cotter, Ph.D., 2012
  • AL Abacus Adding 4+3=7Answer 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+5Children use the abacus while playing this “Go Fish” type game. © Joan A. Cotter, Ph.D., 2012149
  • 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., 2012150
  • Go to the Dump GameThe ways to partition 10. © Joan A. Cotter, Ph.D., 2012151
  • Go to the Dump Game StartingA game viewed from above. © Joan A. Cotter, Ph.D., 2012152
  • Go to the Dump Game 72 7 9 5 72 1 3 8 4 6 34 9 StartingEach player takes 5 cards. © Joan A. Cotter, Ph.D., 2012153
  • Go to the Dump Game 72 7 9 5 72 1 3 8 4 6 34 9 Finding pairsDoes YellowCap have any pairs? [no] © Joan A. Cotter, Ph.D., 2012154
  • Go to the Dump Game 72 7 9 5 72 1 3 8 4 6 34 9 Finding pairsDoes BlueCap have any pairs? [yes, 1] © Joan A. Cotter, Ph.D., 2012155
  • Go to the Dump Game 72 7 9 5 72 1 3 8 4 6 34 9 Finding pairsDoes BlueCap have any pairs? [yes, 1] © Joan A. Cotter, Ph.D., 2012156
  • Go to the Dump Game 72 7 9 5 4 6 72 1 3 8 34 9 Finding pairsDoes BlueCap have any pairs? [yes, 1] © Joan A. Cotter, Ph.D., 2012157
  • Go to the Dump Game 72 7 9 5 4 6 72 1 3 8 34 9 Finding pairsDoes PinkCap have any pairs? [yes, 2] © Joan A. Cotter, Ph.D., 2012158
  • Go to the Dump Game 72 7 9 5 4 6 72 1 3 8 34 9 Finding pairsDoes PinkCap have any pairs? [yes, 2] © Joan A. Cotter, Ph.D., 2012159
  • Go to the Dump Game 72 7 9 5 7 3 4 6 2 1 8 34 9 Finding pairsDoes PinkCap have any pairs? [yes, 2] © Joan A. Cotter, Ph.D., 2012160
  • Go to the Dump Game 72 7 9 5 2 8 4 6 1 34 9 Finding pairsDoes PinkCap have any pairs? [yes, 2] © Joan A. Cotter, Ph.D., 2012161
  • Go to the Dump Game 72 7 9 5 2 8 4 6 1 34 9 PlayingThe player asks the player on her left. © Joan A. Cotter, Ph.D., 2012162
  • Go to the Dump Game BlueCap, do you have an3? have a 3? 72 7 9 5 2 8 4 6 1 34 9 PlayingThe player asks the player on her left. © Joan A. Cotter, Ph.D., 2012163
  • Go to the Dump Game BlueCap, do you have an3? have a 3? 72 7 9 5 3 2 8 4 6 1 4 9 PlayingThe player asks the player on her left. © Joan A. Cotter, Ph.D., 2012164
  • Go to the Dump Game 7 3 BlueCap, do you have an3? have a 3? 2 7 9 5 2 8 4 6 1 4 9 Playing165 © Joan A. Cotter, Ph.D., 2012
  • Go to the Dump Game 7 3 BlueCap, do you have an3? have a 8? 2 7 9 5 2 8 4 6 1 4 9 PlayingYellowCap gets another turn. © Joan A. Cotter, Ph.D., 2012166
  • Go to the Dump Game 7 3 BlueCap, do you have an3? have a 8? 2 7 9 5 2 8 4 6 1 4 9 Go to the dump. PlayingYellowCap gets another turn. © Joan A. Cotter, Ph.D., 2012167
  • Go to the Dump Game 7 3 BlueCap, do you have an3? have a 8? 2 2 7 9 5 2 8 4 6 1 4 9 Go to the dump. Playing168 © Joan A. Cotter, Ph.D., 2012
  • Go to the Dump Game 7 3 2 2 7 9 5 2 8 4 6 1 4 9 Playing169 © Joan A. Cotter, Ph.D., 2012
  • Go to the Dump Game 7 3 2 2 7 9 5 2 8 4 6 1 4 9 PinkCap, do you Playing have a 6?170 © Joan A. Cotter, Ph.D., 2012
  • Go to the Dump Game 7 3 2 2 7 9 5 2 8 4 6 1 4 9 PinkCap, do you Go to the dump. Playing have a 6?171 © Joan A. Cotter, Ph.D., 2012
  • Go to the Dump Game 7 3 2 2 7 9 5 2 8 4 6 1 5 4 9 Playing172 © Joan A. Cotter, Ph.D., 2012
  • Go to the Dump Game 7 3 2 2 7 9 5 2 8 4 6 1 5 4 9 Playing173 © Joan A. Cotter, Ph.D., 2012
  • Go to the Dump Game 7 3 2 2 7 9 5 2 8 4 6 1 5 4 9 YellowCap, do you have a 9? Playing174 © Joan A. Cotter, Ph.D., 2012
  • Go to the Dump Game 7 3 2 2 7 5 2 8 4 6 1 5 4 9 YellowCap, do you have a 9? Playing175 © Joan A. Cotter, Ph.D., 2012
  • Go to the Dump Game 7 3 2 2 7 5 2 8 4 6 19 5 4 9 YellowCap, do you have a 9? Playing176 © Joan A. Cotter, Ph.D., 2012
  • Go to the Dump Game 7 3 2 2 7 5 2 1 8 9 4 6 5 4 9 Playing177 © Joan A. Cotter, Ph.D., 2012
  • Go to the Dump Game 7 3 2 2 7 5 2 1 8 9 4 6 2 9 1 7 7 5 4 9 PlayingPinkCap is not out of the game. Her turn ends, but she takes 5 more cards.Joan A. Cotter, Ph.D., 2012 ©178
  • Go to the Dump Game 9 1 4 6 5 5 Winner?179 © Joan A. Cotter, Ph.D., 2012
  • Go to the Dump Game 9 1 4 6 5 Winner?No counting. Combine both stacks. © Joan A. Cotter, Ph.D., 2012180
  • Go to the Dump Game 9 1 4 6 5 Winner?Whose stack is the highest? © Joan A. Cotter, Ph.D., 2012181
  • Go to the Dump Game Next gameNo shuffling needed for next game. © Joan A. Cotter, Ph.D., 2012182
  • “Math” Way of Naming Numbers183 © Joan A. Cotter, Ph.D., 2012
  • “Math” Way of Naming Numbers 11 = ten 1184 © Joan A. Cotter, Ph.D., 2012
  • “Math” Way of Naming Numbers 11 = ten 1 12 = ten 2185 © Joan A. Cotter, Ph.D., 2012
  • “Math” Way of Naming Numbers 11 = ten 1 12 = ten 2 13 = ten 3186 © Joan A. Cotter, Ph.D., 2012
  • “Math” Way of Naming Numbers 11 = ten 1 12 = ten 2 13 = ten 3 14 = ten 4187 © 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 9188 © Joan A. Cotter, Ph.D., 2012
  • “Math” Way of Naming Numbers 11 = ten 1 20 = 2-ten 12 = ten 2 13 = ten 3 14 = ten 4 .... 19 = ten 9Don’t say “2-tens.” We don’t say 3 hundreds eleven for 311. © Joan A. Cotter, Ph.D., 2012189
  • “Math” Way of Naming Numbers 11 = ten 1 20 = 2-ten 12 = ten 2 21 = 2-ten 1 13 = ten 3 14 = ten 4 .... 19 = ten 9Don’t say “2-tens.” We don’t say 3 hundreds eleven for 311. © Joan A. Cotter, Ph.D., 2012190
  • “Math” Way of Naming Numbers 11 = ten 1 20 = 2-ten 12 = ten 2 21 = 2-ten 1 13 = ten 3 22 = 2-ten 2 14 = ten 4 .... 19 = ten 9Don’t say “2-tens.” We don’t say 3 hundreds eleven for 311. © Joan A. Cotter, Ph.D., 2012191
  • “Math” Way of Naming Numbers 11 = ten 1 20 = 2-ten 12 = ten 2 21 = 2-ten 1 13 = ten 3 22 = 2-ten 2 14 = ten 4 23 = 2-ten 3 .... 19 = ten 9Don’t say “2-tens.” We don’t say 3 hundreds eleven for 311. © Joan A. Cotter, Ph.D., 2012192
  • “Math” Way of Naming Numbers 11 = ten 1 20 = 2-ten 12 = ten 2 21 = 2-ten 1 13 = ten 3 22 = 2-ten 2 14 = ten 4 23 = 2-ten 3 .... .... 19 = ten 9 .... 99 = 9-ten 9193 © Joan A. Cotter, Ph.D., 2012
  • “Math” Way of Naming Numbers 137 = 1 hundred 3-ten 7Only numbers under 100 need to be said the “math” way. © Joan A. Cotter, Ph.D., 2012194
  • “Math” Way of Naming Numbers 137 = 1 hundred 3-ten 7 or 137 = 1 hundred and 3-ten 7Only numbers under 100 need to be said the “math” way. © Joan A. Cotter, Ph.D., 2012195
  • “Math” Way of Naming Numbers 100 Chinese Average Highest Number Counted U.S. 90 Korean formal [math way] Korean informal [not explicit] 80 70 60 50 40 30 20 10 0 4 5 6 Ages (yrs.) Song, M., & Ginsburg, H. (1988). p. 326. The effect of the Korean number system on young childrens counting: A natural experiment in numerical bilingualism. International Journal of Psychology, 23, 319-332.Shows how far children from 3 countries can count at ages 4, 5, and 6. © Joan A. Cotter, Ph.D., 2012196
  • “Math” Way of Naming Numbers 100 Chinese Average Highest Number Counted U.S. 90 Korean formal [math way] Korean informal [not explicit] 80 70 60 50 40 30 20 10 0 4 5 6 Ages (yrs.) Song, M., & Ginsburg, H. (1988). p. 326. The effect of the Korean number system on young childrens counting: A natural experiment in numerical bilingualism. International Journal of Psychology, 23, 319-332.Purple is Chinese. Note jump between ages 5 and 6. © Joan A. Cotter, Ph.D., 2012197
  • “Math” Way of Naming Numbers 100 Chinese Average Highest Number Counted U.S. 90 Korean formal [math way] Korean informal [not explicit] 80 70 60 50 40 30 20 10 0 4 5 6 Ages (yrs.) Song, M., & Ginsburg, H. (1988). p. 326. The effect of the Korean number system on young childrens counting: A natural experiment in numerical bilingualism. International Journal of Psychology, 23, 319-332.Dark green is Korean “math” way. © Joan A. Cotter, Ph.D., 2012198
  • “Math” Way of Naming Numbers 100 Chinese Average Highest Number Counted U.S. 90 Korean formal [math way] Korean informal [not explicit] 80 70 60 50 40 30 20 10 0 4 5 6 Ages (yrs.) Song, M., & Ginsburg, H. (1988). p. 326. The effect of the Korean number system on young childrens counting: A natural experiment in numerical bilingualism. International Journal of Psychology, 23, 319-332.Dotted green is everyday Korean; notice smaller jump between ages 5 and Joan A. Cotter, Ph.D., 2012 © 6.199
  • “Math” Way of Naming Numbers 100 Chinese Average Highest Number Counted U.S. 90 Korean formal [math way] Korean informal [not explicit] 80 70 60 50 40 30 20 10 0 4 5 6 Ages (yrs.) Song, M., & Ginsburg, H. (1988). p. 326. The effect of the Korean number system on young childrens counting: A natural experiment in numerical bilingualism. International Journal of Psychology, 23, 319-332.Red is English speakers. They learn same amount between ages 4-5 and©5-6. Cotter, Ph.D., 2012 Joan A.200
  • 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.)201 © 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.202 © 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.203 © 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.204 © Joan A. Cotter, Ph.D., 2012
  • Math Way of Naming Numbers Compared to reading:205 © Joan A. Cotter, Ph.D., 2012
  • Math Way of Naming Numbers Compared to reading: • Just as reciting the alphabet doesn’t teach reading, counting doesn’t teach arithmetic.206 © Joan A. Cotter, Ph.D., 2012
  • Math Way of Naming Numbers Compared to reading: • 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).207 © Joan A. Cotter, Ph.D., 2012
  • Math Way of Naming Numbers Compared to reading: • 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.208 © 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 Researchers209 © Joan A. Cotter, Ph.D., 2012
  • Math Way of Naming Numbers Research task: Using 10s and 1s, ask the child to construct 48.210 © Joan A. Cotter, Ph.D., 2012
  • Math Way of Naming Numbers Research task: Using 10s and 1s, ask the child to construct 48. Then ask the child to subtract 14.211 © Joan A. Cotter, Ph.D., 2012
  • Math Way of Naming Numbers Research task: Using 10s and 1s, ask the child to construct 48. Then ask the child to subtract 14. Children thinking of 14 as 14 ones count 14.212 © Joan A. Cotter, Ph.D., 2012
  • Math Way of Naming Numbers Research task: Using 10s and 1s, ask the child to construct 48. Then ask the child to subtract 14. Children thinking of 14 as 14 ones counted 14.213 © Joan A. Cotter, Ph.D., 2012
  • Math Way of Naming Numbers Research task: Using 10s and 1s, ask the child to construct 48. Then ask the child to subtract 14. Children thinking of 14 as 14 ones counted 14.214 © Joan A. Cotter, Ph.D., 2012
  • Math Way of Naming Numbers Research task: Using 10s and 1s, ask the child to construct 48. Then ask the child to subtract 14. Children thinking of 14 as 14 ones counted 14.215 © Joan A. Cotter, Ph.D., 2012
  • Math Way of Naming Numbers Research task: Using 10s and 1s, ask the child to construct 48. Then ask the child to subtract 14. Children thinking of 14 as 14 ones counted 14.216 © Joan A. Cotter, Ph.D., 2012
  • Math Way of Naming Numbers Research task: Using 10s and 1s, ask the child to construct 48. Then ask the child to subtract 14. Children thinking of 14 as 14 ones counted 14.217 © Joan A. Cotter, Ph.D., 2012
  • Math Way of Naming Numbers Research task: Using 10s and 1s, ask the child to construct 48. Then ask the child to subtract 14. Children thinking of 14 as 14 ones counted 14.218 © Joan A. Cotter, Ph.D., 2012
  • Math Way of Naming Numbers Research task: Using 10s and 1s, ask the child to construct 48. Then ask the child to subtract 14. Children thinking of 14 as 14 ones counted 14.219 © Joan A. Cotter, Ph.D., 2012
  • Math Way of Naming Numbers Research task: Using 10s and 1s, ask the child to construct 48. Then ask the child to subtract 14. Children who understand tens remove a ten and 4 ones.220 © Joan A. Cotter, Ph.D., 2012
  • Math Way of Naming Numbers Research task: Using 10s and 1s, ask the child to construct 48. Then ask the child to subtract 14. Children who understand tens remove a ten and 4 ones.221 © Joan A. Cotter, Ph.D., 2012
  • Math Way of Naming Numbers Research task: Using 10s and 1s, ask the child to construct 48. Then ask the child to subtract 14. Children who understand tens remove a ten and 4 ones.222 © Joan A. Cotter, Ph.D., 2012
  • Math Way of Naming Numbers Traditional names4-ten =fortyThe “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 names6-ten = sixtyThe “ty”means tens. © Joan A. Cotter, Ph.D., 2012
  • Math Way of Naming Numbers Traditional names3-ten = thirty“Thir” alsoused in 1/3,13 and 30. © Joan A. Cotter, Ph.D., 2012
  • Math Way of Naming Numbers Traditional names5-ten = fifty“Fif” alsoused in 1/5,15 and 50. © Joan A. Cotter, Ph.D., 2012
  • Math Way of Naming Numbers Traditional names2-ten = twentyTwo used to bepronounced“twoo.” © Joan A. Cotter, Ph.D., 2012
  • Math Way of Naming Numbers Traditional names A word game fireplace place-fireSay 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 newspaper paper-newsSay 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 newspaper paper-news box-mail mailboxSay 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” alsomeans ten. © Joan A. Cotter, Ph.D., 2012
  • Math Way of Naming Numbers Traditional names ten 4 teen 4“Teen” alsomeans ten. © Joan A. Cotter, Ph.D., 2012
  • Math Way of Naming Numbers Traditional names ten 4 teen 4 fourtee n“Teen” alsomeans 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 leftTwopronounced“twoo.” © Joan A. Cotter, Ph.D., 2012
  • Math Way of Naming Numbers Traditional names two left twelveTwopronounced“twoo.” © Joan A. Cotter, Ph.D., 2012
  • Composing Numbers3-ten © Joan A. Cotter, Ph.D., 2012
  • Composing Numbers3-ten © Joan A. Cotter, Ph.D., 2012
  • Composing Numbers3-ten30 © Joan A. Cotter, Ph.D., 2012
  • Composing Numbers 3-ten 30Point to the 3 and say 3. © Joan A. Cotter, Ph.D., 2012
  • Composing Numbers 3-ten 30Point to 0 and say 10. The 0 makes 3 a ten. © Joan A. Cotter, Ph.D., 2012
  • Composing Numbers3-ten 730 © Joan A. Cotter, Ph.D., 2012
  • Composing Numbers3-ten 730 © Joan A. Cotter, Ph.D., 2012
  • Composing Numbers3-ten 730 7 © Joan A. Cotter, Ph.D., 2012
  • Composing Numbers 3-ten 7 30 7Place the 7 on top of the 0 of the 30. © Joan A. Cotter, Ph.D., 2012
  • Composing Numbers 3-ten 7 30 7Notice the way we say the number, represent thenumber, and write the number all correspond. © Joan A. Cotter, Ph.D., 2012
  • Composing Numbers 7-ten 6 78 6Another example. © Joan A. Cotter, Ph.D., 2012
  • Composing Numbers 7-ten 6 78 6 In the UK, pupils are expected to know the amount remaining: 24, that is 100 – 76.Another example. © Joan A. Cotter, Ph.D., 2012
  • Composing Numbers10-ten © Joan A. Cotter, Ph.D., 2012
  • Composing Numbers10-ten100 © Joan A. Cotter, Ph.D., 2012
  • Composing Numbers10-ten100 © Joan A. Cotter, Ph.D., 2012
  • Composing Numbers10-ten100 © Joan A. Cotter, Ph.D., 2012
  • Composing Numbers1 hundred © Joan A. Cotter, Ph.D., 2012
  • Composing Numbers1 hundred100 © Joan A. Cotter, Ph.D., 2012
  • Composing Numbers 1 hundred 100Of course, we can also read it as one-hun-dred. © Joan A. Cotter, Ph.D., 2012
  • Composing Numbers 1 hundred 100Of course, we can also read it as one-hun-dred. © Joan A. Cotter, Ph.D., 2012
  • Composing Numbers 1 hundred 100Of course, we can also read it as one-hun-dred. © Joan A. Cotter, Ph.D., 2012
  • Composing Numbers Reading numbers backwardTo read a number, students are ofteninstructed to start at the right (onescolumn), contrary to normal readingof numbers and text: 4258 © Joan A. Cotter, Ph.D., 2012
  • Composing Numbers Reading numbers backward To read a number, students are often instructed to start at the right (ones column), contrary to normal reading of numbers and text: 4258262 © Joan A. Cotter, Ph.D., 2012
  • Composing Numbers Reading numbers backward To read a number, students are often instructed to start at the right (ones column), contrary to normal reading of numbers and text: 4258263 © Joan A. Cotter, Ph.D., 2012
  • Composing Numbers Reading numbers backward To read a number, students are often instructed to start at the right (ones column), contrary to normal reading of numbers and text: 4258264 © Joan A. Cotter, Ph.D., 2012
  • Composing Numbers Reading numbers backward To read a number, students are often instructed to start at the right (ones column), contrary to normal reading of numbers and text: 4258 The Decimal Cards encourage reading numbers in the normal order.265 © Joan A. Cotter, Ph.D., 2012
  • Composing Numbers Scientific Notation 3 4000 = 4 x 10 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.)266 © Joan A. Cotter, Ph.D., 2012
  • Fact Strategies267 © Joan A. Cotter, Ph.D., 2012
  • Fact Strategies • A strategy is a way to learn a new fact or recall a forgotten fact.268 © 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.269 © Joan A. Cotter, Ph.D., 2012
  • Fact Strategies Complete the Ten9+5= © Joan A. Cotter, Ph.D., 2012
  • Fact Strategies Complete the Ten9+5= © Joan A. Cotter, Ph.D., 2012
  • Fact Strategies Complete the Ten9+5= © Joan A. Cotter, Ph.D., 2012
  • Fact Strategies Complete the Ten 9+5=Take 1 fromthe 5 and giveit 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 fromthe 5 and giveit to the 9. © Joan A. Cotter, Ph.D., 2012
  • Fact Strategies Complete the Ten 9 + 5 = 14Take 1 fromthe 5 and giveit to the 9. © Joan A. Cotter, Ph.D., 2012
  • Fact Strategies Two Fives8+6= © Joan A. Cotter, Ph.D., 2012
  • Fact Strategies Two Fives8+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 = 14Just 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 Fives7+5= © Joan A. Cotter, Ph.D., 2012
  • Fact Strategies Two Fives7 + 5 = 12 © Joan A. Cotter, Ph.D., 2012
  • Fact Strategies Going Down15 – 9 = © Joan A. Cotter, Ph.D., 2012
  • Fact Strategies Difference 7–4=Subtract 4 from5; then add 2. © Joan A. Cotter, Ph.D., 2012
  • Fact Strategies Going Down15 – 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 = 6Subtract 5;then 4. © Joan A. Cotter, Ph.D., 2012
  • Fact Strategies Subtract from 1015 – 9 = © Joan A. Cotter, Ph.D., 2012
  • Fact Strategies Subtract from 10 15 – 9 =Subtract 9from 10. © Joan A. Cotter, Ph.D., 2012
  • Fact Strategies Subtract from 10 15 – 9 =Subtract 9from 10. © Joan A. Cotter, Ph.D., 2012
  • Fact Strategies Subtract from 10 15 – 9 =Subtract 9from 10. © Joan A. Cotter, Ph.D., 2012
  • Fact Strategies Subtract from 10 15 – 9 = 6Subtract 9from 10. © Joan A. Cotter, Ph.D., 2012
  • Fact Strategies Going Up13 – 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=4Start with 9;go up to 13. © Joan A. Cotter, Ph.D., 2012
  • MoneyPenny © Joan A. Cotter, Ph.D., 2012
  • MoneyNickel © Joan A. Cotter, Ph.D., 2012
  • Money Dime © Joan A. Cotter, Ph.D., 2012
  • MoneyQuarter © Joan A. Cotter, Ph.D., 2012
  • MoneyQuarter © Joan A. Cotter, Ph.D., 2012
  • MoneyQuarter © Joan A. Cotter, Ph.D., 2012
  • MoneyQuarter © Joan A. Cotter, Ph.D., 2012
  • Base-10 Picture Cards One310 © Joan A. Cotter, Ph.D., 2012
  • Base-10 Picture Cards Ten One311 © Joan A. Cotter, Ph.D., 2012
  • Base-10 Picture Cards Hundred Ten One312 © Joan A. Cotter, Ph.D., 2012
  • Base-10 Picture Cards Thousand Hundred Ten One313 © Joan A. Cotter, Ph.D., 2012
  • Base-10 Picture Cards Add using the base-10 picture cards. 3658 +2724314 © Joan A. Cotter, Ph.D., 2012
  • Base-10 Picture Cards 6 5 8 3 0 0 0315 © Joan A. Cotter, Ph.D., 2012
  • Base-10 Picture Cards 6 5 8 3 0 0 0316 © Joan A. Cotter, Ph.D., 2012
  • Base-10 Picture Cards 6 5 8 3 0 0 0317 © Joan A. Cotter, Ph.D., 2012
  • Base-10 Picture Cards 6 5 8 3 0 0 0318 © Joan A. Cotter, Ph.D., 2012
  • Base-10 Picture Cards 7 2 4 2 0 0 0319 © Joan A. Cotter, Ph.D., 2012
  • Base-10 Picture Cards 7 2 4 2 0 0 0320 © Joan A. Cotter, Ph.D., 2012
  • Base-10 Picture Cards 6 5 8 3 0 0 0 7 2 4 2 0 0 0 Add them together.321 © Joan A. Cotter, Ph.D., 2012
  • Base-10 Picture Cards 6 5 8 3 0 0 0 7 2 4 2 0 0 0322 © Joan A. Cotter, Ph.D., 2012
  • Base-10 Picture Cards 6 5 8 3 0 0 0 7 2 4 2 0 0 0 Trade 10 ones for 1 ten.323 © Joan A. Cotter, Ph.D., 2012
  • Base-10 Picture Cards 6 5 8 3 0 0 0 7 2 4 2 0 0 0 Trade 10 ones for 1 ten.324 © Joan A. Cotter, Ph.D., 2012
  • Base-10 Picture Cards 6 5 8 3 0 0 0 7 2 4 2 0 0 0 Trade 10 ones for 1 ten.325 © Joan A. Cotter, Ph.D., 2012
  • Base-10 Picture Cards 6 5 8 3 0 0 0 7 2 4 2 0 0 0326 © Joan A. Cotter, Ph.D., 2012
  • Base-10 Picture Cards 6 5 8 3 0 0 0 7 2 4 2 0 0 0 Trade 10 hundreds for 1 thousand.327 © Joan A. Cotter, Ph.D., 2012
  • Base-10 Picture Cards 6 5 8 3 0 0 0 7 2 4 2 0 0 0 Trade 10 hundreds for 1 thousand.328 © Joan A. Cotter, Ph.D., 2012
  • Base-10 Picture Cards 6 5 8 3 0 0 0 7 2 4 2 0 0 0 Trade 10 hundreds for 1 thousand.329 © Joan A. Cotter, Ph.D., 2012
  • Base-10 Picture Cards 6 5 8 3 0 0 0 7 2 4 2 0 0 0 Trade 10 hundreds for 1 thousand.330 © Joan A. Cotter, Ph.D., 2012
  • Base-10 Picture Cards 6 5 8 3 0 0 0 7 2 4 2 0 0 0 3 8 2 6 0 0 0331 © Joan A. Cotter, Ph.D., 2012
  • Bead Frame 1 10 100 1000332 © Joan A. Cotter, Ph.D., 2012
  • Bead Frame 1 8 10 +6 100 1000333 © Joan A. Cotter, Ph.D., 2012
  • Bead Frame 1 8 10 +6 100 1000334 © Joan A. Cotter, Ph.D., 2012
  • Bead Frame 1 8 10 +6 100 1000335 © Joan A. Cotter, Ph.D., 2012
  • Bead Frame 1 8 10 +6 100 1000336 © Joan A. Cotter, Ph.D., 2012
  • Bead Frame 1 8 10 +6 100 1000337 © Joan A. Cotter, Ph.D., 2012
  • Bead Frame 1 8 10 +6 100 1000338 © Joan A. Cotter, Ph.D., 2012
  • Bead Frame 1 8 10 +6 100 1000339 © Joan A. Cotter, Ph.D., 2012
  • Bead Frame 1 8 10 +6 100 1000340 © Joan A. Cotter, Ph.D., 2012
  • Bead Frame 1 8 10 +6 100 1000341 © Joan A. Cotter, Ph.D., 2012
  • Bead Frame 1 8 10 +6 100 14 1000342 © Joan A. Cotter, Ph.D., 2012
  • Bead Frame Difficulties for the child343 © Joan A. Cotter, Ph.D., 2012
  • Bead Frame Difficulties for the child • Distracting: Room is visible through the frame.344 © Joan A. Cotter, Ph.D., 2012
  • Bead Frame Difficulties for the child • Distracting: Room is visible through the frame. • Not visualizable: Beads need to be grouped in fives.345 © Joan A. Cotter, Ph.D., 2012
  • Bead Frame Difficulties for the child • Distracting: Room is visible through the frame. • Not visualizable: Beads need to be grouped in fives. • Inconsistent with equation order when beads are moved right: Beads need to be moved left.346 © Joan A. Cotter, Ph.D., 2012
  • Bead Frame Difficulties for the child • Distracting: Room is visible through the frame. • Not visualizable: Beads need to be grouped in fives. • Inconsistent with equation order when beads are moved right: Beads need to be moved left. • Hierarchies of numbers represented sideways: They need to be in vertical columns.347 © Joan A. Cotter, Ph.D., 2012
  • Bead Frame Difficulties for the child • Distracting: Room is visible through the frame. • Not visualizable: Beads need to be grouped in fives. • Inconsistent with equation order when beads are moved right: 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 combined before trading.348 © Joan A. Cotter, Ph.D., 2012
  • Bead Frame Difficulties for the child • Distracting: Room is visible through the frame. • Not visualizable: Beads need to be grouped in fives. • Inconsistent with equation order when beads are moved right: 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 combined before trading. • Answer is read going up: We read top to bottom.349 © Joan A. Cotter, Ph.D., 2012
  • Trading Side Cleared1000 100 10 1 © Joan A. Cotter, Ph.D., 2012
  • Trading Side Thousands1000 100 10 1 © Joan A. Cotter, Ph.D., 2012
  • Trading Side Hundreds 1000 100 10 1The third wire from each end is not used. © Joan A. Cotter, Ph.D., 2012
  • Trading Side Tens 1000 100 10 1The third wire from each end is not used. © Joan A. Cotter, Ph.D., 2012
  • Trading Side Ones 1000 100 10 1The third wire from each end is not used. © Joan A. Cotter, Ph.D., 2012
  • Trading Side Adding1000 100 10 1 8 +6 © Joan A. Cotter, Ph.D., 2012
  • Trading Side Adding1000 100 10 1 8 +6 © Joan A. Cotter, Ph.D., 2012
  • Trading Side Adding1000 100 10 1 8 +6 © Joan A. Cotter, Ph.D., 2012
  • Trading Side Adding1000 100 10 1 8 +6 © Joan A. Cotter, Ph.D., 2012
  • Trading Side Adding1000 100 10 1 8 +6 14 © Joan A. Cotter, Ph.D., 2012
  • Trading Side Adding 1000 100 10 1 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
  • Trading Side Adding1000 100 10 1 8 +6 14 Too many ones; trade 10 ones for 1 ten. © Joan A. Cotter, Ph.D., 2012
  • Trading Side Adding1000 100 10 1 8 +6 14 Too many ones; trade 10 ones for 1 ten. © Joan A. Cotter, Ph.D., 2012
  • Trading Side Adding1000 100 10 1 8 +6 14 Same answer before and after trading. © Joan A. Cotter, Ph.D., 2012
  • Trading Side Cleared1000 100 10 1 © Joan A. Cotter, Ph.D., 2012
  • Trading Side Bead Trading game1000 100 10 1 Object: To get a high score by adding numbers on the green cards. © Joan A. Cotter, Ph.D., 2012
  • Trading Side Bead Trading game1000 100 10 1 7 Object: To get a high score by adding numbers on the green cards. © Joan A. Cotter, Ph.D., 2012
  • Trading Side Bead Trading game1000 100 10 1 7 Object: To get a high score by adding numbers on the green cards. © Joan A. Cotter, Ph.D., 2012
  • Trading Side Bead Trading game 1000 100 10 1 6Turn over another card. Enter 6 beads. Do we need to trade? © Joan A. Cotter, Ph.D., 2012
  • Trading Side Bead Trading game 1000 100 10 1 6Turn over another card. Enter 6 beads. Do we need to trade? © Joan A. Cotter, Ph.D., 2012
  • Trading Side Bead Trading game 1000 100 10 1 6Turn over another card. Enter 6 beads. Do we need to trade? © Joan A. Cotter, Ph.D., 2012
  • Trading Side Bead Trading game1000 100 10 1 6 Trade 10 ones for 1 ten. © Joan A. Cotter, Ph.D., 2012
  • Trading Side Bead Trading game1000 100 10 1 6 © Joan A. Cotter, Ph.D., 2012
  • Trading Side Bead Trading game1000 100 10 1 6 © Joan A. Cotter, Ph.D., 2012
  • Trading Side Bead Trading game1000 100 10 1 9 © Joan A. Cotter, Ph.D., 2012
  • Trading Side Bead Trading game1000 100 10 1 9 © Joan A. Cotter, Ph.D., 2012
  • Trading Side Bead Trading game1000 100 10 1 9 Another trade. © Joan A. Cotter, Ph.D., 2012
  • Trading Side Bead Trading game1000 100 10 1 9 Another trade. © Joan A. Cotter, Ph.D., 2012
  • Trading Side Bead Trading game1000 100 10 1 3 © Joan A. Cotter, Ph.D., 2012
  • Trading Side Bead Trading game1000 100 10 1 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 thegreater 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 thegreater value of each column from left to right.• To detect a pattern, there must be at least threeexamples in the sequence. (Place value is a pattern.) © Joan A. Cotter, Ph.D., 2012
  • Trading Side Adding 4-digit numbers1000 100 10 1 3658 + 2738 © Joan A. Cotter, Ph.D., 2012
  • Trading Side Adding 4-digit numbers1000 100 10 1 3658 + 2738 Enter the first number from left to right. © Joan A. Cotter, Ph.D., 2012
  • Trading Side Adding 4-digit numbers1000 100 10 1 3658 + 2738 Enter the first number from left to right. © Joan A. Cotter, Ph.D., 2012
  • Trading Side Adding 4-digit numbers1000 100 10 1 3658 + 2738 Enter the first number from left to right. © Joan A. Cotter, Ph.D., 2012
  • Trading Side Adding 4-digit numbers1000 100 10 1 3658 + 2738 Enter the first number from left to right. © Joan A. Cotter, Ph.D., 2012
  • Trading Side Adding 4-digit numbers1000 100 10 1 3658 + 2738 Enter the first number from left to right. © Joan A. Cotter, Ph.D., 2012
  • Trading Side Adding 4-digit numbers1000 100 10 1 3658 + 2738 Enter the first number from left to right. © Joan A. Cotter, Ph.D., 2012
  • Trading Side Adding 4-digit numbers1000 100 10 1 3658 + 2738 Add starting at the right. Write results after each step. © Joan A. Cotter, Ph.D., 2012
  • Trading Side Adding 4-digit numbers1000 100 10 1 3658 + 2738 Add starting at the right. Write results after each step. © Joan A. Cotter, Ph.D., 2012
  • Trading Side Adding 4-digit numbers1000 100 10 1 3658 + 2738 Add starting at the right. Write results after each step. © Joan A. Cotter, Ph.D., 2012
  • Trading Side Adding 4-digit numbers1000 100 10 1 3658 + 2738 Add starting at the right. Write results after each step. © Joan A. Cotter, Ph.D., 2012
  • Trading Side Adding 4-digit numbers 1000 100 10 1 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
  • Trading Side Adding 4-digit numbers 1000 100 10 1 1 3658 + 2738 6 Add starting at the right. Write results after each step.Is it okay to show the extra ten by writing a 1 above the tens column? © Joan A. Cotter, Ph.D., 2012
  • Trading Side Adding 4-digit numbers1000 100 10 1 1 3658 + 2738 6 Add starting at the right. Write results after each step. © Joan A. Cotter, Ph.D., 2012
  • Trading Side Adding 4-digit numbers 1000 100 10 1 1 3658 + 2738 6 Add starting at the right. Write results after each step.Do we need to trade? [no] © Joan A. Cotter, Ph.D., 2012
  • Trading Side Adding 4-digit numbers1000 100 10 1 1 3658 + 2738 96 Add starting at the right. Write results after each step. © Joan A. Cotter, Ph.D., 2012
  • Trading Side Adding 4-digit numbers1000 100 10 1 1 3658 + 2738 96 Add starting at the right. Write results after each step. © Joan A. Cotter, Ph.D., 2012
  • Trading Side Adding 4-digit numbers 1000 100 10 1 1 3658 + 2738 96 Add starting at the right. Write results after each step.Do we need to trade? [yes] © Joan A. Cotter, Ph.D., 2012
  • Trading Side Adding 4-digit numbers 1000 100 10 1 1 3658 + 2738 96 Add starting at the right. Write results after each step.Notice the number of yellow beads. [3] Notice the number of bluebeads left. [3] Coincidence? No, because 13 – 10 = 3. © Joan A. Cotter, Ph.D., 2012
  • Trading Side Adding 4-digit numbers1000 100 10 1 1 3658 + 2738 96 Add starting at the right. Write results after each step. © Joan A. Cotter, Ph.D., 2012
  • Trading Side Adding 4-digit numbers1000 100 10 1 1 3658 + 2738 396 Add starting at the right. Write results after each step. © Joan A. Cotter, Ph.D., 2012
  • Trading Side Adding 4-digit numbers1000 100 10 1 1 1 3658 + 2738 396 Add starting at the right. Write results after each step. © Joan A. Cotter, Ph.D., 2012
  • Trading Side Adding 4-digit numbers1000 100 10 1 1 1 3658 + 2738 396 Add starting at the right. Write results after each step. © Joan A. Cotter, Ph.D., 2012
  • Trading Side Adding 4-digit numbers1000 100 10 1 1 1 3658 + 2738 396 Add starting at the right. Write results after each step. © Joan A. Cotter, Ph.D., 2012
  • Trading Side Adding 4-digit numbers1000 100 10 1 1 1 3658 + 2738 6396 Add starting at the right. Write results after each step. © Joan A. Cotter, Ph.D., 2012
  • Trading Side Adding 4-digit numbers1000 100 10 1 1 1 3658 + 2738 6396 Add starting at the right. Write results after each step. © Joan A. Cotter, Ph.D., 2012
  • The Stamp Game1000 1000 100 100 10 10 1 11000 1000 100 100 10 10 1 1 100 100 10 10 1 1 100 100 10 10 1 1 100 100 10 10 1 1 100 100 10 100 100 100 100 © 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
  • Multiplication on the AL Abacus For facts > 5 × 5 7×8= This method was used in the Middle Ages, rather than memorize the facts > 5 × 5.432 © Joan A. Cotter, Ph.D., 2012
  • Multiplication on the AL Abacus For facts > 5 × 5 7×8= This method was used in the Middle Ages, rather than memorize the facts > 5 × 5.433 © Joan A. Cotter, Ph.D., 2012
  • Multiplication on the AL Abacus For facts > 5 × 5 7×8= Tens: This method was used in the Middle Ages, rather than memorize the facts > 5 × 5.434 © Joan A. Cotter, Ph.D., 2012
  • Multiplication on the AL Abacus For facts > 5 × 5 7×8= Tens: This method was used in the Middle Ages, rather than memorize the facts > 5 × 5.435 © Joan A. Cotter, Ph.D., 2012
  • Multiplication on the AL Abacus For facts > 5 × 5 7×8= 50 + 6 Tens: 20 Ones: 3 + 30 ×2 50 6 This method was used in the Middle Ages, rather than memorize the facts > 5 × 5.436 © Joan A. Cotter, Ph.D., 2012
  • Multiplication on the AL Abacus For facts > 5 × 5 7×8= 50 + 6 = 56 Tens: 20 Ones: 3 + 30 ×2 50 6 This method was used in the Middle Ages, rather than memorize the facts > 5 × 5.437 © Joan A. Cotter, Ph.D., 2012
  • Multiplication on the AL Abacus For facts > 5 × 5 9×7=438 © Joan A. Cotter, Ph.D., 2012
  • Multiplication on the AL Abacus For facts > 5 × 5 9×7=439 © Joan A. Cotter, Ph.D., 2012
  • Multiplication on the AL Abacus For facts > 5 × 5 9×7= Tens:440 © Joan A. Cotter, Ph.D., 2012
  • Multiplication on the AL Abacus For facts > 5 × 5 9×7= Tens:441 © Joan A. Cotter, Ph.D., 2012
  • Multiplication on the AL Abacus For facts > 5 × 5 9×7= Tens: 40 + 20442 © Joan A. Cotter, Ph.D., 2012
  • Multiplication on the AL Abacus For facts > 5 × 5 9×7= 60 + Tens: 40 + 20 60443 © Joan A. Cotter, Ph.D., 2012
  • Multiplication on the AL Abacus For facts > 5 × 5 9×7= 60 + Tens: 40 Ones: + 20 60444 © Joan A. Cotter, Ph.D., 2012
  • Multiplication on the AL Abacus For facts > 5 × 5 9×7= 60 + Tens: 40 Ones: + 20 60445 © Joan A. Cotter, Ph.D., 2012
  • Multiplication on the AL Abacus For facts > 5 × 5 9×7= 60 + Tens: 40 Ones: 1 + 20 ×3 60446 © Joan A. Cotter, Ph.D., 2012
  • Multiplication on the AL Abacus For facts > 5 × 5 9×7= 60 + 3 Tens: 40 Ones: 1 + 20 ×3 60 3447 © Joan A. Cotter, Ph.D., 2012
  • Multiplication on the AL Abacus For facts > 5 × 5 9×7= 60 + 3 = 63 Tens: 40 Ones: 1 + 20 ×3 60 3448 © Joan A. Cotter, Ph.D., 2012
  • The Multiplication Board 1 2 3 4 5 6 7 8 9 10 6×4 66 x 4 on original multiplication board. © Joan A. Cotter, Ph.D., 2012449
  • The Multiplication Board 1 2 3 4 5 6 7 8 9 10 6×4 6Using two colors. © Joan A. Cotter, Ph.D., 2012450
  • The Multiplication Board 1 2 3 4 5 6 7 8 9 10 7×7 77 x 7 on original multiplication board. © Joan A. Cotter, Ph.D., 2012451
  • The Multiplication Board 1 2 3 4 5 6 7 8 9 10 7×7 7Upper left square is 25, yellow rectangles are 10. So, 25, 35, 45, 49. © Joan A. Cotter, Ph.D., 2012452
  • The Multiplication Board 7×7Less clutter. © Joan A. Cotter, Ph.D., 2012453
  • Multiples Patterns Twos 2 4 6 8 10 12 14 16 18 20Recognizing multiples needed for fractions and algebra. © Joan A. Cotter, Ph.D., 2012454
  • 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., 2012455
  • Multiples Patterns Fours 4 8 12 16 20 24 28 32 36 40 The ones repeat in the second row.456 © 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 80457 © 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 80458 © 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 80Again the ones repeat in the second row. © Joan A. Cotter, Ph.D., 2012459
  • 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 2.460 © 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 2.461 © 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 2.462 © 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 2.463 © 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 2.464 © Joan A. Cotter, Ph.D., 2012
  • Multiples Patterns Sixes and Eights 6 12 18 24 30 6× 4 36 42 48 54 60 8 16 24 32 40 48 56 64 72 80 6 × 4 is the fourth number (multiple).465 © 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).466 © 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 9.467 © Joan A. Cotter, Ph.D., 2012
  • Multiples Patterns Threes 3 6 9 2 15 18 21 24 27 30 The 3s have several patterns: Observe the ones.468 © Joan A. Cotter, Ph.D., 2012
  • Multiples Patterns Threes 3 6 9 2 15 18 21 24 27 30 The 3s have several patterns: Observe the ones.469 © Joan A. Cotter, Ph.D., 2012
  • Multiples Patterns Threes 3 6 9 2 15 18 21 24 27 30 The 3s have several patterns: Observe the ones.470 © 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.471 © 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.472 © 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.473 © 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.474 © 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.475 © 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.476 © 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.477 © 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.478 © 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.479 © 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.480 © 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.481 © 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.482 © 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.”483 © 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.”484 © 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.”485 © 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.”486 © 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.487 © 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.488 © 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.489 © 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.490 © Joan A. Cotter, Ph.D., 2012
  • Multiples Patterns Sevens 7 14 21 28 35 42 49 56 63 70 Look at the tens.491 © Joan A. Cotter, Ph.D., 2012
  • Multiples Patterns Sevens 7 14 21 28 35 42 49 56 63 70 Look at the tens.492 © Joan A. Cotter, Ph.D., 2012
  • Multiples Patterns Sevens 7 14 21 28 35 42 49 56 63 70 Look at the tens.493 © 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 MemoryAim: To help the players learn themultiples patterns.Object of the game: To be the first player to collect all tencards of a multiple in order. © Joan A. Cotter, Ph.D., 2012
  • Multiples Memory 7 14 21 28 35 42 49 56 63 70The 7s envelope contains 10 cards,each with one of the numbers listed. © Joan A. Cotter, Ph.D., 2012
  • Multiples Memory 8 16 24 32 40 48 56 64 72 80The 8s envelope contains 10 cards,each with one of the numbers listed. © Joan A. Cotter, Ph.D., 2012
  • Multiples Memory 7 14 21 28 35 42 8 16 24 32 40 49 56 63 48 56 64 72 80 70 7 14 21 8 16 24 32 40 28 35 42 48 56 64 72 80 49 56 63 70Players may refer to their envelopes at all times. © Joan A. Cotter, Ph.D., 2012
  • Multiples Memory 7 14 21 28 35 42 8 16 24 32 40 49 56 63 48 56 64 72 80 70 7 14 21 8 16 24 32 40 28 35 42 48 56 64 72 80 49 56 63 70Players may refer to their envelopes at all times. © Joan A. Cotter, Ph.D., 2012
  • Multiples Memory 7 14 21 28 35 42 49 56 63 70 7 14 21 8 16 24 32 40 28 35 42 48 56 64 72 80 49 56 63 70Players may refer to their envelopes at all times. © Joan A. Cotter, Ph.D., 2012
  • Multiples Memory 7 14 21 28 35 42 49 56 63 70 7 14 21 8 16 24 32 40 28 35 42 48 56 64 72 80 49 56 63 14 70The 7s player is looking for a 7. © Joan A. Cotter, Ph.D., 2012
  • Multiples Memory 7 14 21 28 35 42 49 56 63 70 7 14 21 8 16 24 32 40 28 35 42 48 56 64 72 80 49 56 63 70Wrong card, so it is turned face down in its original space. © Joan A. Cotter, Ph.D., 2012
  • Multiples Memory 8 16 24 32 40 48 56 64 72 80 7 14 21 8 16 24 32 40 28 35 42 48 56 64 72 80 49 56 63 70The 8s player takes a turn. © Joan A. Cotter, Ph.D., 2012
  • Multiples Memory 8 16 24 32 40 48 56 64 72 80 7 14 21 8 16 24 32 40 28 35 42 48 56 64 72 80 49 56 63 70 40Cannot use this card yet. © Joan A. Cotter, Ph.D., 2012
  • Multiples Memory 8 16 24 32 40 48 56 64 72 80 7 14 21 8 16 24 32 40 28 35 42 48 56 64 72 80 49 56 63 70Card returned. © Joan A. Cotter, Ph.D., 2012
  • Multiples Memory 7 14 21 28 35 42 49 56 63 70 7 14 21 8 16 24 32 4028 35 42 48 56 64 72 8049 56 6370 © Joan A. Cotter, Ph.D., 2012
  • Multiples Memory 7 14 21 28 35 42 49 56 63 70 7 14 21 8 16 24 32 4028 35 42 48 56 64 72 8049 56 6370 8 © Joan A. Cotter, Ph.D., 2012
  • Multiples Memory 7 14 21 28 35 42 49 56 63 70 7 14 21 8 16 24 32 4028 35 42 48 56 64 72 8049 56 6370 © Joan A. Cotter, Ph.D., 2012
  • Multiples Memory 8 16 24 32 40 48 56 64 72 80 7 14 21 8 16 24 32 4028 35 42 48 56 64 72 8049 56 6370 © Joan A. Cotter, Ph.D., 2012
  • Multiples Memory 8 16 24 32 40 48 56 64 72 80 7 14 21 8 16 24 32 4028 35 42 48 56 64 72 8049 56 6370 8 © Joan A. Cotter, Ph.D., 2012
  • Multiples Memory 8 16 24 32 40 48 56 64 72 80 7 14 21 8 16 24 32 40 28 35 42 48 56 64 72 80 49 56 63 70 8The needed card is collected. Receives another turn. © Joan A. Cotter, Ph.D., 2012
  • Multiples Memory 8 16 24 32 40 48 56 64 72 80 7 14 21 8 16 24 32 40 28 35 42 48 56 64 72 80 49 56 63 56 70 8Who needs 56? [both 7s and 8s] At least one card per game is a duplicate.Joan A. Cotter, Ph.D., 2012 ©
  • Multiples Memory 8 16 24 32 40 48 56 64 72 80 7 14 21 8 16 24 32 4028 35 42 48 56 64 72 8049 56 6370 8 © Joan A. Cotter, Ph.D., 2012
  • Multiples Memory 7 14 21 28 35 42 49 56 63 70 7 14 21 8 16 24 32 4028 35 42 48 56 64 72 8049 56 6370 8 © Joan A. Cotter, Ph.D., 2012
  • Multiples Memory 7 14 21 28 35 42 49 56 63 70 7 7 14 21 8 16 24 32 40 28 35 42 48 56 64 72 80 49 56 63 70 8The needed card. © Joan A. Cotter, Ph.D., 2012
  • Multiples Memory 7 14 21 28 35 42 49 56 63 70 7 14 21 8 16 24 32 40 28 35 42 48 56 64 72 80 49 56 63 70 8 7Where is that 14? © Joan A. Cotter, Ph.D., 2012
  • Multiples Memory 7 14 21 28 35 42 49 56 63 70 7 14 21 8 16 24 32 40 28 35 42 48 56 64 72 80 49 56 63 14 70 87 © Joan A. Cotter, Ph.D., 2012
  • Multiples Memory 7 14 21 28 35 42 49 56 63 70 7 14 21 8 16 24 32 40 28 35 42 48 56 64 72 80 49 56 63 70 87 14 © Joan A. Cotter, Ph.D., 2012
  • Multiples Memory 7 14 21 28 35 42 49 56 63 70 24 7 14 21 8 16 24 32 40 28 35 42 48 56 64 72 80 49 56 63 70 8 7 14A another turn. © Joan A. Cotter, Ph.D., 2012
  • Multiples Memory 7 14 21 28 35 42 8 16 24 32 40 49 56 63 48 56 64 72 80 70 7 14 21 8 16 24 32 40 28 35 42 48 56 64 72 80 49 56 63 70 8 7 14We’ll never know who won. © Joan A. Cotter, Ph.D., 2012
  • Multiples Memory 7 14 21 28 35 42 8 16 24 32 40 49 56 63 48 56 64 72 80 70 7 14 21 8 16 24 32 40 28 35 42 48 56 64 72 80 49 56 63 70We’ll never know who won. © Joan A. Cotter, Ph.D., 2012
  • Multiplication Tables © Joan A. Cotter, Ph.D., 2012
  • Multiplication Tables © Joan A. Cotter, Ph.D., 2012
  • Multiplication TablesA rectangle 3 × 6 © Joan A. Cotter, Ph.D., 2012
  • Multiplication TablesA rectangle 3 × 6 © Joan A. Cotter, Ph.D., 2012
  • Multiplication TablesA rectangle 3 × 6 © Joan A. Cotter, Ph.D., 2012
  • Multiplication TablesA rectangle 3 × 6 © Joan A. Cotter, Ph.D., 2012
  • Multiplication TablesA rectangle 3 × 6 © Joan A. Cotter, Ph.D., 2012
  • Multiplication Tables 4×7 © Joan A. Cotter, Ph.D., 2012
  • Multiplication Tables 4×7Grouping in fives makes counting over unnecessary. © Joan A. Cotter, Ph.D., 2012
  • Multiplication TablesRemoving duplicates. © Joan A. Cotter, Ph.D., 2012
  • Multiplication Tables 9×3Removing duplicates. © Joan A. Cotter, Ph.D., 2012
  • Multiplication Tables 9×3Removing duplicates. © Joan A. Cotter, Ph.D., 2012
  • Multiplication Tables 6×6 © Joan A. Cotter, Ph.D., 2012
  • Multiplication Tables 6×6 © Joan A. Cotter, Ph.D., 2012
  • Multiplication Tables 4×7 © Joan A. Cotter, Ph.D., 2012
  • Multiplication Tables 4×7 © Joan A. Cotter, Ph.D., 2012
  • Multiplication Tables 4×7 © Joan A. Cotter, Ph.D., 2012
  • Multiplication Tables 7×9 © Joan A. Cotter, Ph.D., 2012
  • Multiplication Tables 7×9 © Joan A. Cotter, Ph.D., 2012
  • Multiplication Tables 7×9 9×7 © Joan A. Cotter, Ph.D., 2012
  • Multiplication Tables 7×9 9×7 © Joan A. Cotter, Ph.D., 2012
  • Multiplication Tables squares © Joan A. Cotter, Ph.D., 2012
  • Multiplication Tables squares © Joan A. Cotter, Ph.D., 2012
  • Fraction Chart 1 1 1 2 2 1 1 1 3 3 3 1 1 1 1 4 4 4 4 1 1 1 1 1 5 5 5 5 5 1 1 1 1 1 1 6 6 6 6 6 6 1 1 1 1 1 1 1 7 7 7 7 7 7 7 1 1 1 1 1 1 1 1 8 8 8 8 8 8 8 8 1 1 1 1 1 1 1 1 1 9 9 9 9 9 9 9 9 9 1 1 1 1 1 1 1 1 1 1 10 10 10 10 10 10 10 10 10 10Giving the student the big picture. © Joan A. Cotter, Ph.D., 2012
  • Fraction Chart 1 1 1 2 2 1 1 1 3 3 3 1 1 1 1 4 4 4 4 1 1 1 1 1 5 5 5 5 5 1 1 1 1 1 1 6 6 6 6 6 6 1 1 1 1 1 1 1 7 7 7 7 7 7 7 1 1 1 1 1 1 1 1 8 8 8 8 8 8 8 8 1 1 1 1 1 1 1 1 1 9 9 9 9 9 9 9 9 9 1 1 1 1 1 1 1 1 1 1 10 10 10 10 10 10 10 10 10 10 How many fourths in a whole?Giving the child the big picture, a Montessori principle. © Joan A. Cotter, Ph.D., 2012
  • Fraction Chart 1 1 1 2 2 1 1 1 3 3 3 1 1 1 1 4 4 4 4 1 1 1 1 1 5 5 5 5 5 1 1 1 1 1 1 6 6 6 6 6 6 1 1 1 1 1 1 1 7 7 7 7 7 7 7 1 1 1 1 1 1 1 1 8 8 8 8 8 8 8 8 1 1 1 1 1 1 1 1 1 9 9 9 9 9 9 9 9 9 1 1 1 1 1 1 1 1 1 1 10 10 10 10 10 10 10 10 10 10 How many fourths in a whole?Giving the child the big picture, a Montessori principle. © Joan A. Cotter, Ph.D., 2012
  • Fraction Chart 1 1 1 2 2 1 1 1 3 3 3 1 1 1 1 4 4 4 4 1 1 1 1 1 5 5 5 5 5 1 1 1 1 1 1 6 6 6 6 6 6 1 1 1 1 1 1 1 7 7 7 7 7 7 7 1 1 1 1 1 1 1 1 8 8 8 8 8 8 8 8 1 1 1 1 1 1 1 1 1 9 9 9 9 9 9 9 9 9 1 1 1 1 1 1 1 1 1 1 10 10 10 10 10 10 10 10 10 10 How many fourths in a whole?Giving the child the big picture, a Montessori principle. © Joan A. Cotter, Ph.D., 2012
  • Fraction Chart 1 1 1 2 2 1 1 1 3 3 3 1 1 1 1 4 4 4 4 1 1 1 1 1 5 5 5 5 5 1 1 1 1 1 1 6 6 6 6 6 6 1 1 1 1 1 1 1 7 7 7 7 7 7 7 1 1 1 1 1 1 1 1 8 8 8 8 8 8 8 8 1 1 1 1 1 1 1 1 1 9 9 9 9 9 9 9 9 9 1 1 1 1 1 1 1 1 1 1 10 10 10 10 10 10 10 10 10 10 How many fourths in a whole?Giving the child the big picture, a Montessori principle. © Joan A. Cotter, Ph.D., 2012
  • Fraction Chart 1 1 1 2 2 1 1 1 3 3 3 1 1 1 1 4 4 4 4 1 1 1 1 1 5 5 5 5 5 1 1 1 1 1 1 6 6 6 6 6 6 1 1 1 1 1 1 1 7 7 7 7 7 7 7 1 1 1 1 1 1 1 1 8 8 8 8 8 8 8 8 1 1 1 1 1 1 1 1 1 9 9 9 9 9 9 9 9 9 1 1 1 1 1 1 1 1 1 1 10 10 10 10 10 10 10 10 10 10 How many fourths in a whole?Giving the child the big picture, a Montessori principle. © Joan A. Cotter, Ph.D., 2012
  • Fraction Chart 1 1 1 2 2 1 1 1 3 3 3 1 1 1 1 4 4 4 4 1 1 1 1 1 5 5 5 5 5 1 1 1 1 1 1 6 6 6 6 6 6 1 1 1 1 1 1 1 7 7 7 7 7 7 7 1 1 1 1 1 1 1 1 8 8 8 8 8 8 8 8 1 1 1 1 1 1 1 1 1 9 9 9 9 9 9 9 9 9 1 1 1 1 1 1 1 1 1 1 10 10 10 10 10 10 10 10 10 10 How many eighths in a whole?Giving the student the big picture. © Joan A. Cotter, Ph.D., 2012
  • Fraction Chart 1 1 1 2 2 1 1 1 3 3 3 1 1 1 1 4 4 4 4 1 1 1 1 1 5 5 5 5 5 1 1 1 1 1 1 6 6 6 6 6 6 1 1 1 1 1 1 1 7 7 7 7 7 7 7 1 1 1 1 1 1 1 1 8 8 8 8 8 8 8 8 1 1 1 1 1 1 1 1 1 9 9 9 9 9 9 9 9 9 1 1 1 1 1 1 1 1 1 1 10 10 10 10 10 10 10 10 10 10Which is more, 3/4 or 4/5? © Joan A. Cotter, Ph.D., 2012
  • Fraction Chart 1 1 1 2 2 1 1 1 3 3 3 1 1 1 1 4 4 4 4 1 1 1 1 1 5 5 5 5 5 1 1 1 1 1 1 6 6 6 6 6 6 1 1 1 1 1 1 1 7 7 7 7 7 7 7 1 1 1 1 1 1 1 1 8 8 8 8 8 8 8 8 1 1 1 1 1 1 1 1 1 9 9 9 9 9 9 9 9 9 1 1 1 1 1 1 1 1 1 1 10 10 10 10 10 10 10 10 10 10Which is more, 3/4 or 4/5? © Joan A. Cotter, Ph.D., 2012
  • Fraction Chart 1 1 1 2 2 1 1 1 3 3 3 1 1 1 1 4 4 4 4 1 1 1 1 1 5 5 5 5 5 1 1 1 1 1 1 6 6 6 6 6 6 1 1 1 1 1 1 1 7 7 7 7 7 7 7 1 1 1 1 1 1 1 1 8 8 8 8 8 8 8 8 1 1 1 1 1 1 1 1 1 9 9 9 9 9 9 9 9 9 1 1 1 1 1 1 1 1 1 1 10 10 10 10 10 10 10 10 10 10 Which is more, 3/4 or 4/5?Giving the child the big picture. © Joan A. Cotter, Ph.D., 2012
  • Fraction Chart 1 1 1 2 2 1 1 1 3 3 3 1 1 1 1 4 4 4 4 1 1 1 1 1 5 5 5 5 5 1 1 1 1 1 1 6 6 6 6 6 6 1 1 1 1 1 1 1 7 7 7 7 7 7 7 1 1 1 1 1 1 1 1 8 8 8 8 8 8 8 8 1 1 1 1 1 1 1 1 1 9 9 9 9 9 9 9 9 9 1 1 1 1 1 1 1 1 1 1 10 10 10 10 10 10 10 10 10 10 Which is more, 3/4 or 4/5?Giving the child the big picture. © Joan A. Cotter, Ph.D., 2012
  • Fraction Chart1 Stairs (Unit fractions)10 1 9 1 8 1 7 1 6 1 5 1 4 1 3 1 2 1 © Joan A. Cotter, Ph.D., 2012
  • Fraction Chart1 Stairs (Unit fractions)10 1 9 1 8 1 7 1 6 1 5 1 4 1 3 1 2 1 A hyperbola. © Joan A. Cotter, Ph.D., 2012
  • Fraction Chart 1 1 1 2 2 1 1 1 3 3 3 1 1 1 1 4 4 4 4 1 1 1 1 1 5 5 5 5 5 1 1 1 1 1 1 6 6 6 6 6 6 1 1 1 1 1 1 1 7 7 7 7 7 7 7 1 1 1 1 1 1 1 1 1 8 8 8 8 8 8 8 8 8 1 1 1 1 1 1 1 1 1 9 9 9 9 9 9 9 9 91 1 1 1 1 1 1 1 1 110 10 10 10 10 10 10 10 10 10 9/8 is 1 and 1/8. © Joan A. Cotter, Ph.D., 2012
  • Circle ModelAre we comparing angles, arcs, or area? © Joan A. Cotter, Ph.D., 2012
  • Circle Model 1 1 3 3 1 1 2 2 1 3 1 1 1 11 1 5 5 6 64 4 1 1 1 1 6 6 1 1 5 5 4 4 1 1 1 6 6 5Try to compare 4/5 and 5/6 with this model. © Joan A. Cotter, Ph.D., 2012
  • Circle ModelExperts in visual literacy say that comparingquantities in pie charts is difficult becausemost people think linearly. It is easier tocompare along a straight line than comparepie slices. askoxford.com © Joan A. Cotter, Ph.D., 2012
  • Circle ModelExperts in visual literacy say that comparingquantities in pie charts is difficult becausemost people think linearly. It is easier tocompare along a straight line than comparepie slices. askoxford.comSpecialists also suggest refraining from usingmore than one pie chart for comparison. www.statcan.ca © Joan A. Cotter, Ph.D., 2012
  • Circle Model Difficulties564 © Joan A. Cotter, Ph.D., 2012
  • Circle Model Difficulties • Perpetuates cultural myth fractions are < 1.565 © Joan A. Cotter, Ph.D., 2012
  • Circle Model Difficulties • Perpetuates cultural myth fractions are < 1. • Does not give the child the “big picture.”566 © Joan A. Cotter, Ph.D., 2012
  • Circle Model Difficulties • Perpetuates cultural myth fractions are < 1. • Does not give the child the “big picture.” • Limits understanding of fractions: they are more than “a part of a whole or part of a set.”567 © Joan A. Cotter, Ph.D., 2012
  • Circle Model Difficulties • Perpetuates cultural myth fractions are < 1. • Does not give the child the “big picture.” • Limits understanding of fractions: they are more than “a part of a whole or part of a set.” • Makes it difficult for the child to see how fractions relate to each other.568 © Joan A. Cotter, Ph.D., 2012
  • Simplifying Fractions 1 2 3 4 5 6 7 8 9 10 2 4 6 8 10 12 14 16 18 20 3 6 9 12 15 18 21 24 27 30 4 8 12 16 20 24 28 32 36 40 5 10 15 20 25 30 35 40 45 50 6 12 18 24 30 36 42 48 54 60 7 14 21 28 35 42 49 56 63 70 8 16 24 32 40 48 56 64 72 80 9 18 27 36 45 54 63 72 81 90 10 20 30 40 50 60 70 80 90 100 © Joan A. Cotter, Ph.D., 2012
  • Simplifying Fractions 1 2 3 4 5 6 7 8 9 10 2 4 6 8 10 12 14 16 18 20 3 6 9 12 15 18 21 24 27 30 4 8 12 16 20 24 28 32 36 40 5 10 15 20 25 30 35 40 45 50 6 12 18 24 30 36 42 48 54 60 7 14 21 28 35 42 49 56 63 70 8 16 24 32 40 48 56 64 72 80 9 18 27 36 45 54 63 72 81 90 10 20 30 40 50 60 70 80 90 100The fraction 4/8 can be reduced on the multiplication table as 1/2. © Joan A. Cotter, Ph.D., 2012
  • Simplifying Fractions 1 2 3 4 5 6 7 8 9 10 2 4 6 8 10 12 14 16 18 20 3 6 9 12 15 18 21 24 27 30 4 8 12 16 20 24 28 32 36 40 5 10 15 20 25 30 35 40 45 50 6 12 18 24 30 36 42 48 54 60 7 14 21 28 35 42 49 56 63 70 8 16 24 32 40 48 56 64 72 80 9 18 27 36 45 54 63 72 81 90 10 20 30 40 50 60 70 80 90 100The fraction 4/8 can be reduced on the multiplication table as 1/2. © Joan A. Cotter, Ph.D., 2012
  • Simplifying Fractions 1 2 3 4 5 6 7 8 9 10 2 4 6 8 10 12 14 16 18 20 3 6 9 12 15 18 21 24 27 30 21 4 8 12 16 20 24 28 32 36 40 28 5 10 15 20 25 30 35 40 45 50 6 12 18 24 30 36 42 48 54 60 7 14 21 28 35 42 49 56 63 70 8 16 24 32 40 48 56 64 72 80 9 18 27 36 45 54 63 72 81 90 10 20 30 40 50 60 70 80 90 100In what column would you put 21/28? © Joan A. Cotter, Ph.D., 2012
  • Simplifying Fractions 1 2 3 4 5 6 7 8 9 10 2 4 6 8 10 12 14 16 18 20 3 6 9 12 15 18 21 24 27 30 4 8 12 16 20 24 28 32 36 40 5 10 15 20 25 30 35 40 45 50 6 12 18 24 30 36 42 48 54 60 7 14 21 28 35 42 49 56 63 70 8 16 24 32 40 48 56 64 72 80 9 18 27 36 45 54 63 72 81 90 10 20 30 40 50 60 70 80 90 100 © Joan A. Cotter, Ph.D., 2012
  • Simplifying Fractions 1 2 3 4 5 6 7 8 9 10 2 4 6 8 10 12 14 16 18 20 3 6 9 12 15 18 21 24 27 30 21 4 8 12 16 20 24 28 32 36 40 28 5 10 15 20 25 30 35 40 45 50 6 12 18 24 30 36 42 48 54 60 7 14 21 28 35 42 49 56 63 70 8 16 24 32 40 48 56 64 72 80 9 18 27 36 45 54 63 72 81 90 10 20 30 40 50 60 70 80 90 100In what column would you put 21/28? © Joan A. Cotter, Ph.D., 2012
  • Simplifying Fractions 1 2 3 4 5 6 7 8 9 10 2 4 6 8 10 12 14 16 18 20 3 6 9 12 15 18 21 24 27 30 21 4 8 12 16 20 24 28 32 36 40 28 5 10 15 20 25 30 35 40 45 50 6 12 18 24 30 36 42 48 54 60 7 14 21 28 35 42 49 56 63 70 8 16 24 32 40 48 56 64 72 80 9 18 27 36 45 54 63 72 81 90 10 20 30 40 50 60 70 80 90 100In what column would you put 21/28? © Joan A. Cotter, Ph.D., 2012
  • Simplifying Fractions 1 2 3 4 5 6 7 8 9 10 2 4 6 8 10 12 14 16 18 20 3 6 9 12 15 18 21 24 27 30 21 4 8 12 16 20 24 28 32 36 40 28 5 10 15 20 25 30 35 40 45 50 6 12 18 24 30 36 42 48 54 60 45 72 7 14 21 28 35 42 49 56 63 70 8 16 24 32 40 48 56 64 72 80 9 18 27 36 45 54 63 72 81 90 10 20 30 40 50 60 70 80 90 100 © Joan A. Cotter, Ph.D., 2012
  • Simplifying Fractions 1 2 3 4 5 6 7 8 9 10 2 4 6 8 10 12 14 16 18 20 3 6 9 12 15 18 21 24 27 30 21 4 8 12 16 20 24 28 32 36 40 28 5 10 15 20 25 30 35 40 45 50 6 12 18 24 30 36 42 48 54 60 45 72 7 14 21 28 35 42 49 56 63 70 8 16 24 32 40 48 56 64 72 80 9 18 27 36 45 54 63 72 81 90 10 20 30 40 50 60 70 80 90 100 © Joan A. Cotter, Ph.D., 2012
  • Simplifying Fractions 1 2 3 4 5 6 7 8 9 10 2 4 6 8 10 12 14 16 18 20 3 6 9 12 15 18 21 24 27 30 21 4 8 12 16 20 24 28 32 36 40 28 5 10 15 20 25 30 35 40 45 50 6 12 18 24 30 36 42 48 54 60 45 72 7 14 21 28 35 42 49 56 63 70 8 16 24 32 40 48 56 64 72 80 9 18 27 36 45 54 63 72 81 90 10 20 30 40 50 60 70 80 90 100 © Joan A. Cotter, Ph.D., 2012
  • Simplifying Fractions 1 2 3 4 5 6 7 8 9 10 2 4 6 8 10 12 14 16 18 20 3 6 9 12 15 18 21 24 27 30 12 4 8 12 16 20 24 28 32 36 40 16 5 10 15 20 25 30 35 40 45 50 6 12 18 24 30 36 42 48 54 60 7 14 21 28 35 42 49 56 63 70 8 16 24 32 40 48 56 64 72 80 9 18 27 36 45 54 63 72 81 90 10 20 30 40 50 60 70 80 90 100 © Joan A. Cotter, Ph.D., 2012
  • Simplifying Fractions 1 2 3 4 5 6 7 8 9 10 2 4 6 8 10 12 14 16 18 20 3 6 9 12 15 18 21 24 27 30 12 4 8 12 16 20 24 28 32 36 40 16 5 10 15 20 25 30 35 40 45 50 6 12 18 24 30 36 42 48 54 60 7 14 21 28 35 42 49 56 63 70 8 16 24 32 40 48 56 64 72 80 9 18 27 36 45 54 63 72 81 90 10 20 30 40 50 60 70 80 90 100 © Joan A. Cotter, Ph.D., 2012
  • Simplifying Fractions 1 2 3 4 5 6 7 8 9 10 2 4 6 8 10 12 14 16 18 20 3 6 9 12 15 18 21 24 27 30 12 4 8 12 16 20 24 28 32 36 40 16 5 10 15 20 25 30 35 40 45 50 6 12 18 24 30 36 42 48 54 60 7 14 21 28 35 42 49 56 63 70 8 16 24 32 40 48 56 64 72 80 9 18 27 36 45 54 63 72 81 90 10 20 30 40 50 60 70 80 90 1006/8 needs further simplifying. © Joan A. Cotter, Ph.D., 2012
  • Simplifying Fractions 1 2 3 4 5 6 7 8 9 10 2 4 6 8 10 12 14 16 18 20 3 6 9 12 15 18 21 24 27 30 12 4 8 12 16 20 24 28 32 36 40 16 5 10 15 20 25 30 35 40 45 50 6 12 18 24 30 36 42 48 54 60 7 14 21 28 35 42 49 56 63 70 8 16 24 32 40 48 56 64 72 80 9 18 27 36 45 54 63 72 81 90 10 20 30 40 50 60 70 80 90 1006/8 needs further simplifying. © Joan A. Cotter, Ph.D., 2012
  • Simplifying Fractions 1 2 3 4 5 6 7 8 9 10 2 4 6 8 10 12 14 16 18 20 3 6 9 12 15 18 21 24 27 30 12 4 8 12 16 20 24 28 32 36 40 16 5 10 15 20 25 30 35 40 45 50 6 12 18 24 30 36 42 48 54 60 7 14 21 28 35 42 49 56 63 70 8 16 24 32 40 48 56 64 72 80 9 18 27 36 45 54 63 72 81 90 10 20 30 40 50 60 70 80 90 1006/8 needs further simplifying. © Joan A. Cotter, Ph.D., 2012
  • Simplifying Fractions 1 2 3 4 5 6 7 8 9 10 2 4 6 8 10 12 14 16 18 20 3 6 9 12 15 18 21 24 27 30 12 4 8 12 16 20 24 28 32 36 40 16 5 10 15 20 25 30 35 40 45 50 6 12 18 24 30 36 42 48 54 60 7 14 21 28 35 42 49 56 63 70 8 16 24 32 40 48 56 64 72 80 9 18 27 36 45 54 63 72 81 90 10 20 30 40 50 60 70 80 90 10012/16 could have put here originally. © Joan A. Cotter, Ph.D., 2012
  • Enriching Montessori Mathematics with Visualization by Joan A. Cotter, Ph.D. JoanCotter@rightstartmath.com 1000 3 2 5 5 100 10 7 x7 1 NJMAC Conference March 2, 2012 Edison, New Jersey585 Presentations available: rightstartmath.com © Joan A. Cotter, Ph.D., 2012