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How Visualization Enhances   Montessori Mathematics PART 1                      by Joan A. Cotter, Ph.D.                  ...
Counting ModelIn Montessori, counting is pervasive:       • Number Rods       • Spindle Boxes       • Decimal materials   ...
Verbal Counting Model   From a childs perspective                                © Joan A. Cotter, Ph.D., 2012
Verbal Counting Model            From a childs perspectiveBecause we’re so familiar with 1, 2, 3, we’ll use letters.      ...
Verbal Counting Model   From a childs perspective              F             +E                                © Joan A. C...
Verbal Counting Model    From a childs perspective               F              +EA                                 © Joan...
Verbal Counting Model        From a childs perspective                   F                  +EA   B                       ...
Verbal Counting Model        From a childs perspective                   F                  +EA   B    C                  ...
Verbal Counting Model        From a childs perspective                      F                     +EA   B    C   D   E   F...
Verbal Counting Model        From a childs perspective                      F                     +EA   B    C   D   E   F...
Verbal Counting Model        From a childs perspective                      F                     +EA   B    C   D   E   F...
Verbal Counting Model        From a childs perspective                      F                     +EA   B    C   D   E   F...
Verbal Counting Model        From a childs perspective                      F                     +EA   B    C   D   E   F...
Verbal Counting Model        From a childs perspective                      F                     +E                      ...
Verbal Counting Model    From a childs perspective  Now memorize the facts!!               G              +D              ...
Verbal Counting Model    From a childs perspective  Now memorize the facts!!                                         H    ...
Verbal Counting Model    From a childs perspective  Now memorize the facts!!                                         H    ...
Verbal Counting Model    From a childs perspective  Now memorize the facts!!                                         H    ...
Verbal Counting Model        From a childs perspective    Now memorize the facts!!                                        ...
Verbal Counting Model            From a childs perspective                       H                      –ESubtract with yo...
Verbal Counting Model     From a childs perspective                 J                –FSubtract without using your fingers...
Verbal Counting Model      From a childs perspectiveTry skip counting by B’s to T:    B, D, . . . T.                      ...
Verbal Counting Model      From a childs perspectiveTry skip counting by B’s to T:    B, D, . . . T.What is D × E?        ...
Verbal Counting Model     From a childs perspectiveLis written ABbecause it is A Jand B A’s                               ...
Verbal Counting Model     From a childs perspectiveLis written ABbecause it is A Jand B A’s              huh?             ...
Verbal Counting Model     From a childs perspectiveL (twelve)is written ABbecause it is A Jand B A’s                      ...
Verbal Counting Model     From a childs perspectiveL (twelve)is written AB (12)because it is A Jand B A’s                 ...
Verbal Counting Model     From a childs perspectiveL (twelve)is written AB (12)because it is A J (one 10)and B A’s        ...
Verbal Counting Model     From a childs perspectiveL (twelve)is written AB (12)because it is A J (one 10)and B A’s (two 1s...
Calendar Math                © Joan A. Cotter, Ph.D., 2012
Calendar Math        August1   2   3   4   5   6   78   9   10 11 12 13 1415 16 17 18 19 20 2122 23 24 25 26 27 2829 30 31...
Calendar Math    Calendar Counting        August1   2   3   4   5   6   78   9   10 11 12 13 1415 16 17 18 19 20 2122 23 2...
Calendar Math    Calendar Counting        August1   2   3   4   5   6   78   9   10 11 12 13 1415 16 17 18 19 20 2122 23 2...
Calendar Math    Calendar Counting        August1   2   3   4   5   6   78   9   10 11 12 13 1415 16 17 18 19 20 2122 23 2...
Calendar Math  Septemb    Calendar Counting1234567        August891012141   2     113    11921151126288 122820   67527    ...
Calendar Math            Septemb              Calendar Counting          1234567                   August         89101214...
Calendar Math    Partial Calendar        August1   2   3    4   5   6   78   9   10                             © Joan A. ...
Calendar Math              Partial Calendar                 August         1   2   3    4   5   6   7         8   9   10Ch...
Calendar Math               Septemb                 Calendar patterning             1234567                      August   ...
Research on Counting   Karen Wynn’s research                           © Joan A. Cotter, Ph.D., 2012
Research on Counting   Karen Wynn’s research                           © Joan A. Cotter, Ph.D., 2012
Research on Counting   Karen Wynn’s research                           © Joan A. Cotter, Ph.D., 2012
Research on Counting   Karen Wynn’s research                           © Joan A. Cotter, Ph.D., 2012
Research on Counting   Karen Wynn’s research                           © Joan A. Cotter, Ph.D., 2012
Research on Counting   Karen Wynn’s research                           © Joan A. Cotter, Ph.D., 2012
Research on Counting   Karen Wynn’s research                           © Joan A. Cotter, Ph.D., 2012
Research on Counting   Karen Wynn’s research                           © Joan A. Cotter, Ph.D., 2012
Research on Counting      Other research                       © Joan A. Cotter, Ph.D., 2012
Research on Counting                         Other research• Australian Aboriginal children from two tribes.       Brian B...
Research on Counting                         Other research• Australian Aboriginal children from two tribes.       Brian B...
Research on Counting                         Other research• Australian Aboriginal children from two tribes.       Brian B...
Research on Counting                         Other research• Australian Aboriginal children from two tribes.       Brian B...
Research on Counting           In Japanese schools:• Children are discouraged from usingcounting for adding.              ...
Research on Counting            In Japanese schools:• Children are discouraged from usingcounting for adding.• They consis...
Subitizing Quantities(Identifying without counting)                                 © Joan A. Cotter, Ph.D., 2012
Subitizing Quantities    (Identifying without counting)• Five-month-old infants can subitize to 3.                        ...
Subitizing Quantities    (Identifying without counting)• Five-month-old infants can subitize to 3.• Three-year-olds can su...
Subitizing Quantities    (Identifying without counting)• Five-month-old infants can subitize to 3.• Three-year-olds can su...
Subitizing Quantities    (Identifying without counting)• Five-month-old infants can subitize to 3.• Three-year-olds can su...
Research on Counting                   Subitizing• Subitizing “allows the child to grasp the wholeand the elements at the ...
Research on Counting                   Subitizing• Subitizing “allows the child to grasp the wholeand the elements at the ...
Research on Counting                   Subitizing• Subitizing “allows the child to grasp the wholeand the elements at the ...
Research on Counting                    Subitizing• Subitizing “allows the child to grasp the wholeand the elements at the...
Research on Counting                     Subitizing• Subitizing “allows the child to grasp the wholeand the elements at th...
Visualizing Quantities                         © Joan A. Cotter, Ph.D., 2012
Visualizing Quantities“Think in pictures, because thebrain remembers images betterthan it does anything else.”   Ben Pridm...
Visualizing Quantities“The role of physical manipulativeswas to help the child form thosevisual images and thus to elimina...
Visualizing Quantities   Japanese criteria for manipulatives• Representative of structure of numbers.• Easily manipulated ...
Visualizing Quantities      Visualizing also needed in:• Reading• Sports• Creativity• Geography• Engineering• Construction...
Visualizing Quantities      Visualizing also needed in:• Reading            • Architecture• Sports             • Astronomy...
Visualizing Quantities    Ready: How many?                         © Joan A. Cotter, Ph.D., 2012
Visualizing Quantities    Ready: How many?                         © Joan A. Cotter, Ph.D., 2012
Visualizing Quantities   Try again: How many?                          © Joan A. Cotter, Ph.D., 2012
Visualizing Quantities   Try again: How many?                          © Joan A. Cotter, Ph.D., 2012
Visualizing QuantitiesTry to visualize 8 identical apples without grouping.                                               ...
Visualizing QuantitiesTry to visualize 8 identical apples without grouping.                                               ...
Visualizing QuantitiesNow try to visualize 5 as red and 3 as green.                                           © Joan A. Co...
Visualizing QuantitiesNow try to visualize 5 as red and 3 as green.                                           © Joan A. Co...
Visualizing Quantities   Early Roman numerals      1     I       2    II       3    III       4    IIII       5    V      ...
Visualizing Quantities       :   Who could read the music?                               © Joan A. Cotter, Ph.D., 2012
Grouping in Fives                    © Joan A. Cotter, Ph.D., 2012
Grouping in Fives• Grouping in fives extends subitizing.                                      © Joan A. Cotter, Ph.D., 2012
Grouping in Fives              Using fingersGrouping in Fives is a three-period lesson.                                   ...
Grouping in Fives              Using fingersGrouping in Fives is a three-period lesson.                                   ...
Grouping in Fives              Using fingersGrouping in Fives is a three-period lesson.                                   ...
Grouping in Fives              Using fingersGrouping in Fives is a three-period lesson.                                   ...
Grouping in Fives              Using fingersGrouping in Fives is a three-period lesson.                                   ...
Grouping in Fives  Yellow is the Sun   Yellow is the sun.   Six is five and one.   Why is the sky so blue?   Seven is five...
Grouping in Fives   Recognizing 5                    © Joan A. Cotter, Ph.D., 2012
Grouping in Fives   Recognizing 5                    © Joan A. Cotter, Ph.D., 2012
Grouping in Fives      Recognizing 55 has a middle; 4 does not.                              © Joan A. Cotter, Ph.D., 2012
Grouping in Fives    Tally sticks                    © Joan A. Cotter, Ph.D., 2012
Grouping in Fives    Tally sticks                    © Joan A. Cotter, Ph.D., 2012
Grouping in Fives    Tally sticks                    © Joan A. Cotter, Ph.D., 2012
Grouping in Fives    Tally sticks                    © Joan A. Cotter, Ph.D., 2012
Grouping in Fives    Tally sticks                    © Joan A. Cotter, Ph.D., 2012
Grouping in Fives    Tally sticks                    © Joan A. Cotter, Ph.D., 2012
Grouping in Fives                                                      Pairing Finger Cards        QuickTime™ and a       ...
Grouping in Fives                                    Ordering Finger Cards        QuickTime™ and a   TIFF (LZW) decompress...
Grouping in Fives                       Matching Number Cards to Finger Cards        QuickTime™ and a                  Qui...
Grouping in FivesMatching Finger Cards to Number Cards9       1               10                                     4    ...
Grouping in Fives                                    Finger Card Memory game          QuickTime™ and a                   Q...
Grouping in Fives    Number Rods                    © Joan A. Cotter, Ph.D., 2012
Grouping in Fives    Number Rods                    © Joan A. Cotter, Ph.D., 2012
Grouping in Fives    Number Rods                    © Joan A. Cotter, Ph.D., 2012
Grouping in Fives     Spindle Box                    © Joan A. Cotter, Ph.D., 2012
Grouping in Fives     Spindle Box                    © Joan A. Cotter, Ph.D., 2012
Grouping in Fives          Spindle Box0     1       2         3   4                            © Joan A. Cotter, Ph.D., 2012
Grouping in Fives          Spindle Box5     6       7         8   9                            © Joan A. Cotter, Ph.D., 2012
Grouping in Fives          Spindle Box5     6       7         8   9                            © Joan A. Cotter, Ph.D., 2012
Grouping in Fives          Spindle Box5     6       7         8   9                            © Joan A. Cotter, Ph.D., 2012
Grouping in Fives          Spindle Box5     6       7         8   9                            © Joan A. Cotter, Ph.D., 2012
Grouping in Fives          Spindle Box5     6       7         8   9                            © Joan A. Cotter, Ph.D., 2012
Grouping in Fives          Spindle Box5     6       7         8   9                            © Joan A. Cotter, Ph.D., 2012
Grouping in Fives   1000      100     10        1   1000      100     10        1   1000      100     10        1   1000  ...
Grouping in Fives   1000      100     10        1   1000      100     10        1   1000      100     10        1   1000  ...
Grouping in Fives 1000   1000     100   100   10   10   1            1 1000   1000     100   100   10   10   1            ...
Grouping in Fives 1000   1000     100   100   10        1            1 1000   1000     100   100             1            ...
Grouping in Fives 1000   1000     100   100   10        1            1 1000   1000     100   100             1            ...
Grouping in Fives       Black and White Bead Stairs“Grouped in fives so the child does notneed to count.”          A. M. J...
Grouping in Fives  Entering quantities                        © Joan A. Cotter, Ph.D., 2012
Grouping in Fives      Entering quantities3                            © Joan A. Cotter, Ph.D., 2012
Grouping in Fives      Entering quantities5                            © Joan A. Cotter, Ph.D., 2012
Grouping in Fives      Entering quantities7                            © Joan A. Cotter, Ph.D., 2012
Grouping in Fives       Entering quantities10                             © Joan A. Cotter, Ph.D., 2012
Grouping in Fives     The stairs                    © Joan A. Cotter, Ph.D., 2012
Grouping in Fives      Adding                    © Joan A. Cotter, Ph.D., 2012
Grouping in Fives      Adding 4+3=                    © Joan A. Cotter, Ph.D., 2012
Grouping in Fives      Adding 4+3=                    © Joan A. Cotter, Ph.D., 2012
Grouping in Fives      Adding 4+3=                    © Joan A. Cotter, Ph.D., 2012
Grouping in Fives      Adding 4+3=                    © Joan A. Cotter, Ph.D., 2012
Grouping in Fives      Adding 4+3=7                    © Joan A. Cotter, Ph.D., 2012
Math Card Games                  © Joan A. Cotter, Ph.D., 2012
Math Card Games• Provide repetition for learning the facts.                                               © Joan A. Cotter...
Math Card Games• Provide repetition for learning the facts.• Encourage autonomy.                                          ...
Math Card Games• Provide repetition for learning the facts.• Encourage autonomy.• Promote social interaction.             ...
Math Card Games• Provide repetition for learning the facts.• Encourage autonomy.• Promote social interaction.• Are enjoyed...
Go to the Dump GameObjective: To learn the facts that total 10:                1+9                2+8                3+7  ...
Go to the Dump GameObjective: To learn the facts that total 10:                1+9                2+8                3+7  ...
“Math” Way of Naming Numbers                        © Joan A. Cotter, Ph.D., 2012
“Math” Way of Naming Numbers  11 = ten 1                        © Joan A. Cotter, Ph.D., 2012
“Math” Way of Naming Numbers  11 = ten 1  12 = ten 2                        © Joan A. Cotter, Ph.D., 2012
“Math” Way of Naming Numbers  11 = ten 1  12 = ten 2  13 = ten 3                        © Joan A. Cotter, Ph.D., 2012
“Math” Way of Naming Numbers  11 = ten 1  12 = ten 2  13 = ten 3  14 = ten 4                        © Joan A. Cotter, Ph.D...
“Math” Way of Naming Numbers  11 = ten 1  12 = ten 2  13 = ten 3  14 = ten 4   ....  19 = ten 9                        © J...
“Math” Way of Naming Numbers  11 = ten 1   20 = 2-ten  12 = ten 2  13 = ten 3  14 = ten 4   ....  19 = ten 9              ...
“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 ...
“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   ...
“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   ...
“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   ...
“Math” Way of Naming Numbers    137 = 1 hundred 3-ten 7                              © Joan A. Cotter, Ph.D., 2012
“Math” Way of Naming Numbers    137 = 1 hundred 3-ten 7              or  137 = 1 hundred and 3-ten 7                      ...
“Math” Way of Naming Numbers                                 100             Chinese                                      ...
“Math” Way of Naming Numbers                                 100             Chinese                                      ...
“Math” Way of Naming Numbers                                 100             Chinese                                      ...
“Math” Way of Naming Numbers                                 100             Chinese                                      ...
“Math” Way of Naming Numbers                                 100             Chinese                                      ...
Math Way of Naming Numbers • Only 11 words are needed to count to 100 the math way, 28 in English. (All Indo-European lang...
Math Way of Naming Numbers • Only 11 words are needed to count to 100 the math way, 28 in English. (All Indo-European lang...
Math Way of Naming Numbers • Only 11 words are needed to count to 100 the math way, 28 in English. (All Indo-European lang...
Math Way of Naming Numbers • Only 11 words are needed to count to 100 the math way, 28 in English. (All Indo-European lang...
Math Way of Naming Numbers       Compared to reading:                              © Joan A. Cotter, Ph.D., 2012
Math Way of Naming Numbers                Compared to reading:• Just as reciting the alphabet doesn’t teach reading,counti...
Math Way of Naming Numbers                Compared to reading:• Just as reciting the alphabet doesn’t teach reading,counti...
Math Way of Naming Numbers                Compared to reading:• Just as reciting the alphabet doesn’t teach reading,counti...
Math Way of Naming Numbers“Rather, the increased gap between Chinese andU.S. students and that of Chinese Americans andCau...
Math Way of Naming Numbers              Traditional names4-ten =fortyThe “ty”means tens.                                  ...
Math Way of Naming Numbers              Traditional names4-ten =fortyThe “ty”means tens.                                  ...
Math Way of Naming Numbers              Traditional names6-ten = sixtyThe “ty”means tens.                                 ...
Math Way of Naming Numbers               Traditional names3-ten = thirty“Thir” alsoused in 1/3,13 and 30.                 ...
Math Way of Naming Numbers                Traditional names5-ten = fifty“Fif” alsoused in 1/5,15 and 50.                  ...
Math Way of Naming Numbers            Traditional names2-ten = twentyTwo used to bepronounced“twoo.”                      ...
Math Way of Naming Numbers          Traditional names A word game   fireplace          place-fire                         ...
Math Way of Naming Numbers          Traditional names A word game   fireplace          place-fire   newspaper          pap...
Math Way of Naming Numbers          Traditional names A word game   fireplace          place-fire   newspaper          pap...
Math Way of Naming Numbers                  Traditional names              ten 4“Teen” alsomeans ten.                     ...
Math Way of Naming Numbers                  Traditional names              ten 4      teen 4“Teen” alsomeans ten.         ...
Math Way of Naming Numbers                  Traditional names              ten 4      teen 4     fourtee                  ...
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...
Math Way of Naming Numbers        Traditional names     a one left     a left-one   eleven                                ...
Math Way of Naming Numbers                Traditional names             two leftTwo saidas “twoo.”                        ...
Math Way of Naming Numbers                Traditional names             two left    twelveTwo saidas “twoo.”              ...
Composing Numbers3-ten                       © Joan A. Cotter, Ph.D., 2012
Composing Numbers3-ten                       © Joan A. Cotter, Ph.D., 2012
Composing Numbers3-ten3030                       © Joan A. Cotter, Ph.D., 2012
Composing Numbers3-ten3030                       © Joan A. Cotter, Ph.D., 2012
Composing Numbers3-ten3030                       © Joan A. Cotter, Ph.D., 2012
Composing Numbers3-ten 73030                       © Joan A. Cotter, Ph.D., 2012
Composing Numbers3-ten 73030                       © Joan A. Cotter, Ph.D., 2012
Composing Numbers3-ten 73030 7 7                       © Joan A. Cotter, Ph.D., 2012
Composing Numbers3-ten 73037 0 7                       © Joan A. Cotter, Ph.D., 2012
Composing Numbers  3-ten 7  30  37   0   7Note the congruence in how we say the number,represent the number, and write the...
Composing Numbers1-ten1010        Another example.                           © Joan A. Cotter, Ph.D., 2012
Composing Numbers1-ten 81010                       © Joan A. Cotter, Ph.D., 2012
Composing Numbers1-ten 81010                       © Joan A. Cotter, Ph.D., 2012
Composing Numbers1-ten 81010 8 8                       © Joan A. Cotter, Ph.D., 2012
Composing Numbers1-ten 81818                       © Joan A. Cotter, Ph.D., 2012
Composing Numbers10-ten                      © Joan A. Cotter, Ph.D., 2012
Composing Numbers10-ten100100                      © Joan A. Cotter, Ph.D., 2012
Composing Numbers10-ten100100                      © Joan A. Cotter, Ph.D., 2012
Composing Numbers10-ten100100                      © Joan A. Cotter, Ph.D., 2012
Composing Numbers1 hundred                      © Joan A. Cotter, Ph.D., 2012
Composing Numbers1 hundred100100                      © Joan A. Cotter, Ph.D., 2012
Composing Numbers1 hundred100100                      © Joan A. Cotter, Ph.D., 2012
Composing Numbers1 hundred100100                      © Joan A. Cotter, Ph.D., 2012
Composing Numbers1 hundred100100                      © Joan A. Cotter, Ph.D., 2012
Composing Numbers2 hundred                      © Joan A. Cotter, Ph.D., 2012
Composing Numbers2 hundred                      © Joan A. Cotter, Ph.D., 2012
Composing Numbers2 hundred200200                      © Joan A. Cotter, Ph.D., 2012
Evens and Odds                 © Joan A. Cotter, Ph.D., 2012
Evens and Odds     Evens                 © Joan A. Cotter, Ph.D., 2012
Evens and Odds     Evens             Use two fingers             and touch each             pair in succession.           ...
Evens and Odds     Evens             Use two fingers             and touch each             pair in succession.           ...
Evens and Odds     Evens             Use two fingers             and touch each             pair in succession.           ...
Evens and Odds     Evens             Use two fingers             and touch each             pair in succession.           ...
Evens and Odds     Odds            Use two fingers            and touch each            pair in succession.               ...
Evens and Odds     Odds            Use two fingers            and touch each            pair in succession.               ...
Evens and Odds     Odds            Use two fingers            and touch each            pair in succession.               ...
Evens and Odds     Odds            Use two fingers            and touch each            pair in succession.               ...
Evens and Odds     Odds            Use two fingers            and touch each            pair in succession.               ...
Learning the Facts                     © Joan A. Cotter, Ph.D., 2012
Learning the FactsLimited success when:• Based on counting.    Whether dots, fingers, number lines, or    counting words. ...
Learning the FactsLimited success when:• Based on counting.    Whether dots, fingers, number lines, or    counting words.•...
Learning the FactsLimited success when:• Based on counting.    Whether dots, fingers, number lines, or    counting words.•...
Fact Strategies                  © 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.                                © Jo...
Fact Strategies             Complete the Ten 9+5=Take 1 fromthe 5 and giveit to the 9.                                © Jo...
Fact Strategies             Complete the Ten 9+5=Take 1 fromthe 5 and giveit to the 9.                                © Jo...
Fact Strategies              Complete the Ten 9 + 5 = 14Take 1 fromthe 5 and giveit to the 9.                             ...
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 Fives8+6=                         © Joan A. Cotter, Ph.D., 2012
Fact Strategies           Two Fives8+6=                         © Joan A. Cotter, Ph.D., 2012
Fact Strategies              Two Fives8+6=10 + 4 = 14                          © Joan A. Cotter, Ph.D., 2012
Fact Strategies           Going Down15 – 9 =                         © 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., 2...
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,...
Fact Strategies              Subtract from 10 15 – 9 =Subtract 9from 10.                                 © Joan A. Cotter,...
Fact Strategies              Subtract from 10 15 – 9 =Subtract 9from 10.                                 © Joan A. Cotter,...
Fact Strategies              Subtract from 10 15 – 9 = 6Subtract 9from 10.                                 © Joan A. Cotte...
Fact Strategies           Going Up15 – 9 =                         © Joan A. Cotter, Ph.D., 2012
Fact Strategies                Going Up 15 – 9 =Start with 9;go up to 15.                             © Joan A. Cotter, Ph...
Fact Strategies                Going Up 15 – 9 =Start with 9;go up to 15.                             © Joan A. Cotter, Ph...
Fact Strategies                Going Up 15 – 9 =Start with 9;go up to 15.                             © Joan A. Cotter, Ph...
Fact Strategies                Going Up 15 – 9 =Start with 9;go up to 15.                             © Joan A. Cotter, Ph...
Fact Strategies                Going Up 15 – 9 = 1+5=6Start with 9;go up to 15.                             © Joan A. Cott...
Rows and Columns GameObjective:  To find a total of 15 by adding 2, 3, or 4cards in a row or in a column.                 ...
Rows and Columns GameObjective:  To find a total of 15 by adding 2, 3, or 4cards in a row or in a column.Object of the gam...
Rows and Columns Game       8   7   1   9       6   4   3   3       2   2   5   6       6   3   8   8                     ...
Rows and Columns Game       8   7   1   9       6   4   3   3       2   2   5   6       6   3   8   8                     ...
Rows and Columns Game       8   7   1   9       6   4   3   3       2   2   5   6       6   3   8   8                     ...
Rows and Columns Game               1   9       6   4   3   3       6   3   8   8                       © Joan A. Cotter, ...
Rows and Columns Game       7   6   1   9       6   4   3   3       2   1   5   1       6   3   8   8                     ...
Rows and Columns Game       7   6   1   9       6   4   3   3       2   1   5   1       6   3   8   8                     ...
Rows and Columns Game       7   6   1   9       6   4   3   3       2   1   5   1       6   3   8   8                     ...
Rows and Columns Game               1       6   4   3   3           1   5   1           3   8   8                       © ...
Rows and Columns Game                   © 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
Place Value Two aspects               © Joan A. Cotter, Ph.D., 2012
Place Value          Two aspectsStatic                        © Joan A. Cotter, Ph.D., 2012
Place Value                  Two aspectsStatic  • Value of a digit is determined by position                              ...
Place Value                  Two aspectsStatic  • Value of a digit is determined by position.  • No position may have more...
Place Value                  Two aspectsStatic  • Value of a digit is determined by position.  • No position may have more...
Place Value                  Two aspectsStatic  • Value of a digit is determined by position.  • No position may have more...
Place Value                  Two aspectsStatic  • Value of a digit is determined by position.  • No position may have more...
Place Value                  Two aspectsStatic  • Value of a digit is determined by position.  • No position may have more...
Place Value                  Two aspectsStatic  • Value of a digit is determined by position.  • No position may have more...
Exchanging1000   100   10   1                      © Joan A. Cotter, Ph.D., 2012
Exchanging              Thousands1000   100   10   1                          © Joan A. Cotter, Ph.D., 2012
Exchanging                  Hundreds1000   100   10     1                             © Joan A. Cotter, Ph.D., 2012
Exchanging                  Tens1000   100   10   1                         © Joan A. Cotter, Ph.D., 2012
Exchanging                  Ones1000   100   10   1                         © Joan A. Cotter, Ph.D., 2012
Exchanging                  Adding1000   100   10    1                            8                           +6          ...
Exchanging                  Adding1000   100   10    1                            8                           +6          ...
Exchanging                  Adding1000   100   10    1                            8                           +6          ...
Exchanging                  Adding1000   100   10    1                            8                           +6          ...
Exchanging                  Adding1000   100   10    1                            8                           +6          ...
Exchanging                  Adding1000   100   10    1                                8                               +6  ...
Exchanging                  Adding1000   100   10    1                                8                               +6  ...
Exchanging                  Adding1000   100   10    1                                8                               +6  ...
Exchanging                  Adding1000   100   10    1                                8                               +6  ...
Bead Frame   1  101001000                    © Joan A. Cotter, Ph.D., 2012
Bead Frame   1                    8  10                   +61001000                    © Joan A. Cotter, Ph.D., 2012
Bead Frame   1                    8  10                   +61001000                    © Joan A. Cotter, Ph.D., 2012
Bead Frame   1                    8  10                   +61001000                    © Joan A. Cotter, Ph.D., 2012
Bead Frame   1                    8  10                   +61001000                    © Joan A. Cotter, Ph.D., 2012
Bead Frame   1                    8  10                   +61001000                    © Joan A. Cotter, Ph.D., 2012
Bead Frame   1                    8  10                   +61001000                    © Joan A. Cotter, Ph.D., 2012
Bead Frame   1                    8  10                   +61001000                    © Joan A. Cotter, Ph.D., 2012
Bead Frame   1                    8  10                   +61001000                    © Joan A. Cotter, Ph.D., 2012
Bead Frame   1                    8  10                   +61001000                    © Joan A. Cotter, Ph.D., 2012
Bead Frame   1                    8  10                   +6100                    141000                    © Joan A. Cot...
1 Bead Frame                              10                             100                             1000Difficulties ...
1               Bead Frame                                               10                                              1...
1               Bead Frame                                               10                                              1...
1                Bead Frame                                               10                                              ...
1                Bead Frame                                               10                                              ...
1                Bead Frame                                               10                                              ...
1                Bead Frame                                                10                                             ...
Exchanging       Adding 4-digit numbers1000    100   10   1                          3658                        + 2738   ...
Exchanging       Adding 4-digit numbers1000    100   10   1                          3658                        + 2738   ...
Exchanging       Adding 4-digit numbers1000    100   10   1                          3658                        + 2738   ...
Exchanging       Adding 4-digit numbers1000    100   10   1                          3658                        + 2738   ...
Exchanging       Adding 4-digit numbers1000    100   10   1                          3658                        + 2738   ...
Exchanging       Adding 4-digit numbers1000    100   10   1                          3658                        + 2738   ...
Exchanging       Adding 4-digit numbers1000    100   10   1                          3658                        + 2738   ...
Exchanging       Adding 4-digit numbers1000    100   10   1                          3658                        + 2738   ...
Exchanging       Adding 4-digit numbers1000    100   10   1                          3658                        + 2738   ...
Exchanging       Adding 4-digit numbers1000    100   10   1                          3658                        + 2738   ...
Exchanging       Adding 4-digit numbers1000    100   10   1                          3658                        + 2738   ...
Exchanging       Adding 4-digit numbers1000    100   10   1                          3658                        + 2738   ...
Exchanging       Adding 4-digit numbers1000    100   10   1            1                          3658                    ...
Exchanging       Adding 4-digit numbers1000    100   10   1            1                          3658                    ...
Exchanging       Adding 4-digit numbers1000    100   10   1            1                          3658                    ...
IMF: Visualizing and Montessori Math PART 1
IMF: Visualizing and Montessori Math PART 1
IMF: Visualizing and Montessori Math PART 1
IMF: Visualizing and Montessori Math PART 1
IMF: Visualizing and Montessori Math PART 1
IMF: Visualizing and Montessori Math PART 1
IMF: Visualizing and Montessori Math PART 1
IMF: Visualizing and Montessori Math PART 1
IMF: Visualizing and Montessori Math PART 1
IMF: Visualizing and Montessori Math PART 1
IMF: Visualizing and Montessori Math PART 1
IMF: Visualizing and Montessori Math PART 1
IMF: Visualizing and Montessori Math PART 1
IMF: Visualizing and Montessori Math PART 1
IMF: Visualizing and Montessori Math PART 1
IMF: Visualizing and Montessori Math PART 1
IMF: Visualizing and Montessori Math PART 1
IMF: Visualizing and Montessori Math PART 1
IMF: Visualizing and Montessori Math PART 1
IMF: Visualizing and Montessori Math PART 1
IMF: Visualizing and Montessori Math PART 1
IMF: Visualizing and Montessori Math PART 1
IMF: Visualizing and Montessori Math PART 1
IMF: Visualizing and Montessori Math PART 1
IMF: Visualizing and Montessori Math PART 1
IMF: Visualizing and Montessori Math PART 1
IMF: Visualizing and Montessori Math PART 1
IMF: Visualizing and Montessori Math PART 1
IMF: Visualizing and Montessori Math PART 1
IMF: Visualizing and Montessori Math PART 1
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  • Transcript of "IMF: Visualizing and Montessori Math PART 1"

    1. 1. How Visualization Enhances Montessori Mathematics PART 1 by Joan A. Cotter, Ph.D. JoanCotter@RightStartMath.com Montessori Foundation 30 30 Conference 77 Friday, Nov 2, 2012 Sarasota, Florida 30 370 7 1000 100 10 1 7 7 7 3 3 3 PowerPoint PresentationRightStartMath.com >Resources © Joan A. Cotter, Ph.D., 2012
    2. 2. Counting ModelIn Montessori, counting is pervasive: • Number Rods • Spindle Boxes • Decimal materials • Snake Game • Dot Game • Stamp Game • Multiplication Board • Bead Frame © Joan A. Cotter, Ph.D., 2012
    3. 3. Verbal Counting Model From a childs perspective © Joan A. Cotter, Ph.D., 2012
    4. 4. Verbal Counting Model From a childs perspectiveBecause we’re so familiar with 1, 2, 3, we’ll use letters. A=1 B=2 C=3 D=4 E = 5, and so forth © Joan A. Cotter, Ph.D., 2012
    5. 5. Verbal Counting Model From a childs perspective F +E © Joan A. Cotter, Ph.D., 2012
    6. 6. Verbal Counting Model From a childs perspective F +EA © Joan A. Cotter, Ph.D., 2012
    7. 7. Verbal Counting Model From a childs perspective F +EA B © Joan A. Cotter, Ph.D., 2012
    8. 8. Verbal Counting Model From a childs perspective F +EA B C © Joan A. Cotter, Ph.D., 2012
    9. 9. Verbal Counting Model From a childs perspective F +EA B C D E F © Joan A. Cotter, Ph.D., 2012
    10. 10. Verbal Counting Model From a childs perspective F +EA B C D E F A © Joan A. Cotter, Ph.D., 2012
    11. 11. Verbal Counting Model From a childs perspective F +EA B C D E F A B © Joan A. Cotter, Ph.D., 2012
    12. 12. Verbal Counting Model From a childs perspective F +EA B C D E F A B C D E © Joan A. Cotter, Ph.D., 2012
    13. 13. Verbal Counting Model From a childs perspective F +EA B C D E F A B C D E What is the sum? (It must be a letter.) © Joan A. Cotter, Ph.D., 2012
    14. 14. Verbal Counting Model From a childs perspective F +E KA B C D E F G H I J K © Joan A. Cotter, Ph.D., 2012
    15. 15. Verbal Counting Model From a childs perspective Now memorize the facts!! G +D © Joan A. Cotter, Ph.D., 2012
    16. 16. Verbal Counting Model From a childs perspective Now memorize the facts!! H + G F +D © Joan A. Cotter, Ph.D., 2012
    17. 17. Verbal Counting Model From a childs perspective Now memorize the facts!! H + G F +D D +C © Joan A. Cotter, Ph.D., 2012
    18. 18. Verbal Counting Model From a childs perspective Now memorize the facts!! H + G F +D D C +C +G © Joan A. Cotter, Ph.D., 2012
    19. 19. Verbal Counting Model From a childs perspective Now memorize the facts!! H +E G F I+ +D D C +C +G © Joan A. Cotter, Ph.D., 2012
    20. 20. Verbal Counting Model From a childs perspective H –ESubtract with your fingers by counting backward. © Joan A. Cotter, Ph.D., 2012
    21. 21. Verbal Counting Model From a childs perspective J –FSubtract without using your fingers. © Joan A. Cotter, Ph.D., 2012
    22. 22. Verbal Counting Model From a childs perspectiveTry skip counting by B’s to T: B, D, . . . T. © Joan A. Cotter, Ph.D., 2012
    23. 23. Verbal Counting Model From a childs perspectiveTry skip counting by B’s to T: B, D, . . . T.What is D × E? © Joan A. Cotter, Ph.D., 2012
    24. 24. Verbal Counting Model From a childs perspectiveLis written ABbecause it is A Jand B A’s © Joan A. Cotter, Ph.D., 2012
    25. 25. Verbal Counting Model From a childs perspectiveLis written ABbecause it is A Jand B A’s huh? © Joan A. Cotter, Ph.D., 2012
    26. 26. Verbal Counting Model From a childs perspectiveL (twelve)is written ABbecause it is A Jand B A’s © Joan A. Cotter, Ph.D., 2012
    27. 27. Verbal Counting Model From a childs perspectiveL (twelve)is written AB (12)because it is A Jand B A’s © Joan A. Cotter, Ph.D., 2012
    28. 28. Verbal Counting Model From a childs perspectiveL (twelve)is written AB (12)because it is A J (one 10)and B A’s © Joan A. Cotter, Ph.D., 2012
    29. 29. Verbal Counting Model From a childs perspectiveL (twelve)is written AB (12)because it is A J (one 10)and B A’s (two 1s). © Joan A. Cotter, Ph.D., 2012
    30. 30. Calendar Math © Joan A. Cotter, Ph.D., 2012
    31. 31. Calendar Math August1 2 3 4 5 6 78 9 10 11 12 13 1415 16 17 18 19 20 2122 23 24 25 26 27 2829 30 31 © Joan A. Cotter, Ph.D., 2012
    32. 32. Calendar Math Calendar Counting August1 2 3 4 5 6 78 9 10 11 12 13 1415 16 17 18 19 20 2122 23 24 25 26 27 2829 30 31 © Joan A. Cotter, Ph.D., 2012
    33. 33. Calendar Math Calendar Counting August1 2 3 4 5 6 78 9 10 11 12 13 1415 16 17 18 19 20 2122 23 24 25 26 27 2829 30 31 © Joan A. Cotter, Ph.D., 2012
    34. 34. Calendar Math Calendar Counting August1 2 3 4 5 6 78 9 10 11 12 13 1415 16 17 18 19 20 2122 23 24 25 26 27 2829 30 31 © Joan A. Cotter, Ph.D., 2012
    35. 35. Calendar Math Septemb Calendar Counting1234567 August891012141 2 113 11921151126288 122820 67527 9 3 4 5 6 10 11 12 13 14 72234 2015 16 17 18 19 20 2129 322 23 24 25 26 27 2829 30 31 © Joan A. Cotter, Ph.D., 2012
    36. 36. Calendar Math Septemb Calendar Counting 1234567 August 89101214 1 113 11921 2 15112628 122820 8 67527 9 3 4 5 6 10 11 12 13 14 7 2234 20 15 16 17 18 19 20 21 29 3 22 23 24 25 26 27 28 29 30 31This is ordinal counting, not cardinal counting. © Joan A. Cotter, Ph.D., 2012
    37. 37. Calendar Math Partial Calendar August1 2 3 4 5 6 78 9 10 © Joan A. Cotter, Ph.D., 2012
    38. 38. Calendar Math Partial Calendar August 1 2 3 4 5 6 7 8 9 10Children need the whole month to plan ahead. © Joan A. Cotter, Ph.D., 2012
    39. 39. Calendar Math Septemb Calendar patterning 1234567 August 89101214 1 2 113 11921 15112628 8 122820 67527 9 3 4 5 6 10 11 12 13 14 7 2234 20 15 16 17 18 19 20 21 29 3 22 23 24 25 26 27 28 29 30 31Patterns are rarely based on 7s or proceed row by row.Patterns go on forever; they don’t stop at 31. © Joan A. Cotter, Ph.D., 2012
    40. 40. Research on Counting Karen Wynn’s research © Joan A. Cotter, Ph.D., 2012
    41. 41. Research on Counting Karen Wynn’s research © Joan A. Cotter, Ph.D., 2012
    42. 42. Research on Counting Karen Wynn’s research © Joan A. Cotter, Ph.D., 2012
    43. 43. Research on Counting Karen Wynn’s research © Joan A. Cotter, Ph.D., 2012
    44. 44. Research on Counting Karen Wynn’s research © Joan A. Cotter, Ph.D., 2012
    45. 45. Research on Counting Karen Wynn’s research © Joan A. Cotter, Ph.D., 2012
    46. 46. Research on Counting Karen Wynn’s research © Joan A. Cotter, Ph.D., 2012
    47. 47. Research on Counting Karen Wynn’s research © Joan A. Cotter, Ph.D., 2012
    48. 48. Research on Counting Other research © Joan A. Cotter, Ph.D., 2012
    49. 49. Research on Counting Other research• Australian Aboriginal children from two tribes. Brian Butterworth, University College London, 2008. © Joan A. Cotter, Ph.D., 2012
    50. 50. 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. © Joan A. Cotter, Ph.D., 2012
    51. 51. 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. © Joan A. Cotter, Ph.D., 2012
    52. 52. 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. © Joan A. Cotter, Ph.D., 2012
    53. 53. Research on Counting In Japanese schools:• Children are discouraged from usingcounting for adding. © Joan A. Cotter, Ph.D., 2012
    54. 54. Research on Counting In Japanese schools:• Children are discouraged from usingcounting for adding.• They consistently group in 5s. © Joan A. Cotter, Ph.D., 2012
    55. 55. Subitizing Quantities(Identifying without counting) © Joan A. Cotter, Ph.D., 2012
    56. 56. Subitizing Quantities (Identifying without counting)• Five-month-old infants can subitize to 3. © Joan A. Cotter, Ph.D., 2012
    57. 57. Subitizing Quantities (Identifying without counting)• Five-month-old infants can subitize to 3.• Three-year-olds can subitize to 5. © Joan A. Cotter, Ph.D., 2012
    58. 58. Subitizing Quantities (Identifying without counting)• Five-month-old infants can subitize to 3.• Three-year-olds can subitize to 5.• Four-year-olds can subitize 6 to 10 byusing five as a subbase. © Joan A. Cotter, Ph.D., 2012
    59. 59. Subitizing Quantities (Identifying without counting)• Five-month-old infants can subitize to 3.• Three-year-olds can subitize to 5.• Four-year-olds can subitize 6 to 10 byusing five as a subbase.• Counting is like sounding out each letter;subitizing is recognizing the quantity. © Joan A. Cotter, Ph.D., 2012
    60. 60. Research on Counting Subitizing• Subitizing “allows the child to grasp the wholeand the elements at the same time.”—Benoit © Joan A. Cotter, Ph.D., 2012
    61. 61. Research on Counting Subitizing• Subitizing “allows the child to grasp the wholeand the elements at the same time.”—Benoit• Subitizing seems to be a necessary skill forunderstanding what the counting process means.—Glasersfeld © Joan A. Cotter, Ph.D., 2012
    62. 62. Research on Counting Subitizing• Subitizing “allows the child to grasp the wholeand the elements at the same time.”—Benoit• Subitizing seems to be a necessary skill forunderstanding what the counting process means.—Glasersfeld• Children who can subitize perform better inmathematics long term.—Butterworth © Joan A. Cotter, Ph.D., 2012
    63. 63. Research on Counting Subitizing• Subitizing “allows the child to grasp the wholeand the elements at the same time.”—Benoit• Subitizing seems to be a necessary skill forunderstanding what the counting process means.—Glasersfeld• Children who can subitize perform better inmathematics long term.—Butterworth• Counting-on is a difficult skill for many children.—Journal for Res. in Math Ed. Nov. 2011 © Joan A. Cotter, Ph.D., 2012
    64. 64. Research on Counting Subitizing• Subitizing “allows the child to grasp the wholeand the elements at the same time.”—Benoit• Subitizing seems to be a necessary skill forunderstanding what the counting process means.—Glasersfeld• Children who can subitize perform better inmathematics long term.—Butterworth• Counting-on is a difficult skill for many children.—Journal for Res. in Math Ed. Nov. 2011• Math anxiety affects counting ability, butnot subitizing ability. © Joan A. Cotter, Ph.D., 2012
    65. 65. Visualizing Quantities © Joan A. Cotter, Ph.D., 2012
    66. 66. Visualizing Quantities“Think in pictures, because thebrain remembers images betterthan it does anything else.” Ben Pridmore, World Memory Champion, 2009 © Joan A. Cotter, Ph.D., 2012
    67. 67. Visualizing Quantities“The role of physical manipulativeswas to help the child form thosevisual images and thus to eliminatethe need for the physicalmanipulatives.” Ginsberg and others © Joan A. Cotter, Ph.D., 2012
    68. 68. Visualizing Quantities 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
    69. 69. Visualizing Quantities Visualizing also needed in:• Reading• Sports• Creativity• Geography• Engineering• Construction © Joan A. Cotter, Ph.D., 2012
    70. 70. Visualizing Quantities Visualizing also needed in:• Reading • Architecture• Sports • Astronomy• Creativity • Archeology• Geography • Chemistry• Engineering • Physics• Construction • Surgery © Joan A. Cotter, Ph.D., 2012
    71. 71. Visualizing Quantities Ready: How many? © Joan A. Cotter, Ph.D., 2012
    72. 72. Visualizing Quantities Ready: How many? © Joan A. Cotter, Ph.D., 2012
    73. 73. Visualizing Quantities Try again: How many? © Joan A. Cotter, Ph.D., 2012
    74. 74. Visualizing Quantities Try again: How many? © Joan A. Cotter, Ph.D., 2012
    75. 75. Visualizing QuantitiesTry to visualize 8 identical apples without grouping. © Joan A. Cotter, Ph.D., 2012
    76. 76. Visualizing QuantitiesTry to visualize 8 identical apples without grouping. © Joan A. Cotter, Ph.D., 2012
    77. 77. Visualizing QuantitiesNow try to visualize 5 as red and 3 as green. © Joan A. Cotter, Ph.D., 2012
    78. 78. Visualizing QuantitiesNow try to visualize 5 as red and 3 as green. © Joan A. Cotter, Ph.D., 2012
    79. 79. Visualizing Quantities Early Roman numerals 1 I 2 II 3 III 4 IIII 5 V 8 VIII © Joan A. Cotter, Ph.D., 2012
    80. 80. Visualizing Quantities : Who could read the music? © Joan A. Cotter, Ph.D., 2012
    81. 81. Grouping in Fives © Joan A. Cotter, Ph.D., 2012
    82. 82. Grouping in Fives• Grouping in fives extends subitizing. © Joan A. Cotter, Ph.D., 2012
    83. 83. Grouping in Fives Using fingersGrouping in Fives is a three-period lesson. © Joan A. Cotter, Ph.D., 2012
    84. 84. Grouping in Fives Using fingersGrouping in Fives is a three-period lesson. © Joan A. Cotter, Ph.D., 2012
    85. 85. Grouping in Fives Using fingersGrouping in Fives is a three-period lesson. © Joan A. Cotter, Ph.D., 2012
    86. 86. Grouping in Fives Using fingersGrouping in Fives is a three-period lesson. © Joan A. Cotter, Ph.D., 2012
    87. 87. Grouping in Fives Using fingersGrouping in Fives is a three-period lesson. © Joan A. Cotter, Ph.D., 2012
    88. 88. Grouping in Fives 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. Cotter © Joan A. Cotter, Ph.D., 2012
    89. 89. Grouping in Fives Recognizing 5 © Joan A. Cotter, Ph.D., 2012
    90. 90. Grouping in Fives Recognizing 5 © Joan A. Cotter, Ph.D., 2012
    91. 91. Grouping in Fives Recognizing 55 has a middle; 4 does not. © Joan A. Cotter, Ph.D., 2012
    92. 92. Grouping in Fives Tally sticks © Joan A. Cotter, Ph.D., 2012
    93. 93. Grouping in Fives Tally sticks © Joan A. Cotter, Ph.D., 2012
    94. 94. Grouping in Fives Tally sticks © Joan A. Cotter, Ph.D., 2012
    95. 95. Grouping in Fives Tally sticks © Joan A. Cotter, Ph.D., 2012
    96. 96. Grouping in Fives Tally sticks © Joan A. Cotter, Ph.D., 2012
    97. 97. Grouping in Fives Tally sticks © Joan A. Cotter, Ph.D., 2012
    98. 98. Grouping in Fives Pairing Finger Cards QuickTime™ and a QuickTime™ and a QuickTime™ and a QuickTime™ and a QuickTime™ and a TIFF (LZW) decompressor TIFF (LZW) decompressor TIFF (LZW) decompressor TIFF (LZW) decompressor TIFF (LZW) decompressorare 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. QuickTime™ and a TIFF (LZW) decompressor are needed to see this picture. QuickTime™ and a QuickTime™ and a QuickTime™ and a QuickTime™ and a QuickTime™ and a TIFF (LZW) decompressor TIFF (LZW) decompressor TIFF (LZW) decompressor TIFF (LZW) decompressor TIFF (LZW) decompressorare 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. QuickTime™ and a TIFF (LZW) decompressor are needed to see this picture. QuickTime™ and aa TIFFQuickTime™and aa QuickTime™ and QuickTime™ and TIFF(LZW) decompressor areTIFF (LZW) decompressor TIFF (LZW) decompressor are needed toto seethisa picture. needed(LZW)seedecompressor see this (LZW) and QuickTime™ are needed toseedecompressorpicture. are neededto seethis picture. TIFF to are needed this picture. picture. this © Joan A. Cotter, Ph.D., 2012
    99. 99. Grouping in Fives Ordering Finger Cards QuickTime™ and a TIFF (LZW) decompressorare needed to see this picture. QuickTime™ and a TIFF (LZW) decompressor QuickTime™ and a are needed to see this picture. TIFF (LZW) decompressor are needed to see this picture. QuickTime™ and a TIFF (LZW) decompressorare needed to see this picture. QuickTime™ and a TIFF (LZW) decompressor are needed to see this picture. QuickTime™ and a TIFF (LZW) decompressor are needed to see this picture. QuickTime™ and a TIFF (LZW) decompressor are needed to see this picture. QuickTime™ and a TIFF (LZW) decompressor are needed to see this picture. QuickTime™ and a TIFF (LZW) decompressor are needed to see this picture. QuickTime™ and a TIFF (LZW) decompressor are needed to see this picture. © Joan A. Cotter, Ph.D., 2012
    100. 100. Grouping in Fives Matching Number Cards to Finger Cards QuickTime™ and a QuickTime™ and a QuickTime™ and a QuickTime™ and a QuickTime™ and a TIFF (LZW) decompressor TIFF (LZW) decompressor TIFF (LZW) decompressor TIFF (LZW) decompressor TIFF (LZW) decompressorare 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 QuickTime™ and a QuickTime™ and a QuickTime™ and a QuickTime™ and a QuickTime™ and a TIFF (LZW) decompressor TIFF (LZW) decompressor TIFF (LZW) decompressor TIFF (LZW) decompressor TIFF (LZW) decompressorare 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. 10 © Joan A. Cotter, Ph.D., 2012
    101. 101. Grouping in FivesMatching Finger Cards to Number Cards9 1 10 4 6 QuickTime™ and a TIFF (LZW) decompressor are needed to see this picture.2 3 7 8 5 QuickTime™ and a TIFF (LZW) decompressor are needed to see this picture. QuickTime™ and aa QuickTime™ and a QuickTime™ and a TIFF (LZW) decompressor a QuickTime™ and TIFF (LZW) decompressor QuickTime™ and are needed (LZW)this picture. a TIFF (LZW)decompressor QuickTime™ and are neededtotosee this picture. TIFF tosee decompressor are needed (LZW)decompressor TIFF (LZW)this picture. are needed tosee this picture. TIFF see decompressor are needed to see this picture. are needed to see this picture. © Joan A. Cotter, Ph.D., 2012
    102. 102. Grouping in Fives Finger Card Memory game QuickTime™ and a QuickTime™ and a QuickTime™ and a QuickTime™ 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. QuickTime™ and a QuickTime™ and a QuickTime™ and a QuickTime™ 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. QuickTime™ and a QuickTime™ and a QuickTime™ and a QuickTime™ 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. QuickTime™ and a QuickTime™ and a QuickTime™ and a QuickTime™ 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. QuickTime™ and a QuickTime™ and a QuickTime™ and a QuickTime™ and a TIFF (LZW) decompressor TIFF (LZW) decompressor TIFF (LZW) decompressor TIFF (LZW) decompressorare needed to see this picture. are needed to see this picture. are needed to see this picture. are needed to see this picture. © Joan A. Cotter, Ph.D., 2012
    103. 103. Grouping in Fives Number Rods © Joan A. Cotter, Ph.D., 2012
    104. 104. Grouping in Fives Number Rods © Joan A. Cotter, Ph.D., 2012
    105. 105. Grouping in Fives Number Rods © Joan A. Cotter, Ph.D., 2012
    106. 106. Grouping in Fives Spindle Box © Joan A. Cotter, Ph.D., 2012
    107. 107. Grouping in Fives Spindle Box © Joan A. Cotter, Ph.D., 2012
    108. 108. Grouping in Fives Spindle Box0 1 2 3 4 © Joan A. Cotter, Ph.D., 2012
    109. 109. Grouping in Fives Spindle Box5 6 7 8 9 © Joan A. Cotter, Ph.D., 2012
    110. 110. Grouping in Fives Spindle Box5 6 7 8 9 © Joan A. Cotter, Ph.D., 2012
    111. 111. Grouping in Fives Spindle Box5 6 7 8 9 © Joan A. Cotter, Ph.D., 2012
    112. 112. Grouping in Fives Spindle Box5 6 7 8 9 © Joan A. Cotter, Ph.D., 2012
    113. 113. Grouping in Fives Spindle Box5 6 7 8 9 © Joan A. Cotter, Ph.D., 2012
    114. 114. Grouping in Fives Spindle Box5 6 7 8 9 © Joan A. Cotter, Ph.D., 2012
    115. 115. Grouping in Fives 1000 100 10 1 1000 100 10 1 1000 100 10 1 1000 100 10 1 100 10 1 100 10 1 100 10 1 100 1Stamp Game © Joan A. Cotter, Ph.D., 2012
    116. 116. Grouping in Fives 1000 100 10 1 1000 100 10 1 1000 100 10 1 1000 100 10 1 100 10 1 100 10 1 100 10 1 100 1Stamp Game © Joan A. Cotter, Ph.D., 2012
    117. 117. Grouping in Fives 1000 1000 100 100 10 10 1 1 1000 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 100Stamp Game © Joan A. Cotter, Ph.D., 2012
    118. 118. Grouping in Fives 1000 1000 100 100 10 1 1 1000 1000 100 100 1 1 10 10 100 100 1 1 10 10 100 100 1 1 10 10 100 100 1 1 10 10 100 100 10 10 100 100 100 100Stamp Game © Joan A. Cotter, Ph.D., 2012
    119. 119. Grouping in Fives 1000 1000 100 100 10 1 1 1000 1000 100 100 1 1 10 10 100 100 1 1 10 10 1 1 100 100 10 10 1 1 100 100 10 10 100 100 10 10 100 100Stamp Game 100 100 © Joan A. Cotter, Ph.D., 2012
    120. 120. Grouping in Fives Black and White Bead Stairs“Grouped in fives so the child does notneed to count.” A. M. Joosten © Joan A. Cotter, Ph.D., 2012
    121. 121. Grouping in Fives Entering quantities © Joan A. Cotter, Ph.D., 2012
    122. 122. Grouping in Fives Entering quantities3 © Joan A. Cotter, Ph.D., 2012
    123. 123. Grouping in Fives Entering quantities5 © Joan A. Cotter, Ph.D., 2012
    124. 124. Grouping in Fives Entering quantities7 © Joan A. Cotter, Ph.D., 2012
    125. 125. Grouping in Fives Entering quantities10 © Joan A. Cotter, Ph.D., 2012
    126. 126. Grouping in Fives The stairs © Joan A. Cotter, Ph.D., 2012
    127. 127. Grouping in Fives Adding © Joan A. Cotter, Ph.D., 2012
    128. 128. Grouping in Fives Adding 4+3= © Joan A. Cotter, Ph.D., 2012
    129. 129. Grouping in Fives Adding 4+3= © Joan A. Cotter, Ph.D., 2012
    130. 130. Grouping in Fives Adding 4+3= © Joan A. Cotter, Ph.D., 2012
    131. 131. Grouping in Fives Adding 4+3= © Joan A. Cotter, Ph.D., 2012
    132. 132. Grouping in Fives Adding 4+3=7 © Joan A. Cotter, Ph.D., 2012
    133. 133. Math Card Games © Joan A. Cotter, Ph.D., 2012
    134. 134. Math Card Games• Provide repetition for learning the facts. © Joan A. Cotter, Ph.D., 2012
    135. 135. Math Card Games• Provide repetition for learning the facts.• Encourage autonomy. © Joan A. Cotter, Ph.D., 2012
    136. 136. Math Card Games• Provide repetition for learning the facts.• Encourage autonomy.• Promote social interaction. © Joan A. Cotter, Ph.D., 2012
    137. 137. Math Card Games• Provide repetition for learning the facts.• Encourage autonomy.• Promote social interaction.• Are enjoyed by the children. © Joan A. Cotter, Ph.D., 2012
    138. 138. Go to the Dump GameObjective: To learn the facts that total 10: 1+9 2+8 3+7 4+6 5+5 © Joan A. Cotter, Ph.D., 2012
    139. 139. Go to the Dump GameObjective: To learn the facts that total 10: 1+9 2+8 3+7 4+6 5+5Object of the game: To collect the most pairs that equal ten. © Joan A. Cotter, Ph.D., 2012
    140. 140. “Math” Way of Naming Numbers © Joan A. Cotter, Ph.D., 2012
    141. 141. “Math” Way of Naming Numbers 11 = ten 1 © Joan A. Cotter, Ph.D., 2012
    142. 142. “Math” Way of Naming Numbers 11 = ten 1 12 = ten 2 © Joan A. Cotter, Ph.D., 2012
    143. 143. “Math” Way of Naming Numbers 11 = ten 1 12 = ten 2 13 = ten 3 © Joan A. Cotter, Ph.D., 2012
    144. 144. “Math” Way of Naming Numbers 11 = ten 1 12 = ten 2 13 = ten 3 14 = ten 4 © Joan A. Cotter, Ph.D., 2012
    145. 145. “Math” Way of Naming Numbers 11 = ten 1 12 = ten 2 13 = ten 3 14 = ten 4 .... 19 = ten 9 © Joan A. Cotter, Ph.D., 2012
    146. 146. “Math” Way of Naming Numbers 11 = ten 1 20 = 2-ten 12 = ten 2 13 = ten 3 14 = ten 4 .... 19 = ten 9 © Joan A. Cotter, Ph.D., 2012
    147. 147. “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 9 © Joan A. Cotter, Ph.D., 2012
    148. 148. “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 9 © Joan A. Cotter, Ph.D., 2012
    149. 149. “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 © Joan A. Cotter, Ph.D., 2012
    150. 150. “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 9 © Joan A. Cotter, Ph.D., 2012
    151. 151. “Math” Way of Naming Numbers 137 = 1 hundred 3-ten 7 © Joan A. Cotter, Ph.D., 2012
    152. 152. “Math” Way of Naming Numbers 137 = 1 hundred 3-ten 7 or 137 = 1 hundred and 3-ten 7 © Joan A. Cotter, Ph.D., 2012
    153. 153. “Math” Way of Naming Numbers 100 Chinese U.S. Average Highest Number Counted 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. © Joan A. Cotter, Ph.D., 2012
    154. 154. “Math” Way of Naming Numbers 100 Chinese U.S. Average Highest Number Counted 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. © Joan A. Cotter, Ph.D., 2012
    155. 155. “Math” Way of Naming Numbers 100 Chinese U.S. Average Highest Number Counted 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. © Joan A. Cotter, Ph.D., 2012
    156. 156. “Math” Way of Naming Numbers 100 Chinese U.S. Average Highest Number Counted 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. © Joan A. Cotter, Ph.D., 2012
    157. 157. “Math” Way of Naming Numbers 100 Chinese U.S. Average Highest Number Counted 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. © Joan A. Cotter, Ph.D., 2012
    158. 158. Math Way of Naming Numbers • Only 11 words are needed to count to 100 the math way, 28 in English. (All Indo-European languages are non-standard in number naming.) © Joan A. Cotter, Ph.D., 2012
    159. 159. Math Way of Naming Numbers • Only 11 words are needed to count to 100 the math way, 28 in English. (All Indo-European languages are non-standard in number naming.) • Asian children learn mathematics using the math way of counting. © Joan A. Cotter, Ph.D., 2012
    160. 160. Math Way of Naming Numbers • Only 11 words are needed to count to 100 the math way, 28 in English. (All Indo-European languages are non-standard in number naming.) • Asian children learn mathematics using the math way of counting. • They understand place value in first grade; only half of U.S. children understand place value at the end of fourth grade. © Joan A. Cotter, Ph.D., 2012
    161. 161. Math Way of Naming Numbers • Only 11 words are needed to count to 100 the math way, 28 in English. (All Indo-European languages are non-standard in number naming.) • Asian children learn mathematics using the math way of counting. • They understand place value in first grade; only half of U.S. children understand place value at the end of fourth grade. • Mathematics is the science of patterns. The patterned math way of counting greatly helps children learn number sense. © Joan A. Cotter, Ph.D., 2012
    162. 162. Math Way of Naming Numbers Compared to reading: © Joan A. Cotter, Ph.D., 2012
    163. 163. Math Way of Naming Numbers Compared to reading:• Just as reciting the alphabet doesn’t teach reading,counting doesn’t teach arithmetic. © Joan A. Cotter, Ph.D., 2012
    164. 164. 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 mustfirst teach the name of the quantity (math way). © Joan A. Cotter, Ph.D., 2012
    165. 165. 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 mustfirst teach the name of the quantity (math way).• Montessorians need to use the math way of namingnumbers for a longer period of time. © Joan A. Cotter, Ph.D., 2012
    166. 166. Math Way of Naming Numbers“Rather, the increased gap between Chinese andU.S. students and that of Chinese Americans andCaucasian Americans may be due primarily to thenature of their initial gap prior to formal schooling,such as counting efficiency and base-ten numbersense.” Jian Wang and Emily Lin, 2005 Researchers © Joan A. Cotter, Ph.D., 2012
    167. 167. Math Way of Naming Numbers Traditional names4-ten =fortyThe “ty”means tens. © Joan A. Cotter, Ph.D., 2012
    168. 168. Math Way of Naming Numbers Traditional names4-ten =fortyThe “ty”means tens. © Joan A. Cotter, Ph.D., 2012
    169. 169. Math Way of Naming Numbers Traditional names6-ten = sixtyThe “ty”means tens. © Joan A. Cotter, Ph.D., 2012
    170. 170. Math Way of Naming Numbers Traditional names3-ten = thirty“Thir” alsoused in 1/3,13 and 30. © Joan A. Cotter, Ph.D., 2012
    171. 171. Math Way of Naming Numbers Traditional names5-ten = fifty“Fif” alsoused in 1/5,15 and 50. © Joan A. Cotter, Ph.D., 2012
    172. 172. Math Way of Naming Numbers Traditional names2-ten = twentyTwo used to bepronounced“twoo.” © Joan A. Cotter, Ph.D., 2012
    173. 173. Math Way of Naming Numbers Traditional names A word game fireplace place-fire © Joan A. Cotter, Ph.D., 2012
    174. 174. Math Way of Naming Numbers Traditional names A word game fireplace place-fire newspaper paper-news © Joan A. Cotter, Ph.D., 2012
    175. 175. Math Way of Naming Numbers Traditional names A word game fireplace place-fire newspaper paper-news box-mail mailbox © Joan A. Cotter, Ph.D., 2012
    176. 176. Math Way of Naming Numbers Traditional names ten 4“Teen” alsomeans ten. © Joan A. Cotter, Ph.D., 2012
    177. 177. Math Way of Naming Numbers Traditional names ten 4 teen 4“Teen” alsomeans ten. © Joan A. Cotter, Ph.D., 2012
    178. 178. Math Way of Naming Numbers Traditional names ten 4 teen 4 fourtee n“Teen” alsomeans ten. © Joan A. Cotter, Ph.D., 2012
    179. 179. Math Way of Naming Numbers Traditional names a one left © Joan A. Cotter, Ph.D., 2012
    180. 180. Math Way of Naming Numbers Traditional names a one left a left-one © Joan A. Cotter, Ph.D., 2012
    181. 181. Math Way of Naming Numbers Traditional names a one left a left-one eleven © Joan A. Cotter, Ph.D., 2012
    182. 182. Math Way of Naming Numbers Traditional names two leftTwo saidas “twoo.” © Joan A. Cotter, Ph.D., 2012
    183. 183. Math Way of Naming Numbers Traditional names two left twelveTwo saidas “twoo.” © Joan A. Cotter, Ph.D., 2012
    184. 184. Composing Numbers3-ten © Joan A. Cotter, Ph.D., 2012
    185. 185. Composing Numbers3-ten © Joan A. Cotter, Ph.D., 2012
    186. 186. Composing Numbers3-ten3030 © Joan A. Cotter, Ph.D., 2012
    187. 187. Composing Numbers3-ten3030 © Joan A. Cotter, Ph.D., 2012
    188. 188. Composing Numbers3-ten3030 © Joan A. Cotter, Ph.D., 2012
    189. 189. Composing Numbers3-ten 73030 © Joan A. Cotter, Ph.D., 2012
    190. 190. Composing Numbers3-ten 73030 © Joan A. Cotter, Ph.D., 2012
    191. 191. Composing Numbers3-ten 73030 7 7 © Joan A. Cotter, Ph.D., 2012
    192. 192. Composing Numbers3-ten 73037 0 7 © Joan A. Cotter, Ph.D., 2012
    193. 193. Composing Numbers 3-ten 7 30 37 0 7Note the congruence in how we say the number,represent the number, and write the number. © Joan A. Cotter, Ph.D., 2012
    194. 194. Composing Numbers1-ten1010 Another example. © Joan A. Cotter, Ph.D., 2012
    195. 195. Composing Numbers1-ten 81010 © Joan A. Cotter, Ph.D., 2012
    196. 196. Composing Numbers1-ten 81010 © Joan A. Cotter, Ph.D., 2012
    197. 197. Composing Numbers1-ten 81010 8 8 © Joan A. Cotter, Ph.D., 2012
    198. 198. Composing Numbers1-ten 81818 © Joan A. Cotter, Ph.D., 2012
    199. 199. Composing Numbers10-ten © Joan A. Cotter, Ph.D., 2012
    200. 200. Composing Numbers10-ten100100 © Joan A. Cotter, Ph.D., 2012
    201. 201. Composing Numbers10-ten100100 © Joan A. Cotter, Ph.D., 2012
    202. 202. Composing Numbers10-ten100100 © Joan A. Cotter, Ph.D., 2012
    203. 203. Composing Numbers1 hundred © Joan A. Cotter, Ph.D., 2012
    204. 204. Composing Numbers1 hundred100100 © Joan A. Cotter, Ph.D., 2012
    205. 205. Composing Numbers1 hundred100100 © Joan A. Cotter, Ph.D., 2012
    206. 206. Composing Numbers1 hundred100100 © Joan A. Cotter, Ph.D., 2012
    207. 207. Composing Numbers1 hundred100100 © Joan A. Cotter, Ph.D., 2012
    208. 208. Composing Numbers2 hundred © Joan A. Cotter, Ph.D., 2012
    209. 209. Composing Numbers2 hundred © Joan A. Cotter, Ph.D., 2012
    210. 210. Composing Numbers2 hundred200200 © Joan A. Cotter, Ph.D., 2012
    211. 211. Evens and Odds © Joan A. Cotter, Ph.D., 2012
    212. 212. Evens and Odds Evens © Joan A. Cotter, Ph.D., 2012
    213. 213. Evens and Odds Evens Use two fingers and touch each pair in succession. © Joan A. Cotter, Ph.D., 2012
    214. 214. Evens and Odds Evens Use two fingers and touch each pair in succession. © Joan A. Cotter, Ph.D., 2012
    215. 215. Evens and Odds Evens Use two fingers and touch each pair in succession. © Joan A. Cotter, Ph.D., 2012
    216. 216. Evens and Odds Evens Use two fingers and touch each pair in succession. EVEN! © Joan A. Cotter, Ph.D., 2012
    217. 217. Evens and Odds Odds Use two fingers and touch each pair in succession. © Joan A. Cotter, Ph.D., 2012
    218. 218. Evens and Odds Odds Use two fingers and touch each pair in succession. © Joan A. Cotter, Ph.D., 2012
    219. 219. Evens and Odds Odds Use two fingers and touch each pair in succession. © Joan A. Cotter, Ph.D., 2012
    220. 220. Evens and Odds Odds Use two fingers and touch each pair in succession. © Joan A. Cotter, Ph.D., 2012
    221. 221. Evens and Odds Odds Use two fingers and touch each pair in succession. ODD! © Joan A. Cotter, Ph.D., 2012
    222. 222. Learning the Facts © Joan A. Cotter, Ph.D., 2012
    223. 223. Learning the FactsLimited success when:• Based on counting. Whether dots, fingers, number lines, or counting words. © Joan A. Cotter, Ph.D., 2012
    224. 224. Learning the FactsLimited success when:• Based on counting. Whether dots, fingers, number lines, or counting words.• Based on rote memory. Whether by flash cards or timed tests. © Joan A. Cotter, Ph.D., 2012
    225. 225. Learning the FactsLimited success when:• Based on counting. Whether dots, fingers, number lines, or counting words.• Based on rote memory. Whether by flash cards or timed tests.• Based on skip counting for multiplication facts. © Joan A. Cotter, Ph.D., 2012
    226. 226. Fact Strategies © Joan A. Cotter, Ph.D., 2012
    227. 227. Fact Strategies Complete the Ten9+5= © Joan A. Cotter, Ph.D., 2012
    228. 228. Fact Strategies Complete the Ten9+5= © Joan A. Cotter, Ph.D., 2012
    229. 229. Fact Strategies Complete the Ten9+5= © Joan A. Cotter, Ph.D., 2012
    230. 230. Fact Strategies Complete the Ten 9+5=Take 1 fromthe 5 and giveit to the 9. © Joan A. Cotter, Ph.D., 2012
    231. 231. Fact Strategies Complete the Ten 9+5=Take 1 fromthe 5 and giveit to the 9. © Joan A. Cotter, Ph.D., 2012
    232. 232. Fact Strategies Complete the Ten 9+5=Take 1 fromthe 5 and giveit to the 9. © Joan A. Cotter, Ph.D., 2012
    233. 233. Fact Strategies Complete the Ten 9 + 5 = 14Take 1 fromthe 5 and giveit to the 9. © Joan A. Cotter, Ph.D., 2012
    234. 234. Fact Strategies Two Fives8+6= © Joan A. Cotter, Ph.D., 2012
    235. 235. Fact Strategies Two Fives8+6= © Joan A. Cotter, Ph.D., 2012
    236. 236. Fact Strategies Two Fives8+6= © Joan A. Cotter, Ph.D., 2012
    237. 237. Fact Strategies Two Fives8+6= © Joan A. Cotter, Ph.D., 2012
    238. 238. Fact Strategies Two Fives8+6=10 + 4 = 14 © Joan A. Cotter, Ph.D., 2012
    239. 239. Fact Strategies Going Down15 – 9 = © Joan A. Cotter, Ph.D., 2012
    240. 240. Fact Strategies Going Down15 – 9 = © Joan A. Cotter, Ph.D., 2012
    241. 241. Fact Strategies Going Down 15 – 9 =Subtract 5;then 4. © Joan A. Cotter, Ph.D., 2012
    242. 242. Fact Strategies Going Down 15 – 9 =Subtract 5;then 4. © Joan A. Cotter, Ph.D., 2012
    243. 243. Fact Strategies Going Down 15 – 9 =Subtract 5;then 4. © Joan A. Cotter, Ph.D., 2012
    244. 244. Fact Strategies Going Down 15 – 9 = 6Subtract 5;then 4. © Joan A. Cotter, Ph.D., 2012
    245. 245. Fact Strategies Subtract from 1015 – 9 = © Joan A. Cotter, Ph.D., 2012
    246. 246. Fact Strategies Subtract from 10 15 – 9 =Subtract 9from 10. © Joan A. Cotter, Ph.D., 2012
    247. 247. Fact Strategies Subtract from 10 15 – 9 =Subtract 9from 10. © Joan A. Cotter, Ph.D., 2012
    248. 248. Fact Strategies Subtract from 10 15 – 9 =Subtract 9from 10. © Joan A. Cotter, Ph.D., 2012
    249. 249. Fact Strategies Subtract from 10 15 – 9 = 6Subtract 9from 10. © Joan A. Cotter, Ph.D., 2012
    250. 250. Fact Strategies Going Up15 – 9 = © Joan A. Cotter, Ph.D., 2012
    251. 251. Fact Strategies Going Up 15 – 9 =Start with 9;go up to 15. © Joan A. Cotter, Ph.D., 2012
    252. 252. Fact Strategies Going Up 15 – 9 =Start with 9;go up to 15. © Joan A. Cotter, Ph.D., 2012
    253. 253. Fact Strategies Going Up 15 – 9 =Start with 9;go up to 15. © Joan A. Cotter, Ph.D., 2012
    254. 254. Fact Strategies Going Up 15 – 9 =Start with 9;go up to 15. © Joan A. Cotter, Ph.D., 2012
    255. 255. Fact Strategies Going Up 15 – 9 = 1+5=6Start with 9;go up to 15. © Joan A. Cotter, Ph.D., 2012
    256. 256. Rows and Columns GameObjective: To find a total of 15 by adding 2, 3, or 4cards in a row or in a column. © Joan A. Cotter, Ph.D., 2012
    257. 257. Rows and Columns GameObjective: To find a total of 15 by adding 2, 3, or 4cards in a row or in a column.Object of the game: To collect the most cards. © Joan A. Cotter, Ph.D., 2012
    258. 258. Rows and Columns Game 8 7 1 9 6 4 3 3 2 2 5 6 6 3 8 8 © Joan A. Cotter, Ph.D., 2012
    259. 259. Rows and Columns Game 8 7 1 9 6 4 3 3 2 2 5 6 6 3 8 8 © Joan A. Cotter, Ph.D., 2012
    260. 260. Rows and Columns Game 8 7 1 9 6 4 3 3 2 2 5 6 6 3 8 8 © Joan A. Cotter, Ph.D., 2012
    261. 261. Rows and Columns Game 1 9 6 4 3 3 6 3 8 8 © Joan A. Cotter, Ph.D., 2012
    262. 262. Rows and Columns Game 7 6 1 9 6 4 3 3 2 1 5 1 6 3 8 8 © Joan A. Cotter, Ph.D., 2012
    263. 263. Rows and Columns Game 7 6 1 9 6 4 3 3 2 1 5 1 6 3 8 8 © Joan A. Cotter, Ph.D., 2012
    264. 264. Rows and Columns Game 7 6 1 9 6 4 3 3 2 1 5 1 6 3 8 8 © Joan A. Cotter, Ph.D., 2012
    265. 265. Rows and Columns Game 1 6 4 3 3 1 5 1 3 8 8 © Joan A. Cotter, Ph.D., 2012
    266. 266. Rows and Columns Game © Joan A. Cotter, Ph.D., 2012
    267. 267. MoneyPenny © Joan A. Cotter, Ph.D., 2012
    268. 268. MoneyNickel © Joan A. Cotter, Ph.D., 2012
    269. 269. Money Dime © Joan A. Cotter, Ph.D., 2012
    270. 270. MoneyQuarter © Joan A. Cotter, Ph.D., 2012
    271. 271. MoneyQuarter © Joan A. Cotter, Ph.D., 2012
    272. 272. MoneyQuarter © Joan A. Cotter, Ph.D., 2012
    273. 273. MoneyQuarter © Joan A. Cotter, Ph.D., 2012
    274. 274. Place Value Two aspects © Joan A. Cotter, Ph.D., 2012
    275. 275. Place Value Two aspectsStatic © Joan A. Cotter, Ph.D., 2012
    276. 276. Place Value Two aspectsStatic • Value of a digit is determined by position © Joan A. Cotter, Ph.D., 2012
    277. 277. Place Value Two aspectsStatic • Value of a digit is determined by position. • No position may have more than nine. © Joan A. Cotter, Ph.D., 2012
    278. 278. Place Value Two aspectsStatic • Value of a digit is determined by position. • No position may have more than nine. • As you progress to the left, value at each position is ten times greater than previous position. © Joan A. Cotter, Ph.D., 2012
    279. 279. Place Value Two aspectsStatic • Value of a digit is determined by position. • No position may have more than nine. • As you progress to the left, value at each position is ten times greater than previous position.(Shown by the Decimal Cards.) © Joan A. Cotter, Ph.D., 2012
    280. 280. Place Value Two aspectsStatic • Value of a digit is determined by position. • No position may have more than nine. • As you progress to the left, value at each position is ten times greater than previous position.(Shown by the Decimal Cards.)Dynamic © Joan A. Cotter, Ph.D., 2012
    281. 281. Place Value Two aspectsStatic • Value of a digit is determined by position. • No position may have more than nine. • As you progress to the left, value at each position is ten times greater than previous position.(Shown by the Decimal Cards.)Dynamic • 10 ones = 1 ten; 10 tens = 1 hundred; 10 hundreds = 1 thousand, …. © Joan A. Cotter, Ph.D., 2012
    282. 282. Place Value Two aspectsStatic • Value of a digit is determined by position. • No position may have more than nine. • As you progress to the left, value at each position is ten times greater than previous position.(Shown by the Decimal Cards.)Dynamic • 10 ones = 1 ten; 10 tens = 1 hundred; 10 hundreds = 1 thousand, ….(Represented on the Abacus and other materials.) © Joan A. Cotter, Ph.D., 2012
    283. 283. Exchanging1000 100 10 1 © Joan A. Cotter, Ph.D., 2012
    284. 284. Exchanging Thousands1000 100 10 1 © Joan A. Cotter, Ph.D., 2012
    285. 285. Exchanging Hundreds1000 100 10 1 © Joan A. Cotter, Ph.D., 2012
    286. 286. Exchanging Tens1000 100 10 1 © Joan A. Cotter, Ph.D., 2012
    287. 287. Exchanging Ones1000 100 10 1 © Joan A. Cotter, Ph.D., 2012
    288. 288. Exchanging Adding1000 100 10 1 8 +6 © Joan A. Cotter, Ph.D., 2012
    289. 289. Exchanging Adding1000 100 10 1 8 +6 © Joan A. Cotter, Ph.D., 2012
    290. 290. Exchanging Adding1000 100 10 1 8 +6 © Joan A. Cotter, Ph.D., 2012
    291. 291. Exchanging Adding1000 100 10 1 8 +6 © Joan A. Cotter, Ph.D., 2012
    292. 292. Exchanging Adding1000 100 10 1 8 +6 14 © Joan A. Cotter, Ph.D., 2012
    293. 293. Exchanging Adding1000 100 10 1 8 +6 14 Too many ones; trade 10 ones for 1 ten. © Joan A. Cotter, Ph.D., 2012
    294. 294. Exchanging Adding1000 100 10 1 8 +6 14 Too many ones; trade 10 ones for 1 ten. © Joan A. Cotter, Ph.D., 2012
    295. 295. Exchanging Adding1000 100 10 1 8 +6 14 Too many ones; trade 10 ones for 1 ten. © Joan A. Cotter, Ph.D., 2012
    296. 296. Exchanging Adding1000 100 10 1 8 +6 14 Same answer before and after exchanging. © Joan A. Cotter, Ph.D., 2012
    297. 297. Bead Frame 1 101001000 © Joan A. Cotter, Ph.D., 2012
    298. 298. Bead Frame 1 8 10 +61001000 © Joan A. Cotter, Ph.D., 2012
    299. 299. Bead Frame 1 8 10 +61001000 © Joan A. Cotter, Ph.D., 2012
    300. 300. Bead Frame 1 8 10 +61001000 © Joan A. Cotter, Ph.D., 2012
    301. 301. Bead Frame 1 8 10 +61001000 © Joan A. Cotter, Ph.D., 2012
    302. 302. Bead Frame 1 8 10 +61001000 © Joan A. Cotter, Ph.D., 2012
    303. 303. Bead Frame 1 8 10 +61001000 © Joan A. Cotter, Ph.D., 2012
    304. 304. Bead Frame 1 8 10 +61001000 © Joan A. Cotter, Ph.D., 2012
    305. 305. Bead Frame 1 8 10 +61001000 © Joan A. Cotter, Ph.D., 2012
    306. 306. Bead Frame 1 8 10 +61001000 © Joan A. Cotter, Ph.D., 2012
    307. 307. Bead Frame 1 8 10 +6100 141000 © Joan A. Cotter, Ph.D., 2012
    308. 308. 1 Bead Frame 10 100 1000Difficulties for the child © Joan A. Cotter, Ph.D., 2012
    309. 309. 1 Bead Frame 10 100 1000 Difficulties for the child• Not visualizable: Beads need to be grouped in fives. © Joan A. Cotter, Ph.D., 2012
    310. 310. 1 Bead Frame 10 100 1000 Difficulties for the child• Not visualizable: Beads need to be grouped in fives.• When beads are moved right, inconsistent withequation order: Beads need to be moved left. © Joan A. Cotter, Ph.D., 2012
    311. 311. 1 Bead Frame 10 100 1000 Difficulties for the child• Not visualizable: Beads need to be grouped in fives.• When beads are moved right, inconsistent withequation order: Beads need to be moved left.• Hierarchies of numbers represented sideways:They need to be in vertical columns. © Joan A. Cotter, Ph.D., 2012
    312. 312. 1 Bead Frame 10 100 1000 Difficulties for the child• Not visualizable: Beads need to be grouped in fives.• When beads are moved right, inconsistent withequation order: Beads need to be moved left.• Hierarchies of numbers represented sideways:They need to be in vertical columns.• Exchanging done before second number iscompletely added: Addends need to be combined beforeexchanging. © Joan A. Cotter, Ph.D., 2012
    313. 313. 1 Bead Frame 10 100 1000 Difficulties for the child• Not visualizable: Beads need to be grouped in fives.• When beads are moved right, inconsistent withequation order: Beads need to be moved left.• Hierarchies of numbers represented sideways:They need to be in vertical columns.• Exchanging done before second number iscompletely added: Addends need to be combined beforeexchanging.• Answer is read going up: We read top to bottom. © Joan A. Cotter, Ph.D., 2012
    314. 314. 1 Bead Frame 10 100 1000 Difficulties for the child• Not visualizable: Beads need to be grouped in fives.• When beads are moved right, inconsistent withequation order: Beads need to be moved left.• Hierarchies of numbers represented sideways:They need to be in vertical columns.• Exchanging before second number is completelyadded: Addends need to be combined before exchanging.• Answer is read going up: We read top to bottom.• Distracting: Room is visible through the frame. © Joan A. Cotter, Ph.D., 2012
    315. 315. Exchanging Adding 4-digit numbers1000 100 10 1 3658 + 2738 © Joan A. Cotter, Ph.D., 2012
    316. 316. Exchanging Adding 4-digit numbers1000 100 10 1 3658 + 2738 Enter the first number from left to right. © Joan A. Cotter, Ph.D., 2012
    317. 317. Exchanging Adding 4-digit numbers1000 100 10 1 3658 + 2738 Enter the first number from left to right. © Joan A. Cotter, Ph.D., 2012
    318. 318. Exchanging Adding 4-digit numbers1000 100 10 1 3658 + 2738 Enter the first number from left to right. © Joan A. Cotter, Ph.D., 2012
    319. 319. Exchanging Adding 4-digit numbers1000 100 10 1 3658 + 2738 Enter the first number from left to right. © Joan A. Cotter, Ph.D., 2012
    320. 320. Exchanging Adding 4-digit numbers1000 100 10 1 3658 + 2738 Enter the first number from left to right. © Joan A. Cotter, Ph.D., 2012
    321. 321. Exchanging Adding 4-digit numbers1000 100 10 1 3658 + 2738 Enter the first number from left to right. © Joan A. Cotter, Ph.D., 2012
    322. 322. Exchanging 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
    323. 323. Exchanging 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
    324. 324. Exchanging 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
    325. 325. Exchanging 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
    326. 326. Exchanging Adding 4-digit numbers1000 100 10 1 3658 + 2738 6 Add starting at the right. Write results after each step. © Joan A. Cotter, Ph.D., 2012
    327. 327. Exchanging 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
    328. 328. Exchanging 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
    329. 329. Exchanging 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
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