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The Future of Primary Math: More Understanding/Less Counting                    by Joan A. Cotter, Ph.D.                Jo...
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 letter...
Verbal Counting Model       From a childs perspective                  F                 +E4                              ...
Verbal Counting Model        From a childs perspective                   F                  +E    A5                      ...
Verbal Counting Model            From a childs perspective                       F                      +E    A   B6      ...
Verbal Counting Model            From a childs perspective                       F                      +E    A   B    C7 ...
Verbal Counting Model            From a childs perspective                          F                         +E    A   B ...
Verbal Counting Model            From a childs perspective                          F                         +E    A   B ...
Verbal Counting Model             From a childs perspective                           F                          +E     A ...
Verbal Counting Model             From a childs perspective                           F                          +E     A ...
Verbal Counting Model             From a childs perspective                           F                          +E     A ...
Verbal Counting Model             From a childs perspective                           F                          +E       ...
Verbal Counting Model         From a childs perspective        Now memorize the facts!!                    G              ...
Verbal Counting Model         From a childs perspective        Now memorize the facts!!                                   ...
Verbal Counting Model         From a childs perspective        Now memorize the facts!!                                   ...
Verbal Counting Model         From a childs perspective        Now memorize the facts!!                                   ...
Verbal Counting Model             From a childs perspective         Now memorize the facts!!                              ...
Verbal Counting Model                 From a childs perspective                            H                           –E ...
Verbal Counting Model          From a childs perspective                      J                     –F     Subtract withou...
Verbal Counting Model           From a childs perspective     Try skip counting by B’s to T:         B, D, . . . T.21     ...
Verbal Counting Model           From a childs perspective     Try skip counting by B’s to T:         B, D, . . . T.     Wh...
Verbal Counting Model          From a childs perspective     L     is written AB     because it is A J     and B A’s23    ...
Verbal Counting Model          From a childs perspective     L     is written AB     because it is A J     and B A’s      ...
Verbal Counting Model          From a childs perspective     L (twelve)     is written AB     because it is A J     and B ...
Verbal Counting Model          From a childs perspective     L (twelve)     is written AB (12)     because it is A J     a...
Verbal Counting Model          From a childs perspective     L (twelve)     is written AB (12)     because it is A J (one ...
Verbal Counting Model          From a childs perspective     L (twelve)     is written AB (12)     because it is A J (one ...
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 ...
Calendar Math         Calendar Counting             August     1   2   3   4   5   6   7     8   9   10 11 12 13 14     15...
Calendar Math         Calendar Counting             August     1   2   3   4   5   6   7     8   9   10 11 12 13 14     15...
Calendar Math         Calendar Counting             August     1   2   3   4   5   6   7     8   9   10 11 12 13 14     15...
Calendar Math       Septemb         Calendar Counting     1234567             August     89101214     1   2          113  ...
Calendar Math                 Septemb                   Calendar Counting               1234567                        Aug...
Calendar Math         Partial Calendar             August     1   2   3    4   5   6   7     8   9   1035                 ...
Calendar Math                   Partial Calendar                      August              1   2   3    4   5   6   7      ...
Calendar Math                    Septemb                      Calendar Patterning                  1234567                ...
Minnesota Standards                      Number Sense     K: Represent quantities using whole numbers and     understand r...
Minnesota Standards                       Kindergarten     Represent quantities using whole numbers and     understand rel...
Minnesota Standards                         Kindergarten     Represent quantities using whole numbers and     understand r...
Minnesota Standards                          Kindergarten     Represent quantities using whole numbers and     understand ...
Minnesota Standards                          Kindergarten     Represent quantities using whole numbers and     understand ...
Minnesota Standards                          Kindergarten     Represent quantities using whole numbers and     understand ...
Minnesota Standards                          Kindergarten     Represent quantities using whole numbers and     understand ...
Minnesota Standards                          Kindergarten     Represent quantities using whole numbers and     understand ...
Minnesota Standards                        Grade 1     Understand place value and relationships among     whole numbers.46...
Minnesota Standards                          Grade 1     Understand place value and relationships among     whole numbers....
Minnesota Standards                           Grade 1     Understand place value and relationships among     whole numbers...
Minnesota Standards                           Grade 1     Understand place value and relationships among     whole numbers...
Minnesota Standards                           Grade 1     Understand place value and relationships among     whole numbers...
Minnesota Standards                           Grade 1     Understand place value and relationships among     whole numbers...
Minnesota Standards                        Grade 2     Understand place value and relationships among     whole numbers.52...
Minnesota Standards                          Grade 2     Understand place value and relationships among     whole numbers....
Minnesota Standards                          Grade 2     Understand place value and relationships among     whole numbers....
Minnesota Standards                           Grade 2     Understand place value and relationships among     whole numbers...
Minnesota Standards                           Grade 2     Understand place value and relationships among     whole numbers...
Minnesota Standards                           Grade 2     Understand place value and relationships among     whole numbers...
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 research60                              © Joan A. Cotter, Ph.D., 2012
Research on Counting        Karen Wynn’s research61                              © Joan A. Cotter, Ph.D., 2012
Research on Counting        Karen Wynn’s research62                              © Joan A. Cotter, Ph.D., 2012
Research on Counting        Karen Wynn’s research63                              © Joan A. Cotter, Ph.D., 2012
Research on Counting        Karen Wynn’s research64                              © Joan A. Cotter, Ph.D., 2012
Research on Counting        Karen Wynn’s research65                              © Joan A. Cotter, Ph.D., 2012
Research on Counting           Other research66                          © Joan A. Cotter, Ph.D., 2012
Research on Counting                              Other research     • Australian Aboriginal children from two tribes.    ...
Research on Counting                              Other research     • Australian Aboriginal children from two tribes.    ...
Research on Counting                              Other research     • Australian Aboriginal children from two tribes.    ...
Research on Counting                              Other research     • Australian Aboriginal children from two tribes.    ...
Research on Counting                In Japanese schools:     • Children are discouraged from using     counting for adding...
Research on Counting                 In Japanese schools:     • Children are discouraged from using     counting for addin...
Research on Counting                         Subitizing     • Subitizing is quick recognition of quantity     without coun...
Research on Counting                         Subitizing     • Subitizing is quick recognition of quantity     without coun...
Research on Counting                         Subitizing     • Subitizing is quick recognition of quantity     without coun...
Research on Counting                         Subitizing     • Subitizing is quick recognition of quantity     without coun...
Research on Counting                         Subitizing     • Subitizing is quick recognition of quantity     without coun...
Visualizing Quantities78                            © Joan A. Cotter, Ph.D., 2012
Visualizing Quantities     “Think in pictures, because the     brain remembers images better     than it does anything els...
Visualizing Quantities     “The role of physical manipulatives     was to help the child form those     visual images and ...
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?93                                  © Joan A. Cotter, ...
AN ALTERNATIVE      to learning place value:     Subitizing (groups of five)     Math Way (of number naming)     Place Val...
Grouping in Fives    Using fingers                    © Joan A. Cotter, Ph.D., 2012
Grouping in Fives    Using fingers                    © Joan A. Cotter, Ph.D., 2012
Grouping in Fives         Using fingers97                       © Joan A. Cotter, Ph.D., 2012
Grouping in Fives         Using fingers98                       © Joan A. Cotter, Ph.D., 2012
Grouping in Fives         Using fingers99                       © Joan A. Cotter, Ph.D., 2012
Grouping in Fives          Using fingers100                       © Joan A. Cotter, Ph.D., 2012
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 sticks106                       © Joan A. Cotter, Ph.D., 2012
Grouping in Fives          Tally sticks107                       © Joan A. Cotter, Ph.D., 2012
Grouping in Fives          Tally sticks108                       © Joan A. Cotter, Ph.D., 2012
Grouping in Fives          Tally sticks109                       © Joan A. Cotter, Ph.D., 2012
Grouping in Fives          Tally sticks110                       © Joan A. Cotter, Ph.D., 2012
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 quantities      5113                               © Joan A. Cotter, Ph.D., 2012
Grouping in Fives            Entering quantities      7114                               © Joan A. Cotter, Ph.D., 2012
Grouping in Fives             Entering quantities      10115                                © Joan A. Cotter, Ph.D., 2012
Grouping in Fives           The stairs116                       © 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
Grouping in Fives      Adding 4+3=                    © Joan A. Cotter, Ph.D., 2012
Go to the Dump Game      Objective:       To learn the facts that total 10:                      1+9                      ...
Go to the Dump Game      Objective:       To learn the facts that total 10:                      1+9                      ...
Go to the Dump Game         6+   = 10126                         © Joan A. Cotter, Ph.D., 2012
“Math” Way of Naming Numbers127                           © Joan A. Cotter, Ph.D., 2012
“Math” Way of Naming Numbers        11 = ten 1128                           © Joan A. Cotter, Ph.D., 2012
“Math” Way of Naming Numbers        11 = ten 1        12 = ten 2129                           © Joan A. Cotter, Ph.D., 2012
“Math” Way of Naming Numbers        11 = ten 1        12 = ten 2        13 = ten 3130                           © Joan A. ...
“Math” Way of Naming Numbers        11 = ten 1        12 = ten 2        13 = ten 3        14 = ten 4131                   ...
“Math” Way of Naming Numbers        11 = ten 1        12 = ten 2        13 = ten 3        14 = ten 4         ....        1...
“Math” Way of Naming Numbers        11 = ten 1   20 = 2-ten        12 = ten 2        13 = ten 3        14 = ten 4         ...
“Math” Way of Naming Numbers        11 = ten 1   20 = 2-ten        12 = ten 2   21 = 2-ten 1        13 = ten 3        14 =...
“Math” Way of Naming Numbers        11 = ten 1   20 = 2-ten        12 = ten 2   21 = 2-ten 1        13 = ten 3   22 = 2-te...
“Math” Way of Naming Numbers        11 = ten 1   20 = 2-ten        12 = ten 2   21 = 2-ten 1        13 = ten 3   22 = 2-te...
“Math” Way of Naming Numbers        11 = ten 1   20 = 2-ten        12 = ten 2   21 = 2-ten 1        13 = ten 3   22 = 2-te...
“Math” Way of Naming Numbers          137 = 1 hundred 3-ten 7138                                 © Joan A. Cotter, Ph.D., ...
“Math” Way of Naming Numbers          137 = 1 hundred 3-ten 7                    or        137 = 1 hundred and 3-ten 7139 ...
“Math” Way of Naming Numbers                                    100                Chinese                 Average Highest...
“Math” Way of Naming Numbers                                    100                Chinese                 Average Highest...
“Math” Way of Naming Numbers                                    100                Chinese                 Average Highest...
“Math” Way of Naming Numbers                                    100                Chinese                 Average Highest...
“Math” Way of Naming Numbers                                    100                Chinese                 Average Highest...
Math Way of Naming Numbers       • Only 11 words are needed to count to 100 the       math way, 28 in English. (All Indo-E...
Math Way of Naming Numbers       • Only 11 words are needed to count to 100 the       math way, 28 in English. (All Indo-E...
Math Way of Naming Numbers       • Only 11 words are needed to count to 100 the       math way, 28 in English. (All Indo-E...
Math Way of Naming Numbers       • Only 11 words are needed to count to 100 the       math way, 28 in English. (All Indo-E...
Math Way of Naming Numbers             Compared to reading:149                                 © Joan A. Cotter, Ph.D., 2012
Math Way of Naming Numbers                      Compared to reading:      • Just as reciting the alphabet doesn’t teach re...
Math Way of Naming Numbers                      Compared to reading:      • Just as reciting the alphabet doesn’t teach re...
Math Way of Naming Numbers      “Rather, the increased gap between Chinese and      U.S. students and that of Chinese Amer...
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 leftTwopronounced“twoo.”                      ...
Math Way of Naming Numbers                Traditional names             two left    twelveTwopronounced“twoo.”            ...
Composing Numbers3-ten                       © 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 Numbers3-ten30                       © Joan A. Cotter, Ph.D., 2012
Composing Numbers3-ten30                       © 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 Numbers3-ten 730 7                       © 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 correspo...
Composing Numbers7-ten70        Another example.                           © Joan A. Cotter, Ph.D., 2012
Composing Numbers7-ten 870                       © Joan A. Cotter, Ph.D., 2012
Composing Numbers7-ten 870                       © Joan A. Cotter, Ph.D., 2012
Composing Numbers7-ten 870  8                       © Joan A. Cotter, Ph.D., 2012
Composing Numbers7-ten 878                       © 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 Numbers1 hundred100                      © Joan A. Cotter, Ph.D., 2012
Composing Numbers1 hundred100                      © Joan A. Cotter, Ph.D., 2012
Composing Numbers1 hundred100                      © 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 hundred200                      © Joan A. Cotter, Ph.D., 2012
Counting by 2s and 5s                        © Joan A. Cotter, Ph.D., 2012
Counting by 2s and 5s     Counting by 2s                        © Joan A. Cotter, Ph.D., 2012
Counting by 2s and 5s         Counting by 2s2                            © Joan A. Cotter, Ph.D., 2012
Counting by 2s and 5s             Counting by 2s2   4                                © Joan A. Cotter, Ph.D., 2012
Counting by 2s and 5s             Counting by 2s2   4    6                                © Joan A. Cotter, Ph.D., 2012
Counting by 2s and 5s                 Counting by 2s2   4    6   8                                  © Joan A. Cotter, Ph.D...
Counting by 2s and 5s               Counting by 2s2   4    6   8 10                                © Joan A. Cotter, Ph.D....
Counting by 2s and 5s                Counting by 2s 2   4    6   8 1012                                 © Joan A. Cotter, ...
Counting by 2s and 5s               Counting by 2s 2 4     6   8 1012 14                                © Joan A. Cotter, ...
Counting by 2s and 5s             Counting by 2s 2 4 6     8 1012 14 16                              © Joan A. Cotter, Ph....
Counting by 2s and 5s         Counting by 2s 2 4 6 8 1012 14 16 18                            © Joan A. Cotter, Ph.D., 2012
Counting by 2s and 5s            Counting by 2s 2 4 6 8 1012 14 16 18 20                             © Joan A. Cotter, Ph....
Counting by 2s and 5s     Counting by 5s                        © Joan A. Cotter, Ph.D., 2012
Counting by 2s and 5s     Counting by 5s5                        © Joan A. Cotter, Ph.D., 2012
Counting by 2s and 5s         Counting by 5s5   10                          © Joan A. Cotter, Ph.D., 2012
Counting by 2s and 5s          Counting by 5s 5   1015                           © Joan A. Cotter, Ph.D., 2012
Counting by 2s and 5s          Counting by 5s 5   1015   20                           © Joan A. Cotter, Ph.D., 2012
Counting by 2s and 5s          Counting by 5s 5   1015   2025                           © Joan A. Cotter, Ph.D., 2012
Counting by 2s and 5s          Counting by 5s 5   1015   2025   30                           © 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.               ...
Fact Strategies227                     © 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...
Fact Strategies                Multiplication6× 4=(6 taken 4 times)                                 © Joan A. Cotter, Ph.D...
Fact Strategies                Multiplication6× 4=(6 taken 4 times)                                 © Joan A. Cotter, Ph.D...
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...
Trading1000   100   10   1                        © Joan A. Cotter, Ph.D., 2012
Trading              Thousands1000   100   10   1                          © Joan A. Cotter, Ph.D., 2012
Trading                  Hundreds1000   100   10     1                             © Joan A. Cotter, Ph.D., 2012
Trading                  Tens1000   100   10   1                         © Joan A. Cotter, Ph.D., 2012
Trading                  Ones1000   100   10   1                         © Joan A. Cotter, Ph.D., 2012
Trading                  Adding1000   100   10    1                            8                           +6             ...
Trading                  Adding1000   100   10    1                            8                           +6             ...
Trading                  Adding1000   100   10    1                            8                           +6             ...
Trading                  Adding1000   100   10    1                            8                           +6             ...
Trading                  Adding1000   100   10    1                            8                           +6             ...
Trading                  Adding1000   100   10    1                                8                               +6     ...
Trading                  Adding1000   100   10    1                                8                               +6     ...
Trading                  Adding1000   100   10    1                                8                               +6     ...
Trading                  Adding1000   100   10    1                                8                               +6     ...
Trading       Bead Trading Activity1000    100   10   1                               © Joan A. Cotter, Ph.D., 2012
Trading       Bead Trading Activity1000    100   10   1                         Object: To get a                         h...
Trading       Bead Trading Activity1000    100   10   1                            7                         Object: To ge...
Trading       Bead Trading Activity1000    100   10   1                            7                         Object: To ge...
Trading       Bead Trading Activity1000    100   10   1                          6                               © Joan A....
Trading       Bead Trading Activity1000    100   10   1                          6                               © Joan A....
Trading       Bead Trading Activity1000    100   10   1                          6                               © Joan A....
Trading       Bead Trading Activity1000    100   10   1                            6                         Trade 10 ones...
Trading       Bead Trading Activity1000    100   10   1                          6                               © Joan A....
Trading       Bead Trading Activity1000    100   10   1                          6                               © Joan A....
Trading       Bead Trading Activity1000    100   10   1                          9                               © Joan A....
Trading       Bead Trading Activity1000    100   10   1                           9                               © Joan A...
Trading       Bead Trading Activity1000    100   10   1                            9                         Another trade...
Trading       Bead Trading Activity1000    100   10   1                            9                         Another trade...
Trading       Bead Trading Activity1000    100   10   1                          3                               © Joan A....
Trading       Bead Trading Activity1000    100   10   1                          3                               © Joan A....
Trading            Bead Trading Activity• In the Bead Trading activity trading    10 ones for 1 ten occurs frequently;    ...
Trading            Bead Trading Activity• In the Bead Trading activity trading    10 ones for 1 ten occurs frequently;    ...
Trading            Bead Trading Activity• In the Bead Trading activity trading    10 ones for 1 ten occurs frequently;    ...
Trading            Bead Trading Activity• In the Bead Trading activity trading    10 ones for 1 ten occurs frequently;    ...
Trading            Bead Trading Activity• In the Bead Trading activity trading    10 ones for 1 ten occurs frequently;    ...
Trading       Adding 4-digit numbers1000    100   10   1                           3658                         + 2738    ...
Trading       Adding 4-digit numbers1000    100   10   1                           3658                         + 2738    ...
Trading       Adding 4-digit numbers1000    100   10   1                           3658                         + 2738    ...
Trading       Adding 4-digit numbers1000    100   10   1                           3658                         + 2738    ...
Trading       Adding 4-digit numbers1000    100   10   1                           3658                         + 2738    ...
Trading       Adding 4-digit numbers1000    100   10   1                           3658                         + 2738    ...
Trading       Adding 4-digit numbers1000    100   10   1                           3658                         + 2738    ...
Trading       Adding 4-digit numbers1000    100   10   1                           3658                         + 2738    ...
Trading       Adding 4-digit numbers1000    100   10   1                           3658                         + 2738    ...
Trading       Adding 4-digit numbers1000    100   10   1                           3658                         + 2738    ...
Trading       Adding 4-digit numbers1000    100   10   1                           3658                         + 2738    ...
Trading       Adding 4-digit numbers1000    100   10   1                           3658                         + 2738    ...
Trading       Adding 4-digit numbers1000    100   10   1            1                           3658                      ...
Trading       Adding 4-digit numbers1000    100   10   1            1                           3658                      ...
Trading       Adding 4-digit numbers1000    100   10   1            1                           3658                      ...
Trading       Adding 4-digit numbers1000    100   10   1            1                           3658                      ...
Trading       Adding 4-digit numbers1000    100   10   1            1                           3658                      ...
Trading       Adding 4-digit numbers1000    100   10   1            1                           3658                      ...
Trading       Adding 4-digit numbers1000    100   10   1            1                           3658                      ...
Trading       Adding 4-digit numbers1000    100   10   1            1                           3658                      ...
Trading       Adding 4-digit numbers1000    100   10   1            1                           3658                      ...
Trading       Adding 4-digit numbers1000    100   10   1         1   1                           3658                     ...
Trading       Adding 4-digit numbers1000    100   10   1         1   1                           3658                     ...
Trading       Adding 4-digit numbers1000    100   10   1         1   1                           3658                     ...
Trading       Adding 4-digit numbers1000    100   10   1         1   1                           3658                     ...
Trading       Adding 4-digit numbers1000    100   10   1         1   1                           3658                     ...
Minnesota Standards                        Number Sense      K: Represent quantities using whole numbers and      understa...
Minnesota Standards                          Kindergarten      Represent quantities using whole numbers and      understan...
Minnesota Standards                           Kindergarten      Represent quantities using whole numbers and      understa...
Minnesota Standards                           Kindergarten      Represent quantities using whole numbers and      understa...
Minnesota Standards                           Kindergarten      Represent quantities using whole numbers and      understa...
Minnesota Standards                           Kindergarten      Represent quantities using whole numbers and      understa...
Minnesota Standards                           Kindergarten      Represent quantities using whole numbers and      understa...
Minnesota Standards                           Grade 1      Understand place value and relationships among      whole numbe...
Minnesota Standards                            Grade 1      Understand place value and relationships among      whole numb...
Minnesota Standards                            Grade 1      Understand place value and relationships among      whole numb...
Minnesota Standards                            Grade 1      Understand place value and relationships among      whole numb...
Minnesota Standards                            Grade 1      Understand place value and relationships among      whole numb...
Minnesota Standards                           Grade 2      Understand place value and relationships among      whole numbe...
Minnesota Standards                           Grade 2      Understand place value and relationships among      whole numbe...
Minnesota Standards                            Grade 2      Understand place value and relationships among      whole numb...
Minnesota Standards                            Grade 2      Understand place value and relationships among      whole numb...
Minnesota Standards                            Grade 2      Understand place value and relationships among      whole numb...
Research Highlights345                         © Joan A. Cotter, Ph.D., 2012
MCTM Future Primary Math
MCTM Future Primary Math
MCTM Future Primary Math
MCTM Future Primary Math
MCTM Future Primary Math
MCTM Future Primary Math
MCTM Future Primary Math
MCTM Future Primary Math
MCTM Future Primary Math
MCTM Future Primary Math
MCTM Future Primary Math
MCTM Future Primary Math
MCTM Future Primary Math
MCTM Future Primary Math
MCTM Future Primary Math
MCTM Future Primary Math
MCTM Future Primary Math
MCTM Future Primary Math
MCTM Future Primary Math
MCTM Future Primary Math
MCTM Future Primary Math
MCTM Future Primary Math
MCTM Future Primary Math
MCTM Future Primary Math
MCTM Future Primary Math
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MCTM Future Primary Math

  1. 1. The Future of Primary Math: More Understanding/Less Counting by Joan A. Cotter, Ph.D. JoanCotter@RightStartMath.com MCTM Saturday, May 5, 2012 Duluth, Minnesota 1000 100 10 1 30 7 30 7 PowerPoint PresentationRightStartMath.com >Resources © Joan A. Cotter, Ph.D., 2012
  2. 2. Verbal Counting Model2 © Joan A. Cotter, Ph.D., 2012
  3. 3. 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
  4. 4. Verbal Counting Model From a childs perspective F +E4 © Joan A. Cotter, Ph.D., 2012
  5. 5. Verbal Counting Model From a childs perspective F +E A5 © Joan A. Cotter, Ph.D., 2012
  6. 6. Verbal Counting Model From a childs perspective F +E A B6 © Joan A. Cotter, Ph.D., 2012
  7. 7. Verbal Counting Model From a childs perspective F +E A B C7 © Joan A. Cotter, Ph.D., 2012
  8. 8. Verbal Counting Model From a childs perspective F +E A B C D E F8 © Joan A. Cotter, Ph.D., 2012
  9. 9. Verbal Counting Model From a childs perspective F +E A B C D E F A9 © Joan A. Cotter, Ph.D., 2012
  10. 10. Verbal Counting Model From a childs perspective F +E A B C D E F A B10 © Joan A. Cotter, Ph.D., 2012
  11. 11. 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
  12. 12. 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
  13. 13. 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
  14. 14. Verbal Counting Model From a childs perspective Now memorize the facts!! G +D14 © Joan A. Cotter, Ph.D., 2012
  15. 15. Verbal Counting Model From a childs perspective Now memorize the facts!! H + G F +D15 © Joan A. Cotter, Ph.D., 2012
  16. 16. Verbal Counting Model From a childs perspective Now memorize the facts!! H + G F +D D +C16 © 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 +C +G17 © Joan A. Cotter, Ph.D., 2012
  18. 18. 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
  19. 19. Verbal Counting Model From a childs perspective H –E Subtract with your fingers by counting backward.19 © Joan A. Cotter, Ph.D., 2012
  20. 20. Verbal Counting Model From a childs perspective J –F Subtract without using your fingers.20 © Joan A. Cotter, Ph.D., 2012
  21. 21. Verbal Counting Model From a childs perspective Try skip counting by B’s to T: B, D, . . . T.21 © Joan A. Cotter, Ph.D., 2012
  22. 22. Verbal Counting Model From a childs perspective Try skip counting by B’s to T: B, D, . . . T. What is D × E?22 © Joan A. Cotter, Ph.D., 2012
  23. 23. Verbal Counting Model From a childs perspective L is written AB because it is A J and B A’s23 © Joan A. Cotter, Ph.D., 2012
  24. 24. Verbal Counting Model From a childs perspective L is written AB because it is A J and B A’s huh?24 © Joan A. Cotter, Ph.D., 2012
  25. 25. Verbal Counting Model From a childs perspective L (twelve) is written AB because it is A J and B A’s25 © Joan A. Cotter, Ph.D., 2012
  26. 26. Verbal Counting Model From a childs perspective L (twelve) is written AB (12) because it is A J and B A’s26 © Joan A. Cotter, Ph.D., 2012
  27. 27. Verbal Counting Model From a childs perspective L (twelve) is written AB (12) because it is A J (one 10) and B A’s27 © Joan A. Cotter, Ph.D., 2012
  28. 28. 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).28 © Joan A. Cotter, Ph.D., 2012
  29. 29. 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 3129 © Joan A. Cotter, Ph.D., 2012
  30. 30. Calendar Math Calendar Counting 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 3130 © Joan A. Cotter, Ph.D., 2012
  31. 31. Calendar Math Calendar Counting 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 3131 © Joan A. Cotter, Ph.D., 2012
  32. 32. Calendar Math Calendar Counting 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 3132 © Joan A. Cotter, Ph.D., 2012
  33. 33. Calendar Math Septemb Calendar Counting 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 3133 © Joan A. Cotter, Ph.D., 2012
  34. 34. 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 31 This is ordinal counting, not cardinal counting.34 © Joan A. Cotter, Ph.D., 2012
  35. 35. Calendar Math Partial Calendar August 1 2 3 4 5 6 7 8 9 1035 © Joan A. Cotter, Ph.D., 2012
  36. 36. Calendar Math Partial Calendar August 1 2 3 4 5 6 7 8 9 10 Children need the whole month to plan ahead.36 © Joan A. Cotter, Ph.D., 2012
  37. 37. 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 31 Patterns are rarely based on 7s or proceed row by row. Patterns go on forever; they don’t stop at 31.37 © Joan A. Cotter, Ph.D., 2012
  38. 38. Minnesota Standards Number Sense K: Represent quantities using whole numbers and understand relationships among whole numbers. 1–2: Understand place value and relationships among whole numbers. With the counting model, how difficult are the associated benchmarks for children to master?38 © Joan A. Cotter, Ph.D., 2012
  39. 39. Minnesota Standards Kindergarten Represent quantities using whole numbers and understand relationships among whole numbers.39 © Joan A. Cotter, Ph.D., 2012
  40. 40. Minnesota Standards Kindergarten Represent quantities using whole numbers and understand relationships among whole numbers. • Count forward to 31, backward from 10.40 © Joan A. Cotter, Ph.D., 2012
  41. 41. Minnesota Standards Kindergarten Represent quantities using whole numbers and understand relationships among whole numbers. • Count forward to 31, backward from 10. • Count number of objects and identify the quantity.41 © Joan A. Cotter, Ph.D., 2012
  42. 42. Minnesota Standards Kindergarten Represent quantities using whole numbers and understand relationships among whole numbers. • Count forward to 31, backward from 10. • Count number of objects and identify the quantity. • Compare the number of objects in two or more sets.42 © Joan A. Cotter, Ph.D., 2012
  43. 43. Minnesota Standards Kindergarten Represent quantities using whole numbers and understand relationships among whole numbers. • Count forward to 31, backward from 10. • Count number of objects and identify the quantity. • Compare the number of objects in two or more sets. • Given a number, identify one more or one less.43 © Joan A. Cotter, Ph.D., 2012
  44. 44. Minnesota Standards Kindergarten Represent quantities using whole numbers and understand relationships among whole numbers. • Count forward to 31, backward from 10. • Count number of objects and identify the quantity. • Compare the number of objects in two or more sets. • Given a number, identify one more or one less. • Recognize number of objects up to 6, without counting.44 © Joan A. Cotter, Ph.D., 2012
  45. 45. Minnesota Standards Kindergarten Represent quantities using whole numbers and understand relationships among whole numbers. • Count forward to 31, backward from 10. • Count number of objects and identify the quantity. • Compare the number of objects in two or more sets. • Given a number, identify one more or one less. • Recognize number of objects up to 6, without counting. • Add and subtract whole numbers up to 6, using objects.45 © Joan A. Cotter, Ph.D., 2012
  46. 46. Minnesota Standards Grade 1 Understand place value and relationships among whole numbers.46 © Joan A. Cotter, Ph.D., 2012
  47. 47. Minnesota Standards Grade 1 Understand place value and relationships among whole numbers. • Read, write, compare and order numbers to 120.47 © Joan A. Cotter, Ph.D., 2012
  48. 48. Minnesota Standards Grade 1 Understand place value and relationships among whole numbers. • Read, write, compare and order numbers to 120. • Count by 2s to 30 and by 5s to 120.48 © Joan A. Cotter, Ph.D., 2012
  49. 49. Minnesota Standards Grade 1 Understand place value and relationships among whole numbers. • Read, write, compare and order numbers to 120. • Count by 2s to 30 and by 5s to 120. • Count backwards from 30.49 © Joan A. Cotter, Ph.D., 2012
  50. 50. Minnesota Standards Grade 1 Understand place value and relationships among whole numbers. • Read, write, compare and order numbers to 120. • Count by 2s to 30 and by 5s to 120. • Count backwards from 30. • Demonstrate understanding of odd and even to 12.50 © Joan A. Cotter, Ph.D., 2012
  51. 51. Minnesota Standards Grade 1 Understand place value and relationships among whole numbers. • Read, write, compare and order numbers to 120. • Count by 2s to 30 and by 5s to 120. • Count backwards from 30. • Demonstrate understanding of odd and even to 12. • Represent whole numbers up to 20 in various ways.51 © Joan A. Cotter, Ph.D., 2012
  52. 52. Minnesota Standards Grade 2 Understand place value and relationships among whole numbers.52 © Joan A. Cotter, Ph.D., 2012
  53. 53. Minnesota Standards Grade 2 Understand place value and relationships among whole numbers. • Read, write, compare and order numbers to 999.53 © Joan A. Cotter, Ph.D., 2012
  54. 54. Minnesota Standards Grade 2 Understand place value and relationships among whole numbers. • Read, write, compare and order numbers to 999. • Count by 2s, 5s, 10s from any given whole number.54 © Joan A. Cotter, Ph.D., 2012
  55. 55. Minnesota Standards Grade 2 Understand place value and relationships among whole numbers. • Read, write, compare and order numbers to 999. • Count by 2s, 5s, 10s from any given whole number. • Understand the significance of groups of ten.55 © Joan A. Cotter, Ph.D., 2012
  56. 56. Minnesota Standards Grade 2 Understand place value and relationships among whole numbers. • Read, write, compare and order numbers to 999. • Count by 2s, 5s, 10s from any given whole number. • Understand the significance of groups of ten. • Demonstrate understanding of odd and even up to 12.56 © Joan A. Cotter, Ph.D., 2012
  57. 57. Minnesota Standards Grade 2 Understand place value and relationships among whole numbers. • Read, write, compare and order numbers to 999. • Count by 2s, 5s, 10s from any given whole number. • Understand the significance of groups of ten. • Demonstrate understanding of odd and even up to 12. • Represent whole numbers up to 20 in various ways.57 © Joan A. Cotter, Ph.D., 2012
  58. 58. Research on Counting Karen Wynn’s research © Joan A. Cotter, Ph.D., 2012
  59. 59. Research on Counting Karen Wynn’s research © Joan A. Cotter, Ph.D., 2012
  60. 60. Research on Counting Karen Wynn’s research60 © Joan A. Cotter, Ph.D., 2012
  61. 61. Research on Counting Karen Wynn’s research61 © Joan A. Cotter, Ph.D., 2012
  62. 62. Research on Counting Karen Wynn’s research62 © Joan A. Cotter, Ph.D., 2012
  63. 63. Research on Counting Karen Wynn’s research63 © Joan A. Cotter, Ph.D., 2012
  64. 64. Research on Counting Karen Wynn’s research64 © Joan A. Cotter, Ph.D., 2012
  65. 65. Research on Counting Karen Wynn’s research65 © Joan A. Cotter, Ph.D., 2012
  66. 66. Research on Counting Other research66 © Joan A. Cotter, Ph.D., 2012
  67. 67. Research on Counting Other research • Australian Aboriginal children from two tribes. Brian Butterworth, University College London, 2008.67 © Joan A. Cotter, Ph.D., 2012
  68. 68. 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, 2008.68 © Joan A. Cotter, Ph.D., 2012
  69. 69. 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, 2008.69 © Joan A. Cotter, Ph.D., 2012
  70. 70. 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, 2009.70 © Joan A. Cotter, Ph.D., 2012
  71. 71. Research on Counting In Japanese schools: • Children are discouraged from using counting for adding.71 © Joan A. Cotter, Ph.D., 2012
  72. 72. Research on Counting In Japanese schools: • Children are discouraged from using counting for adding. • They consistently group in 5s.72 © Joan A. Cotter, Ph.D., 2012
  73. 73. Research on Counting Subitizing • Subitizing is quick recognition of quantity without counting.73 © Joan A. Cotter, Ph.D., 2012
  74. 74. Research on Counting Subitizing • Subitizing is quick recognition of quantity without counting. • Human babies and some animals can subitize small quantities at birth.74 © Joan A. Cotter, Ph.D., 2012
  75. 75. 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.—Butterworth75 © Joan A. Cotter, Ph.D., 2012
  76. 76. 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.”—Benoit76 © Joan A. Cotter, Ph.D., 2012
  77. 77. 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.— Glasersfeld77 © Joan A. Cotter, Ph.D., 2012
  78. 78. Visualizing Quantities78 © Joan A. Cotter, Ph.D., 2012
  79. 79. Visualizing Quantities “Think in pictures, because the brain remembers images better than it does anything else.” Ben Pridmore, World Memory Champion, 200979 © Joan A. Cotter, Ph.D., 2012
  80. 80. Visualizing Quantities “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 others80 © Joan A. Cotter, Ph.D., 2012
  81. 81. 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
  82. 82. Visualizing Quantities Visualizing also needed in:• Reading• Sports• Creativity• Geography• Engineering• Construction © Joan A. Cotter, Ph.D., 2012
  83. 83. Visualizing Quantities Visualizing also needed in:• Reading • Architecture• Sports • Astronomy• Creativity • Archeology• Geography • Chemistry• Engineering • Physics• Construction • Surgery © Joan A. Cotter, Ph.D., 2012
  84. 84. Visualizing Quantities Ready: How many? © Joan A. Cotter, Ph.D., 2012
  85. 85. Visualizing Quantities Ready: How many? © Joan A. Cotter, Ph.D., 2012
  86. 86. Visualizing Quantities Try again: How many? © Joan A. Cotter, Ph.D., 2012
  87. 87. Visualizing Quantities Try again: How many? © Joan A. Cotter, Ph.D., 2012
  88. 88. Visualizing QuantitiesTry to visualize 8 identical apples without grouping. © Joan A. Cotter, Ph.D., 2012
  89. 89. Visualizing QuantitiesTry to visualize 8 identical apples without grouping. © Joan A. Cotter, Ph.D., 2012
  90. 90. Visualizing QuantitiesNow try to visualize 5 as red and 3 as green. © Joan A. Cotter, Ph.D., 2012
  91. 91. Visualizing QuantitiesNow try to visualize 5 as red and 3 as green. © Joan A. Cotter, Ph.D., 2012
  92. 92. Visualizing Quantities Early Roman numerals 1 I 2 II 3 III 4 IIII 5 V 8 VIII © Joan A. Cotter, Ph.D., 2012
  93. 93. Visualizing Quantities : Who could read the music?93 © Joan A. Cotter, Ph.D., 2012
  94. 94. AN ALTERNATIVE to learning place value: Subitizing (groups of five) Math Way (of number naming) Place Value Cards Trading (with 4-digit numbers)94 © Joan A. Cotter, Ph.D., 2012
  95. 95. Grouping in Fives Using fingers © Joan A. Cotter, Ph.D., 2012
  96. 96. Grouping in Fives Using fingers © Joan A. Cotter, Ph.D., 2012
  97. 97. Grouping in Fives Using fingers97 © Joan A. Cotter, Ph.D., 2012
  98. 98. Grouping in Fives Using fingers98 © Joan A. Cotter, Ph.D., 2012
  99. 99. Grouping in Fives Using fingers99 © Joan A. Cotter, Ph.D., 2012
  100. 100. Grouping in Fives Using fingers100 © Joan A. Cotter, Ph.D., 2012
  101. 101. 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
  102. 102. Grouping in Fives Recognizing 5 © Joan A. Cotter, Ph.D., 2012
  103. 103. Grouping in Fives Recognizing 5 © Joan A. Cotter, Ph.D., 2012
  104. 104. Grouping in Fives Recognizing 55 has a middle; 4 does not. © Joan A. Cotter, Ph.D., 2012
  105. 105. Grouping in Fives Tally sticks © Joan A. Cotter, Ph.D., 2012
  106. 106. Grouping in Fives Tally sticks106 © Joan A. Cotter, Ph.D., 2012
  107. 107. Grouping in Fives Tally sticks107 © Joan A. Cotter, Ph.D., 2012
  108. 108. Grouping in Fives Tally sticks108 © Joan A. Cotter, Ph.D., 2012
  109. 109. Grouping in Fives Tally sticks109 © Joan A. Cotter, Ph.D., 2012
  110. 110. Grouping in Fives Tally sticks110 © Joan A. Cotter, Ph.D., 2012
  111. 111. Grouping in Fives Entering quantities © Joan A. Cotter, Ph.D., 2012
  112. 112. Grouping in Fives Entering quantities3 © Joan A. Cotter, Ph.D., 2012
  113. 113. Grouping in Fives Entering quantities 5113 © Joan A. Cotter, Ph.D., 2012
  114. 114. Grouping in Fives Entering quantities 7114 © Joan A. Cotter, Ph.D., 2012
  115. 115. Grouping in Fives Entering quantities 10115 © Joan A. Cotter, Ph.D., 2012
  116. 116. Grouping in Fives The stairs116 © Joan A. Cotter, Ph.D., 2012
  117. 117. Grouping in Fives Adding © Joan A. Cotter, Ph.D., 2012
  118. 118. Grouping in Fives Adding 4+3= © Joan A. Cotter, Ph.D., 2012
  119. 119. Grouping in Fives Adding 4+3= © Joan A. Cotter, Ph.D., 2012
  120. 120. Grouping in Fives Adding 4+3= © Joan A. Cotter, Ph.D., 2012
  121. 121. Grouping in Fives Adding 4+3= © Joan A. Cotter, Ph.D., 2012
  122. 122. Grouping in Fives Adding 4+3=7 © Joan A. Cotter, Ph.D., 2012
  123. 123. Grouping in Fives Adding 4+3= © Joan A. Cotter, Ph.D., 2012
  124. 124. Go to the Dump Game Objective: To learn the facts that total 10: 1+9 2+8 3+7 4+6 5+5124 © Joan A. Cotter, Ph.D., 2012
  125. 125. Go to the Dump Game Objective: 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.125 © Joan A. Cotter, Ph.D., 2012
  126. 126. Go to the Dump Game 6+ = 10126 © Joan A. Cotter, Ph.D., 2012
  127. 127. “Math” Way of Naming Numbers127 © Joan A. Cotter, Ph.D., 2012
  128. 128. “Math” Way of Naming Numbers 11 = ten 1128 © Joan A. Cotter, Ph.D., 2012
  129. 129. “Math” Way of Naming Numbers 11 = ten 1 12 = ten 2129 © Joan A. Cotter, Ph.D., 2012
  130. 130. “Math” Way of Naming Numbers 11 = ten 1 12 = ten 2 13 = ten 3130 © Joan A. Cotter, Ph.D., 2012
  131. 131. “Math” Way of Naming Numbers 11 = ten 1 12 = ten 2 13 = ten 3 14 = ten 4131 © Joan A. Cotter, Ph.D., 2012
  132. 132. “Math” Way of Naming Numbers 11 = ten 1 12 = ten 2 13 = ten 3 14 = ten 4 .... 19 = ten 9132 © Joan A. Cotter, Ph.D., 2012
  133. 133. “Math” Way of Naming Numbers 11 = ten 1 20 = 2-ten 12 = ten 2 13 = ten 3 14 = ten 4 .... 19 = ten 9133 © Joan A. Cotter, Ph.D., 2012
  134. 134. “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 9134 © Joan A. Cotter, Ph.D., 2012
  135. 135. “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 9135 © Joan A. Cotter, Ph.D., 2012
  136. 136. “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 9136 © Joan A. Cotter, Ph.D., 2012
  137. 137. “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 9137 © Joan A. Cotter, Ph.D., 2012
  138. 138. “Math” Way of Naming Numbers 137 = 1 hundred 3-ten 7138 © Joan A. Cotter, Ph.D., 2012
  139. 139. “Math” Way of Naming Numbers 137 = 1 hundred 3-ten 7 or 137 = 1 hundred and 3-ten 7139 © Joan A. Cotter, Ph.D., 2012
  140. 140. “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, 319-332.140 © Joan A. Cotter, Ph.D., 2012
  141. 141. “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, 319-332.141 © Joan A. Cotter, Ph.D., 2012
  142. 142. “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, 319-332.142 © Joan A. Cotter, Ph.D., 2012
  143. 143. “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, 319-332.143 © Joan A. Cotter, Ph.D., 2012
  144. 144. “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, 319-332.144 © Joan A. Cotter, Ph.D., 2012
  145. 145. 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.)145 © Joan A. Cotter, Ph.D., 2012
  146. 146. 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.146 © Joan A. Cotter, Ph.D., 2012
  147. 147. 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.147 © Joan A. Cotter, Ph.D., 2012
  148. 148. 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.148 © Joan A. Cotter, Ph.D., 2012
  149. 149. Math Way of Naming Numbers Compared to reading:149 © Joan A. Cotter, Ph.D., 2012
  150. 150. Math Way of Naming Numbers Compared to reading: • Just as reciting the alphabet doesn’t teach reading, counting doesn’t teach arithmetic.150 © Joan A. Cotter, Ph.D., 2012
  151. 151. 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).151 © Joan A. Cotter, Ph.D., 2012
  152. 152. 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 Researchers152 © Joan A. Cotter, Ph.D., 2012
  153. 153. Math Way of Naming Numbers Traditional names4-ten =fortyThe “ty”means tens. © Joan A. Cotter, Ph.D., 2012
  154. 154. Math Way of Naming Numbers Traditional names4-ten =fortyThe “ty”means tens. © Joan A. Cotter, Ph.D., 2012
  155. 155. Math Way of Naming Numbers Traditional names6-ten = sixtyThe “ty”means tens. © Joan A. Cotter, Ph.D., 2012
  156. 156. Math Way of Naming Numbers Traditional names3-ten = thirty“Thir” alsoused in 1/3,13 and 30. © Joan A. Cotter, Ph.D., 2012
  157. 157. Math Way of Naming Numbers Traditional names5-ten = fifty“Fif” alsoused in 1/5,15 and 50. © Joan A. Cotter, Ph.D., 2012
  158. 158. Math Way of Naming Numbers Traditional names2-ten = twentyTwo used to bepronounced“twoo.” © Joan A. Cotter, Ph.D., 2012
  159. 159. Math Way of Naming Numbers Traditional names A word game fireplace place-fire © Joan A. Cotter, Ph.D., 2012
  160. 160. Math Way of Naming Numbers Traditional names A word game fireplace place-fire newspaper paper-news © Joan A. Cotter, Ph.D., 2012
  161. 161. 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
  162. 162. Math Way of Naming Numbers Traditional names ten 4“Teen” alsomeans ten. © Joan A. Cotter, Ph.D., 2012
  163. 163. Math Way of Naming Numbers Traditional names ten 4 teen 4“Teen” alsomeans ten. © Joan A. Cotter, Ph.D., 2012
  164. 164. Math Way of Naming Numbers Traditional names ten 4 teen 4 fourtee n“Teen” alsomeans ten. © Joan A. Cotter, Ph.D., 2012
  165. 165. Math Way of Naming Numbers Traditional names a one left © Joan A. Cotter, Ph.D., 2012
  166. 166. Math Way of Naming Numbers Traditional names a one left a left-one © Joan A. Cotter, Ph.D., 2012
  167. 167. Math Way of Naming Numbers Traditional names a one left a left-one eleven © Joan A. Cotter, Ph.D., 2012
  168. 168. Math Way of Naming Numbers Traditional names two leftTwopronounced“twoo.” © Joan A. Cotter, Ph.D., 2012
  169. 169. Math Way of Naming Numbers Traditional names two left twelveTwopronounced“twoo.” © Joan A. Cotter, Ph.D., 2012
  170. 170. Composing Numbers3-ten © Joan A. Cotter, Ph.D., 2012
  171. 171. Composing Numbers3-ten © Joan A. Cotter, Ph.D., 2012
  172. 172. Composing Numbers3-ten © Joan A. Cotter, Ph.D., 2012
  173. 173. Composing Numbers3-ten30 © Joan A. Cotter, Ph.D., 2012
  174. 174. Composing Numbers3-ten30 © Joan A. Cotter, Ph.D., 2012
  175. 175. Composing Numbers3-ten30 © Joan A. Cotter, Ph.D., 2012
  176. 176. Composing Numbers3-ten 730 © Joan A. Cotter, Ph.D., 2012
  177. 177. Composing Numbers3-ten 730 © Joan A. Cotter, Ph.D., 2012
  178. 178. Composing Numbers3-ten 730 7 © Joan A. Cotter, Ph.D., 2012
  179. 179. Composing Numbers3-ten 730 7 © Joan A. Cotter, Ph.D., 2012
  180. 180. 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
  181. 181. Composing Numbers7-ten70 Another example. © Joan A. Cotter, Ph.D., 2012
  182. 182. Composing Numbers7-ten 870 © Joan A. Cotter, Ph.D., 2012
  183. 183. Composing Numbers7-ten 870 © Joan A. Cotter, Ph.D., 2012
  184. 184. Composing Numbers7-ten 870 8 © Joan A. Cotter, Ph.D., 2012
  185. 185. Composing Numbers7-ten 878 © Joan A. Cotter, Ph.D., 2012
  186. 186. Composing Numbers10-ten © Joan A. Cotter, Ph.D., 2012
  187. 187. Composing Numbers10-ten100 © Joan A. Cotter, Ph.D., 2012
  188. 188. Composing Numbers10-ten100 © Joan A. Cotter, Ph.D., 2012
  189. 189. Composing Numbers10-ten100 © Joan A. Cotter, Ph.D., 2012
  190. 190. Composing Numbers1 hundred © Joan A. Cotter, Ph.D., 2012
  191. 191. Composing Numbers1 hundred100 © Joan A. Cotter, Ph.D., 2012
  192. 192. Composing Numbers1 hundred100 © Joan A. Cotter, Ph.D., 2012
  193. 193. Composing Numbers1 hundred100 © Joan A. Cotter, Ph.D., 2012
  194. 194. Composing Numbers1 hundred100 © Joan A. Cotter, Ph.D., 2012
  195. 195. Composing Numbers2 hundred © Joan A. Cotter, Ph.D., 2012
  196. 196. Composing Numbers2 hundred © Joan A. Cotter, Ph.D., 2012
  197. 197. Composing Numbers2 hundred200 © Joan A. Cotter, Ph.D., 2012
  198. 198. Counting by 2s and 5s © Joan A. Cotter, Ph.D., 2012
  199. 199. Counting by 2s and 5s Counting by 2s © Joan A. Cotter, Ph.D., 2012
  200. 200. Counting by 2s and 5s Counting by 2s2 © Joan A. Cotter, Ph.D., 2012
  201. 201. Counting by 2s and 5s Counting by 2s2 4 © Joan A. Cotter, Ph.D., 2012
  202. 202. Counting by 2s and 5s Counting by 2s2 4 6 © Joan A. Cotter, Ph.D., 2012
  203. 203. Counting by 2s and 5s Counting by 2s2 4 6 8 © Joan A. Cotter, Ph.D., 2012
  204. 204. Counting by 2s and 5s Counting by 2s2 4 6 8 10 © Joan A. Cotter, Ph.D., 2012
  205. 205. Counting by 2s and 5s Counting by 2s 2 4 6 8 1012 © Joan A. Cotter, Ph.D., 2012
  206. 206. Counting by 2s and 5s Counting by 2s 2 4 6 8 1012 14 © Joan A. Cotter, Ph.D., 2012
  207. 207. Counting by 2s and 5s Counting by 2s 2 4 6 8 1012 14 16 © Joan A. Cotter, Ph.D., 2012
  208. 208. Counting by 2s and 5s Counting by 2s 2 4 6 8 1012 14 16 18 © Joan A. Cotter, Ph.D., 2012
  209. 209. Counting by 2s and 5s Counting by 2s 2 4 6 8 1012 14 16 18 20 © Joan A. Cotter, Ph.D., 2012
  210. 210. Counting by 2s and 5s Counting by 5s © Joan A. Cotter, Ph.D., 2012
  211. 211. Counting by 2s and 5s Counting by 5s5 © Joan A. Cotter, Ph.D., 2012
  212. 212. Counting by 2s and 5s Counting by 5s5 10 © Joan A. Cotter, Ph.D., 2012
  213. 213. Counting by 2s and 5s Counting by 5s 5 1015 © Joan A. Cotter, Ph.D., 2012
  214. 214. Counting by 2s and 5s Counting by 5s 5 1015 20 © Joan A. Cotter, Ph.D., 2012
  215. 215. Counting by 2s and 5s Counting by 5s 5 1015 2025 © Joan A. Cotter, Ph.D., 2012
  216. 216. Counting by 2s and 5s Counting by 5s 5 1015 2025 30 © Joan A. Cotter, Ph.D., 2012
  217. 217. Evens and Odds Evens © Joan A. Cotter, Ph.D., 2012
  218. 218. Evens and Odds Evens Use two fingers and touch each pair in succession. © Joan A. Cotter, Ph.D., 2012
  219. 219. Evens and Odds Evens Use two fingers and touch each pair in succession. © Joan A. Cotter, Ph.D., 2012
  220. 220. Evens and Odds Evens Use two fingers and touch each pair in succession. © Joan A. Cotter, Ph.D., 2012
  221. 221. Evens and Odds Evens Use two fingers and touch each pair in succession. EVEN! © Joan A. Cotter, Ph.D., 2012
  222. 222. Evens and Odds Odds Use two fingers and touch each pair in succession. © Joan A. Cotter, Ph.D., 2012
  223. 223. Evens and Odds Odds Use two fingers and touch each pair in succession. © Joan A. Cotter, Ph.D., 2012
  224. 224. Evens and Odds Odds Use two fingers and touch each pair in succession. © Joan A. Cotter, Ph.D., 2012
  225. 225. Evens and Odds Odds Use two fingers and touch each pair in succession. © Joan A. Cotter, Ph.D., 2012
  226. 226. Evens and Odds Odds Use two fingers and touch each pair in succession. ODD! © Joan A. Cotter, Ph.D., 2012
  227. 227. Fact Strategies227 © 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 Ten9+5= © 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=Take 1 fromthe 5 and giveit to the 9. © Joan A. Cotter, Ph.D., 2012
  234. 234. Fact Strategies Complete the Ten 9 + 5 = 14Take 1 fromthe 5 and giveit to the 9. © 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= © Joan A. Cotter, Ph.D., 2012
  239. 239. Fact Strategies Two Fives8+6=10 + 4 = 14 © Joan A. Cotter, Ph.D., 2012
  240. 240. Fact Strategies Going Down15 – 9 = © Joan A. Cotter, Ph.D., 2012
  241. 241. Fact Strategies Going Down15 – 9 = © 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 =Subtract 5;then 4. © Joan A. Cotter, Ph.D., 2012
  245. 245. Fact Strategies Going Down 15 – 9 = 6Subtract 5;then 4. © Joan A. Cotter, Ph.D., 2012
  246. 246. Fact Strategies Subtract from 1015 – 9 = © 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 =Subtract 9from 10. © Joan A. Cotter, Ph.D., 2012
  250. 250. Fact Strategies Subtract from 10 15 – 9 = 6Subtract 9from 10. © Joan A. Cotter, Ph.D., 2012
  251. 251. Fact Strategies Going Up15 – 9 = © 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 =Start with 9;go up to 15. © Joan A. Cotter, Ph.D., 2012
  256. 256. Fact Strategies Going Up 15 – 9 = 1+5=6Start with 9;go up to 15. © Joan A. Cotter, Ph.D., 2012
  257. 257. Fact Strategies Multiplication6× 4=(6 taken 4 times) © Joan A. Cotter, Ph.D., 2012
  258. 258. Fact Strategies Multiplication6× 4=(6 taken 4 times) © Joan A. Cotter, Ph.D., 2012
  259. 259. Place Value Two aspects © Joan A. Cotter, Ph.D., 2012
  260. 260. Place Value Two aspectsStatic © Joan A. Cotter, Ph.D., 2012
  261. 261. Place Value Two aspectsStatic • Value of a digit is determined by position © Joan A. Cotter, Ph.D., 2012
  262. 262. 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
  263. 263. 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
  264. 264. 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. • Place value cards show this aspect. © Joan A. Cotter, Ph.D., 2012
  265. 265. 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. • Place value cards show this aspect.Dynamic © Joan A. Cotter, Ph.D., 2012
  266. 266. 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. • Place value cards show this aspect.Dynamic • Ten ones = 1 ten; ten tens = 1 hundred; ten hundreds = 1 thousand, …. © Joan A. Cotter, Ph.D., 2012
  267. 267. Trading1000 100 10 1 © Joan A. Cotter, Ph.D., 2012
  268. 268. Trading Thousands1000 100 10 1 © Joan A. Cotter, Ph.D., 2012
  269. 269. Trading Hundreds1000 100 10 1 © Joan A. Cotter, Ph.D., 2012
  270. 270. Trading Tens1000 100 10 1 © Joan A. Cotter, Ph.D., 2012
  271. 271. Trading Ones1000 100 10 1 © Joan A. Cotter, Ph.D., 2012
  272. 272. Trading Adding1000 100 10 1 8 +6 © Joan A. Cotter, Ph.D., 2012
  273. 273. Trading Adding1000 100 10 1 8 +6 © Joan A. Cotter, Ph.D., 2012
  274. 274. Trading Adding1000 100 10 1 8 +6 © Joan A. Cotter, Ph.D., 2012
  275. 275. Trading Adding1000 100 10 1 8 +6 © Joan A. Cotter, Ph.D., 2012
  276. 276. Trading Adding1000 100 10 1 8 +6 14 © Joan A. Cotter, Ph.D., 2012
  277. 277. Trading Adding1000 100 10 1 8 +6 14 Too many ones; trade 10 ones for 1 ten. © Joan A. Cotter, Ph.D., 2012
  278. 278. Trading Adding1000 100 10 1 8 +6 14 Too many ones; trade 10 ones for 1 ten. © Joan A. Cotter, Ph.D., 2012
  279. 279. Trading Adding1000 100 10 1 8 +6 14 Too many ones; trade 10 ones for 1 ten. © Joan A. Cotter, Ph.D., 2012
  280. 280. Trading Adding1000 100 10 1 8 +6 14 Same answer before and after trading. © Joan A. Cotter, Ph.D., 2012
  281. 281. Trading Bead Trading Activity1000 100 10 1 © Joan A. Cotter, Ph.D., 2012
  282. 282. Trading Bead Trading Activity1000 100 10 1 Object: To get a high score by adding numbers on the green cards. © Joan A. Cotter, Ph.D., 2012
  283. 283. Trading Bead Trading Activity1000 100 10 1 7 Object: To get a high score by adding numbers on the green cards. © Joan A. Cotter, Ph.D., 2012
  284. 284. Trading Bead Trading Activity1000 100 10 1 7 Object: To get a high score by adding numbers on the green cards. © Joan A. Cotter, Ph.D., 2012
  285. 285. Trading Bead Trading Activity1000 100 10 1 6 © Joan A. Cotter, Ph.D., 2012
  286. 286. Trading Bead Trading Activity1000 100 10 1 6 © Joan A. Cotter, Ph.D., 2012
  287. 287. Trading Bead Trading Activity1000 100 10 1 6 © Joan A. Cotter, Ph.D., 2012
  288. 288. Trading Bead Trading Activity1000 100 10 1 6 Trade 10 ones for 1 ten. © Joan A. Cotter, Ph.D., 2012
  289. 289. Trading Bead Trading Activity1000 100 10 1 6 © Joan A. Cotter, Ph.D., 2012
  290. 290. Trading Bead Trading Activity1000 100 10 1 6 © Joan A. Cotter, Ph.D., 2012
  291. 291. Trading Bead Trading Activity1000 100 10 1 9 © Joan A. Cotter, Ph.D., 2012
  292. 292. Trading Bead Trading Activity1000 100 10 1 9 © Joan A. Cotter, Ph.D., 2012
  293. 293. Trading Bead Trading Activity1000 100 10 1 9 Another trade. © Joan A. Cotter, Ph.D., 2012
  294. 294. Trading Bead Trading Activity1000 100 10 1 9 Another trade. © Joan A. Cotter, Ph.D., 2012
  295. 295. Trading Bead Trading Activity1000 100 10 1 3 © Joan A. Cotter, Ph.D., 2012
  296. 296. Trading Bead Trading Activity1000 100 10 1 3 © Joan A. Cotter, Ph.D., 2012
  297. 297. Trading Bead Trading Activity• In the Bead Trading activity trading 10 ones for 1 ten occurs frequently; © Joan A. Cotter, Ph.D., 2012
  298. 298. Trading Bead Trading Activity• In the Bead Trading activity trading 10 ones for 1 ten occurs frequently; 10 tens for 1 hundred, less often; © Joan A. Cotter, Ph.D., 2012
  299. 299. Trading Bead Trading Activity• In the Bead Trading activity trading 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
  300. 300. Trading Bead Trading Activity• In the Bead Trading activity trading 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
  301. 301. Trading Bead Trading Activity• In the Bead Trading activity trading 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
  302. 302. Trading Adding 4-digit numbers1000 100 10 1 3658 + 2738 © Joan A. Cotter, Ph.D., 2012
  303. 303. Trading Adding 4-digit numbers1000 100 10 1 3658 + 2738 Enter the first number from left to right. © Joan A. Cotter, Ph.D., 2012
  304. 304. Trading Adding 4-digit numbers1000 100 10 1 3658 + 2738 Enter the first number from left to right. © Joan A. Cotter, Ph.D., 2012
  305. 305. Trading Adding 4-digit numbers1000 100 10 1 3658 + 2738 Enter the first number from left to right. © Joan A. Cotter, Ph.D., 2012
  306. 306. Trading Adding 4-digit numbers1000 100 10 1 3658 + 2738 Enter the first number from left to right. © Joan A. Cotter, Ph.D., 2012
  307. 307. Trading Adding 4-digit numbers1000 100 10 1 3658 + 2738 Enter the first number from left to right. © Joan A. Cotter, Ph.D., 2012
  308. 308. Trading Adding 4-digit numbers1000 100 10 1 3658 + 2738 Enter the first number from left to right. © Joan A. Cotter, Ph.D., 2012
  309. 309. Trading 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
  310. 310. Trading 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
  311. 311. Trading 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
  312. 312. Trading 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
  313. 313. Trading 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
  314. 314. Trading 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
  315. 315. Trading 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
  316. 316. Trading 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
  317. 317. Trading 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
  318. 318. Trading 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
  319. 319. Trading 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
  320. 320. Trading 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
  321. 321. Trading 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
  322. 322. Trading 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
  323. 323. Trading 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
  324. 324. Trading 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
  325. 325. Trading 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
  326. 326. Trading 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
  327. 327. Trading 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
  328. 328. Minnesota Standards Number Sense K: Represent quantities using whole numbers and understand relationships among whole numbers. 1–2: Understand place value and relationships among whole numbers. With this alternate model, how difficult are the associated benchmarks for children to master?328 © Joan A. Cotter, Ph.D., 2012
  329. 329. Minnesota Standards Kindergarten Represent quantities using whole numbers and understand relationships among whole numbers. • Count forward to 31, backward from 10.329 © Joan A. Cotter, Ph.D., 2012
  330. 330. Minnesota Standards Kindergarten Represent quantities using whole numbers and understand relationships among whole numbers. • Count forward to 31, backward from 10. • Count number of objects and identify the quantity.330 © Joan A. Cotter, Ph.D., 2012
  331. 331. Minnesota Standards Kindergarten Represent quantities using whole numbers and understand relationships among whole numbers. • Count forward to 31, backward from 10. • Count number of objects and identify the quantity. • Compare the number of objects in two or more sets.331 © Joan A. Cotter, Ph.D., 2012
  332. 332. Minnesota Standards Kindergarten Represent quantities using whole numbers and understand relationships among whole numbers. • Count forward to 31, backward from 10. • Count number of objects and identify the quantity. • Compare the number of objects in two or more sets. • Given a number, identify one more or one less.332 © Joan A. Cotter, Ph.D., 2012
  333. 333. Minnesota Standards Kindergarten Represent quantities using whole numbers and understand relationships among whole numbers. • Count forward to 31, backward from 10. • Count number of objects and identify the quantity. • Compare the number of objects in two or more sets. • Given a number, identify one more or one less. • Recognize number of objects up to 6, without counting.333 © Joan A. Cotter, Ph.D., 2012
  334. 334. Minnesota Standards Kindergarten Represent quantities using whole numbers and understand relationships among whole numbers. • Count forward to 31, backward from 10. • Count number of objects and identify the quantity. • Compare the number of objects in two or more sets. • Given a number, identify one more or one less. • Recognize number of objects up to 6, without counting. • Add and subtract whole numbers up to 6, using objects.334 © Joan A. Cotter, Ph.D., 2012
  335. 335. Minnesota Standards Grade 1 Understand place value and relationships among whole numbers. • Read, write, compare and order numbers to 120.335 © Joan A. Cotter, Ph.D., 2012
  336. 336. Minnesota Standards Grade 1 Understand place value and relationships among whole numbers. • Read, write, compare and order numbers to 120. • Count by 2s to 30 and by 5s to 120.336 © Joan A. Cotter, Ph.D., 2012
  337. 337. Minnesota Standards Grade 1 Understand place value and relationships among whole numbers. • Read, write, compare and order numbers to 120. • Count by 2s to 30 and by 5s to 120. • Count backwards from 30.337 © Joan A. Cotter, Ph.D., 2012
  338. 338. Minnesota Standards Grade 1 Understand place value and relationships among whole numbers. • Read, write, compare and order numbers to 120. • Count by 2s to 30 and by 5s to 120. • Count backwards from 30. • Demonstrate understanding of odd and even to 12.338 © Joan A. Cotter, Ph.D., 2012
  339. 339. Minnesota Standards Grade 1 Understand place value and relationships among whole numbers. • Read, write, compare and order numbers to 120. • Count by 2s to 30 and by 5s to 120. • Count backwards from 30. • Demonstrate understanding of odd and even to 12. • Represent whole numbers up to 20 in various ways.339 © Joan A. Cotter, Ph.D., 2012
  340. 340. Minnesota Standards Grade 2 Understand place value and relationships among whole numbers. • Read, write, compare and order numbers to 999.340 © Joan A. Cotter, Ph.D., 2012
  341. 341. Minnesota Standards Grade 2 Understand place value and relationships among whole numbers. • Read, write, compare and order numbers to 999. • Count by 2s, 5s, 10s from any given whole number.341 © Joan A. Cotter, Ph.D., 2012
  342. 342. Minnesota Standards Grade 2 Understand place value and relationships among whole numbers. • Read, write, compare and order numbers to 999. • Count by 2s, 5s, 10s from any given whole number. • Understand the significance of groups of ten.342 © Joan A. Cotter, Ph.D., 2012
  343. 343. Minnesota Standards Grade 2 Understand place value and relationships among whole numbers. • Read, write, compare and order numbers to 999. • Count by 2s, 5s, 10s from any given whole number. • Understand the significance of groups of ten. • Demonstrate understanding of odd and even up to 12.343 © Joan A. Cotter, Ph.D., 2012
  344. 344. Minnesota Standards Grade 2 Understand place value and relationships among whole numbers. • Read, write, compare and order numbers to 999. • Count by 2s, 5s, 10s from any given whole number. • Understand the significance of groups of ten. • Demonstrate understanding of odd and even up to 12. • Represent whole numbers up to 20 in various ways.344 © Joan A. Cotter, Ph.D., 2012
  345. 345. Research Highlights345 © Joan A. Cotter, Ph.D., 2012

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