RightStart™ Mathematics in a        Montessori Environment                    by Joan A. Cotter, Ph.D.                 Joa...
National Math Crisis                       © Joan A. Cotter, Ph.D., 2012
National Math Crisis• 25% of college freshmen take remedial math.                                         © Joan A. Cotter...
National Math Crisis• 25% of college freshmen take remedial math.• In 2009, of the 1.5 million students who tookthe ACT te...
National Math Crisis• 25% of college freshmen take remedial math.• In 2009, of the 1.5 million students who tookthe ACT te...
National Math Crisis• 25% of college freshmen take remedial math.• In 2009, of the 1.5 million students who tookthe ACT te...
National Math Crisis• 25% of college freshmen take remedial math.• In 2009, of the 1.5 million students who tookthe ACT te...
National Math Crisis• 25% of college freshmen take remedial math.• In 2009, of the 1.5 million students who tookthe ACT te...
Math Education is Changing                       © Joan A. Cotter, Ph.D., 2012
Math Education is Changing• The field of mathematics is doubling every 7years.                                          © ...
Math Education is Changing• The field of mathematics is doubling every 7years.• Math is used in many new ways. Theworkplac...
Math Education is Changing• The field of mathematics is doubling every 7years.• Math is used in many new ways. Theworkplac...
Math Education is Changing• The field of mathematics is doubling every 7years.• Math is used in many new ways. Theworkplac...
Math Education is Changing• The field of mathematics is doubling every 7years.• Math is used in many new ways. Theworkplac...
Math Education is Changing• The field of mathematics is doubling every 7years.• Math is used in many new ways. Theworkplac...
Counting Model                 © Joan A. Cotter, Ph.D., 2012
Counting Model       From a childs perspectiveBecause we’re so familiar with 1, 2, 3, we’ll use                   letters....
Counting ModelFrom a childs perspective            F           +E                             © Joan A. Cotter, Ph.D., 2012
Counting Model    From a childs perspective                F               +EA                                 © Joan A. C...
Counting Model    From a childs perspective                F               +EA   B                                 © Joan ...
Counting Model    From a childs perspective                F               +EA   B   C                                 © J...
Counting Model    From a childs perspective                     F                    +EA   B   C   D   E   F              ...
Counting Model    From a childs perspective                     F                    +EA   B   C   D   E   F    A         ...
Counting Model    From a childs perspective                     F                    +EA   B   C   D   E   F    A   B     ...
Counting Model    From a childs perspective                     F                    +EA   B   C   D   E   F    A   B   C ...
Counting Model    From a childs perspective                     F                    +EA   B   C   D   E   F    A   B    C...
Counting Model    From a childs perspective                     F                    +E                     KA   B   C   D...
Counting Model From a childs perspectiveNow memorize the facts!!            G           +D                              © ...
Counting Model From a childs perspectiveNow memorize the facts!!            G           +D                              © ...
Counting Model From a childs perspectiveNow memorize the facts!!            G           +D D+C                            ...
Counting Model From a childs perspectiveNow memorize the facts!!            G           +D D                    C+C       ...
Counting Model From a childs perspectiveNow memorize the facts!!            G           +D D                    C+C       ...
Counting Model  From a childs perspectiveTry subtracting    Hby ―taking away‖ – E                               © Joan A. ...
Counting Model     From a childs perspectiveTry skip counting by B’s to T:    B, D, . . . T.                              ...
Counting Model     From a childs perspectiveTry skip counting by B’s to T:    B, D, . . . T.What is D ´ E?                ...
Counting Model    From a childs perspectiveLis written ABbecause it is A Jand B A’s                                 © Joan...
Counting Model    From a childs perspectiveLis written ABbecause it is A Jand B A’s             huh?                      ...
Counting Model    From a childs perspectiveL (twelve)is written ABbecause it is A Jand B A’s                              ...
Counting Model    From a childs perspectiveL (twelve)is written AB (12)because it is A Jand B A’s                         ...
Counting Model    From a childs perspectiveL (twelve)is written AB (12)                (one 10)because it is A Jand B A’s ...
Counting Model    From a childs perspectiveL (twelve)is written AB (12)                (one 10)because it is A Jand B A’s ...
Counting ModelIn Montessori, counting is pervasive:        • Number Rods        • Spindle Boxes        • Decimal materials...
Counting Model    Summary                 © Joan A. Cotter, Ph.D., 2012
Counting Model                Summary• Is not natural; it takes years ofpractice.                                      © J...
Counting Model              Summary• Is not natural; it takes years ofpractice.• Provides poor concept of quantity.       ...
Counting Model              Summary• Is not natural; it takes years ofpractice.• Provides poor concept of quantity.• Ignor...
Counting Model              Summary• Is not natural; it takes years ofpractice.• Provides poor concept of quantity.• Ignor...
Counting Model              Summary• Is not natural; it takes years ofpractice.• Provides poor concept of quantity.• Ignor...
Counting Model              Summary• Is not natural; it takes years ofpractice.• Provides poor concept of quantity.• Ignor...
Calendar Math                                    August                        1    2     3     4   5   6   7             ...
Calendar Math                                    August                        1    2     3     4   5   6   7             ...
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                                           August                             1      2     3       4     5   ...
Calendar Math                                       August                          1     2     3      4     5     6      ...
Calendar Math                                            August                             1      2       3     4     5  ...
Calendar Math                                    August                        1     2     3     4    5     6     7       ...
Calendar MathThe calendar is not a number line.  • No quantity is involved.  • Numbers are in spaces, not at lines like a ...
Calendar MathThe calendar is not a number line.  • No quantity is involved.  • Numbers are in spaces, not at lines like a ...
Calendar MathThe calendar is not a number line.  • No quantity is involved.  • Numbers are in spaces, not at lines like a ...
Memorizing Math                 Percentage Recall          Immediatel After 1 day After 4 wks              yRote         3...
Memorizing Math                 Percentage Recall          Immediatel After 1 day After 4 wks              yRote         3...
Memorizing Math                 Percentage Recall          Immediatel After 1 day After 4 wks              yRote         3...
Memorizing Math                 Percentage Recall          Immediatel After 1 day After 4 wks              yRote         3...
Memorizing Math                 Percentage Recall          Immediatel After 1 day After 4 wks              yRote         3...
Memorizing Math                 Percentage Recall          Immediatel After 1 day After 4 wks              yRote         3...
Memorizing Math                 Percentage Recall          Immediatel After 1 day After 4 wks              yRote         3...
9Memorizing Math         +7    Flash cards:                   © Joan A. Cotter, Ph.D., 2012
9        Memorizing Math                +7                Flash cards:• Are often used to teach rote.                     ...
9        Memorizing Math                   +7                Flash cards:• Are often used to teach rote.• Are liked only b...
9        Memorizing Math                     +7                Flash cards:• Are often used to teach rote.• Are liked only...
9        Memorizing Math                     +7                Flash cards:• Are often used to teach rote.• Are liked only...
9        Memorizing Math                     +7                Flash cards:• Are often used to teach rote.• Are liked only...
9        Memorizing Math                     +7                Flash cards:• Are often used to teach rote.• Are liked only...
Research on Counting                        Karen Wynn’s researchShow the baby two teddy bears.                  © Joan A....
Research on Counting                        Karen Wynn’s researchThen hide them with a screen.                   © Joan A....
Research on Counting                         Karen Wynn’s researchShow the baby a third teddy bear and put it behind the s...
Research on Counting                         Karen Wynn’s researchShow the baby a third teddy bear and put it behind the s...
Research on Counting                         Karen Wynn’s researchRaise screen. Baby seeing 3 won’t look long because it i...
Research on Counting                        Karen Wynn’s researchResearcher can change the number of teddy bears behind th...
Research on Counting                            Karen Wynn’s researchA baby seeing 1 teddy bear will look much longer, bec...
Research on Counting     Other research                       © 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 usingcounting for adding.                ...
Research on Counting         In Japanese schools:• Children are discouraged from usingcounting for adding.• They consisten...
Research on Counting                    Subitizing• Subitizing is quick recognition of quantitywithout counting.          ...
Research on Counting                    Subitizing• Subitizing is quick recognition of quantitywithout counting.• Human ba...
Research on Counting                    Subitizing• Subitizing is quick recognition of quantitywithout counting.• Human ba...
Research on Counting                    Subitizing• Subitizing is quick recognition of quantitywithout counting.• Human ba...
Research on Counting                    Subitizing• Subitizing is quick recognition of quantitywithout counting.• Human ba...
Research on Counting                Finger gnosia• Finger gnosia is the ability to know whichfingers can been lightly touc...
Research on Counting                Finger gnosia• Finger gnosia is the ability to know whichfingers can been lightly touc...
Research on Counting                Finger gnosia• Finger gnosia is the ability to know whichfingers can been lightly touc...
Visualizing Mathematics                      © Joan A. Cotter, Ph.D., 2012
Visualizing Mathematics―In our concern about the memorizationof math facts or solving problems, wemust not forget that the...
Visualizing Mathematics―Think in pictures, because thebrain remembers images betterthan it does anything else.‖Ben Pridmor...
Visualizing Mathematics―Mathematics is the activity ofcreating relationships, many ofwhich are based in visual imagery.”  ...
Visualizing Mathematics―The process of connectingsymbols to imagery is at the heartof mathematics learning.‖              ...
Visualizing Mathematics―The role of physicalmanipulatives was to help thechild form those visual imagesand thus to elimina...
Visualizing Mathematics        Japanese criteria for           manipulatives• Representative of structure ofnumbers.• Easi...
Visualizing Mathematics     Visualizing also needed in:• Reading• Sports• Creativity• Geography• Engineering• Construction...
Visualizing Mathematics     Visualizing also needed in:• Reading           • Architecture• Sports            • Astronomy• ...
Visualizing Mathematics    Ready: How many?                       © Joan A. Cotter, Ph.D., 2012
Visualizing Mathematics    Ready: How many?                       © Joan A. Cotter, Ph.D., 2012
Visualizing Mathematics   Try again: How many?                          © Joan A. Cotter, Ph.D., 2012
Visualizing Mathematics   Try again: How many?                          © Joan A. Cotter, Ph.D., 2012
Visualizing Mathematics   Try again: How many?                          © Joan A. Cotter, Ph.D., 2012
Visualizing Mathematics    Ready: How many?                       © Joan A. Cotter, Ph.D., 2012
Visualizing Mathematics   Try again: How many?                          © Joan A. Cotter, Ph.D., 2012
Visualizing MathematicsTry to visualize 8 identical apples without                 grouping.                              ...
Visualizing MathematicsTry to visualize 8 identical apples without grouping.                                             ©...
Visualizing MathematicsNow try to visualize 5 as red and 3 as                green.                                     © ...
Visualizing MathematicsNow try to visualize 5 as red and 3 as                green.                                     © ...
Visualizing Mathematics                       Early Roman numerals                                1        I              ...
Visualizing Mathematics                               :                       Who could read the music?Music needs 10 line...
Research on Counting               Teach Counting• Finger gnosia is the ability to know whichfingers can been lightly touc...
Very Early Computation                  NumeralsIn English there are two ways of writing numbers:                     3578...
Very Early Computation                    Numerals In English there are two ways of writing numbers:                      ...
Very Early Computation           Calculating rodsBecause their characters arecumbersome to use for computing, theChinese u...
Very Early Computation      Calculating rods                         © Joan A. Cotter, Ph.D., 2012
Very Early Computation               Calculating rodsNumerals for Ones and Hundreds (Even Powers of                     Te...
Very Early Computation               Calculating rodsNumerals for Ones and Hundreds (Even Powers of                     Te...
Very Early Computation               Calculating rodsNumerals for Ones and Hundreds (Odd Powers of                     Ten...
Very Early Computation       Calculating rods3578                          © Joan A. Cotter, Ph.D., 2012
Very Early Computation                Calculating rods   3578                  3578,3578They grouped, not in thousands, bu...
Naming Quantities    Using fingers                    © Joan A. Cotter, Ph.D., 2012
Naming Quantities           Using fingersNaming quantities is a three-period           lesson.                            ...
Naming Quantities                                   Using fingersUse left hand for 1-5 because we read from left to right....
Naming Quantities    Using fingers                    © Joan A. Cotter, Ph.D., 2012
Naming Quantities    Using fingers                    © Joan A. Cotter, Ph.D., 2012
Naming Quantities                                 Using fingersAlways show 7 as 5 and 2, not for example, as 4 and 3.   © ...
Naming Quantities    Using fingers                    © Joan A. Cotter, Ph.D., 2012
Naming Quantities                            Yellow is the Sun                              Yellow is the sun.            ...
Naming Quantities   Recognizing 5                    © Joan A. Cotter, Ph.D., 2012
Naming Quantities   Recognizing 5                    © Joan A. Cotter, Ph.D., 2012
Naming Quantities                                   Recognizing 5                   5 has a middle; 4 does not.Look at you...
Naming Quantities                                    Tally sticksLay the sticks flat on a surface, about 1 inch (2.5 cm) a...
Naming Quantities    Tally sticks                    © Joan A. Cotter, Ph.D., 2012
Naming Quantities    Tally sticks                    © Joan A. Cotter, Ph.D., 2012
Naming Quantities                                   Tally sticksStick is horizontal, because it won’t fit diagonally and y...
Naming Quantities    Tally sticks                    © Joan A. Cotter, Ph.D., 2012
Naming Quantities                                 Tally sticksStart a new row for every ten.                  © Joan A. Co...
Naming Quantities                Solving a problem without counting  What is 4 apples plus 3 more apples?How would you fin...
Naming Quantities                   Solving a problem without counting  What is 4 apples plus 3 more apples?To remember 4 ...
Naming QuantitiesNumber            1Chart             2             3             4             5                         ...
Naming Quantities Numbe r             1 Chart               2To help the    3child learnthesymbols        4               ...
Naming Quantities Numbe r             1              6 Chart               2              7To help the    3              8...
Naming Quantities                           Pairing Finger CardsUse two sets of finger cards and match them.      © Joan A...
Naming Quantities                            Ordering Finger CardsPutting the finger cards in order.                  © Jo...
Naming Quantities                 Matching Numbers to Finger Cards             5                          1               ...
Naming Quantities                 Matching Fingers to Number Cards             9             1           10   4        6  ...
Naming Quantities                      Finger Card Memory gameUse two sets of finger cards and play Memory.   © Joan A. Co...
Naming Quantities    Number Rods                    © Joan A. Cotter, Ph.D., 2012
Naming Quantities    Number Rods                    © Joan A. Cotter, Ph.D., 2012
Naming Quantities                          Number RodsUsing different colors.                   © Joan A. Cotter, Ph.D., 2...
Naming Quantities                                   Spindle Box45 dark-colored and 10 light-colored spindles. Could be in ...
Naming Quantities                                   Spindle Box45 dark-colored and 10 light-colored spindles in two contai...
Naming Quantities                                    Spindle Box              0              1             2              ...
Naming Quantities                                    Spindle Box              5              6             7              ...
Naming Quantities                                    Spindle Box              5              6             7              ...
Naming Quantities                                    Spindle Box              5              6             7              ...
Naming Quantities                                    Spindle Box              5              6             7              ...
Naming Quantities                                    Spindle Box              5              6             7              ...
Naming Quantities                                    Spindle Box              5              6             7              ...
Naming Quantities                      Black and White Bead Stairs            ―Grouped in fives so the child does         ...
AL Abacus                                                   1000   100   10           1Double-sided AL abacus. Side 1 is g...
AL Abacus  Cleared            © Joan A. Cotter, Ph.D., 2012
AL Abacus                            Entering quantities      3Quantities are entered all at once, not counted.   © Joan A...
AL Abacus                              Entering quantities           5Relate quantities to hands.                         ...
AL Abacus    Entering quantities7                          © Joan A. Cotter, Ph.D., 2012
AL Abacus     Entering quantities10                           © Joan A. Cotter, Ph.D., 2012
AL Abacus                                    The stairsCan use to ―count‖ 1 to 10. Also read quantities on the right side....
AL Abacus  Adding            © Joan A. Cotter, Ph.D., 2012
AL Abacus   Adding4+3=             © Joan A. Cotter, Ph.D., 2012
AL Abacus   Adding4+3=             © Joan A. Cotter, Ph.D., 2012
AL Abacus   Adding4+3=             © Joan A. Cotter, Ph.D., 2012
AL Abacus   Adding4+3=             © Joan A. Cotter, Ph.D., 2012
AL Abacus                                    Adding                      4+3=7Answer is seen immediately, no counting need...
Go to the Dump Game             Aim:             To learn the facts that total 10:                             1+9        ...
Go to the Dump Game             Aim:                 To learn the facts that total 10:                               1+9  ...
Go to the Dump GameThe ways to partition 10.         © Joan A. Cotter, Ph.D., 2012
Go to the Dump Game                            StartingA game viewed from above.              © Joan A. Cotter, Ph.D., 2012
Go to the Dump Game                              72 7 9 5                   72 1 3 8              4 6 34 9                ...
Go to the Dump Game                                      72 7 9 5                  72 1 3 8                       4 6 34 9...
Go to the Dump Game                                   72 7 9 5                  72 1 3 8                       4 6 34 9   ...
Go to the Dump Game                                   72 7 9 5                  72 1 3 8                       4 6 34 9   ...
Go to the Dump Game                                   72 7 9 5                                                 4    6     ...
Go to the Dump Game                                   72 7 9 5                                                 4    6     ...
Go to the Dump Game                                   72 7 9 5                                                 4    6     ...
Go to the Dump Game                                   72 7 9 5                   7    3                        4    6     ...
Go to the Dump Game                                   72 7 9 5                   2    8                        4    6     ...
Go to the Dump Game                                      72 7 9 5                    2     8                         4    ...
Go to the Dump Game                                                          BlueCap, do you                              ...
Go to the Dump Game                                                        BlueCap, do you                                ...
Go to the Dump Game                   7   3   BlueCap, do you                              have an3?                      ...
Go to the Dump Game                                         7   3   BlueCap, do you                                       ...
Go to the Dump Game                                         7   3     BlueCap, do you                                     ...
Go to the Dump Game                     7   3     BlueCap, do you                                  have an3?              ...
Go to the Dump Game                     7   3         2 2 7 9 5 2   8                   4   6     1                       ...
Go to the Dump Game                     7   3         2 2 7 9 5 2   8                   4     6     1                     ...
Go to the Dump Game                              7   3                  2 2 7 9 5         2   8                    4     6...
Go to the Dump Game                     7   3         2 2 7 9 5 2   8                   4    6     1                      ...
Go to the Dump Game                     7   3         2 2 7 9 5 2   8                   4    6     1                      ...
Go to the Dump Game                            7   3                2 2 7 9 5        2   8                   4    6       ...
Go to the Dump Game                               7   3                2 2 7      5        2   8                      4   ...
Go to the Dump Game                                7   3                 2 2 7      5        2   8                       4...
Go to the Dump Game                        7   3         2 2 7      5 2 1   8     9                      4    6           ...
Go to the Dump Game                                                          7     3                                      ...
Go to the Dump Game                   9   1 4   6                 5   5         Winner?                               © Jo...
Go to the Dump Game                                              9                                              1         ...
Go to the Dump Game                                  9                                  1                              4  ...
Go to the Dump Game                                     Next gameNo shuffling needed for next game.               © Joan A...
―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...
―Math‖ Way of Naming Numbers            11 = ten 1                          20 = 2-ten            12 = ten 2              ...
―Math‖ Way of Naming Numbers            11 = ten 1                          20 = 2-ten            12 = ten 2              ...
―Math‖ Way of Naming Numbers            11 = ten 1                          20 = 2-ten            12 = ten 2              ...
―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 7Only numbers under 100 need to be said the ―math‖ way.   ©...
―Math‖ Way of Naming Numbers          137 = 1 hundred 3-ten 7                    or        137 = 1 hundred and 3-ten 7Only...
―Math‖ Way of Naming Numbers                                                     100   Chinese                    Average ...
―Math‖ Way of Naming Numbers                                                     100   Chinese                    Average ...
―Math‖ Way of Naming Numbers                                                     100   Chinese                    Average ...
―Math‖ Way of Naming Numbers                                                      100   Chinese                     Averag...
―Math‖ Way of Naming Numbers                                                      100   Chinese                     Averag...
Math Way of Naming Numbers • Only 11 words are needed to count to 100 the math way, 28 in English. (All Indo- European lan...
Math Way of Naming Numbers • Only 11 words are needed to count to 100 the math way, 28 in English. (All Indo- European lan...
Math Way of Naming Numbers • Only 11 words are needed to count to 100 the math way, 28 in English. (All Indo- European lan...
Math Way of Naming Numbers • Only 11 words are needed to count to 100 the math way, 28 in English. (All Indo- European lan...
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 teachreading, counting...
Math Way of Naming Numbers               Compared to reading:• Just as reciting the alphabet doesn’t teachreading, countin...
Math Way of Naming Numbers               Compared to reading:• Just as reciting the alphabet doesn’t teachreading, countin...
Math Way of Naming Numbers―Rather, the increased gap between Chineseand U.S. students and that of ChineseAmericans and Cau...
Math Way of Naming NumbersResearch task:Using 10s and 1s, askthe child to construct48.                         © Joan A. C...
Math Way of Naming NumbersResearch task:Using 10s and 1s, askthe child to construct48.Then ask the child tosubtract 14.   ...
Math Way of Naming Numbers Research task: Using 10s and 1s, ask the child to construct 48. Then ask the child to subtract ...
Math Way of Naming NumbersResearch task: Using 10s and 1s, ask the child to construct 48. Then ask the child to subtract 1...
Math Way of Naming NumbersResearch task: Using 10s and 1s, ask the child to construct 48. Then ask the child to subtract 1...
Math Way of Naming NumbersResearch task: Using 10s and 1s, ask the child to construct 48. Then ask the child to subtract 1...
Math Way of Naming NumbersResearch task: Using 10s and 1s, ask the child to construct 48. Then ask the child to subtract 1...
Math Way of Naming NumbersResearch task: Using 10s and 1s, ask the child to construct 48. Then ask the child to subtract 1...
Math Way of Naming NumbersResearch task: Using 10s and 1s, ask the child to construct 48. Then ask the child to subtract 1...
Math Way of Naming NumbersResearch task: Using 10s and 1s, ask the child to construct 48. Then ask the child to subtract 1...
Math Way of Naming NumbersResearch task: Using 10s and 1s, ask the child to construct 48. Then ask the child to subtract 1...
Math Way of Naming NumbersResearch task: Using 10s and 1s, ask the child to construct 48. Then ask the child to subtract 1...
Math Way of Naming NumbersResearch task: Using 10s and 1s, ask the child to construct 48. Then ask the child to subtract 1...
Math Way of Naming Numbers            Traditional names4-ten = fortyThe ―ty‖meanstens.                                © Jo...
Math Way of Naming Numbers                             Traditional names        4-ten = forty         The ―ty‖         mea...
Math Way of Naming Numbers           Traditional names6-ten = sixtyThe ―ty‖meanstens.                               © Joan...
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 tobepronounced―twoo.‖                        ...
Math Way of Naming Numbers                           Traditional names           A word game                fireplace     ...
Math Way of Naming Numbers                           Traditional names           A word game                fireplace     ...
Math Way of Naming Numbers                           Traditional names           A word game                fireplace     ...
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    fourteen―Teen‖ alsomeans te...
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-ten30                      © Joan A. Cotter, Ph.D., 2012
Composing Numbers             3-ten             30Point to the 3 and say 3.             © Joan A. Cotter, Ph.D., 2012
Composing Numbers             3-ten             30Point to 0 and say 10. The 0 makes 3 a ten.   © Joan A. Cotter, Ph.D., 2...
Composing Numbers3-ten730                      © Joan A. Cotter, Ph.D., 2012
Composing Numbers3-ten730                      © Joan A. Cotter, Ph.D., 2012
Composing Numbers3-ten730      7                      © Joan A. Cotter, Ph.D., 2012
Composing Numbers             3-ten             7              30               7Place the 7 on top of the 0 of the 30.   ...
Composing Numbers  3-ten  7  30   7Notice the way we say the number,represent the number, and write the numberall correspo...
Composing Numbers            7-ten            8            78Another example.                       © Joan A. Cotter, Ph.D...
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 Numbers1hundred                      © Joan A. Cotter, Ph.D., 2012
Composing Numbers1hundred100                      © Joan A. Cotter, Ph.D., 2012
Composing Numbers             1             hundred             100Of course, we can also read it as one-hun-dred.   © Joa...
Composing Numbers             1             hundred             100Of course, we can also read it as one-hun-dred.   © Joa...
Composing Numbers             1             hundred             100Of course, we can also read it as one-hun-dred.   © Joa...
Composing Numbers     Reading numbers backwardTo read a number, students areoften instructed to start at theright (ones co...
Composing Numbers     Reading numbers backwardTo read a number, students areoften instructed to start at theright (ones co...
Composing Numbers     Reading numbers backwardTo read a number, students areoften instructed to start at theright (ones co...
Composing Numbers     Reading numbers backwardTo read a number, students areoften instructed to start at theright (ones co...
Composing Numbers       Reading numbers backwardTo read a number, students areoften instructed to start at theright (ones ...
Composing Numbers          Scientific Notation                                3        4000 = 4 x 10In scientific notation...
Fact Strategies                  © Joan A. Cotter, Ph.D., 2012
Fact Strategies• A strategy is a way to learn a new factor recall a forgotten fact.                                   © Jo...
Fact Strategies• A strategy is a way to learn a new factor recall a forgotten fact.• A visualizable representation is part...
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                Ten9+5=Take 1 fromthe 5 and giveit to the 9.                      ...
Fact Strategies                             Complete the                                Ten          9+5=        Take 1 fr...
Fact Strategies             Complete the                Ten9+5=Take 1 fromthe 5 and giveit to the 9.                      ...
Fact Strategies             Complete the                Ten9 + 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 Fives          8+6=Two fives make 10.                     © Joan A. Cotter, Ph....
Fact Strategies                               Two Fives           8+6=Just add the ―leftovers.‖                     © Joan...
Fact Strategies                               Two Fives           8+6=           10 + 4 = 14Just add the ―leftovers.‖     ...
Fact Strategies                      Two Fives          7+5=Another example.                     © Joan A. Cotter, Ph.D., ...
Fact Strategies          Two Fives7+5=                         © Joan A. Cotter, Ph.D., 2012
Fact Strategies             Two Fives7 + 5 = 12                         © Joan A. Cotter, Ph.D., 2012
Fact Strategies           Going Down15 – 9 =                         © Joan A. Cotter, Ph.D., 2012
Fact Strategies               Difference7–4=Subtract 4from 5; thenadd 2.                            © Joan A. Cotter, Ph.D...
Fact Strategies           Going Down15 – 9 =                         © Joan A. Cotter, Ph.D., 2012
Fact Strategies              Going Down 15 – 9 =Subtract 5;then 4.                           © Joan A. Cotter, Ph.D., 2012
Fact Strategies              Going Down 15 – 9 =Subtract 5;then 4.                           © Joan A. Cotter, Ph.D., 2012
Fact Strategies              Going Down 15 – 9 =Subtract 5;then 4.                           © Joan A. Cotter, Ph.D., 2012
Fact Strategies              Going Down 15 – 9 = 6Subtract 5;then 4.                           © Joan A. Cotter, Ph.D., 2012
Fact Strategies           Subtract from 1015 – 9 =                              © Joan A. Cotter, Ph.D., 2012
Fact Strategies             Subtract from 1015 – 9 =Subtract 9from 10.                                © Joan A. Cotter, Ph...
Fact Strategies             Subtract from 1015 – 9 =Subtract 9from 10.                                © Joan A. Cotter, Ph...
Fact Strategies             Subtract from 1015 – 9 =Subtract 9from 10.                                © Joan A. Cotter, Ph...
Fact Strategies             Subtract from 1015 – 9 = 6Subtract 9from 10.                                © Joan A. Cotter, ...
Fact Strategies           Going Up13 – 9 =                         © Joan A. Cotter, Ph.D., 2012
Fact Strategies                Going Up 13 – 9 =Start with 9;go up to 13.                            © Joan A. Cotter, Ph....
Fact Strategies                Going Up 13 – 9 =Start with 9;go up to 13.                            © Joan A. Cotter, Ph....
Fact Strategies                Going Up 13 – 9 =Start with 9;go up to 13.                            © Joan A. Cotter, Ph....
Fact Strategies                Going Up 13 – 9 =Start with 9;go up to 13.                            © Joan A. Cotter, Ph....
Fact Strategies                Going Up 13 – 9 = 1+3=4Start with 9;go up to 13.                            © Joan A. Cotte...
MoneyPenny        © Joan A. Cotter, Ph.D., 2012
MoneyNickel         © Joan A. Cotter, Ph.D., 2012
Money Dime        © Joan A. Cotter, Ph.D., 2012
MoneyQuarter          © Joan A. Cotter, Ph.D., 2012
MoneyQuarter          © Joan A. Cotter, Ph.D., 2012
MoneyQuarter          © Joan A. Cotter, Ph.D., 2012
MoneyQuarter          © Joan A. Cotter, Ph.D., 2012
Base-10 Picture Cards                        One                        © Joan A. Cotter, Ph.D., 2012
Base-10 Picture Cards             Ten        One                        © Joan A. Cotter, Ph.D., 2012
Base-10 Picture Cards   Hundred   Ten        One                        © Joan A. Cotter, Ph.D., 2012
Base-10 Picture CardsThousan   Hundred   Ten      One   d                             © Joan A. Cotter, Ph.D., 2012
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NDMA 2012

  1. 1. RightStart™ Mathematics in a Montessori Environment by Joan A. Cotter, Ph.D. JoanCotter@RightStartMath.com 7x7 3 2 5 5 New DiscoveriesMontessori Academy August 31, 2012Hutchinson, Minnesota Other presentations available: rightstartmath.com Cotter, Ph.D., 2012 © Joan A.
  2. 2. National Math Crisis © Joan A. Cotter, Ph.D., 2012
  3. 3. National Math Crisis• 25% of college freshmen take remedial math. © Joan A. Cotter, Ph.D., 2012
  4. 4. National Math Crisis• 25% of college freshmen take remedial math.• In 2009, of the 1.5 million students who tookthe ACT test, only 42% are ready for collegealgebra. © Joan A. Cotter, Ph.D., 2012
  5. 5. National Math Crisis• 25% of college freshmen take remedial math.• In 2009, of the 1.5 million students who tookthe ACT test, only 42% are ready for collegealgebra.• A generation ago, the US produced 30% of theworld’s college grads; today it’s 14%. (CSM 2006) © Joan A. Cotter, Ph.D., 2012
  6. 6. National Math Crisis• 25% of college freshmen take remedial math.• In 2009, of the 1.5 million students who tookthe ACT test, only 42% are ready for collegealgebra.• A generation ago, the US produced 30% of theworld’s college grads; today it’s 14%. (CSM 2006)• Two-thirds of 4-year degrees in Japan andChina are in science and engineering; one-thirdin the U.S. © Joan A. Cotter, Ph.D., 2012
  7. 7. National Math Crisis• 25% of college freshmen take remedial math.• In 2009, of the 1.5 million students who tookthe ACT test, only 42% are ready for collegealgebra.• A generation ago, the US produced 30% of theworld’s college grads; today it’s 14%. (CSM 2006)• Two-thirds of 4-year degrees in Japan andChina are in science and engineering; one-thirdin the U.S.• U.S. students, compared to the world, scorehigh at 4th grade, average at 8th, and nearbottom at 12th. © Joan A. Cotter, Ph.D., 2012
  8. 8. National Math Crisis• 25% of college freshmen take remedial math.• In 2009, of the 1.5 million students who tookthe ACT test, only 42% are ready for collegealgebra.• A generation ago, the US produced 30% of theworld’s college grads; today it’s 14%. (CSM 2006)• Two-thirds of 4-year degrees in Japan andChina are in science and engineering; one-thirdin the U.S.• U.S. students, compared to the world, scorehigh at 4th grade, average at 8th, and nearbottom at 12th.• Ready, Willing, and Unable to Serve says that75% of 17 to 24 year-olds are unfit for militaryservice. (2010) © Joan A. Cotter, Ph.D., 2012
  9. 9. Math Education is Changing © Joan A. Cotter, Ph.D., 2012
  10. 10. Math Education is Changing• The field of mathematics is doubling every 7years. © Joan A. Cotter, Ph.D., 2012
  11. 11. Math Education is Changing• The field of mathematics is doubling every 7years.• Math is used in many new ways. Theworkplace needs analytical thinkers andproblem solvers. © Joan A. Cotter, Ph.D., 2012
  12. 12. Math Education is Changing• The field of mathematics is doubling every 7years.• Math is used in many new ways. Theworkplace needs analytical thinkers andproblem solvers.• State exams require more than arithmetic:including geometry, algebra, probability, andstatistics. © Joan A. Cotter, Ph.D., 2012
  13. 13. Math Education is Changing• The field of mathematics is doubling every 7years.• Math is used in many new ways. Theworkplace needs analytical thinkers andproblem solvers.• State exams require more than arithmetic:including geometry, algebra, probability, andstatistics.• Brain research is providing clues on how tobetter facilitate learning, including math. © Joan A. Cotter, Ph.D., 2012
  14. 14. Math Education is Changing• The field of mathematics is doubling every 7years.• Math is used in many new ways. Theworkplace needs analytical thinkers andproblem solvers.• State exams require more than arithmetic:including geometry, algebra, probability, andstatistics.• Brain research is providing clues on how tobetter facilitate learning, including math.• Calculators and computers have madecomputation with many digits an unneeded skill. © Joan A. Cotter, Ph.D., 2012
  15. 15. Math Education is Changing• The field of mathematics is doubling every 7years.• Math is used in many new ways. Theworkplace needs analytical thinkers andproblem solvers.• State exams require more than arithmetic:including geometry, algebra, probability, andstatistics.• Brain research is providing clues on how tobetter facilitate learning, including math.• Calculators and computers have madecomputation with many digits an unneeded skill. © Joan A. Cotter, Ph.D., 2012
  16. 16. Counting Model © Joan A. Cotter, Ph.D., 2012
  17. 17. 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
  18. 18. Counting ModelFrom a childs perspective F +E © Joan A. Cotter, Ph.D., 2012
  19. 19. Counting Model From a childs perspective F +EA © Joan A. Cotter, Ph.D., 2012
  20. 20. Counting Model From a childs perspective F +EA B © Joan A. Cotter, Ph.D., 2012
  21. 21. Counting Model From a childs perspective F +EA B C © Joan A. Cotter, Ph.D., 2012
  22. 22. Counting Model From a childs perspective F +EA B C D E F © Joan A. Cotter, Ph.D., 2012
  23. 23. Counting Model From a childs perspective F +EA B C D E F A © Joan A. Cotter, Ph.D., 2012
  24. 24. Counting Model From a childs perspective F +EA B C D E F A B © Joan A. Cotter, Ph.D., 2012
  25. 25. Counting Model From a childs perspective F +EA B C D E F A B C D E © Joan A. Cotter, Ph.D., 2012
  26. 26. 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
  27. 27. 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
  28. 28. Counting Model From a childs perspectiveNow memorize the facts!! G +D © Joan A. Cotter, Ph.D., 2012
  29. 29. Counting Model From a childs perspectiveNow memorize the facts!! G +D © Joan A. Cotter, Ph.D., 2012
  30. 30. Counting Model From a childs perspectiveNow memorize the facts!! G +D D+C © Joan A. Cotter, Ph.D., 2012
  31. 31. Counting Model From a childs perspectiveNow memorize the facts!! G +D D C+C +G © Joan A. Cotter, Ph.D., 2012
  32. 32. Counting Model From a childs perspectiveNow memorize the facts!! G +D D C+C +G © Joan A. Cotter, Ph.D., 2012
  33. 33. Counting Model From a childs perspectiveTry subtracting Hby ―taking away‖ – E © Joan A. Cotter, Ph.D., 2012
  34. 34. Counting Model From a childs perspectiveTry skip counting by B’s to T: B, D, . . . T. © Joan A. Cotter, Ph.D., 2012
  35. 35. 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
  36. 36. Counting Model From a childs perspectiveLis written ABbecause it is A Jand B A’s © Joan A. Cotter, Ph.D., 2012
  37. 37. Counting Model From a childs perspectiveLis written ABbecause it is A Jand B A’s huh? © Joan A. Cotter, Ph.D., 2012
  38. 38. Counting Model From a childs perspectiveL (twelve)is written ABbecause it is A Jand B A’s © Joan A. Cotter, Ph.D., 2012
  39. 39. 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
  40. 40. Counting Model From a childs perspectiveL (twelve)is written AB (12) (one 10)because it is A Jand B A’s © Joan A. Cotter, Ph.D., 2012
  41. 41. Counting Model From a childs perspectiveL (twelve)is written AB (12) (one 10)because it is A Jand B A’s (two 1s). © Joan A. Cotter, Ph.D., 2012
  42. 42. 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
  43. 43. Counting Model Summary © Joan A. Cotter, Ph.D., 2012
  44. 44. Counting Model Summary• Is not natural; it takes years ofpractice. © Joan A. Cotter, Ph.D., 2012
  45. 45. Counting Model Summary• Is not natural; it takes years ofpractice.• Provides poor concept of quantity. © Joan A. Cotter, Ph.D., 2012
  46. 46. Counting Model Summary• Is not natural; it takes years ofpractice.• Provides poor concept of quantity.• Ignores place value. © Joan A. Cotter, Ph.D., 2012
  47. 47. Counting Model Summary• Is not natural; it takes years ofpractice.• Provides poor concept of quantity.• Ignores place value.• Is very error prone. © Joan A. Cotter, Ph.D., 2012
  48. 48. Counting Model Summary• Is not natural; it takes years ofpractice.• Provides poor concept of quantity.• Ignores place value.• Is very error prone.• Is tedious and time-consuming. © Joan A. Cotter, Ph.D., 2012
  49. 49. Counting Model Summary• Is not natural; it takes years ofpractice.• Provides poor concept of quantity.• Ignores place value.• Is very error prone.• Is tedious and time-consuming.• Does not provide an efficientway to master the facts. © Joan A. Cotter, Ph.D., 2012
  50. 50. Calendar Math August 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31Sometimes calendars are used for counting. © Joan A. Cotter, Ph.D., 2012
  51. 51. Calendar Math August 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31Sometimes calendars are used for counting. © Joan A. Cotter, Ph.D., 2012
  52. 52. 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
  53. 53. Calendar Math August 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31This is ordinal, not cardinal counting. The 3 doesn’t include the 1 and the© 2. A. Cotter, Ph.D., 2012 Joan
  54. 54. Calendar Math August 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31This is ordinal, not cardinal counting. The 4 doesn’t include 1, 2 and 3. © Joan A. Cotter, Ph.D., 2012
  55. 55. 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 31 1 2 3 4 5 6A calendar is NOT a ruler. On a ruler the numbers are not in the spaces. © Joan A. Cotter, Ph.D., 2012
  56. 56. Calendar Math August 1 2 3 4 5 6 7 8 9 10Always show the whole calendar. A child needs to see the wholebefore the parts. Children also need to learn to plan ahead. © Joan A. Cotter, Ph.D., 2012
  57. 57. Calendar MathThe calendar is not a number line. • No quantity is involved. • Numbers are in spaces, not at lines like a ruler. © Joan A. Cotter, Ph.D., 2012
  58. 58. Calendar MathThe calendar is not a number line. • No quantity is involved. • Numbers are in spaces, not at lines like a ruler.Children need to see the whole month, not justpart. • Purpose of calendar is to plan ahead. • Many ways to show the current date. © Joan A. Cotter, Ph.D., 2012
  59. 59. Calendar MathThe calendar is not a number line. • No quantity is involved. • Numbers are in spaces, not at lines like a ruler.Children need to see the whole month, not justpart. • Purpose of calendar is to plan ahead. • Many ways to show the current date.Calendars give a narrow view of patterning. • Patterns do not necessarily involve numbers. © Joan A. Cotter, Ph.D., 2012
  60. 60. Memorizing Math Percentage Recall Immediatel After 1 day After 4 wks yRote 32 23 8 69 69 58Concept © Joan A. Cotter, Ph.D., 2012
  61. 61. Memorizing Math Percentage Recall Immediatel After 1 day After 4 wks yRote 32 23 8 69 69 58Concept © Joan A. Cotter, Ph.D., 2012
  62. 62. Memorizing Math Percentage Recall Immediatel After 1 day After 4 wks yRote 32 23 8 69 69 58Concept © Joan A. Cotter, Ph.D., 2012
  63. 63. Memorizing Math Percentage Recall Immediatel After 1 day After 4 wks yRote 32 23 8 69 69 58Concept © Joan A. Cotter, Ph.D., 2012
  64. 64. Memorizing Math Percentage Recall Immediatel After 1 day After 4 wks yRote 32 23 8 69 69 58Concept © Joan A. Cotter, Ph.D., 2012
  65. 65. Memorizing Math Percentage Recall Immediatel After 1 day After 4 wks yRote 32 23 8 69 69 58Concept © Joan A. Cotter, Ph.D., 2012
  66. 66. Memorizing Math Percentage Recall Immediatel After 1 day After 4 wks yRote 32 23 8 69 69 58Concept Math needs to be taught so 95% is understood and only 5% memorized. Richard Skemp © Joan A. Cotter, Ph.D., 2012
  67. 67. 9Memorizing Math +7 Flash cards: © Joan A. Cotter, Ph.D., 2012
  68. 68. 9 Memorizing Math +7 Flash cards:• Are often used to teach rote. © Joan A. Cotter, Ph.D., 2012
  69. 69. 9 Memorizing Math +7 Flash cards:• Are often used to teach rote.• Are liked only by those who don’t needthem. © Joan A. Cotter, Ph.D., 2012
  70. 70. 9 Memorizing Math +7 Flash cards:• Are often used to teach rote.• Are liked only by those who don’t needthem.• Don’t work for those with learningdisabilities. © Joan A. Cotter, Ph.D., 2012
  71. 71. 9 Memorizing Math +7 Flash cards:• Are often used to teach rote.• Are liked only by those who don’t needthem.• Don’t work for those with learningdisabilities.• Give the false impression that math isn’tabout thinking. © Joan A. Cotter, Ph.D., 2012
  72. 72. 9 Memorizing Math +7 Flash cards:• Are often used to teach rote.• Are liked only by those who don’t needthem.• Don’t work for those with learningdisabilities.• Give the false impression that math isn’tabout thinking.• Often produce stress – children understress stop learning. © Joan A. Cotter, Ph.D., 2012
  73. 73. 9 Memorizing Math +7 Flash cards:• Are often used to teach rote.• Are liked only by those who don’t needthem.• Don’t work for those with learningdisabilities.• Give the false impression that math isn’tabout thinking.• Often produce stress – children understress stop learning. © Joan A. Cotter, Ph.D., 2012
  74. 74. Research on Counting Karen Wynn’s researchShow the baby two teddy bears. © Joan A. Cotter, Ph.D., 2012
  75. 75. Research on Counting Karen Wynn’s researchThen hide them with a screen. © Joan A. Cotter, Ph.D., 2012
  76. 76. Research on Counting Karen Wynn’s researchShow the baby a third teddy bear and put it behind the screen. © Joan A. Cotter, Ph.D., 2012
  77. 77. Research on Counting Karen Wynn’s researchShow the baby a third teddy bear and put it behind the screen. © Joan A. Cotter, Ph.D., 2012
  78. 78. Research on Counting Karen Wynn’s researchRaise screen. Baby seeing 3 won’t look long because it is expected. © Joan A. Cotter, Ph.D., 2012
  79. 79. Research on Counting Karen Wynn’s researchResearcher can change the number of teddy bears behind the screen. © Joan A. Cotter, Ph.D., 2012
  80. 80. Research on Counting Karen Wynn’s researchA baby seeing 1 teddy bear will look much longer, because it’s unexpected. A. Cotter, Ph.D., 2012 © Joan
  81. 81. Research on Counting Other research © Joan A. Cotter, Ph.D., 2012
  82. 82. Research on Counting Other research • Australian Aboriginal children from two tribes. Brian Butterworth, University College London, 2008.These groups matched quantities without using counting words. © Joan A. Cotter, Ph.D., 2012
  83. 83. Research on Counting Other research • Australian Aboriginal children from two tribes. Brian Butterworth, University College London, 2008. • Adult Pirahã from Amazon region. Edward Gibson and Michael Frank, MIT, 2008.These groups matched quantities without using counting words. © Joan A. Cotter, Ph.D., 2012
  84. 84. Research on Counting Other research • Australian Aboriginal children from two tribes. Brian Butterworth, University College London, 2008. • Adult Pirahã from Amazon region. Edward Gibson and Michael Frank, MIT, 2008. • Adults, ages 18-50, from Boston. Edward Gibson and Michael Frank, MIT, 2008.These groups matched quantities without using counting words. © Joan A. Cotter, Ph.D., 2012
  85. 85. Research on Counting Other research • Australian Aboriginal children from two tribes. Brian Butterworth, University College London, 2008. • Adult Pirahã from Amazon region. Edward Gibson and Michael Frank, MIT, 2008. • Adults, ages 18-50, from Boston. Edward Gibson and Michael Frank, MIT, 2008. • Baby chicks from Italy. Lucia Regolin, University of Padova, 2009.These groups matched quantities without using counting words. © Joan A. Cotter, Ph.D., 2012
  86. 86. Research on Counting In Japanese schools:• Children are discouraged from usingcounting for adding. © Joan A. Cotter, Ph.D., 2012
  87. 87. Research on Counting In Japanese schools:• Children are discouraged from usingcounting for adding.• They consistently group in 5s. © Joan A. Cotter, Ph.D., 2012
  88. 88. Research on Counting Subitizing• Subitizing is quick recognition of quantitywithout counting. © Joan A. Cotter, Ph.D., 2012
  89. 89. Research on Counting Subitizing• Subitizing is quick recognition of quantitywithout counting.• Human babies and some animals cansubitize small quantities at birth. © Joan A. Cotter, Ph.D., 2012
  90. 90. Research on Counting Subitizing• Subitizing is quick recognition of quantitywithout counting.• Human babies and some animals cansubitize small quantities at birth.• Children who can subitize perform better inmathematics.—Butterworth © Joan A. Cotter, Ph.D., 2012
  91. 91. Research on Counting Subitizing• Subitizing is quick recognition of quantitywithout counting.• Human babies and some animals cansubitize small quantities at birth.• Children who can subitize perform better inmathematics.—Butterworth• Subitizing ―allows the child to grasp thewhole and the elements at the same time.‖—Benoit © Joan A. Cotter, Ph.D., 2012
  92. 92. Research on Counting Subitizing• Subitizing is quick recognition of quantitywithout counting.• Human babies and some animals cansubitize small quantities at birth.• Children who can subitize perform better inmathematics.—Butterworth• Subitizing ―allows the child to grasp thewhole and the elements at the same time.‖—Benoit• Subitizing seems to be a necessary skill forunderstanding what the counting processmeans.—Glasersfeld © Joan A. Cotter, Ph.D., 2012
  93. 93. Research on Counting Finger gnosia• Finger gnosia is the ability to know whichfingers can been lightly touched withoutlooking. © Joan A. Cotter, Ph.D., 2012
  94. 94. Research on Counting Finger gnosia• Finger gnosia is the ability to know whichfingers can been lightly touched withoutlooking.• Part of the brain controlling fingers isadjacent to math part of the brain. © Joan A. Cotter, Ph.D., 2012
  95. 95. Research on Counting Finger gnosia• Finger gnosia is the ability to know whichfingers can been lightly touched withoutlooking.• Part of the brain controlling fingers isadjacent to math part of the brain.• Children who use their fingers asrepresentational tools perform better inmathematics—Butterworth © Joan A. Cotter, Ph.D., 2012
  96. 96. Visualizing Mathematics © Joan A. Cotter, Ph.D., 2012
  97. 97. Visualizing Mathematics―In our concern about the memorizationof math facts or solving problems, wemust not forget that the root ofmathematical study is the creation ofmental pictures in the imagination andmanipulating those images andrelationships using the power of reasonand logic.‖ Mindy Holte (E1) © Joan A. Cotter, Ph.D., 2012
  98. 98. Visualizing Mathematics―Think in pictures, because thebrain remembers images betterthan it does anything else.‖Ben Pridmore, World Memory Champion, 2009 © Joan A. Cotter, Ph.D., 2012
  99. 99. Visualizing Mathematics―Mathematics is the activity ofcreating relationships, many ofwhich are based in visual imagery.” Wheatley and Cobb © Joan A. Cotter, Ph.D., 2012
  100. 100. Visualizing Mathematics―The process of connectingsymbols to imagery is at the heartof mathematics learning.‖ Dienes © Joan A. Cotter, Ph.D., 2012
  101. 101. Visualizing Mathematics―The role of physicalmanipulatives was to help thechild form those visual imagesand thus to eliminate the need forthe physical manipulatives.‖ Ginsberg and others © Joan A. Cotter, Ph.D., 2012
  102. 102. Visualizing Mathematics Japanese criteria for manipulatives• Representative of structure ofnumbers.• Easily manipulated by children.• Imaginable mentally. Japanese Council of Mathematics Education © Joan A. Cotter, Ph.D., 2012
  103. 103. Visualizing Mathematics Visualizing also needed in:• Reading• Sports• Creativity• Geography• Engineering• Construction © Joan A. Cotter, Ph.D., 2012
  104. 104. Visualizing Mathematics Visualizing also needed in:• Reading • Architecture• Sports • Astronomy• Creativity • Archeology• Geography • Chemistry• Engineering • Physics• Construction • Surgery © Joan A. Cotter, Ph.D., 2012
  105. 105. Visualizing Mathematics Ready: How many? © Joan A. Cotter, Ph.D., 2012
  106. 106. Visualizing Mathematics Ready: How many? © Joan A. Cotter, Ph.D., 2012
  107. 107. Visualizing Mathematics Try again: How many? © Joan A. Cotter, Ph.D., 2012
  108. 108. Visualizing Mathematics Try again: How many? © Joan A. Cotter, Ph.D., 2012
  109. 109. Visualizing Mathematics Try again: How many? © Joan A. Cotter, Ph.D., 2012
  110. 110. Visualizing Mathematics Ready: How many? © Joan A. Cotter, Ph.D., 2012
  111. 111. Visualizing Mathematics Try again: How many? © Joan A. Cotter, Ph.D., 2012
  112. 112. Visualizing MathematicsTry to visualize 8 identical apples without grouping. © Joan A. Cotter, Ph.D., 2012
  113. 113. Visualizing MathematicsTry to visualize 8 identical apples without grouping. © Joan A. Cotter, Ph.D., 2012
  114. 114. Visualizing MathematicsNow try to visualize 5 as red and 3 as green. © Joan A. Cotter, Ph.D., 2012
  115. 115. Visualizing MathematicsNow try to visualize 5 as red and 3 as green. © Joan A. Cotter, Ph.D., 2012
  116. 116. Visualizing Mathematics Early Roman numerals 1 I 2 II 3 III 4 IIII 5 V 8 VIIIRomans grouped in fives. Notice 8 is 5 and 3. © Joan A. Cotter, Ph.D., 2012
  117. 117. Visualizing Mathematics : Who could read the music?Music needs 10 lines, two groups of five. © Joan A. Cotter, Ph.D., 2012
  118. 118. Research on Counting Teach Counting• Finger gnosia is the ability to know whichfingers can been lightly touched withoutlooking.• Part of the brain controlling fingers isadjacent to math part of the brain.• Children who use their fingers asrepresentational tools perform better inmathematics—Butterworth © Joan A. Cotter, Ph.D., 2012
  119. 119. Very Early Computation NumeralsIn English there are two ways of writing numbers: 3578 Three thousand five hundred seventy eight © Joan A. Cotter, Ph.D., 2012
  120. 120. Very Early Computation Numerals In English there are two ways of writing numbers: 3578 Three thousand five hundred seventy eightIn Chinese there is only one way of writing numbers: 3 Th 5 H 7 T 8 U (8 characters) © Joan A. Cotter, Ph.D., 2012
  121. 121. Very Early Computation Calculating rodsBecause their characters arecumbersome to use for computing, theChinese used calculating rods,beginning in the 4th century BC. © Joan A. Cotter, Ph.D., 2012
  122. 122. Very Early Computation Calculating rods © Joan A. Cotter, Ph.D., 2012
  123. 123. Very Early Computation Calculating rodsNumerals for Ones and Hundreds (Even Powers of Ten) © Joan A. Cotter, Ph.D., 2012
  124. 124. Very Early Computation Calculating rodsNumerals for Ones and Hundreds (Even Powers of Ten) © Joan A. Cotter, Ph.D., 2012
  125. 125. Very Early Computation Calculating rodsNumerals for Ones and Hundreds (Odd Powers of Ten)Numerals for Tens and Thousands (Odd Powers of Ten) © Joan A. Cotter, Ph.D., 2012
  126. 126. Very Early Computation Calculating rods3578 © Joan A. Cotter, Ph.D., 2012
  127. 127. Very Early Computation Calculating rods 3578 3578,3578They grouped, not in thousands, but ten-thousands! © Joan A. Cotter, Ph.D., 2012
  128. 128. Naming Quantities Using fingers © Joan A. Cotter, Ph.D., 2012
  129. 129. Naming Quantities Using fingersNaming quantities is a three-period lesson. © Joan A. Cotter, Ph.D., 2012
  130. 130. Naming Quantities Using fingersUse left hand for 1-5 because we read from left to right. © Joan A. Cotter, Ph.D., 2012
  131. 131. Naming Quantities Using fingers © Joan A. Cotter, Ph.D., 2012
  132. 132. Naming Quantities Using fingers © Joan A. Cotter, Ph.D., 2012
  133. 133. Naming Quantities Using fingersAlways show 7 as 5 and 2, not for example, as 4 and 3. © Joan A. Cotter, Ph.D., 2012
  134. 134. Naming Quantities Using fingers © Joan A. Cotter, Ph.D., 2012
  135. 135. Naming Quantities Yellow is the Sun Yellow is the sun. Six is five and one. Why is the sky so blue? Seven is five and two. Salty is the sea. Eight is five and three. Hear the thunder roar. Nine is five and four. Ducks will swim and dive. Ten is five and five. –Joan A. CotterAlso set to music. Listen and download sheet music from Web site. © Joan A. Cotter, Ph.D., 2012
  136. 136. Naming Quantities Recognizing 5 © Joan A. Cotter, Ph.D., 2012
  137. 137. Naming Quantities Recognizing 5 © Joan A. Cotter, Ph.D., 2012
  138. 138. Naming Quantities Recognizing 5 5 has a middle; 4 does not.Look at your hand; your middle finger is longer to remind you 5 has a middle.A. Cotter, Ph.D., 2012 © Joan
  139. 139. Naming Quantities Tally sticksLay the sticks flat on a surface, about 1 inch (2.5 cm) apart. © Joan A. Cotter, Ph.D., 2012
  140. 140. Naming Quantities Tally sticks © Joan A. Cotter, Ph.D., 2012
  141. 141. Naming Quantities Tally sticks © Joan A. Cotter, Ph.D., 2012
  142. 142. Naming Quantities Tally sticksStick is horizontal, because it won’t fit diagonally and young children haveproblems with diagonals. © Joan A. Cotter, Ph.D., 2012
  143. 143. Naming Quantities Tally sticks © Joan A. Cotter, Ph.D., 2012
  144. 144. Naming Quantities Tally sticksStart a new row for every ten. © Joan A. Cotter, Ph.D., 2012
  145. 145. Naming Quantities Solving a problem without counting What is 4 apples plus 3 more apples?How would you find the answer without counting? © Joan A. Cotter, Ph.D., 2012
  146. 146. Naming Quantities Solving a problem without counting What is 4 apples plus 3 more apples?To remember 4 + 3, the Japanese child is taught to visualize 4 and 3. Thentake 1 from the 3 and give it to the 4 to make 5 and 2. © Joan A. Cotter, Ph.D., 2012
  147. 147. Naming QuantitiesNumber 1Chart 2 3 4 5 © Joan A. Cotter, Ph.D., 2012
  148. 148. Naming Quantities Numbe r 1 Chart 2To help the 3child learnthesymbols 4 5 © Joan A. Cotter, Ph.D., 2012
  149. 149. Naming Quantities Numbe r 1 6 Chart 2 7To help the 3 8child learnthesymbols 4 9 5 10 © Joan A. Cotter, Ph.D., 2012
  150. 150. Naming Quantities Pairing Finger CardsUse two sets of finger cards and match them. © Joan A. Cotter, Ph.D., 2012
  151. 151. Naming Quantities Ordering Finger CardsPutting the finger cards in order. © Joan A. Cotter, Ph.D., 2012
  152. 152. Naming Quantities Matching Numbers to Finger Cards 5 1 10Match the number to the finger card. © Joan A. Cotter, Ph.D., 2012
  153. 153. Naming Quantities Matching Fingers to Number Cards 9 1 10 4 6 2 3 7 8 5Match the finger card to the number. © Joan A. Cotter, Ph.D., 2012
  154. 154. Naming Quantities Finger Card Memory gameUse two sets of finger cards and play Memory. © Joan A. Cotter, Ph.D., 2012
  155. 155. Naming Quantities Number Rods © Joan A. Cotter, Ph.D., 2012
  156. 156. Naming Quantities Number Rods © Joan A. Cotter, Ph.D., 2012
  157. 157. Naming Quantities Number RodsUsing different colors. © Joan A. Cotter, Ph.D., 2012
  158. 158. Naming Quantities Spindle Box45 dark-colored and 10 light-colored spindles. Could be in separate containers. Ph.D., 2012 © Joan A. Cotter,
  159. 159. Naming Quantities Spindle Box45 dark-colored and 10 light-colored spindles in two containers. © Joan A. Cotter, Ph.D., 2012
  160. 160. Naming Quantities Spindle Box 0 1 2 3 4The child takes blue spindles with left hand and yellow with right. © Joan A. Cotter, Ph.D., 2012
  161. 161. Naming Quantities Spindle Box 5 6 7 8 9The child takes blue spindles with left hand and yellow with right. © Joan A. Cotter, Ph.D., 2012
  162. 162. Naming Quantities Spindle Box 5 6 7 8 9The child takes blue spindles with left hand and yellow with right. © Joan A. Cotter, Ph.D., 2012
  163. 163. Naming Quantities Spindle Box 5 6 7 8 9The child takes blue spindles with left hand and yellow with right. © Joan A. Cotter, Ph.D., 2012
  164. 164. Naming Quantities Spindle Box 5 6 7 8 9The child takes blue spindles with left hand and yellow with right. © Joan A. Cotter, Ph.D., 2012
  165. 165. Naming Quantities Spindle Box 5 6 7 8 9The child takes blue spindles with left hand and yellow with right. © Joan A. Cotter, Ph.D., 2012
  166. 166. Naming Quantities Spindle Box 5 6 7 8 9The child takes blue spindles with left hand and yellow with right. © Joan A. Cotter, Ph.D., 2012
  167. 167. Naming Quantities Black and White Bead Stairs ―Grouped in fives so the child does not need to count.‖ A. M. JoostenThis was the inspiration to group in 5s. © Joan A. Cotter, Ph.D., 2012
  168. 168. AL Abacus 1000 100 10 1Double-sided AL abacus. Side 1 is grouped in 5s.Trading Side introduces algorithms with trading. © Joan A. Cotter, Ph.D., 2012
  169. 169. AL Abacus Cleared © Joan A. Cotter, Ph.D., 2012
  170. 170. AL Abacus Entering quantities 3Quantities are entered all at once, not counted. © Joan A. Cotter, Ph.D., 2012
  171. 171. AL Abacus Entering quantities 5Relate quantities to hands. © Joan A. Cotter, Ph.D., 2012
  172. 172. AL Abacus Entering quantities7 © Joan A. Cotter, Ph.D., 2012
  173. 173. AL Abacus Entering quantities10 © Joan A. Cotter, Ph.D., 2012
  174. 174. AL Abacus The stairsCan use to ―count‖ 1 to 10. Also read quantities on the right side. © Joan A. Cotter, Ph.D., 2012
  175. 175. AL Abacus Adding © Joan A. Cotter, Ph.D., 2012
  176. 176. AL Abacus Adding4+3= © Joan A. Cotter, Ph.D., 2012
  177. 177. AL Abacus Adding4+3= © Joan A. Cotter, Ph.D., 2012
  178. 178. AL Abacus Adding4+3= © Joan A. Cotter, Ph.D., 2012
  179. 179. AL Abacus Adding4+3= © Joan A. Cotter, Ph.D., 2012
  180. 180. AL Abacus Adding 4+3=7Answer is seen immediately, no counting needed. © Joan A. Cotter, Ph.D., 2012
  181. 181. Go to the Dump Game Aim: To learn the facts that total 10: 1+9 2+8 3+7 4+6 5+5Children use the abacus while playing this ―Go Fish‖ type game. © Joan A. Cotter, Ph.D., 2012
  182. 182. Go to the Dump Game Aim: To learn the facts that total 10: 1+9 2+8 3+7 4+6 5+5 Object of the game: To collect the most pairs that equal ten.Children use the abacus while playing this ―Go Fish‖ type game. © Joan A. Cotter, Ph.D., 2012
  183. 183. Go to the Dump GameThe ways to partition 10. © Joan A. Cotter, Ph.D., 2012
  184. 184. Go to the Dump Game StartingA game viewed from above. © Joan A. Cotter, Ph.D., 2012
  185. 185. Go to the Dump Game 72 7 9 5 72 1 3 8 4 6 34 9 StartingEach player takes 5 cards. © Joan A. Cotter, Ph.D., 2012
  186. 186. Go to the Dump Game 72 7 9 5 72 1 3 8 4 6 34 9 Finding pairsDoes YellowCap have any pairs? [no] © Joan A. Cotter, Ph.D., 2012
  187. 187. Go to the Dump Game 72 7 9 5 72 1 3 8 4 6 34 9 Finding pairsDoes BlueCap have any pairs? [yes, 1] © Joan A. Cotter, Ph.D., 2012
  188. 188. Go to the Dump Game 72 7 9 5 72 1 3 8 4 6 34 9 Finding pairsDoes BlueCap have any pairs? [yes, 1] © Joan A. Cotter, Ph.D., 2012
  189. 189. Go to the Dump Game 72 7 9 5 4 6 72 1 3 8 34 9 Finding pairsDoes BlueCap have any pairs? [yes, 1] © Joan A. Cotter, Ph.D., 2012
  190. 190. Go to the Dump Game 72 7 9 5 4 6 72 1 3 8 34 9 Finding pairsDoes PinkCap have any pairs? [yes, 2] © Joan A. Cotter, Ph.D., 2012
  191. 191. Go to the Dump Game 72 7 9 5 4 6 72 1 3 8 34 9 Finding pairsDoes PinkCap have any pairs? [yes, 2] © Joan A. Cotter, Ph.D., 2012
  192. 192. Go to the Dump Game 72 7 9 5 7 3 4 6 2 1 8 34 9 Finding pairsDoes PinkCap have any pairs? [yes, 2] © Joan A. Cotter, Ph.D., 2012
  193. 193. Go to the Dump Game 72 7 9 5 2 8 4 6 1 34 9 Finding pairsDoes PinkCap have any pairs? [yes, 2] © Joan A. Cotter, Ph.D., 2012
  194. 194. Go to the Dump Game 72 7 9 5 2 8 4 6 1 34 9 PlayingThe player asks the player on her left. © Joan A. Cotter, Ph.D., 2012
  195. 195. Go to the Dump Game BlueCap, do you have an3? have a 3? 72 7 9 5 2 8 4 6 1 34 9 PlayingThe player asks the player on her left. © Joan A. Cotter, Ph.D., 2012
  196. 196. Go to the Dump Game BlueCap, do you have an3? have a 3? 72 7 9 5 3 2 8 4 6 1 4 9 PlayingThe player asks the player on her left. © Joan A. Cotter, Ph.D., 2012
  197. 197. Go to the Dump Game 7 3 BlueCap, do you have an3? have a 3? 2 7 9 5 2 8 4 6 1 4 9 Playing © Joan A. Cotter, Ph.D., 2012
  198. 198. Go to the Dump Game 7 3 BlueCap, do you have an3? have a 8? 2 7 9 5 2 8 4 6 1 4 9 PlayingYellowCap gets another turn. © Joan A. Cotter, Ph.D., 2012
  199. 199. Go to the Dump Game 7 3 BlueCap, do you have an3? have a 8? 2 7 9 5 2 8 4 6 1 4 9 Go to the dump. PlayingYellowCap gets another turn. © Joan A. Cotter, Ph.D., 2012
  200. 200. Go to the Dump Game 7 3 BlueCap, do you have an3? have a 8? 2 2 7 9 5 2 8 4 6 1 4 9 Go to the dump. Playing © Joan A. Cotter, Ph.D., 2012
  201. 201. Go to the Dump Game 7 3 2 2 7 9 5 2 8 4 6 1 4 9 Playing © Joan A. Cotter, Ph.D., 2012
  202. 202. Go to the Dump Game 7 3 2 2 7 9 5 2 8 4 6 1 4 9 PinkCap, do you Playing have a 6? © Joan A. Cotter, Ph.D., 2012
  203. 203. Go to the Dump Game 7 3 2 2 7 9 5 2 8 4 6 1 4 9 PinkCap, do youGo to the dump. Playing have a 6? © Joan A. Cotter, Ph.D., 2012
  204. 204. Go to the Dump Game 7 3 2 2 7 9 5 2 8 4 6 1 5 4 9 Playing © Joan A. Cotter, Ph.D., 2012
  205. 205. Go to the Dump Game 7 3 2 2 7 9 5 2 8 4 6 1 5 4 9 Playing © Joan A. Cotter, Ph.D., 2012
  206. 206. Go to the Dump Game 7 3 2 2 7 9 5 2 8 4 6 1 5 4 9YellowCap, doyou have a 9? Playing © Joan A. Cotter, Ph.D., 2012
  207. 207. Go to the Dump Game 7 3 2 2 7 5 2 8 4 6 1 5 4 9YellowCap, doyou have a 9? Playing © Joan A. Cotter, Ph.D., 2012
  208. 208. Go to the Dump Game 7 3 2 2 7 5 2 8 4 6 19 5 4 9YellowCap, doyou have a 9? Playing © Joan A. Cotter, Ph.D., 2012
  209. 209. Go to the Dump Game 7 3 2 2 7 5 2 1 8 9 4 6 5 4 9 Playing © Joan A. Cotter, Ph.D., 2012
  210. 210. Go to the Dump Game 7 3 2 2 7 5 2 1 8 9 4 6 2 9 1 7 7 5 4 9 PlayingPinkCap is not out of the game. Her turn ends, but she takes 5 more cards. A. Cotter, Ph.D., 2012 © Joan
  211. 211. Go to the Dump Game 9 1 4 6 5 5 Winner? © Joan A. Cotter, Ph.D., 2012
  212. 212. Go to the Dump Game 9 1 4 6 5 Winner?No counting. Combine both stacks. © Joan A. Cotter, Ph.D., 2012
  213. 213. Go to the Dump Game 9 1 4 6 5 Winner?Whose stack is the highest? © Joan A. Cotter, Ph.D., 2012
  214. 214. Go to the Dump Game Next gameNo shuffling needed for next game. © Joan A. Cotter, Ph.D., 2012
  215. 215. ―Math‖ Way of Naming Numbers © Joan A. Cotter, Ph.D., 2012
  216. 216. ―Math‖ Way of Naming Numbers 11 = ten 1 © Joan A. Cotter, Ph.D., 2012
  217. 217. ―Math‖ Way of Naming Numbers 11 = ten 1 12 = ten 2 © Joan A. Cotter, Ph.D., 2012
  218. 218. ―Math‖ Way of Naming Numbers 11 = ten 1 12 = ten 2 13 = ten 3 © Joan A. Cotter, Ph.D., 2012
  219. 219. ―Math‖ Way of Naming Numbers 11 = ten 1 12 = ten 2 13 = ten 3 14 = ten 4 © Joan A. Cotter, Ph.D., 2012
  220. 220. ―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
  221. 221. ―Math‖ Way of Naming Numbers 11 = ten 1 20 = 2-ten 12 = ten 2 13 = ten 3 14 = ten 4 .... 19 = ten 9Don’t say ―2-tens.‖ We don’t say 3 hundreds eleven for 311. © Joan A. Cotter, Ph.D., 2012
  222. 222. ―Math‖ Way of Naming Numbers 11 = ten 1 20 = 2-ten 12 = ten 2 21 = 2-ten 1 13 = ten 3 14 = ten 4 .... 19 = ten 9Don’t say ―2-tens.‖ We don’t say 3 hundreds eleven for 311. © Joan A. Cotter, Ph.D., 2012
  223. 223. ―Math‖ Way of Naming Numbers 11 = ten 1 20 = 2-ten 12 = ten 2 21 = 2-ten 1 13 = ten 3 22 = 2-ten 2 14 = ten 4 .... 19 = ten 9Don’t say ―2-tens.‖ We don’t say 3 hundreds eleven for 311. © Joan A. Cotter, Ph.D., 2012
  224. 224. ―Math‖ Way of Naming Numbers 11 = ten 1 20 = 2-ten 12 = ten 2 21 = 2-ten 1 13 = ten 3 22 = 2-ten 2 14 = ten 4 23 = 2-ten 3 .... 19 = ten 9Don’t say ―2-tens.‖ We don’t say 3 hundreds eleven for 311. © Joan A. Cotter, Ph.D., 2012
  225. 225. ―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
  226. 226. ―Math‖ Way of Naming Numbers 137 = 1 hundred 3-ten 7Only numbers under 100 need to be said the ―math‖ way. © Joan A. Cotter, Ph.D., 2012
  227. 227. ―Math‖ Way of Naming Numbers 137 = 1 hundred 3-ten 7 or 137 = 1 hundred and 3-ten 7Only numbers under 100 need to be said the ―math‖ way. © Joan A. Cotter, Ph.D., 2012
  228. 228. ―Math‖ Way of Naming Numbers 100 Chinese Average Highest Number Counted U.S. 90 Korean formal [math way] Korean informal [not explicit] 80 70 60 50 40 30 20 10 0 4 5 6 Ages (yrs.) Song, M., & Ginsburg, H. (1988). p. 326. The effect of the Korean number system on young childrens counting: A natural experiment in numerical bilingualism. International Journal of Psychology, 23, 319-332.Shows how far children from 3 countries can count at ages 4, 5, and 6. © Joan A. Cotter, Ph.D., 2012
  229. 229. ―Math‖ Way of Naming Numbers 100 Chinese Average Highest Number Counted U.S. 90 Korean formal [math way] Korean informal [not explicit] 80 70 60 50 40 30 20 10 0 4 5 6 Ages (yrs.) Song, M., & Ginsburg, H. (1988). p. 326. The effect of the Korean number system on young childrens counting: A natural experiment in numerical bilingualism. International Journal of Psychology, 23, 319-332.Purple is Chinese. Note jump between ages 5 and 6. © Joan A. Cotter, Ph.D., 2012
  230. 230. ―Math‖ Way of Naming Numbers 100 Chinese Average Highest Number Counted U.S. 90 Korean formal [math way] Korean informal [not explicit] 80 70 60 50 40 30 20 10 0 4 5 6 Ages (yrs.) Song, M., & Ginsburg, H. (1988). p. 326. The effect of the Korean number system on young childrens counting: A natural experiment in numerical bilingualism. International Journal of Psychology, 23, 319-332.Dark green is Korean ―math‖ way. © Joan A. Cotter, Ph.D., 2012
  231. 231. ―Math‖ Way of Naming Numbers 100 Chinese Average Highest Number Counted U.S. 90 Korean formal [math way] Korean informal [not explicit] 80 70 60 50 40 30 20 10 0 4 5 6 Ages (yrs.) Song, M., & Ginsburg, H. (1988). p. 326. The effect of the Korean number system on young childrens counting: A natural experiment in numerical bilingualism. International Journal of Psychology, 23, 319-332.Dotted green is everyday Korean; notice smaller jump between ages 5 and 6. Cotter, Ph.D., 2012 © Joan A.
  232. 232. ―Math‖ Way of Naming Numbers 100 Chinese Average Highest Number Counted U.S. 90 Korean formal [math way] Korean informal [not explicit] 80 70 60 50 40 30 20 10 0 4 5 6 Ages (yrs.) Song, M., & Ginsburg, H. (1988). p. 326. The effect of the Korean number system on young childrens counting: A natural experiment in numerical bilingualism. International Journal of Psychology, 23, 319-332.Red is English speakers. They learn same amount between ages 4-5 andJoan A. Cotter, Ph.D., 2012 © 5-6.
  233. 233. 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
  234. 234. 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
  235. 235. 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
  236. 236. 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
  237. 237. Math Way of Naming Numbers Compared to reading: © Joan A. Cotter, Ph.D., 2012
  238. 238. Math Way of Naming Numbers Compared to reading:• Just as reciting the alphabet doesn’t teachreading, counting doesn’t teach arithmetic. © Joan A. Cotter, Ph.D., 2012
  239. 239. Math Way of Naming Numbers Compared to reading:• Just as reciting the alphabet doesn’t teachreading, 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). © Joan A. Cotter, Ph.D., 2012
  240. 240. Math Way of Naming Numbers Compared to reading:• Just as reciting the alphabet doesn’t teachreading, counting doesn’t teach arithmetic.• Just as we first teach the sound of the letters,we must first teach the name of the quantity(math way).• Montessorians do use the math way of namingnumbers but are too quick to switch to traditionalnames. Use the math way for a longer period oftime. © Joan A. Cotter, Ph.D., 2012
  241. 241. Math Way of Naming Numbers―Rather, the increased gap between Chineseand U.S. students and that of ChineseAmericans and Caucasian Americans may bedue primarily to the nature of their initial gapprior to formal schooling, such as countingefficiency and base-ten number sense.‖ Jian Wang and Emily Lin, 2005 Researchers © Joan A. Cotter, Ph.D., 2012
  242. 242. Math Way of Naming NumbersResearch task:Using 10s and 1s, askthe child to construct48. © Joan A. Cotter, Ph.D., 2012
  243. 243. Math Way of Naming NumbersResearch task:Using 10s and 1s, askthe child to construct48.Then ask the child tosubtract 14. © Joan A. Cotter, Ph.D., 2012
  244. 244. Math Way of Naming Numbers Research task: Using 10s and 1s, ask the child to construct 48. Then ask the child to subtract 14.Children thinking of 14 as 14 ones count 14. © Joan A. Cotter, Ph.D., 2012
  245. 245. Math Way of Naming NumbersResearch task: Using 10s and 1s, ask the child to construct 48. Then ask the child to subtract 14.Children thinking of 14 as 14 ones counted 14. © Joan A. Cotter, Ph.D., 2012
  246. 246. Math Way of Naming NumbersResearch task: Using 10s and 1s, ask the child to construct 48. Then ask the child to subtract 14.Children thinking of 14 as 14 ones counted 14. © Joan A. Cotter, Ph.D., 2012
  247. 247. Math Way of Naming NumbersResearch task: Using 10s and 1s, ask the child to construct 48. Then ask the child to subtract 14.Children thinking of 14 as 14 ones counted 14. © Joan A. Cotter, Ph.D., 2012
  248. 248. Math Way of Naming NumbersResearch task: Using 10s and 1s, ask the child to construct 48. Then ask the child to subtract 14.Children thinking of 14 as 14 ones counted 14. © Joan A. Cotter, Ph.D., 2012
  249. 249. Math Way of Naming NumbersResearch task: Using 10s and 1s, ask the child to construct 48. Then ask the child to subtract 14.Children thinking of 14 as 14 ones counted 14. © Joan A. Cotter, Ph.D., 2012
  250. 250. Math Way of Naming NumbersResearch task: Using 10s and 1s, ask the child to construct 48. Then ask the child to subtract 14.Children thinking of 14 as 14 ones counted 14. © Joan A. Cotter, Ph.D., 2012
  251. 251. Math Way of Naming NumbersResearch task: Using 10s and 1s, ask the child to construct 48. Then ask the child to subtract 14.Children thinking of 14 as 14 ones counted 14. © Joan A. Cotter, Ph.D., 2012
  252. 252. Math Way of Naming NumbersResearch task: Using 10s and 1s, ask the child to construct 48. Then ask the child to subtract 14.Children who understand tens remove a ten and 4 ones. © Joan A. Cotter, Ph.D., 2012
  253. 253. Math Way of Naming NumbersResearch task: Using 10s and 1s, ask the child to construct 48. Then ask the child to subtract 14.Children who understand tens remove a ten and 4 ones. © Joan A. Cotter, Ph.D., 2012
  254. 254. Math Way of Naming NumbersResearch task: Using 10s and 1s, ask the child to construct 48. Then ask the child to subtract 14.Children who understand tens remove a ten and 4 ones. © Joan A. Cotter, Ph.D., 2012
  255. 255. Math Way of Naming Numbers Traditional names4-ten = fortyThe ―ty‖meanstens. © Joan A. Cotter, Ph.D., 2012
  256. 256. Math Way of Naming Numbers Traditional names 4-ten = forty The ―ty‖ means tens.The traditional names for 40, 60, 70, 80, and 90 follow a pattern. © Joan A. Cotter, Ph.D., 2012
  257. 257. Math Way of Naming Numbers Traditional names6-ten = sixtyThe ―ty‖meanstens. © Joan A. Cotter, Ph.D., 2012
  258. 258. Math Way of Naming Numbers Traditional names3-ten = thirty―Thir‖ alsoused in 1/3,13 and 30. © Joan A. Cotter, Ph.D., 2012
  259. 259. Math Way of Naming Numbers Traditional names5-ten = fifty―Fif‖ alsoused in 1/5,15 and 50. © Joan A. Cotter, Ph.D., 2012
  260. 260. Math Way of Naming Numbers Traditional names2-ten = twentyTwo used tobepronounced―twoo.‖ © Joan A. Cotter, Ph.D., 2012
  261. 261. Math Way of Naming Numbers Traditional names A word game fireplace place-fireSay the syllables backward. This is how we say the teen numbers. © Joan A. Cotter, Ph.D., 2012
  262. 262. Math Way of Naming Numbers Traditional names A word game fireplace place-fire newspaper paper-newsSay the syllables backward. This is how we say the teen numbers. © Joan A. Cotter, Ph.D., 2012
  263. 263. Math Way of Naming Numbers Traditional names A word game fireplace place-fire newspaper paper-news box-mail mailboxSay the syllables backward. This is how we say the teen numbers. © Joan A. Cotter, Ph.D., 2012
  264. 264. Math Way of Naming Numbers Traditional names ten 4―Teen‖ alsomeans ten. © Joan A. Cotter, Ph.D., 2012
  265. 265. Math Way of Naming Numbers Traditional names ten 4 teen 4―Teen‖ alsomeans ten. © Joan A. Cotter, Ph.D., 2012
  266. 266. Math Way of Naming Numbers Traditional names ten 4 teen 4 fourteen―Teen‖ alsomeans ten. © Joan A. Cotter, Ph.D., 2012
  267. 267. Math Way of Naming Numbers Traditional names a one left © Joan A. Cotter, Ph.D., 2012
  268. 268. Math Way of Naming Numbers Traditional names a one left a left-one © Joan A. Cotter, Ph.D., 2012
  269. 269. Math Way of Naming Numbers Traditional names a one left a left-one eleven © Joan A. Cotter, Ph.D., 2012
  270. 270. Math Way of Naming Numbers Traditional names two leftTwopronounced ―twoo.‖ © Joan A. Cotter, Ph.D., 2012
  271. 271. Math Way of Naming Numbers Traditional names two left twelveTwopronounced ―twoo.‖ © Joan A. Cotter, Ph.D., 2012
  272. 272. Composing Numbers3-ten © Joan A. Cotter, Ph.D., 2012
  273. 273. Composing Numbers3-ten © Joan A. Cotter, Ph.D., 2012
  274. 274. Composing Numbers3-ten30 © Joan A. Cotter, Ph.D., 2012
  275. 275. Composing Numbers 3-ten 30Point to the 3 and say 3. © Joan A. Cotter, Ph.D., 2012
  276. 276. Composing Numbers 3-ten 30Point to 0 and say 10. The 0 makes 3 a ten. © Joan A. Cotter, Ph.D., 2012
  277. 277. Composing Numbers3-ten730 © Joan A. Cotter, Ph.D., 2012
  278. 278. Composing Numbers3-ten730 © Joan A. Cotter, Ph.D., 2012
  279. 279. Composing Numbers3-ten730 7 © Joan A. Cotter, Ph.D., 2012
  280. 280. Composing Numbers 3-ten 7 30 7Place the 7 on top of the 0 of the 30. © Joan A. Cotter, Ph.D., 2012
  281. 281. Composing Numbers 3-ten 7 30 7Notice the way we say the number,represent the number, and write the numberall correspond. © Joan A. Cotter, Ph.D., 2012
  282. 282. Composing Numbers 7-ten 8 78Another example. © Joan A. Cotter, Ph.D., 2012
  283. 283. Composing Numbers10-ten © Joan A. Cotter, Ph.D., 2012
  284. 284. Composing Numbers10-ten100 © Joan A. Cotter, Ph.D., 2012
  285. 285. Composing Numbers10-ten100 © Joan A. Cotter, Ph.D., 2012
  286. 286. Composing Numbers10-ten100 © Joan A. Cotter, Ph.D., 2012
  287. 287. Composing Numbers1hundred © Joan A. Cotter, Ph.D., 2012
  288. 288. Composing Numbers1hundred100 © Joan A. Cotter, Ph.D., 2012
  289. 289. Composing Numbers 1 hundred 100Of course, we can also read it as one-hun-dred. © Joan A. Cotter, Ph.D., 2012
  290. 290. Composing Numbers 1 hundred 100Of course, we can also read it as one-hun-dred. © Joan A. Cotter, Ph.D., 2012
  291. 291. Composing Numbers 1 hundred 100Of course, we can also read it as one-hun-dred. © Joan A. Cotter, Ph.D., 2012
  292. 292. Composing Numbers Reading numbers backwardTo read a number, students areoften instructed to start at theright (ones column), contrary tonormal reading of numbers andtext: 4258 © Joan A. Cotter, Ph.D., 2012
  293. 293. Composing Numbers Reading numbers backwardTo read a number, students areoften instructed to start at theright (ones column), contrary tonormal reading of numbers andtext: 4258 © Joan A. Cotter, Ph.D., 2012
  294. 294. Composing Numbers Reading numbers backwardTo read a number, students areoften instructed to start at theright (ones column), contrary tonormal reading of numbers andtext: 4258 © Joan A. Cotter, Ph.D., 2012
  295. 295. Composing Numbers Reading numbers backwardTo read a number, students areoften instructed to start at theright (ones column), contrary tonormal reading of numbers andtext: 4258 © Joan A. Cotter, Ph.D., 2012
  296. 296. Composing Numbers Reading numbers backwardTo read a number, students areoften instructed to start at theright (ones column), contrary tonormal reading of numbers andtext: 4258The Decimal Cards encourage readingnumbers in the normal order. © Joan A. Cotter, Ph.D., 2012
  297. 297. Composing Numbers Scientific Notation 3 4000 = 4 x 10In scientific notation, we ―stand‖on the left digit and note thenumber of digits to the right.(That’s why we shouldn’t refer tothe 4 as the 4th column.) © Joan A. Cotter, Ph.D., 2012
  298. 298. Fact Strategies © Joan A. Cotter, Ph.D., 2012
  299. 299. Fact Strategies• A strategy is a way to learn a new factor recall a forgotten fact. © Joan A. Cotter, Ph.D., 2012
  300. 300. Fact Strategies• A strategy is a way to learn a new factor recall a forgotten fact.• A visualizable representation is partof a powerful strategy. © Joan A. Cotter, Ph.D., 2012
  301. 301. Fact Strategies Complete the Ten9+5= © Joan A. Cotter, Ph.D., 2012
  302. 302. Fact Strategies Complete the Ten9+5= © Joan A. Cotter, Ph.D., 2012
  303. 303. Fact Strategies Complete the Ten9+5= © Joan A. Cotter, Ph.D., 2012
  304. 304. Fact Strategies Complete the Ten9+5=Take 1 fromthe 5 and giveit to the 9. © Joan A. Cotter, Ph.D., 2012
  305. 305. Fact Strategies Complete the Ten 9+5= Take 1 from the 5 and give it to the 9.Use two hands and move the beads simultaneously. © Joan A. Cotter, Ph.D., 2012
  306. 306. Fact Strategies Complete the Ten9+5=Take 1 fromthe 5 and giveit to the 9. © Joan A. Cotter, Ph.D., 2012
  307. 307. Fact Strategies Complete the Ten9 + 5 = 14Take 1 fromthe 5 and giveit to the 9. © Joan A. Cotter, Ph.D., 2012
  308. 308. Fact Strategies Two Fives8+6= © Joan A. Cotter, Ph.D., 2012
  309. 309. Fact Strategies Two Fives8+6= © Joan A. Cotter, Ph.D., 2012
  310. 310. Fact Strategies Two Fives 8+6=Two fives make 10. © Joan A. Cotter, Ph.D., 2012
  311. 311. Fact Strategies Two Fives 8+6=Just add the ―leftovers.‖ © Joan A. Cotter, Ph.D., 2012
  312. 312. Fact Strategies Two Fives 8+6= 10 + 4 = 14Just add the ―leftovers.‖ © Joan A. Cotter, Ph.D., 2012
  313. 313. Fact Strategies Two Fives 7+5=Another example. © Joan A. Cotter, Ph.D., 2012
  314. 314. Fact Strategies Two Fives7+5= © Joan A. Cotter, Ph.D., 2012
  315. 315. Fact Strategies Two Fives7 + 5 = 12 © Joan A. Cotter, Ph.D., 2012
  316. 316. Fact Strategies Going Down15 – 9 = © Joan A. Cotter, Ph.D., 2012
  317. 317. Fact Strategies Difference7–4=Subtract 4from 5; thenadd 2. © Joan A. Cotter, Ph.D., 2012
  318. 318. Fact Strategies Going Down15 – 9 = © Joan A. Cotter, Ph.D., 2012
  319. 319. Fact Strategies Going Down 15 – 9 =Subtract 5;then 4. © Joan A. Cotter, Ph.D., 2012
  320. 320. Fact Strategies Going Down 15 – 9 =Subtract 5;then 4. © Joan A. Cotter, Ph.D., 2012
  321. 321. Fact Strategies Going Down 15 – 9 =Subtract 5;then 4. © Joan A. Cotter, Ph.D., 2012
  322. 322. Fact Strategies Going Down 15 – 9 = 6Subtract 5;then 4. © Joan A. Cotter, Ph.D., 2012
  323. 323. Fact Strategies Subtract from 1015 – 9 = © Joan A. Cotter, Ph.D., 2012
  324. 324. Fact Strategies Subtract from 1015 – 9 =Subtract 9from 10. © Joan A. Cotter, Ph.D., 2012
  325. 325. Fact Strategies Subtract from 1015 – 9 =Subtract 9from 10. © Joan A. Cotter, Ph.D., 2012
  326. 326. Fact Strategies Subtract from 1015 – 9 =Subtract 9from 10. © Joan A. Cotter, Ph.D., 2012
  327. 327. Fact Strategies Subtract from 1015 – 9 = 6Subtract 9from 10. © Joan A. Cotter, Ph.D., 2012
  328. 328. Fact Strategies Going Up13 – 9 = © Joan A. Cotter, Ph.D., 2012
  329. 329. Fact Strategies Going Up 13 – 9 =Start with 9;go up to 13. © Joan A. Cotter, Ph.D., 2012
  330. 330. Fact Strategies Going Up 13 – 9 =Start with 9;go up to 13. © Joan A. Cotter, Ph.D., 2012
  331. 331. Fact Strategies Going Up 13 – 9 =Start with 9;go up to 13. © Joan A. Cotter, Ph.D., 2012
  332. 332. Fact Strategies Going Up 13 – 9 =Start with 9;go up to 13. © Joan A. Cotter, Ph.D., 2012
  333. 333. Fact Strategies Going Up 13 – 9 = 1+3=4Start with 9;go up to 13. © Joan A. Cotter, Ph.D., 2012
  334. 334. MoneyPenny © Joan A. Cotter, Ph.D., 2012
  335. 335. MoneyNickel © Joan A. Cotter, Ph.D., 2012
  336. 336. Money Dime © Joan A. Cotter, Ph.D., 2012
  337. 337. MoneyQuarter © Joan A. Cotter, Ph.D., 2012
  338. 338. MoneyQuarter © Joan A. Cotter, Ph.D., 2012
  339. 339. MoneyQuarter © Joan A. Cotter, Ph.D., 2012
  340. 340. MoneyQuarter © Joan A. Cotter, Ph.D., 2012
  341. 341. Base-10 Picture Cards One © Joan A. Cotter, Ph.D., 2012
  342. 342. Base-10 Picture Cards Ten One © Joan A. Cotter, Ph.D., 2012
  343. 343. Base-10 Picture Cards Hundred Ten One © Joan A. Cotter, Ph.D., 2012
  344. 344. Base-10 Picture CardsThousan Hundred Ten One d © Joan A. Cotter, Ph.D., 2012

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