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Teaching Math to Childrenwith Special NeedsJoan A. Cotter, Ph.D.JoanCotter@RightStartMath.comSioux Empire Christian Home EducatorsHomeschool ConferenceSioux Falls, SDSaturday, May 2, 20153:00 p.m.– 4:00 p.m. Joan A. Cotter, Ph.D., 2015

Myths about LD People with LD have lower intelligence.(33% are gifted.) They are lazy or stubborn. Children with LD can be cured or will outgrow it. Boys are more likely to be affected.(Girls are equally affected.) Dyslexia and learning disability are the same thing. Students with LD and ADHD cannot succeed in higher education. Joan A. Cotter, Ph.D., 2015

Characteristics of LDChild usually: Is more creative. Cannot learn by rote. Must understand and make sense of a concept inorder to remember it. Is more visual and hands-on. Dislikes worksheets. Finds it very difficult to unlearn. Joan A. Cotter, Ph.D., 2015

Problems Occurring with MathDyscalculia Reversals in writing numbers Poor number sense Slow fact retrieval Errors in computation Difficulty in solving word problemsDyscalculia mainly affects arithmetic, not otherbranches of math. Joan A. Cotter, Ph.D., 2015

How Math is Traditionally Taught Counting Learn sequence (number names by heart) One-to-one correspondence (one count per object) Cardinality principle (last number tells how many) Memorizing facts Flash cards and timed tests Rhymes and songs Memorizing algorithms (procedures) Using key words to solve story problems Joan A. Cotter, Ph.D., 2015

Traditional CountingFrom a child’s perspectiveBecause we’re so familiar with 1, 2, 3,we’ll use letters of the alphabet.A 1B 2C 3D 4E 5, and so forth Joan A. Cotter, Ph.D., 2015

Traditional CountingFrom a child’s perspectiveF E ABCDEFABCDEWhat is the sum? Joan A. Cotter, Ph.D., 2015

Traditional CountingFrom a child’s perspectiveF E KABCDEFGHIJK Joan A. Cotter, Ph.D., 2015

Traditional CountingFrom a child’s perspectiveNow memorize the facts!!EH G DD CH FF F Joan A. Cotter, Ph.D., 2015

Traditional CountingFrom a child’s perspectiveH–C Try subtracting by “taking away.” Joan A. Cotter, Ph.D., 2015

Traditional CountingFrom a child’s perspectiveTry skip counting by B’s to T:B, D, . . . , T.What is D E? Joan A. Cotter, Ph.D., 2015

Compared to ReadingJust as reciting the alphabet doesn’t teachreading, counting doesn’t teach arithmetic. Joan A. Cotter, Ph.D., 2015

Counting-Based Arithmetic Rote counting to 100 in kindergarten. Calendars misused to teach counting. Counting on for addition. (Jack and Jill) Counting back for subtraction. Number lines are abstract counting. Skip counting for multiplication facts. Does not work well for money or fractions orreading graphs. Joan A. Cotter, Ph.D., 2015

Memorizing MathPercentage RecallImmediately After 1 day After 4 weeksRote32238Concept696958Math needs to be taught so 95% isunderstood and only 5% memorized.–Richard Skemp Joan A. Cotter, Ph.D., 2015

Memorizing MathAccording to a study with college students,it took them: 93 minutes to learn 200 nonsense syllables. 24 minutes to learn 200 words of text. 10 minutes to learn 200 words of poetry. Joan A. Cotter, Ph.D., 2015

Flash Cards and Timed Tests Often used to teach rote. Liked only by those who don’t need them. Give the false impression that math does notrequire thinking. Often produce stress – children under stress stoplearning. Not concrete – use abstract symbols. Cause stress (may become physically ill). Result in short-term learning. May lead to math anxiety. Joan A. Cotter, Ph.D., 2015

dehsiPubl4102May,VisualizingA pilot study of the effects of RightStartinstruction on early numeracy skills ofchildren with specific language impairmentRiikka Mononen, Pirjo Aunio, Tuire KoponenAbstract:.The children with SLI [specific languageimpairment] began kindergarten with significantlyweaker early numeracy skills compared to their peers.Immediately after the instruction phase, there was nosignificant difference between the groups in countingskills. Joan A. Cotter, Ph.D., 2015

Visualizing Visual is related to seeing. Visualize is to form a mental image. Joan A. Cotter, Ph.D., 2015

VisualizingVisualizing is also needed in other fields: Reading Architecture Sports Astronomy Arts Archeology Geography Chemistry Engineering Physics Construction Surgery Biology History Joan A. Cotter, Ph.D., 2015

VisualizingTry to visualize 8 identical apples without grouping. Joan A. Cotter, Ph.D., 2015

VisualizingNow try to visualize 8 apples: 5 red and 3 green. Joan A. Cotter, Ph.D., 2015

Grouping in FivesEarly Roman numerals123458IIIIIIIIIIVVIII Joan A. Cotter, Ph.D., 2015

Grouping in FivesMusical staff Joan A. Cotter, Ph.D., 2015

Grouping in FivesClocks and nickels Joan A. Cotter, Ph.D., 2015

Grouping in FivesTally marks Joan A. Cotter, Ph.D., 2015

Grouping in FivesSubitizing Instant recognition of quantity is calledsubitizing. Grouping in fives extends subitizing beyondfive. Joan A. Cotter, Ph.D., 2015

Subitizing Five-month-old infants can subitize to 1–3. Three-year-olds can subitize to 1–5. Four-year-olds can subitize 1–10 by groupingwith five. Joan A. Cotter, Ph.D., 2015

Research on SubitizingKaren Wynn’s research Joan A. Cotter, Ph.D., 2015

Research on SubitizingKaren Wynn’s research Joan A. Cotter, Ph.D., 2015

Research on SubitizingKaren Wynn’s research Joan A. Cotter, Ph.D., 2015

Research on SubitizingKaren Wynn’s research Joan A. Cotter, Ph.D., 2015

Research on SubitizingKaren Wynn’s researchYou could say subitizing is much more“natural” than counting. Joan A. Cotter, Ph.D., 2015

Research on SubitizingIn Japanese schools Children are discouraged from usingcounting for adding. They consistently group in fives. Joan A. Cotter, Ph.D., 2015

Quantities 1–10Using fingers Joan A. Cotter, Ph.D., 2015

Quantities 1–10Subitizing five Joan A. Cotter, Ph.D., 2015

Quantities 1–10Subitizing five5 has a middle; 4 does not. Joan A. Cotter, Ph.D., 2015

Quantities 1–10Tally sticksFive as a group. Joan A. Cotter, Ph.D., 2015

Quantities 1–10Tally sticks Joan A. Cotter, Ph.D., 2015

Quantities 1–10Number chart for remembering numerals12345!6!7!8!9!10! Joan A. Cotter, Ph.D., 2015

Quantities 1–10Entering quantities3 Joan A. Cotter, Ph.D., 2015

Quantities 1–10Entering quantities7 Joan A. Cotter, Ph.D., 2015

Quantities 1–10Stairs Joan A. Cotter, Ph.D., 2015

Quantities 1–10Adding Joan A. Cotter, Ph.D., 2015

Quantities 1–10Adding4 3 Joan A. Cotter, Ph.D., 2015

Quantities 1–10Adding4 3 7Japanese children learn to do this mentally. Joan A. Cotter, Ph.D., 2015

GamesGamesMathBooks Reading Games provide interesting repetition neededfor automatic responses in a social setting. Games provide an application for newinformation. Games provide instant feedback. Joan A. Cotter, Ph.D., 2015

GamesGo to the DumpObjective: To learn the facts that total 10:1 92 83 74 65 5It is played similar to Go Fish. Joan A. Cotter, Ph.D., 2015

Place ValueThe problem English-speaking children often think of 14as 14 ones, not 10 and 4 ones. The pattern that is needed to make sense oftens and ones is hidden! Joan A. Cotter, Ph.D., 2015

Place ValueIts importance Place value is the foundation of modernarithmetic. It is critical for understanding algorithms. It must be taught, not left for discovery. Children need the big picture, not tinysnapshots. Joan A. Cotter, Ph.D., 2015

Transparent Number Naming11 ten 112 ten 213 ten 314 ten 4. . . .19 ten 920 2-ten21 2-ten 122 2-ten 223 2-ten 3. . . . . . .99 9-ten 9 Joan A. Cotter, Ph.D., 2015

Transparent Number Naming137 1 hundred 3-ten 7or137 1 hundred and 3-ten 7 Joan A. Cotter, Ph.D., 2015

Average Highest Number CountedTransparent Number Naming1009080ChineseU.S.transparentKorean formal[math way]non-transparentKorean informal[not explicit]70605040302010045Ages (years)6Song, M., & Ginsburg, H. (1988). p. 326. The effect of the Korean number system on young children's counting:A natural experiment in numerical bilingualism. International Journal of Psychology, 23, 319-332.! Joan A. Cotter, Ph.D., 2015

Math Way of Number Naming Only 11 words are needed to count to 100 themath way, 28 in English. (All Indo-Europeanlanguages are non-standard in number naming.) Asian children learn mathematics usingtransparent number naming. Mathematics is the science of patterns. Numbernames need to be an example. Children who are hearing-impaired can distinguishbetween 14 and 40, and 13 and 30. Learning two languages helps brain development. Joan A. Cotter, Ph.D., 2015

Compared to ReadingJust as we first teach the sound of theletters, we must first teach the name of thequantity, the math way. Joan A. Cotter, Ph.D., 2015

Math Way of Number NamingRegular names4-ten fortyThe “ty”means tens. Joan A. Cotter, Ph.D., 2015

Math Way of Number NamingRegular names6-ten sixtyThe “ty”means tens. Joan A. Cotter, Ph.D., 2015

Math Way of Number NamingRegular names3-ten thirty“Thir” also usedin 1/3, 13 and 30. Joan A. Cotter, Ph.D., 2015

Math Way of Number NamingRegular names2-ten twentyTwo used tobe pronounced“twoo.” Joan A. Cotter, Ph.D., 2015

Math Way of Number NamingRegular namesA word mailbox Joan A. Cotter, Ph.D., 2015

Math Way of Number NamingRegular namesten 4teen 4fourteenSuffix -teenmeans ten. Joan A. Cotter, Ph.D., 2015

Math Way of Number NamingRegular namesa one lefta left-oneeleven Joan A. Cotter, Ph.D., 2015

Math Way of Number NamingRegular namestwo lefttwelveTwo said as“twoo.” Joan A. Cotter, Ph.D., 2015

Composing Numbers3-ten30 Joan A. Cotter, Ph.D., 2015

Composing Numbers3-ten 7307 Joan A. Cotter, Ph.D., 2015

Composing Numbers3-ten 7307Note the congruence in how we say the number,represent the number, and write the number. Joan A. Cotter, Ph.D., 2015

Composing Numbers1-ten 8108 Joan A. Cotter, Ph.D., 2015

Composing Numbers1-ten 8108 Joan A. Cotter, Ph.D., 2015

Composing Numbers10-ten100 Joan A. Cotter, Ph.D., 2015

Place-Value Cards303-ten3003 hun-dred30003 th-ou-sand Joan A. Cotter, Ph.D., 2015

Place-Value Cards400020050400020050742005077 Joan A. Cotter, Ph.D., 2015

Place-Value Cards300083000830008 Joan A. Cotter, Ph.D., 2015

Place ValueTwo aspectsStatic (Recording) Value of a digit is determined by position. No position may have more than nine. As you progress to the left, value at each positionis ten times greater than previous position. Represented by the place-value cards.Dynamic (Trading) 10 ones 1 ten;10 tens 1 hundred;10 hundreds 1 thousand, . Represented with abacus and other materials. Joan A. Cotter, Ph.D., 2015

Learning the FactsLimited success, especially for strugglingchildren, when learning is: Based on counting: whether dots, fingers,number lines, or counting words. Based on rote memory: whether flash cards,timed tests, or computer games. Based on skip counting: whether fingers orsongs.A child is considered to know a fact if they cangive it in 2–3 seconds. Joan A. Cotter, Ph.D., 2015

Fact StrategiesComplete the Ten9 5 Take 1 fromthe 5 and giveit to the 9. Joan A. Cotter, Ph.D., 2015

Fact StrategiesComplete the Ten9 5 Take 1 fromthe 5 and giveit to the 9. Joan A. Cotter, Ph.D., 2015

Fact StrategiesComplete the Ten9 5 14Take 1 fromthe 5 and giveit to the 9. Joan A. Cotter, Ph.D., 2015

Fact StrategiesTwo Fives8 6 Joan A. Cotter, Ph.D., 2015

Fact StrategiesTwo Fives8 6 10 4 14The two fivesmake 10. Joan A. Cotter, Ph.D., 2015

Fact StrategiesMissing Addend9 15Start with 9;go up to 15. Joan A. Cotter, Ph.D., 2015

Fact StrategiesMissing Addend9 6 15Start with 9;go up to 15. Joan A. Cotter, Ph.D., 2015

Fact StrategiesSubtracting Part from Ten15 – 9 Subtract 5 from 5and 4 from 10. Joan A. Cotter, Ph.D., 2015

Fact StrategiesSubtracting Part from Ten15 – 9 6Subtract 5 from 5and 4 from 10. Joan A. Cotter, Ph.D., 2015

Fact StrategiesSubtracting All from Ten15 – 9 Subtract 9from 10. Joan A. Cotter, Ph.D., 2015

Fact StrategiesSubtracting All from Ten15 – 9 6Subtract 9from 10. Joan A. Cotter, Ph.D., 2015

MoneyPenny Joan A. Cotter, Ph.D., 2015

MoneyNickel Joan A. Cotter, Ph.D., 2015

MoneyDime Joan A. Cotter, Ph.D., 2015

MoneyQuarter Joan A. Cotter, Ph.D., 2015