1. An Exercise to Aid in Learning The Multiplication Table Learning Disabilities – SPE 5590 Francis Kisner

2. Learning Multiplication by Creating the Multiplication Table Counting Rectangles This project began when I noticed a property of the standard multiplication table: You can group the small squares of the table into rectangles. Create a rectangle on top of the multiplication table such that one corner includes the 1 x 1 = 1 square with its opposite corner anywhere on the field. That lower right corner will have in it the total number of squares in the rectangle you just created.

3. Here is the complete table as usually shown

4. Here I have highlighted two rectangles The red rectangle shows 7 X 8 = 56. It contains 56 squares. The blue rectangle shows 11 X 6 = 66. It contains 66 squares.

5. How can this fact be used to aid instruction? Is it possible for a student to Create the multiplication table by discovering this property them self? In the examples which follow, I will use the convention of multiplying the number across the top by the number down the side.

6. Starting at the upper left corner, with the small numbers, have the student count and fill in the table. A cut-out could be used to help them focus. 2 x 1 = 2 3 x 1 = 3

7. Here are the related facts: 2 x 1 = 2 and 1 x 2 = 2

8. Continuing in the same way, the student would create the table to 12 x 1= 12 and 1 x 12= 12

9. After 2 x 1 = 2 and 1 x 2 = 2, 2 x 2 = 4

10. In this way, the student would work their way through the set to 12 x 2 = 24 and 2 x 12 = 24

11. This exercise would take considerable time for mathematics deficient students and should be spread over several sessions to avoid fatigue. Also, it could be repeated in several ways and with extra suggestions if needed. If the student has not already noticed it, they could be taught “Skip Counting” or “Counting By” strategy.

12. “Counting By” or “Skip Counting” Strategy is just what it sounds like. Counting by twos. Counting by threes. Eventually counting by twelves. The technique uses information already known and decreases the load while encouraging the student who arrives at the answer sooner.

13. Example: Counting by 6 to reach 6 x 7 6, 12, 18, 24, 30, 36, 42 (Direction is top to bottom)

14. Example: Counting by 7 to reach 6 x 7 7, 14, 21, 28, 35, 42 (Direction is Left to Right)

15. “Counting Up” Strategy. Instead of starting from the upper corner each time, in the Counting Up Strategy, the student is encouraged to start with a number which is already known, already on the table and count up from it to arrive at the answer. Like the Counting By Strategy, this uses information already known, decreases cognitive load and encourages the student who arrives at the answer sooner.

16. Counting up to go from 9 x 3 to 9 x 4 Red numbers being counted by the student but not written down. Section from 1x4 to 8x4 would have been covered up or numbers not yet filled in for this exercise.

17. The big question is Will It Work? Will this technique actually help mathematics deficient students to gain number sense, aid retrieval of information from memory, improve counting strategies, and teach procedural calculation? These are among the characteristics associated with mathematics deficiency.

18. Research has shown that multisensory exercises are more effective than worksheets. To do this exercise using physical materials, students could be given counters, markers, square tiles, cubes, etc which they could then arrange in the rectangular patterns on top of a large copy of the worksheet. After the actual counting, they would write the write their answer on the table they were creating.

19. In researching this concept, I did not find any articles detailing similar activities for students creating the Times Table. The following pages give references on Number Theory, Strategies, Devices, and Methods which may be useful.

20. References - Number Concept Bobis, J Understanding the Empty Number Line http://www.nlnw.nsw.edu.au/videos09/lo_Bobis_Understanding/lo_Bobis_Under standing_00.htm Use of the Empty Number Line can include counting on and skip counting. Box, K., Scott, P. Early Concepts of Number and Counting Australian Mathematics Teacher 60(4) 2-6 Early in the sense of primative. What did early people use for counting? Body parts, marks on sticks, knots on string.

21. References - Number Concept Despete, A (2009) Mathematics and metacognition in adolescents and adults with learning disabilities International Electronic Journal of Elementary Educatin 2(1) 82-100 Download at www.iejee.com Study mainly with adults compares the results of number concept trials for people with math disability as compared with those with math and reading disability Dowker, A. (2001) Numeracy recovery: a pilot scheme for early intervention with young children with numeracy difficulties Support for Learning 16(1) 6-10 Eight components of early numeracy are identified. When early students have difficulties with any of these, they are given weekly intervention to correct the problem. Good results.

22. Number Concept References Jordan, N.C. (2007) The Need for Number Sense Educational Leadership 65(2) 63-66 The foundation of math learning disabilites is often lack of number sense. Early detection. Sadler, F.H. (2009) Help! They Still Don't Understand Counting TEACHING Exceptional Children Plus 6(1) Article 3. Retrieved [date] from http://escholarship.bc.edu.education/tecplus/vol6/iss1/art3 Details stages in development of counting and problem solving

23. Number Concept References Sood, S., Jitendra, A.K. (2007) A Comparative Analysis of Number Sense Instruction in Reform-Based and Traditional Mathematics Textbooks The Journal of Special Education 41(3) 145-157 Traditional textbooks included more opportunities for number relationship tasks but the Reform-based text emphasized more real-world connections and promoted relational understanding and spatial relationships. van den Heuvel-Panhuizen,M. (2008) Learning From "Didactikids": An Impetus for Revisiting the Empty Number Line Mathematics Education Research Journal, 20 (3) 6-31 Wing, R.E., Beal, C.R. (2004) Young Children's Judgments About the Relative Size of Shared Portions: The Role of Material Type Mathematical Thinking and Learning 6(1) 1-14 Experiments with 5-7 year old children on ability to judge a relative amount by havles and thirds.

24. References - Strategies Bjorklund, D.F., Hubertz, M.J., Reubens, A.C. (2004) Young children's arithmetic strategies in social context: How parents contribute to children's strategy development while playing games International Journal of Behavioral Development 28(4)347-357 http://www.tandf.co.uk/journals/pp01650254.html Makes reference to counting on. Camos, V. (2003) Counting strategies from 5 years to adulthood: Adaption to structural features European Journal of Psychology of Education 18(3) 251- 265 Includes counting on and skip counting strategies. Carruthers, E., Worthington, M (2004) Young Children Exploring Early Calculation Mathematics Teaching 187 30-34 Children's own symbolic representation of number and calculation.

25. References - Strategies Lucangeli, D., Tressoldi, P.E., Bendotti, M., Bonamoni, M., Siegel, L.S. (2003) Effective Strategies for Mental And Written Arithmetic Calculation from the Third to the Fifth Grade Educational Psychology 23(5) 507-520 Strategies used to solve mental and written multidigit arithmetical problems. A strategy was considered effective if it resulted in the correct solution at least 75% of the time. For mental addition and subtraction, counting on the fingers and counting-on (mental counting from a specific point), and the more sophisiticated 1010 strategy (solution of the calculation problem using tens and units separately) were more effective than the strategies learned at school. Murphy, C. (2004) How Do Children Come To Use A Taught Mental Calculatin Strategy? Educational Studies in Mathematics 56 (1) 3-18 Three children who used different calculation approaches were interviewed after direct instructin of a calculation strategy.

26. References - Strategies Ostad, S.A. (1998) Developmental Differences in Solving Simple Arithmetic Word Problems and Simple Number-fact Problems: A Comparison of Mathematically Normal and Mathematically Disabled Children Mathematical Cognition 4(1) 1-19 The MD children's performance showed a peak in Grade 2. Their use of material strategies showed they had both fact-retrieval problems and working-memory problems. It indicated absence of an adequate domain- specific knowledge base of task-specific strategies. Saxton, M., Cakir, K. (2006) Counting-On, Trading and Partitioning: Effects of Training and Prior Knowledge on Performance on Base-10 Tasks Child Development 77(3) 767-785 Development of counting-on strategy.

27. References - Strategies Schopman, E.A.M., Van Luit, J.E.H (1999) Counting strategies among kintergartners with special educational needs: and exploratory study European Journal of Special Needs Education 14(1) 61-69 Investigates the effect of a mathematics intervention on counting strategies and levels of performance of young children with special educational needs. Results of the study suggested that the particular apporach used in the experimental group was no better than the usual method. Trundley, R (2008) The Value of Two Mathematics Teaching Incorporating Micro Math Nov 211 17-21 Mentions but does not list that the author identified 50 different elements involved in the development of counting. She then mapped the development of these in her twin children.

28. References - Devises Bell, G., Henderson, C. (2004) 30 Track games for young children Australian Primary Mathematics Classroom 9(2) 19-24 Use of a meter long device which sits on the teacher's lap and has 30 blocks which rotate to reveal the corresponding numerals. Jones, M. (2004) Fractions with the Counting Stick Mathematics Teaching 186 34-35 Using a marked stick to help teach fractions.

29. References - Methods Ramani, G.B., Siegler, R.S. (2008) Promoting Broad and Stable Improvements in Low-Income Children's Numerical Knowledge Through Playing Number Board Games Child Development 79(2) 375-394 Playing linear number board games is shown to improve preschoolers' proficiency on four numerical tasks. Tournaki, N. (2008) Rekenrek: A Manipulative Used to Teach Addition and Subtraction to Students with Learning Disabilities Learning Disabilities: A Contemporary Journal 6(2) 41-59 The Rekenrek device looks superficially like a kind of abacus but is based on a five-structure with rows of ten beads broken into two sets of five by color. This encourages the young students to understand the number concepts using their fingers and the beads. Significant results from using the device.

30. References - Methods Bottge, B.A., Rueda, E., LaRoque, P.T., Serlin, R.C., Kwon, J. (2007) Integrating Reform-Oriented Math Instruction in Special Education Settings Learning Disabilities Research & Practice 22(2) 96-109 Emphasis on having the students discover the concepts which allows them to understand in their own terms first. Fuchs, L, Powell, S, Seethaler, P, Fuchs, D, Hamlett, C, 2010 A Framework for Remediating Number Combination Deficits Exceptional Children 76(2) 135- 156 Graham, L., Bellert, A., Pegg, J. (2007) Supporting Students in the Middle School Years with Learning Difficulties in Mathematics: Research into Classroom Practice Australasian Journal of Special Education 31(2) 171- 182 Reviews the use of the QuickSmart mathematics program in New South Wales.

31. References - Methods Gu, W. (2001) The Lattice Method Used in Teaching Multiplication with Whole Numbers and Decimals to Students with Learning Disabilities retrieved from ERIC Harries, T., Barmby, P. (2008) Representing Multiplication Mathematics Teaching Incorporating Micromath 206 37-41 Use of arrays to help with multiplication concepts. Allows skip counting and counting-on.

32. References - Methods Rousselle, L., Noel, M. (2008) Mental Arithmetic in Children with Mathematics Learning Disabilities - The Adaptive Use of Approximate Calculation in an Addition Verification Task Journal of Learning Disabilities 41(6) 498-513 Deals with differences in perception of correctness of answers between math learning disabled and regular students. Scott, K.S. (1993) Multisensory Mathematics for Children With Mild Disabilities Exceptionality 4(2) 97-111 Examines the use of a program (Touch Math) which uses touching raised dots and circles on numerals to reinforce the concepts.

33. References - Methods Wadlington, E., Wadlington, P., (2008) Helping Students With Mathematical Disabilities to Succeed Preventing School Failure 53(1) 2-7 Discussion of characteristics of dyscalculia and successful interventions. Wong, M., Evans, D., (2007) Improving Basic Multiplication Fact Recall for Primary School Students Mathematics Education Research Journal 19(1) 89-106 Pencil and Paper work was shown to be more effective than computer based instruction. This may have been due to the fact that the test was done on paper rather than on computer. Includes a chart of Calculation strategies for Whole-number, Multiplicative problems.

34. Other References Bryant, B.R., Bryant, D.P. (2008) Introduction to the Special Series: Mathematics and Learning Disabilities Learning Disability Quarterly 31 (1) 3-8 Includes a table Ranked Mathematical Difficulties Exhibited by Students with Learning Disabilities and Math Weaknesses. Garcia, A.I., Jimenez, J.E., Hess, S. (2006) Solving Arithmetic Word Problems: An Analysis of Classification as a Function of Difficulty in Children With and Without Arithmetic LD Journal of Learning Disabilities 39(3) 270-281 The position of the unknown quantity had a greater influence on the level of difficulty of story problems than other variables. When the unknown term was in the first place, problems were more difficult for ALD and typically achieving students.

35. Other References Micallef, S., Prior, M. (2004) Arithmetic Learning Difficulties in Children Educational Psychology 24(2) 175-200 Study of children with Arithmetic Learning Difficulties to determine the relative contribution of reading impairments to arithmetic performance. Findings suggest that children with ALD appear to show delayed development of arithmetic skills rather than specific processing deficits or abnormatlities when comparison is made with their chronological age and arithmetic-matched normal peers. Morin, J.E., Franks, D.J. (2010) Why Do Some Children Have Difficulty Learning Mathematics? Looking at Language for Answers Preventing School Failure 54(2) 111-118 Language deficit linked to math disability.