Specially Designed Instruction in Math PDU Session Two

1. Specially Designed Instruction in Math PDU Session Two Oct 30, 2012 4:30-6:30

2. Text Chapter Two: Learning to Calculate

3. Outcomes for Session One Participants will review the foundational knowledge of the psychological processes of math through the development of number sense Participants will deepen their understanding between natural number sense and invented calculation tasks to solve problems that begins with multiplication

4. Review the math psychological processes Magnitude of number Perceptual subitizing Conceptual subitizing Counting Cardinality Development of an Internal Number line Addition and Subtraction

5. “The human brain has serious problems with calculations. Nothing in its evolution prepared it for the task of memorizing dozens of multiplication facts or for carrying out the multiple step operations required for two-digit subtraction. Our ability to approximate number quantities may be imbedded in our genes, but dealing with exact symbolic calculation can be in error prone ordeal.” - Sousa

6. Example of CRA with Multiplication Rita’s class is having a bake sale. Each student in Rita’s class will bring 12 treats to sell. There are 25 students in Rita’s class. How many treats will the class bring in altogether? 12 x 25

7. Concrete Materials needed -Base Ten Blocks -Masking Tape

8. Concrete Fill in the intersection of each row and column using a block that as the same dimensions

9. Representation

10. Abstract 25 x 12

11. By age 4, children have created, refined, and selected algorithms for basic arithmetic (Griffin 2002) Every two years the child’s number sense is reorganized based on experience to understand more complex tasks. This progression is predictable in about 80% of the population. 20% are slower and 20% are faster.

12. 4 year olds Operational Sense Global Quantity Schema Initial Counting Schema more than 1  2  3  4  5 less than Requires Subitizing Requires one-on-one Correspondence

13. 6 year olds Operational Sense Internal Number line has been developed 1 10 20 30 40 50 a little a lot This developmental stage is a major turning point because children come to understand that mathematics is not just something that occurs out in the environment but can also occur inside their own heads.

14. 8 year olds Operational Sense Double internal number line has been loosley developed to allow for two digit operational problem solving 1 10 20 30 40 50 a little a lot 1 10 20 30 40 50 a little a lot Loosely coordinated number line is developed to allow for understanding of place value and solving double digit additional problems.

15. 8 year old can now…

16. 10 year olds Operational Sense Double internal number line has been well developed to allow for two digit operational problem solving effeciently 1 10 20 30 40 50 a little a lot 1 10 20 30 40 50 a little a lot These two well developed number lines allow for the capability of doing two digit addition calculations mentally.

17. A 10 year old can …

18. Multiplication: Natural or Invented?

19. What is multiplication?

20. Imaging studies show that the brain recruits more neural networks during multiplication than during subtraction. 3X3= 12 3+3= 9

21. Natural vs Invented + - x The big idea is that addition and subtraction are somewhat natural skills that are accused for survival, however multiplication is an invented tool and requires additional work to conquer the skill.

22. Multiplication and Mistakes • The average adult makes multiplication mistakes about 10% of the time. • Some multiplication facts such as 8x7 and 9x7 can take up to 2 seconds longer to solve and has an error rate of 25%.

23. Why? Three factors … • Memory • Pattern and associations • Language

24. Memory

25. Memory and Multiplication In a study in 1978 by Ashcroft they determined that memory plays a critical factor when doing calculations. It took less than a second to determine the results of 2+3 or 2x3, but about 1.3 second two solve 8+7 or 8x7. Why?

26. Because… • The accuracy of our mental representations of numerocity drops quickly with increasing number size. • We remember best what we learned first. When we begin learning our arithmetic fact, we started was simple problems containing small digits. • Because small digits appear more frequently than larger ones, we most likely receive much less practice with multiplication problems involving larger numbers.

27. “Children in the primary grades encounter a sudden shift from their intuitive understanding of numerical quantities and counting strategies to the rote learning of arithmetic facts. Unfortunately, most children lose their intuition about arithmetic in the process.” -Sousa

28. What is the tip on this bill? 20% tip on 76.10 Did you memorize the tip table?

29. Memorization of multiplication facts; Is this intuitive? NO Ones and Tens are easiest because it aligns with our intuitive understanding of number; base ten system. It still doesn’t solve the problem…

30. Memorization of multiplication facts; Is this intuitive? this leaves 64 left to memorize HOW? It still doesn’t solve the problem…

31. Memorization of multiplication facts; Is this intuitive? This cuts the number to only 32 to memorize. This still doesn’t solve the problem. It still doesn’t solve the problem…

32. So what does this mean for your teaching?

33. Pattern Recognition and Associations

34. Associative Memory Recital

35. Associative Memory three

36. Associative Memory 6x7 ? ? ?

37. Associations interfere with multiplication mastery 6x9is54 7x8is56 8x8is64 Listen to a partner say these facts out loud. What kind of linguistic information do you hear? Rhythm?

38. The brains strong pattern-seeking ability detects the rhythm of these entities when said aloud, thus making it difficult to keep these three expressions separate.

39. Carl Dennis lives on Allen Brian Avenue Carl Gary lives on Brian Allen Avenue Gary Edwards lives on Carl Edward Avenue Who lives on Allen Brian Avenue? Where does Gary Edwards live?

40. 3x4=12 Carol Dennis lives on Allen Brian Avenue 3x7=21 Carol Gary lives on Brian Allen Avenue 7x5= 35 Gary Edwards lives on Carol Edward Avenue The brain’s ability to recognize patterns interferes with the child’s ability to learn their multiplication facts. Learning multiplication actually interferes with understanding addition. In 1998 a study by Miller discovered that students in third grade took more time to perform addition when they started learning the multiplication tables, and errors like 2+3= 6 began to occur.

41. So what does this mean for your teaching?

42. Language

43. Russian English Bilinguals 1 full second longer to solve a two digit problem presented in the second language

44. Language and Multiplication 25 x 30= Exact Approximate This does not mean that students who struggle with language processing will automatically struggle with mathematics, however language plays a critical role in learning multiplication.

45. There is strong evidence to suggest that the power of language and verbal memory can greatly enhance the child’s mastery of multiplication; using poems and music are examples of this power of language.

46. So what does this mean for your teaching?

47. Next time we meet… 11-27-12 Chapter 7 How the Brain Learns Mathematics; Assessing math learning disability; Best practice for remediation; CRA strategies