P/J Mathematics

1. P/J Mathematics Session 15 Week: November 11 - 15

2. Next Presentation • Wednesday, Nov. 13 • Sydney, Bryce, Andrew, Sara, Bailey • Topic: Measurement

3. Number Sense: A Consideration of Whole Numbers, Basic Facts, and “Drill” Whole Numbers: Naming Them and Using Them: Place Value and Operations—Big Ideas

4. Number… • What do we use ‘number’ for? • Number tells us about “quantity” • For example--- • “Muchness”: How much? • “Manyness”: How many?

5. Let’s Just Make Sure… • Question: What are whole numbers? • Do they include numbers like ½ or 4.7? • Do they include numbers like 4; 932; and 60 001(or is it 60,001?) • Do they include 0 (zero)? • Do they include numbers like -153 and -1? • You are strongly encouraged to read chapters 8 to 10 (Small, 2013) • [chapters 6 to 8 (Small, 2009)]

6. Three Big Ideas • Chapter 9: (A Sense of Quantity with larger Whole Numbers) • The position of the digits in numbers determines what they represent—which size group they count. This is the major principle of place-valuenumeration. • Patterns are inherent in our numeration system because each place value is 10 times the value of the place to the right. • A number has many different ‘forms.’ The groupings of ones, tens, and hundreds, etc., can be taken apart in different ways. • For example, 256 can be 2 hundreds, 5 tens, and 6 ones, but also 1 hundred, 14 tens, and 16 ones. • Taking numbers apart and then combining them in flexible ways is a significant skill for computation. • We can refer to this as “Composing” and “decomposing”

7. Now... Basic Facts and the Place of Drill Are there basic facts? Does drill have any place?

8. Considering the So-called “Basic Facts” • What are we considering the “basic facts”? • 0-9 addition and subtraction facts • 0-9 multiplication and division facts • Let’s consider some sources in our review: • 2005 1-8 mathematics curriculum • Popular Math Education Authors: • Small (2013) • Van de Walle, Lovin, Karp, Bay-Williams (2014) • Teaching student-centered mathematics: Developmentally appropriate instruction for grades pre-K – 2 (2nd ed.)

9. What do we mean by “drill”? • Drill in recent years has come, very often, to be considered as synonymous with memorized, rote learning of facts and rules, rather than conceptual and relational understanding. [So, “drill and kill”] • Van de Walle, Lovin, Karp, and Bay-Williams (2014) define “drill” as “repetitive non-problem-based activity” (p. 168).

10. Grades 1-8 Math curriculum (2005) • It makes no mention of the “basic facts” of addition and multiplication (that is, the use of this expression) • It makes no mention of drill • It assumes, from earliest grades, a strong emphasis on teaching through problem solving—an inquiry, investigative approach; attention to big ideas; rich problems; varied forms of representation; varied strategies. • But it does support “meaningful practice both inside and outside the classroom.” • [“Teaching Approaches,” pp. 24-25]

11. Curriculum: Some Specific Expectations • Grade 1 • Compose and decompose numbers up to 20 in a variety of ways, using concrete materials . . . {Note, Kindergarten to 10} • Solve a variety of problems involving the addition and subtraction of whole numbers to 20 . . . {Note: Kindergarten - some investigation of addition and subtraction in everyday activities} • Solve problems involving the addition and subtraction of single-digit whole numbers, using a variety of mental strategies (e.g., one more than, one less than, counting on, counting back, doubles) [NOTE: no limit placed on which single digit numbers in addition]

12. Curriculum: Some Specific Expectations • Grade 2: • Solve problems involving the addition and subtraction of whole numbers to 18, using a variety of mental strategies . . . • Solve problems involving the addition and subtraction of two-digit numbers, with and without regrouping, using concrete materials . . . • [Note that we are already well beyond a focus on the A & S basic facts, at the official level of curriculum document specifications]

13. Curriculum: Some Specific Expectations • Grade 3: • Add and subtract three-digit numbers . . . • Multiply to 7 x 7 and divide to 49 ÷ 7, using a variety of mental strategies (e.g., doubles, doubles plus another set, skip counting) • Grade 4: • Multiply to 9 x 9 and divide to 81 ÷ 9, using a variety of mental strategies (e.g., doubles, doubles plus another set, skip counting)

14. The Grade 1-8 Math Curriculum • Notice the emphasis throughout on either problem solving, or the development of mental strategies. • When the “basic addition and subtraction facts” are addressed in grade 1, suggest why “multiplication and division basic facts” might not be fully addressed until grade 4? • [DF comment: some reasons—(1) in some sense, addition and subtraction are more ‘fundamental actions’ and are experienced more readily early in life as combining and taking away; (2) one way to view--multiplication and division may be thought of as building on addition and subtraction and thus follow when addition and subtraction are better understood.]

15. The Authors: Small (2013) • Small (especially chapter 8) • does not refer to the use of basic facts charts • stresses underlying “principles” for adding and subtracting, multiplying and dividing. • emphasizes “fact families” [sometimes also known as “companion statements”]: • Addition and subtraction are “related inverse operations”: they undo each other • 5 – 2 = 3; 3 + 2 = 5 • And 5 – 3 = 2; 2 + 3 = 5 • The commutative property applies to addition (but not to subtraction) • 2 + 3 = 5; 3 + 2 = 5 So: 2 + 3 = 3 + 2 • Multiplication and division are “related inverse operations”: they also undo each other • 6 ÷ 2 = 3; 3 x 2 = 6 • 6 ÷ 3 = 2; 2 x 3 = 6 • The commutative property applies to multiplication (but not to division) • 2 x 3 = 6; 3 x 2 = 6 So: 2 x 3 = 3 x 2

16. The Authors: Small (2013) • Based on extensive sets of principles for the four operations, Small offers a wide range of strategies and activities for students to develop their basic facts. • Small does not refer to “drill.” • Small does, however, support practice when well used, and she notes, “before students can work efficiently with algorithms, they must know their addition and subtraction facts” (p. 218). Similarly, they need to know their multiplication and division facts. • [DF: Be sure you understand what is meant by “algorithm.”]

17. Van de Walle et al. (2014) on DRILL • These authors are more explicit in their views on drill for basic facts. They refer to “drill” as “repetitive non-problem-based activity” (p. 168). Drill does have a place, as it can strengthen memory and retrieval capabilities. So...

18. Van de Walle et al. (2014) on DRILL • Drill is important to consider once children have effectively used reasoning strategies that they understand, but that they have not yet memorized-practice putting those strategies to work [DF: understand strategies; use them to learn basic facts, and ultimately to memorize them. But do not start with memory work] • For drill to be effective, pacing and focus are crucial. • Too often children become frustrated and overwhelmed because drill includes too many facts too quickly. • Children will progress at different rates—gifted children tend to be good at memorizing whereas children with intellectual disabilities have difficulty memorizing. • Make drill enjoyable. • Drill in short segments.

19. Van de Walle et al. (2014) on “Basic Facts” charts • These authors make use of 0-9 addition and multiplication facts charts, but primarily as a basis for relating “reasoning strategies” (which could be taught explicitly) to sets of facts to which the strategies apply. • In other words, use them as a means of supporting an understanding of the strategies and operations relationships, and not strictly as a memorization tool.

20. “Facts” Strategies • Many of the principles and related strategies that Small describes are the same as discussed by Van de Walle, Lovin, Karp, & Bay-Williams. • Key among these strategies is recognizing patterns, and understanding mathematics relationships such commutativity, and the inverse relationship between addition and subtraction and between multiplication and division.

21. “Addition Facts” Strategies • One More Than • Two More Than • Adding Zero • Doubles • Near Doubles (Doubles Plus One, Doubles Plus Two) • Make Ten (limit to working with One More Than and Two More Than) • Up Over Ten or Make Ten Extended

22. “Multiplication Facts” Strategies • Doubles • Fives Facts • Zeros and Ones • “Nifty Nines”

23. YOUR TASK! • On the blank addition and multiplication charts, identify which columns and/or rows each of the strategies apply to. • Use different shading or colours. • After marking off the above strategies, what facts are left unmarked? • How does understanding these strategies facilitate basic facts understanding? • Does recognizing a pattern of symmetry help? What mathematics property is involved? • Focus on activities that support the development of each of these strategies.

24. The basic facts strategies • Go to “Notices” under EDUC4274 (Franks website) and look for the file “basic facts strategies” There you will find a discussion and chart displays that directly relate to this activity.