Whole Number Operations: The Seminar Handout

1. Whole Number Operations:The Seminar Handout

2. Whole Number Operations • Remember: • A general sequence of concrete to pictorial to symbolic levels of representation is important. • An algorithm is a standard procedure for performing a given process (such as addition, division, etc.)

3. Whole Number Operations • Part 1: Addition and Subtraction • The use of different concrete materials: • Base ten blocks, popsicle sticks, four wire abacus. • ‘Trading’ and ‘regrouping’ are the preferred current terms, instead of ‘borrowing’ and ‘carrying’. Their use grows out of concrete model (e.g., base ten blocks) use. • What advantages and disadvantages? • Can the symbolic level operation be modelled by the concrete model?

4. Whole Number Operations • A left to right subtraction algorithm: • Don’t panic! This is not “the way we have to teach subtraction now”! • BUT: As the teacher you need to be aware that there are other correct ways to perform operations, not just the ‘usual way’ (right to left algorithm). • Be open to students’ ideas, and look at them carefully before offering opinion on them. [“student-generated algorithms,” “invented algorithms.” See Van de Walle and Math curr. doc.] • You may also find that children from other systems, countries, have learned different operation algorithms. Be prepared to learn about them and accept them.

5. Whole Number Operations • Remember: • what is important for students is learning and understanding an appropriate procedure for performing a given operation, not being required to know and use only the “established” approach. • This does not mean that standard algorithms should not be taught! • It does mean that it is not wise to impose a single method on all children or to deny children the opportunity to use an appropriate alternative.

6. Whole Number Operations • It will be very important that you understand place value thoroughly, and its relation to the various algorithms. • Spend time understanding place value as evident in the various algorithms. • Don’t rush to teach the standard algorithms

7. Whole Number Operations • Van de Walle writes extensively about “Invented Strategies” and encourages their development/practice in class. (chap. 13) • He describes them as any strategy other than the traditional algorithm and that does not involve the use of physical materials or counting by ones. (p. 227) • He recommends that such strategies be supported by written records of thinking (sharing and focusing) by students.

8. Whole Number Operations • About Invented Strategies Van de Walle states: • Number oriented rather than digit oriented • Left- rather than right-handed • Flexible rather than rigid (methods vary with the number, e.g. 465 + 230 and 526 + 98) • Enhance base ten concepts • Built on student understanding • Fewer errors made • Basis for mental computation and estimation • Serve students at least as well on standard tests • (pp. 227-228)

9. Whole Number Operations • Part 2: Multiplication and Division • Multiplication • Think: 6 x 3 as ‘six groups of three’ • Can be thought of as • Repeated addition: 3 + 3 + 3 + 3 + 3 + 3 • An array: (6 rows of 3) • An area representation (think of a rectangle 3 units wide and 6 units long—How many units enclosed?) • Try modeling each of these forms with concrete materials (esp. base 10 materials).

10. Whole Number Operations • Multiplication. • Partial products: What are they? • “Product” is the name given to the result of a multiplication. • If I multiply the following two, two-digit numbers, I have a series of intermediate “products”. These are “partial products”... Not yet the final answer or product…….

11. 34X 26 204 (partial product: 34 x 6)680 (partial product: 34 x 20) 884 BUT ALSO…..

12. 34 = 30 + 4X 26 = 20 + 6 24 (partial product: 6 x4) 180 (partial product: 6 X 30) 80 (partial product: 20 X 4)600 (partial product: 20 X 30) 884

13. Whole Number Operations • Division • Think: 30 ÷ 5 as “In 30 there are how many groups of 5?” • Can think of as repeated subtraction [Repeatedly remove or subtract groups of 5 until you can no longer do so. How many groups of 5 did you subtract?] • In written contexts (story problems) it is useful to be aware of: partitive contexts and measurement contexts. For example…..

14. Whole Number Operations • Partitive and measurement contexts • Partitive (sharing): • Example: Determining how many items a known number of friends will each receive: “Tammy wanted to share 30 candies equally among her 5 friends. How many candies did each friend receive?” [30 ÷ 5 = ?] • Think in terms of distributing or sharing them equally until they are all out.

15. Whole Number Operations • Partitive and measurement contexts • Measurement (think “unit” quantity) • Example: Determining how many friends will get equal amounts of an item: “Tammy had 30 candies in small bags of 5 candies each. How many of her friends will get a bag of candies?” [30 ÷ 5 = ?] • Notice the partitive and measurement contexts are quite different, while the symbolic level equation is the same in each case.

16. Whole Number Operations • Partitive and measurement contexts • Research has shown that students tend to have more difficulty with measurement contexts than partitive contexts in determining how to solve these story problems.