1. Alternative Algorithms for Multiplication and Division If we don’t teach them the standard way, how will they learn to compute? Kathy Cheval and Kathy Bowers, Salem-Keizer Public Schools, Salem, OR http://is.salkeiz.k12.or.us
2. If we can convince students that mathematics is figure-out-able, that it is more than memorization, then we can increase student buy-in and confidence. If we can get students to think in class, instead of just trying to memorize series after series of steps, we can save time and decrease frustration because by building on understanding, we will have fewer misapplied and mixed-up rules. Why Numeracy for Secondary Students Harris & Pope, 2005
3. How has this student misapplied the rules for multiplying? Based upon the work above, what understandings and misunderstandings does this student have?
4. Multi-digit Multiplication and DivisionWhat are the goals for students? Develop conceptual understanding Develop computational fluency
5. What is Computational Fluency? Fluency demands more of students than does memorizing a procedure. Fluency rests on a well-build mathematical foundation that involves: Understandingimplies that the student brings meaning to the operation being carried out. The student can explain the “why” of each step taken to solve the problem. Efficiencyimplies that the student does not get bogged down in the steps or lose track of the logic of the strategy. An efficient strategy is one that the student can carry out easily. Accuracy depends on careful recording, knowledge of basic number combinations and other important number relationships, as well as verifying the results. Flexibility requires the knowledge of more than one approach to solving a problem. Students need to be flexible to choose an appropriate strategy for a specific problem.
7. Multiplicative Thinking Multiplication is more complex than addition because the two numbers (factors) in the problem take different roles. 12 cars with 4 wheels each. How many wheels? 12 x 4 = 48 cars wheels/car wheels (groups) (items per group) (total number of items) (multiplier) (multiplicand) (product)
8. Multi-digit Multiplication Strategies12 cars with 4 wheels each. How many wheels? Additive Strategies Direct Modeling Repeated Addition Doubling Multiplicative Strategies
9. Multi-digit Multiplication Strategies52 cards per deck. 18 decks of cards. How many cards? Multiplicative Strategies Single Number Partitioning Both Number Partitioning Compensating
10. Multi-digit Multiplication StrategiesAs you look at student work, try to identify the kinds of strategies you see students using. While this list is not comprehensive, it will give you a place to begin. Often you will see evidence of more than one strategy being used. Multiplicative Strategies Single Number Partitioning Both Number Partitioning Compensating Additive Strategies Direct Modeling Repeated Addition Doubling
11. There are 18 ants with 6 legs each. How many legs altogether? Sample 1 Sample 2
12. Students collected cans to recycle. Each box holds 12 cans. They filled 38 boxes with cans. How many cans did they collect? Sample 3 Sample 4
13. There are 62 fifth graders. It costs $38 per student for outdoor school. How much do the fifth graders need to earn so everyone can go? Sample 5 Sample 6
14. There are 62 fifth graders. It costs $38 per student for outdoor school. How much do the fifth graders need to earn so everyone can go? Sample 6 Sample 7
15. Teacher’s Role Provide rich problems to build understanding Encourage the use of “thinking tools” (manipulatives like snap cubes or 300 charts) when needed Guide student thinking Provide multiple opportunities for students to share strategies Help students complete their approximations Model ways of recording strategies Press students toward more efficient strategies
16. Two Contexts for Division Measurement Division (number of groups unknown) There are 54 children on a full bus. Each seat can hold 3 children. How many seats are there on the bus? Partition Division (size of groups unknown) There are 54 children on a full bus. There are 18 seats. How many children are sitting on each seat?
17. Multi-digit Division StrategiesThe strategies students use for division will be very similar to those they used for multiplication. As you look at student work, try to identify the kinds of strategies you see students using. This is not a comprehensive list, and often you will see evidence of more than one strategy being used. Multiplicative Strategies Single Number Partitioning Both Number Partitioning Compensating Additive Strategies Direct Modeling Repeated Addition/Subtraction Doubling
18. There are 54 children on a full bus. Each seat can hold 3 children. How many seats are there on the bus? Sample 1 Sample 2
19. There are 54 children on a full bus. There are 18 seats. How many children are sitting on each seat? Sample 3 Sample 4
22. Teaching a “standard” way? Delay! Delay! Delay! Spend most of your time on invented strategies. The understanding students gain from working with invented strategies will make it much easier for them to understand a standard algorithm. For most students, this means delaying the teaching of a standard way of multiplying and dividing until 5th grade. Students who don’t clearly understand the way should be allow to use a way that make sense to them.
23. Which “standard” way? Partial Products for 52 x 18 (modeled by an open array) 52 52 x 18x 18 16 16 400 400 20 416 500 . 20 936 436 500 936 50 2 500 20 10 8 400 18
24. Which “standard” way? Partial Products for 936 18 18 936 18 936 100 x 18 = 1800 180 10 x 18 900 50 756 36 360 20 x 18 36 2 396 0 52 360 20 x 18 36 36 2 x 18 0 52 x 18
