1. Chapter 11 To accompany Helping Children Learn Math Cdn Ed, Reys et al. ©2010 John Wiley & Sons Canada Ltd.

2. Guiding Questions 1. What is a computational algorithm? How and why are manipulative materials useful in helping children develop understanding of algorithms? 2. How can teachers help children develop the addition algorithm? Do all children need to use the same addition algorithm? 3. What are two standard subtraction algorithms and how did they develop? 4. How does the distributive property support the development of the multiplication algorithm?

3. Computational Algorithms A computational algorithms is a computational skill with paper-and-pencil procedures. Focus has shifted to more attention on what children construct or develop for themselves. Computation has become a problem-solving process, one in which children are encouraged to reason their way to answers, rather than merely memorizing procedures that the teacher says are correct.

4. Balancing Conceptual Understanding and Computational Proficiency Recommendations for the teaching of computation include the following: – Fostering a solid understanding of and proficiency with simple calculations – Abandoning the teaching of tedious calculations using paper-and- pencil algorithms in favor of exploring more mathematics – Fostering the use of a wide variety of computation and estimation techniques—ranging from quick mental calculation, to paper-and- pencil work, to using calculators or computers—suited to different mathematical settings – Developing the skills necessary to use appropriate technology and then translating computed results to the problem setting – Providing students with ways to check the reasonableness of computations (number and algorithmic sense, estimation skills)

5. Modelling Written Algorithms with Concrete Materials Use base-ten blocks and/or other concrete materials to model the "common" algorithms for addition, subtraction, multiplication, and division in the following slides. Compare your models with those used in our text and those of your classmates.

6. Addition Algorithms Standard Addition Algorithm Partial-Sum Addition Algorithm Higher-Decade Addition

7. Addition: Standard Algorithm When students attempt to use the standard algorithm without understanding why it works, they may be more prone to errors than when they do addition in ways that intuitively make sense to them. Example: 27 + 35

8. Addition: Partial-Sum The Partial-Sum Algorithm adds from left to right (tens first, then ones), as is done when reading text. The partial-sum algorithm can be used as an alternative algorithm for addition or it can be useful as a transitional algorithm on the way to learning the standard algorithm. Children should be encouraged to work with whichever procedure they find easiest to understand.

9. Addition: Higher-Decade Combinations such as 17 + 4 or 47 + 8 or 3 + 28, called higher- decade combinations, are used in a strategy sometimes referred to as “adding by endings.” The adjacent activity focuses attention on the relationship of 9 + 5, 19 + 5, 29 + 5, and so on. As a result of this activity, children realize the sum will have a 4 in the ones place because 9 + 5 = 14, and the tens place will always have 1 more ten.

10. Subtraction Standard Subtraction Algorithm Partial-Difference Subtraction Algorithm

11. Subtraction: Standard Algorithm The standard subtraction algorithm taught in most of North America for the past 50 or 60 years is the decomposition algorithm. It involves a logical process of decomposing or renaming the sum (the number you are subtracting from). In the following example, 9 tens and 1 one is renamed as 8 tens and 11 ones:

12. Subtraction: Partial-Difference Take a moment to examine this “algorithm.” What is going on?

13. Subtraction: Partial-Difference When children have been taught to subtract without understanding, they simply do the subtraction in each column separately, always subtracting the larger number from the smaller. To remedy this, it is important to have students return to modelling some simple problems with manipulatives.

14. Multiplication Multiplication with One-Digit Multipliers Multiplication by 10 and Multiples of 10 Multiplication with Zeros Multiplication with Two-Digit Multipliers Multiplication with Large Numbers

15. Multiplication with One-Digit Multipliers 2 × 14 = 2 × (10 + 4) = (2 × 10) + (2 × 4) = 20 + 8 = 28 Array of 2 x 14 Array of Distributive Property of Multiplication 14 + 14

16. Multiplication by 10 and Multiples of 10 Multiplying by 10 comes easily to most children, and is readily extended to multiplying by 100 and 1000 as children gain an understanding of larger numbers. Multiplying by 20, 30, 200, 300, and so on is an extension of multiplying by 10 and 100. Emphasize what happens across examples and generalize from the pattern. For example, have children consider 3 x 50: 3 x 5 = 15 3 x 5 tens = 15 tens = 150 3 x 50 = 150 Then have them consider 4 x 50: 4 x 5 = 20 4 x 5 tens = ____ tens = _____ 4 x 50 = ______

17. Multiplication with Zeros When zeros appear in the factor being multiplied, particular attention needs to be given to the effect on the product or partial product. Many children are prone to ignore the zero. When an estimate is made first, children have a way of determining whether their answer is in the ballpark.

18. Multiplication with Two-Digit Multipliers Arrays or grids offer one way to bridge the gap from concrete materials to symbols and they also help illustrate, once again, why the partial-products algorithm makes sense.

19. Lattice Multiplication Use the lattice method to solve the problem 275 x 92 275 9 2

20. Multiplication with Large Numbers As children experiment with using a calculator for multiplication, there will come a time when they overload the calculator. Sometimes the number to be entered contains more digits than the display will show. At other times the factors can be entered, but the product will be too big for the display. When this happens, children should be encouraged to estimate an answer and then use the distributive property plus mental computation along with the calculator.

21. Division The traditional division algorithm is without doubt the most difficult of the algorithms for children to master, for several reasons: – Computation begins at the left rather than at the right as for the other operations. – The algorithm involves not only basic division facts, but also subtraction and multiplication. – There are several interactions in the algorithm, but their pattern moves from one spot to another. – Trial quotients, involving estimation, must be used and may not always be successful at the first attempt—or even the second.

22. Division Contemporary Canadian curriculum documents suggest that in today’s technological society, helping students understand when division is appropriate and how to estimate answers to division problems is an extremely important goal. Instruction should focus on teaching children how division is done through one and two-digit divisors. The calculator does the job of multi-digit division for most adults, so there is little reason to have children spend months or years mastering it. Other mathematics is of more importance for children to learn.

23. Division Division with One-Digit Divisors Division with Two-Digit Divisors Making Sense of Division and Remainders

24. Division: Standard Algorithm The distributive algorithm is considered the standard algorithm in North America.

25. Division: Subtractive Algorithm The subtractive algorithm has also commonly been used in North America. Many believe it to be a more intuitive and straightforward method for helping children learn to divide. Here are three examples.

26. Making Sense of Division and Remainders Try these problems: Pass out 17 candies to 3 children. (Each child receives 5 candies with 2 candies left over. Or, if the candies can be cut into pieces, each child could have 5 candies plus 2/3 of a candy.) If 17 children are going on the class trip and 3 children can ride in each car, how many cars are needed? (You will need 6 cars. With 5 cars you could seat only 15 children, with 2 children still waiting for a ride. With 6 cars, you can seat all 17 children, with 1 seat left over.) Note that the remainder is handled differently in each of these real-world problems.

27. Checking The calculator can serve many other functions, but its use in checking has not been overlooked by teachers. Nevertheless, the calculator should not be used primarily to check paper-and-pencil computation. Encourage estimation extensively, both as a means of identifying the ballpark for the answer and as a means of ascertaining the correctness of the calculator answer.

28. Choosing Appropriate Methods Children must learn to choose an appropriate means of calculating. Sometimes paper and pencil is better; sometimes mental computation is more efficient. Other times, use of a calculator is better than either, and sometimes only an estimate is needed. Encouraging students to defend their answers often yields valuable insight into their thinking. Children need to discuss when each method or tool is appropriate, and they need practice in making the choice, followed by more discussion, so that a rationale for their choice is clear.

29. Building Computational Proficiency Computational fluency with addition, subtraction, multiplication, and division is an important part of mathematics education in the elementary grades; however, “developing fluency requires a balance and connection between conceptual understanding and computational proficiency” (NCTM, 2000, p. 35).

30. How Many Strategies? Examine the problems in the following slide. How many different ways can you solve them using mental computation and/or written computation? Compare your strategies with those of your classmates or those used in our text.

31. 74 -58 28 +36 14 x 8 Problems ___ 4 )52

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