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Algorithms for Addition and Subtraction. If we don’t teach them the standard way, how will they learn to compute?.
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If we don’t teach them the standard way, how will they learn to compute? Children’s first methods are admittedly inefficient. However, if they are free to do their own thinking, they invent increasingly efficient procedures just as our ancestors did. By trying to bypass the constructive process, we prevent them from making sense of arithmetic. Kamii & Livingston
Algorithms in Arithmetic • An algorithm is a set of rules for solving a math problem • generally involve repeating a series of steps over and over • A variety of algorithms exist for addition, subtraction, multiplication and division.
Arithmetic Today • The learning of the algorithms of arithmetic hasbeen the core of mathematics programs in elementary schools • Today’s society demands more from its citizens than knowledge of basic arithmetic skills. • There is general agreement among mathematics educators that more emphasis be placed on areas like geometry, measurement, data analysis, probability and problem solving
Current Traditional Algorithm Addition 1 47 +28 75 “7 + 8 = 15. Put down the 5 and carry the 1. 4 + 2 + 1 = 7” Subtraction 7 13 83 - 37 46 “I can’t do 3 – 7. So I borrow from the 8 and make it a 7. The 3 becomes 13. 13 – 7 = 6. 7 – 3 = 4.”
Time to do some computing! Solve the following problems. Here are the rules: • You may NOT use a calculator • You may NOT use the traditional algorithm • Record your thinking and be prepared to share • You may solve the problems in any order you choose. Try to solve at least two of them. 658 + 253 = 297 + 366 = 76 + 27 = 314 + 428 =
Sharing Strategies Think about how you solved the equations and the strategies that others in the group shared. • Did you use the same strategy for each equation? • Are some strategies more efficient for certain problems than others? • How did you decide what to do to find a solution? • Did you think about the numbers or digits?
Number Line Method Add on Tens, Then Add Ones 46 + 38 46 + 30 = 76 76 + 8 = 76 + 4 + 4 76 + 4 = 80 80 + 4 = 84
Lattice Method • First arrange the numbers in a column-like fashion. • Next, create squares directly under each column of numbers. • Then split each box diagonally from the bottom-left corner to the top-right corner. This is called the lattice. • Now add down the columns and place the sum in the respective box, making the tens place in the upper box and the ones place in the lower box. • Lastly, add the diagonals, carrying when necessary.
Strategies • In contrast to the traditional algorithm, these alternative algorithms are: • Number oriented rather than digit oriented • Place value is enhanced, not obscured • Often are left handed rather than right handed • Flexible rather than rigid • Try 465 + 230 and 526 + 98 Did you use the same strategy?
Teacher’s Role Traditional Algorithm • Use manipulatives to model the steps • Clearly explain and model the steps without manipulatives • Provide lots of drill for students to practice the steps • Monitor students and reteach as necessary Alternative Algorithms • Provide manipulatives and 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
The reason that one problem can be solved in multiple ways is that… mathematics does NOT consist of isolated rules, but of CONNECTED IDEAS!
Time to do some more computing! Solve the following problems. Here are the rules: • You may NOT use a calculator • You may NOT use the traditional algorithm • Record your thinking and be prepared to share • You may solve the problems in any order you choose. Try to solve at least two of them. 636 - 397 = 221 - 183 = 502 - 256 = 892 - 486 =
Sharing Strategies Think about how you solved the equations and the strategies that others in the group shared. • Did you use the same strategy for each equation? • Are some strategies more efficient for certain problems than others? • How did you decide what to do to find a solution? • Did you think about the numbers or digits?
Nines Complement 827 → 827 - 259 → 740 (nines complement) + 1 (to get the ten's complement) 1568 568 (Drop the leading digit)
Another Look at the Subtraction Problems 636 - 397 = 221 - 183 = 502 - 256 = 892 - 486 = Now that we have discussed some alternative methods for solving subtraction equations, let’s return to the problems we solved earlier. Go back and try to solve one or more of the problems using some of the ways on the subtraction handout. Try using a strategy that is different from what you used earlier.
Summing Up Subtraction • Subtraction can be thought of in different ways: • Finding the difference between two numbers • Finding how far apart two numbers are • Finding how much you have to “add on” to get from the smaller number to the larger number. • Students need to understand a variety of methods for subtraction and be able to use them flexibly with different types of problems. To encourage this: • Write subtraction problems horizontally & vertically • Have students make an estimate first, solve problems in more than one way, and explain why their strategies work.
Benefits of Alternative Algorithms • Place value concepts are enhanced • They are built on student understanding • Students make fewer errors
Suggestions for Using/Teaching Traditional Algorithms • We are not saying that the traditional algorithms are bad. • The problems occur when they are introduced too early, before students have developed adequate number concepts and place value concepts to fully understand the algorithm. • Then they become isolated processes that stop students from thinking.
More and more, people need to apply algorithmic and procedural thinking in order to operate technologically advanced devices. Algorithms beyond arithmetic are increasingly important in theoretical mathematics, in applications of mathematics, in computer science, and in many areas outside of mathematics.
