Creating Problem-Solving Experiences with Ordinary Arithmetic ProcessesAuthor(s): John BernardSource: The Arithmetic Teacher, Vol. 30, No. 1 (September 1982), pp. 52-53Published by: National Council of Teachers of MathematicsStable URL: http://www.jstor.org/stable/41192043 .

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Creating Problem-Solving Experiences with Ordinary

Arithmetic Processes By John Bernard

Table 1 Selected Number-Search Problems

For teachers of arithmetic who are seeking new and interesting ways to provide students with needed compu- tational practice or review, here is a way to cast such practice in a prob- lem-solving context without moving into "word problems." To illustrate by way of an example, suppose you start with the number 99; are allowed only two kinds of legal moves, adding elev- en or subtracting seven (you can switch back and forth or repeat such moves as often as you like); and you are to hit 100. Try this before reading on.

For the problem just posed, a stu- dent's thoughts might go as follows:

I'm at ninety-nine. If I add eleven, that's going to be too much, so I had better subtract seven.

99 - 7 = 92.

That gives me ninety-two. Adding eleven would still give me too much, so I'll subtract seven again.

92 - 7 = 85

That gives me eighty-five. Now if I add eleven I get ninety-six.

85+ 11 =96

Better subtract seven.

96 - 7 = 89

John Bernard is an assistant professor of math- ematics education in the Department of Mathe- matical Sciences at Northern Illinois University in DeKalb, Illinois. He is now involved in research and teacher preparation. Problem solving and the psychology of mathematics are particular areas of interest. His previous pro- fessional experience includes teaching second- ary school mathematics, physics, and chemis- try.

Now I add eleven.

89+ 11 = 100 That's it!

Obviously, your own solution and the one described in the preceding paragraph are only two of several approaches to the problem. There are ten routes from 99 to 100, using the given moves and taking the minimum five steps. See if you can find all ten by tracing the connections in figure 1 . Many more solution paths exist if you allow extra steps. Such variety in so- lutions usually stimulates interesting dialogue between solvers as they compare strategies and solutions.

Now let's look at the problem struc- ture to discover ways you can provide for the interest and needs of your students. First, there is a starting number (like 99), then some legal moves (here, adding eleven and sub- tracting seven), and finally a goal (like 100). This is summarized in the first entry of table 1. From the table, it is easy to see that you can vary the goal, the starting number, or the legal moves. The problems in the table show only two legal moves for each, but problems with three or more moves could be constructed as well. As new mathematical operations - such as squaring and taking square roots - are learned, these processes could also be used as legal moves for certain problems. With algebra, even functions like /(jc) = 2* + 1 could be used for legal moves.

The level of difficulty of problems like this can be controlled in at least two ways. Typically, problems be- come more difficult with an increase in the minimum number of moves. For example, figure 2 shows all of the possible outcomes generated from 80 by adding 11 or subtracting 7, to a limit of four moves. Thus the follow- ing problem could be solved in four moves, but no less than four moves:

Start Legal Moves . Goal

a) 99 add 11, subtract 7 UN" b) 99 add 11, subtract 7 101 c) 17 add 11, subtract 7 19 d) 25 multiply by 2, subtract 7 65 e) 135 multiply by 2, divide by 3 60 f) 321 add 11, divide by 3 54

52 Arithmetic Teacher

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Start. 80 Legal moves. Add 11, subtract 7 Goal. 70

Looking again at figure 2, we see that it does not matter if we first add eleven and then subtract seven or if we subtract seven and then add elev- en. We arrive at 84 either way. We say the operations (moves) are com- mutative. This is not the case for the operations of multiplying by 2 and subtracting 7, as in problem d of table 1. For this reason, problems d, e, and / would tend to be harder than prob- lems a, b, and c of that same table. Mapping out a diagram of the possible routes and solution paths for one of these noncommutative problems might be informative and challenging. (See Bernard 1978 for a theoretical account.)

Let students help you explore the many possibilities. Let them develop and challenge others with problems of their own. Let them attempt some problems with calculators and others without calculators. (See DuRapau and Bernard 1979.) Many interesting problem-solving strategies will begin to emerge. Students develop a sense of direction, going up or down to hit the goal. They discover that it is not al- ways a good idea to divide even if they want to go down. Some even develop more general approaches such as working backwards, develop- ing subgoals, and moving both ends toward the middle (Bernard 1978). Discoveries such as these could be the forerunners of important problem- solving skills in mathematics. (See Polya 1957.)

References

Bernard, John E. "A Theory of Mathematical Problem Solving Derived From General The- ories of Directed Thinking and Problem Solv- ing." Unpublished Doctoral Dissertation, the University of Texas, 1978.

Cohen, Martin P. and John E. Bernard. "Num- ber Search: An Understanding of Problem Solving." PCTM-Pennsylvania Council of Teachers of Mathematics Journal 4 (Spring 1980).

DuRapau, Victor J. and John E. Bernard. "From Games to Mathematical Concepts via the Hand-Held Programmable Calculator." International Journal of Mathematical Edu- cation in Science and Technology 10 (3, 1979).

Polya, George. How to Solve It. Princeton, N.J.: Princeton University Press, 1957. W

September 1982

Fig. 1

121

110 114

99 103 107

92 96 100!

85 89

78

Fig. 2 .^124

113

102 ^106

^91 95^

80 J^84C^ ^88

73C^ ^77

59^

^52

53

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