scope and sequence for a problem-solving curriculum

Scope and Sequence for a Problem-Solving CurriculumAuthor(s): Alexander TobinSource: The Arithmetic Teacher, Vol. 29, No. 6 (February 1982), pp. 62-65Published by: National Council of Teachers of MathematicsStable URL: http://www.jstor.org/stable/41192020 .

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Scope and Sequence for a Problem-Solving Curriculum

By Alexander Tobin

"The development of problem- solving ability should direct the efforts of mathematics educators through the next decade. Performance in problem solving will measure the effectiveness of our personal and national posses- sion of mathematical competency. . . The mathematics curriculum should be organized around problem solving." (NCTM, An Agenda for Action) It was with these things in mind that the School District of Philadelphia and its Division of Mathematics Edu- cation took on as one of its major priorities the creation of a scope and sequence outline for problem solving. We also recognized, from the very inception of this project, that teachers would need activities, direction, guid- . ance, and materials to effectively im- plement the program.

Prior to the creating of our new scope and sequence outline, problem solving was contained within the strands of our existing curriculum. The decision to make problem solving a separate strand in the new outline was made for several reasons. If problem solving was given status as a separate strand, there would be more assurance that it would be taught. Since problem solving was indicated as a major goal of mathematics education, it would be given its proper perspective if handled separately.

Alexander Tobin is director of mathematics education for the School District of Philadel- phia. He has also served as the president of the National Council of Supervisors of Mathemat- ics, the Pennsylvania Council of Teachers of Mathematics, and the Association of Teachers of Mathematics of Philadelphia and Vicinity.

Making problem solving a separate strand would enable teachers to rec- ognize the difference between verbal exercise and true problems. Finally, and probably most important, problem solving would not be overlooked by teachers who considered it an op- tional topic rather than one of su- preme importance.

The School District of Philadelphia has an elementary mathematics program covering the content from grades one through eight. The curriculum for this program has been divided into twenty-one levels. Each level em- bodies carefully delineated areas of learning arranged in progressive stages. Such an arrangment of se- quential skills and subject matter eliminates grade restrictions and pro- motes continuous growth according to the individual student's ability and rate of learning. Under our revised curriculum, the elementary mathematics concepts and skills are developed sequentially in each level in sev- en basic strands:

Section 1.00 Numeration Section 2.00 Whole numbers and

integers: operations Section 3.00 Fractional numbers:

operations Section 4.00 Measurement Section 5.00 Organization and in-

terpretation of data Section 7.00 Geometry Section 8.00 Problem solving

The topics under each section are structured so that a mathematical sequence is developed as the teacher proceeds from one level to another. For example, under Section 2.00,

whole numbers and integers, the topic "multiplication" (2.60) is introduced at level 4 and developed and extended through level 16. Concepts are introduced initially through use of manip- ulative materials and physical models. As insights are gained and discoveries made, pupils are guided through the semiconcrete stage and on to the ab- stract level of understanding.

To make the transition from the current to the new curriculum flow smoothly, we decided to effect the changes over a three-year period. Levels one to five were rewritten, including the problem-solving strand, and distributed throughout the school system in September 1980. Mathemat- ics activity booklets and evaluation tests accompanied each of those revised levels.

Levels six through twenty-one are being revised as this is being written. The upper-level materials will be phased in during the 1981-82 and 1982-83 school years. The three-year period was planned so that children in the upper levels would not be handi- capped by being required to pass tests in content that they may not have studied in the lower levels. In the revised levels, some new content has been introduced, some old content has been deleted, and some content has been placed in different levels.

The emphasis in the new outline is on process and not product. There is no concentration in the early levels on the "right" answer. The emphasis is on analyzing the data, determining the facts, posing questions, and so on. The sequence goes from readiness, to strategies, to analysis of word problems, to brainstorming. The subsec-

62 Arithmetic Teacher



tion on brainstorming is an attempt to encourage youngsters to approach problem solving in a creative fashion. Its intent is to point out that there is no one correct way to solve problems. There are many methods and proce- dures, and a youngster is encouraged to explore as many of these methods as possible. Under the subsection on strategies, the intent is not so much that students be expert in each strategy as it is that students, by their study of the various strategies, will develop a memory bank to fall back on when they are faced with a problem to be solved.

Problem Solving Strand

Level 1

Readiness

a. Provide opportunities for commu- nication in quantitative language. b. Provide practice in matching numbers to pictures/objects.

Level 2

Readiness

a. Develop recognition of number in everyday use. b. Continue matching numbers to pictures/objects. с Continue to develop quantitative language.

Level 3

Readiness

a. Dramatize number situations. b. Match number facts to groupings of pictures/objects.

Analysis of word problems

a. Create a simple word problem for number facts up to 9.

(1) Addition (2) Subtraction

Level 4

Strategies

a. Develop the use of number patterns to solve problems.


a. Identify question being asked in a simple word problem. b. Use concrete or pictorial represen- tation to solve simple word problems. c. Create a number sentence from a simple word problem.

Level 5

Strategies

a. Develop the use of number patterns to solve problems.


a. Extend creation of simple word problems from number facts up to 18, using all four operations. b. Complete unfinished word problem.

(1) Possible questions to be asked (2) Possible number sentences

Level 6

Strategies

a. Develop the use of diagrams to solve problems. b. Extend relating patterns to solution of problems.


a. Choose operation for problem from multiple choice list; perform subsequent computation. b. Introduce reviewing a problem and its solution.

Level 7

Strategies

a. Introduce use of guess-and-verify strategy.


a. Apply previously taught skills to two-step problems.

Level 8

Strategies

a. Develop the use of organized lists to solve problems.


a. Develop the ability to select pertinent data from a problem. b. Introduce the concept of labeling where necessary.

Level 10

Strategies

a. Extend the use of diagrams to solve problems.


a. Continue selection of pertinent data from problems. b. Identify problems with sufficient/ insufficient data. с Review concept of labeling where necessary.

Level 11

Strategies

a. Extend the use of guess and verify.


a. Review identification of sufficient/ insufficient data.

(1) Extend to problems with abun- dant data.

b. Identify explicit/implicit information. с Match verbal problem with corresponding complete number sentence in a multiple-choice format.

Level 12


a. Extend identification of explicit/ implicit information.

(1) Deduced information b. Extend matching of verbal problems with corresponding open sentence in a multiple-choice format.

(1) Compute answer, с Extend reviewing the problems

(1) Reasonableness of solution (2) Checking solution

Level 13

Strategies

a. Continue to use patterns to aid in

February 1982 63



the solution of problems.

Analysis of word problems a. Review choosing correct open sentence for problem.

(1) Extend to choosing two sentences for two-step problems; perform appropriate computation.

b. Continue to review the problem. (1) Reasonableness of answer

Level 14

Strategies a. Use charts and tables to aid in the solution of problems. b. Use diagrams to aid in the solution of problems.

Analysis of word problems a. Review labeling. b. Review and extend choosing the appropriate number sentence.

(1) Extend to choosing one sentence for two-step problems.

Brainstorming

Level 15

Strategies a. Extend strategy of guess and verify to solve problems. b. Review and extend use of charts, tables, and diagrams for problem solving. с Review and extend use of patterns for problem solving.

Analysis of word problems a. Write mathematical sentence for one-step problem, do appropriate computation, label where necessary, and review problem.

Brainstorming

Level 16

Strategies a. Review strategies developed in previous levels. b. Introduce strategy of making a

simpler problem from a given problem.

Analysis of word problems a. Write a mathematical sentence for each step of a two-step problem. Compute and label where necessary. Review problem.

Brainstorming

Level 17

Strategies a. Review previously taught strategies

Analysis of word problems a. Write one mathematical sentence for all steps in a multistep problem

Level 18

Strategies a. Apply all previously developed strategies to problem solving situations.

Brainstorming * * * * *

Levels 19, 20, and 21 are in the process of being written at the time of this writing.

Strategies for Teaching Problem Solving As was mentioned in an earlier part of the article, it is not enough to provide teachers with a scope and sequence chart. It is incumbent upon mathematics educators to give teachers sample activities that illustrate the implemen- tation of the program and to suggest materials they can use in the accom- plishment of their task. Several activities, from different levels of the problem-solving strand, are included here as samples.

Level 4

Develop the use of number patterns

to solve problems.

Using number patterns can be a very effective strategy in problem solving. At a higher level, the recognition of a pattern in a sequence of numbers can help the student to predict an answer and generalize for other problems. At this level, it is important to practice recognizing patterns and to apply that skill to simple problem situations. For example:

Write some partial number sequences on the board.

1, Zr, 3, *t, , , ^___

2, 4, 6, 8, , , - _ 1,3,5,7, , , 1,4,7,10, , , 5, 10, 15, 30, , ,

Ask students to try to determine the next three numbers in each sequence. Discuss with the students the general pattern of each sequence. Other sequences of this type may be found in textbooks or created by teacher and students.

Level 8

Develop the use of organized lists to solve problems.

Many problems involving classifica- tion of numbers, figures, and so on, can be solved more expeditiously by arranging information in some orderly fashion. When students test many cases, the use of an organized list helps to avoid using the same case twice. For example:

(1) Given 3 different digits, how many 3-digit numbers can you name? Students normally begin forming 3-digit numbers without or- ganizing their data according to some rule or procedure. A system- atic listing makes it clear that there are six 3-digit numbers.

12 3 2 13 3 12 13 2 2 3 1 3 2 1

Repeat the procedure using 4 digits. How many 4-digit numbers can you name using 4 different digits? (2) How many ways can you make change for a quarter? Rather than rely on "trial and error" (in some

64 Arithmetic Teacher



problems, this is a good strategy) an organized list is most helpful. dimes nickles pennies - - 25 - 1 20 - 2 15 1 - 15 - 3 10 1 1 10 - 4 5 1 2 5 2-5 1 3 - 2 1 - - 5 -

Level 11

Extend the use of guess and verify. The use of the guess-and- verify strategy can be extended to problems involving arrays of numbers in magic squares. After several trials, students may begin to see a pattern in the solution. For example:

(1) In a 5-by-5 array of squares, fill in each square with the number 1,2, 3, 4, or 5 such that the same number does not appear twice in any hori- zontal, vertical, or diagonal lines.

Once the array is completed, using guess and verify, encourage students to analyze the solution to see^ if any pattern appears. (2) A 3-by-3 magic square is also a good example of a problem that can be approached by the guess and verify method. In this problem, the numbers 1 through 9 are placed in the array, one in each square, so the sum across any row, down any col- umn, and on either diagonal is 15.

If the square is completed by the guess-and- verify method, encourage students to look for a pattern in the solution.

Level 14

Use charts and tables to aid in the solution of problems. The use of a chart is especially helpful in the solution of problems in which deduction is necessary. This can be illustrated with the following problem:

In a certain bank, the positions of

cashier, manager, and teller are held by A, B, and C, but not neces- sarily in that order. (a) The teller, who is an only child, earns the least. (b) C, who married A's sister, earns more than the manager. A chart for recording the information can be set up.

A | В | С Cashier x x yes Teller je, a, b yes x, a, b Manager yes x, b

Statement В tells us that A is not an only child (since he has a sister) and, therefore, not the teller. Put an x in that box. The a and b in the box means we used both statements to deduce the information.

Since С earns more than the manager, he is not the manager. Place an x and b in that box.

С is also not the teller since he earns more than the manager and the teller earns the least. Place an jc, a, and b here.

Now we see that С must be the cashier (write yes). Write an x under A and В for cashier. Now A must be the manager, leaving the teller's job to B.

Conclusion Our problem-solving curriculum is not complete. There is a great deal of work still to be accomplished. Not only must we finish the scope and sequence outline for levels six through twenty-one and provide activities and evaluation tests for each of these, but also we must review what we have already distributed. At the present time, levels one to five are being field tested. We have asked teachers to report to us, as they teach these levels, which material seems to work and which does not. In many cases, we will be asked to add more activities; in other cases, we will be asked to delete materials.

One of our major responsibilities will be the necessary staff development essential to the ultimate success of any curriculum revision. This staff development will take many forms - individual school faculty meetings,

demonstration teaching within class- rooms, small cluster meetings in which schools will come together in a convenient geographic location, dis- trictwide inservice courses, and com- bined efforts of local universities and the school district to provide gradu- ate-level courses designed to meet the needs of the teachers as they imple- ment the program.

The creation of the new mathematics curriculum in Philadelphia ad- dresses what has been one of our major needs in mathematics education for many, many years. It really is "an agenda for action" in the area of problem solving. We are confident that once the momentum has been gener- ated, a great many worthwhile activities will come as a result of this effort.

In conclusion, we encourage read- ers to share with us similar undertak- ings as well as recommendations and suggestions for improving upon this effort, m

February 1982 65

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