Problems and Problem Solving - Ministry of Education

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se c t I o n 1 clArIFyIng pr o B l e m s A n d pr o B l e m so l V I n g Problems and Problem Solving A cursory perusal of the mathematics content outlined in the Revised Primary Curriculum (RPC) reveals that the terms ‘problems’ and ‘problem solving’ are introduced as early as Grade 1 and used with increasing frequency thereafter. This sharp focus on problem solving in the RPC reflects the fact that, in recent times, greater significance has been placed on curricula that produce students who are able to solve problems in the real world. These terms – ‘problem’ and ‘problem solving’ – however, require some clarification. Below, we offer four definitions from various sources. Definition 1 NCTM (2000, 24) sees a problem as: a task for which the solution method is not known in advance. In order to find a solution, students must draw on their knowledge, and through this process, they will often develop new mathematical understandings. Definition 2 Erickson (1999, 516) describes problems as: situations in which no readily known or accessible procedure or algorithm determines the method of solution. The problem may come from a real-world application or a puzzling dilemma. The problem or task should be an activity that focuses students’ attention on a particular mathematics concept, generalization, process, or way of thinking that matches the goals of school mathematics. Definition 3 Hiebert, et al. (1997, 75) explains a problem as: a task or activity for which the students have no prescribed or memorized rules or methods, nor is there a perception by students that there is a specific ‘correct’ solution method. Definition 4 Foong, P.Y. (2001) put forward a definition of a problem situation as: one where thinking takes place when a person is confronted with a [task] that has no immediate solution and that the problem-solver accepts the challenge to tackle it. Bearing in mind the definitions above, note the following points. 8 PROBLEMS AND PROBLEM SOLVING ProblemSolving.indd 8 8/24/12 6:55:33 PM
Bearing in mind the definitions above, note the following points. a) b) Problems allow students to do more than simply recall stored facts and/or procedures in order to obtain answers; rather problem-solvers use critical thinking to devise solution methods by drawing on and applying their knowledge in order to produce results. Problems allow students to make sense of mathematical situations for which they have no well-defined, rehearsed, standard routines or procedures. Problems are tasks that allow students to develop new insights into, or a sophisticated un- derstanding of, mathematical concepts or procedures arising from or associated with these concepts. c) Types of Problems Broadly speaking, there are two types of problems. Routine problems According to Gilfeather and Regato (1999), routine problems are tasks which use up to three steps to facilitate the use of known, or prescribed, procedures in obtaining solutions. Routine problems are generally described as being well-structured – the tasks are clearly formulated and, typically, use traditional ‘worded problems’ to determine students’ ability to use standard algorithms. Algorithms can be seen as: •y rules for calculating; •y computational procedures for deriving a solution to a given problem; and •y logical step-by-step procedures for solving a mathematical problem. Sometimes they involves the repetition of the same steps. Algorithms Algorithms are suitable for solving routine problems such as: a) (2 + 6) × 3 b) What is the area of a rectangle with a length of 12 cm and a width of 9 cm? c) 134 ÷ 14 d) Determine the value of x if 2x + 3 = 9 e) ¹∕₄ + ²⁄₃ For each of these examples, it is likely that students would have been exposed to known algorithms that they could use to obtain solutions. Therefore, in assessing the responses of a class of students who attempted these questions, one expects that most of them would have employed the following algorithms: Ministry of Education 2011 9 ProblemSolving.indd 9 8/24/12 6:55:33 PM
Page 1 and 2: Ministry of Education Problems and
Page 3 and 4: Co n t e n t s Introduction........
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Page 21 and 22: Model solutions Essentially, the ap
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Problem 2 - Cutting Squares Grades:
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Problem 3 - Taking Counters Grade:
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Problem 5 - A Tricky river Crossing
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It can be unfolded to give on one s
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Problem 8 - Bicycles and Tricycles
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Problem 10 - Filling the Pool Grade
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Problem 12 - The Knight’s Dance G
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Problem 13 - Trading Places Grade:
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Problem 16 - Folded Corners Grade:
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Problem 18 - Counting Rectangles Gr
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Problem 19 - Box it Up Grade: 3 or
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Problem 21 - T-Patterns Grades: 5 -
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Problem 23 - Last One Standing Grad
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Ministry of Education 2011 85 Probl
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Problem 26 - Noughts and Crosses Gr
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Problem 27 - A Weighing Problem Gra
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Problem 29 - Frog Hopping Grade: 4
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2. In general, how many squares are
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Problem 33 - Chicken Combos Grades:
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2. Explain how you know the fractio
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Problem 36 - Measuring Jugs Grades:
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Problem 38 - The Trapezium Problem
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Problem 39 - Karen’s Troubles Gra
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Problem 41 - Stand Out Grade: 3 or
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Problem 43 - Mathematics Maze Grade
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Problem 45 - Circling Numbers Grade
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Problem 47 - Triangle Problems Grad
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Problem 48 - Factors Grade: 3 or ab
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Problem 50 - Target Grade: 4 or abo
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Problem 52 - Pond Borders Grade: 5
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Problem 54 - Teacher’s Banner Gra
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Grades: 4 - 6 Strand: Number Studen
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Problem 58 - Patty Please Grade: 4
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Problem 60 - Text-a-holic Grade: 3
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Problem Number 62 - Magic Squares G
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Ap p e n d i x 1 - Se l e c t e d S
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Problem 4 - Perfect squares 1. When
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Problem 11- The 400 Problem 1. Sinc
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When we look at the results recorde
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Problem 18 - Counting Rectangles 1.
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2. We can note that: a) The pattern
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As we go around the circle we may s
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For example, if a 1g weight is plac
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Problem 30 - Chess Board Squares 1.
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Problem 32 - Newspaper Math 1. 2. 3
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Problem 38 - The Trapezium Problem
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Problem 56 - Cross Out 8 The winnin
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Heuristics Strategies employed in p
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Scaffolding A teaching strategy in
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