Mathematical Problem Solving: Linking Theory and Practice

Abstract

Mathematical problem solving is the primary goal of school mathematics curriculum in Singapore. Prospective secondary school mathematics teachers, as part of their teacher education at the National Institute of Education, undertake a 96 hour course called Teaching and Learning of Mathematics. Throughout the course, as part of the study of content and pedagogy of various topics of secondary mathematics, they are engaged in solving mathematical problems. A formal introduction to mathematical problem solving and review of the relevant literature is done at the beginning of the course. As an introduction to mathematical problem solving, we engage our teachers in two tasks, The Circular Flower Bed and Solve 4 Problems, to jump start discussion on mathematical problem solving and bridge theory into practice. The goals of the tasks are as follows. The Circular Flower Bed task engages prospective teachers in problem solving and initiates discussion on the process of finding a solution, specifically the feelings, emotions and regulation of thinking during the process. The Solve 4 Problems task engages prospective teachers in clarifying the definition of a problem, distinguishing heuristics from strategies and making connections with Polya’s (1973) four phases of problem solving.

Appendices

Appendix A

Appendix B

Appendix C

• How many squares are there on a standard (8 ´ 8) chessboard? • Karen has to number the 396 pages in her biology book. How many digits will she have to write? • Into how many different plane regions do n lines, no three of which are concurrent and none of which are parallel, separate the plane? • Find all rectangles with integral sides whose area and perimeter are numerically equal. • What is the maximum number of regions into which n chords divide a circle? • In the “equation”, (he)2 = she, the letters represent digits and the configurations represent numerals in base 10. Find the replacements for the letters that make the statement true. • Find the last three digits of 19951995. • Two circles are concentric. The tangent to the inner circle forms a chord of 12 cm in the larger circle. Find the area of the “ring” between the two circles. • A rectangle 4 by 3 has six squares passed through a diagonal. Find the number of squares passed through by a diagonal for a rectangle of size m by n. • A palindrome is a number that reads the same backwards as forwards. How many 4-digit palindromes are there? Show that all palindromes are divisible by 11. • Tennis balls are packed in cylindrical cans of three. The balls just touch the sides, top and bottom of the can. How does the height of the can compare with the circumference of the top? What is the ratio of the volume of a ball to that of the can? • The Tans are having a party. The first time the doorbell rings, a guest enters. On the second ring, three quests enter. On the third ring, five guests enter, and so on. That is, on each successive ring, the entering group is two larger than the preceding group. How many guests will enter on the 15th ring? How many guests will be present after the 15th ring? • Twelve golf balls appear identical but 11 weigh exactly the same while 1 is either lighter or heavier than the others. Determine the odd ball and whether it is lighter or heavier than the others in as few weighting of the balls on a balance as possible. • Sixty-four cubes are assembled to form a large cube. The face of the large cube is then painted. How many of all the small cubes are untouched by paint? How many of the small cubes have (a) one face, (b) two faces, and (c) three faces painted?

• Two towns lie to the south of a straight road, but they are neither connected to it, nor to one another. The citizens of the two towns decide to build two roads, one from each town, to the existing road. They are cost conscious in the choice of the roads. Find the shortest route connecting the two towns via the existing road. • A curious biological fact is that a male bee has only one parent, a mother, whereas a female bee has both a mother and a father. How many second-generation ancestors (grandparents) does the male bee have? Third-generation ancestors (great-grandparents)? Fourth? Fifth? Tenth? • A cake that is in the form of a cube falls into a large vat of frosting and comes out frosted on all faces. The cake is cut into small cubes of the same size. The cake is cut so that the number of pieces having frosting on three faces will be 1/8 the number of pieces having no frosting at all. Find the total number of small cubes. • Five women are seated around a circular table. Mrs Ong is sitting between Mrs Lim and Miss Mah. Ellen is sitting between Cathy and Mrs Ng. Mrs Lim is between Ellen and Alice. Cathy and Doris are sisters. Betty is seated with Mrs Png on her left and Miss Mah on her right. Match the names with the surnames. • A new school has exactly 1000 lockers and exactly 1000 students. On the first day of school, the students meet outside the building and agree on the following plan: The first student will enter the school and open all of the lockers. The second student will then enter the school and close every locker with an even number (2,4,6,8, etc.). The third student will then “reverse” every third locker. That is, if the locker is closed, he will open it; if the locker is open, he will close it. The fourth student will reverse every fourth locker, and so on until all 1000 students in turn have entered the building and reversed the proper lockers. Which lockers will finally remain open? • After gathering a pile of coconuts one day three sailors on a deserted island agreed to divide the coconuts evenly after a night’s rest. During the night, one sailor got up, divided the coconuts into three equal piles with a remainder of one, which he tossed to a monkey that was conveniently near by, and, secreting his pile, mixed up the others and went back to sleep. The second sailor did the same thing, and so did the third. In the morning, the remaining pile of coconuts (less one) is again divisible by 3. What is the smallest number of coconuts that the original pile could have contained?