Strategy of Solving the - Math Problem Example

Personal Solution to the problem Let X be the total number of pates bought. Since, there are ¼ cracked and chipped plates it means that the cracked plates are: ½ of ¼ = 1/8 (cracked). Hence, the chipped plates are ¼ -1/8 = 1/8 (represented by 2/3) Subtracting these fractions from the total number of plates we must obtain 2 X-(1/8x + 1/8x) = 2 X – ¼ x = 2 ¾ x =2 3x = 8 X = 22/3 plates. Analyzing different solutions obtained The first strategy evident in the three solutions collected is the organization of information in different ways. It is clear that when the information was organized in different ways the approach used to solve the problem also differed significantly from one individual to the other. For example, in solving the problem above the first individual was organized information in a table in order to enhance his understanding by extracting the key information that would help to solve the problem such as the number of plates chipped, the number of plates cracked, the number of plates chipped and cracked and the number of plates without chips and cracks. The second strategy used in solving the above problem was through the guess and check strategy (Blastland and Dilnot, 2008). The other individual used the trial and error method to solve the problem by substituting different values in order to establish the total number of plates bought by Mr. Hash. However, the trial and error method proved cumbersome as the individual had to take numbers at random each time substituting into the pattern of equations determined before. It also proved difficult because only one choice of a number could be chosen. The purpose of using the trial and error method by one of the individuals was to avoid the time consuming and complicated task of thinking logically in order to establish the solution to the problem. The third strategy applied by one of the individuals was the use of logical thinking. This strategy was used after information was organized. The individual made an early identification of the fact that the above problem presented conditional data set which was to be used to arrive at the final solution. In doing so, the individual begun by making an early assumption of the total number of plates as X and proceeding from this point to determine the number of cracked plates, chipped plates and the number of plates that never cracked nor chipped. It is clear from the first strategy that by organizing information in a table form the individual sought to simply the whole problem in order to have a more compressive understanding of the scenario. From the table, the individual was able to single out different rations and arranged them in a given pattern which demonstrated a greater understanding of the whole problem. In the second strategy, the individual demonstrated a low understanding of the problem and hoe to solve thus resorting to guess work. In the third strategy, the solver demonstrated a high understanding of the problem and the methods he could use to solve the problem. In the application of the first strategy of organizing the information given following a given pattern in the table form procedural knowledge was evident because the individual demonstrate his knowledge of associating skills and carrying out procedures. For example, procedural knowledge was evident in that the individual demonstrated that adding the rations given in the problem and equating them to the arbitrary letter (X) taken to represent the total number of plates bought would help in determining the total number of plates bought. In addition, the use of procedural knowledge was also evident in the manner which the individual manipulated specific factors in the problem to arrive at the solution. A part from procedural knowledge the use of conceptual knowledge was also evident because the individual demonstrated his understanding of the concepts involved in solving the problem. For example, the individual demonstrated the fact that adding 2/3 of the chipped plates, ½ of the cracked plates and subtracting the ratio obtained from the total number of plates (X) would give the remainder plates not cracked not chipped. Furthermore, the individual also demonstrated that adding 2/3 and ½ would give a number larger than 1 yet the ratios should total to 1. Consequently, the application of metacognition was also evident in this strategy because the individual kept a proper track of every step used to solve the problem each time going back to an earlier step to relate it to the present step. By arranging the ratios in a pattern, the individual demonstrated the use of a plan in solving the problem. Constant referrals to earlier steps used to solve the problem were a good demonstration of the metacognition aspect used to solve the problem. Additionally, there was evidence of affective responses through out the problem solving session. For example, the individual felt elated when he discovered that he understood a given concept forgotten earlier or ignored. However, the fact that the individual could not solve the problem in the first five minutes prompted a sense of frustration. In the second solution the individual used guesswork as the strategy for solving the problem. Procedural knowledge was also evident in the first few steps of setting apart and identifying the key information required to solve the problem such as identifying the fractions in the problem. Additionally, the individual also demonstrated the use of procedural knowledge in the arrangement of fractions and attempting to arrive at a whole number particularly 1. Procedural knowledge was also evident in the substitution of different fractions and whole numbers in the identified fractions in order to arrive at a whole number. Additionally, the individual also demonstrated the use of conceptual knowledge because in all attempts, the individual sought for numbers that could be substituted in the fractions to result in a whole number. For example, the individual was aware that since 2/3 of the plates were chipped and ½ of the plates were cracked and ¼ of the plates were cracked and chipped then whole fraction was supposed to add up to one. In this regard, the individual made several attempts to add the fractions and subtracting from one. However, the individual demonstrated less use of metacognition skills because in all attempts to arrive at a whole number, the individual did not keep tack of what he was doing which made it difficult for him to realize that adding 2/3 to ½ gave a number more than 1. Hence, the individual did not have a clear plan of what each step entailed in solving the problem. One thing that was evident in this case is that the individual did not check his answer in some way because the failed to realize in the first steps that adding 2/3 and 1/ 2 gave a larger number than one. Towards this end, the individual could not solve the problem in the first five minutes and felt frustrated all through. Although, the individual appeared confidence that he could solve the problem the confidence was frustrated each time the individual reached a dead end (Debbie, 2006). In the third solution, the individual applied the logical reasoning strategy to work out the problem. Procedural knowledge was also evident in identification and the arrangement of fractions. The individual also used his logical skills to establish that ¼ was the total number of cracked and chipped plates and therefore to get the actual fraction that represented cracked plates the individual multiplied ½ with ¼. In order to obtain the remainder of the chipped plates, the individual subtracted the new fraction of cracked plates from ¼. With the new fraction of chipped plates obtained the individual added the two new fractions (chipped and cracked plates) to obtain the total fraction of plates cracked and chipped. All the above processes and steps demonstrated the broad application of procedural knowledge as the individual moved from one step to the other. Additionally, conceptual knowledge was also evident in all the steps used in solving the problem. For example, individual knew that adding 1/12 and 1/6 would yield ¼ which was the number of cracked and chipped plates. In the same perspective, the individual knew that subtracting ¼ from the arbitrary number of plates (X) would give the number of plates which did not have cracks or chips (2). Thus, the individual made use of conceptual knowledge in solving the problem. Metacognition was also exhibited in solving the problem because the individual kept track of all the steps used to solve the problem. The individual also demonstrated that the use of a plan in his work since each step was meant to accomplish a given objective. For example by multiplying ½ with ¼, the objective was to obtain the new fraction of cracked plates. Subtracting the new fraction (1/8) from ¼ the objective was to obtain the new fraction of chipped plates (Harold, 1985). Finally, affective ness was also evident especially when the answer obtained contained a fraction. With the logical knowledge that a plate can not be sold as a fraction the individual felt frustrated by the answer obtained. Throughout the process of solving the problem the individual demonstrated a high level of confidence although he did not solve the problem within the first 5 minutes. By personally solving the mathematical problem above ad observing others solve the same problem, I learned several things. First, people have different approaches of solving problems presented to them even though the differing approaches may not yield the required solution. Second, solving mathematical problems require step by step application of a wide knowledge of knowledge and skills which in many cases is ignored by many people. By solving the mathematical problem above I have a better understanding of solving any other mathematical problem that involves logical reasoning and particularly the application of fractions through multiplication, addition and subtraction. From the above solutions, it is quite evident that individuals have different ways of learning mathematics. For example, some people are very confident when tackling mathematical problems than others. Additionally, some people take long to conceptualize the logical behind mathematical problems. Consequently, while some people attempt to use logical reasoning others take mathematical problems casually which makes it difficult for them to arrive at the required solutions. In this regard, I could assist myself and others to learn mathematics in the followings ways. First, encouraging frequent practice in solving mathematical problems in order to develop the logical reasoning required to solve the problems. Second, to apply procedural, conceptual, metacognition and affective steps and knowledge in solving mathematical problems. Grade 5 solution attached 2/3 + ½ + 2 = X X- 2/ 3 x + ½ x = 2 X -7/6x = 2 6x – 7x = 2 1x= 2 X = 2 Adult solution attached Let X be the total number of plates bought. 1/ 2 of X are cracked and 2/3 of X are chipped. The total number of cracked and chipped plates are 2/3 x + ½ x = 7/6 X. Add the total number of plates not cracked nor chipped (2) 7/6X +2 = 21/6 plates But plates can not fractions so we should discard the fraction in the answer. REFERENCES Blastland, M., & Dilnot, A. (2008). The tiger that isn’t: Seeing through a world of numbers. London: Profile Books. Debbie, D. (2006). Math problem solving strategies. New York: Sage Publishers. Harold, S. (1985). Mathematical problem solving. Department of Mathematics, KTH, 100 44 Stockholm Read More