Abstract
This paper proposes solutions for three learning problems: each of the proposed problems was solved using concrete, that is, taking into account the manipulation of real objects, and abstract, that is, by adopting a mathematical symbolic language, approaches. Polya’s approach of conceptualizing and fragmenting the overall problem was used to solve all three problems. The work also offers a discussion and reflection part, where previously solved problems are placed in a theoretical framework and explored in terms of personal experience with them, respectively.
Introduction
Problem-solving skills might be the most critical skills that students have to develop in order to understand mathematics. Consequently, this report focuses on the solution of three problems that require the problem-solving approach. Each task will be presented with a block of concrete and abstract solutions, followed by an explanation of the logic behind the undertaken assumptions and steps. Then, the provided solutions will be discussed in the light of the problem-solving approach as a whole while also providing the analysis of particular techniques used in the solution development. The overall purpose of the report is to showcase how essential a problem-solving approach might be for the understanding of mathematical concepts.
Solving the Problems
Treasure to Ship
To come to the solution to this task, I needed to grasp the complete picture of the process first. Thus, I decided to approach this task concretely with the strategy of acting it out. I chose two eggs for the pirates and four macaroni pieces to represent the treasure chests (Figure 1). Afterward, I moved two objects simultaneously, with one egg always present in the pair. When all items finally “arrived on the ship,” I wrote down every boat transfer:
- Pirate transfers the chest to the ship.
- Pirate returns to the island.
- Pirate transfers the chest to the ship.
- Pirate returns to the island.
- Pirate transfers the chest to the ship.
- Pirate returns to the island.
- Pirate transfers the chest to the ship.
- Pirate returns to the island.
- Both pirates board the ship.
Thus, in nine trips, pirates can complete the treasure transfer from the island. Then, I looked for patterns to generalize the task and develop the abstract solution. The steps from one to eight are repetitive; each treasure chest will require two steps of loading it onto the ship and then coming back to the island. It implies that a treasure could be labeled as a letter (c, for example) with a multiplier equal to the number of chests. One additional trip is needed to bring the second pirate to the ship; therefore, the abstract equation for this task would be 4c + 1. When I asked myself whether there was another solution, I realized that the second pirate did not have to board the ship at the last step. Moreover, his trip to the ship can occur at any time of the treasure transfer – it would only change the order in the remaining repeating steps.
Three in a Line
The second problem required determining the number of ways that placing three cartons of strawberry milk in a nine-pack could form a single line. To find a specific solution, I took a chessboard and only three pieces to mimic the three milk cartons. Since I did not need the whole chessboard, but only the nine adjacent cells forming a square, I covered the rest of the field with white sheets so they would not get in the way. For this solution I placed the three pieces in different ways and looked for all possible variations that could form a single line: it is worth saying that the line could be placed vertically, horizontally, or diagonally, which increased the number of variations. Meanwhile, I only wanted unique solutions that did not repeat themselves. In other words, in Figure 2 in the Appendix, the first two solutions turned out to be identical, because turning the box 90° would have led to exactly the same result — such solutions are non-unique and had to be counted as only one. As a consequence of my experiments, I realized that there are only three unique solutions for this problem: when the strawberry milk cartons are arranged in one line in the first column, in the second column, and when they are arranged diagonally; all other solutions were repeats of these three when the box was rotated.
For the abstract solution, we need to use a 3×3 matrix in which each cell corresponds to the location of the strawberry milk carton, as shown below. Placing the three objects in column X given the box rotation gives me four options, one of which repeats the placement of the three packages in column Z. Hence, if I place all three milk only vertically in column X, this gives me four non-unique solutions and one unique solution. By analogy, I move my attention to column Y: placing the cartons in it gives me two non-unique solutions or just one unique solution. Finally, placing milk in column Z makes no unique sense because it is a repetition of column X. Then attention should be paid to the diagonals; there are only two. Placing the milk along diagonal XXYYZZ, given the rotation, yields only two solutions, only one of which turns out to be unique. All other arrangements of the three packages either do not form a single line or repeat previously created solutions. For this reason, the abstract solution allows me to obtain the same answer as in the case of the chessboard.
Snails
In this problem it was necessary to find the minimum time for the snail to travel based on the drawn lines with dots. It is known that each dot represents a five-minute pause, and the speed of the snail is 2 cm per minute regardless of the chosen path. For a particular solution, it is necessary to use toothpicks, placing them also to visually repeat the four trajectories of movement (Fig. 3). Then it must be counted the number of toothpicks used to estimate the shortest route. This strategy allowed us to estimate that the second path is the shortest, but without taking into account the stops. If we take into account that the second path has five joints and the first path has only three, it is clear that the first path will be the shortest. For an abstract solution, it has to be assumed that the distance between two fingers will be about two centimeters — for convenience, the minimum distance on the path is taken to be two centimeters. Using this distance between two fingers as an imaginary ruler, it becomes possible to calculate all the values on the path:
- (2+4+6+2)/2+ 5×3=22
- (2+2+2+2+2+2)/2+ 5×5=31
- (4+6+2+4)/2+ 5×3=23
- (2+2+2+2+2+2+2+2+2)/2+ 5×8=49
Now it can be seen that the first path is really the shortest, which means it is the fastest way for the snail to reach the plant.
Discussion
The chosen tasks might seem different on the surface, but they share a common ground in their problem-solving skills requirements. Overall, every task required the approach commonly referred to as Polya’s approach to problem-solving. According to YuMi Deadly Centre (2019), Polya’s method can be classified into four consequent steps. It begins with understanding the problem – for example, to grasp the first task, I had to visualize it with the help of concrete materials concretely. Then comes the development of a plan to follow to solve the task; in the case of the snails, I decided to calculate each path separately and compare the results afterward. After the plan is developed, the third step is to execute it and proceed with the evaluation of acquired results (Berenger, 2018). In the case of the second task, the final evaluation showed the mistakes in the chosen course of action, which motivated the revision and a consequent inclusion of new conditions. Overall, despite the actual solution being uncovered in different steps of the approach, I unintentionally used Polya’s technique on every chosen problem.
Apart from that, I implemented several strategies to help me solve the tasks. In the first task, the strategy of acting it out provided the needed generalization. It allowed me to use the technique of looking for patterns in the generalized vision for the development of the abstract solution: 4c + 1, where c is the number of treasure chests. The pattern search strategy was also useful in the second task, when I needed to understand how many unique solutions there are. Using matrices and numerical expression of the number of repetitions allowed me to be consistent and get the answer. When I gathered the information for the study, I frequently met the notion of creativity development due to the problem-solving attempts (Thapa et al., 2021; Booker et al., 2014; NESA, 2019). Provided tasks required me to use creativity in practice — it is fundamental in both task generalization and pattern searching.
The last strategy I applied was to break down the problem into smaller questions to simplify the process, namely the decision to focus on specific paths in the snail problem. In the specific solution, I modeled the problem using intuitive physical objects of equal length, that is, toothpicks. By measuring their number and comparing the number of joints between the toothpicks, I was able to find the shortest path. Remarkably, the abstract solution also led me to the same conclusion: for it I used the distance between two fingers as a ruler and measured the lengths of the segments. Another important realization was that not every problem requires direct mathematical calculations-the treasure problem is predominantly focused on logic. Özcan & Eren Gümüş (2017) noted that some students have problems when the task requires a more logical approach. In the example with the first problem, it is impossible to find additional solutions to the first problem without a logical approach.
Reflection
I managed to learn a lot from the problem modeling during this assignment. An important condition in solving a mathematical problem is accepting the need for effort and diligence in order to achieve a result (Livy et al., 2018). In this context, the effort I had to put into modeling unexpectedly boosted my will to proceed with the tasks. It revitalized and animated the problem-solving process, making it more personally appealing. An issue students often experience during their education is the absence of motivation; partly, the lack of engagement is responsible for such a tendency (Lisciandro et al., 2018). Consequently, supported by what I experienced myself, the need to incorporate not only the abstract solution might be crucial in developing problem-solving skills. I also learned that problem-solving techniques develop critical thinking. In the line of three tasks, I stumbled upon the solution almost accidentally, which made me question all my consequent assumptions and decisions.
Moreover, I consider the second task to require the most significant effort to be solved. Developing its final abstract solution took the most time out of all three tasks due to the extensions I had to make to understand the pattern finally. In conjunction with the need to literally change the perspective in order to find the correct solution, the second task can be considered the most fruitful in my personal experience of problem-solving. I would also like to mention the simplicity that comes with the concrete solution to the treasure task. The initial task seemed complex at first, which invoked a sense of anxiety in the form of possible solutions that cannot be perceived at first glance. Nevertheless, moving eggs and macaroni exclude the pressure, clearly showing the steps needed to find the answer.
Conclusion
The chosen tasks required a certain amount of effort and creativity to yield the final answer. This tendency showcases the way in which problem-solving techniques can influence students by boosting their critical thinking and imagination. Overall, these skills can be used outside the mathematical field, which provides another argument for their importance in the students’ personal growth. After all, Polya’s comment on the personal desire to find an answer might prove to be the ultimate factor in pursuing knowledge.
References
Berenger, A. (2018). Pre-service teachers’ difficulties with problem solving [PDF document].
Booker, G., Bond, D., Sparrow, L., & Swan, P. (2014). Teaching primary mathematics (5th ed.). Pearson Australia.
Lisciandro, J. G., Jones, A., & Geerlings, P. (2018). Enabling learners starts with knowing them:
Student attitudes, aspiration and anxiety towards science and maths learning in an Australian pre-university enabling program. Australian Journal of Adult Learning, 58(1), 13-40.
Livy, S., Muir, T., & Sullivan, P. (2018). Challenging tasks lead to productive struggle! Australian Primary Mathematics Classroom, 23(1), 19-24.
NESA. (2019). Aim and objectives. Mathematics K-10.
Özcan, Z. Ç., & Eren Gümüş, A. (2019). A modeling study to explain mathematical problem-solving performance through metacognition, self-efficacy, motivation, and anxiety. Australian Journal of Education, 63(1), 116-134.
Thapa, A., Valentine, A., & Hamilton, M. (2021). Investigating creativity in computer science syllabi in Australia [PDF document].
YuMi Deadly Centre. (20191). Problem solving [PDF document].
Appendix
