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Approaches and Methods of Solving Mathematical Problems


Five high school mathematics students were invited to complete this assignment. The students’ identities remain anonymous, but it should be said that each of them was over the age of 14 and had never taken a MA 105 course. The students voiced no discontent about math while also being uninspired by the discipline; in other words, they were ordinarily high school students. All respondents were asked to solve three uncomplicated problems and explain their methodology.

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Reflecting Subtraction Tasks

The first problem in this project was a problem on subtraction skills: students were asked to solve Examples 107-68. The choice of these numbers was not accidental since a more profound knowledge using short-term memory is needed to perform subtraction competently in this case. Interestingly, each of the students produced the correct answer, with only three (1, 2, and 3) completing the entire assignment, while the others ignored the first part, focusing on the second. The first student divides the numbers into components: 107 turns into 100+7, and 68 turns into 70-2. In one case, the first student uses an additional format to represent a whole number and a subtraction format in the other, but it does not seem systematic. For 107, the number 110 (107+3) is much closer, but for this student, the number 100 is probably easier to perceive than 110. Using the complex method, the student arrives at the correct solution and then successfully checks himself by subtracting the column’s target numbers and even using notations. The second student also uses the column and also marks that he has solved the problem several times, getting the same answer. Among these solutions is a complex decomposition of the problem into four different steps, which complicates the procedure and leads to errors. The third and fourth students also use columns, but the fourth student uses a more detailed sequential notation, while the third student does parallel calculations and crosses out the numbers when they are no longer needed. The fifth student chooses an evaluative comparison strategy in which the most specific closest possible number is chosen for each number to simplify the subtraction procedure. Thus, the fifth student uses the analogy of 110-70=40, calling it rounding. In addition, the student shows his calculations using column subtraction but does not provide any additional strokes or marks: it can be concluded that the basic calculations are implemented in mind.

Reflecting on the Multiplication Problem

Students were asked to solve the 14×15 example for the multiplication task and explain the solution. It is noteworthy that the first and second students performed the problem identically, column by individual digits, and their solutions are indistinguishable. However, the fifth student seemed to think more comprehensively and, using the column, multiplied not digit by digit but a number by digit. This method may not always lead to accurate results because multiplying a number by a digit is not always straightforward; hence, the fifth student takes a bit of a risk by ignoring the procedures for multiplying by digits. The fourth student also uses a column like 1 and 2 but divides it into three parts, explaining each step in passing. This is the most detailed solution and takes the longest. It is interesting to highlight the third student’s answer: it is also correct, but it gives the impression that a calculator was used because there is no solution process. It is unlikely that this student could have solved the example in his mind, so it seems that either third-party drafts or a calculator were used. Again, only 1,2, and 3 students answered the first part using different degrees of comparison with round numbers chosen by personal preference.

Reflecting on the Percentage Task

In this part, students were asked to find 75% of the number 12 using either method. Students 1 and 2 showed a similar process using column multiplication but slightly different procedures. Student 1 multiplied the full 12 by each of the numbers 75 as 12×5+12×70 and then separated the decimal point. This seems not always straightforward since multiplying by digits of a number can lead to arithmetic error more often than multiplying a digit by a digit. Student 2 multiplied 12 by each of the digits of 75 but did the calculations according to the principle 12×5+12×7. Student 5, on the other hand, multiplied 75 by the number 12, so his calculations were different, and again used the method of multiplying a number by a digit. This time, the third and fourth students seem to have both used additional tools, as no calculations were shown. Again, only 1,2, and 3 students answered the first part.

General Comparison

The students showed strong dynamics in their methods, but some of the patterns were detectable. The third student was the most likely to walk away from more complicated problems, probably using a calculator. The first two students and the fifth used similar techniques, but the fifth always used them uniquely, sometimes reversing the problem. The fourth student seemed tired toward the end or unable to solve the percentage problem, so only at the end did he turn to a calculator (probably). In general, the columnar solution was always used by students, but 3, 4, and 5 required written steps or problem statements more often than others. To emphasize the overall bottom line, not only did the solution strategies differ between students, but they were also used differently by each student. They all ended up with correct answers, but the task performed showed how differentially the problem could be approached.

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