There are several methods, ranging from the simple to the complex and abstract, that are relied upon and taught as a means of solving math problems. Math problems can be solved by drawing and counting tally marks, finger counting, drawing pictures, using a calculator, applying basic and memorized math rules and performing abstract calculations. One of the traditional approaches toward teaching math in elementary schools is to begin with a concrete, or hands-on and manipulative approach, then progress to a math problem's pictorial representation and conclude with the abstract calculations.
An approach to solving math problems that is recommended, typically at the secondary-education level, is to begin by making sure that the problem is fully understood and that it can be well visualized. The next step is to recall, or look up, all of the relevant rules that apply to the problem. When deciding upon the best approach to obtain the answer, every attempt should be made to disqualify the solution method before finally implementing it. When a solution has passed a rigorous qualification test, it can then be applied to the problem.
In contrast to the traditional approaches to teaching students at the elementary school level to solve math problems, proponents of non-traditional approaches stress the importance of students comparing, assessing and reflecting upon multiple solution methods rather than proceeding on the basis of rote memorization of single solution methods learned sequentially. Findings published in 2007, relating to a study involving 70 students, showed that there are benefits to the new approach. The students who were taught to solve math problems by contrasting and comparing a variety of solution methods were found to have progressed further in gaining conceptual and procedural knowledge, acquiring flexibility and learning how to judge the accuracy and efficiency of various solution methods.