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Here is the short and the most common answer: mental math is preforming math computations mentally without using pencil/pen and paper or electronic devices such as a calculator or other technology-based systems. In this definition, math problems could be solved quickly, and in many cases, be even faster than a calculator with practice and proficiency.
Mental math skills are further enhanced through core principles based on standard math. And just as important, mental math opens the mind to alternative ways or insights in solving math problems through reverse engineering (manipulation of numbers), and the understanding of the relationships between numbers directly involved, and those interconnected with them allowing for additional options to be made available.
The underlining effects of mental math are real. Besides academic benefits not only in math but other essential subjects as well, it increases the processing power of the mind, particularly improvement of logical thinking and reasoning capabilities. In addition, the mind can hold information more efficiently, improving recalling features.