Student thinking has a great affect when working with mathematics. A common mistake I have seen amongst students I have tutored was when solving 3x - 2y=10 for y. The first step done was subtracting both equivalent expressions by 3x, but their yield was 2y = 3x + 10. Also when referring to the original equation and asked for the sign of 2y, the common response was positive. The expression shows a positive number being subtracted, so in the result, the remaining portion of the expression is a positive; the subtraction operation is gone.
With students, the focus is usually on what to do, rather than on how it works. Common mistakes like this are usually caught too late by teachers, or not addressed appropriately early on. As tutors, we must spot and correct these type mistakes as early as possible. There are many techniques to both spot and correct, such as through highly effective questioning and probing.
With another student I recently tutored, (I’ve also seen this many times in the past), the student would make “careless” mistakes like in -4x = 8, divided both sides by 4 and not by -4. Also, would divide by 4x instead of just by 4, or would change the -4x to 1 instead of 1x or just x after dividing by -4. Most students seem to focus on procedure (student thinking), rather than on understanding the concepts which is the purpose of mathematics. Most of the mathematics taught is not actually used in the everyday life. One of the main reasons to study and learn mathematics is to develop our logical thinking and improve our brain muscle in order to be more effective in other subjects and activities.