Given Information about mathematics (What we know):
Mathematics is an academic subject. The definition of success in mathematics is most generally defined as consistently finding the correct answer to any set of given problems. The complexity of math problems very greatly. A first grader might solve problems like sample problems 1 or 2. A seventh grader might be expected to solve problems like number 3 or 4. A tenth grade geometry student might have to solve problems like number sample problem 5. A calculus student whether in high school or college must be able to solve problems like sample problem 6.
Sample Problem #1: Is 3 < 5 ?
Sample Problem #2: 4 + 2 = ?
Sample Problem #3: x – 7 = 9
Sample Problem #4: 12x = -72
Sample Problem #5: Prove that Line AB and Line CD are parallel, given that Line AB is parallel to Line EF and Line CD is parallel to Line EF.
Sample Problem #6: f(x) = sin(cos x) f'(x) = ?
Each of these problems has a differing level of complexity and requires progressively higher levels of understandings of mathematics. A calculus student should be able to solve each of the other problems with ease. A first grader would not be expected to solve any problem above his or her grade level. Expectations for each of these different students is very different, but performance is still measured by the ability to consistently solve problems that students at a similar level should be able to solve. A calculus student who is able to solve problems at a first grade level consistently would not be considered successful. At the same time a first grader who could solve problems at a higher grade level would be considered advanced or a prodigy.
Now that we have generally defined success for a student, it is important to define how success is achieved. Being a successful math student at one level requires having a solid if not proficient understanding of the levels of mathematics that preceded the current one. A student will experience difficulty finding a least common denominator if the student is not comfortable with his or her multiplication tables. How can a student add and subtract terms of a polynomial if they cannot add or subtract integers. Mastery of the most fundamental skills of mathematics are necessary for the mastery and understanding of more advanced concepts.
How does a math student build these fundamental skills? The best method I know for the mastery of mathematics skills at any level is sufficient proper repetition. Notice repetition is preceded by two very important adjectives. Repetition for the sake of repetition is not enough. The repetition must be conducted properly and sufficently.
Solving math problems of any level of complexity must be done properly. Not only will improper mathematics lead to incorrect answers, a false confidence is created and bad habits are formed. When mathematical skills are assessed, the student may have the confidence that they are able to perform the skills sufficiently. However, if their confidence is not well grounded and they perform poorly, they can end up becoming more frustrated because they may have practiced sufficiently, but not properly. Then these bad habits can take even longer to break once they have been established. Not to mention, the motivation of the student can drop significantly making it difficult to convince them to want to practice more.
Now let’s assume that a student is practicing and the problems are being done properly. This is clearly a necessity to being successful. However, many times a student will do a couple problems properly and consider themselves proficient. While this may be the case, many times the student will completely forget how to do the problems a day or two later. There is no set number of the amount of proper repetitions required for a student to master a certain concept or type of problem. However, the proper repetition must be sufficient enough for good habits to form and mastery to be achieved. The pressure of an assessment can lead to doubt and confusion if the student hasn’t practiced sufficiently to truly master the mathematical skill regardless of the circumstances.
This concept of sufficient proper repetition is the key to being successful in mathematics. However, it must start when a student is first learning mathematics and continue as they progress. A student who does not build a proper fundamental base in the most basic mathematical skills and concepts will struggle to grasp and learn any concepts or skills that are more advanced. I have also worked with students who have good skills in a lot of areas but possess a significant gap that causes them to struggle. This is a sign of successful sufficient proper repetition up to a certain point and then a drop off at some point along the line. Maybe it was a poor teacher. Maybe it was significant events in their lives that created a distraction for them and caused them not to be able to focus sufficiently or properly on mathematics. In any case, this gap in their skills is an obstacle to mastering higher level concepts that rely on one or more weak or missing skills.