Our brains contain neurons, and when we learn we build dendrites between the neurons. Dendrites are connections which give thoughts more possible pathways to travel along, allowing more of a chance for a certain piece of information to be recalled correctly. This is why mnemonics work; even though you have to remember more information, you remember both the mnemonic and the information it represents better because of it.
Normally students view learning math memorization of formulas and algorithms. This only uses a limited portion of the brain, so it is not the most effective way to build dendrites. My goal for students is to build connections outside of memorization. As a consequence, formulas will actually be easier to remember.
A teacher will use the method that helps the greatest number of students, but the teaching methods will not necessarily help everyone. A private tutor can help build on strengths that the student already possesses to teach a concept. Some students are better visual learners. Some students are better with hands on learning. Building connections with these strengths will greatly help the student.
Here are some of my methods to create these connections:
1. Before a student attempts to solve a problem, the student should attempt to estimate what the answer should be. Very often I see students punch numbers into a calculator and write the first thing that appears on the screen without any regard to whether
that number makes sense. For example, what is 60% of 80? Before I solve this problem, I can tell that it should be at least 0, at most 80, and that it is probably around 40. Now when I use my calculator and I end up getting 120 for some reason, I know I made
a mistake on the calculator. The goal here is to connect math with intuition, which is crucial to being successful in math.
2. After solving a problem, the student should evaluate whether the answer makes sense not just based on the initial estimate but also from previous experience. For example, if a student is solving a "two trains" problem, and gets an answer like 13,000 hours, it might be time to double check. Again, the goal of this is to connect math with intuition.
3. Oftentimes, the way that we are taught to solve a problem, even if it is valid, is not the best way to solve the problem. This is especially true on SAT/ACT. For example, if the problem you get on a multiple choice test is solve 3x^2-5x-28 = 0, and you feel tempted to whip out the quadratic formula, try simply plugging in the possible answers until you find the right one. You will solve the problem quickly and have more time for more difficult problems.
4. Most formulas can be understood very well in terms of a picture. This helps visual learners very well to understand the more difficult formulas. Consider, for example, cos^2(a)+sin^2(a)=1. This formula makes a lot more sense with a picture (which I cannot draw here) of a right triangle inside a circle of radius 1.