Teaching Sequences

Recently, I tutored someone in algebra. In his math class, he was covering arithmetic sequences as an example of functions. In his textbook, there were some applications, but not many and not very good. The presentation was not the worst, but it certainly was not the best. The book did not give many good examples, it did not contrast arithmetic sequences with anything, it did not relate arithmetic sequences to anything (except, of course, to functions) and it did not motivate the topic by identifying where in reality and human life the concept (and related concepts) come up. There was bad epistemology (see also the Internet Encyclopedia of Philosophy or the Stanford Encyclopedia of Philosophy), bad thinking and teaching methods, all around. Please avoid these mistakes and rectify them when teaching your child or your students!! To teach properly, to train the rational faculty of a child for adult life or student for professional life, we need to put what we do in context of rational, objective principles of epistemology and in context of practical life and human survival. So I discussed arithmetic sequences vs. geometric and Fibonacci sequences, and said there were other types. We looked at examples of each type of sequence, to clearly differentiate the three. We saw they were all similar, and so should be classified together, but we saw what differentiated one from another. We looked at some examples of the sequences in real life — finance, bacterial growth, nuclear half-lives, plant and animal growth — so we knew why they were important and something we should study. We had examples that should be relevant to the student’s adult life (and, depending on the student, we provided enough information so that the math-intelligent student should be able to see how they could use some sequences now) and that showed how sequences were important in human life in general. Then, and only then, we did some work specifically on arithmetic sequences, and did his book work. When we finished, I reviewed some points from the beginning. We should show students how we develop a concept or method from real, practical experience in the real world, how classification — using similarities, differences, narrower groups/classifications, wider groups/classifications — comes into play and is important, and why the idea is important. It just takes a little time, but has profounds results on a person’s confidence, motivation, ability to reason correctly and logically, and success in life. Update: To be able to do this, you need to immerse yourself in the material. You need to be a specialist — which is why you should hire a teacher or tutor, just as you would seek out a specialist in medicine,  mechanics, HVAC, electricity, dance, martial arts. Homeschoolers who want to work with their own children should do so, but should seek out a teacher/tutor for guidance. The teacher/tutor could give a lesson or two every week with you there, and you could handle the teaching the rest of the week. Or you could work with the teacher/tutor on your own, then take back what you learned to your children.