A student I tutor received a 94 on her algebra 2 midterm at one of the top private schools in Houston. Her first ever A in a math final/midterm. Congrats!
Partly my help — but partly her hard work and intelligence, and partly the teacher and the amount of rigor being demanded by the teacher compared to past math courses.
Someone recently asked what a zero exponent would mean in the real world. Here is my basic response.
The important method of research is to ask and answer questions, and the important questions to ask are “how did we get the concept?” and “what does it mean?” We should look for things in the world where we take exponents. Where did we start? As far as I know, people started in history with geometry, with squares and cubes. What did we do from there? Include natural numbers as exponents. What did we do from there? Include zero as an exponent. What did we do from there? Include negatives, fractions and irrational numbers as exponents.
But let’s consider squares and cubes. We would multiply a length by a length to get an area; this is why we “square,” and from where the name comes. We would multiply a length by a length by a length to get a volume; this is why we “cube,” and from where the name comes. So a cube is a volume, a square is an area, a first power is a line, and so we could define a zero power to be a what? A point. (Considering 5^0 meter to not be the same as (5 meter)^0, just as they are different for the square or cube.)
The person suggested thinking about the number of apples to the zero power. Apples are not a good example, as I’m thinking. Why would we square or cube or otherwise exponentiate the number of apples?
Better examples would be things we can square or cube, or situations where we do so. One good example is compound interest (and continuous interest). We find that we can multiply a dollar amount by a factor, e.g., (1 + 0.08/12), which would be 8% interest compounded monthly, to get a dollar amount for next year. We can multiply that by the same factor to get a dollar amount for two years out. When we look, we find we could’ve multiplied the original amount by the factor squared, (1 + 0.08/12)^2, to get a dollar amount two years out. And so on for three or four years: (1 + 0.08/12)^3 and (1 + 0.08/12)^4, respectively.
Recently, I helped an eighth-grade student with an English paper. (We work mostly on math, but work also on grammar, writing, and whatever else comes up.) The paper was a short fictional story.
We had two fun, valuable sessions working on writing skills and thinking skills. We talked first about the big picture, about some perspectives and questions to consider when reading and writing fiction: how do the characters feel about themselves; how do they feel about each other; how do they feel about the world; what does the author (or do you) think about the world. These questions need to be answered to define the characters, define the story, and determine what the story means; they need to be answered to determine what the author is saying or what the writer is trying to say.
Being young, the student was focused more on what was outside than what was inside, on extrospection rather than introspection. I pointed out that he needed to portray what was going on inside the characters’ minds, about what the characters were thinking and how they were feeling. Discussing the internal, what was going on inside the characters’ minds, would add more power and emotion to the story. The internal would help us know why the action was occurring as it was, and how the characters were motivated. And sometimes it is a vital key to knowing why the story is developing as it is. In our sessions, we added a few items discussing the main character’s thoughts and feelings; we added back a line that the student had cut out, a line describing the main character’s feelings, a line that was a critical part of the story. Story improved.
Another important concept we discussed was context: what a reader needs to know before a writer makes a statement, and where to put a descriptor so it makes sense.
Some students I tutor are doing so well in school –- getting 98s, 99s, and other As in math and chemistry — that they canceled tutoring for this week! Congrats on the good grades!!
Understandable to cancel, but not a good idea, I think. Getting good grades would be a good measure of education and preparation for adult life only if the school had a rational curriculum. But, when it does not, as is unfortunately often the case today, then we are measuring by a false standard. There is a lot missing in modern education — modern education in general; there are some good schools and teachers out there! A student would have to be going to a good school to get a full, rational education in all areas of knowledge. (A teacher alone, teaching only one subject, could not provide the full breadth of a rational education in a school context, but, yes, could make a big difference in a student’s life. And this difference is important. Sometimes vital.)
Tutoring should be about more than grades, it should be about making sure a student gets what should be in a proper education and what is missing from modern education: understanding concepts and being trained in objective logic and rational methods. There are a variety of mathematical and scientific (and other) concepts to cover correctly — number, fraction, the infinite, induction, scientific method, reading skills, writing skills — since they are often, unfortunately, not taught correctly today, just as the “why,” the “how,” and the use of some idea are often not taught. Such things should be remedied in tutoring; a good time for this is in “down time” from the demands of school work.
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.
Nice. A student I tutor earned a 2190 on his recent SAT!! That’s up 250 points from the previous time: 1940. (I helped him on both SATs.)
He could have gotten the 2190 in the first place; he should be able to get at least a 2250, but he is good with the 2190.
The recent score was math: 800, writing: 720, reading: 670.
One student I tutor received a 690 on the math section of the SAT, a 650 on the writing, and a 590 on the reading, and another a 610 on the math, a 590 on the writing, and a 690 on the reading. Pretty good, but not as good as they can do! They, as do most of us, need to do more practice and take the test a little more seriously.
A student I tutor earned a 710 on the math section of the SAT, a 690 on the reading, and a 700 on the writing!! Good job!! The test was harsh: he missed only a few in each section. In the nation, he was top 6% or higher, I think it was, for all his scores!! (The percentages were compared to all seniors who took the test last year.)
Using WebEx, we can enjoy live, online classes, but there is the option of taking the classes by watching the videos, if you so desire or if your schedule demands. Those who take the class live will also be able to watch the videos to review class material after class or at a later date. Classes start in mid-September 2012, but you are welcome to drop in for the second semester, starting in January 2013.
Classes will be on Tuesday and Thursdays, for an hour and a half, more or less, each session, totaling more than 80 hours of instruction for the school year. Algebra, Algebra 2, and Geometry will be offered. With interest, we will also hold classes in Arithmetic, Precalculus, Calculus, Probability and Statistics, Physics, Chemistry, or SAT/ACT Prep.
Geometry will be offered TuTh from 9:00 AM till 10:30 AM, CST; Algebra 1 will be TuTh from 11:00 AM till 12:30 PM, CST; Algebra 2 will be TuTh from 1:00 till 2:30 PM, CST. Other classes will be later in the day on TuTh, or be on MW. Note that times given are CST; make adjustments for your time zone as necessary.
What you get: personalized attention and small classes; step-by-step, logical methods; focus on how and why, not just the what (not just content); confidence and subject mastery; a solid education in reasoning and thinking skills; an expert instructor who knows more than just the math, who brings in science, history, law, literature, and logic.
Course description: we will cover the concepts of arithmetic in each sequence with stress on practical application, deriving concepts, as much as possible, from real-life examples, and reasoning things out. We will make sure students know how and why something is true. This is critical for higher mathematics, science, finance, accounting, household calculations, and more. We will cover addition, subtraction, multiplication, division, and more, as appropriate for the age group.
Possible texts:
1. Ray’s Arithmetic by Joseph Ray, available online through various sources such as Mott Media, or free on sites such as Google Books
Course description: we will cover the concepts of algebra with stress on how we know things and on practical application, which are the more important things to get out of algebra. Learning derivations and explanations of algebraic concepts teaches us how to reason and gives us confidence in our ability to understand; not learning the derivations and explanations short-circuits the mind, keeps students from developing the ability to think critically, intelligently, and imaginatively, and stifles their self-confidence. We will cover the real numbers, algebraic expressions, linear equations in one and two variables, systems of equations, inequalities, polynomials, functions, exponents, powers, roots, quadratic equations, rational expressions/equations, radical expressions/equations, and graphing. Time permitting, maybe also other topics like the conic sections, exponential functions, logarithmic functions, probability and statistics.
Possible texts:
1. Introductory Algebra By Keedy & Bittinger, Addison-Wesley Publishing Company, (c) Addison-Wesley Publishing Company, Inc.
2. Elementary Algebra by Larson & Hostetler, Houghton Mifflin Company, (c) Houghton Mifflin Company
3. Algebra: Structure and Method by Dolciani, Brown, Ebos & Cole, Houghton Mifflin Company, (c) Houghton Mifflin Company
A student earned a 93 on his precal final. Another student earned a 94 on his geometry final. Another student earned a 94 on a test. But, darn it, two students earned only a C+ on their math final in December 2011. They, however, go to a top private school, and have a hard teacher. They still pulled out Bs for the semester, which is good for them. Well done.
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