# Blogs

## Precalculus Blogs

Function decomposition can be tricky at first.   Equations used to be y=mx+b.   Soon after, they turned to f(x) = mx+b.   f(x) simply means f, as a function of x.    Then, we started getting into f(x), g(x), and h(x), and started seeing things like f(g(x)) and f(g(h(x))).   If f(x) means that f is a function of x, then f(g(x)) means that f is a function of the function g(x), rather than x.    This means that for every x you see in the function f(x), you will replace it with g(x).     When I was first taught function decomposition, my instructor would have us label the functions f, g, and h, rather than f(x), g(x), and h(x).   Example:   f(x) = x+1. g(x)=x+2. Find f(g(x)).   (rewrite)    f(x)= x+1. g=x+2. Find f(g). (For every x in the equation, replace with g(x).)   f(g) = (x+2)... read more

Infinity is a term with which most people are familiar, but few truly understand. Infinity is not an actual value, like the number 3 -- it is an abstract concept. In math terms, it is used as a "limit", where a value can approach infinity by getting continuously larger, but it will never actually get there. Consider the act of cutting a pizza into slices. You can cut it in half, then cut those halves in half, then cut those halves in half, etc. As the slices get smaller, the number of slices gets larger; therefore, as the size of each slice approaches zero, the number of slices approaches infinity. Again, in math terms, this means that as x approaches zero, the value of 1/x approaches infinity. Some go so far as to say that 1/0 equals infinity, but this would not be entirely correct; nothing can actually "equal" infinity, since it isn't a value, but an abstract limit that can only be "approached". Here's another example. You are standing a... read more

When I was studying to be a teacher, one of the classes I had to take was Literacy in Secondary Education. Since the word literacy is associated to reading and writing by most, it would strike many as a surprise that Math teachers have to take courses on literacy. However, literacy is the most practical and crucial aspect of ANY academic discipline, simply because it involves the ability to read and write in said subject. For mathematics, it could not be anymore important. If you cannot understand the words that I am using, then it is almost as if we were communicating to each other in different languages. So whatever subject you are studying, I suggest you learn its vocabulary. As the helpful tutor that I am, I will share a list of vocabulary terms that was distributed in my literacy class to all of you so that you can check your own vocabulary. Keep in mind that this is considered to be the Mathematics vocab that one should know by the time they finish high school... read more

Humans have a tremendous capacity to learn and adapt. However, we consistently build barriers that hinder our natural ability to change and grow. Many people, regardless of age, perceive themselves as not being talented enough to excel at math and science. They view math and science as the realms in which only scientists, engineers, mathematicians, and geniuses truly soar. Nothing could be further than the truth. Sure, possessing a natural affinity towards these subjects helps. Yet, a supposed lack of talent does not prevent you from learning. The path may be more arduous. The journey may be longer. Nevertheless, you possess within you the fire to endure. Willpower, dedication, self belief, and an open mind can compensate for any lack of ability. Bruce Lee was a legendary martial artist, actor, and philosopher who continues to inspire millions with the sheer intensity which he pursued his endeavors. Frail, sickly, and small as a child, Bruce Lee overcame many physical limitations... read more

SUMMER OPPORTUNITIES Now that students, teachers, parents and tutors have had a chance to catch their breath from final exams, it's time to make use of the weeks we have before school starts back. Consider all that could be accomplished in the next few weeks: Areas of math that students NEVER REALLY GRASPED could be fully explained. This could be elementary skills like adding fractions, middle school topics like systems of equations, or high school areas like sequences and series. Students could have a TREMENDOUS HEAD STARTon topics that will be covered in the first few weeks of school. Imagine your son or daughter being able to raise their hand to answer a question in the first week of school because they had worked several problems just like the ones that the teacher is demonstrating. ENORMOUS PROGRESS could be made in the area of preparation for the standardized tests (PSAT, SAT, ACT and more) that are so important to getting into a great college. STUDY SKILLS... read more

Greetings Wyzant community, prospective students, fellow tutors: I have just returned from my studies abroad and am ready to begin teaching again. Please take a look at my profile. My education ranges from my Masters in Physics, to my undergrad degrees in physics, biology and music. I just completed the coursework for a masters program in peace and conflict resolution as well. Aside from know knowledge and experience teaching, I think I possess a very good ability to understand the different ways students learn. This helps me to engage with them in a way that is most effective for them. Not only does it help to comprehend the material for the subjects they are learning but it also helps them to develop a wisdom and intuition for further (creative) learning and a strategic approach towards test taking. I'm looking forward to working with all of you. Don't hesitate to contact me for any reason...

y = f(x) I can't tell you how many times I've had students come to me profoundly confused about their entire math unit, all because their teachers never fully explained this concept. Teachers throw this equation up on the board without discussion as if it explains everything – which it does, but only if you know what it means. So let's discuss! First off, it's important to remember that this is not just an equation; it's an indication of a larger concept. We'll get to that in a minute, but let's start at the beginning. Imagine that I have a little machine which I set on the table in front of you and turn on. You place a number in the slot in the top, and the machine begins to hum and churn. After a few moments, a drawer opens at the bottom and you pull out a different number. You can repeat this with any number you like, any number of times. Now this is a single-purpose machine, which means it has one rule that it uses to transform the starting number into the final... read more

Over the time that I have tutored, both at Austin Community College and as an independent tutor, the best thing that I have learned is that any student that is having fun while they learn make them actually want to come back and learn more. Although people came in to the tutoring lab at Austin Community College, they were coming in to the tutoring lab reluctantly because they just didn't understand what was going on and weren't really enjoying the subject because of it. My personal goal for any person that I tutor is to not only help someone understand the concepts while they are sitting with me, but also make it more fun. Not only does time fly when you're having fun but, as noted in "Brain & Behavior, An Introduction to Biological Psychology", a person is able to learn better and more efficiently as they have positive memories with those concepts rather than a neutral emotion of repetition. I love learning math and, as I look back, I did well in all my math classes,... read more

I have been working with a few students who are ready to learn math much, MUCH faster than allowed by the traditional classroom model in which math is taught over 6 to 8 years. Based on this experience I believe that many students as young as 4th grade and as old as 8th grade (when starting in the program) can master math in 2 years from simple addition through the first semester of Calculus, with Arithmetic, Algebra 1, Geometry, Algebra 2, Precalculus, Probability, Statistics, and Trigonometry in between. This is significantly faster than the traditional approach and is enabled by a combination of one-on-one teaching and coaching and a variety of media that I assign to students to complete in between our sessions. This is a "leveraged blended learning" approach that makes use of online software, selected games, and selected videos with guided notes that I have created that ensure that students pick up the key points of the videos, and which we discuss later. The result... read more

When someone is interested in a topic, there is a heavy intuitive knowledge associated with that interest. A good musician intuitively knows what would be enjoyable to their target audience; a good fashion designer intuitively knows what would be fashionable for the next season; a good personal trainer knows intuitively how to work a particular person with a particular body and a particular mentality to make that person more active and more healthy. In all of these fields and any field you can think of, there is a certain amount of memorization required, a certain set of rules to follow, but is mainly following personal intuition within that field. Science and math works the same way. Sure, there is a certain amount of memorization involved – terms, history, phrases – but there is a certain amount of intuition involved. There is a logic behind every concept in all of science and math, regardless of terms and phrases. This logic has its beginnings in the appropriate intuition. Think... read more

As the school year begins to wind down, I have noticed that many of the students I help have begun the journey of signing up for next years classes or, better yet, deciding where they will start the next chapter of their life in college. I began to reminisce on my Senior year of high school and how stressful that year was for me. It was so easy to become overwhelmed by all of the choices that (seemed to be) abruptly placed in front of me: what college should I go to? What should I major in? Should I choose a college close to home? Should I rush? Should I go to a college with all of my friends? Will I absolutely hate it? I ended up choosing the wrong college and transferred twice until I finally ended up a college that I love! I say all of this to jump into the idea of NOT stressing about this time of year. Yes, I did say not to stress. College is a time of change. That change, no matter how terrifying it may seem, will take you on a wonderful journey that no one can plan for. If... read more

To My Future and Current Students, I can't stress enough the IMPORTANCE of ALGEBRA! Of all the mathematics I have taken in my lifetime...BELIEVE ME IT'S BEEN A LOT, ALGEBRA is the only course that is WOVEN into every single course. I was lucky enough that my first mathematics teacher in High School (Mr. Large), turned me from a B student into an A student such that I graduated High School with a 4.0 in mathematics. The one piece of advice he gave me that I will share with you is that...I NEED TO CHECK, DOUBLE CHECK AND TRIPLE CHECK ALL OF MY ANSWERS! Algebra is a required course (prerequisite) for many of your other math courses, but most importantly in your High School career it is MANDATORY in order to be successful in Algebra 2. It may seem silly to learn and master Algebra, however, it is an integral part of every math course you will take after that except some geometry courses. Algebra teaches you how to think, be organized and how to prove your answers by checking them... read more

A parent told me recently that her son scored a near 100% on his last test. I was so proud. I feel proud when all my students succeed. The question is what does it mean for a student to be successful. I think it's a mix between the student having more confidence than when I begin working with the student, as well as an increase in the student's grades. Depending on the student and his or her own situation grades may increase immediately and with others it may take a bit of time. I want my students to feel confident about their abilities and also be able to show the world and themselves that they understand what's going on in class. I make a commitment when I take on a student, which is, I will work my hardest to be available and flexible. Your child's success is my success.

When interviewing a prospective tutor, parents should ask about the tutor's skills and experience, and find out if the tutor truly enjoys teaching. When the tutor feels enthusiastic about the subject, and communicates well, the student has an opportunity to learn to enjoy the subject too. I recommend for parents to observe the first lesson to see the tutor's skills in action, and watch/listen carefully to future lessons when possible, to make sure the tutor has an encouraging, supportive attitude at all times. (Tutors should welcome and respond positively to the child's questions, and NEVER make the child feel "stupid," no matter what.) It is most important to have a safe and quiet place for studying, without distractions. I like to find a quiet table at a library, and work with students there. I welcome suggestions from parents, and I am always looking for ways to improve my teaching skills.

Hello to all you Happy Learners! I am just setting up my profile and getting all my qualifying exams completed for subjects I am qualified and experienced in for teaching. There will be new posts as we get things set up here. For astronomy and science tutoring help, I can make my self and my home astronomical observatory available as part of the learning experience. I have multiple telescopes and some basic CCD astrophotography equipment which could be used for an astronomical observing project students may want to do and hand in as a project grade in science or specific astronomy class...this is always a great deal of fun and there is much to learn and do with telescopes, computer control of the telescope, CCD camera and autoguider as well as the reading of star charts and catalogues too! Let me know via my email link if you are interested in this service and we can discuss the possible projects we could do. Yours, Mark

I have noticed a number of students in this area struggling with Algebra I and Algebra II. The students are at different levels of their educational process from High School to adult students re-entering the collegiate experience. In my experience having taught upper division sciences is that students who find that they have deficiencies in their lower level math skills, generally, 6-8th grade math, don't 'see' these deficiencies show up until they move into upper division college programs where more critical, cross-disciplinary thinking is required of them. Here is the problem. There is an aspiration to be a science teacher, for instance, or a doctor of some sort. Because a math deficiency was not addressed earlier, they struggle in Chemistry or Cell Biology. The assumption on the part of the faculty is that the poor grade reflect ineptitude with a given subject. Hence, it is presumed that the student is not 'able to compete' for careers in these kinds of careers. The student,... read more

Greetings, scholars! One of my dad's favorite sayings is, "If something seems too good to be true, it probably is." The website Coursera is an example of why that saying needs the word "probably". The idea of taking real college courses from top-notch instructors at prestigious schools for free sounds impossible, yet students around the world are doing just that. When I first heard of Coursera, I was skeptical. To try it out, I enrolled in some basic undergraduate courses so that I could see how they stacked up against the classes I took at KU and Emporia State University. I am currently taking precalculus at UC Irvine, organic chemistry at Illinois, and calculus at The Ohio State University. All three classes are superlative. The video lectures give me new insights into familiar concepts, and the online quizzes motivate me to practice my skills and keep them sharp and up-to-date. Best of all, they haven't cost me a dime, and I can attend class from the... read more