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# Blogs

## Mathematics Blogs

Hi there and welcome to my blog! This is my first post, with hopefully many to follow.   In my undergraduate years, I learned about a very curious summation discovered by the great Ramanujan. Since then, whenever a student tells me that they hate mathematics and that it is stupid, I show this to them and they almost always see math in a new and enthusiastic light. Here, I will explain the series to you, and hope that it brings you as much excitement and curiousity as it first brought me.   Consider the series 1+2+3+4+5+6+... The series is simple, we simply add two to one, then add three, then four, then 5, and keep going forever. The series is called a "monotonic series", meaning that it is ever increasing. This should be intuitive, since if we look at the first few terms, we have   1 = 1 1 + 2 = 3 1 + 2 + 3 = 6 1 + 2 + 3 + 4 = 10   If we continue the process,... read more

Don’t be stubborn: its The Monty Hall Problem. This is one of the least generally understood problems of all time. My hypothesis: the reason most people fail on The Monty Hall problem is that it isn’t straight, and it involves changing plans. If you don’t know, the way this works is that you are on a game show and must find a prize behind one of three doors. You pick a door and then The Game Show Host reveals that the prize is not behind one of the two remaining doors. With due intellect your supposed to reason that it is always advisable two switch your selection. What isn’t understood during the time the game show hosts open the door is that he will never open a door that has the prize in it. He will always open a null door. Vital information is encoded by the pact the game show host has with the producers and it moves in the transaction between the game show host and you. Think of it as the elements of America being encoded to the writing and voice of Stephen... read more

To whom taking Calculus, it has been a long journey to finish all the prep courses successfully.  Smart, intelligent, so they get excited for this new challenging subject.  There are some students who heard about its difficultiness from their seniors, having trouble of making decision of whether taking or not.  But it is a mandatory subject you have to take to get in any important majors in the colleges.   Yes, Calculus is not an easy subject, and many failed.  However, it is not difficult if you understand well.  It is not the subject that you just memorize the formula and replace numbers to get answers.  It requires a lot of thinking.  Take time, read the topics, understand by trying to get the meaning behind.     For example, when you read the first part of Calculus, it is about the Limits.  It looks easy and they just go.   There are a lot of explanations but seem to be not much things in it.  Then... read more

To many students math is a difficult time consuming process. In many developing countries they learn by rote and memorization. This inefficient teaching method leads to 12+ hour school days. The end result is a student who has less understanding and has learned that math is boring.   I see math as like solving a puzzle and playing detective. Math is how we used to entertain ourselves before video games and smart phones. Ultimately, math is the silent rhythm by which the universe dances. Math is a universal language that transcends historical, cultural and language barriers.

Normally, an equation has a single solution when it contains only one undefined variable.  For example, take the equation 3x + 7 = 19.   3x + 7 = 19     [original equation] 3x = 12     [subtracted 7 from both sides] x = 4     [divided both sides by 3]   This is one case of a larger trend in algebra.  As I've already said, you can solve an equation for one answer when it contains a single variable.  However, this is derived from the larger rule that you can solve a set of equations where there are as many distinct equations as there are variables.  These are called simultaneous equations, and occur any time that two equations are both true over a certain domain.  In the more practical sense, this is what you should do if an exam asks you to solve for a value and gives you two different equations to use.   To solve simultaneous equations, we can use three strategies... read more

Being in the mathematics field, I am constantly looking for online resources that break down problems in an effective and clear way. My favorite online resource that I have found the most useful is Khan Academy. Khan Academy has a variety of video tutorials in multiple content areas, not just mathematics. The videos are well organized into units and concepts which makes them easy to navigate. Having the video tutorials are wonderful because you can pause them if they are going too quickly, rewind to see something again, and replay for more exposure to the content. Khan Academy does a wonderful job at explaining the problems in depth and color coding the steps of the problem as they go through it. I suggest Khan Academy for both enrichment and remediation for anyone.

When it comes to using a legitimate online resource to help with tutoring mathematics, or answering mathematical questions I use Wolfram.com.   This website is very diverse and allows the user to input any mathematical equation, formula etc.   With subject areas of mathematics, such as calculus, Wolfram.com has proved to be extremely beneficial, especially when working with difficult integrals and derivatives.    With the Pro version of this website, which is well worth its value, you will be provided step-by-step instructions on how to solve the particular problem that you have inputted.    Check out this website and explore the countless benefits it has to offer.   Keith

Hey folks,  I am sure many of you have plans of going to college or finishing up that last hectic year of school.  Well with these endeavors comes not only tests and quizzes created by books and your professors/teachers, but you also have to take nation and statewide test in order to pass and/or qualify for a position in a higher learning institute.  Such tests include the SAT, ACT, MCAT, etc.  What you want to remember about taking these tests is that these tests are testing you ability to locate small mistakes and easy to miss information.  They also want you to understand this material.  You have to be prepared for these easy to miss situations.  For example, I am sure you all have done a math question, felt like you did it perfectly correct only to find out that you actually got it incorrect.  Furthermore, the answer you got appeared as one of the answer choices!  Or you were on the right track to answering correctly, but made... read more

Rigor is something that is emphasized frequently in higher levels of mathematics and physics, and it has always been something that I appreciated. Unfortunately, with increased rigor often comes a decreased number of people who can understand an argument. One pedagogical ploy that has been used to great effect has been to offer "proofs" of rather difficult concepts on the basis of certain tricks that are not themselves rigorous. I call these things "lazy proofs", and they suffer from the problem of leading to outright contradictions and nonsense if taken to far. This kind of problem, usually, is swept under the rug by the person (usually a teacher) offering the proof in hopes that the misconceptions that could arise never rear their ugly head. Sometimes they never do. Other times, they cause problems down the road. One example of such a lazy proof is the following argument that the centripetal acceleration is   a = v2/r.  ... read more

In elementary school, mathematics is often taught as a set of rules for counting and computation. Students learn that there is only one right answer and that the teacher knows it. There is no room for judgment or making assumptions. Students are taught that Arithmetic is the way it is because it's the truth, plain and simple. Often students go on to become trapped in this view of the universe. As fairy tales fade from the imagination, so is mathematical creativity lost. There is evidence that Mathematics and Arithmetic existed over 3000 years ago, but only the very well educated leisure class had access to it. The rules for simple computation only were developed recently, so much of the computation of sums and products was much more complicated. Imagine adding and multiplying Roman Numerals for example. Because of this difficulty, computations were laid out only to solve very specific practical problems. Although mathematics was mainly limited to solving... read more

Hello everyone, One of my Calculus students had an interesting Related Rates problem that I had to go home and think about for a while in order to figure out.  The problem was set up as such:   A 25 inch piece of rope needs to be cut into 2 pieces to form a square and a circle. How should the rope be cut so that the combined surface area of the circle and square is as small as possible?     Here's what we'll need to do: 1.  We will have to form equations that relate the length of the perimeter and circumference to the combined surface area.  2.  We will then differentiate to create an equation with the derivative of the surface area with respect to lengths of rope.  3.  Wherever this derivative equals 0 there will be a maxima or minima, and so we will set the derivative = to 0 and determine which critical points are minima... read more

Let me preface this by saying that I completely understand that children don’t develop true abstraction skills until late teens/early 20s — and that some never develop a full ability for such. That being said, guided prompts through a proof seem to work. I have been substitute-teaching in a school district lately (I have loan payments coming up since I am post-graduation, and private tutoring, plus app-revenue doesn’t cover living expenses plus my loans), and I got a job for teaching 6th grade math for the day. There were a total of 5 classes of regular math and one advanced class (since I had to teach Reading and ELA in the morning, too). Let me say that I’m not clueless — I have (essentially) a psych degree completed (only missing the keystone thesis — I took a loooooooot of psych classes because I was always interested in computational brain simulations), as well as 5+ years as a TA and 7+ years as a private tutor. I’ve spent more than my fair... read more

In my work as a teacher, I cannot help but notice that many of the reading selections written for our students include words that are beyond our students' experience. Students simply do not have & could not usually acquire the background knowledge necessary for understanding some words they encounter in subject-specific reading selections, such as social studies & science. Reading instruction in language arts classes cannot adequately address all the words students need to know, as language arts teachers have other specific concerns to address every day. This is why every teacher must be a reading teacher & consider reading an integral part of their subject. Certain subjects are the best place for students to encounter, learn, and understand some of the vocabulary they need to know, while context clues are only useful if students already have the needed background knowledge. In other words, a context clue is not really a clue at all if students do not have the... read more

While I am pursing my doctorate degree, I tutor math and reading with a student. It is true that many students do not like math and reading. But we all do math and reading in our daily living. As we go about our daily walk, we read to understand and we use numbers to calculate things mentally and physically. For instance, when we go shopping, we use math mentally to see if we can pay.  Also, when we drive, we read road signs.  So in our daily walk, we continue to use math and reading subtly because they are omnipresent.  So students need to realize that we encounter math and reading because they are part of our daily living. So I encourage my students to believe and stay focused which will help them to achieve anything they are doing at the moment.

I have worked with students who had difficulty learning math facts (addition, subtraction, multiplication, or division) for years.  Let's face it, it's boring to sit and learn facts, especially with flashcards!  I remember sitting night after night with my mother, her flashing the problems to me over and over and they just wouldn't stick!  I would cry and get so frustrated and I just wanted it to be OVER!   You don't have to do that to your children.  Research states that the best way to teach these skills is through games.  There are a variety of math websites on the internet that can help your child learn their facts by playing fairly easy games.  Sometimes they are more challenging and time your child if they are a bit more advanced, or they initially teach them a fact family at one time.  Either way, playing games on a safe website is a much more effective way than using flashcards.  I can recommend some to you if you... read more

Anyone with the right attitude become good at math. And anyone with the wrong attitude will become terrible at math. “Natural ability” is a subjective concept. But grit has proven itself to be the real determining factor in student success. Whether or not you are a good math student is a state of attitude. - http://scienceray.com/mathematics/attitude-the-difference-between-the-good-math-student-the-bad-math-student/

"SUMMER SLUMP SURVIVAL GUIDE" 1.  GEOGRAPHY...If you travel this summer, that too can be a learning experience.  For example, at some point during your journey, you likely look at a map.  To do this, you must understand north, south, east and west.  Mathematically, you should also experience the relationship between speed of travel (e.g., 65 miles-per-hour), distance and timing.  If you travel inter-state, then you study the geography of the United States.  If your journeys take you internationally, then this is more than a geographic experience. 2.  LANGUAGE...Traveling to another country may require developing the knowledge of how to say HELLO and GOODBYE, how to ask a cab driver for a quote for the cost of a trip, as well as many other details. 3.  TELEVISION...When you are at home, you may find yourself watching television.  Although it has been called the IDIOT BOX, it can be as... read more