[{"title":"Understanding Logarithms","content":"

By Evan Ma<\/a><\/b><\/h3>\r\n

(Math and Physics\u00a0tutor at The Edge Learning Center<\/a>)<\/p>\r\n\r\n\r\n[caption id=\"attachment_4306\" align=\"alignright\" width=\"245\"]\"log-a-rhythms<\/a> Log-a-rhythms (Photo taken from Pinterest)[\/caption]\r\n\r\nThe logarithmic<\/a> function seems to confound many students studying the IB Math SL program<\/a>. In order to understand it properly, let us define\r\n<\/span>\r\n\r\n\"ar-m\"\r\n\r\nwhere\u00a0\"a\"<\/span>, a strictly positive real number, is called the base and\u00a0\"r\"<\/span>\u00a0is the power. For example, we all know the following by heart:<\/span>\r\n\r\n\"10-3-0-001\"\r\n\r\nand so on. The logarithmic function does it backwards \u2013 given a number as the argument, it yields the power subject to a specified base, and therefore it is defined as follows.<\/span>\r\n\r\nIf<\/span>\r\n\r\n\"ar-m-a-0\"\r\n\r\nthen<\/span>\r\n\r\n\"r-loga-m\"\r\n\r\nTherefore, it is clear that<\/span>\r\n\r\n\"log-10-0-0001\"\r\n\r\nand so on. Let\u2019s look at other examples involving other bases:<\/span>\r\n\r\n\"23-8-log2\"\r\n\r\nLet\u2019s complete the following questions as practice: (a)\"log10010000\"<\/span>\u00a0<\/span>, (b)\"log5125\"<\/span>\u00a0<\/span>, (c)<\/span> \"log91729\"<\/span>\u00a0\u00a0, (d)\"log1749\"<\/span>\u00a0<\/span>. The answers can be found at the end of this blog.<\/span>\r\n\r\nFrom the definition of logarithm, we can also see that if we raise the base by the base-<\/span>\"a\"logarithm of a real number\u00a0\"m\"<\/span>, we will get back the number\u00a0\"m\"<\/span>, viz<\/span>\r\n\r\n\"alogamm\"\r\n\r\n[caption id=\"attachment_4307\" align=\"alignright\" width=\"263\"]\"logarithmic-keys\" (Image taken from Wikimedia Commons)[\/caption]\r\n\r\nYou may try to verify this by using your calculator for different valid bases. This identity is important when we try to solve logarithmic equations, examples of which will be given alter.<\/span>\r\n\r\n \r\n\r\nNow, let\u2019s take a look at the rules of the logarithm. Similar to the definition of the logarithm, we use the laws of indices to derive the rules:<\/span>\r\n\r\n1. Logarithm of a product is sum of the logarithms<\/strong>\r\n\r\nProof: Say \"apm\"\u00a0and\u00a0\"aqn\"<\/span>, <\/span>therefore<\/span>\r\n

\"mn-apaq\"<\/h3>\r\n2. Base-\"a\"\u00a0logarithm of Mr is\u00a0\"r\"\u00a0times\u00a0\"logam\"\u00a0<\/strong>\r\n\r\nProof: Say\u00a0\"apm\"<\/span>, therefore<\/span>\r\n\r\n\"mr-apr\"\r\n\u00a0<\/span>3. Logarithm of a quotient is the difference of logarithms<\/strong>\r\n\r\nProof: Say\u00a0\"apm\"<\/span>\u00a0and\u00a0\"aqn\"<\/span>, <\/span>therefore<\/span>\r\n\r\n\"mn-apaq\"\r\n\r\n4. The final rule is called the change-of-base formula.<\/strong> We shall use an example as a tool to derive the formula. Let\u2019s say we want to know what power must 2 be raised in order to get 50. As 50 is not an integer power of 2, the answer is not very obvious. Hence, we write\r\n\r\n\"2r50\"\r\n\r\nand therefore\"rlog250\"<\/span>. In order to find\u00a0\"r\"<\/span>, say we apply base-10 logarithm to both sides of the equation as our calculator may not have the base-2 logarithm key:<\/span>\r\n\r\n\"log102rlog1050\"\r\n\r\nCorrected to 4 significant figures, the value of\u00a0\"r\"<\/span>\u00a0is approximately 5.644. In fact, you can verify your answer by raising 2 to the power of 5.644 to see that the answer is approximately 50. <\/span>\r\n\r\nTo generalize the problem, let\u2019s say we have to find\u00a0\"r\"<\/span>\u00a0such that\"arm\"<\/span>\u00a0and base-\"a\"<\/span>\u00a0logarithm is not at our disposal, we can therefore instead use base-\"b\"<\/span>\u00a0logarithm according to the equation\u00a0\u00a0<\/span>\r\n\r\n\"rlogblogba\"\r\n\r\nand therefore<\/span>\r\n\r\n\"logamlogbmlogba\"\r\n\r\nBy way of example, let\u2019s consider the following. Suppose an amount of $10,000 is deposited at an interest of 2.5% per annum, compounded annually. How long must the money be kept in the account for it to grow to $50,000? <\/span>\r\n\r\nTo answer the question, we are basically trying to find <\/span>n<\/span> such that<\/span>\r\n\r\n\"1000010-025\"\r\n\r\nAs my simple scientific calculator does not allow me to specify a base of 1.025 for the logarithmic key, I will have to rely on the change-of-base formula and use base-10, and hence<\/span>\r\n\r\n\"nlog105\"\r\n\r\nSince interest is compounded annually, it will take 66 years to grow at least 5 times.<\/span>\r\n\r\nHaving explored rules of logarithms, we introduce the natural logarithmic function, or\u00a0<\/span>\"inx\" <\/span>. It is base-\"e\"<\/span>, where\"e\"<\/span>\u00a0is the eminent irrational number<\/a>\u00a0\"e\"<\/span>\u22482.71828<\/span> and whose significance in sciences and mathematics cannot be emphasized enough. Hence, base-\"e\"l<\/span>ogarithm is defined as follows.<\/span>\r\n\r\nIf<\/span>\r\n\r\n\"erm\"\r\n\r\nthen<\/span>\r\n\r\n\"rinm\"\r\n\r\nNext, let us take a look at two examples where common mistakes are made in solving equations of logarithms. See if you can identify the mistake.<\/span>\u00a0 \u00a0\"log2x224\"<\/span>\r\n\r\nWhere is the mistake? Look closer. Of course, in the second step, one cannot \u201csplit\u201d the logarithm across the addition sign. Remember, you can only \u201csplit\u201d a logarithm into a sum if the logarithm is applied to a product, not a sum. The correct steps are therefore, as follows:<\/span>\r\n\r\n\"log2x224\"\r\n\r\nNow, let\u2019s look at the following example, and see if you can identify the mistake:<\/span>\r\n\r\n\"in27x-1in3-2\"\r\n\r\nWhere is the mistake? Yes, it is in the second step \u2013 a quotient<\/a> of logarithms is of course not the logarithm of the quotient. Rather, we can use the change-of-base formula to simplify the first step, as follows:\r\n\r\n\"in27x-1in3\"\r\n\r\nFinally, we will illustrate how to use rules of logarithm to solve the following:<\/span>\r\n\r\n\"logx6x2-94\"\r\n\r\nYou may notice that the unknown <\/span>\"x\"<\/span> appears as the base and a variable in the argument of the logarithm. How can we solve for\u00a0\"x\"<\/span>? The method still depends on applying the rules of logarithm consistently. First, we raise both sides as powers of the base\u00a0\"x\"<\/span>, and hence<\/span>\r\n\r\n\"xlogx6x2-9\"\r\n\r\nYou may recall from the definition of logarithm that the left hand side will just become the argument of the logarithm, and therefore <\/span>\r\n\r\n\"6x2-9x4\"\r\n\r\nBy re-arranging the above equation, we need to solve\r\n\r\n\"x4-6x29\"\r\n\r\nNoticing that this resembles a quadratic<\/a> in\u00a0\"x2\"<\/span>, we solve for\u00a0\"x\"<\/span>\u00a0as follows:<\/span>\r\n\r\n\"x2-32\"\r\n\r\nNow, where is the negative root? Since\u00a0\"x\"<\/span>\u00a0is also the base of the logarithm, the negative root is therefore rejected as a solution. Hence,\u00a0\"x3\"<\/span>\u00a0is the final answer.\u00a0<\/span>\r\n\r\nBy reviewing the above examples, you may see that solving a seemingly difficult logarithmic equation is not hard at all \u2013 simply apply the rules of logarithm consistently and the correct solution can be obtained.<\/span>\r\n\r\nAnswers to questions: (a) 2 (b) 3 (c) -3 (d) -2.<\/span>\r\n\r\n ","excerpt":"By Evan Ma (Math and Physics\u00a0tutor at The Edge Learning Center) The logarithmic function seems to confound many students studying the IB Math SL program. In order to understand it properly, let us define where\u00a0, a strictly positive real number, is called the base and\u00a0\u00a0is the power. For example, we all know the following by […]","link":"http:\/\/theedge.com.hk\/understanding-logarithms\/","category":"Edge Insights","catIcon":"http:\/\/theedge.com.hk\/wp-content\/uploads\/2014\/11\/announcement-logo.png","background":"http:\/\/theedge.com.hk\/wp-content\/uploads\/2015\/03\/EvanMa-300x200.jpg","date":"

By\u00a0Pratik Choudgury<\/a><\/b><\/h3>\r\n

(ACT, IGCSE\/MYP Math & Physics Instructor at The Edge Learning Center<\/a>)<\/p>\r\n

The topic of free fall is integral to all higher secondary physics curriculum. At the very least, the basic fundamentals associated with free fall are introduced to the students at junior higher secondary and more complicated scenarios involving free fall are usually discussed at a higher secondary level. Free fall is also known as one-dimensional motion.<\/span><\/p>\r\n\"free-fall-accerelation\"<\/a>\r\n\r\nAn object is said to be experiencing free fall when it falls solely under the influence of gravity. There are a few conceptual characteristics of a free falling object:<\/span>\r\n

    \r\n \t
  1. Free falling objects are not subjected to air resistance.<\/span><\/li>\r\n \t
  2. A free falling object experiences an acceleration of\u00a0\"9-8msec2\"<\/span>. The acceleration of\u00a0<\/span>\u00a0\"9-8msec2\"is known as <\/span>acceleration due to gravity<\/span><\/i><\/a>. Whether explicitly stated or not in a particular free fall related question, the value of acceleration due to gravity is set to<\/span> \"9-8msec2\"<\/span>\u00a0for any objects experiencing free fall.<\/span><\/li>\r\n \t
  3. If an object is dropped from a particular height, then the initial velocity of the object is always considered to be 0.<\/span><\/li>\r\n \t
  4. From experience we know that when an object is thrown vertically upwards, then it will slow down as it rises upwards, but what many of us do not know that the object will slow down at the same rate of deceleration of <\/span>-\"9-8msec2\"<\/span>\u00a0that it would otherwise accelerate with when dropped from a height. <\/span><\/li>\r\n<\/ol>\r\nTo elucidate further on the aforementioned characteristic, when an object is thrown vertically upward, the object is still subject to gravity. As a result, the velocity of the object will decrease at a rate of\u00a0\"9-8msec2\"<\/span>. When the highest point is reached and the velocity will decrease to 0, it will then start going down and the velocity will subsequently increase at a rate of\u00a0\"9-8msec2\"<\/span>.<\/span>\r\n\r\nThese four characteristics can be combined together to solve problems involving free falling objects.<\/span>\r\n\r\n\"4-characteristics-of-free-falling-objects\"<\/a>\r\n

    The Big Misconception<\/b><\/h3>\r\nOne big misconception is whether acceleration due to gravity is same for all the objects. In other words, \u201cDoesn\u2019t a more massive object accelerate at a greater rate than a less massive object?\u201d \u201cWouldn\u2019t an elephant fall faster than a mouse?\u201d This misconception leads from our personal observation of the rate of fall of a single piece of paper and a textbook. The two objects clearly travel at a different speed towards the ground. The factor that actually plays a role in eluding us is somewhat invisible. As a result, we find it difficult to associate with the role it plays in the motion of a free falling object. The factor is the air resistance that an object encounters when free falling. If the object is very small (e.g a small metal ball) then the air resistance is almost zero. However, if we recollect the five characteristics above then a free falling object cannot be termed as free falling if it is subjected to air resistance. <\/span>\r\n\r\nThe actual explanation of why all objects accelerate at the same rate (if there is negligible air resistance) involves the concept of force and mass, pretty much to do with Newton\u2019s First Law of Motion<\/a>. <\/span>\r\n\r\nNewton\u2019s First Law of Motion states that the Force on an object is directly proportional to mass of the object and the acceleration that it undergoes. In other words, we can say that the acceleration of an object is directly proportional to force that it is subjected to and inversely proportional to its mass. Thus, greater force on a massive object is offset by the inverse influence of greater mass. Subsequently, all objects free fall at the same rate of acceleration, regardless of their mass. \u00a0<\/span>\r\n\r\n\"free-fall-explaination\"<\/span>\r\n\r\nSkydiving is considered free falling, as the air resistance is considered negligible, but not entirely zero. Since, the air resistance is not entirely zero, it plays a vital role in regulating the downward velocity of the object. After a certain period of free falling, the upward force resulting from the air resistance becomes equal to the downward force due to gravity and the mass of the object. At that point in time, the skydiver is said to have reached <\/span>terminal velocity<\/span><\/i><\/a>: a concept that I will be discussing further in my upcoming blog. <\/span>\r\n\r\nThe formulas associated with free fall is tabulated as below:<\/span>\r\n\r\n\"free-fall-formulas\"<\/a>\r\n\r\nA typical free fall-IB examination<\/a> question is demonstrated below:<\/span>\r\n\r\n\"free-fall-problem-throw-up\"","excerpt":"By\u00a0Pratik Choudgury (ACT, IGCSE\/MYP Math & Physics Instructor at The Edge Learning Center) The topic of free fall is integral to all higher secondary physics curriculum. At the very least, the basic fundamentals associated with free fall are introduced to the students at junior higher secondary and more complicated scenarios involving free fall are usually […]","link":"http:\/\/theedge.com.hk\/introduction-to-free-fall\/","category":"Edge Insights","catIcon":"http:\/\/theedge.com.hk\/wp-content\/uploads\/2014\/11\/announcement-logo.png","background":"http:\/\/theedge.com.hk\/wp-content\/uploads\/2015\/10\/IMG_0009-copy-2-300x204.jpeg","date":"

    By Alfred Tang<\/a><\/b><\/h3>\r\n

    (Math & Physics Instructor at The Edge Learning Center<\/a>)<\/p>\r\nMath and physics have a reputation of being hard to understand. \u00a0Whenever I tell people that I am a physicist, their first response is always something like \u201cphysics is hard\u201d or \u201cI did not do well in physics\u201d. \u00a0Similarly, the mere mention of the word \u201ccalculus\u201d melts the heart. \u00a0Of course, young Einstein would not know what I am talking about. \u00a0Even though math and physics are so intimidating, many students still choose to take the challenge because they intuitively know that the process is good for them. \u00a0At the same time, no one wants to fail. \u00a0It is why students seek help to turn challenge into success. \u00a0Private tutoring and tailor-made courses at<\/span> The Edge<\/span><\/a> provide these helps.<\/span>\r\n\r\nThere are basically 3 main teaching styles in math and physics: \u00a0(1) Terse and intelligent short answers, (2) constant massive drills and (3) self-teaching and self-learning. \u00a0\u00a0Teaching style #3 is the most hated by students. \u00a0The motivation behind this teaching method is to force students to develop independence and inductive reasoning. \u00a0The job of the teacher is simply to facilitate learning by providing materials and an environment for the students to discover knowledge for themselves. \u00a0Teachers may give no more instruction than to tell students to \u201cthink like a scientist\u201d. \u00a0This approach works well for graduate students and a few highly gifted teenagers. \u00a0Most students find this teaching style frustrating and completely useless. \u00a0Why do they need school if school is not teaching them anything? \u00a0In developmental theory, students grow through stages. \u00a0In the secondary school stage, method #3 is not effective because the students are not intellectually ready for it. \u00a0However a small dose of self-learning may push the students to the next level of their development. \u00a0Method #1 is characteristic of highly intelligent teachers who are experts of their subject areas. \u00a0\u00a0A famous professor of Christian philosophy<\/span> J. P. Moreland<\/span><\/a> once said that every graduate student should be able to summarize a book in one sentence. \u00a0The idea is that, if you understand something, you should be able to explain it briefly. \u00a0Sometimes when a student has a simple question, they just want a simple answer. \u00a0If the teacher beats around the bush too much, the student becomes annoyed and confused and thinks that the teacher is inapt. \u00a0On the other extreme, bright people who are terse as a habit of thought expect the same thing from other people. \u00a0In the opposite direction of the highway of communication, if a student is fumbling around too much and never be able to get to the point, a very smart teacher may get impatient. \u00a0An impatient teacher stresses out a student. \u00a0At the worst of method #1, the teacher\u2019s answer is too terse and the student does not get enough explanation. \u00a0It is why a teacher needs to constantly observe the student\u2019s micro-expressions to gauge how much explanation he needs to give. \u00a0Teaching style #2 is typical of Asian teachers. \u00a0When I was a kid taught in Chinese school, I did so much homework that my middle finger developed a huge callus that still stays with me today. \u00a0Most of our students go to international schools and do not understand what it really means to have a lot of homework. \u00a0It is true that too much homework is counter-productive. \u00a0But a certain amount of drills and practice are necessary to help the students internalize the knowledge and develop the mental muscle memory needed to do well in tests. \u00a0For the teachers, a lot of homework for the students means a lot of homework for them to grade. \u00a0It is why lazy teachers do not give a lot of homework. \u00a0But then, these teachers are also not doing their job. \u00a0<\/span>\r\n\r\n[caption id=\"attachment_4089\" align=\"alignleft\" width=\"220\"]\"feynman-the-joker\"<\/a> Richard Feynman fooling around with drums.[\/caption]\r\n\r\nAnother teaching style not yet mentioned so far but is becoming more and more sought after in the multi-media age is the ability to speak well in public and be dramatic or \u201cinteresting\u201d (I call it a pseudo method). \u00a0In the TV generation, teenagers are used to sound bites and having all their senses stimulated all at once in multi-media presentations. \u00a0In addition to sight and sound, technocrats are now designing gadgets that generate smell as well. \u00a0Soon, when you see a peach on the computer screen, you will be able to smell peach as well. \u00a0It is the multi media world that our students live in. \u00a0Actors are experts in engaging all the senses and emotions. \u00a0Actors do not get work by being boring. \u00a0Teachers on other hand can get work even if they are boring. \u00a0If math and physics are inherently interesting, it is a crime to bore the students with them. \u00a0All the dynamic math and physics teachers I know have personality, are first rate public speakers and actors in addition to being experts in their fields. \u00a0For instance, physics Nobel laureate Richard Feynman<\/span><\/a> was called the world\u2019s smartest man by New York Times. \u00a0He was notorious for being a joker. \"feynman-the-charismatic-teacher\"<\/a>\u00a0People who did not know him did not know that he was one of the most brilliant minds in history. \u00a0He taught freshman physics at Caltech. \u00a0His lectures were hard to understand because he was too smart. \u00a0His lecture notes become a classic 3 volumes set and food for\u00a0fodder even to the experts. \u00a0Regardless of the challenge, Feynman got the students\u2019 interest because he had something to say. \u00a0His dynamic personality helped a lot too. \u00a0The phenomenon of combining information and personality in teaching is rapidly becoming a norm in a market driven media world.\u00a0<\/span>\r\n\r\nMost of our students follow<\/span> MYP<\/span><\/a>,<\/span> IB Diploma Programs<\/span><\/a> and<\/span> AP<\/span><\/a> courses. \u00a0MYP is traditionally internally assessed\u2014meaning that the teachers decide the students\u2019 grades. \u00a0Because of that, teachers have some freedom to tailor-make the curriculum to their taste. \u00a0As a result, I have seen MYP students learning things way beyond their levels and subsequently struggle. \u00a0MYP is just beginning to implement computer based external assessment this year (2016). \u00a0But most schools are still slow to adapt. \u00a0Until MYP curriculum is standardized by external assessment, we will continue to see some students struggle. \u00a0IB and AP levels math and physics strike fear in the heart of students as well\u2014especially calculus and calculus based physics. \u00a0Math and physics are different than other subjects in that understanding is lauded. \u00a0Rote memorization gets a student only so far. \u00a0Critical thinking and problem solving skills are key. \u00a0These skills are highly sought after in today\u2019s work place because of the complexity and the constantly evolving nature of real life problems. \u00a0It is why companies like to hire math and physics graduates even though their day-to-day operations have little or nothing to do with the subjects. \u00a0Students who do well in math and physics tend to do well in everything else. \u00a0Hence math and physics are good education even though the students do not plan to pursue careers in these fields.<\/span>\r\n\r\n[caption id=\"attachment_4087\" align=\"alignright\" width=\"261\"]\"Physicist<\/a> Physicist Richard Feynman, during on the hearing of the Rogers Commission. It was Feynman who, with a simple demonstration with ice water, exposed the fatal problem with the shuttle rockets' o-rings.[\/caption]\r\n\r\nMath and physics can be very abstract. \u00a0Students sometimes have trouble understanding certain concepts not because they do not understand the math per se but because they do not believe the concepts. \u00a0For this reason, it is especially important for teachers to relate abstract mathematical formalisms to everyday life as much as possible. \u00a0<\/span>It is a misunderstanding that brilliant people are so heavenly minded that they are of no earthly good. \u00a0The truth is the opposite. \u00a0Some of the best teachers are geniuses. \u00a0Richard Feynman as mentioned before is the best example. \u00a0He invented quantum electrodynamics that revolutionizes particle physics and won himself the Nobel Prize. \u00a0On January 28, 1986, space shuttle Challenger exploded 73 seconds after liftoff. \u00a0Feynman was asked by the Rogers Commission to investigate the disaster. Feynman quickly understood the source of the problem as the O-rings losing their elasticity in cold weather. \u00a0He demonstrated this simple physics idea before the panel by using a piece of O-ring, a C-clamp, a Styrofoam cup and some iced water. \u00a0He showed that it did not have to be fancy to explain physics. \u00a0Take for example: \u00a0In my teaching experience, some students have trouble understand the concept of<\/span> momentum<\/span><\/a>. \u00a0The definition of momentum is very straightforward\u2014i.e. momentum is mass times velocity. \u00a0But students cannot visualize it and do not know how to relate it to everyday phenomena. \u00a0So they do not believe it. \u00a0Following the spirit of Feynman, I used a portable fan, some plastic bowls and aluminum tubes to demonstrate the concept of momentum as shown in the video. \u00a0Some people think that expensive tools are indispensable for good teaching. \u00a0Many schools spend a fortune on their teaching labs but the students still do not understand the physics concepts any better. \u00a0Tools are important. \u00a0But intelligence is much more important. \u00a0Students do not care about expensive equipment unless it helps them understand better.<\/span>\r\n\r\n[video width=\"854\" height=\"480\" mp4=\"http:\/\/theedge.com.hk\/wp-content\/uploads\/2016\/11\/Momentum.mp4\"][\/video]\r\n\r\nAs another example, the concept of limit in the beginning of a calculus course sometimes seems mysterious to students. \u00a0Teachers often do not give the students a simple answer but instead go through a long drawn out mathematical process that make the explanation unnecessarily complicated. \u00a0As I said before, \u201cIf you understand something, you should be able to explain it in one sentence.\u201d \u00a0OK, I lie. \u00a0I may need more than one sentence. \u00a0If I were asked by a student to explain the concept of limit, I will simply say that it is based on the concepts of continuity and infinity. \u00a0If I have a collection of coins and keep dividing it in halves and keep half the coins each time, eventually I will always end up with just one coin. \u00a0The process of division stops when it reaches the smallest indivisible unit\u2014i. e. a coin. \u00a0Hence coins are discrete and not continuous. \u00a0Real number on the other hand is continuous because it does not have any smallest indivisible unit. \u00a0For any two numbers, there is always another number between them. \u00a0If I divide a number by 2, I get a number halfway between 0 and the original number. \u00a0I can do it again\u2026and again\u2026and again\u2026until forever. \u00a0No matter how many times I divide, the process never ends. \u00a0Here is the concept of infinity\u2014i. e. something going on forever. \u00a0The answer of my real number division problem will get closer and closer to 0 but is never exactly 0. \u00a0We can either say that the answer approaches 0 or simply say that the limit is 0 for the economy of words. \u00a0Hence \u201climit\u201d is just a word mathematicians invented to facilitate communication. \u00a0Once the students understand the motivation behind the concept, I will teach them how to do calculations with it. \u00a0If a teacher gives a straight answer to a simple question as simply as possible, the students tend to understand it more readily. \u00a0If a student is hanged up on a basic concept from the gecko, he will always be bothered by it even though he can do the calculations in the later stages. Therefore my note to myself is KISS\u2014i.e. Keep It Simple, Stupid. \u00a0Once the students are unhooked from simple things on the lower level, they are comfortable to receive more complicated things on the higher level. \u00a0Math and physics are built on layers of concepts. \u00a0Students need to master one layer before they move on to the next. \u00a0It is why a strong foundation is so important for a student to move on.<\/span>\r\n\r\nThinking well is key to speaking and writing well. \u00a0What I heard from Readers\u2019 Digest ages ago still sticks to my mind: \u00a0In terms of effective communication, the keys words are clarity, brevity and conciseness. \u00a0I cannot always be brief, as evidenced by this 2000 words essay. \u00a0At least I can be clear and concise. \u00a0At the end of the day, what I want to model for students is good thinking by communicating math and physics concepts well to give them understanding and a strong foundation.<\/span>","excerpt":"By Alfred Tang (Math & Physics Instructor at The Edge Learning Center) Math and physics have a reputation of being hard to understand. \u00a0Whenever I tell people that I am a physicist, their first response is always something like \u201cphysics is hard\u201d or \u201cI did not do well in physics\u201d. \u00a0Similarly, the mere mention of […]","link":"http:\/\/theedge.com.hk\/understand-the-concepts-in-math-and-physics\/","category":"Edge Insights","catIcon":"http:\/\/theedge.com.hk\/wp-content\/uploads\/2014\/11\/announcement-logo.png","background":"http:\/\/theedge.com.hk\/wp-content\/uploads\/2015\/09\/alfredIMG_0026-copy-300x195.jpg","date":"

    IGCSE Seminar\u00a0<\/span><\/h1>\r\n\"IGCSE\r\n
    \r\n

    If you or your children are planning to take IGCSE exam with an interest in science, then this seminar is for you. Our Head of Math\u00a0and Sciences, Luke Palmer<\/a>, will be giving a lecture on the basics of the IGCSE as well as the specifics as related to the sciences covered by the curriculum.\u00a0<\/span><\/p>\r\n\r\n