tag:blogger.com,1999:blog-158637112015-09-16T16:59:22.715-07:00CayMathFor the learning of mathematics in the Cayman Islandsdamezumarihttp://www.blogger.com/profile/11339548449089902020noreply@blogger.comBlogger42125tag:blogger.com,1999:blog-15863711.post-1129040937146647322005-10-11T07:23:00.000-07:002005-10-11T07:28:57.153-07:00Learning is stupidly simple<a onblur="try {parent.deselectBloggerImageGracefully();} catch(e) {}" href="http://photos1.blogger.com/blogger/1987/1484/1600/mrbell1.gif"><img style="cursor: pointer;" src="http://photos1.blogger.com/blogger/1987/1484/400/mrbell1.gif" border="0" alt="" /></a><br /><p>In November 2001 I heard an inspiring talk by Russell Bell in the Cayman Islands. Russell Bell is running a learning centre in Kingston, Jamaica, 'Helping Students Realize Their Full Potential'.</p> <p>This is his main idea:</p> <p class="Quote"> 'Learning is stupidly simple. To learn one has to remove obstacles, not to "strive for higher intelligence".'</p> <p>Water can't flow out of a hose if it is blocked. The same goes for learning. His book 'The Learning Process' discusses how we learn, which barriers exist, and how to remove them.<br /></p> <p><a onblur="try {parent.deselectBloggerImageGracefully();} catch(e) {}" href="http://photos1.blogger.com/blogger/1987/1484/1600/tlpbook.jpg"><img style="cursor: pointer;" src="http://photos1.blogger.com/blogger/1987/1484/400/tlpbook.jpg" border="0" alt="" /></a></p> <p>I am looking forward to a promised update of his <a href="http://mrcmath.com/">web site.</a></p>damezumarihttp://www.blogger.com/profile/11339548449089902020noreply@blogger.com0tag:blogger.com,1999:blog-15863711.post-1128908423317039672005-10-10T18:33:00.000-07:002005-10-09T18:40:23.323-07:00Arithmetic without tables<a onblur="try {parent.deselectBloggerImageGracefully();} catch(e) {}" href="http://photos1.blogger.com/blogger/1987/1484/1600/0285629166.02._SCLZZZZZZZ_.jpg"><img style="cursor: pointer;" src="http://photos1.blogger.com/blogger/1987/1484/400/0285629166.02._SCLZZZZZZZ_.jpg" border="0" alt="" /></a><br /><br />When you look for patterns you may find amazing things.<br /><br />2 * 6 = 12<br />4 * 6 = 24<br />6 * 6 = 36<br />8 * 6 = 48<br /><br />Can you see a pattern?<br /><br />The answer's second digit is the number we multiply six with and the first digit is half that number.<br /><br />1 * 6 = 06<br />3 * 6 = 18<br />5 * 6 = 30<br />7 * 6 = 42<br />9 * 6 = 54<br /><br />What is the pattern?<br /><br />Let me take 7 * 6 as an example. Add 5 to 7 to get 12. 2 is the answer's second digit. Half of 7 rounded down is 3, plus the 'carry' from 12 is 4. That gives the answer's first digit.<br /><br />These two patterns can be combined to a rule for multiplying any number by 6:<br /><br />To each digit add half the neighbour to the right rounded down and 5 more if the digit is odd.<br /><br />What is 749 * 6?<br /><br />I will write this as 0749 * 6 and start from the right.<br /><br />9 has no neighbour and is odd: 9 + 5 gives 4 with 1 carry.<br /><br />4's neighbour is 9 and there is a carry: 4 + 4 + 1 gives 9 with no carry.<br /><br />7's neighbour is 4 and is odd: 7 + 2 + 5 gives 4 with 1 carry.<br /><br />0's neighbour is 7 and there is a carry: 3 + 1 gives 4.<br /><br />The answer is 4494.<br /><br />Study multiplication by 7 and come up with a similar rule.<br /><br />These patterns were first discovered by <a href="http://hucellbiol.mdc-berlin.de/%7Emp01mg/oldweb/Tracht.htm">Jakow Trachtenberg</a> while he was a prisoner of war. He discovered that anyone who can take halfs rounded down and add small numbers can multiply.damezumarihttp://www.blogger.com/profile/11339548449089902020noreply@blogger.com0tag:blogger.com,1999:blog-15863711.post-1128783070535179802005-10-08T07:43:00.000-07:002005-10-08T07:55:40.733-07:00I hate school<a onblur="try {parent.deselectBloggerImageGracefully();} catch(e) {}" href="http://photos1.blogger.com/blogger/1987/1484/1600/hateschool7.jpg"><img style="cursor: pointer;" src="http://photos1.blogger.com/blogger/1987/1484/400/hateschool7.jpg" border="0" alt="" /></a><br /><a onblur="try {parent.deselectBloggerImageGracefully();} catch(e) {}" href="http://photos1.blogger.com/blogger/1987/1484/1600/peanuts20024419510051.gif"><br /></a>damezumarihttp://www.blogger.com/profile/11339548449089902020noreply@blogger.com0tag:blogger.com,1999:blog-15863711.post-1128782577213694402005-10-07T07:26:00.000-07:002005-10-08T07:42:57.230-07:00The mathematical experience<a onblur="try {parent.deselectBloggerImageGracefully();} catch(e) {}" href="http://photos1.blogger.com/blogger/1987/1484/1600/DavisHershMathmaticalExperience1.jpeg"><img style="cursor: pointer;" src="http://photos1.blogger.com/blogger/1987/1484/400/DavisHershMathmaticalExperience1.jpeg" border="0" alt="" /></a><br /><br />What is math? A teacher's answer determines how he teaches. 'The Mathematical Experience' is a book that should widen most people's perspective.<br /><br />A <a href="http://www.amazon.com/exec/obidos/ASIN/0395929687/103-6080978-8878249">review </a>from Amazon:<br /><i><br /></i> <p><i>I was going to study history. Math? Who cared about math? Math was for those science-types. I had an image of mathematicians as bespectacled, socially-inept, hunch-shouldered gnomes who lived in universities and ventured out of their burrows for--well, maybe they didn't venture out at all.</i></p> <p><i>The joke's on me. I'm a math major now. This book is one of the reasons.</i></p> <p><i>I've always loved history: the march of events, the ebb and flow of cause and effect and unexpected accident. I didn't realize that math, too, had a history, an ebb and flow. If I'd ever thought about it, I would have realized that an angel didn't come down from the heavens bearing The Big Book of Math, complete with proofs. But that's what it seemed like, until I read about the almost architectural building of theorem upon theorem, idea upon idea. Math wasn't a Big Book; it evolved and grew. Grows still, I should say.</i></p> <p><i>Did numbers exist? Well, of course they existed. Wait a second. What *is* a number anyway? How *does* one exist? Would they exist if there were no people?</i></p> <p><i>And so I learned that math, too, has its philosophies.</i></p> <p><i>Most of all, I learned that mathematicians were and are people, not gnomes in burrows who have nothing to do with the rest of the world. That math is important for more than the homework assignments that plagued my high school evening hours. That math is worth studying.</i></p> <p><i>If you could convey this to heaven knows how many disgruntled and frustrated math students around the world, I wonder if they might like the subject better.</i></p> <i>I sure did.</i>damezumarihttp://www.blogger.com/profile/11339548449089902020noreply@blogger.com0tag:blogger.com,1999:blog-15863711.post-1128711689655403702005-10-06T11:42:00.000-07:002005-10-07T12:10:47.066-07:00The one who has the shoe on<i>It is important that students bring a certain ragamuffin, barefoot irreverence to their studies; they are not here to worship what is known, but to question it.</i> - Jacob Bronowski<br /><br />The IMPACT program, led by <a href="http://extranet.edfac.unimelb.edu.au/DSME/lps/DC/">David Clarke</a>, required pupils to give written responses fortnightly to eight questions.<br /><ol> <li>What was the best thing to happen in Maths during the past two weeks?</li> <li>Write down one new problem which you now can do.</li> <li>What would you most like more help with?</li> <li>What is the biggest worry affecting your work in Maths at the moment?</li> <li>Write down the most important thing you have learnt in Maths during the last two weeks.</li> <li>Write down one particular problem which you still find difficult.</li> <li>How do you feel in Maths classes at the moment? (Circle the words which apply to you.)<br />A. Interested B. Relaxed C. Worried D. Successful E. Confused F. Clever G. Happy H. Bored I. Rushed J. (Write down one word of your own)</li> <li>How could we improve Maths classes?</li> </ol>Sample answers to each question:<br /><ol> <li>We worked hard and learnt.</li> <li>I can't do any problems but I can now do triangles.</li> <li>Fractions but the teacher thinks I know them.</li> <li>Sometimes I'm a bit unsure where to put the decimal point.</li> <li>I'm stupid in class.</li> <li>Algebra a bit, but because I don't understand why we don't use numbers. It would be simpler.</li> <li>Relaxed. Bored. I feel relaxed because I'm bored.</li> <li>Have less work and more learning.<br /> </li> </ol> Source: The Interactive Monitoring of Children's Learning of Mathematics, David Clarke, For the Learning of Mathematics, Feb 1987.<br /><br />If you are a teacher, give these questions to your students. It is suggested that you ask students to write their names, but that their answers will be treated in confidence. If you are a student you are welcome to put your answers in a comment below.<br /><br />It is all too seldom that students are asked how they feel about their math life. However, in some countries the students have to <a href="http://simpler-solutions.net/pmachinefree/comments.php?id=62_0_1_0_C">evaluate </a>their school life once a year via the Internet.<br /><br /><a onblur="try {parent.deselectBloggerImageGracefully();} catch(e) {}" href="http://simpler-solutions.net/pmachinefree/images/uploads/20040415b.jpg"><img style="cursor: pointer; width: 400px;" src="http://simpler-solutions.net/pmachinefree/images/uploads/20040415b.jpg" border="0" alt="" /></a>damezumarihttp://www.blogger.com/profile/11339548449089902020noreply@blogger.com0tag:blogger.com,1999:blog-15863711.post-1128526549223171492005-10-05T08:19:00.000-07:002005-10-05T08:37:49.326-07:00Under my noseMany complain that math is of little use. 'I've never had to divide two fractions after I left school,' is a common sigh. I have three comments to this.<br /><br />1. I've never had to divide two fractions either.<br />2. Mathematical problem solving skills I have used on a daily basis.<br />3. I have found math in unexpected places.<br /><br />A bit more on 3.<br /><br />Today, I had to rename these files on my computer:<br /><br /><a onblur="try {parent.deselectBloggerImageGracefully();} catch(e) {}" href="http://photos1.blogger.com/blogger/1987/1484/1600/files1.jpg"><img style="cursor: pointer;" src="http://photos1.blogger.com/blogger/1987/1484/400/files1.jpg" border="0" alt="" /></a><br /><br />I wanted to rename them in order of date like this:<br /><br /><a onblur="try {parent.deselectBloggerImageGracefully();} catch(e) {}" href="http://photos1.blogger.com/blogger/1987/1484/1600/files2.jpg"><img style="cursor: pointer;" src="http://photos1.blogger.com/blogger/1987/1484/400/files2.jpg" border="0" alt="" /></a><br /><br />But, when I tried to rename one of them, this monster appeared:<br /><br /><a onblur="try {parent.deselectBloggerImageGracefully();} catch(e) {}" href="http://photos1.blogger.com/blogger/1987/1484/1600/files3.jpg"><img style="cursor: pointer;" src="http://photos1.blogger.com/blogger/1987/1484/400/files3.jpg" border="0" alt="" /></a><br /><br />A math problem is born: How few files is it enough to rename to get the job done?<br /><br />I can rename jecengine06.mdb to jecengine10.mdb without any problems, but I can't rename jecengine04.mdb to jegengine09.mdb without first renaming the existing jecengine09.mdb. I would like to rename the existing 09 to 07, but I can't since 07 already exists.<br /><br />You might think, this is silly, who cares how few renames have to be done, just do it any way you like, it won't take long anyway. I agree! The practical value in this case is close to nil, but, like Mount Everest, the problem has been discovered and has to be conquered. Why? Because it is a challenge, it is fun, and it feels good to create something.<br /><br />So, tell me, how few renames do you need?!damezumarihttp://www.blogger.com/profile/11339548449089902020noreply@blogger.com1tag:blogger.com,1999:blog-15863711.post-1128433745293729312005-10-04T06:47:00.000-07:002005-10-04T07:09:22.890-07:00Proof by authority<img src="http://simpler-solutions.net/pmachinefree/images/uploads/20040420b.jpg" border="0" alt="" name="image" width="348" height="155" /><br /><span style="font-size:78%;">Aristotle and Dennis Lindley</span><br /><br />There are several ways to prove that you are right. One is by reasoning, another is by authority. News media these days prefer the latter. Here is a recent example.<br /><br />Professor Dennis Victor Lindley, 80, well-known statistician from University College London, is a prankster or in search of fame. He just launched this formula:<br /><br /><img src="http://simpler-solutions.net/pmachinefree/images/uploads/20040420a.jpg" border="0" alt="" name="image" width="141" height="58" /><br /><br />M is the ideal age to get married. Y is the age you start looking for a partner. X is the age you intend to stop looking. e is 2.718281828..., an irrational constant only second to pi in fame.<br /><br />An example. If you intend to look for the right one from 18 to 50 you should get married at 30, since 18 + (50 - 18)/2.718281828 = 29.8.<br /><br />The idea is very simple, for every year you plan to look for a partner wait about 0.37 years, or four and a half months. In the example above, 32 years searching time gives 32 x 0.37 = 11.8 waiting time. So, if you start at 18, get married at 29.8.<br /><br />Does the professor believe in his formula. I would hope not. Does he justify it? If he does, it is not reported in the media.<br /><br />Is it a joke or a demonstration of stupidity? I go for the first. The use of e is the clue. The precision given by it and that it should crop up in a formula of this kind.<br /><br />I don't think the professor's prank does mathematics any good. People in general have a distorted view of the subject already and the formula makes it worse, not better.<br /><br />Mathematical thinking can clarify things as mathematician John Allen Paulos demonstrates in his column <a href="http://abcnews.go.com/Technology/WhosCounting/">Who's Counting</a> and books.damezumarihttp://www.blogger.com/profile/11339548449089902020noreply@blogger.com0tag:blogger.com,1999:blog-15863711.post-1128127735653502152005-10-03T17:46:00.000-07:002005-10-03T07:51:58.820-07:00Mathematicians talking<a onblur="try {parent.deselectBloggerImageGracefully();} catch(e) {}" href="http://photos1.blogger.com/blogger/1987/1484/1600/singer2.jpg"><img style="cursor: pointer;" src="http://photos1.blogger.com/blogger/1987/1484/400/singer2.jpg" border="0" alt="" /></a><br /><p>Michael Atiyah and Isadore Singer interviewed by Martin Raussen and Christian Skau in May last year.</p> <p>When mathematicians talk about how they work and why they work, I listen. Basically because I like to have supported my ideas about what problem solving is and feels like. This week I read an <a href="http://www.ams.org/notices/200502/comm-interview.pdf">interview</a> (pdf) with Michael Atiyah and Isadore Singer.</p> <p>Some highlights from the interview:</p> <p class="Quote"><i> Atiyah: A theorem is never arrived at in the way that logical thought would lead you to believe or that posterity thinks. It is usually much more accidental, some chance discovery in answer to some kind of question. Eventually you can rationalize it and say that this is how it fits. Discoveries never happen as neatly as that. You can rewrite history and make it look much more logical, but actually it happens quite differently.</i></p> <p class="Quote"><i> Atiyah: Almost all mathematics originally arose from external reality, even numbers and counting. At some point, mathematics then turned to ask internal questions, e.g., the theory of prime numbers, which is not directly related to experience but evolved out of it. There are parts of mathematics about which the human mind asks internal questions just out of curiosity.</i></p> <p class="Quote"><i> Singer: I find it disconcerting speaking to some of my young colleagues, because they have absorbed, reorganized, and simplified a great deal of known material into a new language, much of which I don’t understand. Often I’ll finally say, “Oh; is that all you meant?” Their new conceptual framework allows them to encompass succinctly considerably more than I can express with mine. Though impressed with the progress, I must confess impatience because it takes me so long to understand what is really being said.</i></p> <p class="Quote"><i> Atiyah: My fundamental approach to doing research is always to ask questions. You ask “Why is this true?” when there is something mysterious or if a proof seems very complicated. I used to say— as a kind of joke—that the best ideas come to you during a bad lecture. If somebody gives a terrible lecture—it may be a beautiful result but with terrible proofs—you spend your time trying to find better ones; you do not listen to the lecture. It is all about asking questions—you simply have to have an inquisitive mind! Out of ten questions, nine will lead nowhere, and one leads to something productive. You constantly have to be inquisitive and be prepared to go in any direction. If you go in new directions, then you have to learn new material.</i></p> <p class="Quote"><i> Singer: ... when I try out my ideas, I’m wrong 99% of the time. I learn from that and from studying the ideas, techniques, and procedures of successful methods. My stubbornness wastes lots of time and energy. But on the rare occasion when my internal sense of mathematics is right, I’ve done something different.</i></p> <p class="Quote"><i> Singer: I love to play tennis, and I try to do so two to three times a week. That refreshes me, and I think that it has helped me work hard in mathematics all these years.</i></p> <p class="Quote"><i> Atiyah: I believe that if you do mathematics, you need a good relaxation that is not intellectual—being outside in the open air, climbing a mountain, working in your garden. But you actually do mathematics meanwhile. While you go for a long walk in the hills or you work in your garden, the ideas can still carry on. My wife complains, because when I walk she knows I am thinking of mathematics.</i></p> <p>Other sources for listening to mathematicians talking about what they do is the book 'Mathematical People - Profiles and Interviews' from 1985, with a continuation in 1990 called 'More Mathematical People: Contemporary Conversations.' (ISBN 0817631917 and 0120482509). There is a similar book about computer programmers called 'Programmers at work: interviews with 19 programmers who shaped the computer industry' (ISBN 1556152116).</p> <p>A sigh: when will we get MathConversations, podcasts where mathematicians talk about their work as in the interview above? ITConversations already exist for the computer people. Click <a href="http://www.itconversations.com/shows/detail58.html">here</a> for an interview with the creator of php. The language that helps you bring this blog.</p>damezumarihttp://www.blogger.com/profile/11339548449089902020noreply@blogger.com0tag:blogger.com,1999:blog-15863711.post-1128201488079143412005-10-01T14:16:00.000-07:002005-10-01T14:18:08.086-07:00Knowledge is power<a onblur="try {parent.deselectBloggerImageGracefully();} catch(e) {}" href="http://photos1.blogger.com/blogger/1987/1484/1600/knowledgeispower2.jpg"><img style="cursor: pointer;" src="http://photos1.blogger.com/blogger/1987/1484/400/knowledgeispower2.jpg" border="0" alt="" /></a><br /><br /><a href="http://www.dilbert.com/">More </a>Dilbert cartoons.damezumarihttp://www.blogger.com/profile/11339548449089902020noreply@blogger.com0tag:blogger.com,1999:blog-15863711.post-1128086867597537782005-09-30T06:24:00.000-07:002005-09-30T06:27:47.603-07:00Philosophical maintenance<a href="http://photos1.blogger.com/blogger/1987/1484/1600/zen_hut.jpg"><img style="CURSOR: hand" alt="" src="http://photos1.blogger.com/blogger/1987/1484/400/zen_hut.jpg" border="0" /></a><br /><em>"Laws of nature are human inventions, like ghosts. Laws of logic, of mathematics are also human inventions, like ghosts. The whole blessed thing is a human invention, including the idea that it isn’t a human invention. The world has no existence whatsoever outside the human imagination. It’s all a ghost, and in antiquity was so recognized as a ghost, the whole blessed world we live in. It’s run by ghosts. We see what we see because these ghosts show it to us, ghosts of Moses and Christ and the Buddha, and Plato, and Descartes, and Rousseau and Jefferson and Lincoln, on and on and on. Isaac Newton is a very good ghost. One of the best. Your common sense is nothing more than the voices of thousands and thousands of these ghosts from the past. Ghosts and more ghosts. Ghosts trying to find their place among the living."</em><br /><br />Want to read more? 'Zen and the Art of Motorcycle Maintenance' can be read in its entirety <a href="http://www.design.caltech.edu/Misc/pirsig.html">here</a>.damezumarihttp://www.blogger.com/profile/11339548449089902020noreply@blogger.com0tag:blogger.com,1999:blog-15863711.post-1128000594491848012005-09-29T06:29:00.000-07:002005-09-29T06:32:45.070-07:00What is a proof?<a href="http://photos1.blogger.com/blogger/1987/1484/1600/jd20040730.jpg"><img style="CURSOR: hand" alt="" src="http://photos1.blogger.com/blogger/1987/1484/400/jd20040730.jpg" border="0" /></a><br /><p>Formally, a mathematical proof is a sequence of expressions in a formal language. An expression can be entered in the sequence if it follows from earlier expressions using rules of inference also expressed in the formal language, or if it belongs to a finite set of expressions called axioms. The last expression in the sequence is what the proof proves. It is called a theorem.</p><p>In real life, mathematicians do not prove their theorems in formal languages. Basically this is the accepted definition of a mathematical proof: 'A proof is a proof until proved otherwise'</p><p>To prove things can make you rich. This <a href="http://mathpuzzle.com/prizes.html">page</a> lists puzzles worth between 50 and more than a million dollars. Click <a href="http://mathpuzzle.com/eternity.html">here</a> to read how 'The Eternity Puzzle' made someone 1.6 million richer. One million is waiting for the first to settle Riemann's hypothesis.</p><p>A proof of Riemann hypothesis 'is perhaps the most tantalising goal in mathematics today. If true, it tells us that prime numbers, which are those exactly divisible only by one and themselves, are scattered utterly randomly along the number line. If not, then mathematicians may be able to predict where the prime numbers fall.' - <a href="http://www.newscientist.com/news/news.jsp?id=ns99995104">Source</a></p><p>Some time ago Louis de Branges de Bourcia, a professor of mathematics at Purdue University in Indiana, issued a press release claiming he had proved the Riemann hypothesis is true. He can not collect the prize money straight away: 'To claim the one million dollar prize money put up by the Clay Mathematics Institute in Cambridge, Massachusetts, de Branges must first publish his paper in a journal and then the work must survive two years of scrutiny by the mathematics community.'</p><p>A lot of mathematics deals with infinity in one form or another. Click <a href="http://www.zongrila.net/swirl.htm">here</a> to take it for a spin.</p>damezumarihttp://www.blogger.com/profile/11339548449089902020noreply@blogger.com0tag:blogger.com,1999:blog-15863711.post-1128000280211407742005-09-28T06:20:00.000-07:002005-09-29T06:24:40.216-07:00Mathematics is not mathematics<em>We mathematicians are used to the fact that our subject is widely misunderstood, perhaps more than any other subject (except perhaps linguistics). Misunderstandings come on several levels. </em><br /><em><br />One misunderstanding is that the subject has little relevance to ordinay life. Many people are simply unaware that many of the trappings of the present-day world depend on mathematics in a fundamental way. When we travel by car, train, or airplane, we enter a world that depends on mathematics. When we pick up a telephone, watch television, or go to a movie; when we listen to music on a CD, log on to the Internet, or cook our meal in a microwave oven, we are using the products of mathematics. When we go into hospital, take out insurance, or check the weather forecast, we are reliant on mathematics. Without advanced mathematics, none of these technologies and conveniences would exist. </em><br /><br /><em>Another misunderstanding is that, to most people, mathematics is just numbers and arithmetic. In fact, numbers and arithmetic are only a very small part of the subject. To those of us in the business, the phrase that best describes the subject is "the science of patterns," a definition that only describes the subject properly when accompanied by a discussion of what is meant by "pattern" in this context.</em><br /><em></em><br />Read the entire article by mathematician Keith Devlin <a href="http://www.maa.org/devlin/devlin_03_03.html">here</a>.damezumarihttp://www.blogger.com/profile/11339548449089902020noreply@blogger.com0tag:blogger.com,1999:blog-15863711.post-1127925433492420622005-09-27T09:22:00.000-07:002005-09-28T09:39:35.636-07:00To add or multiply, that is the question.- I am trying to calculate 2 + 3 * 5.<br />- OK.<br />- Tell me, should I add or multiply first?<br />- Does it matter?<br />- Sure it matters!<br />- How do you know?<br />- OK. Let me check it out.<br />- I like this.<br />- What is it that you like?<br />- That you are going to check it out.<br />- If I add first and then multiply I get 25. But if I multiply first and then add I get 17.<br />- So which one is right?<br />- They can't both be right?<br />- Don't be silly!<br />- OK<br />- My calculator gives me the answer 25, but my friend's calculator gives 17.<br />- You know what?<br />- No.<br />- In Poland they would write 23+5* or 235*+.<br />- How much is 23+5*?<br />- They start from the left. First comes a 2, so they remember that number. Then comes a 3 so they remember that number too.<br />- They have to remember a lot?<br />- (Smiles) Yes, but then comes a +, so they add the two numbers they have remembered to get 5. Then comes another 5.<br />- So now they remember two 5s?<br />- Which gives 25 when they see the * sign.<br />- Neat! But what about 235*+?<br />- Why don't you tell me?<br />- OK. I'll have a go. They first remember the numbers 2, 3 and 5.<br />- OK<br />- Then the * tells them to multiply, but which numbers?<br />- What do you think?<br />- The two last ones?<br />- Right.<br />- So they get 15.<br />- And what happens when they see the +?<br />- They add the two numbers on their mind.<br />- Which are?<br />- 2 and 15.<br />- You are smart! You know that?<br />- Can I try another one? One that does not give neither 17 nor 25.<br />- No problem. Try this one: 235+*.<br />- (Thinking for a while) The + gives me 8, and the * means 2 * 8, so the answer is 16.<br />- Give me one more.<br />- Why don't you make up one?<br />- OK! Let me try 23*5+.<br />- Go ahead.<br />- (Very fast) 11.<br />- Excellent!<br />- I like this way to calculate!<br />- Why is that?<br />- Because I never wonder if I should add or multiply first. As soon as I see a + sign I add the two last numbers, and if I see a * I multiply.<br />- (Smiling) I am happy for you!<br />- And one more thing. When I calculate I start from the left reading one thing at a time without worrying about what comes next. What follows does not matter. You know what my father always says?<br />- No, tell me.<br />- It his favourite saying. "Let's cross the bridge when we come to it."<br />- Many calculators and computers convert the expressions to this form before they calculate for that reason. The form is called RPN, or Reverse Polish Notation.<br />- Did Einstein come up with this system?<br />- No, his name was Jan Lukasiewicz.<br />- In the 1920s.<br />- Tell me. If this system is so wonderful, why is it not used all over the place?<br />- I don't know. Tradition maybe.<br />I like talking with you. This Polish things and stuff. It makes me think.<br />- I am glad you feel like that.<br />- But you know what? We haven't solved my problem. In 2 + 3 * 5, should I add or multiply first?<br />- Ah, yes, that one. Where did you get the problem from?<br />- From the textbook.<br />- Did it say "Sue ate 2 pancakes in the morning and 3 in the afternoon for 5 days. How many pancakes did she eat in all?"<br />- Then I would add 2 and 3 before multiplying with 5, but it didn't say that.<br />- Did it say "Sue ate 2 pancakes on Monday and then 3 pancakes every day for 5 days."<br />- I get your point! That would mean that I should first multiply 3 and 5 and afterwards add 2. But it said neither. It just said "2 + 3 * 5 = ?"<br />- The convention is to start with the operations from the left.<br />- So I should add before I times?<br />- Well, only if the operations have the same priority.<br />- Do + and * have the same priority?<br />- Multiplication has a higher priority.<br />- So I should times first?<br />- Yes.<br />- So the answer is 17.<br />- Correct.<br />- So it would be wrong to add first?<br />- Yes. If you want to add first you would have to add some parenthesis. (2 + 3) * 5.<br />- So anything inside paranthesis should be calculated first?<br />- Yep.<br />- You know what? In Poland they have a much simpler system!damezumarihttp://www.blogger.com/profile/11339548449089902020noreply@blogger.com1tag:blogger.com,1999:blog-15863711.post-1127509782906123522005-09-26T14:05:00.000-07:002005-09-26T04:49:46.136-07:00Solving equations a step at a time<a onblur="try {parent.deselectBloggerImageGracefully();} catch(e) {}" href="http://photos1.blogger.com/blogger/1987/1484/1600/equationsolver.jpg"><img style="cursor: pointer;" src="http://photos1.blogger.com/blogger/1987/1484/400/equationsolver.jpg" border="0" alt="" /></a><br /><br /><a href="http://www.studyworksonline.com/cda/content/article/0,,EXP1253_NAV2-7_SAR1248,00.shtml">Here</a> you get help to solve equations one step at a time:<br /><br /><ol> <li><i>Type a number or expression in the yellow textbox. </i></li><li><i>Use the calculator buttons to add, subtract, multiply, or divide from each side of the equation. Hover your mouse over each button to see what it does. </i></li><li><i>Watch the results of your work on the right. </i></li><li><i>Continue until you've solved the equation. </i></li><li><i>Just click on the New Problem button to practice again.</i></li> </ol> To get it to work you have to install a plug-in and it will only work in Internet Explorer.<br /><br />About the people behind the site:<i><br /><br />StudyWorks! Online is a free learning site delivering innovative learning tools to help students develop an understanding of math and science concepts. StudyWorks Online gives students, parents, and teachers access to high quality content, interactive activities, real world examples, and monitored homework help through the <a href="http://collab.mathsoft.com/%7Estudyworks">Homework Help Collaboratory</a>. </i>damezumarihttp://www.blogger.com/profile/11339548449089902020noreply@blogger.com0tag:blogger.com,1999:blog-15863711.post-1127568667895468822005-09-24T06:27:00.000-07:002005-09-24T06:34:18.540-07:00Sally<a onblur="try {parent.deselectBloggerImageGracefully();} catch(e) {}" href="http://photos1.blogger.com/blogger/1987/1484/1600/123.jpg"><img style="cursor: pointer;" src="http://photos1.blogger.com/blogger/1987/1484/400/123.jpg" border="0" alt="" /></a><br /><br /><a href="http://www.comics.com/comics/peanuts/index.html">More Peanut strips</a>.damezumarihttp://www.blogger.com/profile/11339548449089902020noreply@blogger.com0tag:blogger.com,1999:blog-15863711.post-1127506295177956422005-09-23T13:01:00.000-07:002005-09-23T13:11:35.183-07:00Get hooked on codes<a onblur="try {parent.deselectBloggerImageGracefully();} catch(e) {}" href="http://photos1.blogger.com/blogger/1987/1484/1600/codebook1.jpg"><img style="cursor: pointer;" src="http://photos1.blogger.com/blogger/1987/1484/400/codebook1.jpg" border="0" alt="" /></a><br /><br /><a href="http://www.simonsingh.net/The_Code_Book.html">The Code Book</a>, The Secret History of Codes and Code Breaking was written by Simon Singh. You can now download it for free <a href="http://www.sunsite.org.uk/package/simonsingh-codebook">here</a>.<br /><i><br />'The aim of the project was to create an interactive version of The Code Book, so that readers could encrypt, break codes and see how the Enigma machine really works. However, it soon became clear that the CD-ROM had a huge potential for getting young people interested in mathematics.'</i>damezumarihttp://www.blogger.com/profile/11339548449089902020noreply@blogger.com1tag:blogger.com,1999:blog-15863711.post-1127310740729999652005-09-22T06:48:00.000-07:002005-09-22T08:23:03.743-07:00Wanted: Math Teachers!This was in the news a few days ago:<br /><br /><span id="headline" style="font-size:85%;">IBM to Encourage Employees to Be Teachers</span><br /><i><br />International Business Machines Corp., worried the United States is losing its competitive edge, will financially back employees who want to leave the company to become math and science teachers.<br /><br /></i><a href="http://www.newsday.com/technology/wire/sns-ap-ibm-education,0,664925.story?coll=sns-ap-technology-headlines">More</a><br /><br />By the way, if you want to keep track of what's happening in the world of math education try a <a href="http://www.google.com/alerts">Google Alert</a>:<br /><br /><i><span style="font-size:-1;">Google Alerts are email updates of the latest relevant Google results (web, news, etc.) based on your choice of query or topic.</span></i>damezumarihttp://www.blogger.com/profile/11339548449089902020noreply@blogger.com0tag:blogger.com,1999:blog-15863711.post-1127229726602180242005-09-21T07:51:00.000-07:002005-09-21T06:40:43.780-07:00Back to basics<a onblur="try {parent.deselectBloggerImageGracefully();} catch(e) {}" href="http://photos1.blogger.com/blogger/1987/1484/1600/DivineCover-small1.jpg"><img style="cursor: pointer;" src="http://photos1.blogger.com/blogger/1987/1484/400/DivineCover-small1.jpg" border="0" alt="" /></a><br /><br />Distance and angle are two basic terms in mathematics. What would happen if they were replaced by other terms? It is a refreshing question asked by N J Wildberger this fall.<br /><br />For separation of points, <i>quadrance</i> is defined as the distance squared. Easy enough. It eliminates a few square-root signs. Pythagoras' is simply a + b = c when a, b , and c are the quadrances of the sides in a right-angled triangle.<br /><br />For separation of lines, <i>spread</i> is defined as the square of the sine of the acute angle between the lines.<br /><br />From these definitions he demonstrates the triple quad formula, the spread law, the cross law, and the triple spread formula. With these you are armed to solve most basic trigonometry questions.<br /><br />Download the first chapter of his soon to be published book <a href="http://wildegg.com/papers/Chapter1.pdf">here</a>. <a href="http://web.maths.unsw.edu.au/%7Enorman/">This </a>is his home page.<br /><br />Wildberger's fresh ideas is an excellent playground where you can set your students free to develop some math on their own.<br /><br /><a href="http://www.kitchentablemath.net/twiki/bin/view/Kitchen/WebHome">Kitchen Table Math is about doing math with children</a> is the web place where I found this exciting news. Hereby recommended.damezumarihttp://www.blogger.com/profile/11339548449089902020noreply@blogger.com0tag:blogger.com,1999:blog-15863711.post-1127161654185159682005-09-20T13:22:00.000-07:002005-09-20T07:29:23.210-07:00Patterns in election data<a onblur="try {parent.deselectBloggerImageGracefully();} catch(e) {}" href="http://photos1.blogger.com/blogger/1987/1484/1600/boddentown.jpg"><img style="cursor: pointer;" src="http://photos1.blogger.com/blogger/1987/1484/400/boddentown.jpg" border="0" alt="" /></a><br /><br />There were 13,118 registered voters at the election held in May this year. 10,005 of them voted, or 76%.<br /><br />Give students the raw data and they may discover several patterns.<br /><br />This is what I found:<br /><br />1. A candidate who got more than 32.7% of the registered voters' vote was elected to the Legislative Assembly regardless of district.<br /><br />2. A candidate who got more than 42% of the votes cast was elected to the Legislative Assembly regardless of district.<br /><br />The big winner, in my mind, was Anthony Eden. 75% of those who voted in Bodden Town voted for him.<br /><br /><a href="http://myprograms.myblogsite.com/_attachments/842711/Elections.xls">Here</a> is my spreadsheet file. Strip it for columns E and F before you give it to students.damezumarihttp://www.blogger.com/profile/11339548449089902020noreply@blogger.com0tag:blogger.com,1999:blog-15863711.post-1127158456712970752005-09-19T12:08:00.000-07:002005-09-19T12:34:16.716-07:00What does a mathematician look like?<a onblur="try {parent.deselectBloggerImageGracefully();} catch(e) {}" href="http://photos1.blogger.com/blogger/1987/1484/1600/mathematician1.jpg"><img style="cursor: pointer;" src="http://photos1.blogger.com/blogger/1987/1484/400/mathematician.jpg" border="0" alt="" /></a><br /><br />To me, a mathematician is someone who creates mathematics, not one who teaches it. But, what does a mathematician look like?<br /><br />Which of the two gentlemen above is a mathematician? Hint: the other one is a teacher of buddhism. Click <a href="http://www.andrej.com/mathematicians/A/Altenkirch_Thorsten.html">here </a>and <a href="http://www.lama-ole-nydahl.org/">here </a>to see if you guessed correctly, and <a href="http://www.andrej.com/mathematicians/">here </a>to see more pictures of today's mathematicians.damezumarihttp://www.blogger.com/profile/11339548449089902020noreply@blogger.com0tag:blogger.com,1999:blog-15863711.post-1126974312047138682005-09-17T09:22:00.000-07:002005-09-17T09:26:26.033-07:00Mafalda is wondering<p><br /><img src="http://simpler-solutions.net/pmachinefree/images/uploads/mafalda.gif" border="0" alt="image" name="image" width="426" height="316" /><br /><br /></p><p><a href="http://www.clubcultura.com/clubhumor/quino/ingles/index.htm">Quino</a> is the artist beyond Mafalda.</p>damezumarihttp://www.blogger.com/profile/11339548449089902020noreply@blogger.com0tag:blogger.com,1999:blog-15863711.post-1126973105583502112005-09-16T08:41:00.000-07:002005-09-17T09:06:20.966-07:00First name statistics<a onblur="try {parent.deselectBloggerImageGracefully();} catch(e) {}" href="http://photos1.blogger.com/blogger/1987/1484/1600/jan1.jpg"><img style="cursor: pointer;" src="http://photos1.blogger.com/blogger/1987/1484/400/jan1.jpg" border="0" alt="" /></a><br /><br />When I was born in 1952 in Norway, Jan was the most popular name to give a baby. The graph above shows that the name Jan is losing popularity, while Mathias (see below) is the most popular name today.<br /><br /><a onblur="try {parent.deselectBloggerImageGracefully();} catch(e) {}" href="http://photos1.blogger.com/blogger/1987/1484/1600/mathias.jpg"><img style="cursor: pointer;" src="http://photos1.blogger.com/blogger/1987/1484/400/mathias.jpg" border="0" alt="" /></a><br /><br /><a href="http://www.ssb.no/navn/">Source</a>.<br /><br />What is the most popular name in Cayman from year to year? I don't know, and I can not find any website for the government's statistical office.<br /><br />At John Gray High School two years ago Michael and Priscilla were the most popular names for boys and girls respectively. Why not take a statistics at your school and email it to CayMath?damezumarihttp://www.blogger.com/profile/11339548449089902020noreply@blogger.com0tag:blogger.com,1999:blog-15863711.post-1126829034665502842005-09-15T17:00:00.000-07:002005-09-15T17:05:35.206-07:00Musical math comedy<a onblur="try {parent.deselectBloggerImageGracefully();} catch(e) {}" href="http://photos1.blogger.com/blogger/1987/1484/1600/MrOneFile.jpg"><img style="margin: 0pt 10px 10px 0pt; float: left; cursor: pointer;" src="http://photos1.blogger.com/blogger/1987/1484/400/MrOneFile.jpg" border="0" alt="" /></a><br /><br /><br /><br /><br /><br /><br /><br /><br /><br /><br /><br /><br /><br /><i><br />"The Calculating Mr. One is a musical comedy that explores importance of mathematics and problem solving in everyday life and is based on the National Curriculum for Maths and the Numeracy Targets. It's also interactive with the audience required to solve mathematical problems in order for play to progress."</i><br /><br />If you like to get emails with news like the above, send an email to enews@atm-online.org.uk with the word 'subscribe' in the subject box. ATM stands for <a href="http://www.atm.org.uk/">Association of Teachers of Mathematics</a>.damezumarihttp://www.blogger.com/profile/11339548449089902020noreply@blogger.com0tag:blogger.com,1999:blog-15863711.post-1126739147733059072005-09-14T16:02:00.000-07:002005-09-15T04:20:00.416-07:00Black box<a onblur="try {parent.deselectBloggerImageGracefully();} catch(e) {}" href="http://photos1.blogger.com/blogger/1987/1484/1600/blackbox7.jpg"><img style="margin: 0pt 10px 10px 0pt; float: left; cursor: pointer;" src="http://photos1.blogger.com/blogger/1987/1484/400/blackbox7.jpg" border="0" alt="" /></a><br /><p><br /></p> <p><br /></p> <p><br /></p> <p><br /></p> <p><br /></p> <p><br /></p> <p><br /></p> <p><br /></p> <p><br /></p> <p><br /></p> <p><br /></p> <p><a href="http://puzzling.caret.cam.ac.uk/game.php?game=2&age=2">Black Box</a> is a logical game.</p> <p>A black box contains several balls that reflect laser beams. Your task is to fire beams, and based on where the beams come out of the box, deduce where the balls are.</p>damezumarihttp://www.blogger.com/profile/11339548449089902020noreply@blogger.com0tag:blogger.com,1999:blog-15863711.post-1126664569500733912005-09-13T19:04:00.001-07:002005-09-13T19:22:49.506-07:00Inversions<a onblur="try {parent.deselectBloggerImageGracefully();} catch(e) {}" href="http://photos1.blogger.com/blogger/1987/1484/1600/mozart.gif"><img style="cursor: pointer;" src="http://photos1.blogger.com/blogger/1987/1484/400/mozart.gif" border="0" alt="" /></a><br /><br />An <i>inversion</i> is a word or name written so it reads in more than one way. Try to read the text above standing on your head. <a href="http://www.scottkim.com/inversions/index.html">More text inversions</a>.<br /><br /><a onblur="try {parent.deselectBloggerImageGracefully();} catch(e) {}" href="http://photos1.blogger.com/blogger/1987/1484/1600/inversions2.gif"><img style="cursor: pointer;" src="http://photos1.blogger.com/blogger/1987/1484/400/inversions2.gif" border="0" alt="" /></a><br /><br />If you want to keep your youth, don't turn up-side-down! <a href="http://www.almaleh.com/inversions/index.html">More image inversions</a>.<br /><br />Why not try to create an inversion?! Email it to CayMath and we will publish it here.damezumarihttp://www.blogger.com/profile/11339548449089902020noreply@blogger.com0