And some more…
In above institute (except IITs ,NITs) students are paid amount ranging from 6000 to 28000 per month and in coming year it may increase. By qualifying NBHM MSc students can get amount 8000 per month in two year of their MSc. Thus NBHM is a good source of money for those student who are not getting any amount from their institute.
Some tests ask only analysis and algebra while some includes graph theory and topology also… Syllabus and previous year papers of each test will help you.
Principle of Mathematical Analysis By Walter Rudin This book is queen of Analysis.Try to solve at least 50% problem from exercise.
Elementary real analysis by thomson and bruckner
Abstract Algebra by Dummit and foote
starting few chapters only required for above tests
Topics in Algebra by I N Herstein
Linear Algebra by hoffman and kunze
General Topology by john kelly
Introduction to topology and modern analysis by G F Simmons
In above list some subjects are missing. For those subjects you can refer the books mentioned in syllabus by institutes like CMI for their bachelor’s students. Just look at their website….
I personally believe in self preparation and no coaching.
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Among the areas of strength, Maine’s graduation rate increased by 11.4 percentage points in the past ten years, a substantially greater than the national increase of 8.4 percent.
At grade four, the percentage of Maine’s students who scored at proficient or above on the National Assessment of Educational Progress (NAEP) in math (41.1 percent) and reading (35.6 percent) were higher than the national percentages. Also, at grade 8, the percentage of Maine’s students who scored at proficient or above in math (35.4 percent) and in reading (35.6 percent) were higher than the national percentage.
One of Maine’s continued challenges remains the lack of improvement in grades four and eight reading scores, which have been flat since 2003. And while there has been growth in grades four and eight NAEP math scores, Maine still lags behind the national improvement rate.
Acting Education Commissioner Bill Beardsley says, “These rankings are positive indicators of Maine education outcomes and reflect the quality of teaching and a significant commitment of State resources to public education. They also suggest Maine has generally positive trends, as do most other states.”
Beardsley added, “There is no question we need to increase our focus on literacy and numeracy in the early elementary years…. Our Departmental objective marker is to increase the percentage of third graders who read and do math at an appropriate pace to graduate college and career ready.”
In addition to Maine’s lagging on the NAEP scores, the ‘Quality Counts’ report shows Maine, like many other states, also faces a wider poverty gap in the proficiency of our students.
For more information on NAEP, contact Coordinator of NAEP and International Assessments Paula Hutton at paula.hutton@maine.gov or 624-6636.
]]>My grocery store is only
A twenty minute drive,
A statement based on distance,
Related speed and time.
You understand my drive at once,
It causes you no fuss.
Who says you don’t comprehend
Integral calculus.
By Kate Rauner
By virtue of genius appearing in all fields of human accomplishment these articles are naturally varied in style, length and approach. Terence Tao works in pioneering-level pure mathematics, and I’m about as proficient with mathematics as a salamander, so this particular entry is coming from a laymen (nay, idiot’s) point of view. It provides a generalised overview of Tao’s life, briefly covers the origins and significance of mathematics for context (which is actually pretty damn interesting), gives rough insight into the significance of his work, explores his giftedness growing up and how it was developed, and ends with an overview of his personality—which is exceptionally kind and humble—and how it all fits together.
Introduction
The term ‘genius’ is more related to accomplishment than ability, and can be equally applied to painting as it can be to theoretical physics. It has very little to do with IQ (though some take having an IQ above 140 to also qualify a person as a genius). There may be a correlation with IQ scores in many cases, but an IQ score is only indicative of isolated aptitudes (such as memory and logical reasoning). Genius-level accomplishment comes from the interplay between cognitive control and creativity; it’s raw intelligence multiplied by open-minded imagination and wonder. Certain fields display a stronger correlation than others, and based on my research so far, it appears strongest in mathematics and physics. The Nobel Prize-winning physicist Richard Feynman is notoriously used as an example of the irrelevancy of IQ testing, with a tested score of only 125 and a clearly genius-level intellect, but closer inspection reveals that to be a likely product of the specific test he took, which was heavily language-focused. IQ tests are largely irrelevant, by Feynman isn’t the best example.
The kind of thinking required for mathematics and physics is pure logical reasoning and abstraction, with processing speed, braveness (yep, braveness) and imagination being key bonuses. Terence Tao has a tested IQ score of over 220, and by many accounts demonstrates those attributes better than any mathematician alive today. He’s known as the “Mozart of math”, and in the classical sense of the term, he may well be the smartest guy on the planet.
What is Mathematics?
For a better appreciation of Tao I think it helps to understand the broader significance of his field, so without deviating too much, here’s a basic rundown:
We don’t exactly know when it ended, but there was a time in human history when we had no concept of counting. We intuitively understood the concepts of ‘more’ and ‘less’—generalised quantity—but couldn’t differentiate anything in abstract terms. Seeing two antelope and recognising them as more than one antelope was one thing, recognising their quantity as an abstract concept equally applicable to fingers and days on a calendar—the concept of the number 2—was a quantum leap in human thought. The first person to achieve this may well be the most important genius in our ancestry. But we have no idea who it was, or how it came about. Anthropologists theorise that counting started as the tallying of single units, seen as vertical lines drawn on a wall, and that symbols were eventually incorporated to represent larger groups of tallies. In ancient Sumerian culture for example, a small clay cone was used to denote ‘1’, a clay sphere ’10’ and a large clay cone was ’60’. Many different systems of symbols were used across the world before the establishment of 0 – 9, which came out of India after 300BC.
The formation of symbols to represent groups of single units created a new dynamic between each symbol, and with each new dynamic came further symbol sub-systems (like algebra) with their own unique interplay, so that complexity grew exponentially from a mathematical big bang—an outward explosion of theory from the use of the first single unit.
The philosopher Bertrand Russell makes the case in The Principles of Mathematics (1903; not to be confused with his Principia Mathematica released in 1928) that mathematics and logic are the same thing (or at least, come from the same place), which becomes easier to comprehend when we consider that numbers are only representative—different systems (such as roman numerals and binary) yield different kinds of patterns, puzzles and insights, but all are bound by logic to the parameters of the system they belong to. Whether or not logic and mathematics are considered the same is a matter of definition, but thinking of logic as being fundamental to math at least helps us understand its nature from a deeper perspective and ponder the question: what exactly is mathematics? Is it something we’ve discovered, or is it something we’ve created?
I think it makes sense to view logic as a core property of the universe, intrinsic to the way everything exists and functions, and that mathematical theory is a form of logical structuring—an interaction of human concepts with the order of the universe. I have nil expertise and may be way off, but it seems like the 0-9 number system could potentially be replaced by something much more complex; it’s just that it works broadly for our population and is complex enough to describe reality to the level we’re capable of being curious.
So is mathematics just a way to describe reality? The physicist Max Tegmark makes the case in his book Our Mathematical Universe that mathematics not only describes reality, but that reality itself is mathematical in nature:
“The idea that everything is, in some sense, mathematical goes back at least to the Pythagoreans of ancient Greece and has spawned centuries of discussion among physicists and philosophers. In the 17th century, Galileo famously stated that our universe is a “grand book” written in the language of mathematics. More recently, the Nobel laureate Eugene Wigner argued in the 1960s that “the unreasonable effectiveness of mathematics in the natural sciences” demanded an explanation.
We humans have gradually discovered many additional recurring shapes and patterns in nature, involving not only motion and gravity, but also electricity, magnetism, light, heat, chemistry, radioactivity and subatomic particles. These patterns are summarized by what we call our laws of physics. Just like the shape of an ellipse, all these laws can be described using mathematical equations.
Equations aren’t the only hints of mathematics that are built into nature: There are also numbers. As opposed to human creations like the page numbers in this book, I’m now talking about numbers that are basic properties of our physical reality.
For example, how many pencils can you arrange so that they’re all perpendicular (at 90 degrees) to each other? The answer is 3, by placing them along the three edges emanating from a corner of your room. Where did that number 3 come sailing in from? We call this number the dimensionality of our space, but why are there three dimensions rather than four or two or 42?”
The example Tegmark gives is a good illustration of the symbolic nature of numbers, showing there to be a fundamental truth of the universe beneath their representation, but whether or not reality is mathematical in nature is mostly redundant to the field; it’s just helpful when trying to understand why it’s all so important, and therefore, the importance of the work being done by someone like Terence Tao. There may be conjecture around the philosophical nature of mathematics but there’s little debate over the benefit. Without it, our cultural and technological evolution wouldn’t have progressed beyond the spear—every scientific and technological advancement involves mathematics to some degree.
The paradigm shift available through understanding mathematics at a deeper level is also about mathematicians. Where once they appeared as number technicians, it now seems talented mathematicians are actually more tuned-in to the universe than anyone else (especially those making that kind of claim). Like a child who develops language early and is therefore at an advantage with interpersonal relationships, the gifted mathematician has an aptitude with the language of the universe, becoming the core force behind the progression of our species within it.
If Tao really is the world’s most gifted mathematician, he’s more than just a guy who solves hard problems: he’s more fluent with universal language than anyone else alive.
The Child Prodigy
Terence Tao was born in Adelaide, South Australia in 1975 to Billy and Grace, both Chinese natives who had emigrated to Australia in 1972. They’d met a few years previously at Hong Kong university; Billy there to complete a doctorate in paediatrics while Grace became an honours-roll mathematics and physics graduate. They had three sons within a few years of arriving: Terence (known to his friends as Terry), Nigel and Trevor—their westernised names chosen to reflect the culture of the couple’s new home country. All three brothers would eventually become standout intellectuals, with Nigel scoring a 180 IQ and winning bronze at two international mathematics olympiads, and Trevor becoming a national chess champion at age 14 while winning numerous prizes for his classical music compositions; broad achievements made all the more impressive by the fact he has autism.
Tao’s precocity became evident before the age of two, when his parents noticed him arranging an older child’s letter blocks alphabetically; a skill he’d learnt through watching Sesame Street. Things didn’t slow down: when he was 4 he was able to multiply two-digit numbers by two-digit numbers in his head. It was soon decided that regular schooling wouldn’t be suitable, and so he was placed into accelerated learning, which was eventually monitored by the Davidson Institute (Australia’s centre for the development of gifted children). The institute’s Miraca Gross writes:
“A few months after Terry’s second birthday, the Taos found him using a portable typewriter which stood in Dr. Tao’s office; he had copied a whole page of a children’s book laboriously with one finger! At this stage his parents decided that, although they did not want to ‘push’ their brilliant son, it would be foolish to hold him back. They began to borrow and buy books for him and, indeed, found it hard to keep pace with the boy. They encouraged Terry to read and explore but were careful not to introduce him to highly abstract subjects, believing, rather, that their task was to help him develop basic literacy and numerical skills so that he could learn from books by himself and thus develop at his own rate. “Looking back,” says Dr. Tao, “we are sure that it was this capacity for individual learning which helped Terry to progress so fast without ever becoming bogged down by the inability to find a suitable tutor at a crucial time.” By the age of 3, Terry was displaying the reading, writing and mathematical ability of a 6-year-old.”
Research has shown the likelihood of a child prodigy transitioning into an adult genius to be extremely rare. Genius-level intellect isn’t just about talent; it’s about creativity, inventiveness and open-minded intrigue. Tiger mothers forcing a discipline on a child may eventually produce a fantastically able technician in line with the best of a field, but geniuses are generally made through self-interested goals; at the core of true genius is one defining characteristic: self-propelled passion.
Billy and Grace Tao are exceptional parents. Instead of marshalling their son’s progression forcibly, it was Tao’s own interest and maturity that informed each incremental step in his education. His father explains:
“Firstly we realised that no matter how advanced a child’s intellectual development, he is not ready for formal schooling until he has reached a certain level of maturity, and it is folly to try to expose him to this type of education before he has reached that stage. This experience has made us monitor Terry’s educational progress very carefully. Certainly, he has been radically accelerated, but we have been careful to ensure, at each stage, that he is both ready and eager to move on, and that we are not exposing him to social experiences which could be harmful.
Secondly, we have become aware that it is not enough for a school to have a fine reputation and even a principal who is perceptive and supportive of gifted education. The teacher who actually works with the gifted student must be a very flexible type of person who can facilitate and guide the gifted child’s development and who will herself model creative thinking and the love of intellectual activity.
Also, and possibly most importantly, we learned that education cannot be the responsibility of the school alone. Probably for most children, but certain for the highly gifted, the educational program should be designed by the teachers and parents working together, sharing their knowledge of the child’s intellectual growth, his social and emotional development, his relationships with family and friends, his particular needs and interests… that is, all the aspects of his cognitive and affective development. This did not happen during Terry’s first school experience but I am convinced that the subsequent success of his academic program from the age of 5 onwards has been largely due to the quality of the relationships my wife and I have had with his teachers and mentors.”
Contrasting this approach to other accelerated prodigies, the Taos seem to have viewed their son as his own person rather than as an extension of themselves. They cultivated an environment of deep caring and unconditional support around the interests of their children, allowing the spark of internal genius to ignite without the repressive force of projected self-expectation. The Davidson Institute’s Marica Gross continues:
“In November of 1983, at the age of 8 years 3 months, Terry informally took the South Australian Matriculation (university entrance) examination in Mathematics 1 and 2 and passed with scores of 90% and 85%, respectively. In February the following year, on the advice of both his primary and secondary teachers, who felt he was emotionally, as well as academically ready, the Taos agreed that he should begin to attend high school full time. He was based in Grade 8 so that he could be with friends with whom he had undertaken some Grade 7 work the year before, and at this level he took English, French, general studies, art, and physical education. Continuing his integration pattern, however, he also studied Grade 12 physics, Grade 11 chemistry, and Grade 10 geography. He also began studying first-year university mathematics, initially by himself and then, after a few months, with help from a professor of mathematics at the nearby Flinders University of South Australia. In September that year he began to attend tutorials in first-year physics at the university, and 2 months later he passed university entrance physics with a score in the upper 90s. In the same month, finding that he had some time on his hands after the matriculation and internal exams, he started Latin at high school.”
Though Tao’s education was governed by his parents and teachers, the trajectory was entirely driven by himself and was aided dramatically by an attention to his emotional and social maturity. In many respects he was actually held back. He was moved into high school at aged 10, but as noted above, he’d nearly aced university entrance exams two years previously (in Australia high school goes up to grade 12). He spent two thirds of his time with grade 11 and 12 students and the remainder attending 1st and 2nd year university maths and physics classes. This was all down to his parents, who felt strongly about not doing anything simply for appearances sake, and only taking steps when it was in their son’s best interests:
“There is no need for him to rush ahead now. If he were to enter full-time now, just for the sake of being the youngest child to graduate, or indeed for the sake of doing anything ‘first,’ that would simply be a stunt. Much more important is the opportunity to consolidate his education, to build a broader base.
If Terry entered university now he would certainly be able to handle the work but he would have little time to indulge in original exploration. Attending part-time, as he is now, he can progress at a more leisurely rate and more emphasis can be placed on creativity, original thinking, and broader knowledge. Later, when he does enter full time, he will have much more time for research or anything else he finds interesting. He may be a few years older when he graduates but he will be much better prepared for the more rigorous graduate and post-doctoral work.”
Sitting among students nearly twice his age, the young Terry Tao became known for his humble and friendly nature, and by all accounts, was universally liked by teachers, mentors and peers alike. This may be his nature, but being as precocious as he was, his personality was undoubtedly benefited by the unwillingness of his parents to treat him any differently to his brothers (and other children of a similar age in ‘regular’ families). Modesty was a virtue in the Tao household; show-boating and arrogance made as much sense as a clown at a librarian convention. He didn’t care about winning prizes or being the best at anything; he just really loved doing maths, and received the perfect balance of encouragement and structure to reach his full potential without ever feeling superior. He knew he was different, but had no value placed on that difference: everyone else was viewed as a human equal. When the 10 year-old Tao was offered a prize for scoring the highest mark ever on the American SAT for a child of his age, he chose a chocolate bar, and when it was handed to him, broke it in half and shared it with his father!
Professional Career
Tao’s work has achieved everything from progressing prime number and infinity theory to advancing MRI scanning technology—rapidly improving the detection rate of tumours and spinal injuries across the globe. Professor of mathematics at Princeton University Charles Fefferman said in an interview:
“Such is Tao’s reputation that mathematicians now compete to interest him in their problems, and he is becoming a kind of Mr Fix-it for frustrated researchers. If you’re stuck on a problem, then one way out is to interest Terence Tao”
The influence of mathematical advancement on society is almost entirely indirect: it usually functions as a basis to the advancement of other sciences, especially physics, so drawing a clear line between Tao and the broader value of his work quickly becomes convoluted by additional theory and speculation. Not to mention, explaining pure mathematics in laymen’s terms is extremely difficult. The concepts being used are comprised of other concepts that themselves require their own multi-conceptual explanations, all of which are already well beyond the learning level of the average person (myself included). What I do understand though, is that mathematics at an advanced level can be a truly beautiful and creative phenomenon, and for many, an emotional one as well.
It’s been said that most people don’t enjoy math because the schooling curriculum gives a vastly incomplete picture of the subject, analogous to an art class only teaching how to paint a single-coloured wall and never showing a Picasso or Rembrandt. For most of us it’s easy to recognise artistic and social talents as we have our own abilities as a point of reference, allowing us to perceive a distance between our own output and that of the great masters. In the case of mathematics it’s usually a case of viewing some kind of alien language. For example, here’s what Tao has been working on most recently:
“I’ve been meaning to return to fluids for some time now, in order to build upon my construction two years ago of a solution to an averaged Navier-Stokes equation that exhibited finite time blowup.
One of the biggest deficiencies with my previous result is the fact that the averaged Navier-Stokes equation does not enjoy any good equation for the vorticity , in contrast to the true Navier-Stokes equations which, when written in vorticity-stream formulation, become
(Throughout this post we will be working in three spatial dimensions .) So one of my main near-term goals in this area is to exhibit an equation resembling Navier-Stokes as much as possible which enjoys a vorticity equation, and for which there is finite time blowup.
Heuristically, this task should be easier for the Euler equations (i.e. the zero viscosity case of Navier-Stokes) than the viscous Navier-Stokes equation, as one expects the viscosity to only make it easier for the solution to stay regular. Indeed, morally speaking, the assertion that finite time blowup solutions of Navier-Stokes exist should be roughly equivalent to the assertion that finite time blowup solutions of Euler exist which are “Type I” in the sense that all Navier-Stokes-critical and Navier-Stokes-subcritical norms of this solution go to infinity…”
I don’t know about you, but I almost need a lay-down after reading that.
It’s my goal over the next 12 months to both increase my own base understanding of mathematics and to source mathematicians capable of providing effective metaphors to better illustrate the work they’re doing for the rest of us. I’ll post more specifically on the subject then, and will potentially revisit this section to give it some greater context.
Closing
It’s no accident that Tao became passionate about mathematics, and it’s not just a matter of encouragement. His parents instilled him with a positive and compassionate outlook and supported him, but it was ultimately the conscious absence of his parents that helped him the most. The common sense fact is, if someone is good at anything, they’re much more inclined towards it over other activities, especially without there being any pressure around their achievements. The brain naturally releases higher dopamine levels when the mind perceives self-accomplishment easily relative to a common standard, which in Tao’s case, came very early when he was teaching children twice his age how to count before turning 3. His aptitude then went on to connect his developing interest to higher-concept (more elegant and interesting) mathematics much sooner than most professionals in the field, thereby giving him an enormous hook. The message for parents here is a clear one: for a child’s potential to be reached, their talent needs guidance without any pressure and expectation.
The choices and direction of Tao’s parents were paramount to his development. They worked tirelessly in the background to create new and nurturing environments for him to grow in, and in terms of his personal experience, they were largely invisible. They recognised the importance of balance in the growth of modest self-confidence, a concept equally important to all avenues of his life—whether it be at school, at home or among friends.
Most importantly, Tao’s parents understood his genius. His father sums it up:
“I have seen too many situations where the parents did the wrong thing. A brilliant mind is not just a cluster of neurons crunching numbers but a deep pool of creativity, originality, experience and imagination. This is the difference between genius and people who are just bright. The genius will look at things, try things, do things, totally unexpectedly. It’s higher-order thinking. Genius is beyond talent. It’s something very original, very hard to fathom.”
Terence Tao is more than a mathematical genius; he’s a role model for human conduct, a rare example of supreme talent and supreme humility existing in side-by-side unison. We may not be able to learn much directly from his work, or even understand the first thing about it, but I think most of us can learn from his outlook on the world: no matter who you are or how good at something you are, be humble, let your work speak for itself, and be a good and genuine person without motive.
If you haven’t seen him before, here’s a brief interview he had on the Colbert Report a couple of years ago. Note his demeanour and the speed of his brain compared to his speech. He’s one of a kind.
If you’re interested in learning more about the ‘Navier-Stokes’ equation or checking out more of his work, Tao runs his own WordPress blog here.
]]>(By the way, I’m @Nebusj on Twitter. I’m happy to pick up new conversational partners even if I never quite feel right starting to chat with someone.)
Schmidt does assume normal, ordinary, six-sided dice for this. You can work out the problem for four- or eight- or twenty- or whatever-sided dice, with most likely a different answer.
But given that, the problem hasn’t quite got an answer right away. Reasonable people could disagree about what it means to say “if you roll a die four times, what is the probability you create a correct proportion?” For example, do you have to put the die result in a particular order? Or can you take the four numbers you get and arrange them any way at all? This is important. If you have the numbers 1, 4, 2, and 2, then obviously 1/4 = 2/2 is false. But rearrange them to 1/2 = 2/4 and you have something true.
We can reason this out. We can work out how many ways there are to throw a die four times, and so how many different outcomes there are. Then we count the number of outcomes that give us a valid proportion. That count divided by the number of possible outcomes is the probability of a successful outcome. It’s getting a correct count of the desired outcomes that’s tricky.
]]>My rating: 4 of 5 stars
The Cyberiad, by Stanislaw Lem, is a collection of science-fiction stories revolving around two inventors named Trurl and Klapaucius. While Trurl and Klapaucius are quite amiable towards one another most of the time, they are also rivals. They create complex pieces of machinery that can perform strange tasks, such as only creating things that begin with the letter N, or writing poetry. The environment in which they exist is a place where humans are not of the norm and where futuristic technology is merged together with medieval concepts of princes, princesses, knights, and dragons. This book contains stories of mystery, treachery, love, and mathematics, with dynamic characters and suspenseful plots. In addition to its scientific aspects, this book is quite philosophical. It deals with the concepts of utopias, society-building, and the quests for happiness and knowledge. People should read this book because of the way it is formatted and because of the unique themes and stories it is comprised of. Anyone that is interested in the science-fiction genre would enjoy The Cyberiad, as well as someone who is looking for a fantasy storybook with an extraterrestrial twist! Since this book is broken up into stories, it can be read over a long period of time in almost any order. Its unique characters and eccentric plotlines should keep the reader interested throughout the duration of the stories.~ Student: Vera G.
]]>If you’re looking for a fun way to teach number sense, skip counting and patterning, and pre-algebraic thinking, or if you simply want something fun to do with your class on Valentine’s Day, have a look.
Click this link to go to TeachersPayTeachers.Com to check it out. When you’re there, click the free preview button to get a sample puzzle and the full Teacher Resource that explains how to incorporate dot to dot puzzles into your math instruction and assessment.
]]>Far too often in response to this question, I hear the answer from children and adults alike, ‘I hate it and I’m not very good at it’. It makes me very sad that such a rich and interesting subject is perceived by many with such negativity.
I believe the key to our children loving maths is to ensure they perceive themselves as capable mathematicians and that they find learning it is fun. If any of us perceive we are good at something or at least feel we are progressing, we engage and therefore make better progress and become more confident at it. I wonder how many adults reading this would answer the above question with, ‘I love maths’ if only they thought they were good at it and had a little confidence in themselves?
At Rye St Antony we promote the importance of a sound knowledge and understanding of mathematical concepts and procedures, encourage a developing interest in mathematical enquiry and the exploration of patterns and relationships, whilst most importantly, trying to ensure all children feel positive about what they are learning and therefore positive about themselves and their ability.
The topics we teach are, of course, in the maths curriculum of every school; this is not what sets us apart. We aim for every child to feel positive towards maths and to achieve their potential, whether that is to grasp the basics to pass a GCSE or to go on to eventually study maths at university and this requires a unique approach.
Firstly at Rye St Antony our core mathematics curriculum is carefully planned and tailored to the needs of our pupils; including many levels of differentiation to ensure that group and individual needs are met. After all, maths is a bit like a game of Jenga, take out a few vital pieces at the bottom (number facts to 10 not secure) and all of a sudden when the tower is added to (adding and subtracting decimals for example) it all becomes rather wobbly (How can you add 4.7+6.5 quickly if you don’t first instantly know 6+4?). If a child is completing work at the right level we can aim to avoid gaps in knowledge and understanding and the child feels they are achieving.
Secondly, we use a variety of multi-sensory teaching techniques and approaches to ensure every child has the best chance of learning irrespective of how they best learn (visual, kinaesthetic or auditory). We use concrete materials to model, and investigations to explore, ICT including My Maths and Mathletics to rehearse key facts (and to complete some homework), outdoor maths to utilise our fabulous school grounds and we make the most of maths cross-curricular links with our Junior Code Club, science, geography and art lessons to name but a few. We promote maths with participation in UK Maths Challenges and our own fortnightly Maths Junior Jottings Challenges.
Thirdly, we aim to make our lessons engaging and enjoyable for our children whilst promoting understanding of how maths works and fits together in the real world. We specialise in maths games, using a variety of materials such as cards or dice; we are creative there is a maths game for everything that is hard to learn or understand! When children are playing games they often don’t even realise the knowledge and understanding they are rehearsing in their enthusiasm to beat their peers or better still the teacher! We use Maths Storytelling strategies (we have a junior specialist drama teacher). What better way to learn measurement and converting between units of measurement than to be working out how to rescue the princess from her tower at 3.5m with only a centimetre ruler available?
Fourthly, and I feel crucially, the children are encouraged to feed back to the teacher how they feel about what they are doing. Our small classes ensure there is on-going communication between child and teacher throughout the lesson, continuing afterwards when required; children who know they are supported and helped when they are stuck feel confident to have another go and as we say at Rye St Antony, ‘Being stuck is honourable’. How else will you make progress if you don’t get through the stuck?!
We have learnt that by tailoring our curriculum to the needs of our children and making it engaging and fun we can change, ‘I hate maths and I can’t do it’ to ‘I like maths and I’m quite good at it now!’ I believe every child deserves that chance.
Catherine Eadle
Rye St Antony Prep School Mathematics Specialist
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We began the week by getting to know each other and sharing information about ourselves. We did a scavenger hunt, played some ‘getting to know you’ games and made an ‘All About Me’ hot air balloon. Here are some of them:
We also refreshed our memories on Place Value. We explored how numbers can be represented in different ways – as digits, as a picture, in written form, and in expanded form. We watched a video on expanded form that you can view below:
We also played many games that helped to consolidate our knowledge on Place Value.
As Valentine’s Day is this Sunday, we talked about things that we love. We made these Valentine’s Day heart mobiles. I think they look great!
Check back soon to see what else we’ve been up to!
Mathematics owes a huge debt to the extraordinary contributions given by Indian mathematicians over many hundreds of years, however there has been a reluctance to recognise this.
Vedic Period (between 1500 BC and 800 BC)
The earliest expression of mathematical understanding is linked with the origin of Hinduism as mathematics forms an important part of the Sulbasutras (appendices of the Vedas – the original Hindu scriptures). They contained geometrical knowledge showing a development in mathematics, although it was purely for practical religious purposes. Additionally, there is evidence of the use of arithmetic operations including square, cubes and roots.
The Sulbasutras were composed by Baudhayana (around 800 BC), Manava (about 750 BC). Apastamba (about 600 BC) and Katyayana (about 200 BC).
Before the end of this period – around the middle of the 3rd century BC – the Brahmi numerals began to appear. Indian mathematicians refined and perfected the numeral system, particularly with the representation of numerals and, thanks to its dissemination by medieval Arabic mathematicians, they developed into the numerals we use today.
Jaina Mathematics
Jainism was a religion and philosophy which was founded in India around the 6th century BC. The main topics of Jaina mathematics in around 150 BC were the theory of numbers, arithmetical operations, operations with fractions, simple equations, cubic equations, quartic equations, and permutations and combinations.
Furthermore, Jaina mathematicians, such as Yativrsabha, recognised five different types of infinities: infinity in one direction, two directions, in area, infinite everywhere and perpetually infinite.
Astronomy
Mathematical advances were often driven by the study of astronomy as it was the science at that time that required accurate information about the planets and other heavenly bodies.
Yavanesvara (2nd century AD) is credited with translating a Greek astrology text dating from 120 BC. In doing so, he adapted the text to make it work into Indian culture using Hindu images with the Indian caste system integrated into his text, thus popularising astrology in India.
Aryabhata was also an important mathematician. His work was a summary of Jaina mathematics as well as the beginning of the new era for astronomy and mathematics. He headed a research centre for mathematics and astronomy where he set the agenda for research in these areas for many centuries to come.
Brahmagupta (beginning of 7th century AD)
Brahmagupta made major contributions to the development of the numbers systems with his remarkable contributions on negative numbers and zero. The use of zero as a number which could be used in calculations and mathematical investigations, would revolutionise mathematics. He established the mathematical rules for using the number zero (except for the division by zero) as well as establishing negative numbers and the rules for dealing with them, another huge conceptual leap which had profound consequences for future mathematics.
As well as this, he established the formula for the sum of the squares of the first n natural numbers as and the sum of the cubes of the first n natural number as .
He even wrote down his concepts using the initials of the names of colours to represent unknowns in his equations. This one of the earliest intimations of what we now know as algebra.
Additionally, he worked on solutions to general linear equations, quadratic equations and even considered systems of simultaneous equations and solving quadratic equations with two unknowns, something which was not even considered in the West until a thousand years later, when Fermat was considering similar problems in 1657. Furthermore, he dedicated a substantial portion of his work to geometry. His biggest achievements in this area was the formula for the area of a cyclic quadrilateral, now known as Brahmagupta’s Formula, as well as a celebrated theorem on the diagonals of a cyclic quadrilateral, usually referred to as Brahmagupta’s Theorem.
‘Golden Age’ (from 5th to 12th centuries)
In this period, the fundamental advances were made in the theory of trigonometry. They utilised sine, cosine and tangent functions to survey the land around them, navigate seas and chart the skies. For example, Indian astronomers used trigonometry to calculate the relative distances between the Earth and the Moon and the Earth and the Sun. They realised that when the Moon is half full and directly opposite the Sun, then the Sun, Moon and Earth form a right angled triangle. By accurately measuring the angle as ^{1}⁄_{7}°, using their sine tables that gave a ratio of the sides of such triangle as 400:1, it shows that the Sun is 400 times further away from the Earth than the Moon.
Bhaskara II lived in the 12th century and is considered one of the most accomplished of India’s mathematicians. He is credited with explaining that the division by zero – a perviously misunderstood calculation – yielded infinity.
He also made important contributions to many different areas of mathematics including solutions of quadratic, cubic and quartic equations, solutions of Diophantine equations of the second order, mathematical analysis and spherical trigonometry. Some of his discoveries predate similar ones made in Europe by several centuries, and he made important contributions in terms of the systemisation of knowledge and improved methods for known solutions.
This Kerala School of Astronomy and Mathematics school was founded late 14th century by Madhava of Sangamagrama. Madhava also developed an infinite series approximation for π. He did this by realising that by successively adding and subtracting different odd number fractions to infinity, he could establish an exact formula for π, a conclusion that was made by Leibniz in Europe two centuries later. Applying this series, Madhava obtained a value for π correct to 13 decimal places! Using this mathematics he went on to obtain infinite series expressions for sine, cosine, tangent and arctangent. Arguably more remarkable though was the fact that he gave estimates of the correction term, implying that he had an understanding of the limit nature of the infinite series.
In addition, he made contributions to geometry and algebra and laid the foundations for later development of calculus and analysis, such as the differentiation and integration for simple functions. It is argued that these may have been transmitted to Europe via Jesuit missionaries, making it possible that the later European development of calculus was influenced by his work to some extent.
In astronomy, Madhava discovered a procedure to determine the positions of the Moon every 36 minutes and methods to estimate the motions of the planets.
I have only included some of the earlier Indian mathematicians, missing out magnificent mathematicians such as Ramanujan, as I feel that these are the most forgotten. To find out more about other Indian mathematicians, this may be a good starting point.
Hope you enjoyed this post; I was thinking of doing Chinese mathematicians next. Let me know what you think! M x
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I have divided this article in short numbered thoughts in Ludwig Wittgenstein’s manner. As an epilogue there’s a bit my own philosophy about forms.
(1) Greek philosopher Plato considered geometry and number as the most reduced and essential and therefore the ideal philosophical language.
But it is only by virtue of functioning at a certain ‘level’ of reality that geometry and number can become a vehicle for philosophic contemplation.
For Plato reality consisted pure essences or archetypal ideas of which the ideas we perceive are only pale reflections. The Greek word “idea” is also translated as form. These ideas cannot be perceived with senses, but with pure reason alone. This is where geometry steps into picture.
© Rafael Laguillo | Dreamstime Stock Photos
(2) Geometry as contemplative practice is personified by elegant, refined woman, for geometry functions as an intuitive, synthesizing, creative yet exact activity of mind associated with feminine principle. But when these geometric laws come to be applied in the technology of daily life they are represented by the rational, masculine principle: contemplative geometry is transformed into practical geometry.
(3) Angle specified the of celestial earthly events.Today the science verifies that the angular position of moon and planets does affect the electromagnetic and cosmic radiation which impact with the Earth and in turn these these field fluctuations affect many biological processes.
Also the word “angle” has same root as “angel”.
In ancient trigonometry an angle is relationship with two whole numbers. This gets us into musical scale system.
(4) Geometry deals with pure form and philosophical geometry re-enacts unfolding of each form out of preceding one. It is a way which the creative essential mystery is rendered visible.
For the end from the book Thomas Taylor’s thought: “All mathematical forms have a primary subsistence in the soul; so that prior to the sensible she contains self-motive numbers; vital figures prior to such as are apparent; harmonic ratios prior to things harmonized; and invisible circles prior to the bodies that are moved in a circle.”
Epilogue
I have come up with some kind of form philosophy myself too. Why do people produce certain kind of forms? Especially the circle form can be found directly in many instances. Typically a glass for drinking has a circle especially at the top of it.
Perhaps it’s easier to drink from this kind of glass. Or are we forced to create these kind of forms because of our “sacred geometry origin”? :-)
My actual own form philosophy is metaphysical by its nature, where also mental things have geometrical form, that are in interaction with concrete forms creating values together, that have their own forms.
For example it’s not insignificant that from what kind of of “glass” one drinks beer. It is disrespect of beer if one doesn’t drink beer from a pint that retell the mental form of the beer. It’s also of course question of what kind of beer one drinks. Some expensive beers come in a bottle that respect the form of the beer as such, at least almost.
I also have analogical thoughts about wine: It is wrong to drink red wine from a white wine glass. Red wine has its own mental form that the wine glass must retell.
Image courtesy of Madrolli at FreeDigitalPhotos.net
Well, I have probably written too much about my own form philosophy already, but with cosmetics it gets particularly interesting.
The essence is to see the mental being of things and respect that with right kind of concept by paying attention to right kind of combination of make up, jewelry and clothes and so on. The essence is the respect of the whole, all the mental dimensions of it. Respect in very wide concept. Including self respect.
For end observation let me state that for example in the buildings for different kind of forms were payed a lot more attention in the past that nowadays. At least that is how I see it in Finland.
Image courtesy of Tuomas_Lehtinen at FreeDigitalPhotos.net
For example the residential districts outside the centers of the towns that are built in the 1970s are especially boring in Finland…
Some relevant links:
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I teach mathematics to new immigrants to the United States. Though the students are of high school age, some of them had not had any formal schooling in their own country. Yesterday, one student, a West African girl to whom I have been teaching basic arithmetic, was struggling with her twos and fives multiplication tables. We worked for a while with only some success, and then she turned to me brightly and said, “Ask me 379 times 13.” I was a little skeptical, but I wrote it out on the paper in front of us, the 13 beneath the 379. She looked at the example quizzically, and said, “Are you sure that’s how you write it?” So I wrote it across in a line, and she became much happier. “Ahh,” she said, “that’s 4,927.”
I thought about it for a few moments and realized she was correct. I was stunned. A minute ago she was having trouble with five times six; now she was answering this difficult multiplication problem seemingly in her head. I asked her how she knew.
She smiled at me and said, “I saw it in a movie in my country. In the movie, there was a school, and the teacher in the classroom put that example on the board. I have been waiting for someone to ask me 379 x 13 ever since!”
]]>The Thrilling Adventures of Lovelace and Babbage: The (Mostly) True Story of the First Computer by Sydney Padua
Synopsis: In The Thrilling Adventures of Lovelace and Babbage Sydney Padua transforms one of the most compelling scientific collaborations into a hilarious set of adventures.
Meet two of Victorian London’s greatest geniuses… Ada Lovelace, daughter of Lord Byron: mathematician, gambler, and proto-programmer, whose writings contained the first ever appearance of general computing theory, a hundred years before an actual computer was built. And Charles Babbage, eccentric inventor of the Difference Engine, an enormous clockwork calculating machine that would have been the first computer, if he had ever finished it.
But what if things had been different? The Thrilling Adventures of Lovelace and Babbage presents a delightful alternate reality in which Lovelace and Babbage do build the Difference Engine and use it to create runaway economic models, battle the scourge of spelling errors, explore the wider realms of mathematics and, of course, fight crime – for the sake of both London and science. Extremely funny and utterly unusual, The Thrilling Adventures of Lovelace and Babbage comes complete with historical curiosities, extensive footnotes and never-before-seen diagrams of Babbage’s mechanical, steam-powered computer. And ray guns.
2015 WINNER OF THE NEUMANN PRIZE IN THE HISTORY OF MATHEMATICS
Published: April 2015 | ISBN-13: 978-0141981512
Author’s Homepage: http://sydneypadua.com
Author’s Twitter: https://twitter.com/sydneypadua
Publishers Weekly Book Review
Science Museum Group Journal Book Review
Confessions of a Science Librarian Book Review
Kirkus Reviews Book Review
[Image Credit: http://www.themarysue.com/wp-content/uploads/2015/04/BookCover-788×1024.jpg ]
]]>In my school, dismissal is a long process with a lot of down time. If I don’t keep my students occupied and focused, behavior gets unruly. It was in this specific situation where I developed Mental Math.
Mental math is easy to implement and simple for students to learn. The teacher slowly provides students an extended math problem with multiple steps. They need to listen carefully to hear the next step in the problem. There is only one simple rule to Mental Math. Students cannot raise their hand to answer the problem until the teacher does! Once the teacher is done giving out the problem, the teacher raises his/her hand. At that point, students can raise their hands to volunteer to share their final answers. Here are some examples, and remember, a teacher would say these slowly, providing time for students to calculate in their head.
For lower elementary: 3+3+4-5= (teacher raises her hand)
For upper elementary: 6×7+8×4-50= (teacher raises her hand)
What’s great about mental math is that it can be done with any grade and any level. Incorporating square roots, decimals, fractions, and negative numbers are all great ways to adapt the game for stronger math students. Making sequences longer and shorter are other adaptations. The hardest part of Mental Math is that the teacher must keep track of the correct answer, too! Prior planning of math sequences can eliminate this challenge, of course.
Give Mental Math a try today during Brain Breaks, transitions, dismissal, or before or after lunch to manage transitions and to make every minute count!
Dr. Pam
TAKE 5! from DeeperDive Learning, Inc.
Dr. Pamela Bruening, is one of the nation’s preeminent experts in RTI, PBIS, MTSS, and behavior initiatives and has incredible experience as an educator, administrator, strategist, consultant, human resource developer, and content curator. She is also the CEO of DeeperDive Learning, Inc.
@PamelaBruening | pam.bruening@gmail.com | LinkedIn
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