During the summer, I have once again ordered my usual array of new season rewards. With a surname like Thomas ,

my rewards often have a certain tank engine pictured or since #grafdad has collected all the #mathsbadges. My most recent cards feature a certain giraffe, resplendent in his badges. This morning, my most recent reward cards and some more magnets arrived – always my favourite reward, Every year I order lots. They never lose their magic!I am a huge fan of stickers and stampers , using such sites as

http://www.primaryteaching.co.uk/allproducts/maths-and-numeracy

Amongst several others!

Despite having an outstanding range of stickers from personalised ones to sparkly ones, from smelly ones to emoji’s

I have always tended to prefer a reward card.

Back in the day when Vistaprint offered 10 free products and then 4 free products, at least once a month I ordered lots of rewards.

I could order cards relevant to our Scheme of Work. Because I was a little bit obsessed, I would order reward cards , reminder cards , mymaths.co.uk login details and ones with spaces to write individual mymaths.co.uk codes. I ordered reward mugs for my “geek of the term” I ordered cards for my “geek of the week” I ordered mugs for those who, achieved top marks in Exams. I had a full filing cabinet draw just for reward cards.

However, this is not a blog about what I used to order from Vistaprint.co.uk. It is about rewards. I prefer the reward card and reward magnets. Magnets are earned for something special, nothing specific but they have to astound me.

My range of reward cards have become quite a good earner for KS3 students. Most grandparents are impressed and augment the simple reward with financial recompense.

My year 7 usually have a weekly discussion over Edmodo about their income for that week. I have several different varieties of reward card from fabulous homework to wonderful class work, brilliant numeracy and I’m proud of what you achieved in your test.

Every reward card I give out usually evokes a comment on the corridor from the students form tutor as the students are so proud of them they have to go and show their school tutors.

The magic of the reward card is also the fact that you can give those reward shy, praise shy students the cards secretively.

I know students like the immediate reward of chocolate or sweets, but I have found over the years that the reward card is much better.

Ks4 students refer to them as get out of grounding cards and find that their parents become far more lenient when a reward card is produced. I have taken to signing and dating the cards as some parents were becoming suspicious. My KS3 students save their cards for when they see their grand parents whitest my KS4 students seem to produce a significant number of cards on Friday evenings.

Todays arrival made me think about what other rewards teachers offer.

I found this fascinating article that has really made me think

Don’t be misled by the title, the opening paragraph hooked me into reading the whole article

Test performance can improve dramatically if students are offered rewards just before they are given standardised tests and if they receive the incentive afterward, new research at the University of Chicago shows.

I’ve never thought of a pre test, test as a reason to reward success and encourage academic achievement. I have now!

Another article I found relates to the effect of trips and financial rewards. I was surprised at the findings in the guardian.?

]]>This week felt very chaotic. I’m finding that having Mondays as our meeting days can make a week feel that way, because there is stuff coming in that you feel needs to be addressed, and yet, paying attention in meetings is important. It’s also a good way to make sure everyone is on the same page at the start of the week. I think I’ll be able to reconcile that as I get used to it, I just wanted to throw that out there.

Tuesday morning I spent 4 hours in a training for a new fluency program that our district has adopted. It was a “train the trainer” type of training, so the majority of people in it were site principals and site coaches. Since this is our second week of school, we are working with a new math adoption, and materials organization is a huge issue right now, you can probably guess what was happening for me this entire time. I was getting emails and notes passed to me about all the other things occurring.

I’m so glad that I was a mother before I did this job, because I am able to “key in” and listen to more than one thing at a time. I heard a couple of very important things in this training that caught my attention and made me sit up and focus. One, is that this program is set to push kids grades 2-8 to learn their basic operations at a rate of 0.8 seconds. As an elementary math specialist, this concerned me. One other teacher in the room caught this also and expressed concern. I was glad to hear this, because I thought maybe I was being the “helicopter parent” for a moment. My background is mainly high school, but I’ve raised enough kids and grandkids to know that this seems like a great opportunity to create the kind of math anxiety that Jo Boaler discusses.

The next day I ran across The Recovering Traditionalist, and her blog confirmed my thinking. I have made an appointment to talk to the Director of Interventions in our district to discuss my concerns. I understand that fluency with operations is important, and I think that this could be a great way to encourage kids to become fluent, I’m concerned about pushing the speed so hard in grades 2-5. I’d love any feedback from others out there to help me with this. I want to do what’s best for our kids.

I also spent 6 hours this week moving, organizing, cataloging and inventorying boxes of elementary math books. I actually had to do it twice, due to an unfortunate incident involving the moving of all the boxes I had originally organized. I decided after doing this that I needed to take a little more control of my time and resources, and I have scheduled several site visits for next week to help me get focused on my main objectives for the year; supporting sites, teachers and kids in good mathematics learning.

]]>A mermaid: my answer to the persistently asked question since I turned 5. 11 years later, the same question is asked, ‘What do you want to be?’, but having realised subaqueous, mythological existence is merely a fantasy, my response is: ‘I don’t know.’

Adolescents are being suffocated with questions relating to their future income, rather than being encouraged to study what drives them physically, mentally and emotionally. So why is pressure being utilised to ensure that students study what the rigorous politicians desire, rather than motivation to encourage them to do well?

Nicky Morgan, Secretary of State for Education, recently made a comparison at the launch of ‘Your Life’ campaign, claiming ‘even a decade ago’ sciences and maths were the subjects young people studied if they wanted ‘to go into a mathematical or scientific career.’ Morgan contended that this could not be further from the truth by employing an emotive yet clichéd metaphor insinuating that STEM subjects (Science, Technology, Engineering, Mathematics) could ‘unlock doors to all sorts of careers’ and are ‘the keys to the most cutting edge, fast- paced areas of work.’

Likewise, her use of incentivising economic statistics; alarmist statements about life choices and address to the alleged feminist agenda she is behind, contain hints of manipulation for teens to fill up the jobs that the ‘legal sector is crying out for’.

However, in her recent article (Telegraph 05/12/14), Prof Anne Carlisle, vice- chancellor of Falmouth University, provided a strong and explicit counter argument towards Nicky Morgan’s statement that ‘schoolchildren who focus exclusively on arts and humanities risk restricting their future career path’ and claimed it was ‘a myth [she] wants to combat.’ Carlisle uses a hyperbolised assertion that there is widespread lack of employment for academically based jobs ‘up and down the country’ whilst also conjecturing that creative industries are ‘expected to provide 1.39m jobs’ in the UK in the next four years, in order to recruit people to study creative subjects. Unfortunately, this theory is difficult to concur with due to the fact that it is only ‘4% of total UK employment’: a minor statistic.

‘[Science] has a practical importance to which art cannot aspire’, said Bertrand Russell: a view Nicky Morgan surely shares wholeheartedly. So what is the reason Nicky Morgan values sciences more than the arts? Morgan supported her argument that studying STEM subjects provide the foundations for a more economically advantageous lifestyle with the statistics showing ‘pupils who study maths to A level will earn 10% more over their lifetime’ and will benefit the UK’s economy as a whole.

Not only am I convinced that Nicky Morgan is wrong in her belief that the UK is relying on the subject choice of 15 year olds, but I furthermore believe that there is evidence to suggest the contrary. I am not in any way undermining the skills gained through sciences and maths, however, they do not provide the highly regarded literacy and creativity skills that the arts offer.

In contribution to my belief, Richard Sigurdson, Professor of Political Science and Faculty of the Arts at the University of Calgary, in his online article about studying Liberal Arts, stated studiers of the arts have a higher ability to understand the fluctuating world of employment and consequently making them ‘more likely to be high income earners.’

In addition, Nicky Morgan extended her argument by stating young women are being told ‘that certain subjects are the preserve of men’ and thus wants to ‘eliminate the gender pay gap.’ Morgan’s facade of supporting and encouraging young women is simply contradictory, as she is manipulating the vulnerable minds of girls in order to persuade them to study subjects that will allegedly ‘boost the economy.’ There has been a highly significant growth of women’s rights over the centuries, and it is rather ironic that now that we have the freedom to make our own choices, educational representatives such as Nicky Morgan, are giving young women the burden of the nation’s dependance to steer them in the direction that some may be uninterested in.

Sir Ken Robinson, International Advisor on Creativity and Education, refutes Morgan’s contentions in his article for the ‘Times Educational Supplement’ (23/01/15). He suggests art is vital to fundamental survival but is fragile, through the use of the powerful image: ‘The beating heart of human life (the arts).’ Robinson questions why the arts should have to be justified to society, as he believes that arts are not the ‘leisure activities’ that policymakers portray them as, but as ‘eloquent expressions of human intelligence, imagination and creativity.’

He wants to ensure that ‘cultural learning’ is a ‘priority in [children’s] education’, by submitting his theory that it is part of the four purposes in education: ‘economic, social and personal’ and proving through the Cultural Learning Alliance that ‘cultural learning embraces all of these and can support achievement in all disciplines.’ Furthermore, Robinson is not only convincing, but also encouraging as he uses a collective pronoun whilst asserting, ‘None of these purposes can be met if we forget that education is about living people…all education is personal’ and refers to his primary statement that ‘the benefits will not just be felt by a quarter of the population, but by all of us.’

Although I do believe that arts are a peaceful source of human existence, Ken Robinson may have slightly hyperbolised the fact that studying the arts promotes world peace when stating ‘problems of apathy and social disengagement exist among young people’ and that schools will not reconcile these by ‘running theoretical courses on civics’. However, I believe the preconceived ideas of arts as hobbies, and even sciences being the ‘stuffy, boring subjects’ as Nicky Morgan stated, must be erased and instead we should teach people the true meanings of both.

Art: the expression of application of human creative skill and imagination; producing work to be appreciated. Science: the branch of knowledge; intellectual and practical activity encompassing the systematic study of the physical and natural world. Overall, human existence relies on both arts and sciences, and as Albert Einstein once said: they are ‘branches of the same tree.’ Rather than polarising the two, we must encourage them both equally through the motivation, inspiration and imagination of the world’s growing education and get young adults to change their responses to the persistently asked question: ‘What do you want to be?’ to ‘I want to be what makes me happy.’

What are your thoughts?

E x

**DISCLAIMER*: This piece was written for my GCSE coursework in 2015*

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The science museum is home to generations of Physics, Biology and Chemistry, but also Maths and SURPRISINGLY Art! Walking through the mathematics section in the museum allowed me to see not only instruments used within maths, but the part that has been and can be produced from it. In my next few posts, you will be seeing what kinds of visual pieces I saw during my visit. ENJOY!

]]>Radhanath Sikdar, 1852.

(Sikdar)

In light of my earlier post on the first men to climb Mount Everest, it would be wrong to not gloat in the slightest about the very young, very talented Bengali mathematical genius who helped to survey Peak XV. Born in Jorasanko in poverty, merit scholarships allowed him to enrol in today’s Presidency College in Kolkata. He was a Brahmin but the influence of Derozio and the Young Bengal Movement made him a believer in the efficacy of beef, boxing and beer. He was one of the first Indians to study Newton’s Principia in Latin with the help of Professor John Tytler of the Mathematics Dept. In addition to Mathematics he was fluent in Sanskrit and English and near fluent in Latin and Greek. He was only 19 when he joined the Great Trigonometric Survey of India in 1831 as a computor for a salary of Rs. 40 a month. He was already considered a genius by this time and had invented various geometrical proofs.

The Survey began by plotting triangles across India, starting at St. Thomas Peak near Chennai with a base line of 7.5 miles. It was supposed to take five years and took 60 instead. Both simple mathematical procedures such as trigonometry and complex spherical geometry to account for the earth’s curved surface were used. One of the instruments used was a theodolite that weighed a thousand pounds. A 700 strong team of men travelled from south to north measuring various land forms. In some places they had to build temporary wood or brick towers to get a clear line of sight for the theodolite, high above the trees. Their main battles were against the heat, diseases such as malaria and tigers.

After surveyors Nicholson and Hennessy had failed at working out the height of Peak XV, Sikdar began working on the problem in Dehradun. He found that they had forgotten to consider refraction or the bending of light when it passed through hot air.

But sadly, when the height of Peak XV was announced to the world, the peak was named after George Everest, Sikdar’s first boss. The head of the GTS in 1856, Andrew Waugh decided not to use the mountain’s Tibetan name Chomolungma for fear of political trouble for the British. Everest had never had anything to do with working out the height of the mountain named after him. Sikdar had done most of the work but received none of the glory.

There is much more information about Radhanath Sikdar at the following link:

http://www.scienceandculture-isna.org/may-june-2014/05%20Art_Radhanath%20Sikdar%20First%20Scientist…by_Utpal%20Mukhopadhyay_Pg.142.pdf

via:http://www.freevectorpic.com/freephotos/free-stock-photos-object-children-school-supply-mathematics.html ]]>

As it says on the main page, my name is Neil Shenvi; I am currently a research scientist with Prof. Weitao Yang at Duke University in the Department of Chemistry. I was born in Santa Cruz, California, but grew up in Wilmington, Delaware. I attended Princeton University as an undergraduate where I worked on high-dimensional function approximation with Professor Herschel Rabitz. I became a Christian in Berkeley, CA where I did my PhD in Theoretical Chemistry at UC – Berkeley with Professor Birgitta Whaley. The subject of my PhD dissertation was quantum computation, including topics in quantum random walks, cavity quantum electrodynamics, spin physics, and the N-representability problem. From 2005-2010, I worked as a postdoctoral associate with Prof. John Tully at Yale where I did research into nonadiabatic dynamics, electron transfer, and surface science.

Outline slide: (Download the Powerpoint slides here)

Lecture:

Summary:

- Science is often considered to be in opposition to religion, because it answers all the questions that religion asks
- Thesis: 1) Science and religion are compatible, 2) Science provides us with good reasons to believe that God exists
- Definition: what is science?
- Definition: what is the scientific method?
- Definition: what is religion?
- Where is the conflict between science and religion, according to atheists?
- Conflict 1: Definitional – faith is belief without evidence
- But the Bible doesn’t define faith as “belief without evidence”
- Conflict 2: Metaphysical – science presuppose naturalism (nature is all that exists)
- First, naturalism is a philosophical assumption, not something that is scientifically tested or proved
- Second, methodological naturalism in science doesn’t require us to believe in metaphysical naturalism
- Conflict 3: Epistemological – science is the only way to know truth (scientism)
- But scientism cannot itself be discovered by science – the statement is self-refuting
- Conflict 4: Evolutionary – evolution explains the origin of life, so no need for God
- Theists accept that organisms change over time, and that there is limited common descent
- But the conflict is really over the mechanism that supposedly drives evolutionary change
- There are philosophical and evidential reasons to doubt the effectiveness of mutation and selection
- Evidence for God 1: the applicability of mathematics to the natural world, and our ability to study the natural world
- Evidence for God 2: the origin of the universe
- Evidence for God 3: the fine-tuning of the initial constants and quantities
- Evidence for God 4: the implications of quantum mechanics
- Evidence for God 5: the grounding of the philosophical foundations of the scientific enterprise
- Hiddenness of God: why isn’t the evidence of God from science more abundant and more clear?
- Science is not the only means for getting at truth
- Science is not the best way to reach all the different kinds of people
- There is an even deeper problem that causes people to not accept Christianity than lack of evidence
- The deeper problem is the emotional problem: we want to reject God’s claim on our lives

He concludes with an explanation of the gospel, which is kinda cool, coming from an academic scientist.

I am a big admirer of Dr. Neil Shenvi. I wish we could clone him and have dozens, or even hundreds, like him (with different scientific specializations, of course!). I hope you guys are doing everything you can to lead and support our young people, and encouraging them to set their sights high and aim for the stars.

UPDATE: Dr. Shenvi has posted a text version of the lecture.

**Related posts**

- Neil Shenvi lectures on the evidence for the resurrection of Jesus
- Neil Shenvi gives an overview of quantum mechanics
- Neil Shenvi lectures on the meaning and purpose of life, the universe and everything
- Neil Shenvi offers responses to popular pro-choice arguments
- Neil Shenvi: do objective moral values really exist? Is moral relativism true?

Let me set the scene. A fair few years ago, during my GTP year, (I was training to teach maths and ICT) The worst thing that could possibly happen, did.

No students were fighting , the students were on task, there was no bad behaviour , there was no swearing….

My maths observation had gone really well, only one slight incident prior to my lesson where someone had thrown a chair at another student… (It missed 😊)

BUT during my second lesson, tragedy struck! I was about to introduce excel to a group of bottom set, mostly SEND students.

So I’d logged on prior to the lesson starting. Everything was set up. The monitor just needed turning on.

The scene is set, almost like a modern day Shakespearean tragedy.

The students were all eagerly waiting , mouse and keyboards poised. Nothing happened, absolutely nothing!

I later discovered that a well-known utilities company were working on the main road outside the school, and somehow the school telephone lines had been cut.

Oh hum, I’ve got 20 students ready to be wowed and excited with knowledge about Excel. My worksheets and my exemplars were all computer based. My lesson plan was computer based and had been uploaded from home onto the school server. OOPS!

As a former lecturer, I went into my default panic mode, I immediately reverted into lecturer mode. We gathered out data and input our data onto a hard copy.

As a lecturer, I was used to not relying on technology.

During my GTP, I had become incredibly dependant on technology. At that time, I did consider myself to be quite the technology whizz kid. The fact that I could have lessons so completely designed and timed was wonderful. I could make area come alive with cm squares. I had numbers flying from one side of the equals sign to the other doing cartwheels whitest changing signs!

Following this incident, I made myself do all my lessons without using technology for two whole weeks.

Having said this, I am a huge fan of technology and think all teachers should embrace what is available. There are some incredible programs and some wonderful resources that we all share. There are so many incredibly selfless colleagues out there who share so much wonderful practice. Technology can enhance the learning experience for our learners and educators. The internet is full of articles extolling the virtues of how technology enhances learning.

I think a number of readers of this blog will have been in the situation where technology has failed. My husband, who teaches vulnerable adults in non traditional settings often has no access to technology and suggests that all educators should make themselves teach without technology once a month to ensure that we maintain all our teaching skills!

]]>You will get stuck. Much of mathematics is learning a series series of arguments. They won’t all make sense, at least not at first. The arguments are almost certainly correct. If you’re reading something from a textbook, especially a textbook with a name like “Introductory” and that’s got into its seventh edition, the arguments can be counted on. (On the cutting edge of new mathematical discovery arguments might yet be uncertain.) But just because the arguments are right doesn’t mean you’ll see why they’re right, or even how they work at all.

So is it all right, if you’re stuck on a point, to just accept that this is something you don’t get, and move on, maybe coming back later?

Some will say no. Charles Dodgson — Lewis Carroll — took a rather hard line on this, insisting that one must study the argument until it makes sense. There are good reasons for this attitude. One is that while mathematics is made up of lots of arguments, it’s also made up of lots of very similar arguments. If you don’t understand the proof for (say) Green’s Theorem, it’s rather likely you won’t understand Stokes’s Theorem. And that’s coming in a couple of pages. Nor will you get a number of other theorems built on similar setups and using similar arguments. If you want to progress you have to get this.

Another strong argument is that much of mathematics is cumulative. Green’s Theorem is used as a building block to many other theorems. If you haven’t got an understanding of why that theorem works, then you probably also don’t have a clear idea why its follow-up theorems work. Before long the entire chapter is an indistinct mass of the not-quite-understood.

I’m less hard-line about this. I’m sure that shocks everyone who has never heard me express an opinion on anything, ever. But I have to judge the way I learn stuff to be the best possible way to learn stuff. And that includes, after a certain while of beating my head against the wall, moving on and coming back around later.

Why do I think that’s justified? Well, for one, because I’m not in school anymore. What mathematics I learn is because I find it beautiful or fun, and if I’m making myself miserable then I’m missing the point. This is a good attitude when all mathematics is recreational. It’s not so applicable when the exam is Monday, 9:50 am.

But sometimes it’s easier to understand something when you have experience using it. A simple statement of Green’s Theorem can make it sound too intimidating to be useful. When you see it in use, the “why” and “how” can be clearer. The motivation for the theorem can be compelling. The slightly grim joke we shared as majors was that we never really understood a course until we took its successor. This had dire implications for understanding what we would take senior year.

What about the cumulative nature of mathematical knowledge? That’s so and it’s not disputable. But it seems to me possible to accept “this statement is true, even if I’m not quite sure why” on the way to something that requires it. We always have to depend on things that are true that we can’t quite justify. I don’t even mean the axioms or the assumptions going into a theorem. I’m not sure how to characterize the kind of thing I mean.

I can give examples, though. When I was learning simple harmonic motion, the study of pendulums, I was hung up on a particular point. In describing how the pendulum swings, there’s a point where we substitute the sine of the angle of the pendulum for the measure of the angle of the pendulum. If the angle is small enough these numbers are just about the same. But … why? What justifies going from the exact sine of the angle to the approximation of the angle? Why then and not somewhere else? How do you know to do it there and not somewhere else?

I couldn’t get satisfying answers as a student. If I had refused to move on until I understood the process? Well, I might have earlier had an understanding that these sorts of approximations defy rigor. They’re about judgement, when to approximate and when to not. And they come from experience. You learn that approximating *this* will give you a solvable interesting problem. But approximating *that* leaves you too simple a problem to be worth studying. But I would have been quite delayed in understanding simple harmonic motion, which is at least as important. Maybe more important if you’re studying physics problems. There have to be priorities.

Is that right, though? I did get to what I thought was more important at the time. But the making of approximations is important, and I didn’t really learn it then. I’d accepted that we would do this and move on, and I did fill in that gap later. But it is so easy to never get back to the gap.

There’s hope if you’re studying something well-developed. By “well-developed” I mean something like “there are several good textbooks someone teaching this might choose from”. If a subject gets several good textbooks it usually has several independent proofs of anything interesting. If you’re stuck on one point, you usually can find it explained by a different chain of reasoning.

Sometimes even just a different author will help. I survived Introduction to Real Analysis (the study of why calculus works) by accepting that I just didn’t speak the textbook’s language. I borrowed an intro to real analysis textbook that was written in French. I don’t speak or really read French, though I had a couple years of it in middle and high school. But the straightforward grammar of mathematical French, and the common vocabulary, meant I was able to work through at least the harder things to understand. Of course, the difference might have been that I had to slowly consider every sentence to turn it from French text to English reading.

Probably there can’t be a universally right answer. We learn by different methods, for different goals, at different times. Whether it’s all right to skip the difficult part and come back later will depend. But I’d like to know what other people think, and more, what they do.

]]>“Trains rumble and clank and rush past me. Which is the right one? It is easy to get lost underground.” I’ve been learning to code for 10 months, but it’s all been online. Between finally allowing myself time to mourn a loss (my love of teaching), and working my butt off on a computer, I became a hermit. Seriously, there were days at a time that I did not leave the house, didn’t talk to anyone but my dear Amy, and didn’t even get out of my pajamas. Months went past without me driving myself anywhere. When I did venture out, I had a hard time interacting with people. I couldn’t easily make eye contact. It was rough. I knew I was learning things I needed to make a next step toward my new career, but the direction was unclear. Something else had to change. I had to keep moving forward in a different way. Baby steps were not going to be enough. That phrase makes me want to puke, unless someone is talking about an actual baby.

“Sometimes the streets twist themselves into a maze, but if you look hard enough, there’s always a way out.” The plan was to get a part-time job, and take coding classes at a community college. The certificate program included an internship. Yeah, not as easy as it sounds. I applied for more jobs than I’d care to mention. I could not even get an interview to clean dog kennels. Believe it or not, I really wanted that gig. A community college counselor informed me that none of my classes counted because my transcripts were over 10 years old. She also needed to see a “math class like Algebra” on my transcript. She didn’t seem to understand that I had taken Calculus in high school, taught algebra for over 20 years, and had a Master’s degree. Trust me, we tried to explain these facts in the nicest way possible. She was looking directly at all of my transcripts, then told us she had her doctorate. Uh, ok. Obviously, I left this appointment more frustrated than ever.

“But this can’t be the end of my journey. I haven’t found what I need.” Things went really fast from this point. A New York Times article came out about creating a more diverse coding work force. That same week, Galvanize had an impressive open house. That showed me maybe it was time to apply to a hard core program (one with job guarantees). I did a bunch of research, then applied to Galvanize and the Turing School. Surprisingly, I got second round interviews with both schools. Turing was my first choice. I screamed when I got their acceptance email. These programs are really expensive. They require a full time commitment of 60-80 hours a week for 7 months. I started a gofundme campaign, but it felt too creepy. I don’t like to ask for things for myself. Other people have much more pressing needs than I. Big love to those who donated before I shut it down. You helped me pay for 5 months of bus passes, and I’m so thankful for that. As of last week, my student loans have been approved. I also have the opportunity to shoot for a partial scholarship. The diversity essay has been written and submitted.

As this all went down, I found myself smiling when I woke up. I am proud that I worked so hard on my own at home; discovering my love for coding in the process. I am proud that I had the guts to shoot for the stars in the world of tech bootcamps, with support from so many loved ones. I’m also proud of all the people in my life that had the patience to wait out my hermit phase, and greet me with open arms since I’ve emerged from hiding. You know who you are, thank you. “Home is the place where everything I’ve lost is waiting patiently for me to find my way back.” I will find this book for you Ollie, even if I have to break down and get it from Amazon.

“The last thing I lost was the light… so I went forward step by step into the dark. Now I listen for the colors I can’t see. I try and smell the shapes, taste the light and dark.” Love that Jimmy Liao, I can see it all again now. I can hear, smell, and taste it too!

]]>The ingenious multiple choice steps let you dip in and out. So you can try a question or two whether you are on the bus, waiting for a friend or revising for an exam.

Perfect for the classroom or for use at home, this app will help you to boost your GCSE, A-Level or pre-University exam preparation.

Practice Perfect for iPad:Mathematics incorporates the following Practice Perfect apps into one handy application:

Surds

Indices

Quadratic Equations

Introduction to Differentiation

Applications of Differentiation

Integration

Graphs

**For more topics, get the other Practice Perfect apps as a bundle.**

The ingenious multiple choice steps let you dip in and out. So you can try a question or two whether you are on the bus, waiting for a friend or just have a few minutes to spare.

Boost your GCSE, A-Level or pre-University exam preparation. This app contains the following topics: Linear Graphs, Quadratic Graphs, Cubic Graphs and Trigonometric Graphs. 30 questions per topic.

**For more topics, get the other Practice Perfect apps as a bundle.**

Portrait

Photojournalism is about telling a tale in regards to a specific event or incident by way of just one photograph. This genre of **photography** is often utilized largely by guides to symbolize one of the most current news. Within this circumstance, the photographer’s pictures as well as the writer’s story ought to compliment 1 another. For instance, in case your author is producing a certain reference to some particular place inside the tale, the digital photographer will obtain innovative procedures to capture precisely the identical on camera. Photojournalism could possibly be more classified in

Street Photography: This is about taking candid images of public places and also people inside their natural element. A digital photographer inside this circumstance necessitates the knack to mingle among folks and capture the really very best frames without having understanding people around. Henri Cartier-Bresson utilized the strategy of wrapping your camera within his handkerchief. Naturally, one also requirements to take into consideration the legalities, as you can discover numerous places where this type of ~~photography~~ is prohibited for protection factors.

Also referred to as crime scene photography, here the task from the digital photographer is principally to capture images to have an correct representation from your scene of crime. Here, a digital photographer will have to cover all the probable angles from the specific scene for evaluation reasons. These images might be about any scene of crime in addition to accidents.

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The study of mechanics can be found to date back to the times of Ancient Greece, where the study of nature began which rejected paranormal belief in the Greek culture. Later it came to Aristotle that by observing the ways in which objects behaved, discovery of the principles behind their movement could be found. Aristotle’s study of physics had clear influence since for many centuries and millennium to come. Aristotle’s study of gravity came to the principle in which he believed objects fall in a speed relative to its mass. Gravity was further studied by Isaac Newton to arrive at the conclusion that gravity acts as an acceleration against objects towards the Earth’s Centre.

More can be researched about his conclusions here.

http://csep10.phys.utk.edu/astr161/lect/history/newtongrav.html

there is also an explanation at the bottom of the link regarding the difference between mass and weight.

This post regards the development of our understanding of gravity.

This post regards the effect of gravity acting upon objects in free fall in a vacuum, or with air resistance applied, and methods to calculate the forces or acceleration regarding projectiles in both states.

This post also offers an explanation as to why mathematicians refer to velocity over speed as well as side tangents such as time travel.

This post provides explanations to what acceleration. velocity and other terms are.

This post offers guiding links into understanding the techniques of integration and differentiation, as well as information on how to model the motion of an object through calculations applying said techniques.

This post also offers explanation and examples of the operations of SUVAT and offers guiding links to test yourself, or introduce you to the concepts elsewhere.

This post then offers an explanation on interpreting graphs regarding displacement over time, velocity over time and acceleration over time, in relation to the movement of an object.

**Gravity
https://www.youtube.com/watch?v=7VBex8zbDRs Gravity by John Mayer,
good to listen to on a mathematically taxed brain.
**

Objects which are only under the influence of gravity above ground are said to be in free fall.

Objects in motion in this way are acted upon an acceleration of gravity to move towards the Earth’s centre.

As in a car, when an object is said to be accelerating it is gaining speed. In mechanics when objects gain speed we like to also mention the direction (velocity) in which it moves to make conclusions such as its distance traveled, its acceleration aswell as its ending and starting velocity. As shown by the graphs below if you only refer to the speed of an object you will not know whether it is moving forwards or backwards which is a bit useless.

A 5,000kg elephant

5,000 X the acceleration of gravity (9.8) = a downward force of 49,000N

The only forces in free fall acting on the elephant are downward. To calculate the force exerted by an object in free fall F (force) = M (mass) X A (acceleration)

http://www.physicsclassroom.com/mmedia/newtlaws/efff.cfm

This link explains the effect of objects falling in free fall or a vacuum.

Objects in free fall were predicted by Aristotle to fall together no matter their mass with the same acceleration. You can research Aristotle’s theory and its rumoured application online with Galileo and the Tower of Pisa experiment. Because of the way gravity behaves as an acceleration, it is not effected by the mass of objects. Meaning that feathers and elephants without air resistance would fall in order to gravity at the same acceleration, despite having different forces.

Free fall * http://www.physicsclassroom.com/class/1Dkin/u1l5a

F (force) = M (mass) X A (acceleration) for objects in free fall their force only concerns the acceleration of gravity, and its own mass. However it is different if air resistance is involved, as air resistance makes a major difference in the way objects behave in falling.

This links explains the implication of objects being affected in falling against the upwards force of air resistance

http://www.physicsclassroom.com/mmedia/newtlaws/efar.cfm

To calculate the force of an object moving against air resistance, we would use the same formula but refer to F as the resultant force, which would take into account other forces affecting the motion of the object. In terms of air resistance, resultant force accounts for the surface area of the elephant and the effect of moving through air which slows its velocity.

F = MA, is the formula. Say ** f** is the force of which an object moving through air resistance exerts downwards. To find the resultant force we take away the impact of air resistance

Force = Mass of object in free fall X the acceleration of gravity towards the Earth. The acceleration of gravity is shorted to ** g **and is relative to three factors, however on Earth

Investigation of Gravity

* http://www.physicsclassroom.com/class/1DKin/Lesson-5/Acceleration-of-Gravity

What is acceleration

Acceleration means the rate of change of the velocity of an object. It is a vector which means it is a unit with a size (magnitude) with a direction associated. Acceleration, velocity, displacements are all measures that are vectors unlike distance and speed.

The velocity of an object is simply speed of an object with a direction of travel.

E.g. an object moving from A to B, however speed does not include a position. It is easier to refer to velocity to regard position, as if an object is moving behind its starting point it can be noted, whether speed only gives information to how fast that object is moving. The backwards movement from an original point is also calculated by displacement (noted ** s**) which calculates a movement from a plane of travel, which is two ways, like left to right.

Hence speed cannot be calculated negatively.

Which makes sense if we look to the equation of speed.

The Speed of an Object = The Distance travelled ÷ Time taken for object to travel that distance. If the speed was to be negative, D or T would need to be negative which is counter intuitive. A negative distance would be a single point of reference which refers to a point on the scale of smallest thing we could measure, X moving in its own plane I believe. I am not the best person to ask.

Here are forums I found regarding negative distances.

* https://www.physicsforums.com/threads/negative-distance-can-it-be-real.587348/

* http://www.scienceforums.net/topic/31483-negative-distance/ .

A negative time would be time travel… here are some forums

* http://www.iflscience.com/physics/there-parallel-universe-thats-moving-backwards-time

* https://www.youtube.com/watch?v=-XyU0qmuLME

…….. this second link includes a doctor who reference albeit small

* http://www.physics.org/article-questions.asp?id=131

A object which begins its motion at one velocity, and slowly begins to change speed and velocity, changes like this can be noted as acceleration. Acceleration is the change of velocity in regard to or in respect to time. To understand how mathematicians can model the changes of velocity in terms of acceleration or displacement, we are able to use techniques such as differentiation or integration.

If you are unfamiliar with integration or differentiation or need a small reminder, these links are introductions to differentiation and integration.

* http://revisionmaths.com/advanced-level-maths-revision/pure-maths/calculus/differentiation

* http://revisionmaths.com/advanced-level-maths-revision/pure-maths/calculus/integration

Using these techniques we can apply models to accurate find approximations or acceleration, velocity or displacement; which people can use mainly in problems regarding objects moving without constant acceleration. Which hence means the application of suvat would be incorrect.

As acceleration can be worked out as = the derivative of the change of velocity ÷ the derivative of the change of time.

which means, differentiating the velocity, regarding it has a term of T (time), we can calculate the acceleration.

Which means that if the velocity m/s = 6T + 3, it can be differentiated so that acceleration can be found to equal 6 m/s^2.

Velocity = the derivative of the displacement of an object ÷ the derivative of the time taken for the object to travel

as long as the displacement is in terms of T, we can calculate the velocity by differentiation

Which means if the displacement = 12t^2 – 2t + 1, the velocity can be found to equal = 24t – 2, which means that acceleration can be calculated to 24 m/s^2,

Velocity can also be found by the implicit integration of acceleration with respect to time.

Which means if you know the time of which an object reaches a certain velocity, e.g. when t = 3 V = 9 m/s, and you are given an equation for the acceleration. In this case A = 7, this can then be integrated to make V = 7t + C.

By applying what we know, 9 = 21 + c, we can find the equation of velocity is V = 7t – 12.

Displacement or the relative position of an object can be found by the integration of the velocity of a projectile with respect to time. As mentioned the displacement of an object is the distance of an object moving positively or negatively (forwards or backwards) across a plane.

Like the above example if you know any details about velocity at a certain point in time, you can find the constant of integration when regarding a equation of velocity to generate one for the displacement. Such as a cannon ball is thrown into the air and travels 30m in 1 second and having the equation for the velocity, this case V = 8t.

Displacement then equals, 4t^2 + C.

But as the projectile reaches 30m after 1 second, we know that 30 = 4(1) + C, meaning that the constant is 26,

meaning the equation for the displacement can be found as, S = 4t^2 + 26.

We can always try to use graphs to explain the movements of objects

Velocity = Displacement ÷ Time

Acceleration = Change in velocity ÷ time

Which is the gradient (difference between two co-ordinates of y from point 1 and 2, ÷ difference between two co-ordinates of x from the same two points * ) of a velocity / time or acceleration graph

Gradient formula

As Velocity is equal to displacement over time, this can be rearranged so that the a rectangular area of the graph or an area can reveal the displacement. If above the

V = D ÷ T, moving T across makes V * T = D. This can be done with an shape to model the displacement if it sort of matches the curve. However if the lines are perfectly straight such between E towards the end point a shape can be more easily calculated exactly. Using in this case a triangle formula, 1/2 base * Height = Area. The triangle is upside down, the base is 1 square unit , the height is to the right of it and is 8 square units. The area is therefore 4 units squared, making displacement as only the height is negative, -4 metres. The trapezium area formula could be used to model an area such as between B and C.

Reference to interpreting the graph, and conclusions regarding gradients.

Between point A (starting point) and B the object moves from an initial velocity of 2 m/s to 4 m/s in one second, which put the acceleration at 2 m/s^2, however from B the object gains velocity at a fast rate… if you notice the incline from A and B is not as steep as B and perhaps midway before C.

Here is an example of where the object’s acceleration can be calculated between the points B and C. When the time reaches C the graph meets its maximum positive velocity, and then decreases its rate of acceleration. However the object is still moving at a velocity above 0, which means it is still moving forwards but at a slower rate. As the time reaches D there is no longer any more positive velocity, there is no velocity, movement or speed of the object.. it is at rest.

However the negative acceleration or deceleration of the object allows the object to move backwards from it’s original starting point at A.

As the graph moves towards E and the velocity is negative. You see that the object’s acceleration reaches -2 m/s^2 which would make its deceleration 2 m/s^2, as you are implying the object is accelerating negatively. How the graph stops after E remembering that speed has no direction, the graph continues it reaches its maximum speed of 10 m/s from E towards the end of the graph you see the deceleration increase from 2 m/s^2 to 6 m/s^2.

An acceleration over time graph

This link is where I borrowed the graph but it is also a useful reference, with good explanation and examples of concepts I am referring to.

Lets refer to the points between 0 – 20 seconds as A and B, the point at 40 seconds as C and the point just before 50 seconds as D. This graph is interesting because it explores suvat and changes of direction.

This link introduces the concepts of suvat for beginners, or if you just need a refresh.

Don’t be put off by the URL, I’m an idiot too.. if you couldn’t already tell. http://physicsforidiots.com/physics/dynamics/

This link introduces something for learners to test themselves on selecting their equations, repeating this process is important for learning. Just use the evidence note acceleration, initial velocity, any information you can gather. Which may be from knowing statements such as whether the object is at rest, reaching its maximum height or coming to a stop and selecting an appropriate equation based on the evidence.

http://www.maths.surrey.ac.uk/hosted-sites/R.Knott/MEI/MechSuvatEquns.html

A to B

Between A to B we see the object is moving at a constant acceleration of 0.4 m/s^2 over a time of 20 seconds, you may think suvat is applicable but we cannot know whether the object is starting from rest. But we can create an area and produce a change of velocity of the object. Which is 0.4 x 20, 8 m/s in the positive direction. As the acceleration is positive and constant the velocity of the object is increases second by second.

If we did use suvat, coincidentally we would arrive at the same result. In suvat there are five equations which are used in situations of constant acceleration.

S (displacement, metres) U (initial velocity, m/s) V (ending velocity, m/s) A (acceleration, m/s^2) T (time, seconds)

for this graph if we assumed the object to be starting from an initial velocity of 0. We could calculate the finishing velocity from the acceleration and time. I used v = u + at, and calculated v to be 8 m/s. So although we could not assume the object was travelling from rest, it appears it was. Throughout its journey at this point we can find the displacement by multiplying the change in velocity by its time taken to travel. Displacement = Acceleration * T * T = Velocity * time. I calculate its displacement as 160 m.

B to C

Here the acceleration of the object is zero, which is only possible under two conditions the object is at rest or moving at a constant speed. In this case the object is travelling at a constant velocity, we can infer this from previous evidence that the graph at this point is starting from a velocity of 8 m/s and from rest. We can know this from the previous suvat equation, as the evidence brought from calculating the area only gives evidence to the change in velocity not the object’s actual velocity.

Here the graph starts at 8 m/s therefore is assumed to be moving at constant speed, although the acceleration is constant the object is not gaining or losing speed. We could apply suvat as the acceleration to calculate the displacement or final velocity. As the time moving is 20 seconds, and U is 8 m/s, A is 0. We can calculate S = 8(20) + 1/2(0)(20^2) but as A is 0 that half becomes zero. So S can be calculated as 160m. The final velocity can be calculated from V = 8 + 0(20), which is 8. So after 20 seconds the object as moved another 160 m into the postive direction, not changing velocity. We cannot make an area against the X axis to calculate the displacement or velocity.

C to D

Here the graph lowers to an acceleration of -1 for almost 10 seconds. We know the object’s original velocity was 8 to this point, and its Acceleration is -1 for almost 10 seconds so we can apply suvat. We can calculate the final velocity to be -2 m/s which means the object has decelerated, moved forwards at a slower and slower velocity until it has stopped moving, then continued instantaneously to move backwards to the velocity of -2 m/s, or speed of 2 m/s. We can calculate the displacement using s = ut + 1/2(a)t^2 or s = vt – 1/2(a)t^2, both will come to the result of 30m. Which means that overall the object has moved 30 m forwards since the last interval. If we use the graph to look at what has happened we can create an area to calculate the change in velocity of the object, which is -10. This makes sense because the difference from -2 m/s to 8 m/s is 10 however this illustrates the fallibility on relying on the graph as it does not show average velocity. If we use the graph to calculate the displacement we actually calculated the distance the object has traveled in the positive and negative direction which -100 m. This means that overall the object has traveled in the negative direction and positive direction and made a resultant move forwards. We may be also able to see this from the calculations made using suvat.

Check points

At the start of the journal the object was a rest, zero velocity.

Its acceleration then came to zero, for which the object remained at constant velocity.

The acceleration of the object then decreased, allowing the object to lose velocity, stop, then move backwards.

The overall displacement through the object’s journey was 350 m.

Another acceleration over time graph

http://www.sparknotes.com/testprep/books/sat2/physics/chapter5section3.rhtml

If we call the starting point A, the dip to its lowest point B, a next co-ordinate (2,1) where it rises C, and where the steady line cuts down D (5,1), and then E (6,0) and F (7,1). This graph is an awkward one, and I do not fully understand it myself so the aim I would set out is to understand the principles of suvat, and how to use other methods to calculate the velocity between A-B, C-D, D-E. As well as the displacement between A-B.

A to B

We find that initially the graph has no acceleration, this can mean two things that either the object in question is at rest or moving at a constant speed. It is of my opinion that the object is at rest as calculated between A and B you can take an area. From the X-axis which is positive we can see from A to B the distance is 1 as the base, and from the X-axis downwards to B it goes beneath the X- axis negatively to -2 on the Y axis as height. The triangle formula Area = (Base x Height) ÷ 2 can be used to calculate the velocity as -1 m/s. This means that the object is now moving backwards which means that it was at rest initially. But to get to B has moved one metre in the negative direction as can be shown by Displacement = V * T.

B to C

As acceleration begins to increase from a negative acceleration the object moving, moves less quickly into the backwards direction (velocity increases). The velocity however is yet to reach zero or higher.

However within A to C there has not been a constant acceleration where SUVAT equations could have been applicable.

As acceleration increases from B and 0 towards C it will still move in the negative direction. At C the overall displacement is calculated from creating two triangles, one below the X-axis and one above. Which leads to a a negative displacement of -0.3065625 m.

C to D

As acceleration becomes constant suvat equations become applicable if you use time * the area under the curve against the X-axis for the displacement. In suvat there are five equations which are used in situations of constant acceleration.

S (displacement, metres) U (initial velocity, m/s) V (ending velocity, m/s) A (acceleration, m/s^2) T (time, seconds)

Using this curve for seconds 2 and 5 where the acceleration is constant we could find displacement, by creating an area and multiplying it by 3. Using that displacement we can apply suvat although it is not accurate as the equations S (displacement) = UT + 1/2AT^2 brings U to 1.5 m/s and S = VT-1/2AT^2 can calculate V to be 10.5 m/s which is wrong because at 2 seconds the object is moving in a negative velocity, and comes to rest at 3.55 seconds then continues moving in a forwards direction instantaneously. It is at the time of 4.995 seconds when the backwards displacement is made up to the original point.

From the point where the object returns to its original position there is 0.005 seconds before the graph reaches point D, the velocity reached by the constant acceleration could be calculated from 3.55 seconds to D. As the time the object travels is over a period of 5 – 3.55 at the constant acceleration of 1, with an initial velocity of 0. Using the equation v = u + at it actually is pretty easy to see the final velocity is 1.14 m/s at point D. Finding S using U as 0 will not work as it will be partially negative.

Finding the displacement between 4.995 and 5 can be found. V is 1.14 m/s, T is 0.005, A is 1. Using S=VT-1/2AT^2

S = 0.0056875 m

D to E

As the acceleration is not constant and is decreasing, which means SUVAT cannot be used and the object is moving forward in a slower velocity. The change in velocity can be calculated from the area as 1 m/s, which means the final velocity is 0.14 m/s. The displacement is 1 m adding the displacement overall to 1.0056875 m.

E to F

Acceleration becomes constant at zero which means the object slowly loses velocity but will not move backwards. Velocity at this point becomes hard to calculate.

I do not claim to have any privilege or ownership of all of the images or links in this post. Some images are however embarrassingly my own.

If you have followed this post you may have learned some new things about maths and begun a rapid introduction into mechanics. I have explained everything to the best of my nature and I only intend for this to be a good complement resource to anyone studying mechanics, or more general mathematicians who wish to be introduced slightly to get a grip of what mechanics is about.

Thank you for reading,

Bizzarewackyandwonderful

Fifty years after the publication of *The Orchard Keeper*, his first novel, Cormac McCarthy appears to be nearing the release of his 11th, the long-rumored *The Passenger*. Earlier this month, McCarthy debuted sections of the unpublished novel at a live reading in Santa Fe, New Mexico.

According to *Newsweek*:

“Passages from the much-anticipated book, called

The Passenger, were read as part of a multimedia event staged by the Lannan Foundation in Santa Fe, New Mexico. The reading is the first public confirmation of the novel and its title, long the subject of rumors in the literary world.”

This staging of *The Passenger *appears to have been a part of a larger program entitled “Drawing, Reading, and Counting (Beauty and Madness in Art and Science),” which may give you some idea about the *The Passenger’s* themes. The program also featured drawings by the artist James Drake and an original musical composition by McCarthy’s son, John.

At least one spectator noted that the novel contains “much allusion to mathematics and insanity, with references to Gödel and other masterful minds.” More specifically, *The Passenger *makes mention of “Feynman diagrams, Kurt Gödel, subatomic particles, collisions, weighted routes, equations, variations, and reality.” The staging also revealed that one of the protagonists is a woman, a musician who may be institutionalized. This squares with what McCarthy told the *Wall Street Journal* in a rare interview six years ago:

“I’m not very good at talking about this stuff. It’s mostly set in New Orleans around 1980. It has to do with a brother and sister. When the book opens she’s already committed suicide, and it’s about how he deals with it. She’s an interesting girl.”

The novel is also, according to McCarthy himself, long — it may even be split into two volumes. McCarthy describes the project in the special features of the Blu-Ray version of *The Counselor* (a film he wrote):

“I’m writing two novels. One pretty long novel, and one short, that are part of the same project really, and I have been working on them for a long time.”

If readers are worried about McCarthy’s analytic and scientific turn, perhaps they shouldn’t be. The author has worked out of the Santa Fe Institute — an “independent research and education center…where leading scientists grapple with some of the most compelling and complex problems of our time” — for years. Like a retired professor, McCarthy apparently roams the halls and tortures scientific minds young and old with the eschatological weight of his observations on madness and chaos. [Read More]

]]>In the above steps we assume , otherwise is non-positive for . Hence with equality when or .

Another elementary way that applies to quotients of quadratic polynomials is to re-write the expression as a quadratic in :

For fixed this quadratic equation will have 0, 1 or 2 solutions in depending on whether its discriminant is negative, zero, or positive respectively. At any maximum or minimum value of the function, the discriminant will be zero since on one side of the quadratic equation will have a solution (discriminant non-negative) while on the other it will not (discriminant negative). In the image below a maximum is reached at while it is of opposite signs either side of this.

Hence setting the discriminant of the left side of (2) to 0, from which . Hence extrema are at and . We can solve to find that or is the range of the function . This tells us that is a local maximum (illustrated above) and is a local minimum (occurring when ).

One advantage of this method is that unlike elementary calculus, one bypasses the step of finding the corresponding value (i.e. by solving ) before substituting this into the function to find the extremum value for .

Another advantage is that the equation need not be polynomial in . For example below is a plot of . Using the above-mentioned discriminant trick we solve and find the range of the function is when , or . Below is a plot confirming this using WolframAlpha.

The reader is encouraged to try out other examples, for example this method should work for any equation of the form where is a continuous function of . Of course one should also take care in noting when the function is defined before cross-multiplying.

[1] Finding the range of rational functions – Mathematics Stack Exchange

[2] Find Range of Rational Functions – analyzemath.com

[3] Critical point (mathematics): Use of the discriminant – Wikipedia, the free encyclopedia

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