## PhDs in Natural Sciences & Mathematics

Mathematics studies numbers, structure and change and draws its origins from early philosophy. This ancient discipline is commonly used for calculations, counting and measurements. However, mathematics is a complex field which also involves theories, discovery of patterns, development of law, nicknamed of &ldquoThe queen of sciences&rdquo .

Throughout history, a significant amount of mathematicians such as Galileo Galilei, Albert Einstein, Pythagoras, Archimedes and many others brought innovations in mathematics and gave birth to new theories and solutions to analytical problems. Mathematical principles can also be found in disciplines such as medicine, natural sciences, engineering, finance and social sciences.

Students who hold a Bachelor&rsquos degree in mathematics can turn to applied mathematics, statistics, physics or engineering, if they wish to continue their studies. Such a programme develops skills such as knowledge of arithmetic, algebra, trigonometry and strong deductive reasoning. After graduating a Master&rsquos degree in mathematics, students have the choice to be employed as operational researchers, statisticians, aerospace engineers, accountants, software testers or teachers.

## Mathematical Misconceptions : A Guide for Primary Teachers

With contributors comprised of teachers, teacher educators, mathematicians and psychologists, Mathematical Misconceptions brings together information about pupils′ work from four different countries, and looks at how children, from the ages of 3 - 11, think about numbers and use them. It explores the reasons for their successes, misunderstandings and misconceptions, while also broadening the reader′s own mathematical knowledge. Chapters explore:

- the seemingly paradoxical number zero

- children′s perceptions and misconceptions of adding, subtracting, multiplying and dividing

- the ways in which children acquire number concepts.

This unique book will transform the way in which primary school teachers think about mathematics. Fascinating reading for anyone working with children of this age, it will be of particular interest to teachers, trainee teachers and teaching assistants. It will show them how to engage children in the mysteries and delights of numbers.

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### Об авторе (2008)

Anne D. Cockburn is a Professor Emeritus in Early Years Education at the University of East Anglia (UEA). She was educated in Edinburgh before reading Psychology at the University of St Andrews. Subsequently, she trained to be a primary teacher and taught in Scotland. In 1979, she became a Research Fellow at the University of Lancaster, working with Neville Bennett and Charles Desforges. Her PhD was completed in 1986 at the UEA. Following a period of working as a researcher, she took up her first lectureship at UEA in 1989. She became an Associate Fellow of the British Psychological Society in 1994. Initially Anne's teaching focused on pre-service teacher education (BA and PGCE), gradually extending to in-service courses (BPhil and MA) and research (PhD and EdD). Throughout, she continued with her own research, with many of the catalysts for her investigations stemming from the needs and interests of professional practitioners and those with whom they work. More recently, she also started working with MA counselling students. Anne has examined doctoral theses, undergraduate and postgraduate courses at universities across the United Kingdom, Australia and Norway. She was a member of the Economic and Social Research Council Board of Examiners for studentships (2002–2005).

## Everyday Mathematics for Parents

The Everyday Mathematics (EM) program was developed by the University of Chicago School Mathematics Project (UCSMP) and is now used in more than 185,000 classrooms by almost three million students. Its research-based learning delivers the kinds of results that all school districts aspire to. Yet despite that tremendous success, EM often leaves parents perplexed. Learning is accomplished not through rote memorization, but by actually engaging in real-life math tasks. The curriculum isn&rsquot linear, but rather spirals back and forth, weaving concepts in and out of lessons that build overall understanding and long-term retention. It&rsquos no wonder that many parents have difficulty navigating this innovative mathematical and pedagogic terrain.

Now help is here. Inspired by UCSMP&rsquos firsthand experiences with parents and teachers, *Everyday Mathematics for Parents* will equip parents with an understanding of EM and enable them to help their children with homework&mdashthe heart of the great parental adventure of ensuring that children become mathematically proficient.

Featuring accessible explanations of the research-based philosophy and design of the program, and insights into the strengths of EM, this little book provides the big-picture information that parents need. Clear descriptions of how and why this approach is different are paired with illustrative tables that underscore the unique attributes of EM. Detailed guidance for assisting students with homework includes explanations of the key EM concepts that underlie each assignment. Resources for helping students practice math more at home also provide an understanding of the long-term utility of EM. Easy to use, yet jam-packed with knowledge and helpful tips, *Everyday Mathematics for Parents* will become a pocket mentor to parents and teachers new to EM who are ready to step up and help children succeed. With this book in hand, you&rsquoll finally understand that while this may not be the way that you learned math, it&rsquos actually much better.

## 17.6: New Page - Mathematics

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The **National Center for Education Statistics (NCES)** is the primary federal entity for collecting and analyzing data related to education.

This Data Point examines whether teachers were required to help students with their academic or social and emotional needs outside regular school hours in public and private schools in the United States in school year 2017&ndash18, by selected school classification. » More info

Preliminary data files for the 2020-21 school year are now available. » More info

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A unique First Look report released by NCES describes the effects of the COVID-19 pandemic on postsecondary students. » More info

The *Condition of Education* is an annual report to Congress summarizing important developments and trends in the U.S. education system. The report presents 50 indicators on topics ranging from prekindergarten through postsecondary education, as well as labor force outcomes and international comparisons. Discover how you can use the *Condition of Education* to stay informed about the latest education data.

Scores are reported on a scale of 0 to 1,000. See Figure M2b of the TIMSS 2019 U.S. Highlights Results. **SOURCE:** International Association for the Evaluation of Educational Achievement (IEA), Trends in International Mathematics and Science Study (TIMSS), 2019.

**Condition of Education**

Browse key indicators on the condition of education in the United States at all levels, from prekindergarten through postsecondary, as well as labor force outcomes and international comparisons. The indicators summarize important developments and trends using the latest statistics, which are updated throughout the year as new data become available.

## Table of Contents

Winner of a CHOICE Outstanding Academic Title Award for 2017!

What does style mean in mathematics? Style is both how one does something and how one communicates what was done. In this book, the author investigates the worlds of the well-known numbers, the binomial coefficients. The author follows the example of Raymond Queneau's Exercises in Style . Offering the reader 99 stories in various styles. The book celebrates the joy of mathematics and the joy of writing mathematics by exploring the rich properties of this familiar collection of numbers. For any one interested in mathematics, from high school students on up.

The exercises are lucidly written there is much beautiful mathematics to learn and enjoy.

-- Debra K. Borkovitz, Mathematics Teacher

By examining and extending binomial coefficients from seemingly every possible direction, the author provides an amazing concoction of ideas, prompting readers to say "Wow, I forgot that connection," or "Wow, I did not know that," or just "Wow. McCleary's effort is exceptional, as it reaches into the realm of élan, clearly demonstrating the energy and enthusiasm that can pervade mathematical writing and mathematics itself.

## MCQ in Engineering Mathematics Part 4 – Answers

Following is the list of multiple choice questions in this brand new series:

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## Entry requirements

Your application, especially your personal statement, should demonstrate your enthusiasm for studying mathematics. This might include relevant reading, voluntary or work experience, topics within mathematics that particularly interest you or other relevant extra-curricular and co-curricular activities.

Strong performance in Mathematics is essential, both in your entry qualifications and any previous study.

We also recommend you consider taking an additional mathematics test such as STEP, MAT, TMUA or the GCE Advanced Extension Award. If you are studying A levels in Mathematics and Further Mathematics strong performance in one of these tests may qualify you for a reduced alternative offer. In most cases an additional maths test is entirely optional, but it remains compulsory alongside some qualifications including students who study A level Mathematics without Further Mathematics. We publish guidance on how we use these different mathematics tests.

We know that the context in which you are studying can have an impact on your ability to perform your best in exams and coursework, or limit which subjects or qualifications you are able to study at your school or college. **We consider any application based on its merits, including your background and circumstances, including through:**

## Department of Mathematics

**Prerequisites**: It is recommended that students re-take any calculus or precalculus course with a C+ or below before proceeding to the subsequent course. Alternatively, the student may wish to study on their own doing all the homework in the prerequisite's syllabus. See below for the syllabi of our courses listed in order. Students are expected to know all the material on the syllabus regardless of what was covered in their particular sections of the course.

**Review Materials:**

**Syllabi**: Note that these syllabi should be adhered to as much as possible so that these courses coincide exactly with its corequisites and prepare students for future classes. The homework on the syllabi is just recommended. Faculty may copy the source files for these syllabi to create their own webpages where they can post their homework assignments and day-to-day progress.

- [PDF] is the prerequisite to Precalculus. [PDF] [PDF] . [PDF] . [PDF] . [PDF]. [PDF] is often called Calculus III or Vector Calculus at other universities.

**Uniform Final Exam Information:** All students must take and pass a Department uniform final exam in order to pass the course. This final will be given during finals week at the end of the semester. A sample final exam can be found in the Department Office, Room 211 of Gillet Hall. A Sample Final Exam is given in the following links:

## Constructive Mathematics

Note: Some of the symbols used on this page may not display correctly with certain web browsers (usually indicated with either a question mark or box). If this is the case with your browser you can view the intended character(s) by clicking of the symbols. Problem browsers include Internet Explorer and older versions of Netscape.

### What is constructive mathematics?

A general answer to this question is that constructive mathematics is mathematics which, at least in principle, can be implemented on a computer.

There are at least two ways of developing mathematics constructively. In the first way one uses classical (that is, traditional) logic. Unfortunately, that logic allows us to prove theorems that no computer can implement, so in order to do things constructively, we have to work within a strict algorithmic framework such as recursive function theory [ 22 ] or Weihrauch’s Type Two Effectivity theory [ 35 ]. This can make the resulting mathematics appear rather hard to read and certainly different from normal analysis, algebra, or the like.

The second way of approaching constructivity is to replace classical logic by intuitionistic logic, which neatly captures the proof processes used when you work in a rigorously computational manner. This way has the advantage that, once you get used to a logic which does not allow, for example, the application of the law of excluded middle (**LEM** )

you find yourself working in the style of a traditional algebraist, analyst, and so on, without referring continually to a special algorithmic language and symbolism.

### Why chose CM at all? Why would a constructive approach interest people?

Meaningful distinctions deserve to be maintained [ 6 ]

If, however, you are not interested in questions of computability, then you should stick to classical logic. There are even areas of mathematics where the content is so highly nonconstructive that it would make little sense to give up classical logic the higher reaches of modern set theory would seem to be just such an area.

### Are our proofs complicated? Is there any estimate of the complexity of these proofs?

Generally, constructive proofs are quite complicated. This is hardly surprising, since they produce more (computational) information than their classical counterparts (if the latter have any). Consider, for example, the constructive proofs of Picard’s theorem in the following two classically equivalent forms.

Let *ƒ* be a holomorphic function on the punctured disc *D* (0,1) := <*z* ∈ C : 0 < |*z*| < 1> that omits two complex values from its range. Then *ƒ* has a pole of determinate order at 0 .

Let *ƒ* be a holomorphic function on *D*(0,1) that has an essential singularity at 0 , and let ζ, ζ′ be two distinct complex numbers. Then either there exists *z* ∈ *D*(0,1) with *ƒ* (*z*) = ζ or else there exists *z* ∈ *D*(0,1) with *ƒ* (*z*) = ζ′

These two theorems, although classically equivalent, are totally different from a constructive point of view. In **PT**_{p} we use the data comprising the function *ƒ* and the two complex values omitted from its range, to construct an integer *ν* , show that the *ν th* Laurent coefficient of *ƒ* is 0 , and to show that all Laurent coefficients with index less than *ν* are zero. In **PT**_{s} our data consist of the holomorphic function *ƒ* and the two distinct complex numbers ζ, ζ′ , and the constructive proof embodies an algorithm converting those data to solution *z* of one of the equations *ƒ* (*z*) = ζ , *ƒ* (*z*) = ζ′ moreover, the proof shows which equation is actually solved.

Now, it is hardly surprising that the constructive proofs of both **PT**_{p} and **PT**_{s} are rather complicated. For one thing, they rely on delicate estimates involving winding numbers and requiring a number of preliminaries that the classical proof of Picard’s theorem does not require. In addition, those algorithms could actually be extracted from the proofs and implemented on a computer. So we pay more in terms of effort and complexity of proof, but we get more for our money as well.

The complexity of constructive proofs, other than those that use the Church–Markov–Turing thesis as an additional hypothesis (see [ 21 ]) is still largely untouched terrain. However, anecdotal evidence from Bas Spitters suggests that, perhaps contrary to one’s initial expectations, many of the proofs in Bishop’s book are remarkably efficient when implemented on a computer.

### Are the practitioners of CM just rewriting classical results? Has anybody produced a brand new result in CM that hasn't been proved classically?

It all depends on what you mean by “brand new result”. If you take the classical viewpoint that every statement is either true or false and hence that once proved, a result is no longer new, then much of what we are doing will look like rewriting classical results. However, if you interpret a constructive theorem and its proof properly, then it is quite clear that, even if the statement of the theorem looks like something that is well known classically, both the properly–interpreted theorem and its proof are brand new.

Consider yet again the Picard theorems discussed above. The full constructive interpretation of **PT**_{p} is this:

*There is an algorithm which, applied to a holomorphic function ƒ on* *D*(0,1) *and to two complex values omitted from the domain of ƒ , computes the order ν of the pole that ƒ has at* 0 *.*

I know of no proof of this statement other than the constructive one in [ 12 ] the theorem, as presented in my statement, is brand new. Moreover, that proof, while drawing on a classical proof of the classical Picard theorem for inspiration, is also new.

Similarly, we have the constructive interpretation of **PT**_{s} :

*There is an algorithm which, applied to a holomorphic function ƒ on* *D*(0,1) *, the data showing that ƒ has an essential singularity at 0 (that is, the sequence of Laurent coefficients of ƒ which contains infinitely many negatively indexed terms), and two distinct complex numbers ζ and ζ′ , computes a complex number z and shows that either* *ƒ* (*z*) *= ζ* *or* *ƒ* (*z*) *= ζ′* *.*

Once again, this is a brand new theorem, nowhere found (to my knowledge) in the classical literature and once again, its proof is also new.

There are aspects of constructive mathematics that are clearly new, in that the classical mathematician would see nothing to prove where the constructive mathematicians does. For example, many theorems of constructive analysis require a certain subset *S* of a metric space *X* to be *located*, in the sense that the distance

exists (is computable). Proving that *S* is located may be a nontrivial matter. This is related to the failure of the classical least–upper–bound principle. For the constructive existence of the least upper bound of a nonempty subset *S* of R that is bounded above we need the additional condition (it is both necessary and sufficient) that *S* be *order located***:** that is, for all real *α* , *β* with *α* < *β* , either *β* is an upper bound of *S* or else there exists *s* ∈ *S* with *s* > *α* . (Note that the “or” here is decidable: in constructive mathematics, to prove the disjunction *p* ∨ *q* , we must either produce a proof of *p* or else produce a proof of *q* .)

### Does CM have any final products?

Yuk! I hate those management jargon words like “final products”. However, since people use them in questioning what we do, we’d better deal with them, like it or not.

What is the final product of any branch of pure mathematics? Do, for example, set theorists like Hugh Woodin, working with extremely high–level abstractions, have a final product? If the questioner means “something that has applications in the real world”, then it seems totally unreasonable to expect constructive analysis to justify itself by the production of such a final product when that justification is not required of classical pure mathematics. If pushed, however, I would say that the final product of all pure mathematics, constructive and nonconstructive, is a body of results, proofs and techniques that contribute to the higher levels of human culture and that may, (as history shows) frequently will, have significant applications in the future.

### Is program extraction from constructive proofs a real thing?

Yes, it is. Research groups in Japan, the States, the United Kingdom, Sweden and Germany have been active in this area for many years [ 14, 18, 23, 30 ] A constructive proof of (let's use this one again) Picard‘s Theorem **PT**_{s} really does contain an extractable algorithm for computing the point *z* with the properties stated in the conclusion of that theorem. Moreover, the proof is itself a proof that the program is correct—that is, meets its specifications. So the constructive result gives us two things for the price of one: an algorithm and a proof of its correctness. That's a real bargain!

### What's the status of the Axiom of Choice?

On one level, this one is relatively easy to answer: the full Axiom of Choice (AC ),

In view of the Diaconescu–Goodman–Myhill theorem, what did Bishop mean when he said that under the hypotheses of (1),

A choice function exists … because a choice is implied by the very meaning of existence ?

I believe that he meant that the constructive interpretation of the hypothesis in (1) is that there is an *algorithm* which leads us from elements *x* of *X* to elements *y* of *Y* such that *P*(*x,y*) holds. However, to compute the *y* from a given *x* , the algorithm will use not just the data describing *x* itself, but also the data proving that *x* satisfies the conditions for membership of the set *A* . Thus the algorithm will not be a function of *x* but a function of *both x and its certificate of membership of A* . The value at *x* of a genuine function from *X* to *Y* would depend only on *x* and not on its membership certificate.

At a deeper level, the question is tricker to answer, at least if recast in the form, “What, if any, choice axioms are permissible in constructive mathematics?”. Some constructive mathematicians, notably Fred Richman, doubt the constructive validity of even countable choice (and hence of dependent choice). The argument in favour of countable choice is that one has to do no work to show that a natural number *x* belongs to the set N of natural numbers: each natural number is, as it were, its own certificate of membership of N . Thus in the case *X* = N , the choice algorithm implied “by the very meaning of existence” in (1) is, in fact, a genuine function on N . Needless to say, those who distrust even countable choice as a constructive principle do not buy into this argument.