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From Wikipedia
Connected Mathematics is a comprehensive, problemcentered curriculum designed for all students in grades 68 based on the NCTM standards. The curriculum was developed by the [http://connectedmath.msu.edu/Connected Mathematics Project (CMP)] at Michigan State University and funded by the National Science Foundation.
Each grade level curriculum is a fullyear program, and in each of the three grade levels, topics of number, algebra, geometry/measurement, probability and statistics are covered in an increasingly sophisticated manner. The program seeks to make connections within mathematics, between mathematics and other subject areas, and to the real world. The curriculum is divided into units, each of which contains investigations with major problems that the teacher and students explore in class. Extensive problem sets are included for each investigation to help students practice, apply, connect, and extend these understandings.
Connected Mathematics addresses both the content and the process standards of the NCTM. The process standards are: Problem Solving, Reasoning and Proof, Communication, Connections and Representation. For example, in Moving Straight Ahead students construct and interpret concrete, symbolic, graphic, verbal and algorithmic models of quantitative and algebraic relationships, translating information from one model to another.
Like other curricula implementing the NCTM standards, Connected Math has been criticized by supporters of traditional mathematics for not directly teaching standard arithmetic methods.
Research Studies
One 2003 study compared the mathematics achievement of eighth graders in the first three school districts in Missouri to adopt NSFfunded Standardsbased middle grades mathematics curriculum materials (MATH Thematics or Connected Mathematics Project) with students who had similar prior mathematics achievement and family income levels from other districts. Significant differences in achievement were identified between students using Standardsbased curriculum materials for at least 2 years and students from comparison districts using other curriculum materials. All of the significant differences reflected higher achievement of students using Standardsbased materials. Students in each of the three districts using Standardsbased materials scored higher in two content areas (data analysis and algebra), and these differences were significant.
Another study compared statewide standardized test scores of fourthgrade students using Everyday Mathematics and eighthgrade students using Connected Mathematics to test scores of demographically similar students using a mix of traditional curricula. Results indicate that students in schools using either of these standardsbased programs as their primary mathematics curriculum performed significantly better on the 1999 statewide mathematics test than did students in traditional programs attending matched comparison schools. With minor exceptions, differences in favor of the standardsbased programs remained consistent across mathematical strands, question types, and student subpopulations.
Controversy
As one of many widely adopted curricula developed around the NCTM standards, Connected Mathematics has been criticized by advocates of traditional mathematics as being particularly ineffective and incomplete and praised by various researchers who have noted its benefits in promoting deep understanding of mathematical concepts among students. In a review by critic James Milgram, "the program seems to be very incomplete... it is aimed at underachieving students." He observes that "the students should entirely construct their own knowledge.. standard algorithms are never introduced, not even for adding, subtracting, multiplying and dividing fractions." However, studies have shown that students who have used the curriculum have "develop[ed] sophisticated ways of comparing and analyzing data sets, . . . refine[d] problemsolving skills and the ability to distinguish between reasonable and unreasonable solutions to problems involving fractions, . . . exhibit[ed] a deep understanding of how to generalize functions symbolically from patterns of data, . . . [and] exhibited a strong understanding of algebraic concepts and procedures," among other benefits.
Districts in states such as Texas were awarded NSF grants for teacher training to support curricula such as CM. Austin ISD received a $5 million NSF grant for teacher training in 1997. NSF awarded $10 million for "Rural Systemic Initiatives" through West Texas A&M. At the state level, the SSI (Statewide Systemic Initiative), was a federallyfunded program developed by the Dana Center at the University of Texas. Its most important work was directing the implementation of CM in schools across the state. But in 1999, Connected Mathematics was rejected by California's revised standards because it was judged at least two years below grade level and it contained numerous errors . After the 20002001 academic year, state monies can no longer be used to buy Connected Mathematics
The Christian Science Monitor noted parents in Plano Texas who demanded that their schools drop use of CM, while the New York Times reported parents there rebelled against folding fraction strips rather than using common denominators to add fractions. For the improved second edition, it is stated that "Students should be able to add two fractions quickly by finding a common denominator". The letter to parents states that students are also expected to multiply and divide fractions by standard methods.
What parents often do not understand is that students begin with exploratory methods in order to gain a solid conceptual understanding, but finish by learning the standard procedures, sometimes by discovering them under teacher guidance. Largescale studies of reform curricula such as Connected Mathematics have shown that students in such programs learn procedural skills to the same level as those in traditional programs, as measured by traditional standardized tests. Students in standardsbased programs gain conceptual understanding and problemsolving skills at a higher level than those in traditional programs.
Despite disbelief on the part of parents whose textbooks always contained instruction in mathematical methods, it is claimed that the pedagogical benefits of this approach find strong support in the research: "Over the past three to four decades, a growing body of knowledge from the cognitive sciences has supported the notion that students develop their own understanding from their experiences with mathematics."
Examples of criticism
Connected Mathematics treatment of some topics include exercises which some have criticized as being either "subjective" or "having nothing to do with the mathematical concept" or "omit standard methods such as the" formula for arithmetic mean. (See above for discussion of reasons for initial suppression of formulas.) The following examples are from the student textbooks, which is all the parents see. (See discussion below.)
Average
In the first edition, one booklet focuses on a conceptual understanding of median and mean, using manipulatives. The standard algorithm was not presented. Later editions included the algorithm.
Comparing fractions
In the 6th grade u
This article 'Algorithm examples supplementsAlgorithm and Algorithm characterizations.
An example: Algorithm specification of addition m+n
Choice of machine model:
There is no â€œbestâ€�, or â€œpreferredâ€� model. The Turing machine, while considered the standard, is notoriously awkward to use. And different problems seem to require different models to study them. Many researchers have observed these problems, for example:
 â€œThe principal purpose of this paper is to offer a theory which is closely related to Turing's but is more economical in the basic operationsâ€� (Wang (1954) p. 63)
 â€œCertain features of Turing machines have induced later workers to propose alternative devices as embodiments of what is to be meant by effective computability.... a Turing machine has a certain opacity, its workings are known rather than seen. Further a Turing machine is inflexible ... a Turing machine is slow in (hypothetical) operation and, usually complicated. This makes it rather hard to design it, and even harder to investigate such matters as time or storage optimization or a comparison between efficiency of two algorithms.â€� (Melzak (1961) p. 281)
 ShepherdsonSturgis (1963) proposed their registermachine model because â€œthese proofs [using Turing machines] are complicated and tedious to follow for two reasons: (1) A Turing machine has only one head... (2) It has only one tape....â€� They were in search of â€œa form of idealized computer which is sufficiently flexible for one to be able to convert an intuitive computational procedure into a program for such a machineâ€� (p. 218).
 â€œI would prefer something along the lines of the random access computers of Angluin and Valiant [as opposed to the pointer machine of SchÃ¶nhage]â€� (Gurivich 1988 p. 6)
 â€œShowing that a function is Turing computable directly...is rather laborious ... we introduce an ostensibly more flexible kind of idealized machine, an abacus machine...â€� (BoolosBurgessJeffrey 2002 p.45).
About all that one can insist upon is that the algorithmwriter specify in exacting detail (i) the machine model to be used and (ii) its instruction set.
Atomization of the instruction set:
The Turing machine model is primitive, but not as primitive as it can be. As noted in the above quotes this is a source of concern when studying complexity and equivalence of algorithms. Although the observations quoted below concern the Random access machine model â€“ a Turingmachine equivalent â€“ the problem remains for any Turingequivalent model:
 â€œ...there hardly exists such a thing as an â€˜innocentâ€™ extension of the standard RAM model in the uniform time measure; either one only has additive arithmetic, or one might as well include all multiplicative and/or bitwise Boolean instructions on small operands....â€� (van Emde Boas (1992) p. 26)
 â€œSince, however, the computational power of a RAM model seems to depend rather sensitively on the scope of its instruction set, we nevertheless will have to go into detail...
 â€œOne important principle will be to admit only such instructions which can be said to be of an atomistic nature. We will describe two versions of the socalled successor RAM, with the successor function as the only arithmetic operation....the RAM0 version deserves special attention for its extreme simplicity; its instruction set consists of only a few one letter codes, without any (explicit) addressing.â€� (SchÃ¶nhage (1980) p.494)
Example #1: The most general (and original) Turing machine â€“ singletape with leftend, multisymbols, 5tuple instruction format â€“ can be atomized into the Turing machine of BoolosBurgessJeffrey (2002) â€“ singletape with no ends, two "symbols" { B,  } (where B symbolizes "blank square" and  symbolizes "marked square"), and a 4tuple instruction format. This model in turn can be further atomized into a PostTuring machineâ€“ singletape with no ends, two symbols { B,  }, and a 0 and 1parameter instruction set ( e.g. { Left, Right, Mark, Erase, Jumpifmarked to instruction xxx, Jumpifblank to instruction xxx, Halt } ).
Example #2: The RASP can be reduced to a RAM by moving its instructions off the tape and (perhaps with translation) into its finitestate machine â€œtableâ€� of instructions, the RAM stripped of its indirect instruction and reduced to a 2 and 3operand â€œabacusâ€� register machine; the abacus in turn can be reduced to the 1 and 2operand Minsky (1967)/ShepherdsonSturgis (1963) counter machine, which can be further atomized into the 0 and 1operand instructions of SchÃ¶nhage (and even a 0operand SchÃ¶nhagelike instruction set is possible).
Cost of atomization:
Atomization comes at a (usually severe) cost: while the resulting instructions may be â€œsimplerâ€�, atomization (usually) creates more instructions and the need for more computational steps. As shown in the following example the increase in computation steps may be significant (i.e. orders of magnitude â€“ the following example is â€œtameâ€�), and atomization may (but not always, as in the case of the PostTuring model) reduce the usability and readability of â€œthe machine codeâ€�. For more see Turing tarpit.
Example: The single register machine instruction "INC 3" â€“ increment the contents of register #3, i.e. increase its count by 1 â€“ can be atomized into the 0parameter instruction set of SchÃ¶nhage, but with the equivalent number of steps to accomplish the task increasing to 7; this number is directly related to the register number â€œnâ€� i.e. 4+n):
More examples can be found at the pages Register machine and Random access machine where the addition of "convenience instructions" CLR h and COPY h_{1},h_{1} are shown to reduce the number of steps dramatically. Indirect addressing is the other significant example.
Precise specification of Turingmachine algorithm m+n
As described in Algorithm characterizations per the specifications of BoolosBurgessJeffrey (2002) and Sipser (2006), and with a nod to the other characterizations we proceed to specify:
 (i) Number format: unary strings of marked squares (a "marked square" signfied by the symbol 1) separated by single blanks (signified by the symbol B) e.g. â€œ2,3â€� = B11B111B
 (ii) Machine type: Turing machine: singletape leftended or noended, 2symbol { B, 1 }, 4tuple instruction format.
 (iii) Head location: See more at â€œImplementation Descriptionâ€� below. A symbolic representation of the head's location in the tape's symbol string will put the current state to the right of the scanned symbol. Blank squares may be included in this protocol. The state's number will appear with brackets around it, or subscripted. The head is shown as
From Encyclopedia
The invention and ideas of many mathematicians and scientists led to the development of the computer, which today is used for mathematical teaching purposes in the kindergarten to college level classrooms. With its ability to process vast amounts of facts and figures and to solve problems at extremely high speeds, the computer is a valuable asset to solve the complex mathladen research problems of the sciences as well as problems in business and industry. Major applications of computers in the mathematical sciences include their use in mathematical biology, where math is applied to a discipline such as medicine, making use of laboratory animal experiments as surrogates for a human biological system. Mathematical computer programs take the data drawn from the animal study and extrapolate it to fit the human system. Then, mathematical theory answers the question of how far these data can be transformed yet still preserve similarity between species. Mathematical ecology tries to understand the patterns of nature as society increasingly faces shortages in energy and depletion of its limited resources. Computers can also be programmed to develop premium tables for life insurance companies, to examine the likely effects of air pollution on forest productivity, and to simulate mathematical model outcomes that are used to predict areas of disease outbreaks. Mathematical geography computer programs model flows of goods, people, and ideas over space so that commodity exchange, transportation, and population migration patterns can be studied. Largescale computers are used in mathematical physics to solve equations that were previously intractable, and for problems involving a third dimension, numerous computer graphics packages display threedimensional spatial surfaces. A byproduct of the advent of computers is the ability to use this tool to investigate nonlinear methods. As a result, the stability of our solar system has been checked for millions of years to come. In the information age, information needs to be stored, processed, and retrieved in various forms. The field of cryptography is loaded with computer science and mathematics complementing each other to ensure the confidentiality of information transmitted over telephone lines and computer networks. Encoding and decoding operations are computationally intense. Once a message is coded, its security may hinge on the inability of an intruder to solve the mathematical riddle of finding the prime factors of a large number. Economical encoding is required in highresolution television because of the enormous amount of information. Data compression techniques are initially mathematical concepts before becoming electromagnetic signals that emerge as a picture on the TV screen. Mathematical application software routines that solve equations, perform computations, or analyze experimental data are often found in areaspecific subroutine libraries which are written most often in Fortran or C. In order to minimize inconsistencies across different computers, the Institute of Electrical and Electronics Engineers (IEEE) standard is met to govern the precision of numbers with decimal positions. The basic configuration of mathematics learning in the classroom is the usage of standalone personal computers or shared software on networked microcomputers. The computer is valued for its ability to aid students to make connections between the verbal word problem, its symbolic form such as a function, and its graphic form. These multiple representations usually appear simultaneously on the computer screen. For home and school use, publicdomain mathematical software and shareware are readily available on the Internet and there is a gamut of proprietary software written that spans the breadth and depth of the mathematical branches (arithmetic, algebra, geometry, trigonometry, elementary functions, calculus, numerical analysis, numerical partial differential equations, number theory, modern algebra, probability and statistics, modeling, complex variables, etc.). Often software is developed for a definitive mathematical maturity level. In lieu of graphics packages, spreadsheets are useful for plotting data and are most useful when teaching arithmetic and geometric progressions. Mathematics, the science of patterns, is a way of looking at the world in terms of entities that do not exist in the physical world (the numbers, points, lines and planes, functions, geometric figuresâ€”â€”all pure abstractions of the mind) so the mathematician looks to the mathematical proof to explain the physical world. Several attempts have been made to develop theoremproving technology on computers. However, most of these systems are far too advanced for high school use. Nevertheless, the nonmathematician, with the use of computer graphics, can appreciate the sets of Gaston Julia and Benoit B. Mandelbrot for their artistic beauty. To conclude, an intriguing application of mathematics to the computer world lies at the heart of the computer itself, its microprocessor. This chip is essentially a complex array of patterns of propositional logic (p and q, p or q, p implies q, not p, etc.) etched into silicon . see also Data Visualization; Decision Support Systems; Interactive Systems; Physics. Patricia S. Wehman Devlin, Keith. Mathematics: The Science of Patterns. New York: Scientific American Library, 1997. Sangalli, Arturo. The Importance of Being Fuzzy and Other Insights from the Border between Math and Computers. Princeton, NJ: Princeton University Press, 1998.
mathematics deductive study of numbers, geometry, and various abstract constructs, or structures; the latter often "abstract" the features common to several models derived from the empirical, or applied, sciences, although many emerge from purely mathematical or logical considerations. Mathematics is very broadly divided into foundations, algebra, analysis, geometry, and applied mathematics, which includes theoretical computer science. Branches of Mathematics Foundations The term foundations is used to refer to the formulation and analysis of the language, axioms, and logical methods on which all of mathematics rests (see logic ; symbolic logic ). The scope and complexity of modern mathematics requires a very fine analysis of the formal language in which meaningful mathematical statements may be formulated and perhaps be proved true or false. Most apparent mathematical contradictions have been shown to derive from an imprecise and inconsistent use of language. A basic task is to furnish a set of axioms effectively free of contradictions and at the same time rich enough to constitute a deductive source for all of modern mathematics. The modern axiom schemes proposed for this purpose are all couched within the theory of sets , originated by Georg Cantor, which now constitutes a universal mathematical language. Algebra Historically, algebra is the study of solutions of one or several algebraic equations, involving the polynomial functions of one or several variables. The case where all the polynomials have degree one (systems of linear equations) leads to linear algebra. The case of a single equation, in which one studies the roots of one polynomial, leads to field theory and to the socalled Galois theory. The general case of several equations of high degree leads to algebraic geometry, so named because the sets of solutions of such systems are often studied by geometric methods. Modern algebraists have increasingly abstracted and axiomatized the structures and patterns of argument encountered not only in the theory of equations, but in mathematics generally. Examples of these structures include groups (first witnessed in relation to symmetry properties of the roots of a polynomial and now ubiquitous throughout mathematics), rings (of which the integers, or whole numbers, constitute a basic example), and fields (of which the rational, real, and complex numbers are examples). Some of the concepts of modern algebra have found their way into elementary mathematics education in the socalled new mathematics. Some important abstractions recently introduced in algebra are the notions of category and functor, which grew out of socalled homological algebra. Arithmetic and number theory , which are concerned with special properties of the integersâ€”e.g., unique factorization, primes, equations with integer coefficients (Diophantine equations), and congruencesâ€”are also a part of algebra. Analytic number theory, however, also applies the nonalgebraic methods of analysis to such problems. Analysis The essential ingredient of analysis is the use of infinite processes, involving passage to a limit . For example, the area of a circle may be computed as the limiting value of the areas of inscribed regular polygons as the number of sides of the polygons increases indefinitely. The basic branch of analysis is the calculus . The general problem of measuring lengths, areas, volumes, and other quantities as limits by means of approximating polygonal figures leads to the integral calculus. The differential calculus arises similarly from the problem of finding the tangent line to a curve at a point. Other branches of analysis result from the application of the concepts and methods of the calculus to various mathematical entities. For example, vector analysis is the calculus of functions whose variables are vectors. Here various types of derivatives and integrals may be introduced. They lead, among other things, to the theory of differential and integral equations, in which the unknowns are functions rather than numbers, as in algebraic equations. Differential equations are often the most natural way in which to express the laws governing the behavior of various physical systems. Calculus is one of the most powerful and supple tools of mathematics. Its applications, both in pure mathematics and in virtually every scientific domain, are manifold. Geometry The shape, size, and other properties of figures and the nature of space are in the province of geometry. Euclidean geometry is concerned with the axiomatic study of polygons, conic sections, spheres, polyhedra, and related geometric objects in two and three dimensionsâ€”in particular, with the relations of congruence and of similarity between such objects. The unsuccessful attempt to prove the "parallel postulate" from the other axioms of Euclid led in the 19th cent. to the discovery of two different types of nonEuclidean geometry . The 20th cent. has seen an enormous development of topology , which is the study of very general geometric objects, called topological spaces, with respect to relations that are much weaker than congruence and similarity. Other branches of geometry include algebraic geometry and differential geometry , in which the methods of analysis are brought to bear on geometric problems. These fields are now in a vigorous state of development. Applied Mathematics The term applied mathematics loosely designates a wide range of studies with significant current use in the empirical sciences. It includes numerical methods and computer science, which seeks concrete solutions, sometimes approximate, to explicit mathematical problems (e.g., differential equations, large systems of linear equations). It has a major use in technology for modeling and simulation. For example, the huge wind tunnels , formerly used to test expensive prototypes of airplanes, have all but disappeared. The entire design and testing process is now largely carried out by computer simulation, using mathematically tailored software. It also includes mathematical physics, which now strongly interacts with all of the central areas of mathematics. In addition, probability theory and mathematical statistics are often considered parts of applied mathematics. The distinction between pure and applied mathematics is now becoming less significant. Development of Mathematics The earliest records of mathematics show it arising in response to practical needs in agriculture, business, and industry. In Egypt and Mesopotamia, where evidence dates from the 2d and 3d millennia BC, it was used for surveying and mensuration; estimates of the value of Ï€ ( pi ) are found in both locations. There is some evidence of similar developments in India and China during this same period, but few records have survived. This early mathematics is generally empirical, arrived at by trial and error as the best available means for obtaining results, with no proofs given. However, it is now known that the Babylonians were aware of the necessity of proofs prior to the Greeks, who had been presumed the originators of this important step. Greek Contributions A profound change occurred in the nature and approach to mathematics with the contributions of the Greeks. The earlier (Hellenic) period is represented by Thales (6th cent. BC), Pythagoras , Plato , and Aristotle , and by the schools associated with them. The Pythagorean theorem, known earlier in Mesopotamia, was discovered by the Greeks during this period. During the Golden Age (5th cent. BC), Hippocrates of Chios made the beginnings of an axiomatic approach to geometry and Zeno of Elea proposed his famous paradoxes concerning the infinite and the infinitesimal, raising questions about the nature of and relationships among points, lines, and numbers. The discovery through geometry of irrational numbers, such as 2 , also dates from this period. Eudoxus of Cnidus (4th cent. BC) resolved certain of the problems by proposing alternative methods to those involving infinitesimals; he is known for his work on geometric proportions and for his exhaustion theory for determining areas and volumes. The later (Hellenis
What is new and different in the world of mathematics? Where is mathematics going in the future? What questions are mathematicians asking and exploring? Let's compare two of the instruments of calculation widely used during the last 50 years of the twentieth century. The slide rule is a portable calculator once carried by engineers so that they could quickly perform complicated calculations. Compare the slide rule, now a collector's item, to the graphing calculator used by high school students taking basic algebra. The graphing calculator can be used to do much moreâ€”more accurately and more easily. The use of powerful calculators and computers will be an integral part of the mathematical problems and questions investigated in the future. Some classic problems, such as those involved with prime numbers , the geometry of soap bubbles, and the fourcolor theorem about how many colors are needed to distinguish neighboring colors on a map, are being extended to more complex questions, some involving three or more dimensions. Other problems of the future involve newer themes, such as chaos theory and how it can be applied to model various systems and computers and how they can be used to generate proofs. The advances in chaos theory made in the twentieth century will probably be extended into many areas of investigation as mathematical understanding of dynamical systems in biology, physiology, and clinical practice is increased. For example, according to Barry Cipra in What's Happening in the Mathematical Sciences, a yearly publication that reports on the latest mathematical research, mathematicians are working with scientists in a variety of fields to address many of our modern challenges, including how the human immune system works and how to deal with hazardous wastes. Chaos theory is also being used in questions about waves in all their forms, along with turbulence, complex fluid flows, and computational fluid dynamics . Ecology is another active field that is benefiting from the mathematics of chaos. Mathematicians continue to explore the nature of proof itself, and now this involves exploring whether a computer can develop a proof. The abilities of computers, the possible ability of computers to recreate themselves, and the bioengineering of computers continue to fascinate those at the cutting edge of developing technology and mathematics. Throughout the history of mathematics, games and game theory have fascinated mathematicians, and this trend continues today. A computer has defeated a human chess champion, but people continue to search for the "perfect" play in various games. In addition to studying games, mathematicians and many other people continue to work with codes and cryptography as they seek complete security for all types of messagesâ€”particularly those transmitted over computer networks and the Internet. In the 1990s, Andrew Wiles solved what has been called "The World's Most Famous Math Problem" when he proved Fermat's Last Theorem. This accomplishment inspired people beyond the world of theoretical mathematics. Marilyn vos Savant wrote a book about Wiles's work in which she quotes a poem about Fermat's Last Theorem; there was a play presented in New York called Fermat's Last Tango ; and there was a Fermat's Last Theorem Poetry Challenge. Mathematics provided the subject matter for other achievements in the arts: The 2001 Pulitzer Prize for drama went to Proof, a mystery about a famous mathematician, and the book and movie titled A Beautiful Mind, which tells the story of John Nash, the mathematician who won a Nobel Prize in 1994 for his work on game theory. So, solemn or frivolous, mathematics will continue to be used to model situations in every field of human endeavor where patterns and predictability pose challenges. And, at the same time, the search for useful models will continue to expand and benefit from the world of theoretical mathematics. see also Chaos; Computers, Future of; Fermat's Last Theorem; Games; Gardner, Martin; Minimum Surface Area; Puzzles, Number; Slide Rule. Lucia McKay American Mathematical Society.
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Fraction is an important concept and considered to be a prerequisite to other topics in Mathematics.
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Answers:There are quite a few experiments outlined and described in the Journal of Parapsychology archive: http://findarticles.com/p/articles/mi_m2320 Most of these experiments do not demonstrate active Psi energies, but some have some statistical significance. The researchers, for the most part, are sincere in their efforts to come to a realistic, experimentally based conclusion. I hope that you can find something interesting ther TR. *********************************** Followup: Here's a brief description of one experiment I conducted.  2 subjects claimed the ability to perform remote viewing and/or telepathy. The goal of the experiment was to determine whether their claims could be verified. We theorized that we could legitimately test these subjects and come to a reasonable conclusion concerning their claim.  The subjects were separated in different houses, miles from each other. Observers were with the subjects to insure there was not communication between subjects. No cell phones or wireless electronic communication devises were in either location. A wired telephone was in each home, but was not used during the experiment. Nobody left either location during the experiment.  The subjects were asked to attempt to make contact and initiate a conversation on a topic defined by the researchers and hidden from the subject until the experiment began. The subjects had 14 hours to complete the conversation.  After 14 hours, the observers asked the subjects to describe the conversation that they had. The descriptions were documented by the observers and compared.  The results: The conversational records were very similar. The order of the topics discussed were the same. Exact phrases appeared in both summaries in the same order. Though the discussion was less than 10 sentences, both summaries listed the same topics being discussed in the same order in the same number of sentences. Each subject identified the speaker for each phrase, and both subjects connected the same phrases with the same speaker.  Conclusion: With the lack of electronic communication, and the hidden selection of the topic to be discussed by the observers, it was highly unlikely that the subjects had been able to prearrange their responses. The similarity of the responses, especially the length, specific word choices, and identification of the speakers that appeared in the summaries indicated a strong probability that there was a common knowledge between the subjects of what had been discussed. Testing appeared to have been successfully completed in this case. 2 other attempts to verify with the same subject using similar techniques did not show the same results. Postmortem: The experiments could have been better designed to eliminate the possibility of prearranged conversations between the subjects or the possibility that the observers were working together to falsify the results of the experiments. The additional steps would have made for better experimental design and should be employed in the future. Also, one test showing significantly different results from 2 other test cases does not show evidence of repeatable results. The results did not definitively show that there was communication between the individuals, but we did conclude that we were able to test the individuals for this ability. ************************************ My personal opinion in this case (not specifically scientifically based): Knowing the observers in this case and their motivations, I do not believe that there was any "cheating" occuring in these experiments. The results for these cases, though not the result of flawless expermental design, seem to accurately reflect the events. In one trial, the subjects showed strong signs of having communicated from a distance. The similarity of the data in that one case was so strikingly similar that it is unreasonable to believe that it is due to coincidence. If they actually had this ability, it was not an ability that was always available to them or was not consistently accurate. ***************************** This is just one experiment, and not a particularly spectacular one. It gives you a sense of the work that I have done, though it does not reflect how my style has changed over the years. The details aren't all there, but this really isn't the forum to go into *complete* experimental design or examine data. I hope that this information reflects what you were asking about.
Answers:I did exactly this for my GCSE coursework: Let your big grid have size m by n, and your rectangle have size a by b. Now choose a rectangle and let the top left element be x. The top right element is then x + a  1(because the rectangle has width a). The bottom left element is x + (b  1)m (you have to drop down b  1 times from the top left), and the bottom right element is x + (b  1)m + a  1. The quadratic you're after is then: diff = (x + a  1)(x + [b  1]m)  x(x + [b  1]m + a 1 ) If you expand and simplify, the answer turns out to be (a  1)(b  1)m if you try it with your 3 x 2 rectangles in the 10 x 10 grid, the difference is 2 x 1 x 10 = 20.
Answers:you can use the factor theorem or the remainder theorem and find the roots by trial and error.if 'a' is a root of f(x) then f(a)=0 and xa will be a factor or group the terms and factor by taking out common factors you may use identities where possible and factor for higher power equations also you can use the remainder theorem and after finding a factor divide by that to reduce the equation to one of lower power and proceed along similar lines till you have found all the factors example suppose you want to factor a^3+3a^2b+3ab^2+b^3 put a=b the equation reduces to o.therefore a+b is a factor divide by a+b and you get a^2+2ab+b^2 again put a=b and the equation reduces to zero and so a+b is a second factor and when a^2+2ab+b^2 is divided by a+b it gives the third factor also as a+b thus a^3+3a^2b+3ab^2+b^3 is (a+b)(a+b)(a+b) since (a+b)^3=a^3+3a^2b+3ab^2+b^3 using this identity also the factoring can be done grouping the terms a^3+3ab(a+b)+b^3=>a^3+b^3+3ab(a+b) =>(a+b)(a^2ab+b^2)+3ab(a+b) using the identity a^3+b^3 taking out a+b as the common factor this reduces to (a+b)(a^2ab+b^2+3ab)=(a+b)(a^2+2ab+b^2) =>(a+b)(a+b)^2 using the identity (a+b)^2
Answers:The area of study known as the history of mathematics is primarily an investigation into the origin of discoveries in mathematics and, to a lesser extent, an investigation into the mathematical methods and notation of the past. Before the modern age and the worldwide spread of knowledge, written examples of new mathematical developments have come to light only in a few locales. The most ancient mathematical texts available are Plimpton 322 (Babylonian mathematics c. 1900 BC), the Moscow Mathematical Papyrus (Egyptian mathematics c. 1850 BC), the Rhind Mathematical Papyrus (Egyptian mathematics c. 1650 BC), and the Shulba Sutras (Indian mathematics c. 800 BC). All of these texts concern the socalled Pythagorean theorem, which seems to be the most ancient and widespread mathematical development after basic arithmetic and geometry. Egyptian and Babylonian mathematics influenced Greek and Hellenistic mathematics, which greatly refined the methods (especially the introduction of mathematical rigor in proofs) and expanded the subject matter of mathematics.[1] Islamic mathematics, in turn, developed and expanded the mathematics known to these ancient civilizations. Many Greek and Arabic texts on mathematics were then translated into Latin, which led to further development of mathematics in medieval Europe. From ancient times through the Middle Ages, bursts of mathematical creativity were often followed by centuries of stagnation. Beginning in Renaissance Italy in the 16th century, new mathematical developments, interacting with new scientific discoveries, were made at an ever increasing pace, and this continues to the present day.
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