#### • Class 11 Physics Demo

Explore Related Concepts

# example of mathematical investigations

From Wikipedia

Connected Mathematics

Connected Mathematics is a comprehensive, problem-centered curriculum designed for all students in grades 6-8 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 full-year 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 NSF-funded Standards-based 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 Standards-based 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 Standards-based materials. Students in each of the three districts using Standards-based materials scored higher in two content areas (data analysis and algebra), and these differences were significant.

Another study compared statewide standardized test scores of fourth-grade students using Everyday Mathematics and eighth-grade 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 standards-based 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 standards-based programs remained consistent across mathematical strands, question types, and student sub-populations.

## 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] problem-solving 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 federally-funded 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 2000-2001 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. Large-scale 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 standards-based programs gain conceptual understanding and problem-solving 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

Algorithm examples

## 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)
Shepherdson-Sturgis (1963) proposed their register-machine 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€� (Boolos-Burgess-Jeffrey 2002 p.45).

About all that one can insist upon is that the algorithm-writer 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 Turing-machine equivalent â€“ the problem remains for any Turing-equivalent 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 so-called 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€“ single-tape with left-end, multi-symbols, 5-tuple instruction format â€“ can be atomized into the Turing machine of Boolos-Burgess-Jeffrey (2002) â€“ single-tape with no ends, two "symbols" { B, | } (where B symbolizes "blank square" and | symbolizes "marked square"), and a 4-tuple instruction format. This model in turn can be further atomized into a Post-Turing machineâ€“ single-tape with no ends, two symbols { B, | }, and a 0- and 1-parameter instruction set ( e.g. { Left, Right, Mark, Erase, Jump-if-marked to instruction xxx, Jump-if-blank 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 finite-state machine â€œtableâ€� of instructions, the RAM stripped of its indirect instruction and reduced to a 2- and 3-operand â€œabacusâ€� register machine; the abacus in turn can be reduced to the 1- and 2-operand Minsky (1967)/Shepherdson-Sturgis (1963) counter machine, which can be further atomized into the 0- and 1-operand instructions of SchÃ¶nhage (and even a 0-operand SchÃ¶nhage-like 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 Post-Turing 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 0-parameter 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 h1,h1 are shown to reduce the number of steps dramatically. Indirect addressing is the other significant example.

## Precise specification of Turing-machine algorithm m+n

As described in Algorithm characterizations per the specifications of Boolos-Burgess-Jeffrey (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: single-tape left-ended or no-ended, 2-symbol { B, 1 }, 4-tuple 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 sub-scripted. The head is shown as

From Encyclopedia

Mathematics

mathematics

Mathematics, New Trends in

From Digg

AN INVESTIGATION OF FRACTION SENSE AMONG FORM ONE STUDENTS

Fraction is an important concept and considered to be a prerequisite to other topics in Mathematics.

Question:I would like to hear from paranormal investigators whether or not they try to use the scientific method in their investigations. TK studies have been beaten to death already, so I'm inquiring more about poltergeist investigations and so forth. In specific, I'm looking for key components of the scientific method which need to be present in order for the study to be accurately described as scientific. 1) Did you have a specific hypothesis? What was it? 2) Was your hypothesis both directly testable and falsifiable? 3) How would you have been able to falsify the hypothesis? 4) Comments on reproducibility? Statistical methods? 5) What was your conclusion? How did you modify your hypothesis based on the results you obtained? 6) Using your hypothesis, were you able to predict other paranormal events? How? 7) Can investigators skeptical of the paranormal use your methods to obtain the same results? Please see http://teacher.pas.rochester.edu/phy_labs/AppendixE/AppendixE.html Artlogical, the Journal of Paranormal Research up until now has not been peer-reviewed and the quality of the work has not been great. They are going to peer-review now, so perhaps things will change. Thanks Tunsa, but I was looking specifically for examples of people right here using Yahoo Answers that are using the scientific method in studying the paranormal. I've been here a while, and although it is often claimed I have yet to see an example of the use of the scientific method. Surely someone out there has a good example to share.

Question:I am investigating the diagonal differences rectangles give on a 10x10 number grid from 1-100 for example a 3x2 rectangle on a 10x10 number grid could be - 1 2 3 11 12 13 When you multiply the diagonal corners (1x13) and (3x11) you then subtract the answers e.g 33 - 13 = 20. Therefore 3x2 rectangles on a 10x10 number grid produces a difference of 20. However, I now need to create a quadratic equation which would allow me to find the difference of ANY rectangle on any sized grid. Please help.

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.

Question:1. how do you find the roots (factors) of an algebraic cubic equation? 2. how do you find roots of an algebraic equation with powers higher than 3? PS: please answer the 1st question. IMPORTANT!! the 2nd question if anyone can help, plase do so too. but more important is the 1st. thanks!

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 x-a 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^2-ab+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^2-ab+b^2+3ab)=(a+b)(a^2+2ab+b^2) =>(a+b)(a+b)^2 using the identity (a+b)^2

Question:Im suppose to come up with a paper. This is what the instructions are: Remember that you need one mathematical topic and one topic that involves the wider society. Usually they will be from the same time period and culture, but not always. There should be some connection between the two topics. For instance, you might like to research a mathematician and an artist who have no direct connection but are both representative of the intellectual spirit of a particular historical age. You want something that allows you to make some connections of your own. This could involve an ancient or modern development in algebra. The critical thinking would come in connecting it to the wider society. How was it used in a way that affected the world outside mathematics? How did it come to be represented in schools/universities? Algebra is a rich field, so I am sure you can refine this to a suitably limited topic. More info: you could look at the move from rhetorical algebra to symbolic algebra and highlight one application that the change facilitated. Likewise, you could look at the development of abstract algebras (e.g. Lie algebras) and one application that has been facilitated. For instance, the Lie algebras are widely used in physics. I also had to pick a theme which was Algebra. Could someone help me on this? Any help would be appreciated. i just need some topic ideas, because i cant think of anything.

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 so-called 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.