Generalization and Conclusions

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  The Process of Learning Mathematics

  Pergamon International Library of Science, Technology, Engineering and Social Studies

  • L. R. Chapman(Author)
  • 2013(Publication Date)
  • Pergamon

    (Publisher)

  Then the above proof of eqn. (6) can be dis-covered by trial. 2. Generalisation The Concise Oxford Dictionary defines generalisation as the forming of a general notation or proposition obtained by induction. In other words, in general, we take a simple proposition and extend it to a more elaborate theorem of which the simple proposition is a special case. Frequently, mathematical induction is used in the generalisation process. Further, the simple result itself plays a key role in the generalisation proof. Dienes has written freely and penetratingly about the twin processes of generalisation and abstraction. t He defines generalisation as the t Z. P. DIENES, On abstraction and generalisation, Harvard Educational Review (1961); An Experimental Study of Mathematics Learning, Hutchinson, 1963. 98 C. PLUMPTON discovery that a general rule extends beyond the first few known cases; while he states that abstraction is, the awareness that the rule applies in a number of other situations. He has attempted to classify various levels of generalisation in the following manner: 1. extension of the rule from a finite number of cases to an in-finite number; 2. the generalisation from one infinite class to another infinite class; 3. what Dienes terms a mathematical generalisation or the formation of an isomorphism between one class and a sub-class of another class. The following example will make the point clearer. A child given the function y = 2x, where x is a whole number, will find the value of y given a value of x. This will be repeated for a finite number of values of x until suddenly the extension is made that given any value of x, then the corresponding value of y can be found. This is the first and easiest stage. If the coefficient of x is changed to 3, or 4, or 5, or to any whole number, the generalisation is made between the infinite number of cases when, say, y = 2x and y = 3x, with x e {1, 2, 3, 4, 5, ...}.

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  The Notion of mathematical proof: Key rules and considerations

  • Olga Moreira(Author)
  • 2023(Publication Date)
  • Arcler Press

    (Publisher)

  Then, a hypothesis, if formulated, and an expression, are or can be produced. On the other hand, just the arithmetic generalization often uses some recurrent relations and, following that, pupils, using this way of thinking are not able to provide a formula for a general case. However, the least sophisticated are the naive inductions based on probable reasoning. The type of generalization can be related to the type of problem occurring, however, the students equipped with well-connected problem solving schemas are not surprisingly more successful in transferring their knowledge [27]. On the other hand, the algebraic generalization raises specific problems for almost every student, even in the course of studying higher grades [28]. Our recent research findings indicate an ability to produce some general algebraic expressions as a crucial factor in solving open-ended non-routine problems in mathematics, even by high-achievers [29]. For the students the use of justification, proving strategies and techniques of various forms of proofs in mathematics, providing them with a crucial importance of mathematical reasoning, is conclusive. Therefore, such abilities clearly involve obtaining strategic knowledge in the specific areas related to the given problem at hand, as well as knowledge and norms specific for proving and reasoning [30], especially in mathematical open- ended problem-solving situations. Pedemonte [31] stresses the structural gap between the argumentation and proof. It is clear that just the argumentation inferences are based on the content, while in case of any proof they follow a deductive scheme. Within the work in the classroom environment, usually, students mostly experience some memorized algorithmic reasoning [32] or with some formal proofs, a sequence of formulas in the formal language [33] instead of the actual creative mathematical reasoning within a complex mathematical inquiry that enables understanding and construction of a new knowledge.

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  The Philosophy of Management Research

  • Eric W.K. Tsang(Author)
  • 2016(Publication Date)
  • Routledge

    (Publisher)

  2006). This chapter focuses on case studies rather than the broad domain of qualitative research partly because some of the qualitative methods, such as ethnography, are rather rare in management research and partly because case studies are the most popular qualitative method used by management researchers (Welch et al. 2013) and are also increasingly popular among social scientists (Gerring 2007). 2 A classification of induction Following Schwandt (1997: 57), generalization is defined as a “general statement or proposition made by drawing an inference from observation of the particular.” 3 This characterization of generalization is consistent with that proposed by logicians, such as Cohen and Nagel (1934), Copi and Cohen (1990), and Hurley (2003). In empirical research, generalization is an act of inferring from specific, observed instances, such as those in a case setting, to general statements. As discussed below, generalization is sometimes confounded with its closely related concept, induction, which refers to inference from matters of fact that we have observed to those we have not (Cambridge Dictionary of Philosophy 1999: 745). Induction has a broader meaning than generalization because generalization denotes a directional inference from something particular to something more general whereas induction does not. There are at least two forms of induction that are not generalization. One of these is what logicians call “statistical syllogism” (Gensler 2001), for example: P1 Nearly all senior accounting managers in the U.K. have college degrees. P2 Tom is a senior accounting manager in the U.K. C Tom has a college degree. This inference is inductive because it goes from observed matters of fact (P1 and P2) to unobserved matters of fact (C). Yet the inference goes from a general premise and a particular premise to a particular conclusion

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  Mathematics and Plausible Reasoning, Volume 1

  Induction and Analogy in Mathematics

  • G. Polya, George Polya(Authors)
  • 2020(Publication Date)
  • Princeton University Press

    (Publisher)

  II GENERALIZATION, SPECIALIZATION, ANALOGY And I cherish more than anything else the Analogies, my most trustworthy masters. They know all the secrets of Nature, and they ought to be least neglected in Geometry. —KEPLER 1. Generalization, Specialization, Analogy, and Induction. Let us look again at the example of inductive reasoning that we have discussed in some detail (sect. 1.2, 1.3). We started from observing the analogy of the three relations 3 + 7 = 10, 3 + 1 7 = 20, 13 + 17 = 30, we generalized in ascending from 3, 7, 13, and 17 to all primes, from 10, 20, and 30 to all even numbers, and then we specialized again, came down to test particular even numbers such as 6 or 8 or 60. This first example is extremely simple. It illustrates quite correctly the role of generalization, specialization, and analogy in inductive reasoning. Yet we should examine less meager, more colorful illustrations and, before that, we should discuss generalization, specialization, and analogy, these great sources of discovery, for their own sake. 2. Generalization is passing from the consideration of a given set of objects to that of a larger set, containing the given one. For example, we generalize when we pass from the consideration of triangles to that of polygons with an arbitrary number of sides. We generalize also when we pass from the study of the trigonometric functions of an acute angle to the trigonometric functions of an unrestricted angle. It may be observed that in these two examples the generalization was effected in two characteristically different ways. In the first example, in passing from triangles to polygons with n sides, we replace a constant by a variable, the fixed integer 3 by the arbitrary integer n (restricted only by the inequality n _: 3). In the second example, in passing from acute angles to GENERALIZATION, SPECIALIZATION, ANALOGY 13 arbitrary angles a, we remove a restriction, namely the restriction that 0° < a < 90°.

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  The Process of Learning Mathematics

  The Commonwealth and International Library: Mathematical Topics

  • L. R. Chapman(Author)
  • 2016(Publication Date)
  • Pergamon

    (Publisher)

  9 8 C. PLUMPTON discovery that a general rule extends beyond the first few known cases; while he states that abstraction is, the awareness that the rule applies in a number of other situations. He has attempted to classify various levels of generalisation in the following manner: 1. extension of the rule from a finite number of cases to an in-finite number; 2. the generalisation from one infinite class to another infinite class; 3. what Dienes terms a mathematical generalisation or the formation of an isomorphism between one class and a sub-class of another class. The following example will make the point clearer. A child given the function y = 2x, where x is a whole number, will find the value of y given a value of x. This will be repeated for a finite number of values of x until suddenly the extension is made that given any value of x, then the corresponding value of y can be found. This is the first and easiest stage. If the coefficient of x is changed to 3, or 4, or 5, or to any whole number, the generalisation is made between the infinite number of cases when, say, y = 2x and y = 3x 9 with x e {1, 2, 3, 4, 5, . . . } . The process is developed further with the realisation that the coefficient of x can be generalised and we can write y = ax. The third stage in his classification may be illustrated: given [y] = [a] [x] 9 where y] [a] and [ JC ] are 1 x 1 matrices, we can generalise to y = ax 9 where y 9 a and x are n x n matrices. Here the first case is isomorphic to and is a sub-class of the second case. GENERALISATION AND STRUCTURE 9 9 The psychological aspects of the process of generalisation may suggest a partial explanation, or give us a small clue to the build up of intrinsic motivation. Perhaps we can take a tiny step forward to stating why some children like mathematics. Dienes argues that a mathematician experiences the feeling of power and is ex-hilarated when he completes the generalisation.

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  Necessity and Language

  • Morris Lazerowitz, Alice Ambrose(Authors)
  • 2016(Publication Date)
  • Taylor & Francis

    (Publisher)

  6 MATHEMATICAL GENERALITY    

  Induction is the process of discovering general laws by the observation and combination of particular instances. It is used in all sciences, even in mathematics … We may observe, by chance, that 1 + 8 + 27 + 64 = 100 and, recognising the cubes and the square, we may give to the fact we observe the more interesting form

  13 + 23 + 33 + 43 = 102 .

  … Does it often happen that such a sum of successive cubes is a square? … In asking this we are like the naturalist who, impressed by a curious plant or a curious geological formation, conceives of a general question. Our general question is concerned with the successive cubes

  13 + 23 + 33 + 43 + … + n3

  We are led to it by the ‘particular instance’ n = 4…. The special cases n = 2,3 are still simpler, the case n = 5 is the next one…. Arranging neatly all these cases, as a geologist would arrange his specimens of a certain ore, we obtain the following table:

  1                                  = 1 = 12

  1 + 8                            = 9 = 32

  1 + 8 + 27                    = 36 = 62

  1 + 8 + 27 + 64            = 100 = 102

  1 + 8 + 27 + 64 + 125  = 225 = 152 .

  It is hard to believe that all these sums of consecutive cubes are squares by mere chance…. In a similar case, the naturalist would have little doubt that the general law suggested by the special cases heretofore observed is correct. Here the following theorem is strongly suggested by the induction:

  The sum of the first n cubes is a square.

  … In mathematics as in the physical sciences we may use observation and induction to discover general laws. … Many mathematical results were found by induction first and proved later. Mathematics presented with rigour is a systematic deductive science but mathematics in the making is an experimental inductive science.1

  This quotation presents an illustration of the analogies mathematicians find between mathematical and empirical investigations. It makes number theory appear ‘as the natural history of the domain of numbers’,2 and it seems entirely natural that it should do so. Proceeding ‘by induction from numerical examples’,3 a mathematician will frequently describe his examination of special cases as confirming a conjecture, or supporting a generalisation, and go on to predict that subsequent ‘empirical evidence’4 will bear out the conjecture or generalisation. Propositions which have been ‘reached and stated as probably true by induction’5

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  Formalism and Beyond

  On the Nature of Mathematical Discourse

  • Godehard Link(Author)
  • 2014(Publication Date)
  • De Gruyter

    (Publisher)

  Universal Algebra as being a direct contribution to such “interesting generalizations” [53, VIII]. This, importantly, also affects the relationship that holds between the new, general theories within mathematics and the more special ones. Not only philosophers such as Husserl (see his remarks on the different forms of generalization, section III), but also mathematicians such as Hankel [20, p. 12] and philosophers/historians of science commenting on these developments have emphasized the necessity of distinguishing between (in Nagel’s terms; [38, p. 167]) “exhibiting” or exemplifying and “deriving” the principles or the results of mathematics. Both Hankel and Nagel view the first of these methods, that of “exhibiting” the principles, as the mathematically more relevant step.

  The paradigm example of the introduction of the complex numbers, therefore, with the ensuing changes that had to be made regarding the concept of number, of quantity, and thereby of (large parts of) mathematics, allows us to summarize these developments, and to embed them into yet broader contexts. The dialectic inherent in aiming simultaneously at an extension of mathematics and at a thoroughgoing revision of basic concepts in mathematics can, at a very abstract level, be seen back in the development (and in the very term!) of metamathematics, and in the host of new, increasingly more general definitions of mathematics that all go beyond defining mathematics via some notion of quantity.

  This program has been taken up by a surprisingly wide range of authors from many different fields, and from very different backgrounds. The academic landscape of the late 19th century was concerned with – one might even say: infatuated with – the search for a systematization of all the individual sciences that seemed to increase steadily in number and specialized complexity [55]. Two features of these debates are particularly remarkable in the present context: a surprising liberalism that could allow for different types of science (in the broad sense) to coexist harmoniously; and the fascination with the search for ever more general sciences, dealing with ever more general types of objects, but still meaningful according to traditional standards. Examples are Husserlian phenomenology with – specified for the subject matter here at hand – its interest in “forms of theories”; Meinongian “Gegenstandstheorie”; Ostwald’s philosophy of nature, which he combined with his energetics and a strong program in the systematization of the sciences; Cassirer’s theory of relational concepts. All these programs interact on many levels: Ostwald and Whitehead share the program of newly establishing a philosophy of nature while at the same time taking recent developments in mathematics into account ([55, ch.IV.7,VII]; [56]); Russell does indeed read texts by Ostwald in the years preceding his Principles ;247 Carnap meets with Cassirer and quotes virtually all the authors mentioned in the above reconstruction of philosophico-mathematical issues in his Logischer Aufbau ([6, p. 3-4]; [14]). Interestingly, some of the most innovative ideas in 19th

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  Teachers Engaged in Research

  • Stephanie Z. Smith, Marvin E. Smith(Authors)
  • 2012(Publication Date)
  • Information Age Publishing

    (Publisher)

  Their work shows that while children implicitly use basic mathematical properties to solve problems, the properties are not made explicit, and often are not generalized to being true for all numbers. They argue that it is important for children to have opportunities for making implicit gener-alizations explicit. They state: Our goal is to make these properties the explicit focus of attention so that • All students have access to basic mathematical properties; • Students understand why the computation procedures they use work the way they do; • Students apply their procedures flexibly in a variety of contexts; • Students recognize the connections between arithmetic and alge-bra and can use their understanding of arithmetic as a foundation for learning algebra with understanding. (pg. 5) Researchers are taking a closer look at algebra in the elementary school (Ball & Bass, 2000; Bastable & Schifter, 1998; Carpenter, Franke & Levi, 2003; Carpenter & Levi, 1999; Carraher, Schliemann, & Brizuela, 2001; Kaput & Blanton, 1999; Schifter, 1999). Common to all of these research projects is the importance of generalized statements, often referred to as con-jectures, and the justification of those ideas in the form of argument or proof. Carpenter, Franke, and Levi (2003) promote the engagement of ele-mentary students in generating and justifying conjectures. They write, “We want students to learn the importance of expressing [the fundamental properties of arithmetic] precisely and accurately using words and sym-bols” (p. 5). They have found that generating conjectures leads students to think about mathematical ideas and is a basis for mathematical argument. They emphasize the importance of students engaging in the justification of the conjectures. Knuth, (2002) promotes proof as a tool for learning mathematics with meaning.

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  Learning and Motivation in the Classroom

  • Scott G. Paris, Gary M. Olson, Harold W. Stevenson(Authors)
  • 2017(Publication Date)
  • Taylor & Francis

    (Publisher)

  It is significant that a plausible simulation of meaningful learning could be formulated using only knowledge structures of relatively standard kinds. Schemata with characteristics that are standard in knowledge representation systems and learning processes that are commonly proposed for learning from examples were combined in a straightforward way to form a simulation of learning with significant structural understanding. This seems quite encouraging for the prospect of developing definite hypotheses about understanding over a substantial range of instructional tasks.

  UNDERSTANDING A GENERAL FORMAL PRINCIPLE

  The second analysis I present was concerned with a somewhat different aspect of understanding, involving a formal principle. This study, in which I collaborated with Maria Magone, was concerned with geometry students’ understanding of the general concept of proof. This constitutes a metaprinciple in relation to the tasks that students are taught to perform in their study of geometry. The skills that students are required to learn involve constructing proofs. The question addressed in this study is whether students understand what proofs are—that is, whether they know the general features that are required for something to be a proof.

  The concept we focused on can be called the principle of deductive consequence. In a formal proof, each assertion that is made is deductively required by the premises of the problem or by other assertions that have been made explicitly. This imposes a strong criterion for statements to be acceptable for inclusion in a proof. For each statement that is put into a proof, there must be a sufficient basis in earlier statements that the added statement must

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  Exploring Mathematics

  An Engaging Introduction to Proof

  • John Meier, Derek Smith(Authors)
  • 2017(Publication Date)
  • Cambridge University Press

    (Publisher)

  All the evidence you collect, short of a mathematical proof, only shows the plausibility of a statement, not its veracity. In any mathematics book you will find a number of words that usually appear in a special font or style at the start of a paragraph. Often they signal the beginning of a mathematical statement. A definition is a statement that precisely describes the meaning of a word, phrase, or notation. Often the term being defined is presented in italics (or is underlined when you are working at a chalkboard). A conjecture is a mathematical statement that has neither been proven nor disproven. Usually there is some sort of evidence, experimental or intuitive, that strongly supports the idea that the conjecture is true. A theorem is a mathematical statement that has been proven. Directly related to this term are • proposition: a theorem that is not so grand; • lemma: a theorem that is either a significant step in the proof of a theorem or proposition, or a theorem that is used in the proofs of many theorems or propositions; • corollary: a theorem that follows immediately, either by specialization or by a short argument, from a preceding lemma, proposition, or theorem. Typically a chapter or article in mathematics will begin with a brief introduction, followed by one or more definitions. Then there will perhaps be an example or two, which motivate the statement of a theorem. This is followed by a proof of the theorem, and then possibly by some corollaries that might be of interest in their own right. Exercise 2.33 Revisit examples of definitions, theorems, propositions, lemmas, and corollaries that have already occurred in the book to this point. Exercise 2.34 Exercise 1.5 asked you to “guess” a certain connection between two mathematical objects, which you can then use to fill in the blanks for Exercise 2.61. Do you believe that you generated enough evidence in Exercise 1.5 to be able to call your guess a conjecture? 2.8.

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