Drawing Conclusions from Examples

3 Key excerpts on "Drawing Conclusions from Examples"

• 

  eBook - ePub

  Fostering Children's Mathematical Power

  An Investigative Approach To K-8 Mathematics Instruction

  • Arthur Baroody, Arthur J. Baroody, Jesse L.M. Wilkins, Ronald T. Coslick(Authors)
  • 1998(Publication Date)
  • Routledge

    (Publisher)

  Reasoning is a critical tool for many aspects of school and everyday life. Finding patterns and using if-then (deductive) reasoning can be central to science and other content areas. Critically evaluating the arguments of lawyers, advertisers, politicians, journalists, or others’ is essential to make many everyday decisions and to avoid being hoodwinked.

  Limitations and Misuses To evaluate their own and others’ conclusions effectively, students need to know the limitations and pitfalls of each type of reasoning (see Part I on page 2–26 of Investigation 2.4).

  Investigation 2.4: Evaluating Resasoning

  ◆ Reflecting on the limits of reasoning + informally evaluating conclusions ◆ Whole class

  An ability to evaluate their own and others’ conclusion is an essential skill for everyday life as well as mathematics. The questions of Part I are designed to prompt reflection about the limitations of intuitive, inductive, and deductive reasoning. Parts II to IV illustrate informal methods for evaluating a conjecture or a logical conclusion.

  Part I: Limitations of Reasoning (◆ 5–8)

  1. (a) Which straight line to the right is longer? (b) What kind of reasoning would children probably use to draw their conclusion? (c) How could they evaluate (check) their conclusion? (d) What point can this activity demonstrate?
  2. (a) Can you guess the rule used to choose the following numbers: 3,5,7? (b) What kind of reasoning was involved in drawing your conclusion? (c) Compare your conclusion to those of your group or class. Did everyone come to the same conclusion? (d) Even if everyone came to the same conclusion, is it necessarily true? Why or why not? (e) What point can this activity demonstrate?
  3. Consider deductive arguments A and B below, (a) Is each valid (logical)? (b) Is the conclusion of each true (reasonable)? (c) What point about deductive reasoning does this activity demonstrate?

     Argument A:	Argument B:
     Teachers are grossly overpaid. (Premise 1)	Teachers are slaves. (Premise 1)
     Winnie is a teacher. (Premise 2)	Slaves are underpaid. (Premise 2)

  Sign up to read

  Learn more about book
• 

  eBook - PDF

  The Notion of mathematical proof: Key rules and considerations

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

    (Publisher)

  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. The Notion of Mathematical Proof: Key Rules and Considerations 44 The ability to make some generalizations and to provide appropriate reasoning are both considered as high-level mathematical skills. Pedemonte further [34] studied certain relations between the level of the generalization and reasoning skills. However, generalization as such and concurrently as a process appears to be required for construction of mathematical induction, although, on the other hand, some relation was pointed out as a structural distance between them. Therefore, just the open-ended problems are considered suitable ways to develop required reasoning skills by students. Thus, the abductive reasoning serves for passing from the inductive to deductive reasoning [26]. It is not clear how the classifications or characteristics of generalization and justification arising from the studies that have focused on generalizing a number patterns apply to other mathematics topics or tasks. Nor do these frameworks make clear other reasoning processes that enable generalization and justification to occur [35]. Combinatorics has a special place in the field of mathematics and mathematics education. It is considered as a source of frequent and hard difficulties of students at various levels [15]. The fact is that just the combinatorial thinking presents the process of creating various possible combinations of ideas and cognitive operations [36]. The use of combinatorial thinking in combinatorial situations means developing a special ability to create abstract models and to find the structure of a set of outcomes [37].

  Sign up to read

  Learn more about book
• 

  eBook - PDF

  Using Internet Primary Sources to Teach Critical Thinking Skills in Mathematics

  • Evan M. Glazer(Author)
  • 2001(Publication Date)
  • Libraries Unlimited

    (Publisher)

  I USING CRITICAL THINKING SKILLS IN MATHEMATICS CHARACTERISTICS OF CRITICAL THINKING Scholars have varying definitions of critical thinking, and some suggest that we generate a math-specific description because the domain “has different criteria for good reasons from most other fields, because mathe- matics accepts only deductive proof, whereas most fields do not even seek it for the establishment of a final conclusion” (Ennis, 1989, p. 8). I will state a working definition of critical thinking in mathematics based on a synthesis of general definitions, as well as those that relate and apply to mathematics. The purpose of this working definition is to pro- vide a framework and common understanding of the ideas and activities presented in this book. Critical thinking in mathematics is the ability and disposition to incorporate prior knowledge, mathematical reasoning, and cogni- tive strategies to generalize, prove, or evaluate unfamiliar mathematical situa- tions in a reflective manner. Figure 1 is an illustration that describes the general connection between each of these components, yet keep in mind that this process can be more complex. I will elaborate on each of the components of this working definition with examples and connections to the World Wide Web and then conclude with strategies that promote critical thinking in the mathematics curriculum and classroom. An illus- trated classroom experience using Internet primary sources is described at the end of the primary sources chapter. Ability and Disposition Ability refers to a skill or power to demonstrate something. Abilities used in critical thinking are support, inference, clarification, and strategies 14 Using Internet Primary Sources Figure 1 The Relationship Between Critical Thinking Criteria in Mathematics (see Ennis, 1987, for a detailed taxonomy). Support refers to examining, observing, and judging the credibility of a source, such as the level of expertise used in or created by the source.

  Sign up to read

  Learn more about book