Drawing Conclusions from Examples
Drawing conclusions from examples in mathematics involves using specific instances to make generalizations or infer broader principles. By analyzing patterns and relationships within the examples, mathematicians can derive overarching rules or theorems that apply to a wider range of situations. This process of induction is fundamental to the development and understanding of mathematical concepts.
6 Key excerpts on "Drawing Conclusions from Examples"
eBook - ePub- W. James Potter(Author)
- 2019(Publication Date)
- SAGE Publications, Inc(Publisher)
...Chapter 7 Deducing: Reasoning with Logic from General Principles I. The Deduction Algorithm Step 1: Begin With an Observation Step 2: Identify a Relevant General Principle Step 3: Use the Two Premises to Reason Logically to a Conclusion II. Heuristics Heuristic 1: Probability Premise Heuristic 2: Conditional Reasoning III. Avoiding Traps Trap 1: Faulty Major Premise Trap 2: Irrelevant Major Premise Trap 3: Too Complex a Major Premise Is Needed Trap 4: Conditional Reasoning Trap 5: Irrational Reasoning Trap 6: Unwillingness to Build Knowledge Structures IV. Chapter Review Exercises Deduction is the skill of using a few premises to reason logically toward a conclusion. The basic procedure of deduction follows a reasoning process in the form of a syllogism, which is a set of three statements. The first statement in the set is called the major premise ; it is usually a general principle or rule. The second statement is called the minor premise ; it is usually an observation. The third statement is the reasoned conclusion. We use logic to see if the observation fits the rule and then derive the conclusion. Perhaps the most familiar example of a syllogism is the one that uses the following two premises: (1) All men are mortal and (2) Socrates is a man. From this we can conclude that Socrates is mortal. The first premise is the major one; that is, it states a general proposition. The second premise is the minor one; that is, it provides information about something specific (in this case, a specific person) in a way that relates it to the major premise. Using logic, we see that the observation in the second premise fits the rule in the first premise and we conclude that Socrates is mortal. Deduction is the skill that the fictional detective Sherlock Holmes employed so successfully to make sense of clues and solve crimes. He knew a great deal about the physical world and about human behavior; this knowledge was his bank of major premises...
eBook - ePubThought and Knowledge
An Introduction to Critical Thinking
- Diane F. Halpern(Author)
- 2013(Publication Date)
- Psychology Press(Publisher)
...75) A distinction is often made between inductive and deductive reasoning. (See the Thinking as Hypothesis Testing chapter for a related discussion of this topic.) In inductive reasoning, observations are collected that support or suggest a conclusion. It is a way of projecting information from known examples to the unknown (Heit, 2000). For example, if every person you have ever seen has only one head, you would use this evidence to support the conclusion (or suggest the hypothesis) that everyone in the world has only one head. Of course, you cannot be absolutely certain of this fact. It is always possible that someone you have never met has two heads. If you met just one person with two heads, your conclusion must be wrong. Thus, with inductive reasoning you can never prove that your conclusion or hypothesis is correct, but you can disprove it. With inductive reasoning, if the premises are true, the conclusion is probably true. In deductive reasoning, we begin with statements known or believed to be true, like “everyone has only one head,” and then conclude or infer that La Tisha, a woman you have never met, will have only one head. This conclusion follows logically from the earlier statement. If we know that it is true that everyone has only one head, then it MUST be true that any specific person will have only one head. Similarly, if I show you a rectangle that is 2’ by 3’, then the area of the rectangle must be 6 square feet. Deductive reasoning is sometimes described as reasoning “down” from beliefs about the nature of the world to particular instances. Rips (1988) argues that deduction is a general purpose mechanism for cognitive tasks: deduction “enables us to answer questions from information stored in memory, to plan actions according to goals, and to solve certain kinds of puzzles” (p...
eBook - ePubThought and Knowledge
An Introduction to Critical Thinking
- Diane F. Halpern, Dana S. Dunn(Authors)
- 2022(Publication Date)
- Routledge(Publisher)
...75) A distinction is often made between inductive and deductive reasoning. (See Chapter 6 “Thinking as Hypothesis Testing” for a related discussion of this topic.) In inductive reasoning observations are collected that support or suggest a conclusion. It is a way of projecting information from known examples to the unknown (Heit, 2000). For example, if every person you have ever seen has only one head, you would use this evidence to support the conclusion (or suggest the hypothesis) that everyone in the world has only one head. Of course, you cannot be absolutely certain of this fact. It is always possible that someone you have never met has two heads. If you met just one person with two heads, your conclusion must be wrong. Thus, with inductive reasoning you can never prove that your conclusion or hypothesis is correct, but you can disprove it. With inductive reasoning, if the premises are true, the conclusion is probably true. In deductive reasoning, we begin with statements known or believed to be true, like “everyone has only one head,” and then conclude or infer that La Tisha, a woman you have never met, will have only one head. This conclusion follows logically from the earlier statement. If we know that it is true that everyone has only one head, then it MUST be true that any specific person will have only one head. Similarly, if we show you a rectangle that is 2’ by 3’, then the area of the rectangle must be 6 square feet. Deductive reasoning is sometimes described as reasoning “down” from beliefs about the nature of the world to particular instances. Rips (1988) argues that deduction is a general-purpose mechanism for cognitive tasks—deduction “enables us to answer questions from information stored in memory, to plan actions according to goals, and to solve certain kinds of puzzles” (p...
eBook - ePubCritical Thinking Across the Curriculum
A Brief Edition of Thought & Knowledge
- Diane F. Halpern(Author)
- 2014(Publication Date)
- Routledge(Publisher)
...It is a description of a trial for an accused child molester: “Telling an attentive jury in Los Angeles Superior Court that the totality of evidence ‘draws the ring around R. B. and P. B. closer and tighter, to the extent that you should find them guilty.’” Here the idea of drawing smaller and smaller circles was used explicitly in the prosecution's arguments. Once you get used to using diagrams as a thinking aid, you will find that you will use them often. APPLYING THE FRAMEWORK 1. What is the goal? In deductive reasoning, the goal is to determine which conclusions are valid given premises or statements that we believe are true. When you identify your goal as a deductive reasoning task, you will use the reasoning skills presented in this chapter. 2. What is known? In everyday prose, you will have to convert phrases and sentences into a reasoning format. You will have to determine what the premises are before you can decide whether they support a conclusion. Often quantifiers are missing and conclusions are left unstated. Sometimes you will have to consider context to decide if “if, then” really means “if, and only if.” Perhaps, most important, you need to determine if you should be assuming that the premises are true. A conclusion is valid if it follows from the premises, but good reasoning from poor premises will not produce desirable outcomes. 3. Which thinking skill will get you to your goal? The following skills to determine whether a conclusion is valid were presented in this chapter. Review each skill and be sure that you understand how and when to use each one: ● Discriminating between deductive and inductive reasoning. ● Using linear diagrams to solve linear syllogisms. ● Using the principles of linear orderings as an aid to clear communication. ● Reasoning with “if, then” statements. ● Using tree diagrams with “if, then” statements. ● Avoiding the fallacies of confirming the consequent and denying the antecedent. 4...
eBook - ePubWe Reason & We Prove for ALL Mathematics
Building Students' Critical Thinking, Grades 6-12
- Fran Arbaugh, Margaret (Peg) S. Smith, Justin D. Boyle, Gabriel J. Stylianides, Michael D. Steele(Authors)
- 2018(Publication Date)
- Corwin(Publisher)
...For example, if a group of students perceives that empirical arguments are sufficient for proving a conjecture (like Student D in Figure 3.2), knowing that information can help you engage the students in ways that challenge this common misconception (by enacting, for example, the series of tasks from Chapter 2). If a student believes that symbolic algebra is required for a proof to be valid, then you might introduce a conjecture or task situation that is difficult to prove algebraically, like the Squares Problem in Chapter 2, which can be proven via proof by induction using algebra—but that is pretty sophisticated for most secondary students. At the end of this chapter, we discuss how we can prove the conjecture generated from your work on the Squares Problem in a way that is accessible for secondary students. The mathematical component of the framework is a bit different from the learner and pedagogical components. It has multiple parts and answers the question, “What are the major activities involved in reasoning-and-proving?” In Chapter 1, you read about the stages of the reasoning-and-proving process: identifying a pattern, making a conjecture, then providing an argument (which may or may not qualify as a proof). Recall that working through these stages may not be linear—that the mathematical work done during reasoning-and-proving activities may include revisiting previous stages before completing a mathematical argument. We unpack the three stages further in the sections below. As you learn about and make sense of the mathematical component of the framework, it will be helpful throughout these next sections to continually refer to the Reasoning-and-Proving Framework found in Figure 3.4. Look for a printable version of the framework on the companion website. resources.corwin.com/reasonandprove Download a printable version of the Reasoning- and-Proving Framework Identifying a Pattern Children spend a lot of time in elementary school on patterning activities...
eBook - ePubThinking Critically About Critical Thinking
A Workbook to Accompany Halpern's Thought & Knowledge
- Diane F. Halpern, Heidi R. Riggio(Authors)
- 2013(Publication Date)
- Routledge(Publisher)
...The skills developed in this chapter are most similar to those used in college-level mathematics as they provide a single correct answer and require careful consideration and execution of all steps in the process. Deductive reasoning is based on the assumptions that if certain information is true, then there are conclusions that must also be true. Both spatial and verbal strategies are used to help readers practice both modes of thinking. Common biases and errors in deductive reasoning are introduced. Many of these are discussed in several other chapters using different perspectives. The use of deductive reasoning skills in real world contexts is also highlighted so that readers can recognize when deductive reasoning skills are needed and when they are being persuaded with deductive reasoning techniques. Name: Date: Course/Section: Journal Entries Record your thoughts as you reflect on the material in the Deductive Reasoning chapter. The purpose of this journal is to let you step back and reflect on the material that you are learning. It is a place to record your discussions with yourself. Write about topics that are unclear or seem particularly useful to you. This is the time to make your own connections between and within chapters, from class, and the real-life that happens out of class. Use the next page, for your second entry. _______________________________ Date of first entry Name: Date: Course/Section: Journal - Second Entry for Chapter 4: _______________________________ Date of second entry Review of Deductive Reasoning Skills Category description: The skills presented in this chapter are used to determine if a conclusion is valid--that it, it must be true if the premises are true. These skills are used in many contexts including law, medicine, financial projections, and the science. Skill Description Example of Use a...
