Part II
- Computational Thinking at first glance – What, why and how
1.1. Definition of Computational Thinking (CT) Is that a fact? There is still no universally accepted definition of “Computational Thinking”. The concept of computational thinking (CT) was first introduced by an educationalist Seymour Papert in 1967 talking about LOGO, the programming language he developed at MIT (Massachusetts Institute of Technology) to teach programming to children. He was convinced that the use of computers could foster formal thinking in children and, in particular, could allow children to autonomously “construct” their learning and thinking. The concept of CT was then revitalized in 2006 by a computer scientist Jeannette M. Wing who, in the article “Computational Thinking”, argued that it addresses the conundrum of machine intelligence by asking what machines do better than man and what man does better than machines. Wing argues that computational thinking is not simply a procedural coding activity, but is a basic conceptual skill that, along with reading, writing and arithmetic, should be taught to all children. It appears that computational thinking purports to be critical thinking in evaluating situations and an advanced problem-solving ability using computerized tools. If computer science is the science of what can be computerized and how to computerize it, however, computational thinking is not a skill unique to computer scientists. It allows problems to be solved, a system to be designed and human behaviour to be understood in everyday life, in an alternative way, through the fundamental concepts of information technology.
Key Skills for Computational Thinking There are four key skills in computational thinking:
1 Decomposition
2 Pattern Recognition
3 Pattern Abstraction
4 Algorithm Design
1) Decomposition Breaking down big and difficult problems into something much simpler. Often big problems are just many little problems put together. Decomposition is an important life skill to be relied on in the future when students and adults need to take on larger tasks. Students will learn ways to delegate group projects and build time management skills
2) Pattern Recognition Sometimes when a problem is made up of many small bits, you will notice that these bits have something in common. If they do not, they could, however, have strong similarities with the pieces of another problem that has already been solved. If you are able to find these regularities, it will become a lot easier to identify the individual pieces: pattern recognition is simply looking for patterns in the puzzles and determining if any of the problems or solutions we encountered in the past may apply to a present situation. May, what we learned in the past, help us sort out the actual problem? If you have ever built a piece of IKEA furniture, you will understand the importance of patterns. While assembling an IKEA drawer unit, it is likely to take you much longer to assemble the first drawer than the fourth or fifth. When we repeat steps in our assembling process we learn how to solve the instructions more quickly and learn from our mistakes. The painstaking process of assembling that first part teaches us the skills to perform the process more efficiently in the future.
3) Pattern Abstraction Once you have located a pattern, it is possible to abstract (ignore) the details that differentiate the various things and use general techniques for finding solutions that work for more than one problem. Identifying the crucial information in a problem and disregarding the irrelevant information is one of the hardest parts of computational learning.
4) Algorithm design When the solution is ready, it is possible to write it down so it can be executed step by step. This makes easier to obtain the expected results. Algorithm design is setting out the steps and rules needed to follow in order to achieve the same desired outcome every time.
Concepts of computational thinking Computational thinking is a cognitive or thought process involving logical reasoning by which problems are solved and artefacts, procedures and systems are better understood. It embraces:
● the ability to think algorithmically;
● the ability to think in terms of decomposition;
● the ability to think in generalisations, identifying and making use of patterns;
● the ability to think in abstractions, choosing good representations; and
● the ability to think in terms of evaluation. Computational thinking skills enable pupils to access parts of the Computing subject content. Importantly, they relate to thinking skills and problem solving across the whole curriculum and through life in general. Computational thinking can be applied to a wide range of artefacts including: systems, processes, objects, algorithms, problems, solutions, abstractions, and collections of data or information. In the following discussion of concepts, artefact refers to any of these. Logical reasoning Logical reasoning enables pupils to make sense of things by analysing and checking facts through thinking clearly and precisely. It allows pupils to draw on their own knowledge and internal models to make and verify predictions and to draw conclusions. It is used extensively by pupils when they test, debug, and correct algorithms. Logical reasoning is the novel application of the other computational thinking concepts to solve problems.
Design and technology pupils, designing a model of a truck, choose materials for different elements of a project. They are employing generalisation when they recognise that the properties of a material used in one situation make it suitable to use in another completely different context. Being able to divide the new project into different parts, requiring different materials, is an example of decomposition. The pupil is using logical reasoning to design a truck. Pupils use logical reasoning when learning about gravity using a weighted string suspended from the lid of a glass jar. Before tilting the jar, pupils can make predictions about the behaviour of the weighted string. They can then evaluate the results of their tests. They may be able to generalise the behaviour to other situations such as a crane. The novel use in understanding a property of gravity is logical reasoning. Logical reasoning is key in allowing pupils to debug their code. They can work with peers to evaluate each other’s code, to isolate bugs, and to suggest fixes. During this process, they may have opportunities to employ abstraction, evaluation, and algorithmic design. The novel use in correcting mistakes in code requires logical reasoning.
Abstraction in the Computer Science Classroom
Although we have presented relevant characterisations of abstraction, it is still far from clear how an abstraction-oriented perspective could become part of the pedagogical practice. Already in the late 1990s, as a revision of the spreading object-first orientaTable 1
PGK Hierarchy levels from Perrenet et al. (2005) and their mapping to K-5 settings according to Waite et al. (2018). PGK Level Definition K-5 name K-5 question Problem Algorithm perceived as a problem solving strategy problem “What is needed” Object (algorithm) Algorithm understood independently of any specific implementation design “What it should do” Program Algorithmic grasp of the program code “How it is done” Execution Focus on individual runs with specific inputs running the code “What it does” 630 C. Mirolo et al. tion, Machanick (1998) endorsed an abstraction-first instructional approach where the implementation of abstract data types is delayed as much as possible in order to stress an abstract view of the models.
Kramer remarked that abstraction per se is not the subject of any computing course, but that all computing courses “rely on or utilize abstraction to explain, model, specify, reason or solve problems,” so confirming that “abstraction is an essential aspect of computing, but that it must be taught indirectly through other topics” (Kramer, 2007, p. 41). In line with Kramer’s remark, Hazzan (2008) discussed abstraction as a soft idea, “that can be neither rigidly nor formally defined, nor is it possible to guide students as to its precise application.” And although “it is not a trivial matter,” like other soft ideas, abstraction should be taught in a computer science curriculum.
Then, a small number of educators have provided guidelines to teach abstraction at different instruction levels. Hence, this section briefly explores their approaches to foster and assess abstraction skills. 5.1. Teaching to Trigger Abstraction in Computer Science Often instructors aim to develop students’ abstraction skills indirectly, by devising particular learning trajectories that are supposed to foster higher-level thinking and require students to use abstraction to succeed. In a program development project, for example, they could assign refactoring tasks in which learners are asked to look for recurrent patterns of code and to re-organise the code by introducing meaningful procedural and/or data abstractions with the purpose of making the whole program easier to read, debug and modify. In the following paragraphs we will outline a selection of representative approaches to (an implicit) abstraction-oriented instruction. Pattern-oriented instruction. This approach has the aim of improving students’ competencies in algorithmic problem solving (Muller and Haberman, 2008).
An algorithm is indeed seen by these authors as a combination of plan patterns in Soloway’s sense (Soloway, 1986), resulting via sequencing, nesting or merging plans from a repertory of basic algorithmic patterns specifically designed for pedagogical purposes. In Muller and Haberman’s scenario, abstraction plays a crucial role in pattern recognition, chunking, and problem structure identification. Their approach relies on having an appropriate pattern repository, as well as on presenting carefully selected problems of gradually increasing difficulty; teachers should then discuss and compare different solutions to a given problem in terms of pattern composition. Additional guidelines for pattern-oriented instruction include:
(1) patterns should be abstracted from related examples or by generalising a simpler problem,
(2) patterns should be revisited in different contexts, in order to make the identification of common problem features easier, and (3) similarities, differences, and also possible misuses of patterns should be considered. According to Muller and Haberman, comparative studies appear to show that novices exposed to this approach develop enhanced problem solving abilities. Multiple representations perspective. Dealing with multiple representations of a given phenomenon can play a key role in the development of abstract concepts. Ac- cording to Ainsworth (2008), in particular, in order “to construct a deeper understanding of a domain,” if the learners “fail to relate representations, then processes like abstraction cannot occur. Moreover, although learners find it difficult to relate different forms of representations, if the representations are too similar, then abstraction is also unlikely to occur.” She then recommends that teachers should foster abstraction over multiple representations “by providing focused help and support on how to relate representations and giving learners sufficient time to master this process.” In this respect, Gautam et al. (2020) have recently proposed an interesting interdisciplinary approach to integrate science (namely, chemistry) and computational thinking in the curriculum. While abstraction is usually “presented as hierarchical” in terms of (i) extracting important features and ignoring unimportant ones, and (ii) finding commonalities across contexts, in their standpoint “abstraction in science” as well as in computing “requires students to move laterally across different representations of the concepts or actions.” In the reported study, the micro-level process of photosynthesis was modeled by a code snippet, and by discussing commonalities and differences between, e.g., a whiteboard and the code representation of the implied chemical reaction, “the instructor pushed students towards higher-level abstract thinking.” Moreover, they suggest to allow for friction emerging when the students explore different representations, in that it encourages to consider alternative views and “negotiate the elements with one another.” According to the authors, this approach “created meaningful accounts of science phenomenon and the science provided access to how computation embeds ideas.” Exploration of artefacts.
A more recent pedagogical trend in programming education attempts to trigger abstraction through activities inspired by the use-modify-create framework. The idea is that the understanding of artefacts such as programs would gradually progress through three major stages, corresponding to
- exploration via passive use (as a consumer),
- (ii) experimentation of the internal machinery by modifying some features, and finally
- (iii) creation of new, original artefacts to achieve specific goals. While discussing the use-modify-create approach, Lee et al. (2014) observe that abstraction, as well as other computational thinking abilities, are “not explicitly taught but rather [develop] through one’s impetus to create;” nevertheless, in this progression the abilities to modify and, later, to create imply the enhancement of learner’s abstraction skills.
This is a course on discrete mathematics as used in Computer Science. It’s only a one-semester course, so there are a lot of topics that it doesn’t cover or doesn’t cover in much depth. But the hope is that this will give you a foundation of skills that you can build on as you need to, and particularly to give you a bit of mathematical maturity—the basic understanding of what mathematics is and how mathematical definitions and proofs work.
- So why do I need to learn all this nasty mathematics?
Why you should know about mathematics, if you are interested in Computer Science: or, more specifically, why you should take CS202 or a comparable course:
• Computation is something that you can’t see and can’t touch, and yet (thanks to the efforts of generations of hardware engineers) it obeys strict, well-defined rules with astonishing accuracy over long periods of time.
• Computations are too big for you to comprehend all at once. Imagine printing out an execution trace that showed every operation a typical $500 desktop computer executed in one (1) second.
If you could read one operation per second, for eight hours every day, you would die of old age before you got halfway through. Now imagine letting the computer run overnight. So in order to understand computations, we need a language that allows us to reason about things we can’t see and can’t touch, that are too big
for us to understand, but that nonetheless follow strict, simple, well-defined rules. We’d like our reasoning to be consistent: any two people using the language should (barring errors) obtain the same conclusions from the same information. Computer scientists are good at inventing languages, so we could invent a new one for this particular purpose, but we don’t have to: the exact same problem has been vexing philosophers, theologians, and mathematicians for much longer than computers have been around, and they’ve had a lot of time to think about how to make such a language work.
Philosophers and theologians are still working on the consistency part, but mathematicians (mostly) got it in the early 20th-century. Because the first virtue of a computer scientist is laziness, we are going to steal their code.
- But isn’t math hard? Yes and no. The human brain is not really designed to do formal mathematical reasoning, which is why most mathematics was invented in the last few centuries and why even apparently simple things like learning how to count or add require years of training, usually done at an early age so the pain will be forgotten later. But mathematical reasoning is very close to legal reasoning, which we do seem to be very good at.1 There is very little structural difference between the two sentences: 1. If x is in S, then x + 1 is in S. 2. If x is of royal blood, then x’s child is of royal blood.
But because the first is about boring numbers and the second is about fascinating social relationships and rules, most people have a much easier time deducing that to show somebody is royal we need to start with some known royal and follow a chain of descendants than they have deducing that to show that some number is in the set S we need to start with some known element of S and show that repeatedly adding 1 gets us to the number we want. And yet to a logician these are the same processes of reasoning. So why is statement
(1) trickier to think about than statement
(2)? Part of the difference is familiarity—we are all taught from an early age what it means to be somebody’s child, to take on a particular social role, etc. For mathematical concepts, this familiarity comes with exposure and practice, just as with learning any other language. But part of the difference is that 1For a description of some classic experiments that demonstrate this, see http://
Foundations and logic Why: This is the assembly language of mathematics—the stuff at the bottom that everything else compiles to.
• Propositional logic.
• Predicate logic.
• Axioms, theories, and models.
• Proofs.
• Induction and recursion
English Language Arts
To Critical Thinking, the critical person is something like a critical consumer of information; he or she is driven to seek reasons and evidence. Part of this is a matter of mastering certain skills of thought: learning to diagnose invalid forms of argument, knowing how to make and defend distinctions, and so on. Much of the literature in this area, especially early on, seemed to be devoted to lists and taxonomies of what a “critical thinker” should know and be able to do (Ennis 1962, 1980). More recently, however, various authors in this tradition have come to recognize that teaching content and skills is of minor import if learners do not also develop the dispositions or inclination to look at the world through a critical lens. By this, Critical Thinking means that the critical person has not only the capacity (the skills) to seek reasons, truth, and evidence, but also that he or she has the drive (disposition) to seek them. For instance, Ennis claims that a critical person not only should seek reasons and try to be well informed, but that he or she should have a tendency to do such things (Ennis 1987, 1996). Siegel criticizes Ennis somewhat for seeing dispositions simply as what animates the skills of critical thinking, because this fails to distinguish sufficiently the critical thinker from critical thinking. For Siegel, a cluster of dispositions (the “critical spirit”) is more like a deep-seated character trait, something like Scheffler’s notion of “a love of truth and a contempt of lying” (Siegel 1988; Scheffler 1991). It is part of critical thinking itself. Paul also stresses this distinction between skills and dispositions in his distinction between “weak-sense” and “strong-sense” critical thinking. For Paul, the “weak-sense” means that one has learned the skills and can demonstrate them when asked to do so; the “strong-sense” means that one has incorporated these skills into a way of living in which one’s own assumptions are re-examined and questioned as well. According to Paul, a critical thinker in the “strong sense” has a passionate drive for “clarity, accuracy, and fairmindedness” (Paul 1983, 23; see also Paul 1994). This dispositional view of critical thinking has real advantages over the skills-only view. But in important respects it is still limited. First, it is not clear exactly what is entailed by making such dispositions part of critical thinking. In our view it not only broadens the notion of criticality beyond mere “logicality,” but it necessarily requires a greater attention to institutional contexts and social relations than Critical Thinking authors have provided. Both the skills-based view and the skills-plus-dispositions view are still focused on the individual person. But it is only in the context
A second theme in the Critical Thinking literature has been the extent to which critical thinking can be characterized as a set of generalized abilities and dispositions, as opposed to content-specific abilities and dispositions that are learned and expressed differently in different areas of investigation. Can a general “Critical Thinking” course develop abilities and dispositions that will then be applied in any of a range of fields; or should such material be presented specifically in connection to the questions and content of particular fields of study? Is a scientist who is a critical thinker doing the same things as an historian who is a critical thinker? When each evaluates “good evidence,” are they truly thinking about problems in similar ways, or are the differences in interpretation and application dominant? This debate has set John McPeck, the chief advocate of content-specificity, in opposition to a number of other theorists in this area (Norris 1992; Talaska 1992). This issue relates not only to the question of how we might teach critical thinking, but also to how and whether one can test for a general facility in critical thinking (Ennis 1984). A third debate has addressed the question of the degree to which the standards of critical thinking, and the conception of rationality that underlies them, are culturally biased in favor of a particular masculine and/or Western mode of thinking, one that implicitly devalues other “ways of knowing.” Theories of education that stress the primary importance of logic, conceptual clarity, and rigorous adherence to scientific evidence have been challenged by various advocates of cultural and gender diversity who emphasize respect for alternative world views and styles of reasoning. Partly in response to such criticisms, Richard Paul has developed a conception of critical thinking that regards “sociocentrism” as itself a sign of flawed thinking (Paul 1994). Paul believes that, because critical thinking allows us to overcome the sway of our egocentric and sociocentric beliefs, it is “essential to our role as moral agents and as potential shapers of our own nature and destiny”(Paul 1990, 67). For Paul, and for some other Critical Thinking authors as well, part of the method of critical thinking involves fostering dialogue, in which thinking from the perspective of others is also relevant to the assessment of truth claims; a too-hasty imposition of one’s own standards of evidence might result not only in a premature rejection of credible alternative points of view, but might also have the effect of silencing the voices of those who (in the present context) need to be encouraged as much as possible to speak for themselves. In this respect, we see Paul introducing into the very definition of critical thinking some of the sorts of social and contextual factors that Critical Pedagogy writers have emphasized.
http://mediaeducation.org.mt/wp-content/uploads/2013/05/Critical-Thinking-and-Critical-Pedagogy.pdf
CHAPTER TWO BASIC CONCEPTS OF LOGIC
Chapter Overview Logic, as field of study, may be defined as the organized body of knowledge, or science that evaluates arguments. The aim of logic is to develop a system of methods and principles that we may use as criteria for evaluating the arguments of others and as guides in constructing arguments of our own. Argument is a systematic combination of two or more statements, which are classified as a premise or premises and conclusion. A premise refers to the statement, which is claimed to provide a logical support or evidence to the main point of the argument, which h known as conclusion. A conclusion is a statement, which is claimed to follow from the alleged evidence. Depending on the logical and real ability of the premise(s) to support the conclusion, an argument can be either a good argument or a bad argument. However, unlike all kinds of passages, including those that resemble arguments, all arguments purport to prove something. Arguments can generally be divided into deductive and inductive arguments. A deductive argument is an argument in which the premises are claimed to support the conclusion in such a way that it is impossible for the premises to be true and the conclusion false. On the other hand, an inductive argument is an argument in which the premises are claimed to support the conclusion in such a way that it is improbable that the premises be true and the conclusion false. The deductiveness or inductiveness of an argument can be determined by the particular indicator word it might use, the actual strength of the inferential relationship between its component statements, and its argumentative form or structure. A deductive argument can be evaluated by its validity and soundness. Likewise, an inductive argument can be evaluated by its strength and cogency. Depending on its actually ability to successfully maintain its inferential claim, a deductive argument can be either valid or invalid. That is, if the premise(s) of a certain deductive argument actually support its conclusion in such a way that it is impossible for the premises to be true and the conclusion false, then that particular deductive argument is valid. If, however, its premise(s) actually support its conclusion in such a
way that it is possible for the premises to be true and the conclusion false, then that particular deductive argument is invalid. Similarly, an inductive argument can be either strong or weak, depending on its actually ability to successfully maintain its inferential claim. That is, if the premise(s) of a certain inductive argument actually support its conclusion in such a way that it is improbable for the premises to be true and the conclusion false, then that particular inductive argument is strong. If, however, its premise(s) actually support its conclusion in such a way that it is probable for the premises to be true and the conclusion false, then that particular inductive argument is weak. Furthermore, depending on its actually ability to successfully maintain its inferential claim as well as its factual claim, a deductive argument can be either sound or unsound. That is, if a deductive argument actually maintained its inferential claim, (i.e., if it is valid), and its factual claim, (i.e., if all of its premises are true), then that particular deductive argument will be a sound argument. However, if it fails to maintain either of its claims, it will be an unsound argument. Likewise, depending on its actually ability to successfully maintain its inferential claim as well as its factual claim, an inductive argument can be either cogent or uncogent. That is, if an inductive argument actually maintained its inferential claim, (i.e., if it is strong), and its factual claim, (i.e., if all of its premises are probably true), then that particular inductive argument will be a cogent argument. However, if it fails to maintain either of its claims, it will be an uncogent argument. In this chapter, we will discuss logic and its basic concepts, the techniques of distinguishing arguments from non-argumentative passages, and the types of arguments.
Chapter Objectives: Dear learners, after the successful completion of this chapter, you will be able to:
Ø Understand the meaning and basic concepts of logic;
Ø Understand the meaning, components, and types of arguments; and
Ø Recognize the major techniques of recognizing and evaluating arguments
Lesson 1: Basic Concepts of Logic: Arguments, Premises and Conclusions Lesson Overview Logic is generally be defined as a philosophical science that evaluates arguments. An argument is a systematic combination of one or more than one statements, which are claimed to provide a logical support or evidence (i.e., premise(s) to another single statement which is claimed to follow logically from the alleged evidence (i.e., conclusion). An argument can be either good or bad argument, depending on the logical ability of its premise(s) to support its conclusion. The primary aim of logic is to develop a system of methods and principles that we may use as criteria for evaluating the arguments of others and as guides in constructing arguments of our own. The study of logic increases students‘ confidence to criticize the arguments of others and advance arguments of their own. In this lesson, we will discuss the meaning and basic concepts of logic: arguments, premises, and conclusions. Lesson Objectives: After the accomplishment of this lesson, you will be able to:
Ø Understand the meaning.
Ø Identify the subject matter of logic.
Ø Understand the meaning of an argument.
Ø Identify the components of an argument.
Ø Understand the meaning and nature of a premise.
Ø Comprehend the meaning and nature of a conclusion.
Ø Recognize the techniques of identifying the premises and conclusion of an argument.
Conditional Statements
A conditional statement is an ―if . . . then . . .‖ statement.
Example: If you study hard, then you will score „A‟ grade.
Every conditional statement is made up of two component statements. The component statement immediately following the ―if‖ is called the antecedent (if-clause), and the one following the ―then‖ is called the consequent (then-clause). However, there is an occasion that the order of antecedent and consequent is reversed.
That is, when occasionally the word ‗‗then‘‘ is left out, the order of antecedent and consequent is reversed. For example if we left out ―then‖ from the above example the antecedent and consequent is reversed: You will score „A‟ grade if you study hard. In the above example, the antecedent is ―You study hard,‖ and the consequent is ―
You will score „A‟ grade.‖ In this example, there is a meaningful relationship between antecedent and consequent.
However, such a relationship need not exist for a statement to count as conditional. The statement ―If Getaneh Kebede is a singer, then Hawassa is in Mekelle‖ is just as much a conditional statement as that in the above example.
Conditional statements are not arguments, because they fail to meet the criteria given earlier.
In an argument, at least one statement must claim to present evidence, and there must be a claim that this evidence implies something. In a conditional statement, there is no claim that either the antecedent or the consequent presents evidence. In other words, there is no assertion that either the antecedent or the consequent is true. Rather, there is only the assertion that if the antecedent is true, then so is the consequent.
For example, the above example merely asserts that if you study hard, then you will score ‗A‘. It does not assert that you study hard.
Nor does it assert you scored ‗A‘. Of course, a conditional statement as a whole may present evidence because it asserts a relationship between statements. Yet when conditional statements are taken in this sense, there is still no argument, because there is then no separate claim that this evidence implies anything.
Therefore, a single conditional statement is not an argument.
The fact that a statement begins with ―if‖ makes it the idea conditional and not a final reasonable assertion.
That is why also conditional statements are not evaluated as true or false without separately evaluating the antecedent and the consequent. They only claim that if the antecedent is true then so is the consequent. However, some conditional statements are similar to arguments in that they express the outcome of a reasoning process. As such, they may be said to have a certain inferential content. Consider the following example: If destroying a political competitor gives you joy, then you have a low sense of morality
https://wcu.edu.et/FirstYearModule/CRITICAL%20THINKING%20module.pdf
Discrete Mathematics
- The Conditional Statement Before we give a formal definition of the conditional statement, we start with an example so we can understand when a conditional statement should be true. For the example, we need the following notation and terminology:
- Notation 1.1.
If p and q are statements, the conditional of q by p is “if p then q” denoted p → q. We call p the hypothesis of the conditional and q the conclusion.
Example 1.2. Consider the conditional statement, “if I am healthy, I will come to class.” To determine the truth value of this statement, we need to determine when this statement is false, so we consider the four different possibilities for the truth values of p and q.
Let p :=“I am healthy” and q :=“I will come to class”. We shall fill in the following table:
p q p → q T T T T F F F T T F F T
• For case # 1, if I am healthy and I come to class, the conditional is clearly true.
• For case # 2, if I am healthy, but I have decided to stay home and not go to class, the conditional is false – the hypothesis is satisfied, but the conclusion is not satisfied, so the statement cannot possibly be true.
• For case # 3, if I am not healthy, but I have come to class anyway though all the people sitting around may not be happy about it, the conditional statement has not been violated since the hypothesis does not hold i.e. the conditional statement is meaningless since the hypothesis is not true. Therefore, the conditional must be true.
• Likewise, for case # 4, if I am not healthy, and I did not come to class, the conditional statement has not been violated since
the hypothesis does not hold. Therefore, the conditional is true. This example implies that a conditional statement is false only when the hypothesis is true and the conclusion is false. Though it is clear that a conditional statement is false only when the hypothesis is true and the conclusion is false, it is not clear why when the hypothesis is false, the conditional statement is always true. To try to explain why this is this case, we consider another example.
Example 1.3.
Consider the mathematical statement “if n is a perfect square, then n is not prime.” Clearly this is a true statement for any n, so it will be true when we substitute values in for n. Now substitute 3 for n: “if 3 is a perfect square, then 3 is not prime.” As remarked above, this conditional statement is still true yet its hypothesis and conclusion are both false. Similarly, if we substitute 6 into this statement, it becomes “if 6 is a perfect square, then 6 is not prime.” This conditional statement is true yet its hypothesis is false and its conclusion is true.
We can now write down a formal definition for the conditional statement.
Definition 1.4. If p and q are statements, the conditional of q by p is “if p then q” or “p implies q” denoted p → q. A conditional statement is false only when the hypothesis is false and the conclusion is true. The truth table for the conditional statement is as follows:
p q p → q
T T T T F F
F T T F F T
https://faculty.up.edu/wootton/discrete/section1.2.pdf
Conditional reasoning
Conditional reasoning is based on the construction “if 𝑝, then 𝑞” when the premise is true,
the conclusion will be true. However, this leaves open the question of what happens when
𝑝 is false, which means that, in this case, 𝑞 can logically be either true or false. Studies
are abundant about four main conditional inferences: modus ponens, modus tollens,
affirmation of the consequent and denial of the antecedent. Johnson-Laird and Byrne
(2002) discuss that, among all four conditional inferences mentioned in §§3.1.2, only
modus ponens and modus tollens are valid for the conditional interpretation. The following
is an example of modus ponens:
If it rains, then you get wet.
It rains.
Therefore, you get wet.
Critics have claimed that mathematics taught in K-11 and K-12 is nothing more than memorizing the facts rather than computing the method of solving the given problem with a known concept of study.
Topic of study which includes discrete mathematics are Set Theory, Relations and Functions, Principles of Mathematical Induction, Permutation and Combinations, Mathematical Reasoning, Probability and some study about Matrices and Determinants.
Discrete Mathematics course is a core part of computer / information science & technology and it facilitates the study of applications in the field of computer science, especially in the areas of data structures, the theory of computer languages and the analysis of algorithms. In addition, this course also provides students with understanding of applications in engineering and the physical and life sciences, as well as in statistics and the social sciences.
To introduce the student at the high-school level, if not earlier, to the topics and techniques of discrete methods and combinatorial reasoning. Whenever the structures from abstract algebra are required, only the basic theory needed for the application development. Further, the solution of the some applications contribute to the iterative procedures that lead to specific algorithms. The algorithmic approach and solution for the problems is fundamental in discrete mathematics.
Counting concept introduces the basic collection of counting techniques with few motivational examples such as paper folding example, Rubik’s cube problem etc. This provides, count visually distinguishable patterns (Binomial Theorem) for collection of objects with identifiable types of objects, each with several copies are available. Counting the number of distinct elements in a union of possibly non-disjoint sets (inclusion-exclusion formula). Probability theory conceptualize the foundational explanations (event, sample space, independence). Methods of determining the probabilities of events are introduced and the notion of equally likely outcomes are defined. The notion of a random variable is to create a variable whose value is determined by the outcome of a random experiment. Probability distribution described for a particular pattern and a collection of conditional probabilities into a different set of conditional probabilities (Bayes’s Theorem). On higher level, they should be aware of some meta-knowledge and heuristics, and be able to use them appropriately. They should be aware that there are many approaches to achieve the same goal but using the appropriate method of solving the problem and reach the desired result. Influenced by all the examples in discrete mathematics concepts, students shall know it is good to work systematically and in phases, virtually every time when it is possible. Educational targets describes above includes both practical usability and theoretical knowledge. These two aspect shall strength each other systematically, where every student of K-11 and K-12 grade classes efficiently.