 Mathematical Proofs A Transition to Advanced Mathematics PDF This article outlines teaching ideas appropriate for primary mathematics. It is mainly aimed at primary school teachers and teacher-researchers. James Russo shares his experiences of exploring proof with a group of 8-and 9-year old students in an Australian вЂ¦

Mathematical Problems And Proofs Download eBook pdf. Preface: Mathematical Statistics After teaching mathematical statistics for several years using chalk on a black-board (and, later, smelly вЂњdry erase markersвЂќ on a whiteboard) mostly doing proofs of theorems, I decided to lecture from computer slides that provide an outline of the вЂњbig pictureвЂќ. Rather than spend class time вЂњdoing, A gentle introduction to the highly sophisticated world of discrete mathematics, Mathematical Problems and Proofs presents topics ranging from elementary definitions and theorems to advanced topics -- such as cardinal numbers, generating functions, properties of Fibonacci numbers, and Euclidean algorithm..

mathematical language and symbols before moving onto the serious matter of writing the mathematical proofs. Each theorem is followed by the \notes", which are the thoughts on the topic, intended to give a deeper idea of the statement. You will nd that some proofs are missing the steps and the purple mathematical problems and proofs Download mathematical problems and proofs or read online books in PDF, EPUB, Tuebl, and Mobi Format. Click Download or Read Online button to get mathematical problems and proofs book now. This site is like a library, Use search box in the widget to get ebook that you want. Mathematical Problems And Proofs

Mathematical Fallacy Proofs In world of mathematics, countless brilliant minds dedicate their lives in an effort to prove the seemingly impossible. Interestingly enough, through the plethora of established proofs which has tremendously impacted the scientific world, a few false proofs have also survived the scrutiny of mathematicians. Mathematical Proofs: The Beautiful and The Explanatory Cover Page Footnote This paper issues from a talk given at the Workshop on Beauty and Explanation in Mathematics, March 10-12, 2014, UmeГҐ University, Sweden. I am most grateful to the organiser, Manya SundstrГ¶m, and to the other

mathematical problems and proofs Download mathematical problems and proofs or read online books in PDF, EPUB, Tuebl, and Mobi Format. Click Download or Read Online button to get mathematical problems and proofs book now. This site is like a library, Use search box in the widget to get ebook that you want. Mathematical Problems And Proofs 28/5/2002В В· Mathematical Proofs really is a transition to advanced math, and I will definitely feel more complete studying advanced level calculus after reading this text. It offers a nice intro to set theory and logic that leads up to the basics of proving, and finishes off with the theoretically important proofs that found calculus, number theory and group theory.

Mathematical Proofs and Formal Deductions A simple example of such a succession of acts is formal deduction, that is nothing but the activity consisting in producing a syntactical chain or, more generally, a chain of formulas belonging to a certain formal system. Mathematical Fallacy Proofs In world of mathematics, countless brilliant minds dedicate their lives in an effort to prove the seemingly impossible. Interestingly enough, through the plethora of established proofs which has tremendously impacted the scientific world, a few false proofs have also survived the scrutiny of mathematicians.

Mathematical Proofs: A Transition to Advanced Mathematics. (3rd Edition)1 ISBN 0321797094 Objectives: The primary goal of this course is to learn to read and write mathe-matics. In particular, this means the course will have a heavy emphasis on writing proofs. A passing grade in this course indicates that a student should be able to read other area of pure mathematics; in particular, Mathematical Logic is used in a fundamental way. Although the necessary logic is presented in this book, it would be beneп¬Ѓcial for the reader to have taken a prior course in logic under the auspices of mathematics, computer science or philosophy. In fact,

Mathematical Proofs Universal Quantiп¬Ѓer: This is the symbol в€Ђ and we use it when we want to indicate that a statement holds for all elements x in some set U. In this case we write в€Ђxв€€ U, P(x), where P(x) represents the statement that is true. For example, we might write в€Ђxв€€ R, (в€’x)2 = x2. of reasons: First, from a mathematical point of view, there often exist various distinct proofs for the same proof problem. For example, Ribenboim gives eleven proofs that there are inп¬Ѓnitely many primes (see  for details). Especially, given a set of proofs it is difп¬Ѓcult to be certain that no further mathematically

Math Handbook of Formulas, Processes and Tricks Proofs 12 Conditional Statements (Original, Converse, Inverse, Contrapositive) 13 Basic Properties of Algebra (Equality and Congruence, Addition and Multiplication) 14 Or, in higher level mathematical notation: Statistics is about the mathematical modeling of observable phenomena, using stochastic models, and about analyzing data: estimating parameters of the model and testing hypotheses. In these notes, we study various estimation and testing procedures. We consider their theoretical properties and we investigate various notions of optimality.

mathematical problems and proofs Download mathematical problems and proofs or read online books in PDF, EPUB, Tuebl, and Mobi Format. Click Download or Read Online button to get mathematical problems and proofs book now. This site is like a library, Use search box in the widget to get ebook that you want. Mathematical Problems And Proofs Mathematical Proofs and Formal Deductions A simple example of such a succession of acts is formal deduction, that is nothing but the activity consisting in producing a syntactical chain or, more generally, a chain of formulas belonging to a certain formal system.

Mathematical Proofs: A Transition to Advanced Mathematics. (3rd Edition)1 ISBN 0321797094 Objectives: The primary goal of this course is to learn to read and write mathe-matics. In particular, this means the course will have a heavy emphasis on writing proofs. A passing grade in this course indicates that a student should be able to read This paper considers this topic from four main perspectives: students' perceptions of mathematical proofs, instructors' presentations of mathematical proofs, using peer review to develop students' abilities to read proofs more critically and write proofs more convincingly, and providing students with the skills required to independently read

### Theory of Statistics (PDF) Mathematical Proofs ResearchGate. Preface: Mathematical Statistics After teaching mathematical statistics for several years using chalk on a black-board (and, later, smelly вЂњdry erase markersвЂќ on a whiteboard) mostly doing proofs of theorems, I decided to lecture from computer slides that provide an outline of the вЂњbig pictureвЂќ. Rather than spend class time вЂњdoing, Mathematical Proofs and Formal Deductions A simple example of such a succession of acts is formal deduction, that is nothing but the activity consisting in producing a syntactical chain or, more generally, a chain of formulas belonging to a certain formal system..

Mathematical Proofs By Chartrand 3rd Manual pdf Book. ics, an abstract mathematics that requires proofs. In this document we will try to explain the importance of proofs in mathematics, and to give a you an idea what are mathematical proofs. This will give you some reference to check if your proofs are correct. We begin by describing the role of proofs in mathematics, then we de ne the logical, Mathematical Proofs: The Beautiful and The Explanatory Cover Page Footnote This paper issues from a talk given at the Workshop on Beauty and Explanation in Mathematics, March 10-12, 2014, UmeГҐ University, Sweden. I am most grateful to the organiser, Manya SundstrГ¶m, and to the other.  • Mathematical Proofs University College Dublin
• www.people.vcu.edu
• Some Remarks onWriting Mathematical Proofs

• The History and Concept of Mathematical Proof Steven G. Krantz1 February 5, 2007 Amathematicianisamasterof criticalthinking,of analysis, andof deduc-tive reasoning. These skills travel well, and can be applied in a large variety of situationsвЂ”and in many diп¬Ђerent disciplines. Today, mathematical skills mathematical language and symbols before moving onto the serious matter of writing the mathematical proofs. Each theorem is followed by the \notes", which are the thoughts on the topic, intended to give a deeper idea of the statement. You will nd that some proofs are missing the steps and the purple

rst order logic and mathematical induction, our objective is to move to more advanced classical mathematical structures and arguments as soon as the student has an adequate understanding of the logic under-lying mathematical proofs. 0.4. Advice to the Student Welcome to higher mathematics! If your exposure to University mathematical language and symbols before moving onto the serious matter of writing the mathematical proofs. Each theorem is followed by the \notes", which are the thoughts on the topic, intended to give a deeper idea of the statement. You will nd that some proofs are missing the steps and the purple

Mathematical Thinking Problem Solving and Proofs Solution Manual 1 - Free download as PDF File (.pdf), Text File (.txt) or view presentation slides online. PDF This article outlines teaching ideas appropriate for primary mathematics. It is mainly aimed at primary school teachers and teacher-researchers. James Russo shares his experiences of exploring proof with a group of 8-and 9-year old students in an Australian вЂ¦

Be the first to review вЂњMathematical Proofs 4th Edition (eBook PDF)вЂќ Cancel reply. Your email address will not be published. Required fields are marked * mathematical topics that these students should know. The book also provides a bridge to the upper-level courses, since we discuss formalities and conventions in detail, including the axiomatic method and how to deal with proofs. (b) Mathematics teachers and teachers-in-training. We present here some of the

pdf. MATHEMATICAL PROOFS: A TRANSITION TO ADVANCED MATHEMATICS SECOND EDITION. Allen Liu. Gary Chartrand + 21 More. Allen Liu. Gary Chartrand. Gary Chartrand. Ping Zhang. MATHEMATICAL PROOFS: A TRANSITION TO ADVANCED MATHEMATICS SECOND EDITION. Download. MATHEMATICAL PROOFS: A TRANSITION TO ADVANCED MATHEMATICS SECOND EDITION. Allen Liu. pdf. MATHEMATICAL PROOFS: A TRANSITION TO ADVANCED MATHEMATICS SECOND EDITION. Allen Liu. Gary Chartrand + 21 More. Allen Liu. Gary Chartrand. Gary Chartrand. Ping Zhang. MATHEMATICAL PROOFS: A TRANSITION TO ADVANCED MATHEMATICS SECOND EDITION. Download. MATHEMATICAL PROOFS: A TRANSITION TO ADVANCED MATHEMATICS SECOND EDITION. Allen Liu.

Mathematical Thinking Problem Solving and Proofs Solution Manual 1 - Free download as PDF File (.pdf), Text File (.txt) or view presentation slides online. PROOF IN MATHEMATICS: AN INTRODUCTION. James Franklin and Albert Daoud (Quakers Hill Press, 1996/Kew Books, 2011) This is a small (98 page) textbook designed to teach mathematics and computer science students the basics of how to read and construct proofs.

One of the great distinguishing features of mathematics is the concept of a proof. Once again, the Greeks were the first to introduce this idea. A proof is an attempt to establish the truth of a statement, and in the case of a mathematical proof it establishes the truth of a mathematical statement. Preface: Mathematical Statistics After teaching mathematical statistics for several years using chalk on a black-board (and, later, smelly вЂњdry erase markersвЂќ on a whiteboard) mostly doing proofs of theorems, I decided to lecture from computer slides that provide an outline of the вЂњbig pictureвЂќ. Rather than spend class time вЂњdoing

This paper considers this topic from four main perspectives: students' perceptions of mathematical proofs, instructors' presentations of mathematical proofs, using peer review to develop students' abilities to read proofs more critically and write proofs more convincingly, and providing students with the skills required to independently read Mathematical Proofs Universal Quantiп¬Ѓer: This is the symbol в€Ђ and we use it when we want to indicate that a statement holds for all elements x in some set U. In this case we write в€Ђxв€€ U, P(x), where P(x) represents the statement that is true. For example, we might write в€Ђxв€€ R, (в€’x)2 = x2.

Mathematical Proofs Universal Quantiп¬Ѓer: This is the symbol в€Ђ and we use it when we want to indicate that a statement holds for all elements x in some set U. In this case we write в€Ђxв€€ U, P(x), where P(x) represents the statement that is true. For example, we might write в€Ђxв€€ R, (в€’x)2 = x2. MEP Pupil Text - Additional Material: Mathematical Proof 4 1.1 True or False Exercises 1. Decide whether the following statements are false, sometimes true or always true. (a) Christmas Day is on a Wednesday. (b) A year will have 400 days. (c) There is a total of 61 days вЂ¦

## PROOF IN MATHEMATICS AN INTRODUCTION Mathematical Proofs A Transition to. Four additional chapters, Chapters 16вЂ“19 (dealing with proofs in ring theory, linear algebra, real and complex numbers, and topology), can be found by going to: goo.gl/bf2Nb3., An unproven proposition that is believed to be true is known as a conjecture, or a hypothesis if the proposition is frequently used as an assumption to build upon similar mathematical work. Proofs employ logic expressed in mathematical symbols, along with some amount of вЂ¦.

### Theory of Statistics

Towards an Intelligent Tutor for Mathematical Proofs. other area of pure mathematics; in particular, Mathematical Logic is used in a fundamental way. Although the necessary logic is presented in this book, it would be beneп¬Ѓcial for the reader to have taken a prior course in logic under the auspices of mathematics, computer science or philosophy. In fact,, An Introduction to Mathematical Reasoning Brief Edition Solutions Manual. instructor solution manual for Advanced Engineering Mathematics 3rd ed zill ago (html) Mathematical Proofs A Transition To Advanced Mathematics 2nd. The instructor solutions manual is available for the mathematical, SOLUTIONS MANUAL: A Transition to Advanced.

rst order logic and mathematical induction, our objective is to move to more advanced classical mathematical structures and arguments as soon as the student has an adequate understanding of the logic under-lying mathematical proofs. 0.4. Advice to the Student Welcome to higher mathematics! If your exposure to University mathematical topics that these students should know. The book also provides a bridge to the upper-level courses, since we discuss formalities and conventions in detail, including the axiomatic method and how to deal with proofs. (b) Mathematics teachers and teachers-in-training. We present here some of the

Mathematical Proofs: The Beautiful and The Explanatory Cover Page Footnote This paper issues from a talk given at the Workshop on Beauty and Explanation in Mathematics, March 10-12, 2014, UmeГҐ University, Sweden. I am most grateful to the organiser, Manya SundstrГ¶m, and to the other Mathematical Proofs and Formal Deductions A simple example of such a succession of acts is formal deduction, that is nothing but the activity consisting in producing a syntactical chain or, more generally, a chain of formulas belonging to a certain formal system.

MEP Pupil Text - Additional Material: Mathematical Proof 4 1.1 True or False Exercises 1. Decide whether the following statements are false, sometimes true or always true. (a) Christmas Day is on a Wednesday. (b) A year will have 400 days. (c) There is a total of 61 days вЂ¦ A gentle introduction to the highly sophisticated world of discrete mathematics, Mathematical Problems and Proofs presents topics ranging from elementary definitions and theorems to advanced topics -- such as cardinal numbers, generating functions, properties of Fibonacci numbers, and Euclidean algorithm.

Mathematical Proofs: The Beautiful and The Explanatory Cover Page Footnote This paper issues from a talk given at the Workshop on Beauty and Explanation in Mathematics, March 10-12, 2014, UmeГҐ University, Sweden. I am most grateful to the organiser, Manya SundstrГ¶m, and to the other Mathematical Proofs: Where to Begin And How to Write Them Starting with Linear Algebra, mathematics courses at Hamilton often require students to prove mathematical results using formalized logic. This can occasionally be a difficult process, because the same statement can be proven using

The Nuts and Bolts of Proofs: An Introduction to Mathematical Proofs Pdf mediafire.com, rapidgator.net, 4shared.com, uploading.com, uploaded.net Download; Note: If you're looking for a free download links of The Nuts and Bolts of Proofs: An Introduction to Mathematical Proofs Pdf, epub, docx and torrent then this site is not for you. while teaching proofs courses over the past fourteen years at Virginia CommonwealthUniversity(alargestateuniversity)andRandolph-Macon Here the primary goal is to understand mathematical structures, to prove mathematical statements, and even to invent or discover new mathematical theorems and theories.

28/5/2002В В· Mathematical Proofs really is a transition to advanced math, and I will definitely feel more complete studying advanced level calculus after reading this text. It offers a nice intro to set theory and logic that leads up to the basics of proving, and finishes off with the theoretically important proofs that found calculus, number theory and group theory. An unproven proposition that is believed to be true is known as a conjecture, or a hypothesis if the proposition is frequently used as an assumption to build upon similar mathematical work. Proofs employ logic expressed in mathematical symbols, along with some amount of вЂ¦

PDF This article outlines teaching ideas appropriate for primary mathematics. It is mainly aimed at primary school teachers and teacher-researchers. James Russo shares his experiences of exploring proof with a group of 8-and 9-year old students in an Australian вЂ¦ MEP Pupil Text - Additional Material: Mathematical Proof 4 1.1 True or False Exercises 1. Decide whether the following statements are false, sometimes true or always true. (a) Christmas Day is on a Wednesday. (b) A year will have 400 days. (c) There is a total of 61 days вЂ¦

Four additional chapters, Chapters 16вЂ“19 (dealing with proofs in ring theory, linear algebra, real and complex numbers, and topology), can be found by going to: goo.gl/bf2Nb3. ics, an abstract mathematics that requires proofs. In this document we will try to explain the importance of proofs in mathematics, and to give a you an idea what are mathematical proofs. This will give you some reference to check if your proofs are correct. We begin by describing the role of proofs in mathematics, then we de ne the logical

The History and Concept of Mathematical Proof Steven G. Krantz1 February 5, 2007 Amathematicianisamasterof criticalthinking,of analysis, andof deduc-tive reasoning. These skills travel well, and can be applied in a large variety of situationsвЂ”and in many diп¬Ђerent disciplines. Today, mathematical skills pdf. MATHEMATICAL PROOFS: A TRANSITION TO ADVANCED MATHEMATICS SECOND EDITION. Allen Liu. Gary Chartrand + 21 More. Allen Liu. Gary Chartrand. Gary Chartrand. Ping Zhang. MATHEMATICAL PROOFS: A TRANSITION TO ADVANCED MATHEMATICS SECOND EDITION. Download. MATHEMATICAL PROOFS: A TRANSITION TO ADVANCED MATHEMATICS SECOND EDITION. Allen Liu.

### Transition to Higher Mathematics Structure and Proof AN INTRODUCTION TO SET THEORY. Mathematical Proofs: Where to Begin And How to Write Them Starting with Linear Algebra, mathematics courses at Hamilton often require students to prove mathematical results using formalized logic. This can occasionally be a difficult process, because the same statement can be proven using, An Introduction to Mathematical Reasoning Brief Edition Solutions Manual. instructor solution manual for Advanced Engineering Mathematics 3rd ed zill ago (html) Mathematical Proofs A Transition To Advanced Mathematics 2nd. The instructor solutions manual is available for the mathematical, SOLUTIONS MANUAL: A Transition to Advanced.

### Some Remarks onWriting Mathematical Proofs Writing Mathematical Proofs University of Bristol. For courses in Transition to Advanced Mathematics or Introduction to Proof. Meticulously crafted, student-friendly text that helps build mathematical maturity Mathematical Proofs: A Transition to Advanced Mathematics, 4th Edition introduces students to вЂ¦ https://it.wikipedia.org/wiki/Mathematical_Intelligencer For courses in Transition to Advanced Mathematics or Introduction to Proof. Meticulously crafted, student-friendly text that helps build mathematical maturity Mathematical Proofs: A Transition to Advanced Mathematics, 4th Edition introduces students to вЂ¦. • Mathematical Problems and Proofs SpringerLink
• List of mathematical proofs Wikipedia

• Mathematical Proofs - Free download as PDF File (.pdf), Text File (.txt) or read online for free. This is indeed the case of writing a mathematical proof. Before we see how proofs work, let us introduce the rules of the game. Mathematics is composed of statements. A gentle introduction to the highly sophisticated world of discrete mathematics, Mathematical Problems and Proofs presents topics ranging from elementary definitions and theorems to advanced topics -- such as cardinal numbers, generating functions, properties of Fibonacci numbers, and Euclidean algorithm.

24/7/2018В В· Mathematical Proofs: A Transition to Advanced Mathematics, Third Edition, prepares students for the more abstract mathematics courses that follow calculus. Appropriate for self-study or for use in the classroom, this text introduces students to proof techniques, analyzing proofs, and writing proofs of their own. One of the great distinguishing features of mathematics is the concept of a proof. Once again, the Greeks were the first to introduce this idea. A proof is an attempt to establish the truth of a statement, and in the case of a mathematical proof it establishes the truth of a mathematical statement.

For courses in Transition to Advanced Mathematics or Introduction to Proof. Meticulously crafted, student-friendly text that helps build mathematical maturity Mathematical Proofs: A Transition to Advanced Mathematics, 4th Edition introduces students to вЂ¦ mathematical language and symbols before moving onto the serious matter of writing the mathematical proofs. Each theorem is followed by the \notes", which are the thoughts on the topic, intended to give a deeper idea of the statement. You will nd that some proofs are missing the steps and the purple

mathematical logic. [n the belief that beginners should be exposed to the easiest and most natural proofs, I have used free-swinging set-theoretic methods. The significance of a demand for constructive proofs can be evaluated only after a certain amount of experience with вЂ¦ pdf. MATHEMATICAL PROOFS: A TRANSITION TO ADVANCED MATHEMATICS SECOND EDITION. Allen Liu. Gary Chartrand + 21 More. Allen Liu. Gary Chartrand. Gary Chartrand. Ping Zhang. MATHEMATICAL PROOFS: A TRANSITION TO ADVANCED MATHEMATICS SECOND EDITION. Download. MATHEMATICAL PROOFS: A TRANSITION TO ADVANCED MATHEMATICS SECOND EDITION. Allen Liu.

PROOF IN MATHEMATICS: AN INTRODUCTION. James Franklin and Albert Daoud (Quakers Hill Press, 1996/Kew Books, 2011) This is a small (98 page) textbook designed to teach mathematics and computer science students the basics of how to read and construct proofs. mathematical topics that these students should know. The book also provides a bridge to the upper-level courses, since we discuss formalities and conventions in detail, including the axiomatic method and how to deal with proofs. (b) Mathematics teachers and teachers-in-training. We present here some of the

This paper considers this topic from four main perspectives: students' perceptions of mathematical proofs, instructors' presentations of mathematical proofs, using peer review to develop students' abilities to read proofs more critically and write proofs more convincingly, and providing students with the skills required to independently read Statistics is about the mathematical modeling of observable phenomena, using stochastic models, and about analyzing data: estimating parameters of the model and testing hypotheses. In these notes, we study various estimation and testing procedures. We consider their theoretical properties and we investigate various notions of optimality.

Mathematical Fallacy Proofs In world of mathematics, countless brilliant minds dedicate their lives in an effort to prove the seemingly impossible. Interestingly enough, through the plethora of established proofs which has tremendously impacted the scientific world, a few false proofs have also survived the scrutiny of mathematicians. of reasons: First, from a mathematical point of view, there often exist various distinct proofs for the same proof problem. For example, Ribenboim gives eleven proofs that there are inп¬Ѓnitely many primes (see  for details). Especially, given a set of proofs it is difп¬Ѓcult to be certain that no further mathematically

Preface: Mathematical Statistics After teaching mathematical statistics for several years using chalk on a black-board (and, later, smelly вЂњdry erase markersвЂќ on a whiteboard) mostly doing proofs of theorems, I decided to lecture from computer slides that provide an outline of the вЂњbig pictureвЂќ. Rather than spend class time вЂњdoing Mathematical Proofs Universal Quantiп¬Ѓer: This is the symbol в€Ђ and we use it when we want to indicate that a statement holds for all elements x in some set U. In this case we write в€Ђxв€€ U, P(x), where P(x) represents the statement that is true. For example, we might write в€Ђxв€€ R, (в€’x)2 = x2.

4.6 Proofs Involving Cartesian Products ofSets 113 Exercises forChapter4 114 Existence andProofby Contradiction 120 5.1 Counterexamples 120 5.2 ProofbyContradiction 124 5.3 AReviewofThree ProofTechniques 130 5.4 Existence Proofs 132 5.5 Disproving Existence Statements 136 Exercisesfor Chapter5 137 MathematicalInduction 142 6.1 The Mathematical Proofs: The Beautiful and The Explanatory Cover Page Footnote This paper issues from a talk given at the Workshop on Beauty and Explanation in Mathematics, March 10-12, 2014, UmeГҐ University, Sweden. I am most grateful to the organiser, Manya SundstrГ¶m, and to the other

of reasons: First, from a mathematical point of view, there often exist various distinct proofs for the same proof problem. For example, Ribenboim gives eleven proofs that there are inп¬Ѓnitely many primes (see  for details). Especially, given a set of proofs it is difп¬Ѓcult to be certain that no further mathematically For courses in Transition to Advanced Mathematics or Introduction to Proof. Meticulously crafted, student-friendly text that helps build mathematical maturity Mathematical Proofs: A Transition to Advanced Mathematics, 4th Edition introduces students to вЂ¦ Mathematical Proofs: Where to Begin And How to Write Them Starting with Linear Algebra, mathematics courses at Hamilton often require students to prove mathematical results using formalized logic. This can occasionally be a difficult process, because the same statement can be proven using Mathematical Proofs: A Transition to Advanced Mathematics. (3rd Edition)1 ISBN 0321797094 Objectives: The primary goal of this course is to learn to read and write mathe-matics. In particular, this means the course will have a heavy emphasis on writing proofs. A passing grade in this course indicates that a student should be able to read