# 1.S: Introduction to Writing Proofs in Mathematics (Summary)

Important Definitions

• Statement
• Odd integer
• Conditional statement
• Even integer
• Pythagorean triple

Important Number Systems and Their Properties

• The natural numbers, (mathbb{N}); the integers, (mathbb{Z}); the rational numbers, (mathbb{Q}); and the real number, (mathbb{R}).
• Closure Properties of the Number Systems Number System Closed Under Natural numbers, (mathbb{N}) addition and multiplication Integers, (mathbb{Z}) addition, subtraction, and multiplication Rational numbers, (mathbb{Q}) addition, subtraction, and multiplication, and division by nonzero rational numbers Real number, (mathbb{R}) addition, subtraction, and multiplication, and division by nonzero real numbers
• Inverse,commutative,associative, and distributive properties of the real numbers.

Important Theorems and Results

• Exercise (1), Section 1.2
If (m) is an even integer, then (m + 1) is an odd integer.
If (m) is an odd integer, then (m + 1) is an even integer.
• Exercise (2), Section 1.2
If (x) is an even integer and (y) is an even integer, then (x + y) is an even integer.
If (x) is an even integer and (y) is an odd integer, then (x + y) is an odd integer.
If (x) is an odd integer and (y) is an odd integer, then (x + y) is an even integer.
• Exercise (3), Section 1.2.
If (x) is an even integer and (y) is an integer, then (x cdot y) is an even integer.
• Theorem1.8. If (x) is an odd integer and (y) is an odd integer, then (x cdot y) is an odd integer.
• The Pythagorean Theorem. If (a) and (b) are the lengths of the legs of a right triangle and (c) is the length of the hypotenuse, then (a^2 + b^2 = c^2).

## Introduction to Geometry

I thought that this book was really good, and teaching out of it was really fun. It did a much better job of holding interest in Geometry for the kids I taught than my Junior High geometry ever did. The proofs it had kids do were interesting, and they didn&apost require any of that 2-column junk I had to learn. Just good solid explanations as to why things worked out the way they did.

And the book covered a wide variety of Euclidean geometry. Barely touched 3 dimensions, but what can we expect? I thought that this book was really good, and teaching out of it was really fun. It did a much better job of holding interest in Geometry for the kids I taught than my Junior High geometry ever did. The proofs it had kids do were interesting, and they didn't require any of that 2-column junk I had to learn. Just good solid explanations as to why things worked out the way they did.

And the book covered a wide variety of Euclidean geometry. Barely touched 3 dimensions, but what can we expect? . more

## Quintilian on the Arrangement of a Speech

"[W]ith regard to the divisions which I have made, it is not to be understood that that which is to be delivered first is necessary to be contemplated first for we ought to consider, before everything else, of what nature the cause is what is the question in it what may profit or injure it next, what is to be maintained or refuted and then, how the statement of facts should be made. For the statement is preparatory to proof, and cannot be made to advantage, unless it is first settled what it ought to promise as to proof. Last of all, it is to be considered how the judge is to be conciliated for, until all the bearings of the cause be ascertained, we cannot know what sort of feeling it is proper to excite in the judge, whether inclined to severity or gentleness, to violence or laxity, to inflexibility or mercy."
- Quintilian, Institutes of Oratory, 95 AD

## Complex Numbers Primer

Before I get started on this let me first make it clear that this document is not intended to teach you everything there is to know about complex numbers. That is a subject that can (and does) take a whole course to cover. The purpose of this document is to give you a brief overview of complex numbers, notation associated with complex numbers, and some of the basic operations involving complex numbers.

This document has been written with the assumption that you’ve seen complex numbers at some point in the past, know (or at least knew at some point in time) that complex numbers can be solutions to quadratic equations, know (or recall) (i=sqrt<-1>), and that you’ve seen how to do basic arithmetic with complex numbers. If you don’t remember how to do arithmetic I will show an example or two to remind you how to do arithmetic, but I’m going to assume that you don’t need more than that as a reminder.

For most students the assumptions I’ve made above about their exposure to complex numbers is the extent of their exposure. Problems tend to arise however because most instructors seem to assume that either students will see beyond this exposure in some later class or have already seen beyond this in some earlier class. Students are then suddenly expected to know more than basic arithmetic of complex numbers but often haven’t actually seen it anywhere and have to quickly pick it up on their own in order to survive in the class.

That is the purpose of this document. We will go beyond the basics that most students have seen at some point and show you some of the notation and operations involving complex numbers that many students don’t ever see once they learn how to deal with complex numbers as solutions to quadratic equations. We’ll also be seeing a slightly different way of looking at some of the basics that you probably didn’t see when you were first introduced to complex numbers and proving some of the basic facts.

The first section is a more mathematical definition of complex numbers and is not really required for understanding the remainder of the document. It is presented solely for those who might be interested.

The second section (arithmetic) is assumed to be mostly a review for those reading this document and can be read if you need a quick refresher on how to do basic arithmetic with complex numbers. Also included in this section is a more precise definition of subtraction and division than is normally given when a person is first introduced to complex numbers. Again, understanding these definitions is not required for the remainder of the document they are only presented so you can say you’ve seen it.

The remaining sections are the real point of this document and involve the topics that are typically not taught when students are first exposed to complex numbers.

## Introduction to Claim Evidence Warrant

This resource introduces one of the most powerful—and most widely taught—methods for understanding and making arguments in the real world. It has many names: Claim/Evidence/Warrant, the Toulmin system, and others. Whatever you call it, however, learning this approach not only makes your writing stronger but also gives you a tool for critiqueing others’ arguments. It’s an easy-to-use formula for organizing an argument that’s as useful in the workplace as it is in school.

“If everyone’s entitled to their own opinion, why do I need a model for making arguments?”

True, everyone’s equally entitled to their opinions, but that doesn’t mean all opinions are equally persuasive. And it’s often precisely your job to be persuasive in your writing. Claim/Evidence/Warrant (CL/EV/WA) helps you articulate logical—and so persuasive—arguments.

This system has three basic elements (and sometimes three additional elements). Those basic elements are:

 1. Claim What you want your readers to believe the “point” you hope to persuade your reader of 2. Evidence What you will use to support the claim your “proof”—often a direct or indirect quotation from a text, but sometimes a statistic or the like 3. Warrant A general principle that explains why you think your evidence is relevant to your claim

Making a point with evidence in writing resembles conversation where you are trying to persuade someone of something.

## If-then statement

When we previously discussed inductive reasoning we based our reasoning on examples and on data from earlier events. If we instead use facts, rules and definitions then it's called deductive reasoning.

We will explain this by using an example.

If you get good grades then you will get into a good college.

The part after the "if": you get good grades - is called a hypotheses and the part after the "then" - you will get into a good college - is called a conclusion.

Hypotheses followed by a conclusion is called an If-then statement or a conditional statement.

This is read - if p then q.

A conditional statement is false if hypothesis is true and the conclusion is false. The example above would be false if it said "if you get good grades then you will not get into a good college".

If we re-arrange a conditional statement or change parts of it then we have what is called a related conditional.

Our conditional statement is: if a population consists of 50% men then 50% of the population must be women.

If we exchange the position of the hypothesis and the conclusion we get a converse statement: if a population consists of 50% women then 50% of the population must be men.

If both statements are true or if both statements are false then the converse is true. A conditional and its converse do not mean the same thing

If we negate both the hypothesis and the conclusion we get a inverse statement: if a population do not consist of 50% men then the population do not consist of 50% women.

The inverse is not true juest because the conditional is true. The inverse always has the same truth value as the converse.

We could also negate a converse statement, this is called a contrapositive statement: if a population do not consist of 50% women then the population do not consist of 50% men.

The contrapositive does always have the same truth value as the conditional. If the conditional is true then the contrapositive is true.

A pattern of reaoning is a true assumption if it always lead to a true conclusion. The most common patterns of reasoning are detachment and syllogism.

If we turn of the water in the shower, then the water will stop pouring.

If we call the first part p and the second part q then we know that p results in q. This means that if p is true then q will also be true. This is called the law of detachment and is noted:

$left [ (p o q)wedge p ight ] o q$

The law of syllogism tells us that if p → q and q → r then p → r is also true.

$left [ (p o q)wedge (q o r ) ight ] o (p o r)$

If the following statements are true:

If we turn of the water (p), then the water will stop pouring (q). If the water stops pouring (q) then we don't get wet any more (r).

Then the law of syllogism tells us that if we turn of the water (p) then we don't get wet (r) must be true.

## 1.S: Introduction to Writing Proofs in Mathematics (Summary)

PROPOSAL FORMS KIT
Introduction

This Proposal Forms Kit supplements the Grant Proposal Guide (GPG), the NSF Guide that provides guidance for the preparation of unsolicited proposals to NSF. An unsolicited proposal is prepared by a Principal Investigator and submitted by an organization on its own initiative and not in response to a formal written solicitation from NSF. Contact with appropriate NSF program personnel is encouraged to help determine if preparation of a formal proposal is appropriate.

The Kit contains the forms necessary for preparation of unsolicited proposals. While these forms are generally applicable, some NSF programs use specific program announcements which may require additional forms for submission of proposals to NSF or which may modify the general guidance in the GPG.

Please give careful attention to the requirements established in the GPG and the instructions provided on each form. Proposals that do not meet such requirements may be returned without further consideration by NSF. Of particular importance are the page limitations, format and content requirements that are identified on the forms and/or established throughout the GPG. Proposers should ensure that, where required, the submitted documents are signed and dated. A properly prepared proposal will facilitate the administrative processing and merit review that must occur before an award can be made.

Codes:
A =Use of format required
B =Use of format optional

The NSF Proposal Forms Kit includes one copy of each of the following forms:

C. Form Preparation Instructions

Each form is self-explanatory or provides specific completion instructions additional information is located in the GPG as follows:

Follow the instructions in the GPG regarding completion of the following sections of the proposal:

NSF Proposal Forms Kit – Paper copy forms provided in the Proposal Forms Kit may be used. Additional copies of this Kit, in paper and electronic medium, are available as indicated in Section G below.

Computer-generated facsimiles – Computer-generated facsimiles may be substituted for any of the forms contained in the Kit. The categories or information requested, however, should not be rearranged or altered.

Photocopies – Forms may be reproduced but must be clear and readable. NSF two-sided forms may be copied and submitted either on one two-sided sheet of paper or on single separate sheets of paper.

Note: NSF is in the process of recording all the contents of a proposal in electronic format. Therefore the various forms contained in the Proposal Forms Kit now include barcodes as part of the form. All paper proposals submitted to NSF must include the bar codes provided on each NSF required format.

Barcodes are not relevant to proposals prepared via FastLane.

FastLane – The NSF FastLane system uses Internet/Web technology to facilitate the way NSF does business with the research, education, and related communities. The NSF FastLane system is available for proposal preparation submission and status checking project reporting and post-award administrative activities. All FastLane functions are accessed by using a web browser on the Internet. There is a link to FastLane on the NSF Web site, or FastLane can be accessed directly at: <http://www.fastlane.nsf.gov/>.

Access to proposal and post-award functions is limited to staff from FastLane-registered organizations and is secured through the use of Personal Identification Numbers (PINs). To register an organization, authorized organizational representatives must complete the registration form that is available through the

Registration Information hyperlink on the FastLane Web site. Once an organization is registered, individual staff should contact the organization's sponsored projects office (or equivalent) for assignment of a PIN and information about how to access and use the system for most grant-related activities.

In the future, the Federal Commons project <http://www.fedcommons.gov/> will provide a common interface for grantees in accessing electronic business functions from a variety of federal grant-funding agencies. The Foundation is an active participant in the Federal Commons project and as this project develops, FastLane registration and security will be integrated with the Federal Commons. In anticipation of this, the FastLane PIN system will be moving to a password system. At that time, all references to PINs will be changed to passwords.

Detailed information about the FastLane system is available from the FastLane Web site at: <http://www.fastlane.nsf.gov/>.

E. When to Submit Proposals

Many NSF programs accept proposals any time. Other programs, however, establish target dates or deadlines for submission of proposals to allow time for their consideration by review panels which meet periodically. These target dates and deadlines are published in specific program announcements or solicitations, which can be obtained from the NSF Clearinghouse at: [email protected]> or electronically through the NSF Web site at: <https://www.nsf.gov/>. Lists of deadlines and target dates also are available electronically on the NSF Web site.

Unless otherwise specified in a program announcement or solicitation, proposals must be received by the specified date. A proposal received after a deadline, however, may be acceptable if it carries a legible proof-of-mailing date assigned by the carrier and the proof-of-mailing date is not later than one week prior to the deadline date. If the deadline date falls on a weekend, it will be extended to the following Monday if the date falls on a holiday, it will be extended to the following work day. The deadline date will be waived only in extenuating circumstances. Inquiry about submission may also be made to the appropriate program. For information about receipt of proposals submitted electronically, see Section F.

F. How to Submit Proposals

In September 1998, the NSF Director issued Important Notice 123, Working Towards a Paperless Proposal and Award System <https://www.nsf.gov/cgi-bin/getpub?iin123> that describes NSF's vision for the future in electronic business and outlines the schedule for implementation. As stipulated in this Notice, all proposals will be required to be submitted via FastLane effective October 1, 2000. Some NSF programs may require electronic submission of all or part of a proposal, including unsolicited proposals prior to this date. Please check the FastLane Web site prior to proposal submission for a listing of programs and program announcements and solicitations that require submission via FastLane. NSF recommends that all proposers and grantee organizations review Important Notice 123 to be aware of the implementation timelines stipulated in this document.

In the interim, for standard unsolicited proposals, electronic proposal submission via FastLane is the preferred method and is strongly encouraged. Unless otherwise specified by a program or in a program announcement or solicitation, however, proposals may continue to be submitted in paper form.

A proposal needs to be submitted only once to NSF, even if the proposer envisions review by multiple programs. The submission of duplicate or substantially similar proposals concurrently for review by more than one program without prior NSF approval may result in the return of the redundant proposals. (See Section IV.B for further information on proposal return.)

The following are specific instructions regarding the submission and receipt of electronic and paper proposals to NSF:

Electronic submission. A proposal is considered complete when the proposal, including the Project Description, has been submitted to NSF. If the Project Description is included in the electronic submission, unless otherwise specified in a program solicitation, the receipt date will be the date the sponsored projects office transmits the proposal to NSF. The signed proposal Cover Sheet (NSF Form 1207) must be postmarked (or provide a legible proof of mailing date assigned by the carrier) within five working days following the electronic submission of the proposal and forwarded to the following address:

National Science Foundation
DIS-FastLane Cover Sheet
4201 Wilson Blvd.
Arlington, VA 22230

A proposal may not be processed until the complete proposal (including signed Cover Sheet) has been received by NSF.

Unless stated otherwise in a program solicitation, proposals should not be addressed or sent directly to the cognizant Program Officer. If copies of the proposal are mailed or delivered in more than one package, the number of packages and the NSF announcement or solicitation number, if applicable, should be marked on the outside of each package. Proposals must be sent prepaid, not collect. Proposals sent by special messenger or courier should be delivered to the address listed below, weekdays, except Federal holidays, between the hours of 8:30 a.m. and 5:00 p.m. ET. Contact the NSF Mail Room, 703.306.0657, with any questions regarding the mailing or delivery of proposals.

Proposals must be addressed exactly as follows:
Announcement/Solicitation No________________
NATIONAL SCIENCE FOUNDATION PPU
4201 WILSON BLVD. ROOM P60
ARLINGTON VA 22230

When the proposal is assigned to a NSF program, the cognizant program information is available through the FastLane "Proposal Status Inquiry" function for PIs and through the "Recent Proposals" report for sponsored projects offices. Communications about the proposal should be addressed to the cognizant Program Officer with reference to the proposal number. Proposers are encouraged to use FastLane to verify the status of their submission to NSF.

G. Additional Copies of this Proposal Forms Kit

Additional copies of the Proposal Forms Kit (NSF 00-3) or the GPG (NSF 00-2), which contains the Proposal Forms Kit, may be ordered from:

## Examples

Suppose you are taking a course on contemporary communication, and the instructor hands out the following essay assignment: “Discuss the impact of social media on public awareness.” Looking back at your notes, you might start with this working thesis:

Social media impacts public awareness in both positive and negative ways.

You can use the questions above to help you revise this general statement into a stronger thesis.

• Do I answer the question? You can analyze this if you rephrase “discuss the impact” as “what is the impact?” This way, you can see that you’ve answered the question only very generally with the vague “positive and negative ways.”
• Have I taken a position that others might challenge or oppose? Not likely. Only people who maintain that social media has a solely positive or solely negative impact could disagree.
• Is my thesis statement specific enough? No. What are the positive effects? What are the negative effects?
• Does my thesis pass the “how and why?” test? No. Why are they positive? How are they positive? What are their causes? Why are they negative? How are they negative? What are their causes?
• Does my thesis pass the “So what?” test? No. Why should anyone care about the positive and/or negative impact of social media?

After thinking about your answers to these questions, you decide to focus on the one impact you feel strongly about and have strong evidence for:

Because not every voice on social media is reliable, people have become much more critical consumers of information, and thus, more informed voters.

This version is a much stronger thesis! It answers the question, takes a specific position that others can challenge, and it gives a sense of why it matters.

Let’s try another. Suppose your literature professor hands out the following assignment in a class on the American novel: Write an analysis of some aspect of Mark Twain’s novel Huckleberry Finn. “This will be easy,” you think. “I loved Huckleberry Finn!” You grab a pad of paper and write:

Mark Twain’s Huckleberry Finn is a great American novel.

You begin to analyze your thesis:

• Do I answer the question? No. The prompt asks you to analyze some aspect of the novel. Your working thesis is a statement of general appreciation for the entire novel.

Think about aspects of the novel that are important to its structure or meaning—for example, the role of storytelling, the contrasting scenes between the shore and the river, or the relationships between adults and children.
Now you write:

In Huckleberry Finn, Mark Twain develops a contrast between life on the river and life on the shore.

• Do I answer the question? Yes!
• Have I taken a position that others might challenge or oppose? Not really. This contrast is well-known and accepted.
• Is my thesis statement specific enough? It’s getting there–you have highlighted an important aspect of the novel for investigation. However, it’s still not clear what your analysis will reveal.
• Does my thesis pass the “how and why?” test? Not yet. Compare scenes from the book and see what you discover. Free write, make lists, jot down Huck’s actions and reactions and anything else that seems interesting.
• Does my thesis pass the “So what?” test? What’s the point of this contrast? What does it signify?”

After examining the evidence and considering your own insights, you write:

Through its contrasting river and shore scenes, Twain’s Huckleberry Finn suggests that to find the true expression of American democratic ideals, one must leave “civilized” society and go back to nature.

This final thesis statement presents an interpretation of a literary work based on an analysis of its content. Of course, for the essay itself to be successful, you must now present evidence from the novel that will convince the reader of your interpretation.

## Conditional and Indirect Proof

This can be done fairly quickly using IMP and AS to set up a DS, finishing with IMP to return us to the horseshoe in the conclusion:

That’s not too hard, but it’s fairly unintuitive you have to know to just manipulate the squiggles by rules, specifically to change the horseshoe to a wedge in order to change it back. Once you’ve done it several times, you can get used to it. But there is a very intuitive strategy known as Conditional Proof (CP) that you can use when you notice that the statement you need to derive is a conditional statement, as the conclusion of this one is: If it rains we’ll go to the movies.

Conditional Proof (CP) proceeds by letting you make an assumption, which is like saying to yourself, “OK, so what if it does rain, what will happen?,” as long as at the end of your musings from this assumption, you remember to sum it all up with a reference back to the fact that you began from that assumption. Here, you’d say:

Well if it does rain, then we’ll either go to the movies or watch basketball. But since you’re sick of watching basketball, that’s not really an option. So there’s really only the one option: if it rains we’ll go to the movies.

We need a new element of notation to accommodate this. To indicate an assumption is being made, we do two things: 1) Indent the assumed line,or, if the website you’re working on won’t save the indentation, place a vertical line, |, in front the lines that are subject to the assumption, and 2) justify it by the notation “ACP,” which means “Assumption for a Conditional Proof.”

| 3. R ACP We are assuming R, to see what it results in.

| 4. M v B MP 3,1 Very obvious

| 5. M CM, DS 2,4 Also very easy to spot

We’re done, because the assumption R, led to the consequent we wanted it to lead to, M, at line 5. That’s what this line 6 says (“R leads to M,” or “If R then M”). In this particular example, we barely saved any time in comparison to the other approach (in fact we only did because of putting the DS and CM on the same line), but in many cases, you’ll be done with the proof if you use CP faster than you’d even suspect how to get started without it.

Here’s another example, which we can approach in either of these two ways again.

If Imus goes off the air, then if Sharpton is a guest on Bill Maher’s Real Live, they will disagree with each other. Sharpton is a guest on Real Live. So if Imus goes off the air, Sharpton and Maher will disagree with each other.

Here’s the non-CP way to do it:

And here’s the CP approach:

Before we look at the next one, it’s time to talk about how you know what to assume, what to introduce by the line “ACP.” The answer is always easy to find, because it is always the antecedent of the conditional sentence that you want to generate. (It is not always the antecedent of the conclusion, because sometimes you can use this approach to get a line that will help you get the conclusion a little later but 90% of the time, when you use CP, it will in fact be to generate a conditional statement which is the conclusion.)

So, you should be able to tell what the antecedent would be for CP proofs if these statements were the proposed conclusions:

Always the antecedent of the line you’re trying to generate. The whole antecedent, but only the antecedent. (Once in a while, for fun, you might try assuming the negation of the consequent instead, since by TRAN, you can always turn that into the antecedent later!)

Now, you don’t use CP unless you are clear on another point, besides knowing what to bring in as your ACP assumption. The other thing you have to know is where you are going, what to shoot for, if you will. But that’s easy to know too, it’s always the consequent of the conditional that you are trying to generate. (Though, if you take me up on trying to do one by assuming the negation of the consequent of the desired conditional, you’d then be aiming for the negation of the antecedent, so that, at the very end, you could flip it with TRAN.)

Let’s look at another example before we introduce the possibility of multiple CPs:

If you’re the Vice President and your aide is found guilty of lying to federal investigators, then if you don’t go on the record to distance yourself from him, you’ll be under a cloud on the cover of Time. (Let’s pretend we’re talking to Mr. Cheney, so that “you” refers to the right person): Your aide, Scooter, was found guilty of just that, and you didn’t distance yourself from him. So, if you’re the Vice-President, you’re going to be under a cloud on the cover of Time.

What to do? The conclusion is a conditional, so we assume its antecedent, and work until we can get its consequent:

| 8. U MP 6, 7 That’s the consequent we want so, we’re done!

9. V ⊃ U CP 3-8 That’s how to say “we’re done.”

Using CP more than once in a proof

Being clear on how to determine what the assumption should be, and on what the goal of the CP sequence is (i.e., when to quit it) are crucial. And they are not complicated things to see they are usually both staring you in the face: the antecedent and the consequent of the conditional statement that you need to produce. This is why it is actually much easier than it looks at first, to do a CP inside another CP, and maybe to even do another one inside of that one. If example d) above were a conclusion:

The ACP line for this would be C, and we’d be finished when we arrived at the line D ⊃ (F ⊃T).

But one way to get D ⊃ (F ⊃T) is to assume D and work until the line F ⊃ T results.

And, since F ⊃ T is also a conditional statement, it could be possible to assume F and work until T showed up.

That is, we could say, “alright, let’s assume C, and let’s assume D, and let’s assume F, and see what happens.” In a few steps, we find that T happens, so we say “so F makes T happen, and that happened because we assumed D, and that happened because we’d assumed C!”

Here’s what it would look like:

If Hyperion and Starbucks are both closed, then if you insist on having coffee, you’re going to go to MacDonald’s but you’ll wish you hadn’t. If Hyperion is closed, then Starbucks is too. So, if Hyperion is closed, if you insist on having coffee, you’re going to wish you hadn’t.

|3. H ACP Why “H”? Because it’s the conclusion’s antecedent.

| | 4. C ACP Why “C”? Because it’s the antecedent of the consequent of the conditional you’re after: ||C > W, the consequent of the conclusion.

|10. C ⊃ W CP 4-9 This is the end of the second sequence begun with an “ACP”

11. H ⊃ (C ⊃ W) CP 3-10 This is the end of the first sequence begun with an “ACP”

Moral: when the consequent of the conditional is a conditional, then you do a CP within a CP. Two CP’s means you wind up with two horseshoe statements: p > (q > r).

## How To Write An Article Review

An article review essay is a critical analysis or evaluation of literature in a given field through making summary of the article in question, comparison or classification. In case it is a scientific article being reviewed, the writer will be required to use database searches to retrieve the results of the search. It is from the searches the writer chooses the article which is appropriate. It is a common skill that is developed in school, to help the students to establish analytical skills. It means the skills are not just used in the classrooms, but also applies in the work environment. Writing an article review has over time proved to be a difficult assignment for most of the students. The fact that it involves the critical evaluation of the ideas that have been presented in the article, means it is not just a summary. For most of the students, an article review is confused with an article summary. The critical analysis of the ideas in the article means, the author has to go further than summarizing the content of the article.

### 1. Purposes of an Article Review

• The purpose of the article review essay is associated with the writer summarizing the content of the article, evaluating other literature content and then adding their own understanding of the content.
• In most of the cases the article to be reviewed is going to be difficult to understand which requires the writer to read it severally before they are able to understand, which then means they will be in a position to offer insights into the content of the article in the review essay.
• One of the major aspects to understand, in reference to the purpose of article reviews essays writing, is a professional writing skill. This is relative to, article review requires one to have in-depth knowledge of a certain field, from which the article belongs.
• It is this information that helps in the review of the article and helps the audience to better understand the content and the argument of the writer.
• The purpose of an article review is associated with values that the people attach to opinion from others about certain information in a given field.
• It is also a crucial skill for journalists as they have to gather information on a given topic and to further assess how the audience view their work.
• Overall, article reviews build on the skills of the students to become better at analyzing information that interact with and make professional evaluations and recommendations.

### 2. Step-by-step guide on article review writing

The first part of the article review writing process is reading the article to be reviewed for the first time. The first time helps the writer to understand what it is the article is actually talking about. If it is an assigned reading in class, the student may not have come across the article in the past and thus will need to familiarize themselves with the content and the style of the author. The first reading of the article is crucial, as it helps the reader to connect with the author of the article to be reviewed. This also means that the reader will have to take note of the key points of the author, the basic argument presented in the article and note down some of the words, phrases and concepts that are new. These are elements that will then be looked at before going into the next reading. After noting the new information, the reader is supposed to look up the meaning of the phrases, words and read up on the concepts that have been presented in the article. The information on the various words, phrases and concepts can be found online or even at the library. It helps with better understanding the article and information presented especially if the article is in an unfamiliar field. This also sets the premises for reviewing the article in question, given it is not possible to write a review of an article that one does not understand.

This is the followed by a close reading, which is associated with critical reading of the article. In this case the, reader is not just reading to understand but to also form an opinion of the presentation of the information, the style of the author, the ease of understanding the information resented and even an opinion on the accuracy of the information presented. As such, the close reading is associated with analytical skills of the student. The writer is supposed to take notes of the various aspects of the article that they are going to include in the article review essay. It is crucial as it makes the work or writing the review much easier, relative to collated information that is then sorted out for inclusion in the essay.

#### 2.3 Identify Ideas

After identify the main ideas in the article, the writer may now develop the essay using their own words.

#### 2.4 Present a Unique Article Review

The point behind an article review essay to develop the skills to analyze information gathered from the various sources and presenting in a unique manner that reflects the ideas of the essay and presented in a manner that also shows the opinion of the writer. The writer has to be in a position to present the ideas in a professional manner, showing in depth understanding of the information found on the essay.

### 3. Article review writing process

The first part of the article review essay is the title, which informs the readers what the essay is about. The title is supposed to be short and precise, giving all the information that the reader may need to understand what the essay is about. At the same time, the title is supposed to be catchy, such that it gets the readers interested. For example, the title may be phrased as follows: “Evaluating the rise home grown terrorism”.

This is then followed by the correct citation of the article to be reviewed. Ideally it does not only give credit to the author of the article to reviewed, but also gives the readers a chance to check out the article that is being reviewed. It is crucial, as some of the readers may want to first read the article under review or may want to compare and make opinions of their own other than what is established in the review essay. In an example the reference to an article home terrorism may be cited as follows:

Park, S. (2016). Counter-Terrorism in France : Home-Grown Terrorism and the Change of Counter-Terrorism Policy. The Journal Of Peace Studies, 17(3), 117-140. http://dx.doi.org/10.14363/kaps.2016.17.3.117.

Within the text of the essay, however the writer will acknowledge the work of the author as follows (Park, 2016).

When identifying the article, it is crucial that the writer selects an article that talks about a topic that they are familiar with. This will help them to better analyze the article as they have some background information. Choosing an article that the writer is not familiar with, will further complicate the analysis process. If for example the writer chooses an article on calculous and they are not familiar with the field, they will have difficult time analyzing the same and may actually distort the meaning presented in the article.

The introduction of the article review essay should be catchy and inform the readers about the topic that they are going to discuss. This is then followed by the introduction of the article to be reviewed, with close reference to the main ideas and arguments presented by the author. The writer then introduces the author of the article and where the information is available, the credentials of the author and some of their other works. If for example the author of the article is a senior lecturer at the local university and working with local antiterror agency, this should be brought to the attention of the readers as it builds on the credibility and authority of the information presented in the article.

The critique section of the essay is organized into three main parts, the introduction of the field in which the article belongs and the wide issues. This is then followed by the article content and later on the review of the ideas and concepts of the article according to the writer. This means that, this is content that has to be unique to the writer. If for example, the writer do not agree with the strategies of reducing home terrorism, they may state so in this section and support their argument with facts.

In the conclusion, the writer summarizes the main points of the article and also connect the same wit their opinion on the matter.