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How to write a conclusion for a research paper, how to start a scholarship essay, how to focus on homework, how to write a conclusion for an essay: tips and tricks.

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A strong conclusion is just as important as any other essay component, as it’s the last part of the essay that leaves a lasting impression on your reader. Or doesn’t. A well-crafted conclusion does. It ties together your essay’s main points, demonstrates the significance and relevance of your arguments, and leaves the reader with a memorable closing thought. This guide will provide you with tips and tricks on what to do and what to avoid while writing effective essay conclusions.

The Purposes of A Conclusion

The conclusion of an essay has several goals, all of which contribute to the overall quality and success of your work. Although it may look similar to a summary of the essay, it actually is an important component that strengthens your argument and brings closure to your discussion. Understanding the following objectives of a conclusion will help you write one that improves your essay and intensifies its impact. Here are the main goals of a conclusion:

Connect the essay’s main points

A conclusion should briefly tie together the main points of your essay, reminding the reader of the key arguments presented. This helps your professor or other audience get a clear understanding of your main ideas and arguments and how they connect. Summarizing the main points, you reinforce your argument and provide a logical end to your paper.

Prove the Significance and Relevance of Your Arguments

Your conclusion should emphasize why your arguments matter, highlighting their importance and relevance. Summarizing your points isn’t enough for a well-written conclusion. There also should be an explanation of their broader impacts or implications. Demonstrating significance helps the reader understand the value of your argument and why it is worth considering. It also shows the relevance of your points to the overall topic or issue.

Leave the Reader with a Lasting Impression

The final sentences of your conclusion should leave a strong, lasting impression, encouraging the reader to dedicate more time and think deeper about the topic. This can be achieved by ending with a thought-provoking statement, a call to action, or a quote that highlights your main argument. An impactful closing makes your essay memorable and distinguishes it from others.

How to End an Essay

To effectively end your essay, you need to consider all the features that a successful conclusion should have and make sure to present them accurately and in the proper sequence. Only in this way will you not lose but rather deepen your impact and connection with your reader.

Restate your thesis: Begin your conclusion by restating your thesis in a new way. This backs your main argument without simply repeating it directly or paraphrasing what’s already been said in the body paragraphs. Restating the thesis helps remind the reader of your central argument and shows how the evidence and analysis in the body of the essay support this thesis. By rephrasing it, you provide a fresh perspective on your main point, reinforcing its importance.

Summary of synthesis: Summarize the main points of your writing, synthesizing the information instead of repeating it. You can use a summary generator tool for it. Highlight how the evidence you presented supports your thesis. Synthesis involves combining the key points and evidence to show how they interrelate and support your overall argument. 

Provide context: Add context to your conclusion by discussing the broader essence of your argument. Explain how your essay contributes to a larger conversation or addresses an important issue. Providing context helps the reader understand the significance of your argument in a wider scope. It can also involve suggesting areas for further research or highlighting the relevance of your points to real-world situations.

How to Conclude an Essay: Things to Avoid

At the same time, keep in mind that while you’re trying to write the perfect conclusion, you may accidentally make meaningful mistakes and add things that should not be there. Look at the common pitfalls and try to avoid the following mistakes:

Avoid Introducing New Information. Your conclusion is not the place to introduce new arguments or evidence. Stick to summarizing and synthesizing what has already been discussed. Introducing new information can confuse the reader and distract from the main points of your essay. It is important to keep the conclusion focused on wrapping up your argument and reinforcing what has already been said.

Avoid Overly Emotional Statements. While it’s important to leave an impression, avoid using overly emotional language or making sweeping statements that aren’t backed by anything. Emotional language can undermine the credibility of your argument and distract from your main points. Stick to logical and reasoned statements in your essay.

Avoid Repetition. Don’t repeat your introduction or body paragraphs. Instead, rephrase your key points and present them in a fresh way. Repetition can make your conclusion feel redundant and uninspired. By rephrasing your points, you can strengthen your argument in a more engaging way, keeping the reader interested until the very end.

What is a good way to end an essay?

A good way to end an essay is to restate your thesis in a new light, summarize the main points you’ve made, and provide a final thought or call to action that leaves an impression on the reader. You might use a quote, a question that encourages further thinking, or a call to action that emphasizes the relevance of your topic. The goal is to tie everything together and reinforce the significance of your argument.

How to end a conclusion?

To end a conclusion effectively, summarize the main points of your essay and restate your thesis in a rephrased manner. Follow this with a closing thought that underscores the importance of your discussion. This can be a final insight, a broader perspective of your argument, or a suggestion for future research or actions. The final sentence should leave a memorable impression and provide a sense of closure.

What is a good concluding word for an essay?

There are phrases that signal to the reader that you are wrapping up your discussion. Good concluding words or phrases for an essay include:

• In conclusion
• In the final analysis
• All things considered
• To conclude

How to conclude a paragraph?

To conclude a paragraph, you should summarize the main idea discussed in the paragraph and show how it supports your overall thesis. Use a closing sentence that ties the paragraph’s content back to your main argument. This could involve restating the key point of the paragraph in a concise manner or linking it to the next paragraph for a smooth transition.

How do you use closing sentences in an essay?

Closing sentences in an essay reinforce the main points discussed in each paragraph or section and help transition to the next idea. In the body paragraphs, use closing sentences to succinctly summarize the paragraph’s key point and link it to your thesis. In the conclusion, use a closing sentence to leave an impression, emphasize the significance of your arguments, and provide closure to your essay. Effective closing sentences are clear, concise, and relevant to the main argument.

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How to Write a Conclusion for an Essay - Tips and Examples

The conclusion of your essay is like the grand finale of a fireworks display. It's the last impression you leave on your reader, the moment that ties everything together and leaves them with a lasting impact. 

But for many writers, crafting a conclusion can feel like an afterthought, a hurdle to jump after the excitement of developing the main body of their work. Fear not! This article will equip you with the tools and techniques regarding how to write a conclusion for an essay that effectively summarizes your main points, strengthens your argument, and leaves your reader feeling satisfied and engaged.

What Is a Conclusion

In an essay, the conclusion acts as your final curtain call. It's where you revisit your initial claim (thesis), condense your main supporting arguments, and leave the reader with a lasting takeaway. 

Imagine it as the bridge that connects your ideas to a broader significance. A well-crafted conclusion does more than simply summarize; it elevates your points and offers a sense of closure, ensuring the reader leaves with a clear understanding of your argument's impact. In the next section, you will find conclusion ideas that you could use for your essay.

Please note that our online paper writing service can provide you not only with a stand-alone conclusion but with a fully new composition as well!

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Types of Conclusion

Here's a breakdown of various conclusion types, each serving a distinct purpose:

How Long Should a Conclusion Be

The ideal length of a conclusion depends on the overall length of your essay, but there are some general guidelines:

• Shorter Essays (500-750 words): Aim for 3-5 sentences. This ensures you effectively wrap up your points without adding unnecessary content.
• Medium Essays (750-1200 words): Here, you can expand to 5-8 sentences. This provides more space to elaborate on your concluding thought or call to action.
• Longer Essays (1200+ words): For these, you can have a conclusion of 8-10 sentences. This allows for a more comprehensive summary or a more nuanced exploration of the future implications or broader significance of your topic.

Here are some additional factors to consider:

• The complexity of your argument: If your essay explores a multifaceted topic, your conclusion might need to be slightly longer to address all the points adequately.
• Type of conclusion: A call to action or a hypothetical scenario might require a few extra sentences for elaboration compared to a simple summary.

Remember: The most important aspect is ensuring your conclusion effectively summarizes your main points, leaves a lasting impression, and doesn't feel rushed or tacked on.

Here's a helpful rule of thumb:

• Keep it proportional: Your conclusion should be roughly 5-10% of your total essay length.

How many sentences should a conclusion be?

Conclusion Transition Words

Transition words for conclusion act like signposts for your reader. They smoothly guide them from the main body of your essay to your closing thoughts, ensuring a clear and logical flow of ideas. Here are some transition words specifically suited for concluding your essay:

Remember, the best transition word will depend on the specific type of conclusion you're aiming for.

How to Write a Conclusion

Every essay or dissertation writer knows that the toughest part of working on a conclusion can be striking the right balance. You want to effectively summarize your main points without redundancy, leaving a lasting impression that feels fresh and impactful, all within a concise and focused section. Here’s a step-by-step guide to help you write a stunning essay conclusion:

Restate Your Thesis

Briefly remind your reader of your essay's central claim. This doesn't have to be a word-for-word repetition but a concise restatement that refreshes their memory.

Summarize Key Points

In a few sentences, revisit the main arguments you used to support your thesis. When writing a conclusion, don't get bogged down in details, but offer a high-level overview that reinforces your essay's focus.

Leave a Lasting Impression

This is where your knowledge of how to write a good conclusion can shine! Consider a thought-provoking question, a call to action, or a connection to a broader truth—something that lingers in the reader's mind and resonates beyond the final sentence.

Avoid Introducing New Information

The conclusion paragraph shouldn't introduce entirely new ideas. Stick to wrapping up your existing arguments and leaving a final thought.

Ensure Flow and Readability

Transition smoothly from the main body of your essay to the conclusion. Use transition words like "in conclusion," "finally," or "as a result," and ensure your closing sentences feel natural and well-connected to the rest of your work.

Note that you can simply buy essay at any time and focus on other more important assignments or just enjoy your free time.

Conclusion Paragraph Outline

Here's an outline to help you better understand how to write a conclusion paragraph:

• Tailor the length of your conclusion to your essay's overall length (shorter essays: 3-5 sentences, longer essays: 8-10 sentences).
• Ensure a smooth transition from the main body using transition words.
• Avoid introducing new information; focus on wrapping up your existing points.
• Proofread for clarity and ensure your conclusion ties everything together and delivers a final impactful statement.

Read more: Persuasive essay outline . 

Do’s and Don’ts of Essay Conclusion Writing

According to professional term paper writers , a strong conclusion is essential for leaving a lasting impression on your reader. Here's a list of action items you should and shouldn’t do when writing an essay conclusion:

Conclusion Examples

A strong conclusion isn't just an afterthought – it's the capstone of your essay. Here are five examples of conclusion paragraphs for essays showcasing different techniques to craft a powerful closing to make your essay stand out.

1. Call to Action: (Essay About the Importance of Recycling)

In conclusion, the environmental impact of our waste is undeniable. We all have a responsibility to adopt sustainable practices. We can collectively make a significant difference by incorporating simple changes like recycling into our daily routines. Join the movement – choose to reuse, reduce, and recycle.

2. Provocative Question: (Essay Exploring the Potential Consequences of Artificial Intelligence)

As artificial intelligence rapidly evolves, it's crucial to consider its impact on humanity. While AI holds immense potential for progress, will it remain a tool for good, or will it eventually surpass human control? This question demands our collective attention, as the decisions we make today will shape the future of AI and its impact on our world.

3. Universal Insight: (Essay Analyzing a Historical Event)

The study of history offers valuable lessons that transcend time. The events of the [insert historical event] remind us that even small acts of defiance can have a ripple effect, inspiring change and ultimately leading to a brighter future. Every voice has the power to make a difference, and courage can be contagious.

4. Future Implications: (Essay Discussing the Rise of Social Media)

Social media's explosive growth has transformed how we connect and consume information. While these platforms offer undeniable benefits, their long-term effects on social interaction, mental health, and political discourse require careful consideration. As social media continues to evolve, we must remain vigilant and ensure it remains a tool for positive connection and not a source of division.

5. Hypothetical Scenario: (Essay Arguing for the Importance of Space Exploration)

Imagine a world where our understanding of the universe is limited to Earth. We miss out on the potential for groundbreaking discoveries in physics, medicine, and our place in the cosmos. By continuing to venture beyond our planet, we push the boundaries of human knowledge and inspire future generations to reach for the stars.

Recommended for reading: Nursing essay examples .

Difference Between Good and Weak Conclusions

Not all conclusions are created equal. A weak ending can leave your reader feeling stranded, unsure of where your essay has taken them. Conversely, writing a conclusion that is strong acts as a landing pad, summarizing your key points and leaving a lasting impression.

Nailed that essay? Don't blow it with a lame ending! A good conclusion is like the mic drop at the end of a rap song. It reminds the reader of your main points but in a cool new way. Throw in a thought-provoking question, a call to action, or a connection to something bigger, and you'll leave them thinking long after they turn the page.

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How To Write A Conclusion For An Essay?

How to write a good conclusion, how to write a conclusion for a college essay.

Daniel Parker

is a seasoned educational writer focusing on scholarship guidance, research papers, and various forms of academic essays including reflective and narrative essays. His expertise also extends to detailed case studies. A scholar with a background in English Literature and Education, Daniel’s work on EssayPro blog aims to support students in achieving academic excellence and securing scholarships. His hobbies include reading classic literature and participating in academic forums.

is an expert in nursing and healthcare, with a strong background in history, law, and literature. Holding advanced degrees in nursing and public health, his analytical approach and comprehensive knowledge help students navigate complex topics. On EssayPro blog, Adam provides insightful articles on everything from historical analysis to the intricacies of healthcare policies. In his downtime, he enjoys historical documentaries and volunteering at local clinics.

• Updated writing tips.
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• Essay Conclusions | UMGC. (n.d.). University of Maryland Global Campus. https://www.umgc.edu/current-students/learning-resources/writing-center/writing-resources/writing/essay-conclusions
• How to Write a Conclusion for an Essay | BestColleges. (n.d.). BestColleges.com. https://www.bestcolleges.com/blog/how-to-write-a-conclusion/
• Ending the Essay: Conclusions | Harvard College Writing Center. (n.d.). https://writingcenter.fas.harvard.edu/pages/ending-essay-conclusions

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How to End a College Admissions Essay | 4 Winning Strategies

Published on October 16, 2021 by Meredith Testa . Revised on May 31, 2023.

The ending of your college essay should leave your reader with a sense of closure and a strong final impression.

Table of contents

Endings to avoid, option 1: return to the beginning, option 2: look forward, option 3: reveal your main point, option 4: end on an action, other interesting articles, frequently asked questions about college application essays.

A bad conclusion can bring your whole essay down, so make sure to avoid these common mistakes.

Summarizing

Unlike an academic essay, an admissions essay shouldn’t restate your points. Avoid ending with a summary; there’s no need to repeat what you’ve already written.

Phrases like “in conclusion,” “overall,” or “to sum it up” signal that you have nothing to add to what you’ve already written, so an admissions officer may stop reading.

Stating the obvious

Instead of stating the obvious, let your work speak for itself and allow readers to draw their own conclusions. If your essay details various times that you worked tirelessly to go above and beyond, don’t finish it by stating “I’m hardworking.” Admissions officers are smart enough to figure that out on their own.

You should also avoid talking about how you hope to be accepted. Admissions officers know you want to be accepted—that’s why you applied! It’s okay to connect what you discuss in the essay to your potential future career or college experience, but don’t beg for admission. Stay focused on your essay’s core topic.

Prevent plagiarism. Run a free check.

Many successful essays follow a “sandwich,” or full-circle, structure , meaning that they start with some image or idea, veer away from it in the middle, and then return to it at the end.

This structure is clean, self-contained, and satisfying for readers, so it’s a great choice if it works with the topic you’ve chosen.

In the “sandwich” essay outlined below, a student discusses his passion for musical theater. Instead of simply stating that interest, his essay starts with a funny anecdote about a minor fire that erupted on set. At the end, it returns to this anecdote, creating a sense of closure.

• Intro: I may be the world’s worst firefighter.
• Flashback to working on the school musical
• Demonstrate my passion for theatre
• Detail the story of the theater set catching fire
• Show how I made the most of the situation
• Conclusion: I proved my value as a director, an actor, and a writer that week一even if I was a terrible firefighter.

Many successful essays end by looking forward to the future. These endings are generally hopeful and positive—always great qualities in an admissions essay—and often connect the student to the college or their academic goals.

Although these endings can be highly effective, it can be challenging to keep them from sounding cliché. Keep your ending specific to you, and don’t default to generalities, which can make your essay seem bland and unoriginal.

Below are a good and a bad example of how you could write a “looking forward” ending for the musical theater “firefighter” essay.

Sometimes, holding back your main point can be a good strategy. If your essay recounts several experiences, you could save your main message for the conclusion, only explaining what ties all the stories together at the very end.

When done well, this ending leaves the reader thinking about the main point you want them to take from your essay. It’s also a memorable structure that can stand out.

However, if you choose this approach, it can be challenging to keep the essay interesting enough that the reader pays attention throughout.

In the essay outlined below, a student gives us snapshots of her experience of gymnastics at different stages in her life. In the conclusion, she ties the stories together and shares the insight that they taught her about different aspects of her character and values.

• Passionate, excited
• Sister born that day—began to consider people beyond myself
• Realizing that no matter how much I love gymnastics, there are more important things
• I’d been working especially hard to qualify for that level
• It came after many setbacks and failures
• I had to give up time with friends, first homecoming dance of high school, and other activities, and I considered quitting
• Conclusion: I’m still all of those selves: the passionate 7-year-old, the caring 11-year-old, and the determined 15-year-old. Gymnastics has been a constant throughout my life, but beyond the balance beam, it has also shown me how to change and grow.

Ending on an action can be a strong way to wrap up your essay. That might mean including a literal action, dialogue, or continuation of the story.

These endings leave the reader wanting more rather than wishing the essay had ended sooner. They’re interesting and can help you avoid boring your reader.

Here’s an example of how this ending could work for the gymnastics essay.

If you want to know more about academic writing , effective communication , or parts of speech , make sure to check out some of our other articles with explanations and examples.

Academic writing

• Writing process
• Transition words
• Passive voice
• Paraphrasing

 Communication

• How to end an email
• Ms, mrs, miss
• How to start an email
• I hope this email finds you well
• Hope you are doing well

 Parts of speech

• Personal pronouns
• Conjunctions

There are a few strategies you can use for a memorable ending to your college essay :

• Return to the beginning with a “full circle” structure
• Reveal the main point or insight in your story
• Look to the future
• End on an action

The best technique will depend on your topic choice, essay outline, and writing style. You can write several endings using different techniques to see which works best.

Unlike a five-paragraph essay, your admissions essay should not end by summarizing the points you’ve already made. It’s better to be creative and aim for a strong final impression.

You should also avoid stating the obvious (for example, saying that you hope to be accepted).

There are no set rules for how to structure a college application essay , but these are two common structures that work:

• A montage structure, a series of vignettes with a common theme.
• A narrative structure, a single story that shows your personal growth or how you overcame a challenge.

Avoid the five-paragraph essay structure that you learned in high school.

When revising your college essay , first check for big-picture issues regarding message, flow, tone, style , and clarity. Then, focus on eliminating grammar and punctuation errors.

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Testa, M. (2023, May 31). How to End a College Admissions Essay | 4 Winning Strategies. Scribbr. Retrieved June 24, 2024, from https://www.scribbr.com/college-essay/conclusion-college-essay/

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• Writing Tips

How to Write an Essay Conclusion

4-minute read

• 1st October 2022

Regardless of what you’re studying, writing essays is probably a significant part of your work as a student . Taking the time to understand how to write each section of an essay (i.e., introduction, body, and conclusion) can make the entire process easier and ensure that you’ll be successful.

Once you’ve put in the hard work of writing a coherent and compelling essay, it can be tempting to quickly throw together a conclusion without the same attention to detail. However, you won’t leave an impactful final impression on your readers without a strong conclusion.

We’ve compiled a few easy steps to help you write a great conclusion for your next essay . Watch our video, or check out our guide below to learn more!

1. Return to Your Thesis

Similar to how an introduction should capture your reader’s interest and present your argument, a conclusion should show why your argument matters and leave the reader with further curiosity about the topic.

To do this, you should begin by reminding the reader of your thesis statement. While you can use similar language and keywords when referring to your thesis, avoid copying it from the introduction and pasting it into your conclusion.

Try varying your vocabulary and sentence structure and presenting your thesis in a way that demonstrates how your argument has evolved throughout your essay.

2. Review Your Main Points

In addition to revisiting your thesis statement, you should review the main points you presented in your essay to support your argument.

However, a conclusion isn’t simply a summary of your essay . Rather, you should further examine your main points and demonstrate how each is connected.

Try to discuss these points concisely, in just a few sentences, in preparation for demonstrating how they fit in to the bigger picture of the topic.

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3. Show the Significance of Your Essay

Next, it’s time to think about the topic of your essay beyond the scope of your argument. It’s helpful to keep the question “so what?” in mind when you’re doing this. The goal is to demonstrate why your argument matters.

If you need some ideas about what to discuss to show the significance of your essay, consider the following:

• What do your findings contribute to the current understanding of the topic?
• Did your findings raise new questions that would benefit from future research?
• Can you offer practical suggestions for future research or make predictions about the future of the field/topic?
• Are there other contexts, topics, or a broader debate that your ideas can be applied to?

While writing your essay, it can be helpful to keep a list of ideas or insights that you develop about the implications of your work so that you can refer back to it when you write the conclusion.

Making these kinds of connections will leave a memorable impression on the reader and inspire their interest in the topic you’ve written about.

4. Avoid Some Common Mistakes

To ensure you’ve written a strong conclusion that doesn’t leave your reader confused or lacking confidence in your work, avoid:

• Presenting new evidence: Don’t introduce new information or a new argument, as it can distract from your main topic, confuse your reader, and suggest that your essay isn’t organized.
• Undermining your argument: Don’t use statements such as “I’m not an expert,” “I feel,” or “I think,” as lacking confidence in your work will weaken your argument.
• Using generic statements: Don’t use generic concluding statements such as “In summary,” “To sum up,” or “In conclusion,” which are redundant since the reader will be able to see that they’ve reached the end of your essay.

Finally, don’t make the mistake of forgetting to proofread your essay ! Mistakes can be difficult to catch in your own writing, but they can detract from your writing.

Our expert editors can ensure that your essay is clear, concise, and free of spelling and grammar errors. Find out more by submitting a free trial document today!

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In a short paper—even a research paper—you don’t need to provide an exhaustive summary as part of your conclusion. But you do need to make some kind of transition between your final body paragraph and your concluding paragraph. This may come in the form of a few sentences of summary. Or it may come in the form of a sentence that brings your readers back to your thesis or main idea and reminds your readers where you began and how far you have traveled.

So, for example, in a paper about the relationship between ADHD and rejection sensitivity, Vanessa Roser begins by introducing readers to the fact that researchers have studied the relationship between the two conditions and then provides her explanation of that relationship. Here’s her thesis: “While socialization may indeed be an important factor in RS, I argue that individuals with ADHD may also possess a neurological predisposition to RS that is exacerbated by the differing executive and emotional regulation characteristic of ADHD.”

In her final paragraph, Roser reminds us of where she started by echoing her thesis: “This literature demonstrates that, as with many other conditions, ADHD and RS share a delicately intertwined pattern of neurological similarities that is rooted in the innate biology of an individual’s mind, a connection that cannot be explained in full by the behavioral mediation hypothesis.”  

Highlight the “so what”  

At the beginning of your paper, you explain to your readers what’s at stake—why they should care about the argument you’re making. In your conclusion, you can bring readers back to those stakes by reminding them why your argument is important in the first place. You can also draft a few sentences that put those stakes into a new or broader context.

In the conclusion to her paper about ADHD and RS, Roser echoes the stakes she established in her introduction—that research into connections between ADHD and RS has led to contradictory results, raising questions about the “behavioral mediation hypothesis.”

She writes, “as with many other conditions, ADHD and RS share a delicately intertwined pattern of neurological similarities that is rooted in the innate biology of an individual’s mind, a connection that cannot be explained in full by the behavioral mediation hypothesis.”  

Leave your readers with the “now what”  

After the “what” and the “so what,” you should leave your reader with some final thoughts. If you have written a strong introduction, your readers will know why you have been arguing what you have been arguing—and why they should care. And if you’ve made a good case for your thesis, then your readers should be in a position to see things in a new way, understand new questions, or be ready for something that they weren’t ready for before they read your paper.

In her conclusion, Roser offers two “now what” statements. First, she explains that it is important to recognize that the flawed behavioral mediation hypothesis “seems to place a degree of fault on the individual. It implies that individuals with ADHD must have elicited such frequent or intense rejection by virtue of their inadequate social skills, erasing the possibility that they may simply possess a natural sensitivity to emotion.” She then highlights the broader implications for treatment of people with ADHD, noting that recognizing the actual connection between rejection sensitivity and ADHD “has profound implications for understanding how individuals with ADHD might best be treated in educational settings, by counselors, family, peers, or even society as a whole.”

To find your own “now what” for your essay’s conclusion, try asking yourself these questions:

• What can my readers now understand, see in a new light, or grapple with that they would not have understood in the same way before reading my paper? Are we a step closer to understanding a larger phenomenon or to understanding why what was at stake is so important?  
• What questions can I now raise that would not have made sense at the beginning of my paper? Questions for further research? Other ways that this topic could be approached?  
• Are there other applications for my research? Could my questions be asked about different data in a different context? Could I use my methods to answer a different question?  
• What action should be taken in light of this argument? What action do I predict will be taken or could lead to a solution?  
• What larger context might my argument be a part of?  

What to avoid in your conclusion  

• a complete restatement of all that you have said in your paper.  
• a substantial counterargument that you do not have space to refute; you should introduce counterarguments before your conclusion.  
• an apology for what you have not said. If you need to explain the scope of your paper, you should do this sooner—but don’t apologize for what you have not discussed in your paper.  
• fake transitions like “in conclusion” that are followed by sentences that aren’t actually conclusions. (“In conclusion, I have now demonstrated that my thesis is correct.”)
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Learn about the elements of a successful essay conclusion.

The conclusion is a very important part of your essay. Although it is sometimes treated as a roundup of all of the bits that didn’t fit into the paper earlier, it deserves better treatment than that! It's the last thing the reader will see, so it tends to stick in the reader's memory. It's also a great place to remind the reader exactly why your topic is important. A conclusion is more than just "the last paragraph"—it's a working part of the paper. This is the place to push your reader to think about the consequences of your topic for the wider world or for the reader's own life!

A good conclusion should do a few things:

Restate your thesis

Synthesize or summarize your major points

Make the context of your argument clear

Restating Your Thesis

You've already spent time and energy crafting a solid thesis statement for your introduction, and if you've done your job right, your whole paper focuses on that thesis statement. That's why it's so important to address the thesis in your conclusion! Many writers choose to begin the conclusion by restating the thesis, but you can put your thesis into the conclusion anywhere—the first sentence of the paragraph, the last sentence, or in between. Here are a few tips for rephrasing your thesis:

Remind the reader that you've proven this thesis over the course of your paper. For example, if you're arguing that your readers should get their pets from animal shelters rather than pet stores, you might say, "If you were considering that puppy in the pet-shop window, remember that your purchase will support 'puppy mills' instead of rescuing a needy dog, and consider selecting your new friend at your local animal shelter." This example gives the reader not only the thesis of the paper, but a reminder of the most powerful point in the argument!

Revise the thesis statement so that it reflects the relationship you've developed with the reader during the paper. For example, if you've written a paper that targets parents of young children, you can find a way to phrase your thesis to capitalize on that—maybe by beginning your thesis statement with, "As a parent of a young child…"

Don’t repeat your thesis word for word—make sure that your new statement is an independent, fresh sentence!

Summary or Synthesis

This section of the conclusion might come before the thesis statement or after it. Your conclusion should remind the reader of what your paper actually says! The best conclusion will include a synthesis, not just a summary—instead of a mere list of your major points, the best conclusion will draw those points together and relate them to one another so that your reader can apply the information given in the essay. Here are a couple of ways to do that:

Give a list of the major arguments for your thesis (usually, these are the topic sentences of the parts of your essay).

Explain how these parts are connected. For example, in the animal-shelter essay, you might point out that adopting a shelter dog helps more animals because your adoption fee supports the shelter, which makes your choice more socially responsible.

One of the most important functions of the conclusion is to provide context for your argument. Your reader may finish your essay without a problem and understand your argument without understanding why that argument is important. Your introduction might point out the reason your topic matters, but your conclusion should also tackle this questions. Here are some strategies for making your reader see why the topic is important:

Tell the reader what you want him or her to do. Is your essay a call to action? If so, remind the reader of what he/she should do. If not, remember that asking the reader to think a certain way is an action in itself. (In the above examples, the essay asks the reader to adopt a shelter dog—a specific action.)

Explain why this topic is timely or important. For example, the animal-shelter essay might end with a statistic about the number of pets in shelters waiting for adoption.

Remind the readers of why the topic matters to them personally. For example, it doesn’t matter much if you believe in the mission of animal shelters, if you're not planning to get a dog; however, once you're looking for a dog, it is much more important. The conclusion of this essay might say, "Since you’re in the market for a dog, you have a major decision to make: where to get one." This will remind the reader that the argument is personally important!

Conclusion paragraphs

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In Summary: 10 Examples of Essay Conclusions

The conclusion of an essay may be the toughest section to write. Think about it; you're really tired at this point. It's probably the night before your paper is due and you just want to be done . So, the temptation is there to simply rush through it, and hope that your teacher is exhausted once she gets to your paper and doesn't bother to read it fully.

But the conclusion is probably the most important part of the paper. It ties everything together up nicely in the end. Not writing a good conclusion would be like if we never found out if Elizabeth and Mr. Darcy got together or if we never knew what that monster was in the Upside Down in "Stranger Things." Though not every ending has to be 100% conclusive (in fact, most endings never are— think the movie Inception ), it does have to have a well-thought out conclusion.

So, how do you write a good conclusion? What are the key components of a solid conclusion? What does a thorough and effective conclusion look like?

Read on for more information about our conclusion on conclusions.

What are the key components of a good conclusion?

Remember that thesis statement which you wrote in the first or second paragraph of your essay? You know, the one where you stated a claim about something? You argued something about a topic and you used the body paragraphs to prove your thesis statement through all of the research that you've performed.

Now that you've fully explained the research and the support for your thesis statement throughout the body of the paper, it's time to come back to that original idea in the conclusion. The conclusion basically asks us to do a few things:

• Restate the main idea of the paper (why you wrote this entire long piece to begin with).
• Summarize all the key points you made throughout the body of the paper (things that proved your thesis statement).
• Write about why this paper and topic are important, and leave the reader with ideas for additional research or maybe some questions that didn't get answered. The idea is that you want to leave the reader with a long-lasting impression. This is your opportunity to really drive your point home and to use some really interesting language.

Okay, so now that we have a game plan of how we need to write a good conclusion and what components consists of, let's look at a few examples of some sample essay conclusions.

Essay conclusion 1 — Why Ross didn't deserve Rachel on "Friends"

Although viewers always expected Ross and Rachel to reunite at the end of the series, the fact remains that Ross didn't deserve Rachel as a partner. As we saw in the beginning of the series, Ross was unfaithful to Rachel when they had been dating for over a year, and he didn't want to admit his wrongdoing when they tried to get back together after their initial breakup. Additionally, Ross was an extremely jealous and demanding partner, yelling at Rachel in front of all of their friends on several occasions. Finally, and most egregiously, Ross had a terrible reaction when Rachel told Ross she was pregnant after Monica and Chandler's wedding, making him an undesirable romantic partner for her, or any other character on the show for that matter. This conclusion is especially apparent after viewing the show more than 10 years after the final episode aired and having a collectively better understanding of women's rights and domestic abuse in relationships.

Essay conclusion 2 — Should students be allowed to have cell phones in elementary school?

In conclusion, although it's easy to see why allowing an elementary school child to have a cell phone would be convenient for after-school pickups or arranging playdates with friends, there is too much evidence to show that it's generally not a good idea. Children already have a lot of access to media (on average over seven hours per day) and it is the parent's responsibility to monitor their media access, which is more difficult if the child has exclusive cell phone access. Cyber bullying, which is increasingly becoming a problem, is also going to be a risk when your child has unlimited access to a smart phone. Clearly, elementary school-aged children are not emotionally mature enough to handle the responsibility of a smart phone, and the borrowing of a parent's cell phone should be highly monitored to ensure safe and healthful usage.

Essay conclusion 3 — Should sexual education be taught in public schools?

It's clear that sexual education is completely vital to the public-school curriculum. Not only does this lead to a better understanding of human development and human sexuality, but awareness and sex education also reduce the rates of teen pregnancy. Studies have shown that comprehensive sexual education increases the age of when teens have sex for the first time. Learning about contraception and how to use contraception correctly ultimately leads to lower rates of STDs. Lastly, comprehensive sex education also teaches students about consensual sex, and will hopefully lead to healthier sexual relationships and lower rates of sexual assault in the future. Not only should sex education be taught in public schools, but it should be mandatory for all public-school systems.

Essay conclusion 4 — What are the biggest challenges for women in the workplace?

Women have outnumbered men on the payroll in nonfarm jobs since 2010, but even with a majority of females in the office, there are still huge challenges for them at work. One of the biggest issues, which has been widely covered and debated on, is the fact that women still earn less of a wage for the same job as their male counterparts. Now that women are the breadwinners of many families, this is stunting economic growth and opportunity for their children. Additionally, women are less likely to be in charge at work. With less than 6% of Fortune 500 companies with a female CEO, women have a steeper hill to climb at the very top echelon of jobs. With a more level playing field, women's opportunities will increase and the workforce will ultimately be more inviting for all.

Essay conclusion 5 — You're having dinner with your favorite author. What happens? Describe the scene.

Harper Lee puts down her cup of coffee on the table, quietly scanning the room for an exit.

I'm nervous, wondering what to say to end this surreal evening.

"Thank you so much for meeting with me. I know that you're a very private person, and I can't tell you how much this means to me."

She smiles slightly at me and waves at the waiter for the check, which he brings promptly.

Essay conclusion 6 — Should music with curse words be allowed at school dances?

Language can be powerful and sometimes even harmful, but censorship of language is one of the worst things we can do as a society. I believe that the content of the song is more important than a few curse words. If a song's content is designed to provoke, intimidate, or make someone feel inferior, then I believe that is more harmful than a few impolite words in a chorus.

Essay conclusion 7 — What is something that should be taught in school that isn't?

Financial literacy is one of the most important things a person needs to understand as a fully functional adult. It's crucial for someone to be able to know how to purchase a car, open a bank account, invest in a 401k plan, and pay back his or her student debt all while being able to balance paying rent and saving money. Financial literacy should be taught to students while they are still in high school so that they can feel prepared to go out on their own and make a positive contribution to society.

Essay conclusion 8 — Is an increased dependence on technology good for society?

Technology surely isn't going anywhere. If anything, we will become more and more dependent on the capabilities of our smartphones and other devices in the future. However, we have to make sure that this dependence on technology isn't making us lazier or less curious about the world around us. With more knowledge available than ever before with today's technology, people are less discerning about what kind of materials they read and whether or not those materials are factual. People are also less likely to make a personal connection with someone while they're out in the world, which can increase levels of depression, anxiety, and stress. Ultimately, we have to learn how to co-exist with technology in a way that is both healthful and constructive.

Essay conclusion 9 — Should schools start later in the morning?

There are some clear benefits to starting school later in the morning for K-12 students such as better academic performance and improved sleeping schedules. Although it might take a bit of rearranging schedules for parents to take their kids to school later on in the day, it's more important that students perform better academically than for the drop-off to be convenient for the parents on their way to work. To combat this, increased bus routes and crossing guards should be implemented so that parents who have to get to work at a certain time can be assured that their kids are making it to school safely.

Essay conclusion 10 — How do video games affect children and teenagers?

Video games have been an integral part of childhood and adolescence for a few decades now, but the effects on aggression levels and exposure to violence may make us take pause on how much exposure parents should let their kids have to these games. The video game industry is growing exponentially, and as the technology and video quality increase so does the ability to separate virtual reality from reality. Games with violent content are known to cause aggressive and sometimes even violent behavior in teens. Many video games, first-person shooter games in particular, have violent content. When the player is rewarded for violent behavior in the game, it reinforces the subtle idea that violence is acceptable and can be used in real life. With busy schedules and easy access to so much media, it's difficult for parents to be able to oversee everything that their children are exposed to. Video game designers should be held accountable for the violent content in their games, and a push should be made for more parental oversight and rules on video game usage.

In conclusion of conclusions

Conclusions are really just about wrapping things up. You want to be as succinct as possible, you want to reiterate the points you've already made throughout the essay, and you want to be compelling. With a little bit of practice and revision, you should be able to get the process down in no time. And if you need help with revising your conclusion or any other part of your paper, be sure to seek out the advice of a trusted teacher or a writing center, or hire one of our professional editors to give you a second opinion on your paper.

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Mastering Your Argumentative Essay Conclusion: Tips & Techniques

Table of contents

• 1.1 Steps to Writing an Argumentative Essay Conclusion
• 2 How to Restate a Thesis Statement
• 3.1 Know how to structure your paper
• 4.1 Example 1
• 4.2 Example 2
• 4.3 Example 3
• 4.4 Examples 4, 5
• 5 How to Finish an Argumentative Project Conclusion Paragraph

Want to write a perfect conclusion for your paper but don’t know how? Everyone has been there, and it’s never easy. It is the final part of your writing, so by the time you reach it, you have no energy and can’t focus.

Still, the conclusion part is crucial for the success of every paper. You have to give the final answer to the audience by restating your thesis and noting your claims and findings. If you think you can’t write one, you’d better buy an argumentative essay online and solve your problems.

In this article, you will find everything you need to know about a conclusion to an argumentative essay and how to write it.

What to Write in the Conclusion for an Argumentative Essay

To write a conclusion argumentative essay, you first need to recall all the key points of your essay.

Except for those basic points, knowing how to conclude an argumentative essay also requires a few more things:

The first thing to pay attention to is your tone of writing. Make sure it is authoritative yet calm and informative. This way, you will assure readers that your work is essential for the case.

Next is your first sentence. How you start your conclusion does matter. You need to state what you did and why. That will remind the readers once again about what they have read.

After you write it, you will need to point out the key findings of your writing. You must note the important evidence you have written about in your paper. Keep it brief and connect them to your text conclusion.

The last step is to finish the conclusion of your argumentative essay in a meaningful way. Ensure a positive final sentence to make the reader reflect on your work and make them act.

Thus, writing a conclusion for an argumentative essay is a complex process. It can be not easy to come up with a good conclusion on your own, so don’t hesitate to seek  essay assistance if you need it.

Author Note: Make sure not to present any new arguments or claims in the conclusion. This section of your paper is your final opinion. Writing further details, ideas, or irrelevant findings can ruin the text.

Steps to Writing an Argumentative Essay Conclusion

Your conclusion should convincingly summarize your viewpoint. Here’s a simplified way to approach it:

• Briefly restate the importance of your topic.
• Summarize your thesis statement again.
• Acknowledge opposing views to strengthen your argument.
• Suggest actions to be taken or hint at further exploration.

How to Restate a Thesis Statement

• Choose the Right Location:

Place your restated thesis statement where it fits best. Often, it works well at the start of the conclusion, but not always. First, draft a rough conclusion to find the ideal spot. Consider opening with a rhetorical question instead, depending on your paper’s structure.

• Leverage Your Paper’s Content:

Use the full context of your paper to enhance your thesis restatement. Now that your readers have the full information, you can deepen the impact of your argument.

• Clarify the Significance:

Address why your argument matters. Answer the “So what?” question by linking back to the broader implications of your thesis.

• Avoid Common Clichés:

Start your conclusion without clichéd phrases like “In conclusion” or “As this paper has shown.” These phrases can undermine the strength of your restatement.

• Maintain Confidence:

Assert your thesis confidently without hedging with phrases like “It seems like.” However, recognize opposing views respectfully to avoid alienating your readers.

How to Format the Conclusion of an Argumentative Essay?

To format a conclusion, you have to follow a well-established standard. The best essay conclusions include a “lead” (opening statement). Then point out one vital factor from your paragraph. Usually, one point per paragraph, no more, or it will get too bulky. Finally, add an appropriate finale that will serve as a smooth exit of the whole paper, the final sentence.

By using the standard format, you will have an easier time when you have to write an argumentative essay conclusion. You can focus on the facts and tailor them to appeal to readers. That will re-convince them about your point for the case.

Here we can add that the final sentence should not always be smooth and friendly. When your conclusion tone is assertive, write the final part of the finale as a call to action—an attempt to affect the reader and make them want to research. To find out more about the matter or even take a stand with their own opinion.

Know how to structure your paper

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Knowing the exact way to structure a conclusion in an argumentative essay is crucial. Someone may say that it is not important. But this is one of the first things people pay attention to. So, you have to format the paper and its main points properly. In any assignment, the style of the text adheres to strict requirements. Usually, you can find them by asking your professor or checking the educational institution’s website.

In that sense, you must stick to proper formatting when writing a perfect argumentative essay . To get the best grade, you have to use the  recommended formatting style , which can be APA, AP, or other. So remember, following the proper structure and formatting can make the critical points of your work stand out. As a result, your paper will look better, and your paper results will score higher.

Writing a perfect conclusion for your paper can be difficult, especially when you have no energy and can’t focus. Fortunately, PapersOwl.com is here to help. Our experienced writers can provide you with an excellent conclusion for your paper so that you can confidently submit it.

Essay Conclusion Examples

If you are still trying to figure out what your conclusion should look like, check below. We have prepared how-to-end argumentative essay examples . These can give you an idea about the structure and format of your paper’s final point.

In this particular sample, the case is about global warming. So, the essay’s conclusion has to give a compelling reason why the reader and the public should act and prevent the issue. You must remember that what you write depends on the type of paper and should be unique.

“Throughout our text, we pointed out findings about the impact of global warming. Nature cannot sustain itself in the ever-changing climate. The ice caps melt, and the shorelines deteriorate, thus causing the extinction of both flora and fauna. Due to the persisting crisis, we must take action and use the best methods to protect the future of our planet.”

Some papers involve public policies and morals. In such cases, you must write in a tone that will feel morally right but will support and justify your arguments. Usually, you write such papers when your topic is pointing towards persuasion. Below, you can see an argumentative essay conclusion example for such texts.

“As time goes on, technology has changed how we, as a society, receive and use information. Media’s influence has been increasing throughout the social applications we use daily. The said impacts public opinion, as we can see from the participants in our study group. Most have stated that their primary information source is social media. These media get large funds from private entities to filter your content. This way, you see their ideas and become part of their audience. If you like your news free of filtering and want truthful information, you must act now and ensure your rights.”

At one point or another, you will get an assignment to help with your career objectives. Usually, it is connected to your writing as you have to research specific matters. For example, bring out your point of view and make conclusions. You can quickly implement such tasks in essays like the argumentative one. Thus, you have to be ready to write a conclusion of an argumentative essay that can fit well and is decisive.

“Often, when you get the opportunity to launch a new business, you must grab it. Plan business meetings, solve the x, y, and z obstacles, and speed up the process. Business is about profit, producing more revenue, and creating an easily manageable structure. If you choose to act on a different undertaking, there will be risks a or b, which can lead to overstepping the estimated budgets.”

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Examples 4, 5

As seen, the conclusion of an argumentative essay may reflect your moral choices. In other cases, on a figure of speech and even sensitivity towards an issue.

So, some good argumentative essay topics need an emotional appeal to the reader. It’s important to present your views effectively and respectfully. But, no matter your point, it is crucial to state your ideas without offending anyone else.

“The right to give birth or not is fundamental for women. They must have it ensured. Otherwise, they have no control or option in their social relationships. The analysis showcases how an unwanted pregnancy can influence and determine the life of a young woman and her child. So without guaranteed rights, women are forced to use dangerous methods to retake ownership of their body, and that must change.” “Life is not a choice given by someone. It is a fundamental right guaranteed by the law. In that sense, denying an unborn child’s right to life is identical to denying any other person’s rights. Furthermore, studies have long proven that life begins with its inception. Therefore, carrying out policies of pro-choice is like murder. With that in mind, saving the unborn by speaking out for them is like giving their rights a voice.”

How to Finish an Argumentative Project Conclusion Paragraph

How to end an argumentative essay? A strong conclusion is key. The final sentence should deeply impact your reader, often ending smoothly yet with a call to action that compels them to think or act. In other cases, the call to action is intense. It could be smoother, but its main goal is to influence the audience to contemplate and act.

It should look like, “ If we don’t do it now, we won’t be able to act in the future. ” If your sentence cuts the flow of the whole text, it will not appeal to your reader.  If you are having trouble crafting the perfect conclusion for your argumentative essay, you can always pay for essay help from a professional writer to get the job done right.

Now you understand how to write a conclusion for an argumentative essay, but remember to catch up on the whole paper flow and finish it in the same tone. Use the call to action sentence and exit your essay smoothly while giving the readers ideas and making them think about the case. If you can’t, please check our argumentative essay writing services , which can easily tackle the task. Note that by getting it done by a professional, you can learn from examples. Besides, the text can get done in a few hours.

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Conclusion Paragraph Starters

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Conclusion Paragraph Starters: Overview, Tips & Examples

Indeed, writing the conclusion paragraph starters can be daunting as it needs you to think beyond the limits. Crafting an interesting conclusion starter can make all the difference in your essay, article, or another piece of writing. The conclusion paragraph sentence starters provide the end of the conversation and cover the important subjects, giving you a final viewpoint.

You need to be clear with the art of starting your best conclusion paragraph starters correctly to leave an impact on the readers. Thus, The Student Helpline brings you an article that tells how to write good conclusion sentence starters for essays. Here are a few examples for different types of contexts and will give you an idea of increasing the importance of conclusion paragraphs.

Conclusion Paragraph Starters: Why To Add Them?

The starting point of a conclusion is the last chance to leave an impact on your reader. An informative introduction can have an ongoing effect by informing readers about your primary points and the value of your case. Recall, that good starters for conclusion paragraph can tie everything together, and prove that your message offers a sense of fulfilment.

How To Write Conclusion Paragraph Starters?

Writing ideal conclusion paragraph starters is indeed important and you should be perfect at it. The Student Helpline's assignment experts bring you a list of tips that you should understand for creating good conclusion starters. They are:

• Fit the Tone: You should make sure that the opening of your starters for conclusion paragraph should meet the tone of your essay. In fact, it never matters whether you are writing formal or informal. In academic writing, you can use more formal openings such as "In conclusion" or "Therefore."
• Keep it Simple and Easy to Understand: Furthermore, select good conclusion paragraph starters that are clear and point to point. Besides, avoid adding complex and difficult-to-understand starters.
• Make a Summary of the main points: You can even add these starters conclusion paragraph to provide a short and clean explanation of the main theme and factors. Like: "To summarise" and "in essence".
• Reflect to an End: Recall, that your starters for the conclusion paragraph should signal the end of your essay. You can use the phrases like "in the end" and "To sum up" to wrap up the discussion properly.

You Can Also Read:-  How To Write A Good Profile Essay?

Conclusion Paragraph Starters: Examples!

Following are some examples of good starters for a conclusion paragraph that will allow you to add an effective ending to your writing!

Essay Topic- Renewable Energy: Its Importance!

Concluding Paragraph: The conclusion paragraph starters for the topic of Renewal Energy: Its importance should show that shifting to clean energy sources is not an option, but rather an essential for the future of humanity. Thus, by using solar, wind, and water power, humans may dramatically lower their emission levels while avoiding the negative effects of climate change. Lastly, switching to green energy not only protects the planet but also supports the economy and energy security. In short, the broad use of renewable energy offers an essential step towards a cleaner, better, and more prosperous globe. Thus, politicians, companies, and people must all invest in and support the growth of sustainable energy solutions.

Essay Topic- Technology: Its Role in Education!

Concluding Paragraph: To summarise this topic with good staters for conclusion paragraphs, it can be said that technology has changed the educational scene, creating new potential for customised learning and global collaboration. So, with the advent of digital tools and online tools, students can now access plenty of resources and engage with different viewpoints outside of typical classroom settings. In simple terms, adding technology to education produces a context of continuous creativity and innovation, providing students with the skills required to flourish in a fast-changing world. So, building on this research, it is clear that infusing modern technology into education is important to educating future generations about the problems and chances of the present day.

Best Conclusion Paragraph Starters for Body Paragraphs

Here are the top 3 good conclusion paragraph starters for body paragraph:

• " In the end , ..." - You can use the conclusion starter "in the end" to give the end to your topic within a body paragraph. For instance, "In the end, you need to be good with communication to become an operations Manager."
• " Ultimately , ..." - The conclusion starters for body paragraphs like "ultimately" fulfil the same objective as completing the entire essay but in a single paragraph. For example, "Ultimately, understanding customer behaviour is key to successful marketing tactics."
• " On balance , ..." - The starting point is ideal for balancing several points within a body paragraph. By way of example, "On balance, while there are problems with remote learning, its advantages cannot be overlooked."

A List of Conclusion Paragraph Starters for College Essays

Notably, you can go through these top conclusion paragraph starters for college essays help , to make your essays highly impressive:

• " In conclusion , ..." - Although this conclusion starter is commonly used, it can still be powerful offer the content is strong. For instance, "In conclusion, studying overseas provides students with important cultural experiences and global views."
• " Therefore , ..." - The conclusion starter, therefore can emphasise the logic of your contention. For example, "Therefore, preserving academic integrity is key to the educational system."
• " Thus , ..." - One of the conclusion paragraph starters for college essays, that can be used for a strong logical argument is "Thus." For example, "Thus, comprehensive education is essential for reducing teenage stress rates."

Conclusion Paragraph Starters: Summary

In short, good starters for a conclusion paragraph are essential for completing your essays and making an eternal impression on the reader. You should focus on using a number of sentence starters to ensure that your opinions are clear, brief, and appealing. For example, if you are summarising significant points, focus on the relevance of your topic, or offer a concluding point. In that case, these sentence starters for conclusion paragraphs will help you come up with attractive endings that will connect with the readers.

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Have No Idea about the Conclusion Paragraph Starters? We are here to Help

Are you finding it hard to begin your good conclusion paragraph starters? Look no further than The Student Helpline! Our army of experts offers personal training to assist you with developing strong conclusion paragraph starters that will create a lasting impression. We provide specialized guidance in order to assist you in improving your writing skills for writing all your academic projects.

So, avoid putting all your efforts ring empty at the final hurdle. Visit The Student Helpline immediately to ensure that your conclusion paragraph starters are as strong as your beginning. Turn to us, and achieve academic excellence with our qualified support!

1. What Are Starters for Conclusion Paragraphs?

Indeed, the conclusion paragraph starters are mainly the statements that wave towards the end of the analysis. However, it mainly carried the summary of the important pointers and delivers a closure to the discussion.

2. Why Is It Important to Have a Strong Conclusion Paragraph Starter?

Recall that strong conclusion paragraph sentence starters will give a better ending to your entire writing. It mainly helps you to put a full stop to the entire discussion, support the main pointers and leave a strong impact on the readers. In fact, you can look up to The Student Helpline experts for an ideal conclusion paragraph starter.

3. How Does the Choice of a Conclusion Starter Influence the Reader's Perception of Your Writing?

Indeed, you need to select good conclusion paragraph starters as they allow you to improve the accuracy and correctness of your work. Thus, this makes it more impactful for the raiders to understand your perspective rightly.

4. Can a Conclusion Paragraph Starter Affect the Overall Tone of Your Essay? How?

Yes, starters for conclusion paragraph truly affect the overall tone of your essay. Because it can create a tone for your summary. Thus, this makes it formal or informal, thoughtful or decisive and solely depends on your preferences.

5. What Are Some Common Mistakes to Avoid When Choosing a Conclusion Paragraph Starter?

Certainly, you need to be very clear and specific while picking conclusion paragraph starters for your essay. You should avoid standard tropes, excessive complex language, and starts that do not fit the tone or style of your essay. Since they can confuse or distract readers from the main themes.

6. How Do Conclusion Paragraph Starters Differ Between Formal and Informal Writing?

In formal writing, the good starters for a conclusion paragraph are mainly more structured and related to the essay themes. Likewise, in informal writing, these conclusion starters are more likely friendly and casual.

7. Which Conclusion Paragraph Starter Do You Find Most Effective and Why?

There are many best conclusion paragraph starters, but the "in conclusion" can be highly ideal for your essays. They add a certain clarity and directness to your writing and make it easy to understand as a sign for ending your essay.

8. How Can a Well-Chosen Conclusion Paragraph Starter Enhance the Cohesiveness of Your Writing?

All good conclusion paragraph starters mainly start with a focus theme, and draft a simple transition to the summary. Besides, do not forget to add coherence to your essay.

9. What Strategies Can You Use to Choose the Most Appropriate Conclusion Paragraph Starter for Your Essay?

For writing good starters for conclusion paragraphs, you should think about the tone, the quantity of your ideas, and who you want to reach. So, choose a starting that aligns with these features and successfully expresses a conclusion.

10. How Can the Context of Your Essay Influence Your Choice of a Conclusion Paragraph Starter?

The factors such as the subject matter and readers determine if a formal or casual introduction is appropriate. Besides, while writing a sentence starters for conclusion paragraphs ensure that it speaks to the readers.

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• 15 Great Essay Conclusion Examples
• 18 Outstanding Essay Conclusion Examples

What Is a Conclusion of an Essay: Outline and Purpose

How to write a conclusion paragraph.

• 18 Good Conclusion Paragraph Examples
• Argumentative

Narrative Essay

Effective strategies to conclude an essay, bottom line.

One of the crucial parts of a writing piece is an essay conclusion. it is the last paragraph that creates the final impression from a paper. It is not enough just to summarize what was written in the body part. A writer must make the reader want to continue exploring the problem, share the author’s position, or finally get a clear understanding of an issue . It all depends on the essay type. Our  essay writer  team has come up with essay conclusion examples and useful tips to help students master the art of concluding an essay logically and effectively. Check them out!

Conclusion is the last paragraph of any academic writing, no matter whether it is a school five paragraph essay or college research paper. It is a compulsory structural part of an essay that gives a sense of closure. The purpose of writing a conclusion is to restate the main idea, summarize the key points discussed in the body of the paper showing how they support or prove your thesis, and draw a general conclusion .

What to write in a conclusion paragraph? A typical conclusion outline has three structural components:

• Restated thesis statement.
• Summary of the key points.
• General conclusion or ideas for broader implications of an issue.

Don’t know how to start essay conclusions? No worries! We have prepared useful tips to help you write a good conclusion for your essay. Follow these simple steps:

• Restate the thesis statement . Start your essay conclusion with reminding readers of the main idea of your paper. However, do not just copy-paste the sentence from the introduction paragraph . You should present the same claim but using different words.
• Summarize the main points . Proceed with analysis and summary the key ideas you have discussed in the body paragraphs. Show how these arguments support and prove your thesis statement.
• Sum up the whole essay . After analyzing the major ideas of the paper, draw up a general essay conclusion. If you do not know how to do it, try answering the So-what?-question. In case you write a conclusion for a research paper, you may be asked to identify the knowledge gap. Also, you may specify broader implications of the issue in the larger context for future research.

18 Good Conclusion Paragraph Examples 

We have prepared essay conclusions for different types of papers. Check them out for better understanding of how to write a conclusion.

Argumentative 

The purpose of argumentative paper is to take a stand on an issue. Check these argumentative essay conclusion examples to make your essay convincing.

The purpose of this essay type is to persuade the readers. Look through persuasive essay conclusion examples to understand how to write a conclusion that will help you win over the audience.

Critical thinking is required in this essay type. You should be able to analyze the whole piece of writing to create a strong final paragraph. Have a look at these analysis essay conclusion examples to get a general idea.

Wondering how to write a literary analysis ? Check out our guide.

These essays are easy to write. The purpose of the narrative essay conclusion is to sum up everything described and discussed in the essay.

Expository 

Expository essays aim to describe or explain ideas, notions, phenomena, etc. to the reader. Such papers require research to support the ideas and be able to provide evidence. Check out a conclusion sample of an expository essay.

Are you assigned to write an exposition? Check our blog post to find out what is an expository essay and how to write it successfully.

Look at the English essay conclusion example below. It may refer to any type of paper.

There exist several most common approaches that allow to conclude an essay logically and reasonably. Here they are.

• So what? This is the most common strategy. It presupposes summing up the paper by giving an answer to a short question So-what?
• Giving a larger context. This strategy mainly applies to the research papers. The main idea is to mention the areas of the issue that need further investigation.
• Rhetorical question . It is a provocative and intriguing question that does not need an answer. It gives readers food for thought. However, such conclusions might be not very effective in academic papers.

Also, we would like to remind you that there are some details that should not be included in the conclusion paragraph. Avoid:

• Retelling what was written in the paper
• Presenting new ideas
• Introducing facts or arguments that contradict the info discussed in the essay
• Adding in-text citations
• Copy-pasting sentences from the intro or body paragraphs
• Using phrases like in conclusion, in summary, to sum up, etc.

Essay conclusion is an essential part of a paper. If you miss it or make it weak, your essay will be incomplete. Thus, try your best to conclude an essay with a strong and balanced final paragraph. It should resonate with the essay introduction and body paragraphs, summarize the whole paper, and be written using parallel sentence structures. Have a close look at conclusion sentence examples to ensure you are able to conclude an essay appropriately. If you have some questions or need help with your essay conclusion, you may ask for writing assistance. Experienced writers will help you write a logical and reasonable essay conclusion.

1. What are the components of a conclusion?

Essay conclusion usually has three main parts. They are: restated thesis statement, summary of the key points, and general conclusion. Make sure you include these parts in the final part to conclude an essay appropriately. Mind that just repeating the thesis and ideas will not work. Show your ability to analyze.

2. How to begin a conclusion?

In the last paragraph synthesize and summarize your paper. A reasonable conclusion starts with reminding readers the main idea of an essay. Make sure to paraphrase your thesis statement. Otherwise, it might seem you don’t really understand the point. Also, avoid starting the paragraph with such words as in conclusion, in summary, to conclude, to sum up, etc. It looks primitive and unprofessional.

It’s time for essay writing but you do not know how to start, what to write about, and how to organize your work? This article will guide you on how to write a 500 word essay fast, will reveal all the essay writing secrets regarding essay structure, writing process as well as give good examples for ...

A topic sentence is an important part of your essay. Its basic function is to help you organize each paragraph by summing up its information in a brief manner to make it easier for readers to grab your point. Use topic sentence examples to write good topic sentences. Without them, your academic pape...

The essay introduction serves as a quick test of your text: if the first paragraph is well-written, the rest of the text may be superb. If you fail with making a good introduction, there’s no matter how exceptional your body and conclusion are because nobody will ever read it. Every essay writer onl...

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How to Conclude a Paragraph

Last Updated: December 13, 2023 Fact Checked

This article was co-authored by Diane Stubbs and by wikiHow staff writer, Danielle Blinka, MA, MPA . Diane Stubbs is a Secondary English Teacher with over 22 years of experience teaching all high school grade levels and AP courses. She specializes in secondary education, classroom management, and educational technology. Diane earned a Bachelor of Arts in English from the University of Delaware and a Master of Education from Wesley College. There are 11 references cited in this article, which can be found at the bottom of the page. This article has been fact-checked, ensuring the accuracy of any cited facts and confirming the authority of its sources. This article has been viewed 114,508 times.

If you want a body paragraph to be effective, you need to conclude it properly — a closing sentence is as imperative as a conclusion is to an essay or a research paper. Closing or concluding sentences act like a concluding paragraph in an essay and review the points you made in the paragraph. To effectively conclude a paragraph, restate your topic sentence and include what you taught the reader. Tailor the sentence to the type of essay you’re writing, whether it’s a persuasive or compare and contrast essay.

Reviewing Your Paragraph

• Focus on what you lay out in the topic sentence.
• Note your evidence and details.

• If your topic sentence reads, “Cats may be small, but they’re mighty predators,” then your main idea is that cats are big hunters.
• Your concluding statement should show how your paragraph supported the idea that cats are big predators. For example, a closing statement might read, “Based on these statistics, cats are predators who hunt frequently and decrease the area bird population.”

• For example, the closing statement above, “Based on these statistics, cats are predators who hunt frequently and decrease the area bird population,” reminds the reader that the paragraph just provided statistics about how often cats hunt and how they impact the local bird population. These details support the main idea, and the writer has mentioned both.

Drafting a Closing Statement

• Consequently
• As a result

• An example topic sentence might read: “Cats are natural predators because they enjoy hunting and will even hunt for sport.”
• Your concluding statement for this paragraph might read: “As a result of their continued hunts even after they’re domesticated and provided cat food, cats are proven to be natural predators.”

• As an example, “The data shows that cats hunt even when they have steady meals, which proves that they are natural hunters.”

• For example, “As the data shows, feral cats hunt 140% more than domestic cats.”

• For example, “Consequently, homes that own cats have fewer birds living in their yards.”

• For example, “In the end, cats hunt out of instinct.”

• For example, “To conclude, feral cats are more dangerous to birds than house cats because they have more hunting opportunities and kill more birds each year on average.” This sentence supports the main idea that feral cats hunt more than house cats and shows how the two details provided in the sentence link back to the topic sentence.

• For example, your closing statement could read, “In summation, statistics show that cats who wear bell collars are less of a threat to birds because they kill fewer birds even if they have the same number of hunting opportunities.” This signals to the reader that the writer has finished with one main idea and is moving onto another in a new paragraph.

Avoiding Common Errors

• You also want to avoid the word “you.” For example, don’t say, “As you can see” in your concluding sentence.
• There are some exceptions, such as if you’re writing an introductory paragraph or an opinion essay.

• Reread your concluding statement, then compare it to your paragraph. Do you reference one detail but not another? If it does, rewrite the sentence to address the main points, not the subpoints.

• A poor closing statement might read: “As you can see, the evidence suggests that cats like to hunt."
• A better closing statement could read: “Based on the data, cats look for opportunities to hunt for sport, proving they are natural predators."

Expert Q&A

• In some cases, the concluding statements in introductions and conclusions may have a slightly different format. Thanks Helpful 0 Not Helpful 0
• Remember that your goal is to show the reader your ideas. Thanks Helpful 0 Not Helpful 0
• Focus on your main idea. Thanks Helpful 0 Not Helpful 0

• Try to avoid sounding redundant. Thanks Helpful 2 Not Helpful 0
• Don’t just restate your topic sentence. Show how the details you’ve provided contribute to the main idea. Thanks Helpful 3 Not Helpful 1

You Might Also Like

• ↑ https://writingcenter.unc.edu/tips-and-tools/editing-and-proofreading/
• ↑ https://writingcenter.unc.edu/tips-and-tools/paragraphs/
• ↑ https://openoregon.pressbooks.pub/wrd/chapter/writing-summaries/
• ↑ https://writingcenter.unc.edu/tips-and-tools/transitions/
• ↑ https://lsa.umich.edu/sweetland/undergraduates/writing-guides/how-do-i-write-an-intro--conclusion----body-paragraph.html
• ↑ https://owl.purdue.edu/owl/general_writing/academic_writing/essay_writing/argumentative_essays.html
• ↑ https://owl.excelsior.edu/rhetorical-styles/compare-and-contrast-essay/
• ↑ https://owl.excelsior.edu/rhetorical-styles/cause-and-effect-essay/
• ↑ https://www.grammarly.com/blog/how-to-write-an-informative-essay/
• ↑ https://wts.indiana.edu/writing-guides/paragraphs-and-topic-sentences.html
• ↑ https://libguides.usc.edu/writingguide/conclusion

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39 Different Ways to Say ‘In Conclusion’ in an Essay (Rated)

Chris Drew (PhD)

Dr. Chris Drew is the founder of the Helpful Professor. He holds a PhD in education and has published over 20 articles in scholarly journals. He is the former editor of the Journal of Learning Development in Higher Education. [Image Descriptor: Photo of Chris]

Learn about our Editorial Process

The phrase “In conclusion …” sounds reductive, simple and … well, just basic.

You can find better words to conclude an essay than that!

So below I’ve outlined a list of different ways to say in conclusion in an essay using a range of analysis verbs . Each one comes with an explanation of the best time to use each phrase and an example you could consider.

Read Also: How to Write a Conclusion using the 5C’s Method

List of Ways to Say ‘In Conclusion’ in an Essay

The following are the best tips I have for to say in conclusion in an essay.

1. The Weight of the Evidence Suggests…

My Rating: 10/10

Overview: This is a good concluding phrase for an evaluative essay where you need to compare two different positions on a topic then conclude by saying which one has more evidence behind it than the other.

You could also use this phrase for argumentative essays where you’ve put forward all the evidence for your particular case.

Example: “The weight of the evidence suggests that climate change is a real phenomenon.”

2. A Thoughtful Analysis would Conclude…

My Rating: 9/10

Overview: I would use this phrase in either an argumentative essay or a comparison essay. As an argument, it highlights that you think your position is the most logical.

In a comparison essay, it shows that you have (or have intended to) thoughtfully explore the issue by looking at both sides.

Example: “A thoughtful analysis would conclude that there is substantial evidence highlighting that climate change is real.”

Related Article: 17+ Great Ideas For An Essay About Yourself

3. A Balanced Assessment of the Above Information…

Overview: This phrase can be used to show that you have made a thoughtful analysis of the information you found when researching the essay. You’re telling your teacher with this phrase that you have looked at all sides of the argument before coming to your conclusion.

Example: “A balanced assessment of the above information would be that climate change exists and will have a strong impact on the world for centuries to come.”

4. Across the Board…

My Rating: 5/10

Overview: I would use this phrase in a less formal context such as in a creative discussion but would leave it out of a formal third-person essay. To me, the phrase comes across as too colloquial.

Example: “Across the board, there are scientists around the world who consistently provide evidence for human-induced climate change.”

5. Logically…

My Rating: 7/10

Overview: This phrase can be used at the beginning of any paragraph that states out a series of facts that will be backed by clear step-by-step explanations that the reader should be able to follow to a conclusion.

Example: “Logically, the rise of the automobile would speed up economic expansion in the United States. Automobiles allowed goods to flow faster around the economy.

6. After all is Said and Done…

Overview: This is a colloquial term that is more useful in a speech than written text. If you feel that the phrase ‘In conclusion,’ is too basic, then I’d also avoid this term. However, use in speech is common, so if you’re giving a speech, it may be more acceptable.

Example: “After all is said and done, it’s clear that there is more evidence to suggest that climate change is real than a hoax.”

7. All in All…

Overview: ‘All in all’ is a colloquial term that I would use in speech but not in formal academic writing. Colloquialisms can show that you have poor command of the English language. However, I would consider using this phrase in the conclusion of a debate.

Example: “All in all, our debate team has shown that there is insurmountable evidence that our side of the argument is correct.”

8. All Things Considered…

My Rating: 6/10

Overview: This term is a good way of saying ‘I have considered everything above and now my conclusion is..’ However, it is another term that’s more commonly used in speech than writing. Use it in a high school debate, but when it comes to a formal essay, I would leave it out.

Example: “All things considered, there’s no doubt in my mind that climate change is man-made.”

9. As a Final Note…

My Rating: 3/10

Overview: This phrase gives me the impression that the student doesn’t understand the point of a conclusion. It’s not to simply make a ‘final note’, but to summarize and reiterate. So, I would personally avoid this one.

Example: “As a final note, I would say that I do think the automobile was one of the greatest inventions of the 20 th Century.”

10. As Already Stated…

My Rating: 2/10

Overview: I don’t like this phrase. It gives teachers the impression that you’re going around in circles and haven’t organized your essay properly. I would particularly avoid it in the body of an essay because I always think: “If you already stated it, why are you stating it again?” Of course, the conclusion does re-state things, but it also adds value because it also summarizes them. So, add value by using a phrase such as ‘summarizing’ or ‘weighing up’ in your conclusion instead.

Example: “As already stated, I’m going to repeat myself and annoy my teacher.”

11. At present, the Best Evidence Suggests…

My Rating: 8/10

Overview: In essays where the evidence may change in the future. Most fields of study do involve some evolution over time, so this phrase acknowledges that “right now” the best evidence is one thing, but it may change in the future. It also shows that you’ve looked at the latest information on the topic.

Example: “At present, the best evidence suggests that carbon dioxide emissions from power plants is the greatest influence on climate change.”

12. At the Core of the Issue…

Overview: I personally find this phrase to be useful for most essays. It highlights that you are able to identify the most important or central point from everything you have examined. It is slightly less formal than some other phrases on this list, but I also wouldn’t consider it too colloquial for an undergraduate essay.

Example: “At the core of the issue in this essay is the fact scientists have been unable to convince the broader public of the importance of action on climate change.”

13. Despite the shortcomings of…

Overview: This phrase can be useful in an argumentative essay. It shows that there are some limitations to your argument, but , on balance you still think your position is the best. This will allow you to show critical insight and knowledge while coming to your conclusion.

Often, my students make the mistake of thinking they can only take one side in an argumentative essay. On the contrary, you should be able to highlight the limitations of your point-of-view while also stating that it’s the best.

Example: “Despite the shortcomings of globalization, this essay has found that on balance it has been good for many areas in both the developed and developing world.”

14. Finally…

My Rating: 4/10

Overview: While the phrase ‘Finally,’ does indicate that you’re coming to the end of your discussion, it is usually used at the end of a list of ideas rather than in a conclusion. It also implies that you’re adding a point rather that summing up previous points you have made.

Example: “Finally, this essay has highlighted the importance of communication between policy makers and practitioners in order to ensure good policy is put into effect.”

15. Gathering the above points together…

Overview: While this is not a phrase I personally use very often, I do believe it has the effect of indicating that you are “summing up”, which is what you want out of a conclusion.

Example: “Gathering the above points together, it is clear that the weight of evidence highlights the importance of action on climate change.”

16. Given the above information…

Overview: This phrase shows that you are considering the information in the body of the piece when coming to your conclusion. Therefore, I believe it is appropriate for starting a conclusion.

Example: “Given the above information, it is reasonable to conclude that the World Health Organization is an appropriate vehicle for achieving improved health outcomes in the developing world.”

17. In a nutshell…

Overview: This phrase means to say everything in the fewest possible words. However, it is a colloquial phrase that is best used in speech rather than formal academic writing.

Example: “In a nutshell, there are valid arguments on both sides of the debate about socialism vs capitalism.”

18. In closing…

Overview: This phrase is an appropriate synonym for ‘In conclusion’ and I would be perfectly fine with a student using this phrase in their essay. Make sure you follow-up by explaining your position based upon the weight of evidence presented in the body of your piece

Example: “In closing, there is ample evidence to suggest that liberalism has been the greatest force for progress in the past 100 years.”

19. In essence…

Overview: While the phrase ‘In essence’ does suggest you are about to sum up the core findings of your discussion, it is somewhat colloquial and is best left for speech rather than formal academic writing.

Example: “In essence, this essay has shown that cattle farming is an industry that should be protected as an essential service for our country.”

20. In review…

Overview: We usually review someone else’s work, not our own. For example, you could review a book that you read or a film you watched. So, writing “In review” as a replacement for “In conclusion” comes across a little awkward.

Example: “In review, the above information has made a compelling case for compulsory military service in the United States.”

21. In short…

Overview: Personally, I find that this phrase is used more regularly by undergraduate student. As students get more confident with their writing, they tend to use higher-rated phrases from this list. Nevertheless, I would not take grades away from a student for using this phrase.

Example: “In short, this essay has shown the importance of sustainable agriculture for securing a healthy future for our nation.”

22. In Sum…

Overview: Short for “In summary”, the phrase “In sum” sufficiently shows that you are not coming to the moment where you will sum up the essay. It is an appropriate phrase to use instead of “In conclusion”.

But remember to not just summarize but also discuss the implications of your findings in your conclusion.

Example: “In sum, this essay has shown the importance of managers in ensuring efficient operation of medium-to-large enterprises.”

23. In Summary…

Overview: In summary and in sum are the same terms which can be supplemented for “In conclusion”. You will show that you are about to summarize the points you said in the body of the essay, which is what you want from an essay.

Example: “In summary, reflection is a very important metacognitive skill that all teachers need to master in order to improve their pedagogical skills.”

24. It cannot be conclusively stated that…

Overview: While this phrase is not always be a good fit for your essay, when it is, it does show knowledge and skill in writing. You would use this phrase if you are writing an expository essay where you have decided that there is not enough evidence currently to make a firm conclusion on the issue.

Example: “It cannot be conclusively stated that the Big Bang was when the universe began. However, it is the best theory so far, and none of the other theories explored in this essay have as much evidence behind them.”

25. It is apparent that…

Overview: The term ‘ apparent ’ means that something is ‘clear’ or even ‘obvious’. So, you would use this word in an argumentative essay where you think you have put forward a very compelling argument.

Example: “It is apparent that current migration patterns in the Americas are unsustainable and causing significant harm to the most vulnerable people in our society.”

26. Last but not least…

Overview: The phrase “last but not least” is a colloquial idiom that is best used in speech rather than formal academic writing. Furthermore, when you are saying ‘last’, you mean to say you’re making your last point rather than summing up all your points you already made. So, I’d avoid this one.

Example: “Last but not least, this essay has highlighted the importance of empowering patients to exercise choice over their own medical decisions.”

27. Overall…

My Rating: 7.5/10

Overview: This phrase means ‘taking everything into account’, which sounds a lot like what you would want to do in an essay. I don’t consider it to be a top-tier choice (which is why I rated it 7), but in my opinion it is perfectly acceptable to use in an undergraduate essay.

Example: “Overall, religious liberty continues to be threatened across the world, and faces significant threats in the 21 st Century.”

28. The above points illustrate…

Overview: This phrase is a good start to a conclusion paragraph that talks about the implications of the points you made in your essay. Follow it up with a statement that defends your thesis you are putting forward in the essay.

Example: “The above points illustrate that art has had an overwhelmingly positive impact on humanity since the renaissance.”

29. The evidence presented in this essay suggests that…

Overview: I like this phrase because it highlights that you are about to gather together the evidence from the body of the essay to put forward a final thesis statement .

Example: “The evidence presented in this essay suggests that the democratic system of government is the best for securing maximum individual liberty for citizens of a nation.”

30. This essay began by stating…

Overview: This phrase is one that I teach in my YouTube mini-course as an effective one to use in an essay conclusion. If you presented an interesting fact in your introduction , you can return to that point from the beginning of the essay to provide nice symmetry in your writing.

Example: “This essay began by stating that corruption has been growing in the Western world. However, the facts collected in the body of the essay show that institutional checks and balances can sufficiently minimize this corruption in the long-term.”

31. This essay has argued…

Overview: This term can be used effectively in an argumentative essay to provide a summary of your key points. Follow it up with an outline of all your key points, and then a sentence about the implications of the points you made. See the example below.

Example: “This essay has argued that standardized tests are damaging for students’ mental health. Tests like the SATs should therefore be replaced by project-based testing in schools.”

32. To close…

Overview: This is a very literal way of saying “In conclusion”. While it’s suitable and serves its purpose, it does come across as being a sophomoric term. Consider using one of the higher-rated phrases in this list.

Example: “To close, this essay has highlighted both the pros and cons of relational dialectics theory and argued that it is not the best communication theory for the 21 st Century.”

33. To Conclude…

Overview: Like ‘to close’ and ‘in summary’, the phrase ‘to conclude’ is very similar to ‘in conclusion’. It can therefore be used as a sufficient replacement for that term. However, as with the above terms, it’s just okay and you could probably find a better phrase to use.

Example: “To conclude, this essay has highlighted that there are multiple models of communication but there is no one perfect theory to explain each situation.”

34. To make a long story short…

My Rating: 1/10

Overview: This is not a good phrase to use in an academic essay. It is a colloquialism. It also implies that you have been rambling in your writing and you could have said everything more efficiently. I would personally not use this phrase.

Example: “To make a long story short, I don’t have very good command of academic language.”

35. To Sum up…

Overview: This phrase is the same as ‘In summary’. It shows that you have made all of your points and now you’re about to bring them all together in a ‘summary’. Just remember in your conclusion that you need to do more than summarize but also talk about the implications of your findings. So you’ll need to go beyond just a summary.

Example: “In summary, there is ample evidence that linear models of communication like Lasswell’s model are not as good at explaining 21 st Century communication as circular models like the Osgood-Schramm model .”

36. Ultimately…

Overview: While this phrase does say that you are coming to a final point – also known as a conclusion – it’s also a very strong statement that might not be best to use in all situations. I usually accept this phrase from my undergraduates, but for my postgraduates I’d probably suggest simply removing it.

Example: “Ultimately, new media has been bad for the world because it has led to the spread of mistruths around the internet.”

37. Undoubtedly…

Overview: If you are using it in a debate or argumentative essay, it can be helpful. However, in a regular academic essay, I would avoid it. We call this a ‘booster’, which is a term that emphasizes certainty. Unfortunately, certainty is a difficult thing to claim, so you’re better off ‘hedging’ with phrases like ‘It appears’ or ‘The best evidence suggests’.

Example: “Undoubtedly, I know everything about this topic and am one hundred percent certain even though I’m just an undergraduate student.”

38. Weighing up the facts, this essay finds…

Overview: This statement highlights that you are looking at all of the facts both for and against your points of view. It shows you’re not just blindly following one argument but being careful about seeing things from many perspectives.

Example: “Weighing up the facts, this essay finds that reading books is important for developing critical thinking skills in childhood.”

39. With that said…

Overview: This is another phrase that I would avoid. This is a colloquialism that’s best used in speech rather than writing. It is another term that feels sophomoric and is best to avoid. Instead, use a more formal term such as: ‘Weighing up the above points, this essay finds…’

Example: “With that said, this essay disagrees with the statement that you need to go to college to get a good job.”

Do you Need to Say Anything?

Something I often tell my students is: “Can you just remove that phrase?”

Consider this sentence:

• “In conclusion, the majority of scientists concur that climate change exists.”

Would it be possible to simply say:

• “ In conclusion, The majority of scientists concur that climate change exists.”

So, I’d recommend also just considering removing that phrase altogether! Sometimes the best writing is the shortest, simplest writing that gets to the point without any redundant language at all.

How to Write an Effective Conclusion

Before I go, I’d like to bring your attention to my video on ‘how to write an effective conclusion’. I think it would really help you out given that you’re looking for help on how to write a conclusion. It’s under 5 minutes long and has helped literally thousands of students write better conclusions for their essays:

You can also check out these conclusion examples for some copy-and-paste conclusions for your own essay.

In Conclusion…

Well, I had to begin this conclusion with ‘In conclusion…’ I liked the irony in it, and I couldn’t pass up that chance.

Overall, don’t forget that concluding an essay is a way to powerfully summarize what you’ve had to say and leave the reader with a strong impression that you’ve become an authority on the topic you’re researching. 

So, whether you write it as a conclusion, summary, or any other synonym for conclusion, those other ways to say in conclusion are less important than making sure that the message in your conclusion is incredibly strong.

• Chris Drew (PhD) https://helpfulprofessor.com/author/chris-drew-phd/ 19 Top Cognitive Psychology Theories (Explained)
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• 10 Sep 2021
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How to End an Essay: 5 Ways to Perfectly End an Essay

It is not enough to write a good essay, you also need to finish it correctly.

A poorly written and clumsy conclusion will spoil the opinion of even the best-quality text, which means that the grade for it will be lower than expected. Today we're going to look at ways to end an essay so that it stays on top.

Why it's important to finish your essay correctly

There are several reasons for this. As a reminder, an essay is a short written work in which the author expresses his or her own opinion on a topic. It can be emotional, intriguing, written in non-official language - in short, the more creativity you put into it, the better. So why is it so important to understand how to finish an essay correctly?

First, if the whole text was interesting and fascinating, just one bad paragraph at once will spoil the whole impression. Immediately there will be a feeling that the student slacked off and didn't finish it, hurried to finish it as soon as possible, or even rewrote the middle part from another author.

Second, everything needs the right ending: essays, books, films, and symphonies. Even in a conversation, the ending is more important than the middle. It is better remembered than anything else. So the beginning and the end are the most important parts of any work.

And finally, thirdly, very often teachers do not want to read two dozen texts in their entirety and limit themselves to the first and last paragraphs. That way they understand what the student wrote about and what conclusions they came to. It makes sense that if you want a good grade, these parts will have to be worked on.

The importance of a proper ending now leaves no doubt. We’ll talk about most important essay ending tips. With them, you could even start writing essays for money .

 So what are some tricks that will allow you to write a proper and appropriate conclusion?

Types of conclusions for essays

It will be much easier for you to do this work if you know what types of essay endings there are in general. That way, you'll decide what works best for you in terms of topic or style. Here are some types compiled by the best essay editing service :

Of course, bringing in a whole paragraph is not the prettiest solution. But using it as the last sentence or phrase is quite appropriate. In addition, an exact quotation, carefully inserted in the text, does not require any explanation or continuation. A rather elegant move would also be to insert the words of the same author in the epigraph or introduction.

Rhetorical question

Addressing the reader is a great move. It will make you think about the topic of the essay, and it will free you from having to give any specific facts at the end of the text. Think about what kind of question you can ask, and how to approach it smoothly. The right words can be found on almost any topic.

Link to the introduction.

This is easy enough to do if a question has been written in the introduction. In that case, it's time to answer it in the conclusion. Or you can use the same phrase (or quote) in the first and last paragraphs: only in the beginning it should intrigue and set the tone for the whole essay, and in the end, it should serve as a conclusion. But in this case, you have to try to color them differently emotionally at the expense of the correct development of thought.

Incitement to action

Such an ending will not suit all topics. And not all readers may not perceive it well. But, if you know that your teacher supports an active life position, it will be an excellent conclusion of any essay. In it, you need to confidently address the reader and suggest that he or she change something: "start with yourself," try something new, motivate him or her to make some changes. To make it look convincing, the whole essay must be written emotionally and believably. The reader should believe you and be convinced of your rightness.

Summarizing

The easiest, yet the win-win option is how to end your essay. Simply summarize all the conclusions you came to during writing the essay. You don't need to retell everything you've said or start new reflections. Such a way will help you finish the essay, but it is unlikely to leave any outstanding impressions on the reader.

Hints and possible mistakes while writing a conclusion

There are quite a few types of essay endings. But you need not only choose the right type but also write it correctly. And after all, there are a lot of opportunities to make a mistake:

• You can overdo the volume, and then instead of a conclusion, you get another page of text. For an essay, which should be only 500-600 words, this is completely inappropriate.
• Trying to add an unusual conclusion can lead to a different style of a text. Agree, a confident call to action will look strange if the entire essay before it was dry and unemotional.
• In trying to summarize, it is easy to draw completely different conclusions and ask new questions, which will interfere with the understanding of the text.

To avoid this, stick to simple principles:

• The conclusion should flow from the main text.
• The last paragraphs should have a meaningful conclusion to the essay. The text should not end early, and it is rarely possible to beat an open ending beautifully.
• The conclusion should be as short as possible. This is not the section where it is worth going into thought.
• If a good phrase comes to your mind while writing the main part of the essay, write it down and put it aside for the future.
• The conclusion should explain to the reader why he or she read the entire essay and what he or she should have gotten out of it.

What about the formatting?

Since the essay is a small work, it does not have separate sections with headings. Neither the introduction nor the conclusion needs to be separated from the main text. Just move to a new paragraph and start summarizing.

At the same time, the concluding part should not occupy more than 10% of the entire volume. If you have written a two-page essay, the ending should take up less than a quarter of the last one.

As a quality check, you can read only the introduction and conclusion (as the teacher is likely to do). If something in them confuses you, then try to work on the style of the text or the wording of the conclusions. Well, if they are written in a unified style, do not require a thorough study of the entire essay, and do not leave any questions after reading - congratulations, you have done a great job!

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How to Structure an Essay

Essay writing is a fundamental skill, a basic task, that is expected of those who choose to pursue their undergraduate and master’s degrees. It constitutes a key requirement for students to complete a given course credit. However, many students and early career researchers find themselves struggling with the challenge of organizing their thoughts into a coherent, engaging structure. This article is especially for those who see essay writing as a daunting task and face problems in presenting their work in an impactful way.  

Table of Contents

• Writing an essay: basic elements and some key principles  
• Essay structure template 
• Chronological structure 
• Problem-methods-solutions structure 
• Compare and contrast structures 
• Frequently asked questions on essay structure 

Read on as we delve into the basic elements of essay writing, outline key principles for organizing information, and cover some foundational features of writing essays.  

Writing an essay: basic elements and some key principles

Essays are written in a flowing and continuous pattern but with a structure of its own. An introduction, body and conclusion are integral to it. The key is to balance the amount and kind of information to be presented in each part. Various disciplines may have their own conventions or guidelines on the information to be provided in the introduction.  

A clear articulation of the context and background of the study is important, as is the definition of key terms and an outline of specific models or theories used. Readers also need to know the significance of the study and its implications for further research. Most importantly, the thesis or the main proposition should be clearly presented.  

The body of the essay is therefore organized into paragraphs that hold the main ideas and arguments and is presented and analyzed in a logical manner. Ideally, each paragraph of the body focuses on one main point or a distinct topic and must be supported by evidence and analysis. The concluding paragraph should bring back to the reader the key arguments, its significance and food for thought. It is best not to re-state all the points of the essay or introduce a new concept here. 

In other words, certain general guidelines help structure the information in the essay. The information must flow logically with the context or the background information presented in the introductory part of the essay. The arguments are built organically where each paragraph in the body of the essay deals with a different point, yet closely linked to the para preceding and following it. Importantly, when writing essays, early career researchers must be careful in ensuring that each piece of information relates to the main thesis and is a building block to the arguments. 

Essay structure template

• Introduction 
• Provide the context and share significance of the study 
• Clearly articulate the thesis statement 
• Body  
• Paragraph 1 consisting of the first main point, followed by supporting evidence and an analysis of the findings. Transitional words and phrases can be used to move to the next main point. 
• There can be as many paragraphs with the above-mentioned elements as there are points and arguments to support your thesis. 
• Conclusion  
• Bring in key ideas and discuss their significance and relevance 
• Call for action 
• References 

Essay structures

The structure of an essay can be determined by the kind of essay that is required.  

Chronological structure

Also known as the cause-and-effect approach, this is a straightforward way to structure an essay. In such essays, events are discussed sequentially, as they occurred from the earliest to the latest. A chronological structure is useful for discussing a series of events or processes such as historical analyses or narratives of events. The introduction should have the topic sentence. The body of the essay should follow a chorological progression with each para discussing a major aspect of that event with supporting evidence. It ends with a summarizing of the results of the events.  

Problem-methods-solutions structure

Where the essay focuses on a specific problem, the problem-methods-solutions structure can be used to organize the essay. This structure is ideal for essays that address complex issues. It starts with presenting the problem, the context, and thesis statement as introduction to the essay. The major part of the discussion which forms the body of the essay focuses on stating the problem and its significance, the author’s approach or methods adopted to address the problem along with its relevance, and accordingly proposing solution(s) to the identified problem. The concluding part offers a recap of the research problem, methods, and proposed solutions, emphasizing their significance and potential impact. 

Compare and contrast structures

This structure of essay writing is ideally used when two or more key subjects require a comparison of ideas, theories, or phenomena. The three crucial elements, introduction, body, and conclusion, remain the same. The introduction presents the context and the thesis statement. The body of the essay seeks to focus on and highlight differences between the subjects, supported by evidence and analysis. The conclusion is used to summarize the key points of comparison and contrast, offering insights into the significance of the analysis.  

Depending on how the subjects will be discussed, the body of the essay can be organized according to the block method or the alternating method. In the block method, one para discusses one subject and the next para the other subject. In the alternative method, both subjects are discussed in one para based on a particular topic or issue followed by the next para on another issue and so on.  

Frequently asked questions on essay structure

An essay structure serves as a framework for presenting ideas coherently and logically. It comprises three crucial elements: an introduction that communicates the context, topic, and thesis statement; the body focusing on the main points and arguments supported with appropriate evidence followed by its analysis; and a conclusion that ties together the main points and its importance .  

An essay structure well-defined essay structure enhances clarity, coherence, and readability, and is crucial for organizing ideas and arguments to effectively communicate key aspects of a chosen topic. It allows readers to better understand arguments presented and demonstrates the author’s ability to organize and present information systematically. 

Yes, while expert recommend following an essay structure, early career researchers may choose how best to adapt standard essay structures to communicate and share their research in an impactful and engaging way. However, do keep in mind that deviating too far from established structures can hinder comprehension and weaken the overall effectiveness of the essay,  By understanding the basic elements of essay writing and employing appropriate structures such as chronological, problem-methods-solutions, or compare and contrast, researchers can effectively organize their ideas and communicate their findings with clarity and precision. 

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End Legal Slavery in the United States

By Andrew Ross ,  Tommaso Bardelli and Aiyuba Thomas

The writers, members of the New York University Prison Education Program Research Lab, are the authors of “Abolition Labor: The Fight to End Prison Slavery.”

Today we celebrate Juneteenth, the day when word of the Emancipation Proclamation reached the farthest outpost in America. Many people do not realize that Emancipation did not legally end slavery in the United States, however. The 13th Amendment — the culmination of centuries of resistance by enslaved people, a lifetime of abolitionist campaigning and a bloody civil war — prohibited involuntary servitude “except as a punishment for crime whereof the party shall have been duly convicted.”

In the North, that so-called exception clause was interpreted as allowing the private contracting of forced prison labor, which was already underway, and in the ex-Confederacy it gave rise to the much more brutal system by which freed men and women were routinely arrested under false charges and then leased out to plantation owners and industrialists to work off their sentence. Some historians have described this convict leasing system as “worse than slavery,” because there was no incentive to avoid working those people to death.

Over time, courts accepted that all people who are incarcerated lose the protection against slavery or involuntary servitude. The legacy of that legal deference is a grim one. Today, a majority of the 1.2 million Americans locked up in state and federal prisons work under duress in jobs that cover the entire spectrum, from cellblock cleaning to skilled manufacturing, for wages as low as a few cents per hour or, in several states, for nothing at all. And though members of Congress denounce imported goods made with prison labor in places like China’s Xinjiang province, the offices of many government agencies in Washington and elsewhere are stocked with furniture and supplies made by prisoners in this country. In fact, federal agencies are mandated to purchase goods from federal prisons, just as state or municipal agencies, including public schools and universities, often must consider sourcing from state penitentiaries. In many states, prison-made goods are freely available on the open market and shipped overseas.

Labor that people have no meaningful right to refuse and that is enforced under conditions of total control is, unquestionably, slavery. It’s a different model from the chattel slavery over which the Civil War was fought, but by all norms of international law, it is a violation of fundamental human rights.

The nation that deigns to teach the rest of the world lessons in liberty should ban this practice on its own shores rather than integrating its products into the economy. For those who want to work while serving their sentence, we should guarantee fair pay for their labor.

The prisoner rights movement of the late 1960s and early ’70s called for raising prisoners’ hourly pay. One of the top demands during the 1971 Attica uprising was to “apply the New York State minimum wage law to all state institutions.” More radical Black nationalists saw the nation’s overcrowded penitentiaries as akin to modern slave ships and argued that even if they were to offer prevailing wages, collective bargaining and workplace protections, they would still be instruments of racial capture and control.

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More From Forbes

How not to write your college essay.

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If you are looking for the “secret formula” for writing a “winning” college essay, you have come to the wrong place. The reality is there is no silver bullet or strategy to write your way to an acceptance. There is not one topic or approach that will guarantee a favorable outcome.

At the end of the day, every admission office just wants to know more about you, what you value, and what excites you. They want to hear about your experiences through your own words and in your own voice. As you set out to write your essay, you will no doubt get input (both sought-after and unsolicited) on what to write. But how about what NOT to write? There are avoidable blunders that applicants frequently make in drafting their essays. I asked college admission leaders, who have read thousands of submissions, to share their thoughts.

Don’t Go In There

There is wide consensus on this first one, so before you call on your Jedi mind tricks or predictive analytics, listen to the voices of a diverse range of admission deans. Peter Hagan, executive director of admissions at Syracuse University, sums it up best, saying, “I would recommend that students try not to get inside of our heads. He adds, “Too often the focus is on what they think we want.”

Andy Strickler, dean of admission and financial aid at Connecticut College agrees, warning, “Do NOT get caught in the trap of trying to figure out what is going to impress the admission committee. You have NO idea who is going to read your essay and what is going to connect with them. So, don't try to guess that.” Victoria Romero, vice president for enrollment, at Scripps College adds, “Do not write about something you don’t care about.” She says, “I think students try to figure out what an admission officer wants to read, and the reality is the reader begins every next essay with no expectations about the content THEY want to read.” Chrystal Russell, dean of admission at Hampden-Sydney College, agrees, saying, “If you're not interested in writing it, we will not be interested when reading it.” Jay Jacobs, vice provost for enrollment management at the University of Vermont elaborates, advising. “Don’t try to make yourself sound any different than you are.” He says, “The number one goal for admission officers is to better understand the applicant, what they like to do, what they want to do, where they spend the majority of their time, and what makes them tick. If a student stays genuine to that, it will shine through and make an engaging and successful essay.”

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Don’t Be Artificial

The headlines about college admission are dominated by stories about artificial intelligence and the college essay. Let’s set some ground rules–to allow ChatGPT or some other tool to do your work is not only unethical, it is also unintelligent. The only worse mistake you could make is to let another human write your essay for you. Instead of preoccupying yourself with whether or not colleges are using AI detection software (most are not), spend your time focused on how best to express yourself authentically. Rick Clark is the executive director of strategic student success at Georgia Institute of Technology, one of the first institutions to clearly outline their AI policy for applicants. He says, “Much of a college application is devoted to lines, boxes, and numbers. Essays and supplements are the one place to establish connection, personality, and distinction. AI, in its current state, is terrible at all three.” He adds, “My hope is that students will use ChatGPT or other tools for brainstorming and to get started, but then move quickly into crafting an essay that will provide insight and value.”

Don’t Overdo It

Michael Stefanowicz, vice president for enrollment management at Landmark College says, “You can only cover so much detail about yourself in an admission essay, and a lot of students feel pressure to tell their life story or choose their most defining experience to date as an essay topic. Admission professionals know that you’re sharing just one part of your lived experience in the essay.” He adds, “Some of the favorite essays I’ve read have been episodic, reflecting on the way you’ve found meaning in a seemingly ordinary experience, advice you’ve lived out, a mistake you’ve learned from, or a special tradition in your life.” Gary Ross, vice president for admission and financial aid at Colgate University adds, “More than a few applicants each year craft essays that talk about the frustration and struggles they have experienced in identifying a topic for their college application essay. Presenting your college application essay as a smorgasbord of topics that ultimately landed on the cutting room floor does not give us much insight into an applicant.”

Don’t Believe In Magic

Jason Nevinger, senior director of admission at the University of Rochester warns, “Be skeptical of anyone or any company telling you, ‘This is the essay that got me into _____.’ There is no magic topic, approach, sentence structure, or prose that got any student into any institution ever.” Social media is littered with advertisements promising strategic essay help. Don’t waste your time, energy, or money trying to emulate a certain style, topic, or tone. Liz Cheron is chief executive officer for the Coalition for College and former assistant vice president of enrollment & dean of admissions at Northeastern University. She agrees with Nevinger, saying “Don't put pressure on yourself to find the perfect, slam dunk topic. The vast majority of college essays do exactly what they're supposed to do–they are well-written and tell the admission officer more about the student in that student's voice–and that can take many different forms.”

Don’t Over Recycle

Beatrice Atkinson-Myers, associate director of global recruitment at the University of California at Santa Cruz tells students, “Do not use the same response for each university; research and craft your essay to match the program at the university you are interested in studying. Don't waste time telling me things I can read elsewhere in your application. Use your essay to give the admissions officer insights into your motivations, interests, and thinking. Don't make your essay the kitchen sink, focus on one or two examples which demonstrate your depth and creativity.” Her UC colleague, Jim Rawlins, associate vice chancellor of enrollment management at the University of California at San Diego agrees, saying “Answer the question. Not doing so is the surest way we can tell you are simply giving us a snippet of something you actually wrote for a different purpose.”

Don’t Overedit

Emily Roper-Doten, vice president for undergraduate admissions and financial assistance at Clark University warns against “Too many editors!” She says, “Pick a couple of trusted folks to be your sounding board when considering topics and as readers once you have drafts. You don’t want too many voices in your essay to drown you out!” Scripps’ Romero agrees, suggesting, “Ask a good friend, someone you trust and knows you well, to read your essays.” She adds, “The goal is for the admission committee to get to know a little about you and who better to help you create that framework, than a good friend. This may not work for all students because of content but helps them understand it’s important to be themselves.” Whitney Soule, vice provost and dean of admissions at The University of Pennsylvania adds, “Avoid well-meaning editorial interference that might seem to polish your writing but actually takes your own personal ‘shine’ right out of the message.” She says, “As readers, we connect to applicants through their genuine tone and style. Considering editorial advice for flow and message is OK but hold on to the 'you' for what you want to say and how you want to say it.”

Don’t Get Showy

Palmer Muntz, senior regional admissions counselor at the University of Alaska Fairbanks cautions applicants, “Don’t be fancier than you are. You don’t need to put on airs.” He adds, “Yes, proofread your work for grammar and spelling, but be natural. Craft something you’d want to read yourself, which probably means keeping your paragraphs short, using familiar words, and writing in an active voice.” Connecticut College’s Strickler agrees, warning, “Don't try to be someone you are not. If you are not funny, don't try to write a funny essay. If you are not an intellectual, trying to write an intellectual essay is a bad idea.”

Anthony Jones, the vice president of enrollment management at Loyola University New Orleans offers a unique metaphor for thinking about the essay. He says, “In the new world of the hyper-fast college admission process, it's become easy to overlook the essential meaning of the college application. It's meant to reveal Y...O...U, the real you, not some phony digital avatar. Think of the essay as the essence of that voice but in analog. Like the completeness and authenticity captured in a vinyl record, the few lines you're given to explain your view should be a slow walk through unrestrained expression chock full of unapologetic nuances, crevices of emotion, and exactness about how you feel in the moment. Then, and only then, can you give the admissions officer an experience that makes them want to tune in and listen for more.”

Don’t Be A Downer

James Nondorf, vice president and dean of admissions and financial aid at The University of Chicago says, “Don’t be negative about other people, be appreciative of those who have supported you, and be excited about who you are and what you will bring to our campus!” He adds, “While admissions offices want smart students for our classrooms, we also want kind-hearted, caring, and joyous students who will add to our campus communities too.”

Don’t Pattern Match

Alan Ramirez is the dean of admission and financial aid at Sewanee, The University of the South. He explains, “A big concern I have is when students find themselves comparing their writing to other students or past applicants and transform their writing to be more like those individuals as a way to better their chances of offering a more-compelling essay.” He emphasizes that the result is that the “essay is no longer authentic nor the best representation of themselves and the whole point of the essay is lost. Their distinctive voice and viewpoint contribute to the range of voices in the incoming class, enhancing the diversity of perspectives we aim to achieve.” Ramirez simple tells students, “Be yourself, that’s what we want to see, plus there's no one else who can do it better than you!”

Don’t Feel Tied To A Topic

Jessica Ricker is the vice president for enrollment and dean of admissions and financial aid at Skidmore College. She says, “Sometimes students feel they must tell a story of grief or hardship, and then end up reliving that during the essay-writing process in ways that are emotionally detrimental. I encourage students to choose a topic they can reflect upon positively but recommend that if they choose a more challenging experience to write about, they avoid belaboring the details and instead focus on the outcome of that journey.” She adds, "They simply need to name it, frame its impact, and then help us as the reader understand how it has shaped their lens on life and their approach moving forward.”

Landmark College’s Stefanowicz adds, “A lot of students worry about how personal to get in sharing a part of their identity like your race or heritage (recalling last year’s Supreme Court case about race-conscious admissions), a learning difference or other disability, your religious values, LGBTQ identity…the list goes on.” He emphasizes, “This is always your choice, and your essay doesn’t have to be about a defining identity. But I encourage you to be fully yourself as you present yourself to colleges—because the college admission process is about finding a school where your whole self is welcome and you find a setting to flourish!”

Don’t Be Redundant

Hillen Grason Jr., dean of admission at Franklin & Marshall College, advises, “Don't repeat academic or co-curricular information that is easily identifiable within other parts of your application unless the topic is a core tenant of you as an individual.” He adds, “Use your essay, and other parts of your application, wisely. Your essay is the best way to convey who your authentic self is to the schools you apply. If you navigated a situation that led to a dip in your grades or co-curricular involvement, leverage the ‘additional information’ section of the application.

Thomas Marr is a regional manager of admissions for the Americas at The University of St Andrews in Scotland and points out that “Not all international schools use the main college essay as part of their assessment when reviewing student applications.” He says, “At the University of St Andrews, we focus on the supplemental essay and students should avoid the mistake of making the supplemental a repeat of their other essay. The supplemental (called the Personal Statement if using the UCAS application process) is to show the extent of their passion and enthusiasm for the subject/s to which they are applying and we expect about 75% of the content to cover this. They can use the remaining space to mention their interests outside of the classroom. Some students confuse passion for the school with passion for their subject; do not fall into that trap.”

A Few Final Don’ts

Don’t delay. Every college applicant I have ever worked with has wished they had started earlier. You can best avoid the pitfalls above if you give yourself the time and space to write a thoughtful essay and welcome feedback openly but cautiously. Don’t put too much pressure on yourself to be perfect . Do your best, share your voice, and stay true to who you are.

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What Makes for ‘Good’ Mathematics?

February 1, 2024

Peter Greenwood for  Quanta Magazine

Introduction

We tend to think of mathematics as purely logical, but the teaching of math, its values, its usefulness and its workings are packed with nuance. So what is “good” mathematics? In 2007, the mathematician Terence Tao wrote an essay for the Bulletin of the American Mathematical Society that sought to answer this question. Today, as the recipient of a Fields Medal, a Breakthrough Prize in Mathematics and a MacArthur Fellowship, Tao is one of the most honored and prolific mathematicians alive. In this episode, he joins our host and fellow mathematician Steven Strogatz to revisit the makings of good mathematics.

Listen on  Apple Podcasts ,  Spotify ,  Google Podcasts , TuneIn  or your favorite podcasting app, or you can  stream it from  Quanta .

STEVEN STROGATZ : Back in October 2007, way back when the first-generation iPhone was still a hot commodity and the stock market was at an all-time high before the Great Recession, Terence Tao, a professor of math at UCLA, was determined to answer a question that had long been debated among mathematicians: What exactly is good mathematics?

Is it about rigor? Elegance? Real-world utility? Terry wrote a very thoughtful and generous, I would even say openhearted, essay about all the ways that math could be good. But now, more than 15 years later, do we need to rethink what good mathematics is?

I’m Steve Strogatz, and this is “The Joy of Why,” a podcast from Quanta Magazine where my co-host, Janna Levin, and I take turns exploring some of the biggest unanswered questions in math and science today.

( Theme plays )

Here today to revisit the eternal question of what makes math good is Terry Tao himself. Professor Tao has authored more than 300 research papers on an amazingly wide swath of mathematics including harmonic analysis, partial differential equations, combinatorics, number theory, data science, random matrices and much more. He’s been referred to as the “Mozart of Mathematics.” And as the winner of a Fields Medal, a Breakthrough Prize in Mathematics, a MacArthur Fellowship and many other awards, that moniker is certainly well-deserved.

Terry, welcome to “The Joy of Why.”

TERENCE TAO : Pleasure to be here.

STROGATZ : I’m very excited to be able to talk to you about this question of what it is that makes some types of mathematical research good. I can remember pretty vividly flipping through the Bulletin of the American Math Society back in 2007 and coming across your essay about this issue that you posed for us. It’s something that all mathematicians think about. But for people out there who may not be so familiar, could you tell us, how did you land on this question? How did you define good math back at the time?

TAO : Right, yes. It was actually a solicitation. So the editor of the Bulletin at the time had asked me to contribute an article. I think I had a very naive idea of what mathematics was as a student. I kind of had this idea that there was some sort of council of greybeards that would hand out problems for people to work on. And it was kind of a shock to me as a graduate student, realizing that there wasn’t actually this central authority to hand out problems, and people did self-directed research.

I kept going to talks and listening to how other mathematicians talked about what they find exciting and what makes them excited about math, and the fact that each mathematician has a different way of approaching mathematics. Like, some would pursue applications, some by sort of aesthetic beauty, some by just problem solving. They wanted to solve a problem and they would focus on sort of the most difficult, the most challenging tasks. Some would focus on technique; some would try to make things as elegant as possible.

But what struck me when sort of listening to so many of these different mathematicians talk about what they find valuable in mathematics is that, even though we all had sort of different ideals as to what good mathematics should look like, they all kind of tend to converge to the same thing.

If a piece of mathematics is really good, people who pursue beauty will eventually happen across it. People who pursue, who value, you know, technical power or applications will eventually land upon it.

Eugene Wigner had a very famous essay on the unreasonable effectiveness of mathematics in the physical sciences almost a century ago, where he just observed that there were areas of mathematics — for example, Riemannian geometry, the study of curved space — that was initially just a purely theoretical exercise to mathematicians, you know, trying to prove the parallel postulate and so forth, turning out to be precisely what Einstein and Poincaré and Hilbert needed to describe the mathematics of general relativity. And that’s just a phenomenon that occurs.

So it’s not just that mathematics, that [what] mathematicians find intellectually interesting end up being physically important. But even within mathematics, subjects that mathematicians find elegant also happen to provide deep insight.

What I feel like is that, you know, there is some platonic good mathematics out there, and all our different value systems are just different ways of accessing that objective good stuff.

STROGATZ : That’s very interesting. Being a sort of person inclined to platonic thinking myself, I’m tempted to agree. Although I’m a little surprised to hear you say that, because I would have thought where you were going initially seemed to be, like, there are so many different points of view about this. It is an interesting fact, though, kind of an empirical fact, that we do converge on agreeing about what is good or not good, even though, as you say, we come at it from so many different values.

TAO : Right. The convergence may take time. You know, so there are definitely fields, for example, where they look a lot better as measured by one metric than others. Like maybe they have a lot of applications, but their presentation is extremely disgusting, you know.

(Strogatz laughs)

Or things that are very elegant but don’t yet have many good applications in the real world. But I do feel like eventually it will converge.

STROGATZ : Well, let me ask you about this point of contact with the real world. It’s an interesting tension in math. And, you know, as little kids, let’s say, when we first learn about geometry, you might think at that point that triangles are real, or circles or straight lines are real, and that they can tell you about the rectangular shapes you see in buildings out in the world, or that surveyors need to use geometry. And after all, the word comes from the measurement of the Earth, right, “geometry.” And so, there was a time when geometry was empirical.

But what I wanted to ask you has to do with a comment that John von Neumann made. So von Neumann, for anyone not familiar, was himself a great mathematician. And he made this comment in this essay, “ The Mathematician ,” about the relationship between math and the empirical world, the real world, where he says roughly that mathematical ideas originate in empirics, but that at some point, once you get the mathematical ideas, the subject starts to take on a life of its own. And then it’s more like a creative piece of art. Aesthetic criteria become important. But he says that causes danger. That when a subject starts to become too far removed from its empirical source, like especially in its second or third generation, he says that there’s a chance the subject can suffer from too much abstract inbreeding and it’s in danger of degeneration.

Any thoughts about that? I mean, does math have to stay in contact with its empirical source?

TAO: Yes, I think it does have to be grounded. When I say that, empirically, all these different ways of doing mathematics do converge, it’s only because — this only happens when the subject is healthy. So, you know, the good news is that usually it is.

But, for example, mathematicians value short proofs over long ones, all other things being equal. But one could imagine people going overboard and, like, one subfield of mathematics being obsessed with making proofs as short as possible and having these extremely opaque two-line proofs of deep theorems. And they make it kind of this contest, and then it becomes this sort of abstruse game and then you lose all the intuition. You lose maybe deeper understanding because you’re just so obsessed with making all your proofs as short as possible. Now, this doesn’t actually happen in practice. But this is kind of a theoretical example, and I think von Neumann was making a similar point.

And in the sixties and seventies, like, there was an era of mathematics where abstraction was making huge strides in simplifying and unifying a lot of mathematics that was previously very empirical. Especially in algebra, people were realizing, you know, numbers and polynomials and many other objects that were previously treated separately, you could all think of them as members of the same algebraic class, in this case a ring.

And a lot of progress in mathematics was being made by finding the right abstraction, you know, whether it was a topological space or a vector space, whatever, and proving theorems in great generality. And this is sometimes what we call the Bourbaki era in mathematics. And it did veer a little bit too far from being grounded.

We of course had, like, the whole New Math episode in the States, where educators tried to teach math in the Bourbaki style and eventually realized that that was not the appropriate pedagogy at that level.

But now the pendulum has swung back quite a bit. We have kind of — the subject has matured quite a bit and every field of mathematics, geometry, topology, whatever, we have kind of satisfactory formalizations and we kind of know what the right abstractions are. And now the field is again focusing on interconnections and applications. It’s connecting much more to the real world now.

I mean, not just sort of physics, which is a traditional connection, but, you know, computer science, life sciences, social sciences, you know. With the rise of big data, pretty much almost any human discipline now can be mathematized to some extent.

STROGATZ : I’m very interested in the word that you just used a minute ago about “interconnections,” because that seems like a central point for us to discuss. It’s something that you mention in your essay that, along with these, what you call “local” criteria about elegance, or real-world applications, or whatever, you mention this “global” aspect of good mathematics: that good mathematics connects to other good mathematics.

That’s almost key to what makes it good, that it’s integrated with other parts. But it’s interesting because it sounds almost like circular reasoning: that good math is the math that connects to other good math. But it’s a really powerful idea, and I’m just wondering if you could expand on it a little bit more.

TAO : Yeah, so, I mean, what mathematics is about — one of the things that mathematics does is that it makes connections that are very basic and fundamental, but not obvious if you just look at it from the surface level. A very early example of this is Descartes’ invention of Cartesian coordinates that made a fundamental connection between geometry — the study of points and lines and spatial objects — and numbers, algebra.

So, for example, a circle you can think of as a geometric object, but you can also think of it as an equation: x 2 + y 2 = 1 is the equation of a circle. At the time, it was a very revolutionary connection. You know, the ancient Greeks viewed number theory and geometry as almost completely disjointed subjects.

But with Descartes, there was this fundamental connection. And now it’s internalized; you know, the way we teach mathematics. It’s not surprising anymore that if you have a geometric problem, you attack it with numbers. Or if you have a problem with numbers, you may attack it with geometry.

It’s somewhat because both geometry and numbers are aspects of the same mathematical concept. We have an entire field called algebraic geometry, which is neither algebra nor geometry, but it’s a unified subject studying objects that you can either think of as geometric shapes, like lines and circles and so forth, or as equations.

But really, it’s a holistic union of the two that we study. And as the subject has deepened, we’ve realized that that is more fundamental somehow than either algebra or geometry separately, in some ways. So, these connections are helping us discover sort of the real mathematics that initially, somehow, our empirical studies only give us a corner of the subject.

There’s this famous parable of the elephant, I forget where, that if you have… There are four blind men, and they discover an elephant. And one of them feels the leg of the elephant and they think, “Oh, this, it’s very rough. It must be like a tree or something.”

And one of them feels the trunk, and it’s only a lot later that they see that there’s a single elephant object that is explaining all their separate hypotheses. Yeah, so we’re all blind initially, you know. We’re just watching the shadows on Plato’s cave and only later realizing —

STROGATZ : Wow, you are very philosophical here. This is something. I can’t resist now: If you’re going to start talking about the elephant and the blind people, this suggests that you think mathematics is out there — that it is something like the elephant and that we are the blind… Or, you know, we’re trying to see something that exists independent of human beings. Is that really what you believe?

TAO : When you do good mathematics, like, it’s not just pushing symbols around. You do feel like there is some actual object that you’re trying to understand, and all our equations we have are just sort of approximations of that, or shadows.

You can debate the philosophical point of what is actually reality and so forth. I mean, these are things you can actually touch, and the more real things get mathematically, sometimes the less physical they seem. As you said, geometry initially, you know, was a very tangible thing about objects in physical space that you could — you know, you can actually build a circle and a square and so forth.

But in modern geometry, you know, we work in higher dimensions. We can talk about discrete geometries, all kinds of wacky topologies. And, I mean, the subject still deserves to be called geometry, even though there is no Earth being measured anymore. The ancient Greek etymology is very outdated but it’s, but there’s definitely something there. Whether — how real you want to call it. But I guess the point is that for the purpose of actually doing mathematics, it helps to believe it’s real.

STROGATZ : Yeah, isn’t that interesting? It does. It seems like that’s something that goes very deep in the history of math. I was struck by an essay by Archimedes writing to his friend, or at least colleague, Eratosthenes.

We’re talking now, like, 250 B.C. And he makes the remark, he’s discovered a way to find the area of what we would call the segment of a parabola. He’s taking a parabola, he cuts across it with a line segment that’s at an oblique angle to the axis of the parabola, and he figures out this area. He gets a very beautiful result. But he says something to Eratosthenes like, “These results were inherent in the figures all along.” You know, like, they’re there. They’re there. They’re just waiting for him to find.

It’s not like he created them. It’s not like poetry. I mean, it’s interesting, actually, isn’t it? That a lot of great artists — Michelangelo talked about releasing the statue from the stone, you know, as if it were in there to begin with. And it sounds like you and many other great mathematicians have — as you say, it’s very useful to believe this idea, that it’s there waiting for us, waiting for the right minds to discover it.

TAO : Right. Well, I think one manifestation of that is that ideas that are often very complicated to explain when they’re first discovered, they get simplified. I mean, you know, often the reason why something looks very deep or difficult at the beginning is you don’t have the right notation.

For example, we have decimal notation now to manipulate numbers, and it’s very convenient. But in the past, we had like, you know, Roman numerals and then there were even more primitive number systems that were just really, really difficult to work with if you wanted to do mathematics.

Euclid’s Elements , you know — some of the arguments in these ancient texts. Like, there’s one theorem in Euclid’s Elements I think called the Bridge of Fools or something. It’s like the statement that, I think the statement is like an isosceles triangle, the two base angles are equal. Like, this is like a two-line proof in modern geometric texts, you know, with the right axioms. But Euclid had this horrendous way of doing it. And it was where many students of geometry in the classical era just completely gave up on mathematics.

STROGATZ : True. ( laughs )

TAO : But, you know, we now have a much better way of doing that. So often the complications we see in mathematics are artifacts of our own limitations. And, so, as we mature, you know, things become simpler. And it feels more real because of that. We’re not seeing the artifacts. We’re seeing the essence.

STROGATZ : Well, so going back to your essay: When you wrote it, at the time — I mean, this was pretty early in your career, not the very beginning, but still. Why did you feel back then that it was important to try to define what good mathematics was?

TAO : I think… So by that point, I was already starting to advise graduate students, and I was noticing that, you know, there was some misconceptions about, sort of, what is good and what is not. And I was also talking to mathematicians in different fields, and what one’s field valued in mathematics seemed different from others. But yet, somehow we were all studying the same subject.

And sometimes someone would say something that sort of rubbed me the wrong way, you know, like, “This mathematics has no applications, therefore it has no value.” Or “This proof is just too complicated; therefore it has no value,” or something. Or conversely, you know, “This proof is too simple; therefore it is not worth…” You know. Like, there was some, like, sort of snobbery and so forth, sometimes I would encounter.

And in my experience, the best mathematics came when I understood a different point of view, a different way of thinking about mathematics from someone in a different field and applying it to a problem that I cared about. And so my experience of how to use mathematics properly, how to wield it, was so different from these — sort of the “one true way of doing mathematics.”

I felt like this point had to be made somehow. That there’s really a plural way of doing mathematics, but whereas mathematics is still united.

STROGATZ : That’s very revealing, because I had wondered, you know, like, in my introduction I mentioned the many different branches of math that you have explored, and I didn’t even include some. Like, I can remember just a few years ago, your work about this mystery in fluid dynamics, about whether certain equations that we think do a good job of approximating the motions of water and air. I don’t want to go into details too much, but just to say, here you are, people think of you doing number theory or harmonic analysis, and suddenly you’re working on fluid dynamics questions. I mean, I realize it’s partial differential equations. But still, your breadth of interest seems to be related to your breadth of accepting different insights, different valuable ideas from all the different ways of doing good math.

TAO : I forget who said it, but there are two types of mathematicians. There’s hedgehogs and foxes. A fox is someone who knows a little bit about everything. A hedgehog is a creature that knows one thing very, very well. And neither is better than the other. They complement each other. I mean, in mathematics, you need people who are really deep domain experts in one subfield, and they know a subject inside-out. And you need people who can see the connections between one field and another. So I definitely identify as a fox, but I work with a lot of hedgehogs. The work I’m most proud of is often a collaboration like that.

STROGATZ : Oh, yeah. Do they realize that they’re hedgehogs?

TAO : Well, okay, the roles change over time. Like, there are other collaborations where I’m the hedgehog and someone else is the fox. These are sort of not permanent — you know, these are not in your DNA.

STROGATZ : Ah, good point. We can adopt — we can wear both cloaks.

Well, what about, was there a response to the essay at the time? Did people say anything back to you?

TAO : I got a fairly positive response in general. I mean, the Bulletin of the AMS is not a hugely, widely circulated publication, I think. And also, I didn’t really say anything too controversial. Also, this kind of predated social media, so, I think maybe there’s a few math blogs that picked it up, but there was no Twitter. There was nothing to make it go viral.

Yeah, also I think, in general, mathematicians don’t spend much of their time and intellectual capital on speculation. I mean, there’s another mathematician called Minhyong Kim who had this very nice metaphor that, to mathematicians, credibility is like currency, like money. If you prove theorems and you demonstrate that you know the subject, you’re accumulating somehow this currency of credibility in the bank. And once you have enough currency, you can afford to speculate a little bit by being a bit philosophical and saying what might be true rather than what you can actually prove.

But we tend to be conservative, and we don’t want an overdraft in our bank account. You know, you don’t want most of your writing to be speculative and only like one percent to actually prove something.

STROGATZ : Fair enough. So, okay. So, lots of years have passed since then. What are we talking about? It’s more than 15 years.

TAO : Oh yeah, time flies.

STROGATZ : Has your opinion changed? Is there anything we need to revise?

TAO : Well, the culture of math is changing quite a bit. I already had a broad view of mathematics, and now I have an even broader one.

So, one very concrete example is: Computer-assisted proofs were still controversial in 2007. There was a famous conjecture called the Kepler conjecture, which concerns the most efficient way to pack unit balls in three-dimensional space. And there’s a standard packing, I think it’s called the cubic central packing or something, that Kepler conjectured to be the best possible.

This was finally resolved, but the proof was very computer-assisted . It was quite complicated, and [Thomas] Hales eventually actually created a whole computer language to formally verify this particular proof, but it was not accepted as a real proof for many years. But it illustrated how controversial the concept of a proof that you needed computer assistance to verify was.

In the years since, there’s been many, many other examples of proofs where a human can reduce a complicated problem to something which still requires a computer to verify. And then the computer goes ahead and verifies it. We’ve kind of developed practices about how to do this responsibly. You know, how to publish code and data and ways to check and new open-source things and so forth. And now, there’s widespread acceptance of computer-assisted proofs.

Now, I think, the next cultural shift will be whether AI-generated proofs will be accepted . Right now, AI tools are not at the level where they can generate proofs to really advance mathematical problems. Maybe undergraduate-level homework assignments, they can kind of manage, but research mathematics, they’re not at that level yet. But at some point, we’re going to start seeing AI-assisted papers come out and there will be a debate.

The way our culture has changed in some ways… Back in 2007, only a fraction of mathematicians made their preprints available before publishing. Authors would jealously guard their preprints until they had the notification of acceptance from the journal. And then they might share.

But now everyone puts their papers on public servers like the arXiv . There’s a lot more openness to put videos and blog posts, about where the ideas of a paper come from. Because people realize that this is what makes work more influential and more impactful. If you try to not publicize your work and be very secretive about it, it doesn’t make a splash.

Math has become much more collaborative . You know, 50 years ago, I would say that the majority of papers in mathematics were single-author. Now, definitely the majority are two or three or four authors. And we’re just beginning to see really big projects like we do in the sciences, you know, like tens, hundreds of people collaborate. That’s still difficult for mathematicians to do, but I think we’re going to get there.

Concurrently, we’re becoming much more interdisciplinary. We’re working with other sciences a lot more. We’re working between fields of mathematics. And because of the internet, we can collaborate with people across the world. So, the way we do mathematics is definitely changing.

I hope in the future, we will be able to utilize the amateur math community more. There are other fields like astronomy, where astronomers make great use of the amateur astronomy community, like, you know, a lot of comets, for example, are found by amateurs.

But mathematicians… There’s a few isolated areas of mathematics such as like, tiling, two-dimensional tiling, and maybe finding records in prime numbers. There’s some very select fields of mathematics where amateurs do contribute, and they’re welcomed. But there’s a lot of barriers. In most areas of mathematics, you need so much training and internalized or conventional wisdom that we can’t crowd source things. But this may change in the future. Maybe one impact of AI would be to allow amateur mathematicians to contribute meaningfully to mathematics.

STROGATZ : That’s very interesting.

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STROGATZ : So the amateurs might, with the help of AIs, either ask new questions that are good or help with good explorations of existing questions, that sort of thing?

TAO : There are many different modalities — yeah. So, for example, there are now projects to formalize proofs of big theorems in these things called formal proof assistants , which are like computer languages that can 100% verify that a theorem is true or not and — is proven or not. This actually enables large-scale collaboration in mathematics.

So in the past, if you collaborate with 10 other people to prove a theorem, and each one contributes a step, everyone has to verify everyone else’s math. Because the thing about math is that if one step has an error in it, the whole thing can fall apart.

So you need trust, and so — therefore this prevents, this really inhibits really large-scale collaborations in mathematics. But there are now, there have been successful examples of really large theorems being formalized where there’s a huge community, they don’t all know each other, they don’t all trust each other, but they communicate through uploading to some Github repository or something, like, individual proofs of individual steps in the argument. And the formal proof software verifies everything, and so you don’t need to worry about trust. So we are enabling new modes of collaboration, which we haven’t seen really in the past.

STROGATZ : It’s really interesting to hear your vision, Terry. It’s a fascinating thought. You don’t hear the phrase “citizen mathematician.” You hear of citizen science, but why not citizen math?

But I’m just wondering, are there any trends that you are worried about, for example, with computer-assisted proofs or AI-generated proofs? Will we know that certain results are true, but we won’t understand why?

TAO : So that is a problem. I mean, it’s already a problem even before the advent of AI. So, there are many fields where the papers in a subject are getting longer and longer, hundreds of pages. And I’m hopeful that AI can actually conversely help simplify and it can explain as well as prove.

So there’s already experimental software where, like, if you take a proof that has been formalized, you can actually convert it into an interactive human-readable document, where you have the proof and you see the high-level steps and if there’s a sentence you don’t understand you can double-click on it, and it will expand into smaller steps. Soon I think you can also get an AI chatbot sitting next to you while you’re going through the proof, and they can take questions and they can explain each step as if they were the author. I think we’re already very close to that.

There are concerns. We have to change the way we educate our students, particularly now that many of our traditional ways of assigning homework and so forth, we are almost at the point where these AI tools can just instantly answer many of our standard exam questions. And so, we need to teach our students new skills, like how to verify whether an AI-generated output is correct or not and how to get a second opinion.

And we may see the advent of a more experimental side to mathematics, you know. So, mathematics is almost entirely theoretical, whereas most sciences have both a theoretical and experimental component. We may eventually have results that are first only proven by computers and, as you say, we don’t understand. But then once we have the data that the AI, the computer-generated proofs provide, we may be able to run experiments.

There’s a little bit of experimental mathematics now. People do study, like, large data sets of various things, elliptic curves, say. But it could become much bigger in the future.

STROGATZ : Gee, you have a very optimistic view, it sounds like to me. It’s not like the Golden Age is in the past. If I’m hearing you right, you think that there’s a lot of very exciting stuff ahead.

TAO : Yeah, a lot of the new technological tools are very empowering. I mean, AI in general has many complex ups and downsides. And outside the sciences, there’s a lot of possible disruption to the economy, intellectual property rights and so forth. But within math, I think the ratio of good to bad is better than in many other areas.

And, you know, the internet really has transformed the way we do mathematics. I collaborate with a lot of people in a lot of different fields. I could not do this without the internet. The fact that I can go on Wikipedia or whatever and get started learning a subject, and I can email somebody, and we can collaborate online. If I had to do things old-school where I could only talk to people in my department and use physical mail for everything else, I could not do the math that I do now.

STROGATZ : Wow, all right. I just have to underline what you just said, because I never thought in a million years I was going to hear this: Terry Tao reads Wikipedia to learn math?

TAO : As a starting point. I mean, it’s not always Wikipedia, but just to get the keywords, and then I will do a more specialized search of, say, MathSciNet or some other database. But yeah.

STROGATZ : It’s not a criticism. I mean, I do the same thing. Wikipedia is actually, if there’s any criticism of the math on Wikipedia, maybe it’s that sometimes it’s a little too advanced for the readers that it’s intended for, I think. Not always. I mean, it depends. It varies a lot from article to article. But that’s just funny. I love hearing that.

TAO : I mean, these tools, you have to be able to vet the output. You know, so, I mean, the reason why I can use Wikipedia to do mathematics is because I already know enough mathematics that I can smell if a piece of Wikipedia in mathematics is suspicious or not. You know, it may get some sources and one of them is going to be a better source than the other. And I know the authors, and I have an idea of which reference is going to be better for me. If I used Wikipedia to learn about a subject that I had no experience in, then I think it would be more of a random variable.

STROGATZ : Well, so we’ve been talking quite a bit about what it is that makes good mathematics, the possible future for new kinds of good mathematics. But maybe we should address the question: Why does this even matter? Why is it important for math to be good?

TAO : Well, so, first of all, I mean, why do we have mathematicians at all? Why does society value mathematicians and give us the resources to do what we do? You know, it’s because we do provide some value. We can have applications to the real world. There’s intellectual interest, and some of the theories we develop eventually end up providing insight into other phenomena.

And not all mathematics is of equal value. I mean, you could compute more and more digits of pi, but at some point, you don’t learn anything. Any subject needs some sort of value judgment because you have to allocate resources. There’s so much mathematics out there. What advances do you want to highlight and publicize and let other people know about, and which ones maybe should just be sitting quietly on a journal somewhere?

Even if you think of a subject as being completely objective and, you know, there’s only true or false, we still have to make choices. You know, just because time is a limited resource. Attention is a limited resource. Money is a limited resource. So, these are always important questions.

STROGATZ : Well, interesting that you mention about publicizing, because it is something that I think is a distinctive feature of your work, that you’ve also put in a lot of effort to make math publicly accessible through your blog, through various articles you’ve written. I remember discussing one that you wrote in American Scientist about universality and that idea. Why is it important to make math publicly accessible and understandable? I mean, what is it that you’re trying to do?

TAO : It kind of happened organically. Early in my career, the World Wide Web was still very new, and mathematicians started having webpages with various content, but there wasn’t much of a central directory. Before Google and so forth, it was actually hard to find individual resources.

So, I started sort of making little directories on my webpage . And I would also make webpages for my own papers, and I’d make some commentary. Initially, it was more for my own benefit, just as an organizational tool, just to help me find things. As a byproduct, it was available to the public, but I was kind of the primary consumer, or at least so I thought, of my own webpages.

But I remember very distinctly, there was one time when I wrote a paper and I put it on my webpage, and I had a little subpage called “What’s New?” And I just said, “Here’s a paper. There’s a question in it that I still couldn’t answer, and I don’t know how to solve it.” And I just made this comment. And then like two days later, I got an email saying, “Oh, I was just checking your homepage. I know the answer to this. There’s a paper which will solve your problem.”

And it made me realize, first of all, that people were actually visiting my webpage, which I didn’t really know. But that interaction with the community could really — well, it could help me directly solve my questions.

There’s this law called Metcalfe’s law in networking that, you know, if you have n people, and they all talk to each other, there’s about n 2 connections between them. And so, the larger the audience and the larger the forum where everyone can talk to everybody else, the more potential connections you can make and the more good things can happen.

I mean, in my career, so much of the discoveries I’ve made, or the connections I’ve made is because of an unexpected connection. My whole career experience has been sort of the more connections equals just better stuff happening.

STROGATZ : I think a beautiful example of what you’re just referring to, but I’d love to hear you talk about it, is the connections that you made with people in data science who are interested in questions having to do with medical resonance imaging, MRI. Could you tell us a little about that story?

TAO : So, this was about 2006, 2005, I think. So, there was an interdisciplinary program here on campus at UCLA on, I think, multiscale geometric analysis, or something like that, where they were bringing together pure mathematicians who were interested in sort of multiscale type geometry in its own right, and then, you know, people who had very concrete data type problems.

And I had just started working on some problems in random matrix theory, so I was sort of known as someone who could manipulate matrices. And I met someone who I already knew, Emmanuel Candès , because at the time he worked right next door in Caltech. And he and another collaborator, Justin Romberg , they had discovered this unusual phenomena.

So they were looking at MRI images, but they’re very slow. To collect enough really high-resolution image of a human body, or enough to maybe catch a tumor, or whatever medically important feature you want to find, it often takes several minutes because they have to scan all these different angles and then synthesize the data. And this was a problem, actually, because little kids, for example, just to sit still for three minutes in the MRI machine was quite problematic.

So they were experimenting with a different way, using some linear algebra. They were hoping to get a 10%, 20% better performance improvement. You know, a slightly sharper image by tweaking the standard algorithm a little bit.

So the standard algorithm was called least squares approximation, and they were doing something else, called total variation minimization. But then when they ran the computer software, they got like almost perfect reconstruction of their test image. Massive, massive improvement. And they couldn’t explain this.

But Emmanuel was at this program, and we were chatting at tea or something. And he just mentioned this and, actually, my first thought was that you must have made a mistake in your calculation, that what you’re saying is not actually possible. And I remember going back home that night and trying to write down an actual proof that what they were seeing could not actually happen. And then halfway through, I realized I had made an assumption which wasn’t true. And then I realized that actually it could work. And then I figured out what might be the explanation. And then we worked together, and we actually found a good explanation and we published that.

And once we did that, people realized that there were many other situations where you had to take a measurement which normally required lots and lots of data, and in some cases you can take a much smaller amount of data and still get a really high-resolution measurement.

So now, modern MRI machines, for example — a scan that used to take three minutes can now take 30 seconds because this software, this algorithm is hardwired, hard-coded into the machines now.

STROGATZ : It’s a beautiful story, it’s such a great story. I mean, talk about important mathematics that is changing lives, literally, in this context of medical imaging. I love the serendipity of it and your open-mindedness, you know, to hear this idea and then think, well, “this is impossible, I can prove it.” And then realizing, no, actually. Fantastic to see math making such an impact.

Well, okay, I think I better let you go, Terry. It’s been a real pleasure discussing the essence of good mathematics with you. Thanks so much for joining us today.

TAO : Yeah, no, it’s been a pleasure.  

STROGATZ : “The Joy of Why” is a podcast from Quanta Magazine , an editorially independent publication supported by the Simons Foundation. Funding decisions by the Simons Foundation have no influence on the selection of topics, guests or other editorial decisions in this podcast or in Quanta Magazine .

“The Joy of Why” is produced by PRX Productions . The production team is Caitlin Faulds, Livia Brock, Genevieve Sponsler and Merritt Jacob. The executive producer of PRX Productions is Jocelyn Gonzales. Morgan Church and Edwin Ochoa provided additional assistance. From Quanta Magazine , John Rennie and Thomas Lin provided editorial guidance, with support from Matt Carlstrom, Samuel Velasco, Nona Griffin, Arleen Santana and Madison Goldberg.

Our theme music is from APM Music. Julian Lin came up with the podcast name. The episode art is by Peter Greenwood and our logo is by Jaki King and Kristina Armitage. Special thanks to the Columbia Journalism School and Burt Odom-Reed at the Cornell Broadcast Studios.

I’m your host, Steve Strogatz. If you have any questions or comments for us, please email us at [email protected] . Thanks for listening.

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