Found Clunkers, Part 1

white ceramic teacup with saucer near two books above gray floral textile
Photo by Thought Catalog on

I often find some really bad sentences in books I read. I wanted to start a series in which we look at why they are bad and how to fix them.

Unfortunately, I’m a bit stuck on how to proceed. I really want this blog to lift up excellent books and art. I don’t want to tear down people who are probably still learning and getting better. In a year, they might be embarrassed by these sentences themselves and wish their name wasn’t attached to it in some blog post.

I’m sure you can find horrible sentences in each of my books as well.

Hence the issue: use a single sentence without attribution for educational purposes (technically a copyright violation) or attach the name and book to it.

For now, I’m erring on the side of writer anonymity. Here’s the sentence:

That sweet, crooning voice, singing a song I’d been in the room for when he wrote.

We have here a classic example of why I say new writers shouldn’t write in first person. I’m sure this gem popped out in the middle of a writing session, when the author was fully into the “character’s voice.” As it came out, she probably thought: yeah, that’s exactly how he would talk.

It’s important to remember that the way people talk does not make good prose. First, full sentences should be used. The above is a sentence fragment. I wish I could show you more context, but the sentences around this are also fragments.

I’m all for using a sentence fragment judiciously for voice, but I’d never do it more than once per chapter. Too many sentence fragments creates jarring, obnoxious prose. It leads to more confusion than it’s worth.

I once heard it described this way: strong prose leads to a strong voice; weak prose leads to a weak voice. In other words, “breaking the rules,” like forming a sentence fragment, is a weak and cheap way to form narrative voice.

Word order pops out next. There’s something funny about: “I’d been in the room for when he wrote.” Unless I grew up with a strange dialect, I’m pretty sure most native English speakers will hear “when he wrote” and feel it is missing a direct object.

So, it should probably read “when he wrote it.” The sentence also has an implied pause after “for.” Again, I’m not of the old school of thought where you can’t end a sentence with a preposition. But in this case, it creates confusion to have “for when” juxtaposed like that.

All I’m trying to say is that the word order is sloppy, and it could be cleaned up easily by writing something like: “I’d been in the room when he wrote it.”

Of course, this brings us back to the first point. The author put herself into a bind when using a sentence fragment, because she didn’t clearly identify the correct subject of the sentence.

She started with “That sweet, crooning voice,” implying this as the subject, but the second clause wants to use “I” as the subject.

I know people don’t believe me when I say to use third-person and full sentences. I guarantee this mess of a sentence happened because using first person caused her to confuse the narrative voice with the thoughts of the main character (leading to a bunch of sentence fragments).

I’ll reiterate this from previous posts. The prose you use when you write in first person is not some stream-of-consciousness coming from the main character (unless you’ve chosen this on purpose for a piece of experimental writing). In modern commercial writing, thoughts of the main character are usually offset and italicized to be clear about the distinction.

This is what makes first person so difficult.

Anyway, I’ll end my rant about point of view. You can obviously do it if the writing calls for it. You must be extra vigilant, or else sentences like this will pop out and sound good in your head.

Here’s my rewrite. I think the key is to realize there are two separate ideas that should be separated: hearing the voice, and recalling the writing.

His sweet, crooning voice lilted through the speakers. I’d been in the room when he wrote the song that changed his life.

I changed “I’d been” to “I was,” because simple past tense causes no confusion and cleans things up even more.

As always, context will matter. Maybe this doesn’t flow with the sentences around it, and some other alterations will be necessary.


Are the Self-Publishing Gurus Out of Touch?


Let me start by saying that I appreciate all the free content people in the self-publishing world put out. It’s quite generous and high quality. There must be ten or more hours of podcasts each week that come to my phone, not to mention blog posts and youtube and e-mails. That’s just me, meaning it’s only a small fraction of what’s actually produced.

If you wanted to be a student of this stuff, you’d have more class time than a full-time college student with no breaks for summer or winter. And there’s so much to learn that it could probably fill an entire major for a college student.

So, thank you to all those content providers.

That being said, I have two theories I’d like to present.

I’m going to call these people “gurus.” You either know who they are or don’t, but I don’t want to name names (google “podcasts for writers” or something if you really don’t know).

Every person I have in mind left their job at some point to be a full-time self-published writer. I think each of them makes at least six figures. This happened years ago for each of them.

Also, each of them has a “thing” they attribute it to: cover design that perfectly fits the genre, great copywriting on the product description, Amazon keywords and categories, Amazon marketing, Facebook marketing, writing to market, price surging, box sets, e-mail auto-responders, mailing list magnets, promotion services, etc.

Note, I’m not saying they push a well-rounded approach to improving each of the above; I’m saying their entire shtick is that once you get that one thing right, you will take off and have huge success.

This seems weird and crazy, so why would they, at least nominally, believe this? The cynical theory would be that they wanted to corner a niche in the market, so they figured this one thing out and pushed it hard as the expert.

I don’t believe this. I think there’s a much more obvious explanation. Like almost all successful authors, they started to gain traction, little by little. They were experimenting with all the above techniques to see if anything could get them a little edge.

As Gladwell explains in The Tipping Point, they just hit a critical mass of followers and readers at some point. This caused them to shoot from obscurity into prominence.

Human brains being what they are, these writers then attributed their success to the most recent major change they made rather than a natural progression to a tipping point. This is how we get people who are convinced you only need to get that one thing right to get success.

And this is fine as long as you don’t take that claim seriously when listening to them. The advice they give on that one thing is going to be pretty solid and useful. It will help keep you crawling upward toward your own tipping point.

I do think some people get frustrated when they work really hard at that one thing, and they only see marginal gains despite doing everything right.

Here’s my second theory based on the first theory. The gurus out there with the biggest platform have been wildly successful for years, and this actually makes them a bit out of touch with how things really are.

My theory is that they could launch a book to number one in their category by doing none of the advice they give: no ads, no pre-release, no notification of the mailing list, a sloppy and vague product description, a less-than-stellar cover, etc.

They have so many followers that news would spread of their release, and it would make thousands of dollars in the first month and be an Amazon bestseller.

If I’m correct about this, that means they actually have no idea if the advice they’re giving is correct. Don’t take this the wrong way. I’m not saying their advice is incorrect (quite the opposite)—but it’s just a fact of their prominence that they can’t know how much of an effect their advice has anymore.

I’m not sure if there’s much of a point to this post. I guess it’s that you shouldn’t put too much stock in any one thing you hear about self-publishing. Success is going to be a slow growth attributed to hundreds of things.

Writing a better product description might get you five more sales. Improving the cover: five more. An experiment with AMS ads: five more. Suddenly, these have added up to enough that you snatch a true fan that leaves a glowing review.

This review converts to twenty more sales, and the new fan starts you one person further along on the next book.

So it’s all interconnected and not traceable to any one thing. After a bunch of books of doing this, you find yourself starting with a hundred fans buying it on launch day getting you to bestseller status and days of free advertising in your genre. These “organic” sales translate to new fans, etc.

That’s the tipping point. It can look like a sudden spike in success, but it’s not the most recent marketing trick you tried. It’s dozens of things synergizing to create the effect.

So, most of all, take everything the gurus say with a grain of salt and don’t be afraid to experiment with your own ideas. What worked for these gurus several years ago may not be working in today’s market or your genre. Or they might. You’ll have to be the judge of that.

On the Accuracy of Memory: or a Nuanced Approach to Current Events


Trigger Warning: Rape is discussed.

Bias disclaimer: I am strongly opposed to Brett Kavanaugh being on the Supreme Court for many reasons, the most obvious being the overwhelming evidence of criminal activity of the current president. Kavanaugh has taken a firm stance that sitting presidents should not be indicted. I’m also a liberal who has never voted for a Republican for any office, no matter how small and local. Etc, etc.

Current event summary: Christine Blasey Ford has accused Supreme Court nominee Brett Kavanaugh of attempted rape from 36 years ago at a high school party based on her personal memory of the event.

I didn’t want to weigh in on this, but as someone who has written extensively on math, science, Bayesian statistics, cognitive biases, truth, and knowledge, I just didn’t see any articles out there with nuanced, clear thinking on this issue. So here goes.

The world is not black and white. People think there are only two options when it comes to Ford’s accusation against Kavanaugh.

The first is that it is perfectly understandable why she didn’t come forward until now. She has nothing to gain and everything to lose, so she must be telling the truth. The other is that this is an opportunistic, political move and clearly a false accusation.

The truth almost never falls in such partisan terms.

The human brain, and memory in particular, is a complicated thing. How about we ditch partisan politics for five seconds and try to take a nuanced approached to things?

No one wants to believe their memory is faulty. Memory is basically our whole sense of self. To make an attack on the accuracy of memory feels like an attack on our selves. But it’s not.

Try to distance yourself from this for a bit, and let’s examine what the science says from a cool, rational perspective. If you feel yourself getting angry, take a deep breath, and chant the mantra: nuance, nuance, nuance.

In 1984, Jennifer Thompson-Cannino was raped. She paid careful attention to detail, and had vivid, horrifying memories of the event. Fortunately, she was able to identify the assailant with “100% certainty” only a few days later: Ronald Cotton. He spent 10 years in prison for the crime.

Whew. Thank God for memory.

Except that he was released from prison after 10 years because DNA evidence exonerated him of the crime.

In 1985, a woman was raped and murdered in Beatrice, Nebraska. Six people were found guilty. Ada JoAnn Taylor was one of those people, and she confessed to the crime. She still has “vivid” memories of committing the crime. When something that horrifying happens, you never forget the details.

In 2008, DNA evidence exonerated all of them. Ada did not commit the murder, yet she has clear memory of doing it. If you read that sentence without getting goosebumps, read it again and again and again until you fully grasp the significance of it. She has a vivid memory of committing a murder she didn’t commit.

Most people talk about how vivid and clear the details of where they were when September 11, 2001 happened. It was a traumatic event in most of our lives. How could we ever forget such things?

Fortunately, a collective of memory researchers got right on that. They interviewed thousands of people while the memory was fresh. A mere one year later, they asked people to recall the event. A majority of people had high confidence in their accuracy (how could you ever forget such things?), yet they were totally wrong about things as major and fundamental as people with them and their location. In fact, consistency was only at 63%.

The shifting of details only gets worse over time. One can imagine how much will have changed in 30+ years.

Recollection of these short, traumatic experiences are called flashbulb memories, and decades of research show they have something in common: they are vivid, people have high confidence in their accuracy, and they are wildly inaccurate. In other words, the listed examples above are not isolated outliers; they are the norm.

No matter the trauma of the event, memories are notoriously faulty. Sometimes we mis-remember small details, like who was with us when we found out about the September 11 attacks. Sometimes the event happened, and despite a vivid, clear recollection of the perpetrator, this major detail is false.

Sometimes our brains fabricate entire traumatic events, like when someone truly remembers being abducted and abused by aliens.

In all these cases, the person is telling the truth about their experience. In other words, the person isn’t lying or making it up or intentionally falsely accusing. The experience and memory is real. But, unfortunately, this tells us very little about the accuracy of any of it. This is why this sort of testimony doesn’t stand up in court anymore.

Deep breath: nuance, nuance, nuance.

So where are we now? Vividness of a memory does not make it accurate. Confidence in a memory does not make it accurate. Trauma surrounding a memory does not make it accurate. The more time that passes, the less accurate a memory gets.

These are all scientific facts about the human brain. To deny these facts in the service of politics is as bad as Republicans who deny climate change for political reasons. We have to be honest, not partisan, when it comes to scientific facts of the world, no matter how inconvenient.

And please do not post your own traumatic experiences here. I get it. You remember every detail with high confidence like it was yesterday. You’re sure it’s all accurate, because it has played out in your mind everyday since it happened.

I’m sorry that happened to you.

Deep breath: nuance, nuance, nuance.

What is the nuanced approach to the situation we find ourselves in?

It’s to say to Christine Blasey Ford: I believe you. This event happened. You’re not lying. You’re not making it up for political reasons. The memory of the event is crystal clear, and Kavanaugh was the one doing it. We understand why you didn’t come forward until now.

Unfortunately, in the world we live in, we must take the position that this memory alone is not disqualifying for him, because memories, even major details like the perpetrator of a crime, are often wrong.

I think there are plenty of reasons to not confirm Kavanaugh, but let’s not set this one as the precedent.

Anti-empiricism is never progressive. Denying reality is no way to change reality. -John McWhorter


Additional Reading:

Memory Distortion for Traumatic Events: The Role of Mental Imagery 

Trauma, PTSD, and Memory Distortion

A mega-analysis of memory reports from eight peer-reviewed false memory implantation studies

Notes as of 9/24/2018: I wrote this about a week ago, mostly as a way to vent against the black and white thinking I saw. New information has come to light, like a new accusation against Kavanaugh and a date for Ford to give her testimony.

This post is not meant to weigh all the current and cumulative evidence that can come to light in the coming hours and days.

Year of Mysteries, Part 8: The Last Place

Woops. I just realized the title of this series changed from “Year of Mystery Novels” to “Year of Mysteries” at Part 5. Sorry. I try to keep this consistent for searching purposes.

The main point of this series was to examine the mystery novel genre. There are people that read a few a week. There’s something about the genre that keeps people coming back.

I feel like my book choices to this point have basically avoided the “typical mystery.” Pretty much every book in this series is considered a “classic” or has been award winning. In other words, there’s nothing typical about them. Hundreds of new mystery novels come out every month, satisfying readers of the genre without the fancy prose of award-winners.

I went to my local used book store and went to the mystery section. I wanted to pick something at random to get a better feel for what a generic, contemporary mystery novel felt like.

This was a bit hard. The first few I picked up were classified as “suspense thrillers.” I wasn’t opposed to this, but I also couldn’t tell if there was an actual mystery from the back cover. I finally grabbed The Last Place by Laura Lippman.

It was by a bestselling and award-winning author, but this particular novel hadn’t won awards. This made it the perfect candidate.  Unfortunately, it turned out to be the seventh book in the Tess Monaghan series, even though nowhere on or in the book told me this.

I read it anyway.

The novel definitely references some stuff from previous books, but it is perfectly readable as a standalone. I assume this is one of those slow character growth things that happens over multiple books.

Tess is the main character, and she is a private investigator. She has anger management issues and basically messes up a child molester. So she gets assigned a community service project by the courts and has to investigate a police department. She checks up on five unrelated murders to make sure they followed procedure, etc.

As it goes on, she finds out that all five have major issues and are unresolved and are possibly related.

The first thing I noticed was the solid prose style. I often read literary stuff, which tends to be fancy. I also read fantasy, which tends to be wordy. This book had that pristine, clean style that I tend to call “bestseller” style.

I can’t go into all the details here, but it just means few adverbs, active voice, just enough description, pared down sentences, etc. It really makes the book come alive fast in your mind without any excess thinking or confusing sentences.

This brings up the next point. This was a page-turner through and through. I also can’t remember the last time I read something like that. I’d say it’s almost too polished. If you’ve studied plotting and structure, you can see how each scene is crafted to raise the stakes and complicate the plot just enough to push you along.

One thing I think this book did better than any of the other mysteries was to provide a lot of red herrings. At one point, the investigators discuss the possibility that the male killer could actually be a female. They go into depth on how this could be pulled off, despite being married, by referencing Boys Don’t Cry.

I found myself thinking: what? No way! Wait, but maybe. 

It was never meant to be taken all that seriously, but little things like this added up to keeping me unsettled. Whenever I thought maybe I was getting a handle on things, something would come up to remind me that I just didn’t have enough information yet.

I was thinking about characterization, since I recently wrote a post on it. One character has a quirky “thing” they do. This is often one of those cheap techniques that give readers a false sense of depth.

He’s really into movie quotes and says them all the time. Each time it happens, you think, oh yeah, that’s the guy that does the quote thing. This made the character more into a caricature than a fully fleshed out character.

Overall, this was a fast and enjoyable read. I definitely understand why people tear through these at the rate of a couple a week.

If you’re following along, I really only have one book left on my list: The Name of the Rose by Umberto Eco. I’ve been avoiding it, because I’ve heard it has a beastly beginning that’s almost impossible to suffer through. We’ll see…


Less is Awesome

I’m abandoning the embarrassment of having a “…is awesome” set of posts. If I find awesome stuff, I want to share it.

Less is a novel by Andrew Sean Greer, and it won the Pulitzer Prize this year. I’ve been steering clear of prizewinners over the past few years, because the politics of who wins what prize is a bit nauseating.

Eventually, I saw enough good comments by enough people I trust to give it a try. Wow, I’m glad I did. It’s funny, has a unique and strong voice, clever and insightful descriptions, and somehow manages to be tragic and heartwarming at the same time.

It’s awesome!

I want to address the elephant in the room first. Most of the negative reviews (and they have a lot of upvotes) demonstrate a clear failing of the modern education system. Here are some actual verbatim quotes:

I also found Less such a privileged, oblivious character. He is tall, attractive, white, able-bodied, and can afford to travel. His privilege is not examined in any interesting or meaningful way.

It’s heavy and dull with this self absorbed white man mourning his youth…There’s nothing wrong with his life at all except for the fact that he’s way too self-absorbed and does next to nothing for anyone besides himself.

He is literally mid-life, 30 years down and roughly 30 to go. Somehow this is a burden to him, living his privileged, tall, handsome white guy life surrounded by men who really like kissing him. Oh, poor thing.

There seems to be a modern trend to think that one cannot have sympathy or empathy for a main character that is white, male, and “able-bodied.” I feel sorry for these readers, because this lack of empathy on their part shows way more about them than the novel itself.

The whole point of a novel is to learn to see through the main character’s eyes, and understand why they think their problems are real. Someone that has actually read the book should understand that Less has struggles and problems.

Why is Less so preoccupied with being fifty? This seems to be a mystery to a large number of reviewers. It seems trivial to them (I mean, he still has 30 great years left, don’t you know!). Let’s see if the text of the novel gives us a clue:

Arthur Less is the first homosexual ever to grow old…He has never seen another gay man age past fifty, none except Robert. He met them all at forty or so but never saw them make it much beyond; they died of AIDS, that generation. Less’ generation often feels like the first to explore the land beyond fifty.

So, dear reader of this blog: can you come up with any reason, any reason at all, why Less might see turning fifty as a monumental point of his life? Take some time. It’s hard to understand why any privileged white man would see any aspect of his life as difficult or challenging.

I think Millennials and iGen have no conception of history. They think that whatever is happening right now is exactly how it’s always been.

As a gay man, I’ll admit that right now is pretty great. But to think Less has had no struggle in his life is to forget a tragic part of history. Less spent most of his youth thinking he’d never grow old, losing all his friends to a horrific epidemic, and then feeling guilty for surviving it.

The novel opens with Less as a failed novelist, whose lover of nine years has just invited him to his wedding to a different man. The way Less tries to avoid the wedding is to accept invitations to speak around the world at events no prominent novelist would be caught dead at.

(Note how different this is than “can afford to travel.” He’s not taking a leisure trip. These are work trips paid for by the events, not the main character).

If you read this book and think: why’s he even upset? At least he’s not being shot by cops. At least his boss didn’t sexually harass him. He’s a white man with no problems. 

Well, then you’re pretty hopeless, and maybe reading books isn’t the best idea until you work some of that out with a therapist. No one is saying his problems are worse than someone else’s. Just because it’s not a talking point of the day, doesn’t mean it’s not important to the character.

The problems of Less have to do with heartbreak and aging and figuring out what makes a life worth living and being remembered and finding love. These are timeless issues found in great art throughout history. Hopefully we don’t lose these themes merely because a generation of critics view the only worthwhile topics to be what they saw on Vox that day.

Sorry to spend so long on this, but as someone who writes books, I find this growing movement very concerning.

So, what’s so awesome about Less?

A lot came up in that rant already, but one of the greatest parts is how brilliant the descriptions are.

His slim shadow is, in fact, still that of his younger self, but at nearly fifty he is like those bronze statues in public parks that, despite one lucky knee rubbed raw by schoolchildren, discolor beautifully until they match the trees.

If such a sentence occurred in a vacuum of otherwise banal prose, I’d say cut it; it would be too much.

But this voice and style abound throughout the novel and stays quite consistent. The result is a flurry of original images and similes and metaphors that always bring the right emotional resonance to the scene.

Greer has a talent for “breaking the rules” in impeccable ways for a powerful reading experience. I think all writers should read this for examples on how to do description and prose style well.

The Character/Caricature Balance

Image result for caricature

One of the most important things we can do as writers is to acknowledge our weaknesses and then work on them. I think too often we get comfortable with “our thing” and then try to hide our weaknesses by going all in on our strength.

I’ve spent an enormous amount of time over the past five years or so really studying and working on prose style and story structure. I write a book according to something like the Story Grid method. Then I see what works and what doesn’t for me, and then move on to another technique like John Truby‘s method. Iterating this process a dozen times has brought me a rich understanding of structure.

I think there’s no doubt that characterization is my weakest point. I’ve had a philosophy for a long time that characters should be “real.” This means my characters often act in contradictory or paradoxical ways. They have subtle and complicated reasons for doing things that only come out in subtext.

You might be thinking: wow, that sounds great! But then you actually read it and every character sounds the same and has no interesting characteristics. I tend to write an “everyman,” and they all sound boring and similar.

This is especially problematic in fantasy writing. There is a general trend in fantasy to create caricatures instead of characters. This means one feature is exaggerated to the point of becoming a one-dimensional defining feature. These flat characters can be boring and predictable.

For example, a book might have one character’s defining trait as loyalty, the bad guy as greed, the sidekick as humor, etc. I don’t have to tell you why this is a bad thing, but I will make an argument for how it can be good if done properly.

When writing caricatures, it is very clear how the character changes over the course of a novel. The one subplot has the cowardly character, and they have to learn to be braver. Their arc will culminate in a test of bravery, and yay, they succeed! It’s exhilarating and emotional to read these things.

Another reason caricatures can be good is that they create very clear differences in stories with many characters. It’s easy to keep track: that one’s the trickster, that one’s the nerdy one, etc.

Unfortunately, I see a bunch of positive reviews of books that take this too far with claims like “I loved the variety of characters.” Or worse, that the characterization is great. Caricatures can create an illusion of depth by keeping the motivations clear and consistent.

As I’ve started exaggerating certain traits of my characters toward the caricature breaking point, I’ve seen reviews get more and more positive. So people obviously like this.

I’m still struggling with this balance. It’s bad to have no defining traits, because then the character will be boring and spastic. It’s bad to be all caricature, even if you trick some of your audience. So it’s a balance, and one that’s far trickier than I initially thought.

So far, the best method I’ve come across is to have a fleshed out backstory that is the cause of a character flaw. The flaw gives the character a clear sense of growth as they learn to overcome the flaw.

Then, if you use the backstory to inform character actions, the character will have a consistent demeanor without being focused on a single trait. This will allow the character to still have some unpredictable behavior, and since the backstory isn’t explicitly spelled out, it allows you to keep some of the motivation as subtext.

This balance brings out some of the positives of the caricature while maintaining the depth and richness of a true character.

Of course, the main thing I’ve learned over the years is that everyone seems to have a different method for everything. What one person thinks is the only method could not work at all for another person. That’s why I’ll keep reading and exploring to find my own way.


Surviving Upper Division Math

It’s that time of the year. Classes are starting up. You’re nervous and excited to be taking some of your first “real” math classes called things like “Abstract Algebra” or “Real Anaylsis” or “Topology.”

It goes well for the first few weeks as the professor reviews some stuff and gets everyone on the same page. You do the homework and seem to be understanding.

Then, all of a sudden, you find yourself sitting there, watching an hour-long proof of a theorem you can’t even remember the statement of, using techniques you’ve never heard of.

You panic. Is this going to be on the test?

We’ve all been there.

I’ve been that teacher, I’m sad to say, where it’s perfectly clear in my head that the students are not supposed to regurgitate any of this. The proof is merely there for rigor and exposure to some ideas. It’s clear in my head which ideas are the key ones, though I maybe forgot to point it out carefully.

It’s a daunting situation for the best students in the class and a downright nightmare for the weaker ones.

Then it gets worse. Once your eyes glaze over that first time, it seems the class gets more and more abstract as the weeks go by, filled with more and more of these insanely long proofs and no examples to illuminate the ideas.

Here’s some advice for surviving these upper division math classes. I’m sure people told me this dozens of times, but I tended to ignore it. I only learned how effective it was when I got to grad school.

Disclaimer: Everyone is different. Do what works for you. This worked for me and may only end up frustrating someone with a different learning style.

Tip Summary: Examples, examples, examples!

I used to think examples were something given in a textbook to help me work the problems. They gave me a model of how to do things.

What I didn’t realize was that examples are how you’re going to remember everything: proofs, theorems, concepts, problems, and so on.

Every time you come to a major theorem, write out the converse, inverse, switch some quantifiers, remove hypotheses, weaken hyphotheses, strengthen conclusions, and whatever you can think of to mess it up.

When you do this you’ll produce a bunch of propositions that are false! Now come up with examples to show they’re false (and get away from that textbook when you do this!). Maybe some rearrangement of the theorem turns out to be true, and so you can’t figure out a counterexample.

This is good, too! I cannot overstate how much you will drill into your memory by merely trying unsuccessfully to find a counterexample to a true statement. You’ll start to understand and see why it’s probably true, which will help you follow along to the proof.

As someone who has taught these classes, I assure you that a huge amount of problems students have on a test would be solved by doing this. Students try to memorize too much, and then when they get to a test, they start to question: was that a “for every” or “there exists?” Does the theorem go this way or that?

You must make up your own examples, so when you have a question like that, the answer comes immediately. It’s so easy to forget the tiniest little hypothesis under pressure.

It’s astounding the number of times I’ve seen someone get to a point in a proof where it looks like everything is in place, but it’s not. Say you’re at a step where f: X\to Y is a continuous map of topological spaces, and X is connected. You realize you can finish the proof if Y is connected.

You “remember” this is a theorem from the book! You’re done!

Woops. It turns out that f has to be surjective to make that true.

But now imagine, before the test, you read that theorem and you thought: what’s a counterexample if I remove the surjective hypothesis?

The example you came up with was so easy and took no time at all. It’s f: [0,1] \to \{0\} \cup \{1\} given by f(x) = 1. This example being in your head saves you from bombing that question.

If you just try to memorize the examples in the book or that the professor gives you, that’s just more memorization, and you could run into trouble. By going through the effort of making your own examples, you’ll have the confidence and understanding to do it again in a difficult situation.

A lesser talked about benefit is that having a bunch of examples that you understand gives you something concrete to think about when watching these proofs. So when the epsilons and deltas and neighborhoods of functions and uniform convergence and on and on start to make your eyes glaze over, you can picture the examples you’ve already constructed.

Instead of thinking in abstract generality, you can think: why does that step of the proof work or not work if f_n(x) = x^n?

Lastly, half the problems on undergraduate exams are going to be examples. So, if you already know them, you can spend all your time on the “harder” problems.

Other Tip: Partial credit is riskier than in lower division classes.

There’s this thing that a professor will never tell you, but it’s true: saying wrong things on a test is worse than saying nothing at all.

Let me disclaimer again. Being wrong and confused is soooo important to the process of learning math. You have to be unafraid to try things out on homework and quizzes and tests and office hours and on your own.

Then you have to learn why you were wrong. When you’re wrong, make more examples!

Knowing a bunch of examples will make it almost impossible for you to say something wrong.

Here’s the thing. There comes a point every semester where the professor has to make a judgment call on how much you understand. If they know what they’re doing, they’ll wait until the final exam.

The student that spews out a bunch of stuff in the hopes of partial credit is likely to say something wrong. When we’re grading and see something wrong (like misremembering that theorem above), a red flag goes off: this student doesn’t understand that concept.

A student that writes nothing on a problem or only a very small amount that is totally correct will be seen as superior. This is because it’s okay to not be able to do a problem if you understand you didn’t know how to do it. That’s a way to demonstrate you’re understanding. In other words: know what you don’t know.

Now, you shouldn’t be afraid to try, and this is why the first tip is so much more important than this other tip (and will often depend on the instructor/class).

And the best way to avoid using a “theorem” that’s “obviously wrong” is to test any theorem you quote against your arsenal of examples. As you practice this, it will become second-nature and make all of these classes far, far easier.