how long does math 55 homework take

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What We Talk About When We Talk About Math 55

On a Wednesday morning on the fifth floor of the Science Center, 20 or so students mill about while Professor Joseph D. Harris ’72 draws on the chalkboard. Harris has sketched a “three-sheeted covering space,” a looping squiggle.

Harris spends the next hour and 15 minutes of Math 55B: “Studies in Real and Complex analysis” lecturing on the classification of covering spaces. He draws arrows and circles and greek letters on the board. He discusses topologies and neighborhoods and endpoints.

Thirty minutes in, Harris pauses drawing his diagrams, which by now take up the entire chalkboard, and turns to face the class. “Now we get to have some fun,” he says.

Welcome to Math 55, the undergraduate course surrounded by what is perhaps the most intrigue and infamy of any class at Harvard.

In 2006, the Crimson published an article on Math 55 called “ Burden of Proof ,” which characterizes Math 55 as a sort of “cult” or a mathematical bootcamp rather than a freshman lecture course. According to the article, students referred to the room where 55-ers completed their problem sets — the basement of Thayer Hall — as “the war room.”

Even outside of Harvard, Math 55 has garnered outsized attention. The course has its own Wikipedia page and has inspired many TikToks, one of which has racked up over 418,000 views.

And for a while, it seemed like the Math Department itself was in on the mythologization of the course. Until at least 2017, the Math Department’s website described Math 55 as “probably the most difficult undergraduate math class in the country.”

But today, the math department takes a different approach to how they present the course. The very first search result that comes up for “Math 55” is now an article published by the department last year titled, “Demystifying Math 55.”

The article debunks six purported myths about the class, including that “Math 55 is only for high school math geniuses” and that “homework takes between 24 to 60 hours per week.”

Denis Auroux, a professor who has previously taught Math 55, says in the article, “Our slogan is, if you’re reasonably good at math, you love it, and you have lots of time to devote to it, then Math 55 is completely fine for you.”

So, what changed?

When Preston C. Bushnell ’26 mentions that he takes Math 55, “people are shocked, they're absolutely floored,” he says.

“My opinion is, I think it’s a hard class, but it’s also still just a class,” he says. “And there are plenty of other hard classes.”

Zoe Z. Shleifer ’26, another current Math 55 student, also doesn’t get the hype.

“It’s fun,” she says. “It’s just like any other class. You know, we go to lecture, and then we leave lecture, and then we do the problem set.”

Math 55 is officially composed of two parts, Math 55A: “Studies in Algebra and Group Theory” and Math 55B: “Studies in Real and Complex analysis.” The department classifies the class alongside Math 22 and 25 as one of “three introductory courses for people with strong math interests coming into Harvard.”

Even still, Math 55 differs from other math courses in its expectations for students’ prior coursework. Although preparation varies, all Math 55 students I spoke to had extensive experience with college- level mathematics.

Shleifer took calculus as a freshman in high school. As a sophomore, she pursued college-level math, taking a real analysis and linear algebra course at Harvard Extension School. Bushnell had a similar path, taking calculus as a freshman and ultimately switching to math courses at a nearby college for his last two years of high school.

Still, Ashvin A. Swaminathan ’17, a former Math 55 student and current Benjamin Pierce Fellow in the Math Department, cautions against the common conception that 55-ers are more intelligent than other Harvard students..

“Most of the students who take 55 are not geniuses,” he says. “They’'re just interested in math, and they want to learn it at a high level.”

It might seem like the Math Department’s efforts to transform Math 55 are relatively recent. But in fact, they stretch back decades.

The course was originally created in 1960, with the title “Advanced Calculus.” The class continued until 1983, when the Math Department decided to get rid of the course. Math 55 became Math 25, which then-department chair David B. Mumford ’57 described as having “the rigor without the abstraction, matching the preparation of students much better.”

But in 1991, the department backtracked, adding Math 55 back to the course catalog. Students who wished to take 55 had to first enroll in Math 25 and “after two weeks, some students will be invited to join Mathematics 55.” Eventually, the Math Department again allowed students to start in Math 55 from their first day at Harvard.

Until recently, Math 55 professors had great latitude with how they taught the course. There was no formalized syllabus passed down year-to-year, professor-to-professor.

But that’s changed. “We wanted to avoid a situation where some students felt excluded because of the way the course was taught a particular year,” Harris says.

Anastasia Yefremova, the Math Department’s publications coordinator who wrote “Demystifying Math 55,” says that the department chair brought the article idea to her attention.

“I think there are very concerted efforts on part of CAs and faculty — especially these days — to address some of the issues that people may have had with Math 55,” Yefremova says.

“To be as inclusive as possible is one of the things I was told before I walked in the classroom the first day,” says Harris, who first taught the course in 2018. “I think we’ve been pretty successful in that.”

When he arrived at Harvard earlier this year, Jonah C. Karafiol ’26, a Crimson News editor, planned to concentrate in math. He had a background in math research and hoped to explore it further. But his father, a high school math teacher, advised him not to take Math 55.

“He told me that if I took it, I would leave the course hating math,” Karafiol says.

Even so, Karafiol wanted to push himself by taking the course.

“There was a sense of imposter syndrome,” he says. “I thought if I took Math 55, I would be proving to myself I belong here.”

Karafiol says he spent about 30 hours each week completing problem sets, which limited his time for other classes that he enjoyed and kept his social circle to mostly other 55-ers. He felt like a “proof factory.”

“You’ll be brushing your teeth, or showering or literally you’re going about any task throughout the day, and it’s kind of on your mind,” he says about Math 55 coursework.

Ivan O.A. Specht ’24, who took the course in 2019, noticed a similar trend among his classmates.

Math 55 students, he says, “were generally not terribly involved in extracurriculars because so much of their time and so much of their social interactions were being eaten up by Math 55.”

Like Karafiol, Swaminathan was cautioned against taking the course while he was in high school.

He’d read The Crimson’s “Burden of Proof” and reached out to one of the students mentioned in it, Daniel A. Litt ’10. The two met for coffee.

“In no uncertain terms, he told me that I should never take this course,” Swaminathan said. “It would ruin my interest in math, and there’s so many better ways to approach the math concentration.”

Litt, who now works as a math professor at The University of Toronto, doesn’t recall giving Swaminathan that advice, but says he trusts Swaminathan’s account.

When Litt took 55, he “loved it,” but he realizes that the course is “definitely not for everyone.” He believes that “it should be somehow clear that that’s okay.”

Swaminathan is grateful he eschewed Litt’s advice and took the course. He considers Math 55 to have “had the most significant impact” on him “as a student and as a mathematician.” But he also noticed it wasn’t a pleasant experience for everyone.

“I never felt like I was at war in that class,” Swaminathan says. “Although I do know some students who did feel that way.”

Litt compares the rigor and speed of the course to “drinking from a firehose.”

“You’re not more advanced if you’re drinking from a firehose,” he says. “It’s not clear to me that in the end, everyone who drinks from a firehose actually ends up learning more math than people who take Math 25.”

After a semester in the course, Karafiol is no longer planning to concentrate in math. In fact, he is not taking any math courses this semester.

“It didn’t make me hate it,” Karafiol says regarding how the course changed his feelings about the subject. “But I don’t love it as much now as I did.”

“Part of me does wonder,” he says, “would I still be a math concentrator if I’d taken Math 22 or Math 25?”

When Daniel Litt took Math 55 in 2006, there were no female students. During Swaminathan’s time in the course in 2013, there were only two. From 2012 to 2017, less than seven percent of Math 55 students were female.

Now, Shleifer says that there are “at least nine or ten” female students out of the 45 students enrolled in the course.

Dora Woodruff ’24, who took Math 55 during her freshman year and worked as a course assistant as a sophomore, says that the gender imbalance in the course can be “discouraging and intimidating for people who are just starting college.”

While working as a course assistant for the class, Woodruff received comments saying some non-male students felt uncomfortable asking certain questions during office hours for fear that they would seem less prepared or capable. To counteract this, she started holding extra “chill, informal” office hours aimed at non-male students.

But not all non-male students find the majority-male environment unpleasant.

When I ask Shleifer how she feels about Math 55’s male-dominated culture, she says “It’s kind of like asking a fish if there’s anything they don’t like about water.”

“I really don’t have any complaints,” she says about the majority-male environment. “I definitely have critiques of math culture, but I don’t think they’re about maleness.”

In an email, Yefremova wrote, “The math department has been working hard to foster a more inclusive culture around Math 55.”

“The overall feedback we have received from students about the last few iterations of the course suggests that this is beginning to bear fruit,” she continued, referencing reviews for Math 55’s Q guide reviews, which are overwhelmingly positive.

“The math is beautiful and it goes at a fast pace, but not too fast to build good intuition for what is going on,” a student writes. “There is a large team of CAs to help out as well and office hours almost every day, so you won’t be left behind.”

Still, the reason for the course’s low female enrollment isn’t entirely clear. Woodruff wonders if the course’s lore is a potential deterrent for female students. She believes its notoriety may “discourage people who could do very well and get a lot out of the course from taking it.”

But she also thinks some of it is beyond Harvard’s control, stemming from a “pipeline issue.”

“In general, fewer women than men come into college intending to major in math,” she says. “And there isn’t as much that Harvard can do about that.”

But Swaminathan, who serves on the department’s Diversity Committee, says he doesn’t “think it’s just a pipeline problem.”

“I think there’s a reputation of the department being sort of exclusionary,” says Swaminathan. “Getting rid of that reputation is going to take time. It’s going to take a lot of work. It’s going to take a lot of listening. But we’re well on our way to achieving that.”

Part of these changes, Swaminathan reflects, might be altering Math 55’s legendary reputation.

“As math teachers, our objective is to bring mathematics to a wider audience, not to preserve some sort of mysticism about the material that we teach,” he says. “If our goal is to teach more people math and broaden appreciation for math, then these changes are good.”

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Students spend three times longer on homework than average, survey reveals

Sonya Kulkarni and Pallavi Gorantla | Jan 9, 2022

Graphic by Sonya Kulkarni

The National Education Association and the National Parent Teacher Association have suggested that a healthy number of hours that students should be spending can be determined by the “10-minute rule.” This means that each grade level should have a maximum homework time incrementing by 10 minutes depending on their grade level (for instance, ninth-graders would have 90 minutes of homework, 10th-graders should have 100 minutes, and so on).

As ‘finals week’ rapidly approaches, students not only devote effort to attaining their desired exam scores but make a last attempt to keep or change the grade they have for semester one by making up homework assignments.

High schoolers reported doing an average of 2.7 hours of homework per weeknight, according to a study by the Washington Post from 2018 to 2020 of over 50,000 individuals. A survey of approximately 200 Bellaire High School students revealed that some students spend over three times this number.

The demographics of this survey included 34 freshmen, 43 sophomores, 54 juniors and 54 seniors on average.

When asked how many hours students spent on homework in a day on average, answers ranged from zero to more than nine with an average of about four hours. In contrast, polled students said that about one hour of homework would constitute a healthy number of hours.

Junior Claire Zhang said she feels academically pressured in her AP schedule, but not necessarily by the classes.

“The class environment in AP classes can feel pressuring because everyone is always working hard and it makes it difficult to keep up sometimes.” Zhang said.

A total of 93 students reported that the minimum grade they would be satisfied with receiving in a class would be an A. This was followed by 81 students, who responded that a B would be the minimum acceptable grade. 19 students responded with a C and four responded with a D.

“I am happy with the classes I take, but sometimes it can be very stressful to try to keep up,” freshman Allyson Nguyen said. “I feel academically pressured to keep an A in my classes.”

Up to 152 students said that grades are extremely important to them, while 32 said they generally are more apathetic about their academic performance.

Last year, nine valedictorians graduated from Bellaire. They each achieved a grade point average of 5.0. HISD has never seen this amount of valedictorians in one school, and as of now there are 14 valedictorians.

“I feel that it does degrade the title of valedictorian because as long as a student knows how to plan their schedule accordingly and make good grades in the classes, then anyone can be valedictorian,” Zhang said.

Bellaire offers classes like physical education and health in the summer. These summer classes allow students to skip the 4.0 class and not put it on their transcript. Some electives also have a 5.0 grade point average like debate.

Close to 200 students were polled about Bellaire having multiple valedictorians. They primarily answered that they were in favor of Bellaire having multiple valedictorians, which has recently attracted significant acclaim .

Senior Katherine Chen is one of the 14 valedictorians graduating this year and said that she views the class of 2022 as having an extraordinary amount of extremely hardworking individuals.

“I think it was expected since freshman year since most of us knew about the others and were just focused on doing our personal best,” Chen said.

Chen said that each valedictorian achieved the honor on their own and deserves it.

“I’m honestly very happy for the other valedictorians and happy that Bellaire is such a good school,” Chen said. “I don’t feel any less special with 13 other valedictorians.”

Nguyen said that having multiple valedictorians shows just how competitive the school is.

“It’s impressive, yet scary to think about competing against my classmates,” Nguyen said.

Offering 30 AP classes and boasting a significant number of merit-based scholars Bellaire can be considered a competitive school.

“I feel academically challenged but not pressured,” Chen said. “Every class I take helps push me beyond my comfort zone but is not too much to handle.”

Students have the opportunity to have off-periods if they’ve met all their credits and are able to maintain a high level of academic performance. But for freshmen like Nguyen, off periods are considered a privilege. Nguyen said she usually has an hour to five hours worth of work everyday.

“Depending on the day, there can be a lot of work, especially with extra curriculars,” Nguyen said. “Although, I am a freshman, so I feel like it’s not as bad in comparison to higher grades.”

According to the survey of Bellaire students, when asked to evaluate their agreement with the statement “students who get better grades tend to be smarter overall than students who get worse grades,” responders largely disagreed.

Zhang said that for students on the cusp of applying to college, it can sometimes be hard to ignore the mental pressure to attain good grades.

“As a junior, it’s really easy to get extremely anxious about your GPA,” Zhang said. “It’s also a very common but toxic practice to determine your self-worth through your grades but I think that we just need to remember that our mental health should also come first. Sometimes, it’s just not the right day for everyone and one test doesn’t determine our smartness.”

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Comments (8).

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Anonymous • Jul 16, 2024 at 3:27 pm

didnt realy help

Anonymous • Nov 21, 2023 at 10:32 am

It’s not really helping me understand how much.

josh • May 9, 2023 at 9:58 am

Kassie • May 6, 2022 at 12:29 pm

Im using this for an English report. This is great because on of my sources needed to be from another student. Homework drives me insane. Im glad this is very updated too!!

Kaylee Swaim • Jan 25, 2023 at 9:21 pm

I am also using this for an English report. I have to do an argumentative essay about banning homework in schools and this helps sooo much!

Izzy McAvaney • Mar 15, 2023 at 6:43 pm

I am ALSO using this for an English report on cutting down school days, homework drives me insane!!

E. Elliott • Apr 25, 2022 at 6:42 pm

I’m from Louisiana and am actually using this for an English Essay thanks for the information it was very informative.

Nabila Wilson • Jan 10, 2022 at 6:56 pm

Interesting with the polls! I didn’t realize about 14 valedictorians, that’s crazy.

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A rigorous introduction to abstract algebra, including group theory and linear algebra. This course covers the equivalent of Mathematics 25a and Mathematics 122, and prepares students for Mathematics 123 and other advanced courses in number theory and algebra. (A course in analysis such as Mathematics 25b or 55b is recommended for Spring semester.)

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How Much Time Should Be Spent on Homework?

At the elementary level homework should be brief, at your child’s ability level and involve frequent, voluntary and high interest activities. Young students require high levels of feedback and/or supervision to help them complete assignments correctly. Accurate homework completion is influenced by your child’s ability, the difficulty of the task, and the amount of feedback your child receives. When assigning homework, your child’s teachers may struggle to create a balance at this age between ability, task difficulty and feedback. Unfortunately, there are no simple guiding principles.

We can assure you, however, that your input and feedback on a nightly basis is an essential component in helping your child benefit from the homework experience.

What is the recommended time in elementary school?

In first through third grade, students should receive one to three assignments per week, taking them no more than fifteen to twenty minutes. In fourth through sixth grade, students should receive two to four assignments per week, lasting between fifteen and forty-five minutes. At this age, the primarily goal of homework is to help your child develop the independent work and learning skills that will become critical in the higher grades. In the upper grades, the more time spent on homework the greater the achievement gains.

What is the recommended time in middle and high school?

For students in middle and high school grades there are greater overall benefits from time engaged in practicing and thinking about school work. These benefits do not appear to depend as much upon immediate supervision or feedback as they do for elementary students. In seventh through ninth grade we recommend students receive three to five sets of assignments per week, lasting between forty-five and seventy-five minutes per set. In high school students will receive four to five sets of homework per week, taking them between seventy-five and 150 minutes per set to complete.

As children progress through school, homework and the amount of time engaged in homework increases in importance. Due to the significance of homework at the older age levels, it is not surprising that there is more homework assigned. Furthermore, homework is always assigned in college preparatory classes and assigned at least three quarters of the time in special education and vocational training classes. Thus at any age, homework may indicate our academic expectations of children.

Regardless of the amount of homework assigned, many students unsuccessful or struggling in school spend less rather than more time engaged in homework. It is not surprising that students spending less time completing homework may eventually not achieve as consistently as those who complete their homework.

Does this mean that time devoted to homework is the key component necessary for achievement?

We are not completely certain. Some American educators have concluded that if students in America spent as much time doing homework as students in Asian countries they might perform academically as well. It is tempting to assume such a cause and effect relationship.

However, this relationship appears to be an overly simple conclusion. We know that homework is important as one of several influential factors in school success. However, other variables, including student ability, achievement, motivation and teaching quality influence the time students spend with homework tasks. Many students and their parents have told us they experience less difficulty being motivated and completing homework in classes in which they enjoyed the subject, the instruction, the assignments and the teachers.

The benefits from homework are the greatest for students completing the most homework and doing so correctly. Thus, students who devote time to homework are probably on a path to improved achievement. This path also includes higher quality instruction, greater achievement motivation and better skill levels.

Authors: Dr. Sam Goldstein and Dr. Sydney Zentall

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How Much Time Should Be Spent on Homework Based on Grade?

• 18 July 2020
• Posted by: ryan
• Category: Tutoring

A common question that parents always ask is, “How much time should my child dedicate to homework every day?” It’s not an easy question to answer. As we all know, every student learns differently from each other. While some kids do, substantially, better in school, by completing one hour of homework every day. There might be some others, who require two hours of homework, but only see a slight improvement in their grades.

To get to the bottom of this, we went to the experts for the answers! So here’s a break down of how much time your child should spend on homework according to their grade.

What is The Recommended Homework Time in Elementary School?

So before we give you a solid figure. We took a look at the results of a May 2012 study from the Los Angeles Unified School District . (Figure 1 below)

If your child is starting out in kindergarten and they receive some basic worksheets to complete for homework, the standard time they should spend on completing homework is 10 minutes per night.

Keep in mind, kindergarten childen might have shorter attention spans, than older kids, and might need a few intervals in between to complete their homework. So let them do it for 5 minutes, then take a 5 minute break, then continue for another 5 minutes to complete.

Usually,  Grade 1 – 3 students receive one to three homework assignments per week. They suggest that your child spend at least 20 – 30 minutes per night on homework.

Grade 4 – 5 students who receive two to four assignments per week, should focus between 40 – 50 minutes on completing each assignment.

What is The Recommended Homework Time in Middle and High school?

As your child enters middle and high school, naturally, their home work time will increase. As subjects get harder and more information needs to be retained for exams, more time is needed to practice. Here are the home work time estimations for older students from the Los Angeles Unified School District . (Figure 2 below)

Students in middle school are from Grades 6 – 8.  As class subjects require more attention and practice, middle school students get assigned three to five sets of assignments per week. We recommend that your child spend between 45 – 75 minutes per night.

Once your child is in highschool, Grade 9 – 12 students usually receive four to five sets of homework per week. According to Figure 2,  high school students should focus about 25-30 minutes on each subject.

For example, if your child is in Grade 10 and has a Math and English assignment to do for homework, they should spend at least 30 minutes on English and 30 minutes on Math. If they take one or two short breaks, it works out to be 75 – 150 minutes per set to complete both assignments.

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How long to do math each day? [closed]

I have seen some posts math.SE (mkko's answer) indicating that it is the norm for (undergrad?) math majors to study 70-80 hours per week. I'm a little bit shocked by that. For some background on me, I'm not very advanced (only finished calculus 1-3 and taking my first DE class). However, personally if I am trying to solve a tough problem or prove a theorem, I can't work on it for more than an hour at a time without killing my ability to think creatively, which is the most important skill we mathematicians should cultivate, right? After about one or two hours, I can't engage the theorem in deep thought, so trying proving it becomes more of just an unproductive guessing game. If I just come back the next day, I feel like I have digested the problem much better and gained more perspective. When there is new material, I do spend multiple hours just trying to learn the material and become familiar with all the definitions and intricacies. However, I feel that it's most important to focus on problem-solving and proofs.

So my question is, what do these 80 hours a week consist of for typical math students? Is most of that time spent on trying to just learn the material? Is it actually spent on cultivating problem-solving skills but I have just have a really low tolerance for focusing? Is spending no more than 1-2 hours per day on the same problem optimal, but just a luxury that students can't afford once they reach a certain level?

• soft-question

• 1 $\begingroup$ Depends on what you mean by studying. Is simply thinking about a problem or theorem studying? Do I need to read a book to be studying? Do I need to write things? It's not really a clear boundary. But yeah, 70-80 hours per week thinking about math problems is not weird, and I guess can be achieved by most people, for you can do it anywhere. $\endgroup$ –  user304329 Commented Feb 23, 2016 at 8:10
• $\begingroup$ Based on this , the expected workload for students in Europe seems to be 60 ECTS, each credit corresponds to 25-30 hours. This means 1500-1800 hours per academic year. (This includes attending lectures, recitations and individual work.) $\endgroup$ –  Martin Sleziak Commented Feb 23, 2016 at 11:37
• 2 $\begingroup$ 80 hours per week i.e. 10 hours a day every day including sunday? That seems impossibly high, I would expect anyone who puts in this much work to be basically obsessed by math (and probably outstandingly good at it, waaaaay above the undergraduate requirements. Or graduates for what matters). 20 hours per week (excluding lessons) are already very good, in my opinion. Of course there will be a higher workload before finals but it's normal. (And honestly if I have 6-8 hours of class it's highly unlikely I'll keep studying at home) $\endgroup$ –  Ant Commented Feb 23, 2016 at 14:42
• 4 $\begingroup$ In six weeks, I aim for about $10!$ seconds... $\endgroup$ –  Benjamin Dickman Commented Feb 23, 2016 at 18:09
• $\begingroup$ @BenjaminDickman or $\pi E7$ seconds per year $\endgroup$ –  Hagen von Eitzen Commented Feb 23, 2016 at 21:31

6 Answers 6

I doubt anyone needs to put in 70-80 hours a week to get a math BA/BS, some people just really really like math and can't help but study it all the time.

If you find yourself burning out after an hour maybe try a different strategy. I put in probably about 50 hours a week and do micro breaks. Basically periods of deep concentrated study/thought, broken up by 10-15 minute breaks whenever my brain needs a quick rest, usually I'll check reddit or Math.SE or watch a little bit of some show.

Sometimes if I'm trying to understand something especially abstract or conceptually deep I'll go for a walk around the block and just ponder. These walks can be very enjoyable.

At your level you really just need to do lots and lots of exercises, however once you've done a couple thousand proofs, and gotten through the standard graduate level material, then it's more like exploration, and proofs and rigor become less important.

Everyone's different, and the study strategies which work for me may not work for you, but this is what works for me.

• 1 $\begingroup$ By doing a lot of exercises, do you mean for example solving the same type of linear DE 100 times? Is it so that I learn certain techniques well enough so that I can recognize their application when I need it in a proof later on? Is it so that I work on other skills such as recognizing how to factor increasingly complex expressions? $\endgroup$ –  Ovi Commented Feb 23, 2016 at 8:18
• 2 $\begingroup$ @Ovi well no I wouldn't solve the same type of problem a bunch of times. I guess it depends on what type of math you're interested in, but if you do a couple linear DE exercises, then even if you forget exactly how to do them later on you can just look back and refresh your memory. Exercises are there to check understanding and to give you a reason to struggle with the material, failing to solve problems is where most of your learning comes from, since you will have charted out many dead ends which later on may be useful, when it becomes rote it's time to move on to something more difficult. $\endgroup$ –  Set Commented Feb 23, 2016 at 8:24
• $\begingroup$ I'm in my undergrad, so should I learn programming along with math? Because I am trying to be better at math problem solving. $\endgroup$ –  Lucky Chouhan Commented Feb 4, 2023 at 16:19

Since this is kind of a subjective thing, I'll pitch my own experience.

I'm a senior undergraduate taking two graduate level classes. Typically, I work for about 3-4 hours a day books open, laptop open, pen and paper all over the place, basically working on either my thesis, or homework problems for my graduate classes. I do this basically every day, sometimes a little more, sometimes a little less. I have a little bit of ADHD, so I take short, but frequent breaks to do something else, stretch my legs, get a drink, or just scroll Facebook for a minute or two. Off the bat, that's somewhere between 20 and 30 hours each week. I think this is an underestimate for what typical people in two graduate classes go through - last semester I worked much harder than this semester. Maybe as much as 6 hours a day.

But this is not really what I would call my 'studying.' I think that, for me, studying is a much broader thing. I spend all my time walking to and from other lectures thinking informally about problems, homework or not. These comprise another hour a day, again sometimes a little more if I have to walk to my tutoring job that day. So here are maybe 5 or 10 more hours.

But I also do a good number of problems completely separate from any class I'm in. I have a very large reading list for mathematics, that I hope to someday have made real progress on. I'm at that stage in my mathematical development where everything I come across is really cool, and no matter how cursory I need a result, I really want to investigate it and internalize its contents. Right now, I'm working through Sets for Mathematics by Lawvere, and Basic Algebraic Geometry I, by Shafarevich, each with a separate buddy.

These serve purposes for me. I've been seeing a lot of category flavored arguments lately, and so I've been trying to sink my teeth into that subject. I also am really interested in geometry, but have never really looked at anything algebraic outside a first course in algebra. I probably spend about two hours a day reading or working on problems of these types. Sometimes I just skim these books really quickly and make notes on my PC about things that catch my eye. Other times these problems look like my homework problems - I take them very seriously.

I figure these are 15 to 20 hours more each week then.

And of course, during exam season, I spend less time on these side problems (but still not $0$, I find it helpful to take my mind off my courses just like everyone else), and spend a lot more studying. I can't estimate this time well - for my topology class last semester, I probably was studying all day every day for at least a week and a half before the exam, punctuated with my usual breaks.

So... typically it looks like I work on something mathematical either on paper or in my head maybe 7 hours a day, on a typical non-stressful day. That's about 50 hours a week.

Now, if you count the time that I'm in lecture or meeting with my adviser, we can add another 10 hours, so we're up to 60.

So I think 70 is probably a lot. I consider myself a pretty driven student - I really want to be successful at mathematics, even though I'm probably not very good at it compared to my peers. I guess some people probably could do it though.

One last comment, regarding optimal studying. I think that it is not about how much time to spend thinking on a problem. Some days I think all day about the problem, some days not at all. There's at least one or two problems that float in my head for a long time, because I don't have the right notions yet to digest them. But I still think about them and try to connect up what I've been doing lately with them... see if I can have any insight, and this has been fruitful in the past. More importantly than spending any amount of time with material is avoiding burnout and frustration. If you are happy with the state of your work, and you feel you can, raise the bar. If not, it is better to stay happy with mathematics and yourself then to struggle and lose interest. That's why I like to tinker with a bunch of different things at once - I can always put down what I'm doing, and look at something else fascinating too!

Too long for a comment. +1 for recognizing the value of alternating hard thinking with letting your subconscious do the work. This from Poincaré via Hadamard and wikipedia ( https://en.wikipedia.org/wiki/Henri_Poincar%C3%A9#Philosophy )

Poincaré's famous lectures before the Société de Psychologie in Paris (published as Science and Hypothesis, The Value of Science, and Science and Method) were cited by Jacques Hadamard as the source for the idea that creativity and invention consist of two mental stages, first random combinations of possible solutions to a problem, followed by a critical evaluation.[64] Although he most often spoke of a deterministic universe, Poincaré said that the subconscious generation of new possibilities involves chance. It is certain that the combinations which present themselves to the mind in a kind of sudden illumination after a somewhat prolonged period of unconscious work are generally useful and fruitful combinations... all the combinations are formed as a result of the automatic action of the subliminal ego, but those only which are interesting find their way into the field of consciousness... A few only are harmonious, and consequently at once useful and beautiful, and they will be capable of affecting the geometrician's special sensibility I have been speaking of; which, once aroused, will direct our attention upon them, and will thus give them the opportunity of becoming conscious... In the subliminal ego, on the contrary, there reigns what I would call liberty, if one could give this name to the mere absence of discipline and to disorder born of chance.

My personal experience is that breaks are really important. As you mentioned it's really difficult to stay focused for a certain amount of time, especially when someone is engaging harder problems which require a high amount of focus. 70 to 80 hours seems unrealistic if you only count the time you are indeed studying. If it's the time spent at the university it's still very unlikely this is right, or at least I don't see how one is able to study efficiently for than 10 hours a day while doing math.

Speaking as an undergrad from the UK, I am assuming it is all fairly similar, throughout the week I try and get to the library at about $9$ and leave about $7$, then SWITCH OFF(or try). Then on the weekend I will do a little bit,maybe just some thinking or tidying up some notes.

This method,so far, has worked for me (Over 90% average... so for.).

If you do too much you WILL get burned out, you will then need to shut off for a while which is a nightmare.

So I think, if you count the "thinking about problems" then 70 hours is not that far away.

Thoth nailed it though.

In the book Outliers, author Malcolm Gladwell says that it takes roughly ten thousand hours of practice to achieve mastery in a field.

If you want to (Bachelor + Graduate + PhD) this makes roughly 5/6 hours per day. That's it. I find this estimate quite reasonable. If you want to accomplish the same result before graduation You have to switch to the 10 hours mode.

• 1 $\begingroup$ the thing is that 10 hours don't equal 10 hours depending on the way they are distributed. $\endgroup$ –  noctusraid Commented Feb 23, 2016 at 22:50
• $\begingroup$ My point is that 10 hours is not in fact needed $\endgroup$ –  Dac0 Commented Feb 24, 2016 at 5:52

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Calculating Your Solar Payback Period: When Will You Break Even?

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Solar panels  are an expensive investment. When you decide to go solar, you are either committing to a significant upfront cost of tens of thousands of dollars or a long-term plan through several years of monthly payments. The breakeven point, or payback period, is the time it takes to recoup the cost from the initial investment. Once that time is up, the real savings start.      

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Jamie Haenggi , president of ADT Solar, told CNET an average payback period in the US is six to 12 years, with most households leaning closer to the latter. Like Jones-Albertus, she emphasized that it's a moving target.

"People are reluctant to say, 'Well here's the payback,' because the energy market has been so volatile," Haenggi said. 

How to calculate your solar payback period

If you want to get a rough idea of your potential solar payback period, here's a way to do it. Keep in mind, that you'll want to consult the experts (read: solar installers) to make sure you have accurate numbers here. This can help you get an idea:

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• Divide the net cost of the system by the annual bill savings.
• The number you end up with is the number of years it will take for your panels to "pay for themselves."

Here's another look at the formula:  (Total solar system costs - rebates) / Electricity bill savings per year = Payback period in years

In practice, here's what that could look like: Let's say the total system cost for your home is $25,000. You know you qualify for $10,000 in incentives, so now the net cost is $15,000. You also know the panels will help you save about $1,500 a year on electricity bills. So, $15,000 divided by $1,500 is 10. That means your solar payback period is 10 years.

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No two solar systems are the same, and that means no two solar payback periods are the same, either. "It seems like an easy answer, but it's more complicated," Haenggi said. 

Calculating your potential payback period will depend on a lot of variables.

Total solar system cost

The more you pay for your system , the longer it's going to take to recoup your costs. Solar systems can range in price from a few thousand dollars to tens of thousands, depending on where you live, your electricity needs and what type of system you choose to install. A solar battery could easily increase the cost of your system by $10,000 or more. It goes without saying: The steeper the price, the longer the payback period.

Here's a look at how much solar panel systems cost on average for most states, according to data from FindEnergy.com.

Incentives and tax credits

Once you know the total cost of your solar system, you also have to factor in any state or federal rebates you might qualify for. The federal residential clean energy credit , for example, gives you up to 30% back. Your state might also have additional incentives. Those credits can lop off a significant chunk of the money you pay for solar panels, making your payback period shorter.

Your home's energy consumption

Sometimes rooftop solar can completely cover your electricity needs -- reducing your utility bill to $0 -- and sometimes it only covers a portion of it. If you consume a lot of electricity, solar might only translate to a small reduction in your electricity costs, which means it could take longer for you to see a return on your investment. That's why it's important to think about your home's energy efficiency before you consider solar panels -- you can save money on energy and get a smaller solar panel system .

Electricity production of your solar system

You probably never thought much about your roof, but it makes a big difference in how your solar investment will play out. If your roof has room for lots of panels that soak in the sun all day , you'll produce a ton of electricity and see a quicker payback. If you live on a shady lot, and your panels' production is more intermittent, you won't see a payback quite as quickly.

Cost of electricity and rate of increase

This is a huge, but sometimes overlooked, factor in the solar payback period. Basically, the higher the electricity rates where you live, the more lucrative solar can be for you. As utility rates increase, you save more money by relying on your solar panels instead of drawing power from the grid. 

Solar panels and electric vehicle charging could be a useful pairing.

Why knowing your payback period is important 

Now you have your solar payback period. How does that factor into your decision?

"It depends on what is motivating the household to make the decision to [install] solar," Jones-Albertus said. Maybe you just want to help the environment and aren't worried about the costs, but "folks are interested in the resilience aspect and the economic aspect, as well," she said.

If you are interested in the financial aspect, then the payback period is an important number for your decision-making. A payback period of around 10 years is pretty average, and could end up being a solid investment, Haenggi said.

Again, it depends on your goals and your comfort level. If you're planning on moving or selling your home in the near term, for example, that changes the calculus. You might not be in the home to see the payback in the form of electricity savings, but you could see a payoff in a higher sale price for your home. 

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Jones-Albertus and Haenggi agree, there are a few scenarios when installing solar probably doesn't make sense, no matter the payback period. If you know your roof will need to be replaced soon, you'll definitely want to wait until that is done before you install solar panels on top of it. If you have a ton of trees looming over your home, a solar system is unlikely to ever generate a significant return; in that case, Jones-Albertus recommends considering community solar .

How to pay for solar panels

There are a lot of different ways to pay for solar panels, and they all affect the solar payback period.

• Cash: If you simply save up for the purchase (using a high-yield savings account , for example), you'll avoid paying any loan interest, and reduce the overall cost of solar panels. "In the long run, typically the highest rate of return comes from paying for cash for a system," Jones-Albertus said.
• Solar loan: Some banks offer loans specifically designed to fund solar installations. Check-in with your installer or with lenders to see what the options are.
• Home equity loan or line of credit, aka HELOC: Generally speaking, using the equity in your home to fund home improvements can be a solid idea -- especially because solar panels will increase the value of your home.
• Lease or power purchase agreement:  If you want to minimize the upfront investment in solar, you could actually lease the system from the installer. The developer would own the panels and would sell you the electricity generated at a reduced rate, basically negating the idea of a "payback period" altogether.

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