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How many times have you found yourself still staring at your textbook around midnight (or later!) even when you started your homework hours earlier? Those lost hours could be explained by Parkinson’s Law, which states, “Work expands to fill the time available for its completion.” In other words, if you give yourself all night to memorize those geometry formulas for your quiz tomorrow, you’ll inevitably find that a 30 minute task has somehow filled your entire evening.

We know that you have more homework than ever. But even with lots and lots to do, a few tweaks to your study routine could help you spend less time getting more accomplished. Here are 8 steps to make Parkinson’s Law work to your advantage:

1. Make a list

This should be a list of everything that has to be done that evening. And we mean, everything—from re-reading notes from this morning’s history class to quizzing yourself on Spanish vocabulary.

2. Estimate the time needed for each item on your list

You can be a little ruthless here. However long you think a task will take, try shaving off 5 or 10 minutes. But, be realistic. You won’t magically become a speed reader.

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3. Gather all your gear

Collect EVERYTHING you will need for the homework you are working on (like your laptop for writing assignments and pencils for problem sets). Getting up for supplies takes you off course and makes it that much harder to get back to your homework.

The constant blings and beeps from your devices can make it impossible to focus on what you are working on. Switch off or silence your phones and tablets, or leave them in another room until it’s time to take a tech break.

Read More: How to Calculate Your GPA

5. Time yourself

Noting how much time something actually takes will help you estimate better and plan your next study session.

6. Stay on task

If you’re fact checking online, it can be so easy to surf on over to a completely unrelated site. A better strategy is to note what information you need to find online, and do it all at once at the end of the study session.

7. Take plenty of breaks

Most of us need a break between subjects or to break up long stretches of studying. Active breaks are a great way to keep your energy up. Tech breaks can be an awesome way to combat the fear of missing out that might strike while you are buried in your work, but they also tend to stretch much longer than originally intended. Stick to a break schedule of 10 minutes or so.

8. Reward yourself! 

Finish early? If you had allocated 30 minutes for reading a biology chapter and it only took 20, you can apply those extra 10 minutes to a short break—or just move on to your next task. If you stay on track, you might breeze through your work quickly enough to catch up on some Netflix.

Our best piece of advice? Keep at it. The more you use this system, the easier it will become. You’ll be surprised by how much time you can shave off homework just by focusing and committing to a distraction-free study plan.

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What’s the Right Amount of Homework?

Decades of research show that homework has some benefits, especially for students in middle and high school—but there are risks to assigning too much.

Many teachers and parents believe that homework helps students build study skills and review concepts learned in class. Others see homework as disruptive and unnecessary, leading to burnout and turning kids off to school. Decades of research show that the issue is more nuanced and complex than most people think: Homework is beneficial, but only to a degree. Students in high school gain the most, while younger kids benefit much less.

The National PTA and the National Education Association support the “ 10-minute homework guideline ”—a nightly 10 minutes of homework per grade level. But many teachers and parents are quick to point out that what matters is the quality of the homework assigned and how well it meets students’ needs, not the amount of time spent on it.

The guideline doesn’t account for students who may need to spend more—or less—time on assignments. In class, teachers can make adjustments to support struggling students, but at home, an assignment that takes one student 30 minutes to complete may take another twice as much time—often for reasons beyond their control. And homework can widen the achievement gap, putting students from low-income households and students with learning disabilities at a disadvantage.

However, the 10-minute guideline is useful in setting a limit: When kids spend too much time on homework, there are real consequences to consider.

Small Benefits for Elementary Students

As young children begin school, the focus should be on cultivating a love of learning, and assigning too much homework can undermine that goal. And young students often don’t have the study skills to benefit fully from homework, so it may be a poor use of time (Cooper, 1989 ; Cooper et al., 2006 ; Marzano & Pickering, 2007 ). A more effective activity may be nightly reading, especially if parents are involved. The benefits of reading are clear: If students aren’t proficient readers by the end of third grade, they’re less likely to succeed academically and graduate from high school (Fiester, 2013 ).

For second-grade teacher Jacqueline Fiorentino, the minor benefits of homework did not outweigh the potential drawback of turning young children against school at an early age, so she experimented with dropping mandatory homework. “Something surprising happened: They started doing more work at home,” Fiorentino writes . “This inspiring group of 8-year-olds used their newfound free time to explore subjects and topics of interest to them.” She encouraged her students to read at home and offered optional homework to extend classroom lessons and help them review material.

Moderate Benefits for Middle School Students

As students mature and develop the study skills necessary to delve deeply into a topic—and to retain what they learn—they also benefit more from homework. Nightly assignments can help prepare them for scholarly work, and research shows that homework can have moderate benefits for middle school students (Cooper et al., 2006 ). Recent research also shows that online math homework, which can be designed to adapt to students’ levels of understanding, can significantly boost test scores (Roschelle et al., 2016 ).

There are risks to assigning too much, however: A 2015 study found that when middle school students were assigned more than 90 to 100 minutes of daily homework, their math and science test scores began to decline (Fernández-Alonso, Suárez-Álvarez, & Muñiz, 2015 ). Crossing that upper limit can drain student motivation and focus. The researchers recommend that “homework should present a certain level of challenge or difficulty, without being so challenging that it discourages effort.” Teachers should avoid low-effort, repetitive assignments, and assign homework “with the aim of instilling work habits and promoting autonomous, self-directed learning.”

In other words, it’s the quality of homework that matters, not the quantity. Brian Sztabnik, a veteran middle and high school English teacher, suggests that teachers take a step back and ask themselves these five questions :

• How long will it take to complete?
• Have all learners been considered?
• Will an assignment encourage future success?
• Will an assignment place material in a context the classroom cannot?
• Does an assignment offer support when a teacher is not there?

More Benefits for High School Students, but Risks as Well

By the time they reach high school, students should be well on their way to becoming independent learners, so homework does provide a boost to learning at this age, as long as it isn’t overwhelming (Cooper et al., 2006 ; Marzano & Pickering, 2007 ). When students spend too much time on homework—more than two hours each night—it takes up valuable time to rest and spend time with family and friends. A 2013 study found that high school students can experience serious mental and physical health problems, from higher stress levels to sleep deprivation, when assigned too much homework (Galloway, Conner, & Pope, 2013 ).

Homework in high school should always relate to the lesson and be doable without any assistance, and feedback should be clear and explicit.

Teachers should also keep in mind that not all students have equal opportunities to finish their homework at home, so incomplete homework may not be a true reflection of their learning—it may be more a result of issues they face outside of school. They may be hindered by issues such as lack of a quiet space at home, resources such as a computer or broadband connectivity, or parental support (OECD, 2014 ). In such cases, giving low homework scores may be unfair.

Since the quantities of time discussed here are totals, teachers in middle and high school should be aware of how much homework other teachers are assigning. It may seem reasonable to assign 30 minutes of daily homework, but across six subjects, that’s three hours—far above a reasonable amount even for a high school senior. Psychologist Maurice Elias sees this as a common mistake: Individual teachers create homework policies that in aggregate can overwhelm students. He suggests that teachers work together to develop a school-wide homework policy and make it a key topic of back-to-school night and the first parent-teacher conferences of the school year.

Parents Play a Key Role

Homework can be a powerful tool to help parents become more involved in their child’s learning (Walker et al., 2004 ). It can provide insights into a child’s strengths and interests, and can also encourage conversations about a child’s life at school. If a parent has positive attitudes toward homework, their children are more likely to share those same values, promoting academic success.

But it’s also possible for parents to be overbearing, putting too much emphasis on test scores or grades, which can be disruptive for children (Madjar, Shklar, & Moshe, 2015 ). Parents should avoid being overly intrusive or controlling—students report feeling less motivated to learn when they don’t have enough space and autonomy to do their homework (Orkin, May, & Wolf, 2017 ; Patall, Cooper, & Robinson, 2008 ; Silinskas & Kikas, 2017 ). So while homework can encourage parents to be more involved with their kids, it’s important to not make it a source of conflict.

4 Tips for Completing Your Homework On Time

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Homework, a necessary evil according to many teachers, has a lot of students tied up in knots. Some students can never seem to get things turned in on time. In fact, many students do not even realize that they  have  homework until a friend from class texts them or they overhear someone in the halls talking about Ms. So-and-so's terrible, no-good, awful, horrifying worksheet for Chemistry that is due the next day. These five tips for completing your homework on time, however, should help you get that homework finished on time. 

Tip 1: Rely On a Planning System

Most of you by now are well acquainted with a homework planner. It has the dates, the school subjects you are taking, and a whole lot of blank space to write down your homework assignments. Use these planners if you have them. Writing with an actual pencil or pen may seem almost archaic what with technology virtually doing everything for us, but the kinesthetic movement of writing down an assignment into one of those little squares (Language Arts test tomorrow - STUDY TONIGHT), will actually help solidify that homework in your brain.

Plus, when you are packing up to go home at the end of the school day, all you have to do is open up that planner to see which books, folders, and binders need to go home with you so you will not miss out anything that you need to do that evening.

Some people  hate  using planners. They'd rather walk on a pile of crushed glass than actually write something down in a planner. That's quite all right. One student kept a wadded up piece of paper in his pocket where he'd scrawl his assignments. It worked for him, so it was fine. For those of you not keen on planners or crumpled up notes, your phone can come in really handy. Just download a productivity app and type your assignments in there. Or, keep track of all the work due in the notes section of your phone. Or, snap a picture of the homework board in each teacher's class before you head out into the hallway. Or, if you are really dead-set against anything planner-related, then just send yourself a text after each class with your homework assignments for the night.

No matter which planning system you prefer, use it. Check off each item once you get it in your backpack. Your brain can only process so much information at a time, so you absolutely must write your homework down if you plan to complete it on time. 

Tip 2: Prioritize Your Homework Assignments

All assignments are not created equal. It's strongly recommended you use a prioritizing system when you sit down at home with your homework. Try a system a little something like this:

• Examples: Studying for a major test coming up tomorrow. Finishing a major project due tomorrow. Writing an essay worth a LOT of points that is due tomorrow. 
• Examples:  Studying for a quiz coming up tomorrow. Completing a homework sheet that is due tomorrow. Reading a chapter that is due tomorrow. 
• Examples: Studying for a spelling test that will occur on Friday. Writing a blog and posting it on the class board by Friday. Finish a book upon which you will take a quiz on Friday.
• Examples: Reviewing chapters for the midterm exam. Working on an on-going project, research paper, or long assignment due at the end of the quarter. Completing a packet that isn't due for two weeks. 

Once you've prioritized the work you have to do, complete all the 1's first, then the 2's, moving down as you go. That way, if you find yourself pressed for time because Great-Grandma decided to stop over for family dinner and your mom insisted you spend the evening playing bridge with her despite the fact that you have hours of homework ahead of you, then you will not have missed anything vitally important to your grade. 

Tip 3: Get the Worst Assignment Over With First

So, maybe you absolutely hate writing essays (But, why, though when all you have to do is follow these essay tips? ) and you have a major essay staring you in the face that  must  be completed before tomorrow. You also have to study for a major math test, complete a social studies blog by Friday, study for the ACT  next month, and finish up your science worksheet from class. Your "1" assignments would be the essay and the math test. Your "2" assignment is the science worksheet, the "3" assignment is that blog, and the "4" assignment is studying for the ACT. 

Ordinarily, you would start with the science worksheet because you  love  science, but that would be a big mistake. Start with those "1" assignments and knock out that essay first. Why? Because you hate it. And completing the worst assignment first gets it off your mind, out of your homework cache, and makes everything that comes after it appears to be really, really easy. It will be an absolute  joy  to complete that science worksheet once you have written the essay. Why rob yourself of joy? 

Then, once you've completed the stuff due first, you can focus on putting in a little bit of time on the ACT. Easy peasy.

Tip 4: Take Planned Breaks

Some people believe that sitting down to complete homework means that you literally park your behind in a chair and you don't move it for the next four thousand hours or so. That is one of the worst study ideas in history. Your brain only has the capacity to stay focused for about 45 minutes (maybe even less for some of you) before it goes on the fritz and starts wanting to make you get up and dance the Roger Rabbit. So, schedule your study time with breaks actually built in . Work for 45 minutes, then take a 10-minute break to do whatever it is people your age like to do. Then, rinse and repeat. It looks a little something like this:

Homework Time:

• 45 minutes: Work on "1" assignments, starting with the absolute worst.
• 10 minutes: Get a snack, play Pokemon Go!, surf Instagram
• 45 minutes: Work on "1" assignments again. You know you didn't finish.
• 10 minutes: Do some jumping jacks, dance the Macarena, polish your nails.
• 45 minutes: Work on "2" assignments and maybe even finish with any 3s and 4s. Put everything in your backpack.

Completing your homework on time is a learned skill. It requires some discipline and not everyone is naturally disciplined. So, you have to practice checking that you have everything you need for homework when you are still at school, prioritizing your work, plunging into the assignments you loathe, and taking planned breaks. Isn't your grade worth it?

You bet it is. 

• Organize Your Homework With Color Coded Supplies
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How to Make Up a Good Excuse for Your Homework Not Being Finished

Last Updated: March 2, 2024 Fact Checked

Choosing an Excuse

Delivering the excuse, potential consequences, moving forward, expert q&a.

This article was co-authored by wikiHow Staff . Our trained team of editors and researchers validate articles for accuracy and comprehensiveness. wikiHow's Content Management Team carefully monitors the work from our editorial staff to ensure that each article is backed by trusted research and meets our high quality standards. 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 373,707 times. Learn more...

If you did not finish your homework, you may want to find an excuse to avoid being penalized. There are a variety of excuses, from blaming technology to your busy schedule, that sounds like a plausible reason for failing to complete an assignment. When you settle on an excuse, work on delivering the excuse in a believable fashion. However, try to be careful moving forward. You do not want to lie habitually, as this reflects poorly on you as a student. In the future, try to make sure your assignments are done on time.

• This is a great excuse if you had a paper you needed to type and print. It may also work if you have homework you had to do online. You could say you did the whole assignment, but then your internet cut out and you couldn't save anything.
• It might be a bad idea to claim your printer stopped working. Your teacher may request you e-mail him/her the assignment instead, which you won't be able to do if you never did it. Teachers may also suggest you should have printed an assignment at a local library or FedEx instead of coming in with nothing.

• If your parents are divorced, for example, you can claim you were at your mom's last night but left your textbook with your dad this weekend. Many teachers are sympathetic to children from divorced homes. Your teacher may take pity on you if you use an excuse like this. [2] X Research source
• Do you have any younger siblings? You could claim you had to babysit your little sister and she got sick, resulting in your being distracted from your homework. [3] X Research source

• You can try running around in the playground or hallways before class. This can help you look flushed and warm. If you look sick, your teacher will be more likely to believe you.
• However, keep in mind some teachers may require a note from your parents in the event of illness. If your teacher typically demands proof of sickness, you may want to avoid using this excuse.

• Avoid saying you left your homework at home. Your teacher may request you call your mom or dad to have it delivered to the school. This will reveal you are lying.
• Try not to use this excuse more than once or twice a term; otherwise, your teacher may see you as disorganized and be less sympathetic towards you if you need to make other excuses in the future.

• Be careful using this excuse if you're not busy. If you're usually late for classes and do not engage in many extracurricular activities, your teacher may catch onto the fact you're lying.

• Never lie you were absent on the day the homework was set. One glance of the register is all it takes for your teacher to see right through this excuse.

• If your teacher is particularly strict, be prepared to answer a lot of questions. A stricter teacher is likely to grill you, poking holes in your excuse. For example, say you claim you couldn't turn in your online math homework because your internet cut out. A strict teacher might respond with something like, "Then why didn't you go do your homework at a coffee shop?" Have a response ready. Try something like, "My mom was working and there was no one to drive me." [7] X Research source
• Do you know anything about your teacher's personal interests? This can help you gauge what excuse may work for this person. For example, you know your chemistry teacher is the oldest of 7 children. He may be more sympathetic to a story about how watching your younger siblings kept you from getting your work done. [8] X Research source

• Stick to only the important details. For example, say you're planning on claiming your piano recital got out late, and this is why you didn't finish your math homework. Do not go overboard with the details. Simply say, "A few students played their solos too long, so we didn't get done until 9:30 and it was a 45-minute drive home." Do not say, "Chester Mifflin spent 25 minutes on his routine when we were only given 10, and then Lisa Gregory was a little late getting up on stage..." The longer your lie, the more unbelievable it sounds. Most people would not remember this much detail.
• If your teacher presses you for specifics, you can improvise as needed but avoid excessive detail. For example, your teacher might ask, "How long did the recital run over?" Do not say, "It was supposed to run until 8:30 but it was 9:23 when we got out." Instead, say something somewhat vague, like, "I'd say about 45 minutes."

• Take a few deep breaths before going into the room to help yourself stay calm.
• Make eye contact with your teacher most of the time.
• Be conscious of what you're doing with your body. Try to avoid fidgeting excessively.

• Refer to the syllabus for that class. There may be a section about honesty that goes over the consequences of lying to a teacher.
• You should also look at your school's handbook if you have a copy. See if there are any sections about what happens if you breach academic honesty policies.
• Consequences can vary from teacher-to-teacher. In some cases, you may only get a strict talking to. However, some teachers may be required to report these kinds of behaviors to the principal and your parents. This could land you in more serious trouble both at home and at school.

• It may depend on the assignment. Late work may not be accepted, but if the assignment is only worth 10 points, is this really a big deal? However, if the assignment is worth 15% of your grade, it may be worth it to ask for an extension.
• Talk to other students who've had this teacher in the past. How has this teacher reacted to late or missing work? Some teachers may accept late work for lower points. Some teachers may allow you to turn in work late if it's your first time. If this is the case, it may be a good idea to simply admit you didn't do the assignment.

• You can make a pro and con list for each scenario. Write down the possible benefits and possible drawbacks of each option. For example, you can write on the top of a piece of paper "Lying To My Teacher" and then have two columns, one for "pro" and one for "con." Under "pro," you may write something like, "Assignment is worth a lot of points - an extension could help my overall grade." Under "con," you could write, "If Ms. Davies finds out I'm lying, she will report it to the principal and I'll get detention for a week."
• Weigh the pros and cons. If the pros outweigh the cons heavily for one option, this may be the right choice for you.

• Do your homework every day after school. Do not do anything else, like playing video games or playing outside, before finishing your work.
• Write down all the assignments you have to do. Make sure to write down an assignment after a teacher mentions it. This way, you won't forget.

• If you habitually struggle to get your homework done and have an overall inability to concentrate, this can be a symptom of Attention Deficit Disorder. Talk to your parents about getting tested for ADD.

You Might Also Like

• ↑ https://helpfulprofessor.com/homework-excuses/
• ↑ https://www.brighthubeducation.com/study-and-learning-tips/51072-10-best-homework-excuses/
• ↑ https://www.psychologytoday.com/blog/extreme-fear/201005/top-ten-secrets-effective-liars

About This Article

While lying too often could reflect badly on you as a student, if you need a good excuse for your homework not being finished, say you lost it. Just don’t use this excuse often, since your teacher will think you’re unorganized. If the homework was on the computer, try blaming technology. Say your computer crashed or your internet was down. Alternatively, claim you were ill last night and had to rest. Only do this if you don’t think your teacher will call your parents to check or ask for a sick note. If you have a sibling that doesn’t go to your school, you can say they were ill and you had to look after them. Or, if you do a lot of things outside of school and have lots of homework to do, pretend you were too busy and ran out of time. For more tips, including how to make your excuse more convincing, read on! Did this summary help you? Yes No

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Should We Get Rid of Homework?

Some educators are pushing to get rid of homework. Would that be a good thing?

By Jeremy Engle and Michael Gonchar

Do you like doing homework? Do you think it has benefited you educationally?

Has homework ever helped you practice a difficult skill — in math, for example — until you mastered it? Has it helped you learn new concepts in history or science? Has it helped to teach you life skills, such as independence and responsibility? Or, have you had a more negative experience with homework? Does it stress you out, numb your brain from busywork or actually make you fall behind in your classes?

Should we get rid of homework?

In “ The Movement to End Homework Is Wrong, ” published in July, the Times Opinion writer Jay Caspian Kang argues that homework may be imperfect, but it still serves an important purpose in school. The essay begins:

Do students really need to do their homework? As a parent and a former teacher, I have been pondering this question for quite a long time. The teacher side of me can acknowledge that there were assignments I gave out to my students that probably had little to no academic value. But I also imagine that some of my students never would have done their basic reading if they hadn’t been trained to complete expected assignments, which would have made the task of teaching an English class nearly impossible. As a parent, I would rather my daughter not get stuck doing the sort of pointless homework I would occasionally assign, but I also think there’s a lot of value in saying, “Hey, a lot of work you’re going to end up doing in your life is pointless, so why not just get used to it?” I certainly am not the only person wondering about the value of homework. Recently, the sociologist Jessica McCrory Calarco and the mathematics education scholars Ilana Horn and Grace Chen published a paper, “ You Need to Be More Responsible: The Myth of Meritocracy and Teachers’ Accounts of Homework Inequalities .” They argued that while there’s some evidence that homework might help students learn, it also exacerbates inequalities and reinforces what they call the “meritocratic” narrative that says kids who do well in school do so because of “individual competence, effort and responsibility.” The authors believe this meritocratic narrative is a myth and that homework — math homework in particular — further entrenches the myth in the minds of teachers and their students. Calarco, Horn and Chen write, “Research has highlighted inequalities in students’ homework production and linked those inequalities to differences in students’ home lives and in the support students’ families can provide.”

Mr. Kang argues:

But there’s a defense of homework that doesn’t really have much to do with class mobility, equality or any sense of reinforcing the notion of meritocracy. It’s one that became quite clear to me when I was a teacher: Kids need to learn how to practice things. Homework, in many cases, is the only ritualized thing they have to do every day. Even if we could perfectly equalize opportunity in school and empower all students not to be encumbered by the weight of their socioeconomic status or ethnicity, I’m not sure what good it would do if the kids didn’t know how to do something relentlessly, over and over again, until they perfected it. Most teachers know that type of progress is very difficult to achieve inside the classroom, regardless of a student’s background, which is why, I imagine, Calarco, Horn and Chen found that most teachers weren’t thinking in a structural inequalities frame. Holistic ideas of education, in which learning is emphasized and students can explore concepts and ideas, are largely for the types of kids who don’t need to worry about class mobility. A defense of rote practice through homework might seem revanchist at this moment, but if we truly believe that schools should teach children lessons that fall outside the meritocracy, I can’t think of one that matters more than the simple satisfaction of mastering something that you were once bad at. That takes homework and the acknowledgment that sometimes a student can get a question wrong and, with proper instruction, eventually get it right.

Students, read the entire article, then tell us:

Should we get rid of homework? Why, or why not?

Is homework an outdated, ineffective or counterproductive tool for learning? Do you agree with the authors of the paper that homework is harmful and worsens inequalities that exist between students’ home circumstances?

Or do you agree with Mr. Kang that homework still has real educational value?

When you get home after school, how much homework will you do? Do you think the amount is appropriate, too much or too little? Is homework, including the projects and writing assignments you do at home, an important part of your learning experience? Or, in your opinion, is it not a good use of time? Explain.

In these letters to the editor , one reader makes a distinction between elementary school and high school:

Homework’s value is unclear for younger students. But by high school and college, homework is absolutely essential for any student who wishes to excel. There simply isn’t time to digest Dostoyevsky if you only ever read him in class.

What do you think? How much does grade level matter when discussing the value of homework?

Is there a way to make homework more effective?

If you were a teacher, would you assign homework? What kind of assignments would you give and why?

Want more writing prompts? You can find all of our questions in our Student Opinion column . Teachers, check out this guide to learn how you can incorporate them into your classroom.

Students 13 and older in the United States and Britain, and 16 and older elsewhere, are invited to comment. All comments are moderated by the Learning Network staff, but please keep in mind that once your comment is accepted, it will be made public.

Jeremy Engle joined The Learning Network as a staff editor in 2018 after spending more than 20 years as a classroom humanities and documentary-making teacher, professional developer and curriculum designer working with students and teachers across the country. More about Jeremy Engle

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1.3: Problem Solving Strategies

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• Michelle Manes
• University of Hawaii

Think back to the first problem in this chapter, the ABC Problem. What did you do to solve it? Even if you did not figure it out completely by yourself, you probably worked towards a solution and figured out some things that did not work.

Unlike exercises, there is never a simple recipe for solving a problem. You can get better and better at solving problems, both by building up your background knowledge and by simply practicing. As you solve more problems (and learn how other people solved them), you learn strategies and techniques that can be useful. But no single strategy works every time.

How to Solve It

George Pólya was a great champion in the field of teaching effective problem solving skills. He was born in Hungary in 1887, received his Ph.D. at the University of Budapest, and was a professor at Stanford University (among other universities). He wrote many mathematical papers along with three books, most famously, “How to Solve it.” Pólya died at the age 98 in 1985. [1]

George Pólya, circa 1973

• Image of Pólya by Thane Plambeck from Palo Alto, California (Flickr) [CC BY 2.0 ( http://creativecommons.org/licenses/by/2.0 )], via Wikimedia Commons ↵

In 1945, Pólya published the short book How to Solve It , which gave a four-step method for solving mathematical problems:

• First, you have to understand the problem.
• After understanding, then make a plan.
• Carry out the plan.
• Look back on your work. How could it be better?

This is all well and good, but how do you actually do these steps?!?! Steps 1. and 2. are particularly mysterious! How do you “make a plan?” That is where you need some tools in your toolbox, and some experience to draw upon.

Much has been written since 1945 to explain these steps in more detail, but the truth is that they are more art than science. This is where math becomes a creative endeavor (and where it becomes so much fun). We will articulate some useful problem solving strategies, but no such list will ever be complete. This is really just a start to help you on your way. The best way to become a skilled problem solver is to learn the background material well, and then to solve a lot of problems!

We have already seen one problem solving strategy, which we call “Wishful Thinking.” Do not be afraid to change the problem! Ask yourself “what if” questions:

• What if the picture was different?
• What if the numbers were simpler?
• What if I just made up some numbers?

You need to be sure to go back to the original problem at the end, but wishful thinking can be a powerful strategy for getting started.

This brings us to the most important problem solving strategy of all:

Problem Solving Strategy 2 (Try Something!).

If you are really trying to solve a problem, the whole point is that you do not know what to do right out of the starting gate. You need to just try something! Put pencil to paper (or stylus to screen or chalk to board or whatever!) and try something. This is often an important step in understanding the problem; just mess around with it a bit to understand the situation and figure out what is going on.

And equally important: If what you tried first does not work, try something else! Play around with the problem until you have a feel for what is going on.

Last week, Alex borrowed money from several of his friends. He finally got paid at work, so he brought cash to school to pay back his debts. First he saw Brianna, and he gave her 1/4 of the money he had brought to school. Then Alex saw Chris and gave him 1/3 of what he had left after paying Brianna. Finally, Alex saw David and gave him 1/2 of what he had remaining. Who got the most money from Alex?

Think/Pair/Share

After you have worked on the problem on your own for a while, talk through your ideas with a partner (even if you have not solved it). What did you try? What did you figure out about the problem? This problem lends itself to two particular strategies. Did you try either of these as you worked on the problem? If not, read about the strategy and then try it out before watching the solution.

Problem Solving Strategy 3 (Draw a Picture).

Some problems are obviously about a geometric situation, and it is clear you want to draw a picture and mark down all of the given information before you try to solve it. But even for a problem that is not geometric, like this one, thinking visually can help! Can you represent something in the situation by a picture?

Draw a square to represent all of Alex’s money. Then shade 1/4 of the square — that’s what he gave away to Brianna. How can the picture help you finish the problem?

After you have worked on the problem yourself using this strategy (or if you are completely stuck), you can watch someone else’s solution.

Problem Solving Strategy 4 (Make Up Numbers).

Part of what makes this problem difficult is that it is about money, but there are no numbers given. That means the numbers must not be important. So just make them up!

You can work forwards: Assume Alex had some specific amount of money when he showed up at school, say $100. Then figure out how much he gives to each person. Or you can work backwards: suppose he has some specific amount left at the end, like $10. Since he gave Chris half of what he had left, that means he had $20 before running into Chris. Now, work backwards and figure out how much each person got.

Watch the solution only after you tried this strategy for yourself.

If you use the “Make Up Numbers” strategy, it is really important to remember what the original problem was asking! You do not want to answer something like “Everyone got $10.” That is not true in the original problem; that is an artifact of the numbers you made up. So after you work everything out, be sure to re-read the problem and answer what was asked!

(Squares on a Chess Board)

How many squares, of any possible size, are on a 8 × 8 chess board? (The answer is not 64... It’s a lot bigger!)

Remember Pólya’s first step is to understand the problem. If you are not sure what is being asked, or why the answer is not just 64, be sure to ask someone!

Think / Pair / Share

After you have worked on the problem on your own for a while, talk through your ideas with a partner (even if you have not solved it). What did you try? What did you figure out about the problem, even if you have not solved it completely?

It is clear that you want to draw a picture for this problem, but even with the picture it can be hard to know if you have found the correct answer. The numbers get big, and it can be hard to keep track of your work. Your goal at the end is to be absolutely positive that you found the right answer. You should never ask the teacher, “Is this right?” Instead, you should declare, “Here’s my answer, and here is why I know it is correct!”

Problem Solving Strategy 5 (Try a Simpler Problem).

Pólya suggested this strategy: “If you can’t solve a problem, then there is an easier problem you can solve: find it.” He also said: “If you cannot solve the proposed problem, try to solve first some related problem. Could you imagine a more accessible related problem?” In this case, an 8 × 8 chess board is pretty big. Can you solve the problem for smaller boards? Like 1 × 1? 2 × 2? 3 × 3?

Of course the ultimate goal is to solve the original problem. But working with smaller boards might give you some insight and help you devise your plan (that is Pólya’s step (2)).

Problem Solving Strategy 6 (Work Systematically).

If you are working on simpler problems, it is useful to keep track of what you have figured out and what changes as the problem gets more complicated.

For example, in this problem you might keep track of how many 1 × 1 squares are on each board, how many 2 × 2 squares on are each board, how many 3 × 3 squares are on each board, and so on. You could keep track of the information in a table:

Problem Solving Strategy 7 (Use Manipulatives to Help You Investigate).

Sometimes even drawing a picture may not be enough to help you investigate a problem. Having actual materials that you move around can sometimes help a lot!

For example, in this problem it can be difficult to keep track of which squares you have already counted. You might want to cut out 1 × 1 squares, 2 × 2 squares, 3 × 3 squares, and so on. You can actually move the smaller squares across the chess board in a systematic way, making sure that you count everything once and do not count anything twice.

Problem Solving Strategy 8 (Look for and Explain Patterns).

Sometimes the numbers in a problem are so big, there is no way you will actually count everything up by hand. For example, if the problem in this section were about a 100 × 100 chess board, you would not want to go through counting all the squares by hand! It would be much more appealing to find a pattern in the smaller boards and then extend that pattern to solve the problem for a 100 × 100 chess board just with a calculation.

If you have not done so already, extend the table above all the way to an 8 × 8 chess board, filling in all the rows and columns. Use your table to find the total number of squares in an 8 × 8 chess board. Then:

• Describe all of the patterns you see in the table.
• Can you explain and justify any of the patterns you see? How can you be sure they will continue?
• What calculation would you do to find the total number of squares on a 100 × 100 chess board?

(We will come back to this question soon. So if you are not sure right now how to explain and justify the patterns you found, that is OK.)

(Broken Clock)

This clock has been broken into three pieces. If you add the numbers in each piece, the sums are consecutive numbers. ( Consecutive numbers are whole numbers that appear one after the other, such as 1, 2, 3, 4 or 13, 14, 15.)

Can you break another clock into a different number of pieces so that the sums are consecutive numbers? Assume that each piece has at least two numbers and that no number is damaged (e.g. 12 isn’t split into two digits 1 and 2.)

Remember that your first step is to understand the problem. Work out what is going on here. What are the sums of the numbers on each piece? Are they consecutive?

After you have worked on the problem on your own for a while, talk through your ideas with a partner (even if you have not solved it). What did you try? What progress have you made?

Problem Solving Strategy 9 (Find the Math, Remove the Context).

Sometimes the problem has a lot of details in it that are unimportant, or at least unimportant for getting started. The goal is to find the underlying math problem, then come back to the original question and see if you can solve it using the math.

In this case, worrying about the clock and exactly how the pieces break is less important than worrying about finding consecutive numbers that sum to the correct total. Ask yourself:

• What is the sum of all the numbers on the clock’s face?
• Can I find two consecutive numbers that give the correct sum? Or four consecutive numbers? Or some other amount?
• How do I know when I am done? When should I stop looking?

Of course, solving the question about consecutive numbers is not the same as solving the original problem. You have to go back and see if the clock can actually break apart so that each piece gives you one of those consecutive numbers. Maybe you can solve the math problem, but it does not translate into solving the clock problem.

Problem Solving Strategy 10 (Check Your Assumptions).

When solving problems, it is easy to limit your thinking by adding extra assumptions that are not in the problem. Be sure you ask yourself: Am I constraining my thinking too much?

In the clock problem, because the first solution has the clock broken radially (all three pieces meet at the center, so it looks like slicing a pie), many people assume that is how the clock must break. But the problem does not require the clock to break radially. It might break into pieces like this:

Were you assuming the clock would break in a specific way? Try to solve the problem now, if you have not already.

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How In-School Tutoring Benefits Both Attendance and Math Scores

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Tutoring has become a popular prescription for academic recovery, thanks to lots of evidence showing that sustained tutoring blocks at least three times a week can boost students’ improvement trajectories.

Now, two new research studies conclude that one of the critical pieces is making sure it happens during school hours—not outside of them. What’s more, students themselves seem to want to come to school when they know they’re going to receive personalized attention.

In essence, the two findings suggest compounding benefits for embedding extra help in the school day: academic improvement and better attendance.

On March 27, researchers at Chicago University’s Education Lab released preliminary findings from a study of two school districts in Chicago and Georgia showing “meaningful” improvement in math scores for the 2,200 students in grades K-11 who received tutoring.

The gains equaled about two-thirds of a year of math learning—improvement that is in line with other studies on tutoring, even though students on average got less tutoring time than in other studies.

“These gains are strikingly like the ones we’ve seen in previous studies on tutoring. The difference is that the implementation was much lower [from other efforts]. ... It shows that even tutoring done in a sub-optimal way can be beneficial,” said Monica Bhatt, a senior research director at the University of Chicago’s Education Lab, and one of the authors of the study.

Doing in-school tutoring can be challenging. So some principals have devised new support systems and designated on-site coordinators to organize the programs.

Those efforts have helped pay dividends for attendance, too. In the second study , released earlier this month, researchers with Stanford University’s National Student Support Accelerator found that students are 7 percent less likely to be absent on days they have scheduled tutoring sessions. The study, conducted over the 2022-23 school year, examined absenteeism rates of 4,478 students in 141 schools in the District of Columbia.

“There are lots of reasons why students are absent. Being disengaged in school is one reason,” said Nancy Waymack, the director of partnerships and policy at the NSSA."Tutoring is one way that students can have one more meaningful relationship in school. Tutoring can be one tool to move the needle in the right direction.”

Taken together, the studies point out a growing connection between tutoring, student attendance, and test scores. The critical factor: Committing to tutoring at school, not as an add-on.

“It shouldn’t be ad-hoc, or on-demand. It’s not homework help,” said Bhatt. “The message to school leaders is to protect the tutoring block in school.”

Contrasting two types of tutoring

Bhatt’s team initially picked four sites to study tutoring interventions over the 2022-23 school year to contrast two different kinds of tutoring. Fulton County in Ga., and Chicago Public Schools offered tutoring in school, while the other two sites—schools in New Mexico districts and a mid-size urban California school district—respectively offered evening and weekend slots, or after-school sessions.

In New Mexico’s case, only 527 students signed up for evening or weekend classes out of a total 34,000 that were eligible, and only about two thirds actually attended. In California, the urban school district that the researchers wanted to study saw a much lower rate of participation. In both cases, the participant numbers were too low to conduct a random-assignment study.

“The districts found that after-school or weekend programs are dependent on kids or their parents raising their hands. And it’s not the same set of kids who may need this tutoring the most,” noted Bhatt. “If we want to ameliorate education inequalities, we have to meet kids where they are literally, and in terms of where they are academically.”

For the other two districts, Bhatt’s team analyzed test scores for 429 students out of the 548 K-11 students randomized across 13 Chicago schools. One group was offered the tutoring at least three times a week at school. Schools picked which students would be part of the sample and school leaders prioritized students in the bottom quartile in terms of performance. All the students were offered tutoring at least three times a week for 30-minute sessions.

In Fulton County, Ga., the team studied outcomes of 1,163 students out of the 1,500 selected across 17 schools. Reading help was offered to students in grades 3-8, and math help was offered to students in grade 9.

In both cases, the treatment group received a much higher number of tutoring sessions, on average. (The control group also received some tutoring sessions).

As for the Stanford study, it’s part of an ongoing look at tutoring sponsored by the District of Columbia’s Office of the State Superintendent of Education, examining who received tutoring and how much it boosted learning.

“One of the things that stood out was the impact on attendance,” said Wymack. “We were able to compare how often students came to school on days when there was tutoring vs. days there wasn’t a session planned.”

The studies both point towards the benefits of fitting tutoring in during the school day. It means a more diverse set of students will have access to additiona help. And if instruction is differentiated for everybody, Bhatt added, students who need tutoring support the most will feel less stigmatized. And the findings are also good information as recovery dollars begin to dwindle.

“We felt an urgency to release these preliminary results because school leaders are making decisions about where to spend their remaining ESSER dollars,” said Bhatt., referring to the federal aid for schools.

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