What is a Problem?

The following passage is from the opening pages of The Art and Craft of Problem Solving, by Paul Zeitz

An exercise is a question that tests the student’s mastery of a narrowly focused technique, usually one that was recently “covered.” Exercises may be hard or easy, but they are never puzzling, for it is always immediately clear how to proceed. Getting the solution may involve hairy technical work, but the path towards solution is always apparent. In contrast, a problem is a question that cannot be answered immediately. Problems are often open-ended, paradoxical, and sometimes unsolvable, and require investigation before one can come close to a solution. Problems and problem solving are at the heart of mathematics. Research mathematicians do nothing but open-ended problem solving. In industry, being able to solve a poorly defined problem is much more important to an employer than being able to, say, invert a matrix. A computer can do the latter, but not the former.

A good problem solver is not just more employable. Someone who learns how to solve mathematical problems enters the mainstream culture of mathematics; he or she develops great confidence and can inspire others. Best of all, problem solvers have fun; the adept problem solver knows how to play with mathematics, and understands and appreciates beautiful mathematics.

An analogy: The average (non-problem-solver) math student is like someone who goes to a gym three times a week to do lots of repetitions with low weights on various exercise machines. In contrast, the problem solver goes on a long, hard backpacking trip. Both people get stronger. The problem solver gets hot, cold, wet, tired, and hungry. The problem solver gets lost, and has to find his or her way. The problem solver gets blisters. The problem solver climbs to the top of mountains, sees hitherto undreamed of vistas. The problem solver arrives at places of amazing beauty, and experiences ecstasy that is amplified by the effort expended to get there. When the problem solver returns home, he or she is energized by the adventure, and cannot stop gushing about the wonderful experience. Meanwhile, the gym rat has gotten steadily stronger, but has not had much fun, and has little to share with others (Zeitz 3).

Works Cited

Zeitz, Paul.  The Art and Craft of Problem Solving. New York: John Wiley & Sons, 1999.  Print.

Write a comment on this post that answers at least one of the following questions:

  • In your opinion, what is the purpose of mathematics: to do exercises or to solve problems?
  • How do you feel when you’re solving problems?
  • How does this reading apply to life outside of a high school mathematics class?

44 thoughts on “What is a Problem?

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  2. Bryan Tiben

    When I finish a math problem and its difficult I think and think and when I finally get the answer to the problem after leaving lost but finding the answer through thinking I feel like I wanna take my answer and show it off right in someones face and saying HAAAAAAAA i got it

    1. Perla

      same here, but its better if you just keep going because you excitement can cause you to get too excited and you might not get the next one correct .

  3. Lillybeth.

    Q: In your opinion, what is the purpose of mathematics: to do exercises or to solve problems ?

    A: In my opinion the purpose of mathematics is to do mental exercise. To be able to challenge your mind and your knowledge, to learn new things, and to become.. Well, wiser. It teaches you how to be patient, and how to look at things through a different perspective.


  4. Jordan Zuniga

    How do you feel when you’re solving problems?

    Depending on what kind of problem it is, usually math problems I always feel confused and lost like I have no idea what I’m doing. But it’s my fault because my lack of interest in math and also science is very low and with that being said the amount I know in both topics is barely anything overall.

  5. Yadieliz B

    I believe that this reading applies to out side of high school experience because if explains how you can sit down and learn many things from books and you can teach some one else about it as well but whats so fun about it if you haven’t gone out and experience it your self. Its like studying for a surgery procedure with books and knowing everything you have to do after you’ve read and studied the book, but whats the whole point of knowing all of that if you haven’t tried it your self.
    -Yadieliz B

  6. Grace

    As stated in paragraph two, when I am solving a problem or when i learn something new I feel much more confident, and once i learn something new I feel motivated and able to learn anything else in life especially in mathematics.

  7. Dimitri Torres

    The purpose of mathematics is to work with your mind what i mean by this is to challenge your mind to see how far the mind can go. I feel that sometimes i may get mad if i don’t get the problem right but then when i do i feel good about it. It apply’s to life out side of high school because i think that it is a good way to challenge your mind and sometimes even a fun way to do so.

  8. Sean Woods

    When solving a problem given to me, I feel as if I am being entered into a race and like any race you have to get to the finish line before anyone else to get that feeling of victory. It’s not enough to just be first, being able to explain how you got your answer and why you took certain steps, in my book, it adds style points to the finish. At the end of the day you get to walk away as the guy/girl who could solve that problem before anyone.

  9. Justin lessington

    I believe the purpose of mathematics is to have mastery of problem solving. Also mathematics helps you make smarter decisions in every day life.

  10. Melissa Soto

    In my opinion, mathematics is just like exercising. While trying to figure out a problem you’re puzzling yourself to figure out the answer. It’s just like when you’re in the gym trying to lift a certain amount of weights in a certain amount of time trying to get in a certain shape. In math you’ll do whatever you can to get the correct answer. In the gym, while exercising you’ll do anything to get those abs.

    -Melissa Soto.

  11. Christian Persaud

    I believe mathematics can be one of the many ways to solve problems in real world situations. When I solve problems, I feel as if I’m closer to understanding math and how it may relate to a situation I may come across.

  12. Brenda

    This reading applies to life outside of high school mathematics class in a way where if i become someone to masters the solving problem technique It will help me in how to do things in life where I can use my mind and make better decisions that make me smarter. This skill will only better me in the outside math class.

    Brenda Flores

  13. Jose Gomez

    i think that the purpose of mathematics is to do exercises. its not like you will be able to solve a problem without going through the right steps first.

  14. terrence

    How i feel when im solving a problem depends on how difficult the problem is, if the problem is hard and i answer it alone and get it right i feel confident. If the problem is easy i would just give myself a little pat on the back.

  15. ramon mesa

    the thing about life is, is that the decisions you make will effect your life for the better or worse. in this article it tells of two men looking for a work out, the problem solver makes the best of what he can do while the non problem solver plays it safe and and goes to a gym. they both get the benefit of the work out but the problem solver got more. the lesson would be to go further with what you want to do.

  16. Mariam

    In my opinion, Mathematics is mental abuse to humans lol. Math is something we must use in our everyday lives. Without math i think we wouldnt be able to determine 24 hours in a day,7 days a week, money etc…

  17. Kavoun

    How do you feel when you’re solving problems?

    I like solving problems , I like experiencing new problems and learning the steps it takes to get there.

  18. Kassandra Hernandez

    Math problems sometimes can be hard , but they involve alot of you thinking so that you can get the right solution that works for your problem , at times it can be a little frustrating but its always good to find more then one way to fix your problem and math does just that . . . .

  19. Naja ^.^

    When I’m solving a problem , i feel very determined to get my problem answered. When im finished with the problem i feel accomplished and proud of myself for trying so hard to get things done after all the hard work.

  20. Jose M. Florian

    To me math is to exercise the mind for problem solving in the near future during my career hood.

    If I solve a problem quick with the correct answer I feel GREAT, if I take a while I get anxious to learn how to get the correct answer or i’ll feel like i’m not good enough.

  21. Logan The Boss

    I think the purpose of mathematics is to exercise the mind, well at least to me because when i do problems my thinking expands not only for mathematics but for other various things not involving math

  22. Melissa Soto

    When i`m solving a problem that is difficult for me to get the answer to I sometimes just end up giving up, but if i have someones motivation or if i`m working with someone else then I try my best to get it done then i feel really good when i finish it and get the answer correct . Math is a difficult subject since there are so many different things to learn but once you get a hang of it then you`ll get it.

  23. Iselsa Jimenez

    The purpose of math is to solve problems. I feel bad when I dont know how to solve a problem but when I understand it I feel relax. This reading applies to life outside the school in many ways like in college or when your calculating a problem. ^__^

  24. Diamond

    In my opinion the purpose of mathematics is to exercise and to solve problems. For example before a math class starts there would be an exercise to warm up your brain. But it’s also a purpose because in life there are gonna be a problems such as if you are buying something you have to figure out how much money is it and how much you have.

  25. Big Daddy Kelvin ;D

    Q.How do you feel when you’re solving problems?

    When I’m solving a problem I i always feel a different type of way weather the problem is hard or easy.

  26. Darryl Taylor

    In my opinion math can be used in many ways, and for many things.there’s easy math and there’s math that is more complex and requires much more thinking , me personally problem solving can be easy if i really pay attention.

  27. Brandon Garcia

    The purpose of math to me is a little bit of both. A lot of it is working on different exercises and working your mind. Exercises can help you gain knowledge about different types of problems. Solving problems in math is a way of thinking about something and putting numbers, information and whatever is given to solve the problem. When i am problem solving i feel challenged, there is so many things that you have to think about and numbers you have to keep track of. This reading applies to life outside of high school because i can use math to solve problems non school related and it can help me out in many different situations.

  28. Noheli Perez

    in my opinion i believe that problem solving is difficult it can be compared to going to the gym because you really cant get much from just solving them . if you try harder and put effort into your answer you will get to another level

  29. Safiyyah Alli

    I think the purpose of mathematics is to help you solve problems that isn’t familiar to you and to challenge your mind.

  30. Catherine Lantigua

    To me a problem is like a maze it makes you think in many different ways, it makes take different paths to try to get the right answer.

  31. Isatou Sinera

    In your opinion, what is the purpose of mathematics: to do exercises or to solve problems?
    A- in my opinion i think the purpose if mathematics is do exercises, to be able to challenge your mind and yourself on problems you don’t know how to do. Exercises Help set your Mind to Do Things Your mind thinks u can’t do and its the same for the problem solving.

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  33. Jamal Miles

    When I answer a difficult math problem I feel really proud of myself after overcoming such a difficult challenge. Along with everyone else after I complete I also get that “YES I GOT IT CORRECT.”

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