How Can We Teach Problem Solving?

We continue to plan our co-op courses for next fall. Some of the classes I had hoped for will not happen, and my children are going to have to make some tough choices between the remaining topics. Unfortunately, they have not yet mastered the ability to be in two classrooms at once.

I have three math courses to plan, and I think I will focus as much as I can on teaching math through problems, even at the elementary level. These are once-a-week enrichment classes for homeschooled students, so I assume they have a “normal” math program at home. I want to introduce a few topics they might not otherwise see, to deepen their understanding of the topics they have studied, and to give them a taste of that “Aha!” feeling that comes from conquering a challenging problem. Has anybody done something like this, and can you recommend some good resources?

Meanwhile, I am in the mood for some inspriation:

There is a distinction between what may be called a problem and what may be considered an exercise. The latter serves to drill a student in some technique or procedure, and requires little, if any, original thought. In contrast to an exercise, a problem, if it is a good one for its level, should require thought on the part of the student. It is impossible to overstate the importance of problems in mathematics. It is by means of problems that mathematics develops and actually lifts itself by its own bootstraps. Every new discovery in mathematics results from an attempt to solve some problem.

— Howard Eves
quoted by Rosemary Schmalz, Out of the Mouths of Mathematicians

The first and foremost duty of the high school in teaching mathematics is to emphasize methodical work in problem solving…The teacher who wishes to serve equally all his students, future users and nonusers of mathematics, should teach problem solving so that it is about one-third mathematics and two-thirds common sense.

—George Polya
Mathematical Discovery, Volume II

Solving problems is a practical skill like, let us say, swimming. We acquire any practical skill by imitation and practice. Trying to swim, you imitate what other people do with their hands and feet to keep their heads above water, and, finally, you learn to swim by practicing swimming. Trying to solve problems, you have to observe and to imitate what other people do when solving problems, and, finally, you learn to do problems by doing them.

—George Polya
How to Solve It

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17 comments on “How Can We Teach Problem Solving?

  1. I am actually writing a book that you might find to be a useful resource (at least, if I fulfill my goals for it, you will find it to be a useful resource!), but unfortunately it is FAR from being done. =)

    You can find a preview version of the first chapter here, on Boolean algebra and first-order predicate logic. (Although you should note that I’ve actually made lots of changes since then and I’ll probably be putting out a new version of the chapter shortly, along with preliminary versions of the next two chapters.) It’s aimed at the high-school level — I don’t know if you’ll be teaching high school students — but some of the topics I intend to cover include proofs and problem-solving, number theory, sequences and series, abstract algebra, combinatorics, set theory and infinities, maybe a bit of topology… none of it very in-depth, but just trying to give the flavor of some of these topics to interested high-school students, through the lens of problem-solving.

    If you were actually interested in using portions of my book, that would be tremendously exciting and would probably motivate me to write it faster… I imagine I could have a good deal more done by the fall! =)

  2. Hi, Brent! I had already downloaded your chapter 0, and I enjoy your writing style. I look forward to seeing the revision and new chapters. Most of my students are elementary-middle school, but I hope to coax some high school students into one of the classes by offering more advanced material.

    [By the way, I vote "Yes" for including solutions in your book. I sometimes get stumped, especially with high school problems---it's been way too many years since then---but as the teacher, I'm supposed to know the answer. Back-of-the-book to the rescue! Once I see it, I can usually figure out how to get there, but I would be much less confident teaching from a book without answers.]

  3. Hi Denise,

    Thanks! I will be sure to let you know when the revision and new chapters are online. (You will be happy to know that due to many comments similar to yours, I saw the error of my ways and fully intend to include complete solutions!)

  4. Wow!
    So much to be said for math!
    I have to tell you…it is mine and my children’s least liked subject.
    I know that this is my own fault.
    In your opinion, what is the one best way to have your child memorize multiplication?
    We love to read, so we read books on logic and math, but as far as memorizing facts…that is our downfall.

  5. Pingback: Carnival of Mathematics IX « JD2718

  6. Making sure kids have some concrete application for all the decimal & fraction rules they learn is so important! It’s so good for them to get to focus on application because it helps them realize that the rules & steps they learned are a means to an end, not the end itself.

    There are a couple of books I used at the middle school level — Real Life Math: Decimals and Real Life Math: Fractions. You can see them here
    along with some other books that look interesting.

    As for memorizing multiplication facts, there is another book called The Mad Minute that is filled with one minute drills from basic math facts to percents — I loved that book. And there is a book called Speed Mathematics that teaches an interesting way to learn the tables. I haven’t seen it used, but I have been enjoying the book, and it is probably worth a look. Good luck!

  7. Yes, Alane. I am amazed at how many people seem to think that the point of math is to learn the rules and steps. Of course, the algorithms are important, but if there is no foundation of understanding to support them, how can you build the next layer of math? Everything gets mixed up in the student’s mind and the whole structure starts to crumble.

  8. I do something like what it sounds like you are planning with the kids at my children’s small private school. Their regular teacher teaches them the “basics” and I do problem solving with them. Lots of fun. You don’t say what ages you are working with.

    For upper elementary and beginners in middle school, I like George Lenchner’s Creative Problem Solving in School Mathematics.

    For younger kids the best thing I have found is the “Figure it Out” series from Curriculum Associates
    There are small workbooks at “grade” 1-6 levels (they do get harder as they go on, but they’re not sequential, so start at whatever level feels right for your group. I had a mixed group of 7-10yos working on the 4th grade book this year, and some younger kids doing books 1 or 2 (depending on their experience)

    Rapid Math Tricks and Tips by Edward H. Julius is fun for teaching some mental math “tricks” that I believe increase numeracy. For older students, make them explain/prove why the tricks work.

    Challenge Math by Edward Zaccaro is good for grades 3-7 or so, and there’s a primary version as well for younger kids.

    Pascal’s triangle has all sorts of cool patterns to play with for upper elementary and middle school. Some nice lessons here:

    Hope that helps, and have fun!

  9. Hi, mathmom, and welcome back! I’ve missed you. Does this mean you’ll be actively blogging again?

    I have one class for grades 4-6 and two for grades 7+. I was leaning toward the Lenchner book for the younger set, along with number games to increase their mental math skills, and an Art of Problem Solving book for the older ones, along with old Math Counts tests. (Though I am also curious to see what Brent comes up with for the rest of his book.)

    Thanks for the other suggestions—they sound great! I will definitely have to look them up. (Especially the series for younger kids, since I may offer a younger class in the future, and I never know what to do with that age group.) I wonder if my library has any of those in stock…

  10. I’m not sure about acctively blogging again or not. So much to take up my time. But I’m actively reading again in any case, so hopefully you’ll see me commenting at least.

    I agree that the MathCounts materials are great for your older group. (In addition to the tests, they have a handbook with problems that start easy and get harder, which are IMO better than the tests for working with gropus of kids.) Will you take a group to MathCounts? You are allowed to make a team from a homeschool group. I have found, however, that when you start out with middle schoolers who don’t have a lot of problem solving experience, it helps to start with a single topic, rather than the mish-mash you’ll find on a MathCounts test or exercise. (They do have stretches that are good for single-topic work.)

    For my grade 6-8 group, I started the year with a triangle puzzle (triangle made of 6 circles, 3 at the corners and 3 in between — how many ways can you arrange the numbers 1-6 in the circles so the sum along each edge is the same) and a magic square. I asked them to figure out in advance what the sum of each row/column/diagnoal would have to be. I asked them to also record observations about where certain numbers must or could not be placed.

    Other topics I did with this group include: patterns and algebraic reasoning (being sure to introduce triangle numbers), handshake problems (oh, look! triangle numbers again!), prime factorization (then used a Factoring Stretch from Mathcounts 01-02 handbook), Venn Diagrams (there’s a good unit on this in Challenge Math), divisibility rules, mean/median/mode/range, converting repeating decimals to fractions (and what happens when you use that method on 0.99999… and what does *that* mean?), similar triangles, pythagorean theorem, pascal’s triangle, logic grid puzzles (check out As they got experienced with multiple problem types, I often made up mixed problem sets drawing from mathcounts problems and anything else I could find on the Internet. About halfway through the year I started also using intact MathCounts warmups and workouts that were carefully selected to contain at least a couple of questions directly related to topics we’d already covered.

    I’m trying to come up with a 3-year rotation of topics that isn’t too order dependent because most kids will be in that group 3 years, but the group shifts every year. The above selection was what I did this past year.

  11. Thank you for the ideas! I have done several of those things with my math clubs in the past. Last year was my first experience with Math Counts—the $80 fee always kept me away, but somebody else volunteered to pay it, so I discovered what I had been missing. We simply went through the Warm-up and Work-out problems, which was fun, but I don’t know how well the concepts stuck. The kids did well enough on the Chapter test but were completely blown away by the State.

    This year, I am planning to go topically, using mainly the Stretches at the beginning of the year, so the students have more concentrated practice on each new idea. Since I hope to get some high school students, too, I will have the AoPS problems available. My difficulty there is that the more difficult AoPS problems are over my head—but it is always fun to learn along with the kids!

    Unfortunately, I think many of the Stretch problems are easier than the average Math Counts question. The ideal, in my opinion, would be to go through and re-arrange the Handbook problems and old tests topically in order of increasing difficulty. There’s no way I have time to do that! But your idea of coming up with a 3-year rotation sounds good, because then I could gradually tweak it into shape and add more challenges. I definitely prefer tweaking to planning from scratch.

  12. There is a topic index for each of the MathCounts handbooks, but to look for problems on a particular topic I often used the Windows “search for files” containing a particular keyword, believe it or not.

    Oh, there’s AMC stuff arranged by topic here:,0-quizes.html

    I have my middle school group for the last time this year on Monday. Any ideas for something fun to do with them? I have nothing planned yet.

  13. Some of our favorite math club activities have been:
    Function Machine—Prepare a set of cards in advance, so each child gets a chance to be the “machine” while the other students guess.
    Straw Polyhedra—Pinch the short ends of bendable straws and insert into the long ends to make polygons, then tape together with masking tape. Plenty you can do with these: vocabulary (face, vertex, etc.), Platonic solids (why only 5?), names of polyhedra, make up your own combination and name it, look for the pattern V + F = E + 2.
    Compass Constructions—For instance, here.
    Knights and Knaves—Logic puzzles from Raymond Smullyan’s books.

  14. A fun way I have found to get students involved in problem solving is by getting them to play Mastermind, typically as a group. I find it works well as there are a base set of strategies they can use but they will use them slightly differently depending on how the game plays out.

    This illustrates to them nicely that these are not just recipes that you apply to given scenarios but skills that you modify and craft to suit.

  15. Mastermind is a great game for building logical thinking skills. How do you play it as a group? Do the students take individual turns guessing, or do they collaborate on choosing the next guess (sharing strategy)? Or do you mean that you have a whole slew of 2-player games going at the same time?

  16. Yes, I get the class to play as a whole group. I like to use this website on the screen ( I’m at the front controlling the game and placing the pegs on the game board. Anyone can call out their opinions on which pegs to place where or if they disagree with a placement. If anyone disagrees however they must state why they disagree. I’ll probe them with questions, especially when I can see that logically a certain peg is wrong etc. If I can see that a particular student has discovered something but is being quiet I’ll ask them specifically to share with the class. It’s great when they start bouncing ideas off each other and challenging assumptions etc.

    Normally the first game is when they get the hang of it and it’s the second and subsequent games where many students start getting that ‘a ha’ moment and a lot of useful discussion starts happening.

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