We often know a good math problem when we see them — or maybe after working on one for awhile — but it’s hard to say just what makes a problem oh so good, and by “good” we mean engaging and mathematically worthwhile. We have come to value the following features, which we think are essential components of a good problem:
1. The goal is clear and concise.
Like a good story, people should be pulled into a good problem right from the start. If it takes too long to figure out what the problem is asking, or if it is broken up into too many little bites, it becomes difficult to know what the goal is or why you are even exploring the problem in the first place.
A clear and concise statement of the problem and a quick “hook” is the best way to pull people into the problem, and making the problem (i.e., what’s being asked) easy to understand is also a really important aspect of making mathematics accessible to all students.
Here are some examples of problems with clear and concise goals. (They don’t necessarily adhere to the later criteria, but they definitely meet criterion #1.) They involve some background knowledge, but if you have that background knowledge, you can be hooked quite quickly:
Is it possible to cut up any trapezoid and rearrange it into a rectangle?
Why is it that every sum of 5 consecutive numbers is divisible by 5?
Does the curve for y=1.01x ever go above the curve for y=100x2+1?
Which is larger, the red area or the blue area?
Note that the central goal can be general or particular in nature, and the hook can be based on a pattern, a key question, or a visual cue. There is a wide range of possibilities for crafting a hook. What we don’t see here are problems that begin with a page-and-a-half of setup and several sub-questions before we ever get to the ultimate point of it all. To be clear, we are not saying that large amounts of information or contextual background are incompatible with good math problems. Indeed, many good math problems will involve deep dives into relevant contexts and sifting through potentially useful information to decide what needs to be incorporated into the solution, but our point is that, with good problems, we already have the goal in mind early on rather than only at the end (or not at all).
So, in a good problem you shouldn’t have to work too hard on figuring out what’s being asked, but…
2. The solution path is not immediately obvious.
If the answer, or the path to that answer, is really obvious, then it’s not a problem at all, is it? Another way to say it is that, if the first thing a person tries works out right away, then that’s more of a procedural exercise than a problem.
In a good problem, there is no one clear path that leads directly to the answer. There might be several paths that seem plausible and we won’t know if they work until we try to follow them and see how far we get — maybe they’ll all work, which would be nice, but we can’t tell at the start. Or maybe there is no clear path at first and we have to explore a bit in the dark before we find a promising first step. (Exploring in the dark is not always a positive experience, but that’s why having a clear goal is so important — at least you know what you’re exploring in the dark for.)
A corollary of this feature is that good problems will necessarily take at least a couple minutes to solve (or to arrive at a satisfying stopping point). If you can solve it in 30 seconds, we wouldn’t really consider it a good problem. It’s important to spend at least a bit of meaningful time working on the problem because this gives you a chance to think about what you are doing and where you are headed, it gives you an experience to possibly share with other people, and it allows you to invest time and effort into the problem so that it really pays off when you do arrive at a solution.
Working on good problems can sometimes take 2-3 minutes (but it was a meaningful and rewarding 2-3 minutes), or it can take an hour, or even several weeks. These time frames don’t guarantee that it’s a good problem (unfortunately people can waste hours working on bad problems, like doing long division for 12,389,010,270 divided by 112 or trying to find the reduced-row-echelon form of a random 5 × 5 matrix) but we do think good problems take some effort and some time to crack open.
So if most of our efforts are spent searching for a workable solution, it’s important that…
3. The problem allows for multiple fruitful paths.
Sometimes a problem satisfies feature #1 (clear goal) and feature #2 (takes some effort to find the solution) but in the end there is really just one way to break through and solve it. Take the following problem as an example:
You will likely try many things that don’t work at all until you finally luck into the key to unlocking it (hint: notice that the first “addend” is a divisor of the “sum”), and in retrospect it seems as though the problem was overtly misleading (the addition symbol is not actually representing addition).
These problems can be fun to share and there is often a feeling of accomplishment after you do break through, but they can typically serve as a “trick” problem or a brain teaser. People who solve them form an exclusive club of those who figured out the trick. (Some people like to share these problems just to see if others are as “clever” as they are. For instance, here is a trick problem where people can actually get membership cards if they figure out the trick.) But, if you happen to think about things in a different way or haven’t had the experiences that allow you to break through, then you will probably remain stumped for a long time and eventually just break down to “Okay, tell me how to solve it.”
Instead, we prefer problems like this one:
Define a binary operator § such that 1 § 1 = 1 and 2 § 4 = 16. What are some other instances of your binary operator in action?
This problem allows for multiple fruitful paths. One person may think about exponents and define a § b as ab. But other people may think differently, such as a § b = a4 or
a § b = ab when both are odd and a § b = 2ab when at least one is even. All of these ideas actually adhere to the requirements of the problem and can bring up interesting ideas about operations’ properties (e.g., one of the above is commutative) and well-defined-ness (e.g., two instances does not uniquely determine an operation).
These types of problems with multiple fruitful paths allow for diversity of thought and make the problem much more interesting to talk about with others, because we can be genuinely interested in how other people thought about it.
So far, the criteria we’ve laid out could result in a good problem in any subject area, so let’s specify that…
4. A good math problem involves worthwhile mathematical ideas.
What makes a problem worthwhile will depend on the person. Something that is worthwhile to a fourth grader may not be so worthwhile to a calculus student. It’s not just what level of mathematics the person is in, it’s also what aspects of mathematics they value. For example, people committed to using mathematics as a tool for enacting social justice may view mathematical representations that quantify inequities as worthwhile. People who hope to want to become engineers may view differential equations as worthwhile. (And for others, worthwhile mathematics may just hinge on criterion #5; see below).
The math in a good problem is neither too laden in mathematical minutiae nor is it too lost in a “real-world” context. For example, the Daily Brain Teaser problem up in #3 ends up hinging on a fairly convoluted exponential function that is probably not worthwhile for most people. And the comparison problem between y=1.01x and y=100x2+1, although it does have a clear goal, is probably not very worthwhile on its own. But it might be worthwhile if it was extended into the bigger idea about exponential growth versus polynomial growth in general. An example of a problem that is probably worthwhile for many students of school mathematics is the Red/Blue region problem above, because it may seem esoteric at first but actually leads to important ideas about the relationship between triangular area and parallelogram area and about the invariance of the area comparison even as the situation varies continuously.
Problems that meet criterion #4 are in the Goldilocks zone. They involve pertinent and attainable mathematical ideas and practices. Overall, this feature helps to make explicit that a good problem is not necessarily a good problem everywhere and at any time — it is a good problem for specific purposes and with specific people.
And finally, to get a bit subjective here…
5. A good math problem is fun to work on.
There might be problems that satisfy #1 through #4 but are just kind of cumbersome to work on or do not really click with individual preferences. And some problems may be a lot of fun for one person but do not resonate with others. So ultimately there is a bit of personal subjectivity here, but hopefully, if you are looking for good problems, your taste in good problems is at least somewhat similar to the taste of your peers or your students or whomever else may be doing the problems with you.
Conclusion and Resources
One important caveat is that these features are what we have found to be important for good math problems, but this does not extend directly to math curriculum. A curriculum is much more than a set of good problems, so we don’t want anyone to mistake this for an endorsement of a particular kind of curricular or instructional approach. We think good problems are a necessary part of a good math curriculum, but curriculum also has to have well-designed concept development along with procedural practice. It should be relevant to learners’ lives and, ideally, it should be aligned with a goal of improving the world we all live in.
But with regard to good math problems specifically, here are some places where you might find some:
See more at http://bit.ly/GoToMathTasks
Share your good problems (or problems that might be “good” but you’re not sure) in the comments below.
By Samuel Otten (firstname.lastname@example.org) with Zandra de Araujo and Jaepil Han