What is a problem? And what types of problems are most useful for helping students learn to love math?

I struggle to define “problem” despite always having tons of them rattling around in my brain. It seems like a decent way to define it is to offer a bunch of examples. Here are some favorites:

**Split 25
**

(Play With Your Math)

**Self-Aware**

(Play With Your Math)

**Cows in Fields**

(Singh)

**To Cross the Bridge**

(Singh)

**The Census Taker**

During a recent census, a man told the census taker that he had three children. The census taker said that he needed to know their ages, and the man replied that the product of their ages was 36. The census taker, slightly miffed, said he needed to know each of their ages. The man said, “Well the sum of their ages is the same as my house number.” The census taker looked at the house number and complained, “I still can’t tell their ages.” The man said, “Oh, that’s right, the oldest one taught the younger ones to play chess.” The census taker promptly wrote down the ages of the three children. How did he know, and what were the ages?

(Batchelder & Alexander)

**Long Division**

The following long division problem has a unique solution, despite providing just one digit. The Xs can represent any digit, and the problem is an 8-digit number divided by a 3-digit number producing a 5-digit number and dividing evenly.

(Gardner)

**105**

In how many ways can 105 be expressed as the sum of at least two consecutive integers?

(Liljedahl)

**Circle in a Parabola**

There are many circles that will “fit” inside a given parabola. What is the largest circle that will do so? Why?

(Lockhart)

**7.11**

A guy walks into a 7-11 store and selects four items to buy. The clerk at the counter informs the gentleman that the total cost of the four items is $7.11. He was completely surprised that the cost was the same as the name of the store. The clerk informed the man that he simply multiplied the cost of each item and arrived at the total. The customer calmly informed the clerk that the items should be added and not multiplied. The clerk then added the items together and informed the customer that the total was still exactly $7.11.

What are the exact costs of each item? (Assume that they multiply to 7.11 exactly, with no rounding.)

(Brumbaugh)

**No Trigonometry Required!**

(Brilliant)

**November Nonagon
**The figure below shows a square within a regular nonagon. What is the measure of the indicated angle?

**Why Problems? **

I think math is worth learning for lots of reasons. I want students to be quantitatively literate in a world that increasingly requires mathematical knowledge to be an informed citizen. I want students to understand math to open doors for them in the future, as mathematicians or in any number of other disciplines that rely on mathematics. I want students to cultivate skills of abstract reasoning, recognition and generalization of patterns, critical argument, precision, and structure. I want students to see math as a subject full of challenges that they are capable of overcoming, and for math to help them recognize their intellectual potential.

But from my perspective, the most important piece is for students to get a sense of the beauty and joy of mathematics, and to experience the “a-ha moments” that characterize our discipline. English has great literature. Science has the mysteries and wonders of the natural world. History has the gripping narratives of the past. Math has problems.

**Two Things**

I want students to experience the a-ha moments of problem solving as a catalyst to help them understand the discipline of mathematics and their potential as mathematicians. But not all problems are equally useful for creating these moments. I’d like to hypothesize two elements that allow a problem to facilitate students’ love of problem solving.

**Insight vs Experimentation **

On one end of a spectrum are insight problems, like “November Nonagon,” “To Cross the Bridge,” and “No Trigonometry Required!” These problems lend themselves to certain representations and strategies, but the approaches one takes at first are unlikely to be successful. Solving the problem relies largely on an insight: a change of perspective that illuminates a path to a solution. A solver might end up staring at the problem, making no progress, for some time. With some luck, the insight will whisper itself at an opportune moment, and the problem will be solved. On the other end are experimentation problems, like “Circle in a Parabola,” “The Census Taker,” and “Split 25.” These problems lend themselves to trial and error and don’t require any large leaps of logic or intuition. A solver can try a number of different approaches, stepping back to look for patterns as necessary, on a much more well-defined path to a solution. That’s not to say that these problems are easy, just that they are more likely to suggest plausible pathways than dead ends.

**Success**

A second spectrum is how quickly a solver is likely to experience a feeling of success — whether or not they solve the problem, can they make some concrete progress early on? The problems “Cows in Fields,” “Circle in a Parabola,” and “105” allow a solver a quick taste of success, where one or several examples are readily available, although finding all of them still requires a great deal of persistence and ingenuity. These successes can act as springboards to the rest of the problem, rather than experiences of frustration from the beginning. Alternatively, problems “Self-Aware” and “7.11” resist easy wins. One could try a few ideas, but they don’t lend themselves to quick strategies, and a successful solver will likely have to muddle through a significant amount of failure, trying unsuccessful ideas, to get to a solution.

**What Makes a Problem Useful?**

I think that the best problems to teach students a love of problem solving allow for experimentation facilitate early success. Experimentation allows multiple access points, gives students half-formed and informal ideas to share and argue about, and gives a sense that, while the journey may not be easy, it is at least possible. Early success builds motivation; feelings of success help students understand that problems exist for the pleasure of solving, rather than to frustrate and bore them.

These aren’t necessarily static properties of problems. A teacher could facilitate experimentation in “November Nonagon” with the suggestion that a solver try adding auxiliary lines, or in “No Trigonometry Required” with the hint that the angles can be rearranged (without changing their size) to try to make a useful shape. Similarly, “Self-Aware” could be modified to make early success more likely by prompting students for 5- and 7-digit self-aware numbers in addition to 10-digit ones. These small changes, combined with choosing problems thoughtfully given students’ knowledge and motivation, can make a big difference.

This isn’t to say that problems without these characteristics are worthless — they can be fantastic fun for students who have already developed some interest and joy in doing math. But to create that a-ha moment that shifts a student onto the path of being a math lover, I think these two features are critical. Staring at a problem with no clear paths forward or ideas to try is likely to result in frustration for many students. And even when there are clear ideas to try, without some positive reinforcement of early success a student is likely to give up before they get to the good stuff. Not all problems fall neatly on one side or the other and no problem is perfect, but I do think these two features make a problem much more useful for all students, rather than just those who already like math.