# What Makes a Good Math Problem?

## June 24, 2020Kate Gasaway

As a team, we talk a lot about what’s wrong with specific problems and how we can improve them. In The Evolution of a Math Problem, I detailed an example of one of those conversations. I’d prepared that problem as a short talk for our weekly meeting, but that was only one of six problems that the math team presented as examples of problems with problems. They were uniquely, and sometimes hilariously, flawed. This deep dive made us think: It seems like bad math problems are bad in their own ways, but on some level, are good problems all alike?

I thought about this a lot over the last few weeks as I wrote, edited, and reviewed problems. I was particularly interested in general, high-level criteria that any math problem, regardless of grade, unit, standard, or assignment, could be held to. For example, while it’s always important that a math problem should align with the standard or topic of a lesson, you usually can’t evaluate that without a lot of context.

So far, we’ve come up with four broad characteristics that we think are essential for a good math problem: Clarity, diagnosticity, tractability, and joy.

### Clarity

A good math problem communicates the outcome it wants a student to work toward without ambiguity. This was the sticking point with the couch problem from The Evolution of a Math Problem. In all the draft versions of the problem, there was more than one reasonable interpretation of the scenario that could lead to different answers. Whether a student’s interpretation matched my intent would basically have been luck.

Here’s another example of an unclear math problem:

 Gabrielle orders 8 pizzas for her class. She has 21 classmates and estimates that each person will want to eat $$\frac{3}{8}$$ of a pizza. Did Gabrielle order enough pizza?

This trouble with this problem is that you don’t know whether to include Gabrielle or not. If you don’t include Gabrielle, there’s enough pizza. If you do, her order is just short. A student could solve the problem either way and show their understanding of fraction operations. In class this could be an interesting conversation starter, but on a homework assignment or assessment this problem could be a source of worry and frustration. Nevermind how much longer it will take to grade to account for both possible right answers.

A small tweak makes this problem clear while preserving the problem solving:

 Gabrielle orders 8 pizzas for her and her class. Including Gabrielle, there are 21 students in the class. She estimates that each person will want to eat $$\frac{3}{8}$$ of a pizza. Did Gabrielle order enough pizza?

In this version everyone gets pizza, there’s one right answer, and there’s less stress for everyone involved. You could just as easily adjust the original problem so that Gabrielle isn’t included, but why would you do that to poor Gabby?

### Diagnosticity

A good math problem gives meaningful insight into students’ understanding and/or skills. Certainly every problem tells you something, but you want to make sure it tells you what you’re interested in. Take a look at this example:

 Which of the following verbal descriptions matches the expression below? $$(k+5) g$$ The sum of five and $$g$$ times $$k$$ $$g$$ times $$k$$ plus 5 The product of $$g$$ and $$k$$ plus 5 The sum of $$k$$ and $$g$$ times 5 $$g$$ times the sum of $$k$$ and 5

This problem can be solved with logic instead of the skill in question: translating from numerical expressions to verbal phrases. A savvy test taker will know that the right answer has to be unambiguously correct, so answers that can be read in more than one way can’t be right. For example, does option 3 mean $$g(5+k)$$ or $$gk+5$$? You can’t tell, so it can’t be right.

A student who gets this question right might understand the topic, but they might not. You just can’t know.

Re-writing the answer choices for the question like this makes the information you get from it much clearer:

 Which of the following verbal descriptions matches the expression below? $$(k+5) g$$ The product of $$k$$ and the sum of five and $$g$$ $$g$$ times $$k$$, plus 5 The product of $$g$$ and $$k$$, plus 5 The sum of $$k$$ and $$g$$, times 5 $$g$$ times the sum of $$k$$ and 5

Voila! When the answer choices are unambiguous, it’s far more likely that students’ answers are a reflection of their ability to translate between numerical expressions and verbal descriptions.

And yes, I did pick my initials - just like students always do. That or their crush’s initials if you’re really lucky. That’s always fun and/or super disruptive.

### Tractability

A good math problem is solvable with the skills a student has and the information they can reasonably get. This not only means that a problem should assess grade level material, but also that the complexity of the problem should be reasonable. Not only are intractable problems fussy and frustrating, but they can make students feel unsuccessful when they’re actually doing great. This doesn’t mean that good problems shouldn’t be rigorous (they absolutely should be!), just that that rigor should be thoughtful.

Take a look at this example:

 Find the solution to the system of equations below. Show your work using the substitution method. $$3x+5y=11$$ $$7x+4y=13$$

Looks fine, right? Small whole numbers, no fractions, decimals or negative numbers - should be a piece of cake. Well. Five of the constants in this problem are prime numbers. None of them share any factors, so nothing is going to cancel out and students are going to be working for a loooooooong time. (The answer is ($$\frac{21}{23}$$, $$\frac{38}{23}$$), by the way.)

This problem’s tediousness tanks its diagnostic value too. If you wanted to use this problem to learn about students’ ability to use the substitution method to solve systems of equations, you’re probably out of luck. Students are going to spend so much mental effort on manipulating those fractions that their system solving skills will likely suffer. Students who get this problem wrong might actually understand the substitution method. You just don’t know. You will know, however, how well (or poorly) they deal with frustration.

Here is a more tractable version of the system above that lets students’ system solving skills shine:

 Find the solution to the system of equations below. Show your work using the substitution method. $$3x+5y=50$$ $$5x+15y=30$$

The coefficients in this problem are much friendlier despite being larger; 3 and 5 are common factors for most of the terms and 5 can be divided out from the bottom equation simplifying it further. There are also multiple solving avenues that students can explore. Choosing to solve for $$x$$ in the second equation avoids manipulating fractions, but solving for either variable in either equation is reasonable. You can be confident that students’ work on this problem is a reflection of their systems of equations skills and they are far less likely to turn this problem set in as confetti.

Tractability is subjective. What is persnickety and unreasonable for a 7th grader may be just another Tuesday for a high school student. Within a grade one class’s bugbear is another class’s fun challenge problem. You know how your students will respond to challenging situations and when to push them to persevere—the key is to choose wisely and be prepared.

### Joy

When I talked to my fellow math teammates, they universally advocated for a fourth criteria: fun. Good math problems should capture students’ imagination and make them want to solve them. You don’t get any information from a problem students don’t solve.

Consider this example of a clear, diagnostic, tractable problem that doesn’t really spark joy:

 How many times larger is $$(4.2 \times 10^2)$$ than $$(7 \times 10^{-1})$$?

 Nyan Cat travels at $$(4.2 \times 10^2)$$ miles per hour. Grumpy Cat travels at a pokey $$(7 \times 10^{-1})$$ miles per hour. How many times faster is Nyan Cat than Grumpy Cat?

The need to understand scientific notation is preserved and you have a great opportunity to share two of our national treasures with your students. Hitting the right mark with interesting context for students means considering developmental milestones, your and your students’ cultural backgrounds, their comfort with unknown and unfamiliar, and authenticity. Cats made regular appearances in my classroom, so this example resonated with my students and felt genuine, but it could have felt forced in the wrong situation.

When adding creative flair to math problems, make sure you’re also adding support for English language learners and students with disabilities so they can access the problem (and have fun) too. Linking to videos of both cats can reinforce that numbers written in scientific notation with positive exponents represent large values while negative exponents represent values between zero and 1.

Not every math problem can or should be captivating on their own (problems that assess fluency come to mind), but wrapped in classroom routines, competitive spirit, or the satisfaction of a job well done, these can shine too.

We invite you to take a look around the Fishtank curriculum for more examples of clear, diagnostic, tractable, and dare we say, fun math problems. Want even more? A Fishtank Plus subscription unlocks access to tons of additional assessment problems, problem sets, and much more.

Kate Gasaway is the Curriculum Associate for Mathematics. She holds a Bachelor’s of Science degree in Psychology with research and business certificates from the Georgia Institute of Technology, and a Master’s in Effective Teaching from the Sposato Graduate School of Education. She started her teaching career at Neighborhood House Charter School, spending five years teaching 8th grade math and one year teaching 6th grade math.