How Into Math Uses Open Middle Tasks to Build Thinking Classrooms

Open Middle math problems are the key to more safe failure in building thinking classrooms. Yes, failure. Do I have your attention, math teachers? Hear me out.

“Cut the red wire! We only have 10 seconds!”—My impression of every action movie, ever. 

In these cinematic nail-biters, our hero has to get the answer correct on the first try, or something very bad will happen.

So far, real life has presented me with exactly zero red wire countdowns. And yet, this is how our math classrooms sometimes feel: high-pressure and high-stakes.

How does this serve our learners? When they should be thinking, they’re panicking! When they could be exploring, they’re mimicking!

Ideally, math students are like artists: iterative, experimental, and collaborative. In what Peter Liljedahl terms a “thinking classroom,” students should be “getting it wrong.” It’s not life or death, but a liberating and short-lived experience of low-stakes failure before a quick “well, let’s try something else.”

So, math teachers, how can we encourage more failure, more iteration, more “let’s try another wire?”

Building Thinking Classrooms: A quick refresher

In Liljedahl’s Building Thinking Classrooms in Mathematics, the premise is simple to state and hard to swallow: most students in a typical math class aren’t actually thinking. They’re diligently mimicking a process they watched a teacher demonstrate minutes earlier, creatively waiting out the clock, or quietly hoping no one calls on them. They’re avoiding—and often, they’re avoiding failure.

Liljedahl’s solution is a set of 14 practices, grouped into toolkits, designed to build an engaged math classroom where thinking is the default instead of the exception. The first toolkit is the starting place: choosing the right kind of task, forming random collaborative groups, and getting students on their feet, working on vertical, non-permanent surfaces (VNPS)—think whiteboards, chart paper, anything a group can write on and erase together. The point is students teaching each other instead of being passive failure-avoiders.

That first practice, choosing the right task, is where a strong curriculum can do real work for a teacher. Liljedahl is clear that not just any problem will do. Thinking tasks need enough of a challenge to require real reasoning, without so much friction that students give up before they start—and without so much pressure that students would rather anyone else “cut the wire.”

Robert Kaplinsky’s Open Middle problems—and hey, he’s one of ours

In his book Mathematics Tasks for the Thinking Classroom, Grades K–5, Liljedahl himself advocates for Kaplinsky’s Open Middle problems. He defines an Open Middle task as “any task that has a well-defined beginning and a well-defined end, but has a lot of options for how to proceed through the task.” He explains, “Sometimes there is an ideal solution. Sometimes there is not. Regardless, the solution is not the important part. It’s the process that is important. It’s the middle that is important.”

A classic example asks students to arrange the digits 1 through 9 to satisfy some condition in a problem—think Sudoku, but with equations. There’s a right answer, sometimes more than one, but no single prescribed route to get there. Students have to test ideas, notice patterns, fail, and revise.

Here’s where it gets cool: Kaplinsky is a contributing author on Into Math. His Open Middle problems are woven into the curriculum from Grades K–Algebra 1 and are featured in the “Practice on Your Own” sections of all modules. This is right where Liljedahl would advocate for their use as extension problems, because after all, as he says, “One of the main tenets of a thinking classroom is that no one ever gets to be done.”

Open Middle problems are one of the more natural tools already available to a teacher building a thinking classroom—no retrofitting, no searching outside your own materials for a task that fits. Instead, a low-stakes, “I’m not afraid to try because failure is baked right in” problem.

Cutting the wrong wire

Here’s the thing about a “Practice on Your Own” problem: it doesn’t have to mean “you’re finished.” Liljedahl is almost allergic to the idea of “finished.” Students who fly through their independent practice don’t get a gold star and free time—they get another problem that asks more of them.

An Open Middle problem, ready and waiting at the end of a module, is built for exactly that moment.

Picture it: a fourth grader wraps up their practice set early and their hand shoots up: “I’m done!” Instead of “great, sit tight,” the Open Middle problem is right there waiting. Same lesson, same skills, but no survival guide.

And here’s the safe failure: a student who’s already nailed the “original” version of the skill has room to cut the wrong wire, without it costing them the assignment. Low stakes, real thinking, right when they’re ready for it.

You don’t have to wait for the end of a module to use one this way, either. The beauty of teachers knowing best is that we can pull that same problem to the front of a lesson instead. Try putting it up on a whiteboard and setting the stage for safe failure.

Extension problem or launch task, the job is the same: give students something worth getting wrong.

Try an Open Middle problem, tomorrow

The good news: Kaplinsky is a part of our curriculum writing team, and his Open Middle problems—the ones that bring to life Liljedahl’s Building Thinking Classroom tenets—are embedded in our framework.

The great news? You can access Kaplinsky’s Open Middle problems, organized by grade, at openmiddle.com.

Want to build some empathy for safe failure? In the next faculty meeting, turn on the Mission: Impossible theme song (or Pink Panther . . . read the room) and try an Open Middle problem with your fellow math teachers. Not only will you anticipate the pitfalls your students will encounter, but you can remember what it feels like to practice safe failure.

Cut the wrong wire, together.

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HMH’s core math solution Into Math for Grades K–Algebra 1 includes a variety of routines, real-world connections, and problem-solving strategies that deepen students’ mathematical understanding.

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