Open Middle Math: Problems That Unlock Student Thinking, 6-12

Book Preface

In the classroom, we have choices. When it appears that students don’t understand what we’re teaching them, we can explore their thoughts using questions, or we can express frustration about them not understanding the lesson being taught. After all, we were clear in our instructions when we delivered the lesson, so why don’t they get it?
Our desire to uncover and explore students’ thinking instead led Robe1t and me to start creating and collecting Open Middle problems. The problems reveal how deeply students understand what they’re learning, often using fewer problems. They are designed to be more accessible, allowing educators to evaluate how students build conceptual understanding through their various attempts at solving a problem. Open Middle problems allow educators to peek into students’ minds to see what mathematical understandings they have.
As an example, here’s one of my favorite Open Middle problems:
“What is the least number of geometric markings needed to demonstrate that a quadrilateral is a square?”

How do you think a student might approach this problem? For example, a quadiilateral that used eight geometric markings (four markings for right angles and four markings for congruent sides) wotlld be a square. So, we know 8 is a possible answer, but is it the least possible answer? Could a sh1dent demonstrate that a quadrilateral is a square in 7 markings or fewer? What exactly is a “geometric marking” anyway, and how might sh1dents define it? How might that intentional vagueness in the problem’s wording push students’ thinking fu1ther?
I vividly remember presenting this problem at a math educator conference. The discussion was exhilarating as teachers argued how their quadrilaterals used the least number of markings. In h1rn, other educators argued back, trying to convince them that their quadrilateral cotlld be a non-square shape, like a parallelogram without right angles, or a kite.
In the midst of a collaborative discussion, I recall someone mentioning how the problem cotlld be solved using “one marking if we just start with a circle … ” This thoughtful and creative approach instantly set the room ablaze with excitement! I was personally amazed because, as the coauthor of this partictllar problem, not once had I ever considered using a circle as a geometric marking.
With traditional problems, students are told what steps to take and know when they have finished a problem. The bland, rote structure of traditional problems tends to limit flexibility and creativity and tlltimately dampens the fervor of learning in most students. They check out, feel frustrated, or mindlessly follow the steps in their notes instead of thinking about and reflecting on their strategies.
In contrast, Open Middle problems often ask students to find strategically chosen types of answers such as the least or greatest solution or one that’s closest to a ce1tain number. This structure requires students to prove to themselves and others that they have truly fo11nd the best possible answer. Rather than putting down their pencils and saying, “I’m done,” students continue to think, argue, and work. They develop the habit of making multiple attempts to solve the problem, each time wonde1ing if they can come up with an even better solution than their last.
Let’s face reality. We live in a digital age where the average person casually walks around with information at their fingertips from a powe1ful computer in their pocket. This is not what it was like when we were children. As a result, the kinds of skills and knowledge that were necessary and useful when we were growing up are not the same as what sh1dents need now.
Acknowledging this fact begs the question: are we really giving students the skills they need to be successful critical thinkers? If a smartphone can readily answer the problems we give them, are we giving them the 1ight kinds of problem? Open Middle
problems are designed to develop c1itical thinking, analytical reasoning, and problem solving, precisely because they are difficult to answer using a device, trick, or procedure.
The clear and powerful book that you are about to read will equip you with the skills, experience, and mindset required for educators to develop students into problem solvers. It provides you with what you need to compete against a c1llt11re that deems it completely acceptable, and even cool, to say, “I’m not a math person” or “I’m bad at math.” It will help you answer the questions, “How do I know that my sh1dents really learned and can apply what they are learning?” and “Are sh1dents just imitating what I showed them, or do they deeply understand?” It will also guide you in yo11r understanding of both why and how you can sta1t implementing Open Middle problems 1ight away.
I hope that Open Middle Math will excite you to explore what your students know and help them develop deeper mathematical understandings for what’s ahead.

-Nanette Johnson

WHAT DOES AN OPEN MIDDLE CI,ASSROOM LOOK LIKE?

The bell rings and lunch ends. Sweaty students gather outside your classroom and slowly shuffle in, finding their seats. As the chatting softens and class begins, they look at you and the math problem you’ve written on the board. You’re trying something new today and are cautiously optimistic about how this unfamiliar experience will be received. The problem looks different from what they’re used to, and they wait for you to explain what to do.
As you describe the problem, you hear the groans and whispered resistance. Too many of your students believe that math is something where the teacher tells them what to do and then they repeat those steps dozens of times. This problem doesn’t follow that pattern, and they’re not sure what to make of it. Once you’re done explaining the instructions, students begin working on the problem. They don’t solve it on their first attempt, and the lesson begins to feel like many others you’ve taught. It’s what happens next, though, that surprises you. Strangely, many of the students who often give up instead sta1t trying the problem again. The familiar clink of pencils dropping onto the table as students check out is much fainter than usual. Slowly you start to notice a different energy taking over the room. Kids seem to be on a quest to figure out the answer.

Many students begin placing themselves in self-imposed friendly competitions against each other, struggling to see if they can improve upon their previous work. Those you frequently find daydreaming are actually excited to figure out how to get the best answer. Students who have felt comfo1table with years of following the steps in their notes are unsure of what to make of the experience and have not fully bought in, but you’re optimistic that they’re on a path toward making sense of mathematics instead of just getting the answer. All you can see from students who normally finish assignments quickly and complain about being bored is the back of their heads and their ft1riously moving hands. Kids start chatting, going back and fo1th. Initially it sounds like they might be off task, but you realize that they’re talking about the problem, whose answer is better, and how they got it. In fact, they’re sharing math discoveries with each other like they’re the first people to realize them, even though you’ve been telling them those same things for weeks.
Minutes fly by, and the time you’d normally spend keeping kids on task is spent guiding students who need help and facilitating powe1ful classroom conversations around problem-solving strategies. Sh1dents still have plenty of misconceptions, but you see them more easily than you ever have before, and they give you a clearer pict11re as to how you’ll want to adjust ft1h1re instruction

Eventually the bell rings, but almost no one leaves. Instead, they beg you for a little more time to work on the problem. It feels like you’re being pranked, because this can’t possibly be happening. You remind them that they’ll be late to their next class and have to leave. There’s more groaning and whispered resistance, but this time it’s for an entirely different reason. Slowly they get up, telling you that they loved this problem and hope to do another one tomorrow.
Some students are in fact late to their next class, and you get a call from their teacher to ve1ify the outlandish excuse they gave. They said that the reason they were late was because they were working on a math problem in your class and didn’t want to stop. Laughing to yourself, you confirm their reason and explain that it won’t happen again. The other teacher hangs up in disbelief, and you stand there feeling the same way. You had hoped this was possible, but it wasn’t until you saw it happen that you were able to believe it.