### Turning Teaching Upside Down

Dec, 22 2020

*Cathy L. Seeley*

Students learn more when we let them wrestle with a math problem before we teach them how to solve it.

Back in the 20th century, I was taught how to teach mathematics pretty much the same way I had learned it. My

fellow preservice teachers and I were told to prepare our lessons thoroughly, present the intended concept or

procedure clearly (and with enthusiasm!), and guide students as they worked through some examples. Eventually,

we would assign homework, including a few word problems in which students would apply the procedure they

had just learned. We hoped students would ask questions if they didn’t understand.

The way I learned to teach mathematics was not that different from the way teachers learned to teach other subjects.

But in the years since then, we’ve begun to realize that this one-way delivery of information may set students up

for frustration and failure, especially when they’re faced with challenging problems they haven’t been taught how

to solve. If we teach students solely by providing them with specific procedures to solve predictable problems,

how will they learn to deal with problems that don’t look like those at the end of the chapter?

An alternative model that I call upside-down teaching can better prepare students to be strong, flexible

problem solvers. In this model (see fig. 1), teachers don’t wait to assign a problem until they’ve taught

students how to solve it. Instead, upside-down teaching starts with a problem or task that students may

not already know how to solve (Seeley, 2014; 2016).

**Figure 1. Traditional Teaching vs. Upside-Down Teaching**

Traditional Teaching | Upside-Down Teaching |

1. I explain the procedure or concept.2. We work examples together.3. You apply what you just learned tosolve a word problem. | 1. |

**Struggling on Purpose **In the past, many of us tried to protect our students from failure, especially in math class. As a result,

generations of students have learned to give up whenever a problem gets hard. But it turns out that

constructively struggling with mathematical ideas can engage students’ thinking and help them learn to

persevere in problem solving (National Council of Teachers of Mathematics, 2014). Emerging from

the growth mindset research of Carol Dweck (2007), we now know that an individual’s brain grows when

he or she struggles in a productive way with something difficult—like a challenging math problem

(Boaler, 2015).

More and more teachers are confirming that students can benefit from wrestling with problems they

haven’t specifically been taught how to solve. In the process, students learn about the power of effort

and persistence, become more confident problem solvers, and even grow their intelligence.

**Upside-Down Teaching in Action **Unlike the traditional teacher-centered classroom based on lecture, an upside-down classroom is

teacher-structured, but centered on students’ thinking. The goal of the lesson isn’t simply for students

to get the answer to a problem, but rather for students to learn the intended mathematics of the lesson

using the problem as the basis for thinking and discussion. After selecting a problem to start the lesson,

the teacher’s job becomes orchestrating the discourse of the classroom—how students will share their

thinking in ways that lead to the mathematical outcome of the lesson—and helping students connect

the discussion to the mathematical goal. As the teacher circulates through the classroom, she generally

asks questions to help students clarify their thinking or take it to the next level. While doing so, the

teacher is also making decisions in the moment about which students will share their work with the

whole class and in what order they will be called on.

Sometimes students’ discussion will lead directly to the mathematical connection the teacher wants to

make—as in a 2nd grade lesson involving subtraction that I’ll describe further along in this article. Other

times, the teacher may need to guide the conversation more directly toward the math outcome. In either

case, students will have been engaged in thinking about the problem and, consequently, they’re much

more likely to learn the mathematics than if they were simply told what to do.

Let’s consider four classroom examples that illustrate the upside-down concept and demonstrate the

variety of tasks teachers might select to start an upside-down lesson.

**Starting with an Engaging Photo or Video: How Many Cookies?**

In a 2nd grade classroom, students watch a video of a furry hand reaching up behind a kitchen counter

and taking away an unopened package of cookies. After some noisy chewing and rattling, the hand

puts the package back on the counter with some cookies gone. The teacher then asks, “What did you

notice in that video? What did you wonder?” The students talk about their observations and the teacher

helps them focus on the question they finally agree to tackle: How many cookies did the cookie monster

eat?

Students then work in pairs to solve the problem. As the teacher circulates among the pairs, she notices

that students have approached the problem in different ways. One of the teacher’s key roles in this kind

of teaching is deciding who she will call on during the whole-class discussion and in what sequence

students should present their work in order to highlight the different approaches. By the lesson’s end,

the teacher can write on the board a clear mathematical summary of students’ work, helping students

see that a subtraction equation might result from either a take-away situation or a difference situation

and helping them notice that the two resulting equations are related.

This lesson setup is based on the Three-Act Lesson model created by Dan Meyer (2011). View an

edited video of this lesson at Teaching Channel.

**Starting with Real-Life Examples: What Happens with Bigger Tires?**

Some problems might present everyday applications that are likely to engage students’ interest. For

example, in the excerpt of a 12th grade quantitative reasoning lesson shown in this video clip1 The

teacher brings in a tire (the spare from her car). She sets it on the floor and asks students to take note

of the numbers on the tire and discuss what those numbers represent in terms of the tire’s

measurements. She then asks her students to consider what would happen if someone were to replace

their vehicle’s tires with bigger tires.

The class offers ideas, speculating that the tire size would affect how fast they could drive, their gas

mileage, the accuracy of the odometer, whether the vehicle would take up more space on the road or

in a parking spot, and so on. Eventually the teacher narrows down the discussion for students and the

class decides to investigate of the effect on gas mileage if the tire size changes. She chooses this

question so that students will be able to deepen their understanding of proportionality as they learn to

use mathematical modeling in ill-defined problems. She then moves among the groups as they work on

the problem in much the same fashion as the 2nd grade teacher in the “How Many Cookies” example,

and the class culminates with students presenting their findings to the whole group.

**Starting with a Basic Word Problem: How to Make Perfect Purple Paint?**

A 6th grade teacher introduces the concept of ratios by presenting a fairly straightforward word problem.

She shows students that she can achieve the perfect shade of purple paint by mixing 2 cups of blue

paint with 3 cups of red paint. She then asks students to figure out, and to model with colored cubes

and drawings, how many cups of red paint and blue paint would be needed to make 20 cups of perfect

purple paint.

As students work in small groups to come up with pictures and models, the teacher moves through the

classroom, seeing how they are progressing and asking questions to push their thinking. When a group

comes up with three different solutions, the teacher reminds them that they will need to reach a group

consensus. Instead of guiding students to the correct answer, she tells them she will return in a few

minutes to see what they’ve agreed on. In this way, students gain experience in explaining their own

ideas and listening to others’ ideas. You can see an edited video of part of this lesson (from Illustrative

Mathematics, the Smarter Balanced Assessment Consortium and Teaching Channel).

**Starting with a Mistake: Are the Coordinates Correct?**

A pre-calculus teacher puts a graph on the board with some coordinates labeled in two different colors.

The teacher tells students there might be an error in the coordinates shown in red. Students work in

pairs to discuss the posted work, considering whether there is a mistake and determining how they will

make their case to the rest of the class. The teacher then convenes the class for a large-group

discussion in which the students present their thinking to their peers, eventually coming to agreement

about the correct solution. (You can see an edited video of this lesson from PBS Learning Media.)

**What We Can Learn from These Upside-Down Classrooms**

Short edited video excerpts of classrooms like these may not show all the elements of an upside-down

lesson. In some of the full lessons for which excerpts are shown in the video clips above, for example,

we can assume that the teacher helped students crystallize the mathematical conclusion at the end of

the lesson (off screen). What we can notice across these examples, however, are the types of tasks

the teachers have chosen and the ways the teachers orchestrate the classroom discourse.

In each of these classrooms, the teacher sets the stage with the whole group, elaborating the task or

facilitating students in formulating the question they will try to answer. All four types of tasks used in

these examples can readily be adapted to any grade level, and there are likely other types of problems

or tasks that would also work well for upside-down lessons. In choosing tasks for such lessons, teachers

look for “low-floor high-ceiling” tasks. This means looking for tasks with multiple entry points—so that

essentially all students can access the task at some level—but that also allow for considerable depth

or extension (Smith & Stein, 2011).

In terms of orchestrating discourse, the teachers in these classrooms move among students as they

work, asking questions or offering comments like, “I notice that in your group you have three different

models. I’ll be back in a few minutes to see if you have agreed on which model you want to present to

the class,” or “Can you draw on your paper a picture of what you just said?” or “How did you decide to

divide by 7?” When the teacher brings students together after their group work, students present their

findings and solutions to the whole class, with the teacher asking clarifying questions, facilitating further

discussion, and, finally, making explicit the mathematical connection between students’ work and the

mathematical goal of the lesson.

We also notice that sometimes students in these classrooms share answers or approaches that are

incorrect. Teachers have learned that valuable classroom discussions can arise from wrong answers.

Jo Boaler (2015) suggests that we actually learn more from making a mistake than from getting a right

answer. Upside-down teaching helps both students and teachers understand that mistakes will happen,

and that when they do, the class will use the opportunity to dig into the thinking that led to the mistake,

leading to deeper understanding of the mathematics and increasing the likelihood that students will be

able to use what they’ve learned to solve other problems in the future.

Teachers today have access to a growing body of publicly available classroom videos showing this kind

of mathematics teaching, whether labeled upside-down, problem-centered, student-focused, or just

math class. Videos such as those described here provide a great opportunity for individual reflection or

professional discussion among colleagues. In looking at classrooms in real time or analyzing online

videos, educators can ask questions like,

▪ What kind of problem or task does the teacher use to start the lesson?

▪ How does the teacher encourage students’ thinking and stimulate student discourse?

▪ What kinds of questions does the teacher ask?

▪ What do you notice about the roles of the teacher and student?

▪ How does the teacher sequence students’ presentation of their work?

▪ How does the teacher connect the class discussion to the mathematical outcome of the lesson?

▪ How is this classroom similar to or different from your classroom or the other classrooms you

see?

Not all upside-down classrooms will complete a lesson in one class period or follow the same format.

Effective upside-down classrooms differ noticeably in terms of how they’re organized and how they flow.

What they have in common is the focus on students engaging in thinking about a problem they haven’t

already been taught how to solve. The teacher sets the stage with the whole class, sometimes (but not

always) facilitating discussion among students as they narrow down what they will explore in the lesson.

Students then work on the task, individually or in groups, followed a whole-group discussion.

As we analyze such classrooms, we notice that they are heavily teacher-structured, but not teachercentered. The focus is on students coming up with ideas, solutions, approaches, and models, even as the teacher facilitates the discussion and makes explicit the mathematical concepts revealed by

students’ work.

**Making the Classroom a Safe Place to Share**

When we look into classrooms that use an upside-down approach to problem solving, it becomes

obvious that the students seem quite willing to share their thoughts and ideas. Regardless of their grade

level, they don’t seem to be anxious about the possibility of making a mistake or having an incorrect

idea. They’ve come to expect that open discussion is what happens in math class.

This willingness to risk sharing their ideas doesn’t come easily to students, and it doesn’t happen

overnight, especially for older students who over the years may have developed a reluctance to speak

up in math class. The teachers in these examples have spent deliberate time making their classrooms

conducive to respectful, open conversation. They have worked with students to create positive class

norms that encourage every student to participate—norms based on respect and recognition of the

value of everyone’s ideas (Boaler, 2015; Chapin, O’Connor, & Anderson, 2013; Kazemi & Hintz, 2014).

A culture in which all students appreciate the contributions of every other student not only promotes

problem solving, but also makes a strong statement that equity is valued in our classrooms.

**Upside-Down Teaching Every Day?**

It may or may not make sense to use this kind of upside-down, problem-centered, student-focused

approach every day. The main idea is to prioritize student thinking, reasoning, and problem solving

every day and to structure classrooms where those outcomes are consistently valued. Some teachers

will use an upside-down approach every day. Others may choose to include an occasional teacher-led

lecture or presentation of an interesting mathematical idea or observation. Whatever way a teacher

structures lessons, students should come to expect that when they walk in the door to math class,

they’re going to have lots of opportunities to talk about their thinking and share their reasoning as they

take on challenging, interesting problems.

**References**

Boaler, J. (2015). *Mathematical mindsets: Unleashing students’ potential through creative math,*

*inspiring messages and innovative teaching*. San Francisco, CA: Jossey-Bass.

Chapin, S. H., O’Connor, C., & Anderson, N. C. (2013). *Talk moves: A teacher’s guide for using*

*classroom discussions in math (3rd ed.)*. Sausalito, CA: Math Solutions.

Dweck, C. (2007). Mindset: *The new psychology of success.* New York: Ballantine Books.

Kazemi, E., & Hintz, A. (2014). *Intentional talk: How to structure and lead productive mathematical*

*discussions*. Portland, ME: Stenhouse Publishers.

Meyer, D. (2011). *The three acts of a mathematical story.* Retrieved April 27, 2017,

from http://blog.mrmeyer.com/2011/the-three-acts-of-a-mathematical-story/

National Council of Teachers of Mathematics. (2014). *Principles to actions: Ensuring mathematical*

*success for all*. Reston, VA: National Council of Teachers of Mathematics.

Seeley, C. L. (2014).* “Upside-Down Teaching” in Smarter than we think: More messages about math,*

*teaching, and learning in the 21st century.* Sausalito, CA: Math Solutions.

Seeley, C. L. (2016). *Making sense of math: How to help every student become a mathematical thinker*

*and problem solver.* Alexandria, VA: ASCD.

Smith, M. S., & Stein, M. K. (2011). *5 practices for orchestrating mathematics discussions.* Reston, VA:

National Council of Teachers of Mathematics.

**Endnote**

1. *This lesson and video comes from Advanced Quantitative Reasoning, a course developed by the Texas Association of*

*Supervisors of Mathematics working with the Charles A. Dana Center at the University of Texas, Austin.*

*Cathy L. Seeley (www.cathyseeley.com) is a mathematics educator, speaker, and writer in Austin,*

*Texas, and former president of the National Council of Teachers of Mathematics. She is the author*

*of Making Sense of Math: How to Help Every Student Become a Mathematical Thinker and Problem*

*Solver (ASCD, 2016) and Smarter Than We Think: More Messages about Math, Teaching, and*

*Learning in the 21st Century (Math Solutions, 2014). Follow her on Twitter.*