Turning Teaching Upside Down

 

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 TeachingUpside-Down Teaching
1. I explain the procedure or concept.
2. We work examples together.
3. You apply what you just learned to
solve a word problem.
 

1. You tackle a problem you may not
know how to solve yet.
2. We talk together about your thinking
and your work.
3. I help connect the class discussion to
the goal of the lesson.

 

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.