There might be lots of different kinds
of money in there. And the other
question is about the number of
dollars…like if you counted each one.
TEACHER: Exactly. Kade is asking
about how much it’s worth, and
Keisha wants to know how many bills
are in the whole roll.
This routine of asking students
what they notice and wonder about
problems, images, and patterns was
developed by the Math Forum at the
National Council for Teachers of
Mathematics and is a helpful way to
support students in identifying the
mathematical elements of contextualized situations. If you look back at
the students’ questions, some of them
can be answered using mathematics
(How much money is there? How
many dollars are there?) whereas
others require my response (Why did
you do that? Is it your money?). Recognizing the differences between these
questions is important for students
to consider. Question formulation
and exploration are key parts of the
problem-solving process and are often
overlooked in math class.
Act 2
The majority of three-act tasks have a
focal question that is most compelling
and surfaces as the one question students want to answer. Creators of
three-act tasks often upload their Act
1 media to a site called 101 Questions
( www.101qs.com) where anyone can
view the media and write the first
question that comes to mind. This
helps creators learn what users find
most appealing and interesting. Once
that focal question is identified, cre-
ators then work to provide sufficient
resources that students will need
during Act 2 to answer that question.
I had used this process for my task,
and “How many bills are in the roll?”
emerged as the focal question.
Likewise, when I tried the task with
this class, almost all of the students
wanted to know how many bills were
in the roll, so it was time for us to
move to Act 2, the richest part of the
process. I asked the students what
information we’d need to answer the
question. Josie said we needed to know
how long the roll is, while Alli wanted
to know how long a dollar is. Efran
asked how they are stuck together—in
other words, if they overlapped or if
they were stuck together end-to-end.
Lila offered that we needed to know if
they were all dollar bills. When I asked
why, she responded, “Are all bills the
same size?”
The class pondered this for a
moment. Some immediately said yes,
but others furrowed their brows as
they thought. I gave them a moment to
talk it out. One student ran to her desk
and brought out a pouch where she
had a one-dollar bill and a five-dollar
bill. She showed they are the same
size by holding them up together. This
moment gave students an opportunity
to work through some ambiguity
together. In real-life problem solving,
we are not given all the information
we need in a nice, neat package—we
must gather and sort it ourselves.
In response to the students’ requests
for information, I let them know that
each bill was a one-dollar bill and that
the bills were taped end-to-end. I then
showed them an image of one dollar
with eight connecting cubes on top
(fig. 3) and an image of the roll laid
out with 179 cubes covering it (fig. 4).
Some students noticed that the
dollar was just over eight “cubes” long
because there was a little bit of space
left that wasn’t covered by cubes. This
seemed important to some students
and less so with others. Once the
students had all this information, I
invited them to work with their math
partners on the problem of how many
bills there were in all.
I walked from group to group
observing their work, listening to conversations, and at times asking probing
questions to get a sense of what
students were thinking. I refrained
from any direct teaching because the
purpose of three-act tasks is to allow
students to navigate their own solution
pathways. Some students approached
the task numerically while others
created visual representations. There
was also a range in mathematical
approaches. Some students used
division and multiplication; others
used repeated addition or subtraction.
When I encountered a group that had
an incorrect solution, I made note of
it, but didn’t intervene to help them
fix it. The emphasis of three-act tasks
is much more on the problem-solving
process than on the solution, and part
of the process is working through
what to do when you have multiple
solutions.
Act 3
The power of multiple solutions and
solution pathways is that they allow
for rich conversations during Act 3,
when students discuss and revise their
approaches. Each task is a class effort;
Question formulation and exploration are
key parts of the problem-solving process
and are often overlooked in math class.
