As an example of a P-R mathematics task,
consider the following question in relation to the
classroom scenario described earlier:
What would be a
mathematically
appropri-
ate way in which the teacher could respond
to Jane’s question about whether the class
could use Mark’s method every time they
had to find a fraction between two given,
nonequivalent positive fractions?
March
2014
Notices of the AMS
269

Figure 2. An algebraic proof of a general method
for finding a fraction between two given positive
and nonequivalent fractions.
We will henceforth refer to this task as the
Fractions
Task
. A solution to the Fractions Task would build
on
the
“course
of
action”
that
we
discussed
earlier under possibility 2, which is the desirable
possibility. According to this course of action, the
teacher would engage the class in the discussion
of a proof that would not only be valid but also
accessible to the group of seventh-graders.
Feature 1: A mathematical focus
P-R mathematics tasks have a mathematical fo-
cus that relates to one or more mathematical ideas
that theory, research, or practice suggested are
important for teachers to know. The mathematical
focus is intended to engage prospective teachers
in mathematical activity. In the Fractions Task, the
mathematical focus is the mathematical evalua-
tion of Mark’s method, which can be expressed
algebraically as follows:
Given two fractions
a
b
and
c
d
where
a, b, c, d >
0
and
a
b
<
c
d
,
a
b
<
a
+
c
b
+
d
<
c
d
.
Feature 2: A substantial pedagogical context
In addition to the mathematical focus, a P-R
mathematics task has a substantial pedagogical
context that is an integral part of the task and
essential for its solution. The pedagogical con-
text situates prospective teachers’ mathematical
activity in a particular teaching scenario and helps
prospective teachers engage with the mathematics
from the perspective of a teacher.
In the Fractions Task the pedagogical context
describes the teacher’s need to formulate a re-
sponse to Jane’s question about whether the class
could use Mark’s method when asked to find a
fraction between two positive and nonequivalent
fractions. According to this context, the event
happened in a seventh-grade class, which allows
the solvers of the task (prospective teachers) to
make certain assumptions about what the students
in the class might know or be able to understand.
Thus a solution to the task must not only satisfy
mathematical considerations but also needs to
take into account pedagogical considerations. Next
we discuss four points related to feature 2 of P-R
mathematics tasks.
First, the pedagogical context in which a P-R
mathematics task is situated determines to a great
extent what counts as an acceptable/appropriate
solution
to
the
task,
because
it
provides
(or
suggests) a set of conditions a possible solution to
the task needs to satisfy. In the Fractions Task, for
example, an algebraic proof of Mark’s method like
the one in Figure 2, though mathematically valid,
would likely not be within the conceptual reach
of students in a seventh-grade class. A proof can