
The teacher of mathematics
should pose tasks that are based on
sound and significant mathematics;
knowledge of students' understandings, interests, and experiences;
knowledge of the range of ways that diverse students learn mathematics;
and that
engage students' intellect;
develop students' mathematical understandings and skills;
stimulate students to make connections and develop a coherent
framework for mathematical ideas;
call for problem formulation, problem solving, and mathematical
reasoning;
promote communication about mathematics;
represent mathematics as an ongoing human activity;
display sensitivity to, and draw on, students' diverse background
experiences and dispositions;
promote the development of all students' dispositions to do mathematics.
Elaboration
Teachers are responsible
for the quality of the mathematical tasks in which students engage.
A wide range of materials exists for teaching mathematics: problem
booklets, computer software, practice sheets, puzzles, manipulative
materials, calculators, textbooks, and so on. These materials contain
tasks from which teachers can choose. Also, teachers often create
their own tasks for students: projects, problems, worksheets, and
the like. Some tasks grow out of students' conjectures or questions.
Teachers should choose and develop tasks that are likely to promote
the development of students' understandings of concepts and procedures
in a way that also fosters their ability to solve problems and to
reason and communicate mathematically. Good tasks are ones that
do not separate mathematical thinking from mathematical concepts
or skills, that capture students' curiosity, and that invite them
to speculate and to pursue their hunches. Many such tasks can be
approached in more than one interesting and legitimate way; some
have more than one reasonable solution. These tasks, consequently,
facilitate significant classroom discourse, for they require that
students reason about different strategies and outcomes, weigh the
pros and cons of alternatives, and pursue particular paths.
In selecting, adapting,
or generating mathematical tasks, teachers must base their decisions
on three areas of concern: the mathematical content, the students,
and the ways in which students learn mathematics.
In considering the mathematical
content of a task, teachers should consider how appropriately the
task represents the concepts and procedures entailed. For example,
if students are to gather, summarize, and interpret data, are the
statistics they are expected to generate appropriate? Does it make
sense to calculate a mean? If there is an explanation of a procedure,
such as calculating a mean, does that explanation focus on the underlying
concepts or is it merely mechanical? Teachers must also use a curricular
perspective, considering the potential of a task to help students
progress in their cumulative understanding of a particular domain
and to make connections among ideas they have studied in the past
and those they will encounter in the future.
A second content consideration
is to assess what the task conveys about what is entailed in doing
mathematics. Some tasks, although they deal nicely with the concepts
and procedures, involve students in simply producing right answers.
Others require students to speculate, to pursue alternatives, to
face decisions about whether or not their approaches are valid.
For example, one task might require students to find means, medians,
and modes for given sets of data. Another might require them to
decide whether to calculate means, medians, or modes as the best
measures of central tendency, given particular sets of data and
particular claims they would like to make about the data, then to
calculate those statistics, and finally to explain and defend their
decisions. Like the first task, the second would offer students
the opportunity to practice finding means, medians, and modes. Only
the second, however, conveys the important point that summarizing
data involves decisions related to the data and the purposes for
which the analysis is being used. Tasks should foster students'
sense that mathematics is a changing and evolving domain, one in
which ideas grow and develop over time and to which many cultural
groups have contributed. Drawing on the history of mathematics can
help teachers to portray this idea: exploring alternative numeration
systems or investigating nonEuclidean geometries, for example.
Fractions evolved out of the Egyptians' attempts to divide quantities
four things shared among ten people. This fact could provide the
explicit basis for a teacher's approach to introducing fractions.
A third content consideration
centers on the development of appropriate skill and automaticity.
Teachers must assess the extent to which skills play a role in the
context of particular mathematical topics. A goal is to create contexts
that foster skill development even as students engage in problem
solving and reasoning. For example, elementary school students should
develop rapid facility with addition and multiplication combinations.
Rolling pairs of dice as part of an investigation of probability
can simultaneously provide students with practice with addition.
Trying to figure out how many ways 36 desks can be arranged in equalsized
groups  and whether there are more or fewer possible groupings
with 36, 37, 38, 39, or 40 desks presses students to produce each
number's factors quickly. As they work on this problem, students
have concurrent opportunities to practice multiplication facts and
to develop a sense of what factors are. Further, the problem may
provoke interesting questions: How many factors does a number have?
Do larger numbers necessarily have more factors? Is there a number
that has more factors than 36? Even as students pursue such questions,
they practice and use multiplication facts, for skill plays a role
in problem solving at all levels. Teachers of algebra and geometry
must similarly consider which skills are essential and why and seek
ways to develop essential skills in the contexts in which they matter.
What do students need to memorize? How can that be facilitated?
The content is unquestionably
a crucial consideration in appraising the value of a particular
task. Defensible reasoning about the mathematics of a task must
be based on a thoughtful understanding of the topic at hand as well
as of the goals and purposes of carrying out particular mathematical
processes.
Teachers must also consider
the students in deciding on the appropriateness of a given task.
They must consider what they know about their particular students
as well as what they know more generally about students from psychological,
cultural, sociological, and political perspectives. For example,
teachers should consider gender issues in selecting tasks, deliberating
about ways in which the tasks may be an advantage either to boys
or to girls  and a disadvantage to the others  in some systematic
way.
In thinking about their
particular students, teachers must weigh several factors. One centers
on what their students already know and can do, what they need to
work on, and how much they seem ready to stretch intellectually.
Wellchosen tasks afford teachers opportunities to learn about their
students' understandings even as the tasks also press the students
forward. Another factor is their students' interests, dispositions,
and experiences. Teachers should aim for tasks that are likely to
engage their students' interests. Sometimes this means choosing
familiar application contexts: for example, having students explore
issues related to the finances of a school store or something in
the students' community. Not always, however, should concern for
"interest" limit the teacher to tasks that relate to the
familiar everyday worlds of the students; theoretical or fanciful
tasks that challenge students intellectually are also interesting:
number theory problems, for instance. When teachers work with groups
of students for whom the notion of "argument" is uncomfortable
or at variance with community norms of interaction, teachers must
consider carefully the ways in which they help students to engage
in mathematical discourse. Defensible reasoning about students must
be based on the assumption that all students can learn and do mathematics,
that each one is worthy of being challenged intellectually. Sensitivity
to the diversity of students' backgrounds and experiences is crucial
in selecting worthwhile tasks.
Knowledge about ways in
which students learn mathematics is a third basis for appraising
tasks. The mode of activity, the kind of thinking required, and
the way in which students are led to explore the particular content
all contribute to the kind of learning opportunity afforded by the
task. Knowing that students need opportunities to model concepts
concretely and pictorially, for example, might lead a teacher to
select a task that involves such representations. An awareness of
common student confusions or misconceptions around a certain mathematical
topic would help a teacher to select tasks that engage students
in exploring critical ideas that often underlie those confusions.
Understanding that writing about one's ideas helps to clarify and
develop one's understandings would make a task that requires students
to write explanations look attractive. Teachers' understandings
about how students learn mathematics should be informed by research
as well as their own experience. Just as teachers can learn more
about students' understandings from the tasks they provide students,
so, too, can they gain insights into how students learn mathematics.
To capitalize on the opportunity, teachers should deliberately select
tasks that provide them with windows on students' thinking.
Vignettes

The
teacher analyzes the content and how to approach it, and she considers
how it connects with other mathematical ideas. 
1.1
Mrs. Jackson is thinking about how to help her students learn about
perimeter and area. She realizes that learning about perimeter and
area entails developing concepts, procedures, and skills. Students
need to understand that the perimeter is the distance around a region
and the area is the amount of space inside the region and that length
and area are two fundamentally different kinds of measure. They need
to realize that perimeter and area are not directly related  that,
for instance, two figures can have the same perimeter but different
areas. Students also need to be able to figure out the perimeter and
the area of a given region. At the same time, they should relate these
to other measures with which they are familiar, such as measures of
volume or weight. 
Task 1 requires
little more than remembering what "perimeter" and "area"
refer to and the formulas for calculating each. Nothing about thin
task requires students to ponder the relationship between perimeter
and area. This task is not likely to engage students intellectually;
it does not entail reasoning or problem solving.
This task can
engage students intellectually because it challenges them to search
for something. Although accessible to even young students, the problem
is not immediately solvable. Neither is it clear how best to approach
it. A question that students confront as they work on the problem
is how to determine that they have indeed found the largest or the
smallest play space. Being able to justify an answer and to show
that a problem is solved are critical components of mathematical
reasoning and problem solving. The problem yields to a variety of
tools  drawings on graph paper, constructions with rulers or compasses,
tables, calculators  and lets students develop their understandings
of the concept of area and its relationship to perimeter. They can
investigate the patterns that emerge in the dimensions and the relationship
between those dimensions and the area. This problem may also prompt
the question of what "largest" or "smallest,"
"most" or "least" mean, setting the stage for
making connections in other measurement contexts.

Mrs. Jackson examines two
tasks designed to help upper elementarygrade students learn about
perimeter and area. She wants to compare what each has to offer.
TASK 1:
Find the area and perimeter
of each rectangle:
TASK 2:
Suppose you had 64 meters
of fence with which you were going to build a pen for your large
dog, Bones. What are some different pens you can make if you use
all the fencing? What is the pen with the least play space? What
is the biggest pen you can make  the one that allows Bones the
most play space? Which would be best for running?

Many
beginning and experienced teachers are in the same position as this
teacher: having to follow a textbook quite closely. Appraising and
deciding how to use textbook material is critical. 
1.2
Ms. Pierce is a firstyear teacher in a large middle school. She uses
a mathematics textbook, published about ten years ago, that her department
requires her to follow closely. In the middle of a unit on fractions
with her seventh graders, Ms. Pierce is examining her textbook's treatment
of division with fractions. She is trying to decide what its strengths
and weaknesses are and whether and how she should use it to help her
students understand division with fractions. 
The
teacher wants her students to understand what it means to divide by
a fraction, not just learn the mechanics of the procedure. 
She
notices that the textbook's emphasis is on the mechanics of carrying
out the procedure ("dividing by a number is the same as multiplying
by its reciprocal"). The text tells students that they "can
use reciprocals to help" them divide by fractions and gives them
a few examples of the procedure. 
The
teacher senses that the idea of "using the reciprocal" is
introduced almost as a trick, lacking any real rationale or connection
to the pictures of necklaces. Furthermore, division with fractions
seems to be presented as a new topic, unconnected to anything the
students might already know, such as division of whole numbers. 
The
picture at the top of one of the pages shows some beads of a necklace
lined up next to a ruler  an attempt to represent, for example, that
there are twentyfour 3/4inch beads and fortyeight 3/8inch beads
in an eighteeninch necklace. Ms. Pierce sees that this does represent
what it means to divide by 3/4 or by 3/8  that the question is, "How
many threefourths or threeeighths are there in eighteen?" Still,
when she considers what would help her students understand this, she
does not think that this representation is adequate. She also suspects
that students may not take this section seriously, for they tend to
believe that mathematics means memorizing rules rather than understanding
why the rules work. 
The practice exercises
involve dividing one fraction by another, and the "problems"
at the end do not involve reasoning or problem solving.
The teacher considers
what she knows about her students  what they know and what is likely
to interest them.
The model used
is a linear one rather than the pie or pizza diagrams most often
used to represent fractions. The teacher sees the need for students
to develop varied representations. Also, different representations
make sense to different students. The teacher wants the task to
help students make connections  in this example, between multiplication
and division and between division of whole numbers and division
of fractions.
Writing stories
to go with the division sentences may help students to focus on
the meaning of the procedure.
The teacher keeps
her eye on the bigger curricular picture as she selects and adapts
tasks. Juxtaposing whole number and fraction division will help
her students review division and make connections.

Ms. Pierce is concerned
that these pages are likely to reinforce that impression. She doesn't
see anything in the task that would emphasize the value of understanding
why, nor that would promote mathematical discourse.
Thinking about her students,
Ms. Pierce judges that these two pages require computational skills
that most of her students do have (i.e., being able to produce the
reciprocal of a number, being able to multiply fractions) but that
the exercises on the pages would not be interesting to them. Nothing
here would engage their thinking.
Looking at the pictures
of the necklaces gives Ms. Pierce an idea. She decides that she
can use this idea, so she copies the drawing only. She will include
at least one picture with beads of some whole number length  2inch
beads, for example. She will ask students to examine the pictures
and try to write some kind of number sentence that represents what
they see. For example, this 7inch bracelet has 14 halfinch beads:
This could be represented
as 7 ¸1/2 or 7 ´ 2. She will try to help them to think
about the reciprocal relationship between multiplication and division
and the meaning of dividing something by a fraction or by a whole
number. Then, she thinks, she could use some of the exercises on
the second page but, instead of just having the students compute
the answers, she will ask them, in pairs, to write stories for each
of about five exercises.
She decides she will also
provide a couple of other examples that involve whole number divisors:
28 ¸8 and 80 ¸16, for example.
Ms. Pierce feels encouraged
from her experience with planning this lesson and thinks that revising
other textbook lessons will be feasible. Despite the fact that she
is supposed to be following the text closely, Ms. Pierce now thinks
that she will be able to adapt the text in ways that will significantly
improve what she can do with her students this year.
1.3 After recently completing
a unit on multiplication and division, a fourthgrade class has
just begun to learn about factors and multiples. Their teacher is
using the calculator as a tool for this topic. This approach is
new for her. The school has just purchased for the first time a
set of calculators, which all the classrooms share. She and many
of her colleagues attended a workshop recently on different uses
of calculators.

The
teacher uses this exploratory task to spur students' mathematical
thinking. She knows that the initial task is likely to generate further,
more focused tasks based on the students' conjectures. The calculators
help the students in looking for patterns. 
Using
the automatic constant feature of their calculators (that is, that
pressing 5 + = = = ... yields 5, 10, 15, 20,....on the display), the
fourth graders have generated lists of the multiples of different
numbers. They have also used the calculator to explore the factors
of different numbers. To encourage the students to deepen their understanding
of numbers, the teacher has urged them to look for patterns and to
make conjectures. She asked them, "Do you see any patterns in
the lists you are making? Can you make any guesses about any of those
patterns?" 
All
year, this teacher has encouraged her students to take intellectual
risks by asking questions. 
Two students have raised
a question that has attracted the interest of the whole class:
Are there more multiples
of 3 or more multiples of 8?

Judging
that this question is a fruitful one, the teacher picks up on the
students' idea and uses it to further the direction of the class's
exploration, even bringing up questions about infinity. 
The
teacher encourages them to pursue the question, for she sees that
this question can engage them in the concept of multiples as well
as provide a fruitful context for making mathematical arguments. She
realizes that the question holds rich mathematical potential and even
brings up questions about infinity. "What do the rest of you
think?" she asks. "How could you investigate this question?
Go ahead and work on this a bit on your own or with a partner and
then let's discuss what you come up with." 
The
question promotes mathematical reasoning, eliciting at least three
competing and, to fourth graders, compelling mathematical arguments.
Students are actively engaged in trying to persuade other members
of the class of the validity of their argument. 
The
children pursue the question excitedly. The calculators are useful
once more as they generate lists of the multiples of 3 and the multiples
of 8. Groups are forming around particular arguments. One group of
children argues that there are more multiples of 3 because in the
interval between 0 and 20 there are more multiples of 3 than multiples
of 8. Another group is convinced that the multiples of 3 are "just
as many as the multiples of 8 because they go on forever." A
few children, thinking there should be more multiples of 8 because
8 is greater than 3, form a new conjecture about numbers  that the
larger the number, the more factors it has. 
The
task has stimulated students to formulate a new problem. The idea
that lessons can raise questions for students to pursue is part of
an emphasis on mathematical inquiry. 
The
teacher is pleased with the ways in which opportunities for mathematical
reasoning are growing out of the initial exploration. She likes the
way in which they are making connections between multiples and factors.
She also notes that students already seem quite fluent using the terms
multiple and factor. 
The
teacher provides a context far dealing with students' conjectures.
She is also able to formulate tasks out of the students' ideas and
questions when it seems fruitful. 
Although it is nearing the
end of class, the teacher invites them to present to the rest of
the class their conjecture that the larger the number, the more
factors it has. She suggests that the students record it in their
notebooks and discuss it in class tomorrow. Pausing for a moment
before she sends them out to recess, she decides to provoke their
thinking a little and remarks, "That's an interesting conjecture.
Let's just think about it for a sec. How many factors does, say,
3 have?"

The
teacher provides practice in multiplication facts at the same time
that she engages the students in considering their peers' conjecture. 
"Two," call out
several students.
"What are they?"
she probes. "Yes, Deng?"
"l and 3," replies
Deng quickly.
"Let's try another
one," continues the teacher. "What about 20?"
After a moment, several
hands shoot up. She pauses to allow students to think and asks,
"Natasha?"

The teacher does
not want to give them a key to challenging the conjecture, but she
does want to get them into investigating it.
She tries to spur
them on to pursuing this idea on their own.
The teacher deliberately
leaves the question unanswered. She wants to encourage them to persevere
and not expect her to give the answers.

"Six 1 and 20, 2 and
10, 4 and 5," answers Natasha with confidence.
The teacher throws out a
couple more numbers  9 and 15. She is conscious of trying to use
only numbers that fit the conjecture. With satisfaction, she notes
that most of the students are quickly able to produce all the factors
for the numbers she gives them. Some used paper and pencil, some
used calculators, and some did a combination of both. As she looks
up at the clock, one child asks, "But what about 17? It doesn't
seem to work."
"That's one of the
things that you could examine for tomorrow. I want all of you to
see if you can find out if this conjecture always holds."
"I don't think it'll
work for odd numbers," says one child.
"Check into it,"
smiles the teacher. "We'll discuss it tomorrow."
Summary: Tasks
The teacher
is responsible for shaping and directing students' activities so
that they have opportunities to engage meaningfully in mathematics.
Textbooks can be useful resources for teachers, but teachers must
also be free to adapt or depart from texts if students' ideas and
conjectures are to help shape teachers' navigation of the content.
The tasks in which students engage must encourage them to reason
about mathematical ideas, to make connections, and to formulate,
grapple with, and solve problems. Students also need skills. Good
tasks nest skill development in the context of problem solving.
In practice, students' actual opportunities for learning depend
on the kind of discourse that the teacher orchestrates, an
issue we examine in the next section.

