Investigations in Number, Data, and Space
sessions are intended to
teachers with the mathematical content and pedagogy of the Investigations
curriculum. All sessions are 2 hours, except as noted.
The Role of Games
Participants work with
number games from different grade levels of Investigations.They
play the games with an eye toward recognizing the underlying mathematics,
and the way in which teachers: 1) "talk mathematics" with
children as game play proceeds; 2) use games for assesment purposes;
3) adapt games to challenge children at different levels. Participants
discuss the mathematical learning that is addressed through Investigations
Overview and Structure of an Investigations Unit
After staff and participants
introduce themselves, the leader explains the goals and pedagogy of
the Investigations curriculum and the structure and purpose of
the units. Participants then work in grade level groups with each reviewing
an individual unit, looking at the introductory material and the components
of an investigation and a session, and reading teacher notes, dialogue
boxes, and assessments.
and Ten-Minute Math in Investigations
The leader explains how
Classroom Routines (K-grade 2) and Ten-Minute Math (grades 3-5) fit
into the Investigations curriculum, and their uses in supporting
practice in arithmetic and also in geometry and data analysis. Then
participants try a number of the activities: Today's Number, Broken
Calculator, Visual Images of Number, Nearest Answer, Estimation Game,
and Collecting Data.
Addition and Subtraction
In this session participants
look closely at how students develop strategies to combine and compare
amounts. They use mental strategies to solve addition and subtraction
problems and generalize these strategies to discover what is happening
mathematically in each one. Then they try out strategies that may not
be as familiar to them. Participants look at activities from kindergarten
through grade 5 and discuss the knowledge of number and operations that
is developed through this number strand. The role of speed and efficiency
in computation is discussed in regard to addition and subtraction.
Multiplication and Division
This is the first of two
sessions on multiplication and division in Investigations. It
focuses on the early grades of Investigations, but also includes
mental arithmetic and counting around which are done throughout the
grades. Participants look at the development of multiplication and division
in grades K-2. Participants first consider how multiplication and division
problems can be interpreted. Then they look at samples of children's
work from grades 1 and 2 that show students using counting and adding
to solve multiplication and division story problems. They see a variety
of other multiplication and division tasks in K-2 Investigations
and discuss what knowledge their solutions require. Participants then
do some mathematics for themselves: mental arithmetic and counting around
to look at how knowledge of landmark numbers and patterns in the multiplication
tables provide foundation for multiplication and division.
Image Dots" (grade 1); "Counting Around the Class" (grade
Multiplication and Division
(2.5 - 3 hours)
This is the second
of two sessions on multiplication and division in Investigations.
Participants try activities from grades 3-5 that support students' understanding
of the operations of multiplication and division: factoring numbers,
and using arrays and cluster problems. They look at the work of fourth
and fifth grade students to consider the role that knowledge of the
number system and operations can play in students' computational strategies.
2-D Geometry in
(Allow an extra
30 minutes for optional GeoLogo.)
Participants first brainstorm
uses of geometry. Then they look at work in area through the grades.
They do activities from grades K-2 using pattern blocks, activities
from grade 3 using triangles and squares, and from grade 4 using geoboards.
If computers are available, participants explore the GeoLogo Shapes
Geometry in Investigations
their own skills in 3-D geometry as they examine problems using materials
that kindergarten through grade 5 students use in Investigations.They
examine and describe 3-dimensional shapes while focusing on specific attributes
of these shapes and how they relate to one another. Participants explore
the connections between two- and three-dimensional shapes by building
shapes based on drawings, diagrams, and written instructions. They see
an overview of the geometry units
Fractions in Investigations
Participants split rectangular
"brownies" into equal shares and discuss different approaches. Then
they assess student work on a similar problem. They decide what percents
are equivalent to various common fractions and compare fractions to
the landmarks, one half and one whole. They work in groups to show a
number of different ways to compare fractions or to do a computation
problem. They see an overview of the fractions units and consider the
importance of geometrical thinking in working with fractions.
Optional Video episode: "Comparing Fractions" (grade 4)
In this session,
an introduction to goals for learning about data is presented. Participants
collect data about how many roofs each has lived under, represent the
data in a line plot and describe and interpret their data. In small
groups, participants list games they played as children, form categories
to sort them, and talk about criteria for effective categorization.
They discuss the use of various measures of typicality.
of Algebra in Investigations
Participants work with problems
from Investigations that require them to reason about quantities
whose value they do not know, but about which they have some information.
They consider how to represent these quantities symbolically. They investigate
how basic properties of the number system (e.g. commutative and associative
properties) appear in student work as early as first grade. They also
use coordinate graphs to explore properties of rectangles.
of Change in Investigations
are introduced to the Math of Change as a strand in mathematics, and look
at possible explanations for a graph of change over time. Then they do
activities from units from grades 2-5. They find different combinations
of changes that have the same net change. They make a time line through
a day and interpret graphs of plant growth to consider the issue of representing
elapsed time with a regular time scale. In the final activity, they make
tables and graphs to fit a story of changing speed of motion along a straight
line and discuss similarities between graphs and between tables that fit
the same stories.
In this session participants
are introduced to the important role of Choice Time in the Investigations
curriculum. This session is designed around the videotape "Choice
Time, grade 1" in which students are engaged in three mathematically
related activities. Teachers reflect on their own experience with Choice
Time, both in and outside of math classes. Participants try out the
activities themselves and then discuss the mathematics that students
will be engaged in as they participate in these activities. Participants
consider the role of the teacher and the role of students during Choice
Time, how the teacher can use Choice Time for assessment, and the value
of Choice Time for the students.
view a videotape of Choice Time in a fourth grade class. They discuss
the issues involved in planning for Choice Time, such as selecting an
appropriate range of activities and in setting expectations for students.
They consider the use of Choice Time in the Investigations curriculum
and theteacher's and students' roles during choice time. Participants
then do two of the Choice Time activities and talk about the important
mathematical ideas in the activities, the procedures or content that might
require clarification and intervention during the session, and the ways
in which they might modify the games to challenge children at different
Assessment in Investigations,
Participants explore an
approach to assessment emphasized in reform curricula. After viewing
the video, "Assesment in Investigations" (grade 2)
and student work, participants discuss criteria to determine the understanding
of knowledge of several students. A second grade teacher discusses ways
she assesses students and how her teaching is informed by observation
and evaluation of student work. Participants consider the strengths
and challenges of this and other approaches to assessment. A brief overview
is provided of the types of support for assessment offered in Investigations.
session begins with a brief discussion of types of assessment that students
typically encounter in mathematics, and what teachers can learn about
their students from each. Next, participants look through samples of student
work on a multiplication problem, and view a video clip,"Assesment
in Investigations" (grade 4)in which a teacher observes and
talks with her students as they work on the problem and explains how she
drew on her observations to help inform her analysis of their papers.
Following the video, participants look again at the students' work and
discuss how this kind of assessment can be used to gather information
for grading. Finally, a brief overview is provided of the types of support
for assessment offered in Investigations.
This session contains three
parts to introduce teachers to three pieces of software in Investigations: Shapes Software, GeoLogo, and Trips. It is expected that leaders will
choose two of the three to do in a 2-hour session.
Investigations Students for
In this session participants
consider ways to prepare students for external tests in ways that are
in concert with the Investigations curriculum. They consider
how to prepare students to recognize vocabulary, computational formats,
and other problem presentations so that they can make sense of what
is asked and still answer the problems in their own way. They evaluate
students' work on open-ended problems.
of Investigations Classroom Episodes
The following video clips can be used with Investigations sessions
"Quick Image Dots" (grade 1);
Subtraction" (grade 1);
"Capture 5" (grade 2);
Around the Class" (grade 2);
"Division with Remainders"
"Comparing Fractions" (grade 4);
Game" (grade 5).