SimCalc Classroom Connectivity Project
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Understanding Math Classroom Affordances
of Networked Hand-Held Devices
PROJECT RESULTS TO DATE
To date, we will describe our advances in five
categories—technological, classroom activity structures,
student-change results, teacher-change results, and the refinement of
issues needing further investigation.
1. Technology advances are based in the following work:
- Modification of our (now commercially available) version of
MathWorlds for the TI-83+ graphing calculator to exploit the TI
Navigator prototype classroom network in as seamless a way as
possible. After log-in to the network, it is possible to send and
receive TI-83+ MathWorlds documents (in the form of
"application variables") from inside the MathWorlds application
using an extension of its menu system.
- Retrofitting a Macintosh-only version of MathWorlds to
pre-prototype new aggregation activity structures (described below)
to accept and aggregate student functions sent from the graphing
calculator to the teacher’s workstation running the
Macintosh-only version of MathWorlds.
- Based on classroom tests of the pre-prototype application,
modifying the cross-platform Java MathWorlds to support aggregation
on a standard computer network, including (a) a MathWorlds Server to
collect and distribute Java MathWorlds functions and documents, and
(b) means by which a teacher can easily manage the import and
display of multiple student functions for pedagogical
purposes.
- Modifying the cross-platform Java MathWorlds to support
importing of TI-83+ MathWorlds functions and documents into the
computer-based MathWorlds in as direct and easy-to-use a way as
possible.
- Creating a prototype version of MathWorlds on the Palm to
exploit peer-peer beaming and its use in new student interaction
activities for the SRI site.
These technological advances supported directly the
two major classroom-based teaching experiments that, as planned, have dominated our
work in Year 2 to date. These advances constitute, we believe, a unique level of
connectivity across hardware types that directly support classroom instruction in both
ordinary classroom contexts with mixed device types, and in computer-intensive environments
such as a computer laboratory. It includes support for students doing homework on hand-helds
and sending it in to the teacher for evaluation or classroom discussion.
2. New Activity Structures: As planned, our focused
interventions are generating new activity structures that exploit
different varieties of classroom connectivity, and we are testing the
viability and impact of those structures. Importantly, each category is
highly generative, scalable, and flexible in terms of being applicable
to serve a wide variety of curricular objectives. And each applies well
beyond the content that we are working with. We will describe these
categories, beginning with activities supported by student-student
beaming between hand-helds (Palm Pilots were used, although earlier
prototyping was done with iMacs).
Peer-Peer Information-Exchange
Challenges. The structure of this activity involves one student (A),
sending a mathematical object to a second student (B), where B’s
view of the object is limited or different from A’s in some
important way by the given configuration of the software. B is to guess
A’s object in the terms of A’s view based on successive
carefully crafted clues from A given in response to incorrect guesses
from B. Typically, after a correct guess, A and B exchange roles. This
activity structure engenders intense focus on the mathematical issues at
hand, two-way translation between mathematical notations and their
natural language formulation, and abundant opportunity to fuel
purposeful cognitive activity with affectively driven peer-peer
engagement.
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In one case worked on, A
creates a function in an algebraic view and beams it to B who sees
it only in a graphical view. Based on his/her interpretation of the
graph, B guesses the algebraic form of the function. If not
correct, then A offers a carefully designed clue to move B closer to
a correct answer, but without giving away the answer. Clearly, Both
A and B must work very hard to translate knowledge about the
relations between algebraic and graphical views of functions, either
into verbal clues (A), or from verbal clues (B). This can be
thought of as a graphical version of the classic "Guess My
Rule" activity. By varying the functions in type and
complexity, this activity can serve a wide range of topics that
involve relations between algebraic and graphical views of
functions.
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Another frequently used pair of versions of this activity structure
involved the relation between a position or velocity vs. time graph
of a motion and the motion itself. A makes and sends a position or
velocity vs. time graph and sends it to B. B can run the motion
determined by the graph, but not see the graph, and must guess the
graph. In this case, if the graph is defined piecewise, then B must
describe the graph in great detail. The activity engenders intense
focus on the relation between a graph and its associated motion, at
a highly detailed level.
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A third pair of versions of this activity involved, respectively, A
sending a position vs. time graph to B and B seeing only its
velocity vs. time graph (so B has to integrate the velocity
function), or, A sending a velocity vs. time graph to B and B seeing
only its position vs. time graph (so B has to determine the
slope–or derivative–of the received position vs. time
function).
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A variation on the previous example involves B using the additional
information provided by the motion associated with the received
position or velocity graph. This helps illustrates how this
activity structure is subject to variation and elaboration. One can
finely tune, in the software configuration, the information that B
has available on which to fashion a guess.
The next series of categories involves individual students or small
groups of students sending a mathematical object to the teacher where it
is subject to processing of one sort or another, and public display
using the teacher’s computational device (most often, but not
necessarily a workstation of some kind).
Creating and Sharing a Personally Meaningful Mathematical
Object with the Class–Mathematical Performances. This is a
relatively simple type of activity, but one that we feel has enormous
pedagogical potential because of the ways it taps into adolescent
students’ personal identity, their need for recognition, and their
creativity in expressing their unique personal experience. It also
serves to focus class attention, which leads to opportunity for
follow-up engagement by the teacher to exploit issues raised, for
pedagogical and curricular purposes. We provide an example with links
to the instructional material as well as graphics and student scripts
illustrating the activity.
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Create an Exciting Sack Race: We
provide the graph of a constant velocity position vs. time function
which controls the (horizontal) screen motion of one object (A) and ask
the student (1) to write a race-script for an "exciting race"
with A and (2) to create a position vs. time graph for B that enacts the
race. The student then sends the race-data to the teacher who replays
the race in front of the class on a large-screen display while the
student author of the race "calls the race" by reading their
narrative script. The target mathematical content of this activity is
the key notion of slope-as-rate-change. This idea is central to using
functions to model situations and phenomena and to interpreting graphs
of functions of any origin. A secondary content target is the idea of
simultaneous conditions underlying simultaneous equations–the basis
for developing and solving such equations. We have seen both a large
variety of uniquely personal student creations in response to this task
and clear indicators of the "performance" aspects of the
task–for example, in most cases, the classroom audience breaks into
spontaneous applause when the race and story are complete. See
here
for the teacher’s instructions and a
teacher-led example of this activity. This links further to some brief
scripts (from student dialog) and animations of students’ scripts
and races.
Aggregation and Display of Systematically Varied
Student Constructions to Expose and Examine Important Mathematical
Structures and Relationships, and to Elevate the Abstraction-Level of
Mathematical Attention. The underlying idea of this very general
application of classroom connectivity is to engage students in building
mathematical objects that systematically vary in ways that depend on the
students’ identity, and then to upload and aggregate these in a
common classroom display in order to examine key mathematical structures
and relationships. This affords an additional mathematical
opportunity–to raise the level of mathematical attention from the
level of the object produced by the individual student to the level of
the aggregate object produced by the class. One obvious example is the
elevation from functions to families of functions. This vertical
flexibility is a powerful pedagogical resource not only for supporting
abstraction to parametrized families of objects but for more general
purposes.
Typically, the class is subdivided into groups, where the size of the
group is determined by the teacher or activity designer to fit both the
given size of the class and the mathematical activity (so the group
might simply be the whole class, or each group might have only two
members, meaning students are organized in pairs). Then the students
Count-Off inside the group. In this way, each student has a two-number
identity that then serves as the value of a "personal
parameter" that thus systematically varies across students. The
students then create mathematical objects that depend in some critical
way on their respective parameter values and then upload these to the
teacher where they are aggregated and displayed to the class. We will
off a series of examples intending to illustrate the strategy’s
flexibility, its power to focus attention, and its power to tap into
student personal identity as well as their identity as a member of a
group with social dimensions (e.g., as colleagues, classmates, friends,
fellow-sufferers, etc.) Again, we offer links to graphic examples drawn
from real classroom episodes. To streamline the presentation, let us
assume for these examples that students are formed into groups with 3-5
members, so each student has a Count-Off Number ranging from 1 to 5, and
the number of groups will depend on the size of the class.
We can vary the Group Number, the Count-Off Number, or both. Assuming
members of a group are physically adjacent, then varying the Count-Off
Number allows students to see the variation in their group’s
productions. On the other hand, if we vary Group Number and not the
Count-Off Number, then group members are creating the same object and
can help each other. Choice of which to vary depends on the goals of
the activity, of course.
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Basic Linear Functions—The "Staggered-Start, Staggered Finish
Races" (varying a single parameter):
In the simplest cases,
students make a linear position vs. time Y = mX + b function where
either m or b is their Count-Off Number. In the latter, they make a 2
ft/sec motion defined by the position vs. time function Y = 2X + b where
"b" is their Count-Off Number. The resulting set of parallel
lines and staggered starting points help reveal the invariance of slope
(2 in this case), and how the systematically varying y-intercept relates
to initial position. A companion activity involves using their Group
Number as a starting point, so everyone in a group travels side-by-side,
as shown in the linked figure, where we see the
screen after 3 seconds
of the 5-second race, and the groups are clearly traveling together.
Furthermore, the position vs. time graphs of a given group are
coincident, while the respective graphs of the 6 groups are all
parallel. Lastly, in the third graphic, we can see the equation of
each function and
hence the parametric variation reflected in the seven values of
"b" in Y = 2X + b.
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Linear Functions—The "Staggered-Start, Simultaneous Finish
Race":
In this activity, one dot (A) starts at 0 m and travels
at 2 m/sec for 6 seconds. Each student starts at 3 times their Group
Number and is to finish in a tie with A. Here each student in a group
is solving the same problem, but may do so in many different ways.
Furthermore, since the Group Numbers vary from 1 to 6, the starting
points vary from 3 to 18, which means that the slopes of the graphs
[see graph]
vary from positive, through 0, to
negative, with all members of a group traveling together. In particular
in the next graphic
we can see how the coefficients of X vary,
along with b. Note the special case of Group 4 starting at the
"finish line" (3 * 4 = 12), having zero
velocity, having X coefficient of 0, and having formula given by Y = 0X
+ 12. This strongly contextualizes Y = 12 in a family of functions in
three ways–algebraically, graphically and in terms of motion (where
slope as rate of change is likewise in a central role).
- "Constant of
Integration": Each student starts at their Group Number &
travels at 2 m/s for 5 sec. We display both position and velocity vs.
time graphs for the aggregated functions
[see graph].
Note that there is only a single, constant
velocity vs. time graph visible since they are all coincident, whereas
the respective groups have parallel position vs. time graphs.
The Where Am I? Aggregation Activity Structure. In this genre of
activities, both group and Count-Off Numbers typically are allowed to
vary, so each student in the class produces and sends up a unique
object. However, the display of the aggregate is deliberately
ambiguated to put the student in the position of needing to focus and
reason in generally predictable ways to "find themselves" in
the common display. We see two sources of pedagogical power in this
type of activity: (1) The control of mathematical focus and reasoning
based on the specific design of the activity (usually through the
variation of representational elements), and (2) The engagement of the
student’s personal identity at the mathematical heart of the
activity via the student’s personal projection of their identity
into the publicly visible display–students and their peers quickly
come to refer to the objects as directly indexing the members of the
class, referring to a dot via a person’s name, rather than
indirectly. For example, a direct reference would use a phrases such as
"John is ahead of Mary," or "Is that you?" whereas
an indirect reference would use phrases such as "John’s dot is
ahead of Mary’s dot," or "Is that your dot?"
Our repeated experience with this activity structure convinces us that
it has enormous power to energize a class, infusing it with affect, and
to focus students’ attention on specific and important mathematical
relationships. We begin with a couple of simple examples with linear
functions before illustrating some activities involving
Position-Velocity connections.
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Linear Functions–Varying Starting Position (Group Number)
and Velocity (Count-Off Number): Start at your Group Number & go for
5 seconds at a velocity (whose numeric value is) equal to your Count-Off
Number.
(a) Which graph is yours? Explain your reasoning.
[see graph]
(b) Based on your motion only, Where Are You? Explain your reasoning.
(c) Which formula is yours? Explain your reasoning.
In versions (a) and (c), respectively, students are must relate the
given initial position and velocity information to vertical intercept
and slope of the graphs, or the constants in the formulas. In (b) they
must relate the given initial position and velocity information to the
motion, with the graphs hidden. Note that the teacher has control of
what information that is visible to the students, hence can hide the
graphs. In this figure, we have displayed all the functions and
representational elements simultaneously. However, we could display the
motion with "Marks" dropped on a per-second basis,
as shown
here. The teacher can even scramble the order of
the objects as needed. In the previous figure, we have included an
"outlier." The potential role of errors is enhanced, although
so is the potential for student embarrassment–hence the teacher has
the option of not displaying any functions she chooses. We have seen
great excitement and excellent logical reasoning occur as students
attempt to track down the author of an erroneously produced object.
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Linear Functions–Varying Starting Position (Group Number) and Velocity (Count-Off Number – 2):
Start at your Group Number & go for 5 seconds at a velocity equal to your Count-Off Number – 2.
(a) Which graph is yours? Explain your reasoning.
[see graph]
(b) Based on your motion only, Where Are You? Explain your reasoning.
(c) Which formula is yours? Explain your reasoning.
This example illustrates the flexibility in the strategy to manipulate
problem features to serve particular objectives. Here, through the
device of requiring the velocity to be the student’s Count-Off
Number — 2, we introduce zero and negative slopes within the same
group, which usually leads to fruitful conversation as students sort out
who is who within a group, and provide feedback to one another when an
erroneous function is produced.
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Velocity→Position: Make a Velocity Function–Where Are You?:
Starting at zero, make a velocity graph so that you travel at a velocity
equal to your Count-Off Number (in m/sec) for 3 seconds, and then for 3
seconds at a velocity equal to your Group Number – 2. We will
display your position graph.
(a) Which position vs. time graph is
yours? Explain your reasoning. [see graph]
(b) Based on your motion only, Where Are
You? Explain your reasoning. [see graph]
This example shares features with the prior examples, but with a focus
on velocity-position connections. Worthy of note is the fact that the
reasoning that we wish to occur in terms of relating velocity to slope
of position graph is exactly what occurs as students try to find
themselves in the fan-shaped first segment of the position graphs
associated with the velocity graphs that the students construct–and
again in the second segment, which provides further branching. Quite
clearly, we can vary the conditions as needed to raise a wide range of
conceptual issues. For example we can enforce a requirement such as a
velocity equal to 2*(-1)Group # where
different groups travel in opposite directions. The reader is invited
to generate possibilities.
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Dot-Buddies–Adopt-a-Dot: Given a motion for a set of dots whose
motion varies systematically in some way, make a motion for yourself so
that you move alongside the dot you adopted.
This can be based on making the motion with position or velocity
functions, depending on the learning objective. This uses the facility
under the control of the teacher to order the sequence of dots on the
screen so that an imported dot can be put alongside its adopter. In
effect, this turns around the identity, where the original dot on the
screen is without identity until a class member "adopts"
it.
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Guess My Rule–By Adopting My Graph and My Dot: Half the class makes
an algebraic formula (within some family specified in advance, further
structured by group or Count-Off Number if desired) and sends it up
where its algebraic identity is hidden and it is graphed. The other
half of the class is to make formulas which fit the respective graphs
that they individually adopt. If needed, the dots can be animated
side-by-side.
Clearly, just as with the student-student challenges based on peer-peer
beaming outlined above, we can vary the representational givens for
pedagogical purposes as needed. And further, hints can be allowed as
well–although this is most easily handled using a group structure,
where an entire group is responsible for a given target graph and
another group is to "guess that group’s rule." This
reduces the number of targets to a manageable number–usually
4—6.
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Uploading Student Motions to Create Collective Motions (Marches, dances, story-enactments, …):
(a) Via synthetically defined functions (either graphs or formulas), or
(b) Via physical motions imported using CBR and then uploaded and aggregated, or
(c) Via a combination of synthetic and physically based functions.
Using the MathWorlds capacity to import, aggregate and then animate
multiple student-created motions, a wide variety of exciting activities
can be produced. These can be done using groups as responsible for a
single object per group (e.g., to create a dance of groups), or by
varying the objects within a group (e.g., to create a within-group
dance). Importantly, the quantitative planning to create motions that
relate to other motions is typically exactly the kind of reasoning that
we would normally attempt to stimulate in more ordinary activities,
except that here the context is strongly social. The common display
acts as an arena for the development and testing of ideas in a shared
space controlled by the teacher–although subject to control by
others as well. As noted, these can be created using any combination of
synthetic and physical means.
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Creating Collective Motions Across Different Classrooms or Schools:
While we have not yet done this, we expect that in the coming year we
will be able to do versions of the previous activity across different
classrooms in the same school or across different schools.
3. Student Learning and Achievement: As planned, we utilized
focused interventions targeted on particular mathematical content
associated with core algebra concepts and skills. One each was done in
Spring 2002, based at schools near, respectively SRI International and
UMass-Dartmouth.
Preliminary pre-post test data from the Massachusetts
intervention has been analyzed with very positive results. The course
met 5 weeks for 3 afternoons a week in a local high school adjacent to a
middle school from which the middle schoolers were drawn. It was taught
by a high school teacher, a SimCalc novice teacher "lightly"
assisted by two other high school teachers (who mainly handled logistics
and field-notes), all trained by SimCalc staff prior to the
intervention. Data on the twenty four 7-9th grade students
who took both a pre- and post-test show significant gains across each
grade level. One third of the sample were volunteers from
7th and 8th grade while the other two thirds were
students who had failed or nearly failed the 8th grade
required state test and were "strongly recommended " into the
course by the department chair. The
test consisted of approximately two thirds 10th
grade Massachusetts Comprehensive Assessment System items and one third
a mix of SimCalc constructed items and items taken from AP Calculus and
NAEP items. The statistical results
show that all students gained on almost all items, and
statistically strong gains summed across items for each of the groups.
Of special note were strong gains on open-ended modeling items which
most students find especially difficult. We are optimistic that these
results can be replicated in subsequent iterations since many of the
activities and the technologies were being class-tested for the first
time.
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