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Response to a Using a Virtual Mathematics
Manipulative in the Classroom
Introduction:
The amount of educational material on the Internet increases everyday.
Between lesson plans, activities and interactive manipulatives, there
is plethora of material waiting to be used. This paper discusses the results
of one attempt to integrate material from the World Wide Web into a lesson
plan for a fourth grade classroom. In my lesson, I used a web-based, interactive
math manipulative that was essentially a virtual version of tangrams.
In the following reflection, I will first give a brief explanation of
the simulation. From there, I will discuss why and how I used this activity.
Finally, I will examine the results of using this web-activity and reflect
on the possibility of using similar materials from the Internet in the
future.
Description of the Simulation
The simulation,
located at
http://matti.usu.edu/nlvm/nav/category_g_2_t_3.html,allowed students
to manipulate the seven tangram shapes on the computer screen. Tangrams
are originally a Chinese puzzle game in which seven geometric shapes are
manipulated to create different designs. The image to the right shows
the seven tangram pieces.
In the simulation there is a main screen that contains the seven virtual tangram pieces, These shapes can be clicked and dragged around the screen with the mouse. The shape being manipulated at the current moment is shaded so that the user can keep track of which piece he or she is moving. The user can also rotate any of the seven tangram pieces by clicking on and holding down the mouse button on the corner of the piece. Unlike in many other tangram simulations, when pieces are rotated here, they snap into preset angles of rotation (every 30 degrees). Therefore, the user does not have to make sure that the pieces are lined up exactly, because there are only a small finite number of positions for each piece. Similarly, when the user brings two pieces close together, the simulation snaps them together automatically, instead of forcing the viewer to line them up perfectly.
To the right of the main screen is a smaller screen with a series of
buttons. One button allows students to flip the tangram pieces over a
vertical axis. This is crucial for the parallelogram, which is not symmetrical.
In addition, there are a series of buttons which allow the user to change
the color of the selected tangram piece.
Underneath the main frame there is another frame with a series of buttons illustrating several puzzles that students can fill in. Clicking on the button for each puzzle brings up an outline of the puzzle to be solved on the main screen. The user can fit the shapes into this outlined puzzle. There is also a button called “activities” which calls up a window with a selection of activities to do with the tangrams, including building one shape out of a combination of the others others and manipulating the pieces to make squares and rectangles with varying numbers of tangram pieces.
Making Decisions: Planning the Lesson
In thinking about how to use a web-based manipulative, I needed to make several decisions. In the first place, what manipulative would I use? In what context would I use this manipulative? What did I want students to get out of using the simulation in this context? Finally, how would I use a computer-based manipulative in a classroom of twenty-two students and three computers?
Given the context of this assignment, which was to be completed within
a narrow frame of time, I made the choice to find an activity that fit
in with the topics I had already planned to cover, instead of creating
a new unit or lesson based around the manipulative. We had been covering
a unit on geometry, which was quite sequential, with each lesson building
on the one before it. While there were many interesting sounding geometry
manipulatives on the web, many of them involved concepts that the class
would not have covered yet, and I did not want to jump around in the sequence
of lessons I had planned.
So, I decided to use a tangram manipulative for its flexibility, before
actually finalizing the lesson in which I would use it. One of the lessons
we would cover before this assignment was due was on types of quadrilaterals.
I thought it might be fun for the students to use the tangrams to build
different quadrilaterals, reinforcing their understanding of the properties
of different quadrilaterals.
I encountered many different interactive web-based tangram simulations
on the World Wide Web. I decided to use this particular simulation came
for a couple of different reasons. This site was nice in that it already
came with some puzzles that students could fill in and also had a few
other interesting activities. Furthermore,because several of the ideas
on the website already lent themselves to the task I wanted to cover,
using this particular website seemed like a good idea. Furthermore, as
I noted in my introduction, the simulation was very easy to use and the
pieces were very easy to assembling cleanly and evenly. I thought that
this was an important feature in choosing an activity for fourth graders.
However, I still needed to reconcile the fact that there were twenty-two
students in a room with three computers. I toyed with the idea of using
the library for this lesson, since there were twelve computers hooked
up to the Internet there. However, I couldn't use the library for the
whole class at the time I would have liked it, since the whole school
was sharing the library space. While I could have sent groups of students
to use some of the computers as additional stations, I really wanted to
keep an eye on all of the students as we worked on the manipulative. While
all three computers in the room had Internet connections and software
that supported the use of this Internet manipulative, I could only conceivably
have 6 students working on the computer at a time.
Instead, I decided to rotate students at the computer. I would spend
the first half of the math lesson working as a class, without the computers,
to come up with properties of rhombi, parallelograms, squares, rectangles
and trapezoids. I broke the class into four groups, each of which worked
to decipher the properties of one of the shape. I gave each group three
examples of their shape (no group worked with the rhombus) on geo-boards
and asked them to think about whether or not their shape had parallel
sides or equal sides, and what types of angles it had. After talking about
what each group found, we would come up with the characteristics of each
shape and then I would introduce the rhombus. We would discuss the way
in which some shapes can be classified as several different types of quadrilaterals
(such as a square, which is a rectangle, a parallelogram and rhombus).
After this session, the students would break for individual work. Some
students would complete a worksheet in which they filled in sentences
with blanks about relationships between the different quadrilateral shapes.
Some students would work with physical tangrams to build various quadrilaterals.
Finally, six or seven students at a time would work at the computers,
in pairs of two, using the virtual tangram manipulative. These students
would work to build two of the puzzles on the website: a square out of
seven pieces and a parallelogram out of seven pieces. Both of these puzzles
provided the outline of the shape for students to work within and fit
pieces into. In theory, students would rotate between these independent
activities for the rest of the lesson, or for approximately twenty minutes.
The activities that students would do with the physical tangrams would
be slightly different than those that the students on the computers would
do, to ensure that students could continue to rotate at the activities
without repetition of activity.
What happened during the lesson:
Students got extremely excited about the option of going onto the computers
to work with the tangrams. Students who did not get to go to the computers
in the first round of students were quite disappointed, even though I
suggested that they could still work with the actual physical tangrams
to build the same puzzles. However, the students who were working on the
computer actually never really left. No group was able to finish both
puzzles at the computer, while the other students rotated among the other
two activities easily. One pair of students eventually solved one puzzle
and vacated allowing other students to try the computers out. One group
stopped working because they could not agree on the placement of the tangrams,
and quickly became frustrated. Over the next few paragraphs, I will examine
where and why this activity did not work as well as I had hoped. This
reflection will move into a discussion of how this activity, and other
activities with internet-manipulatives, could be improved.
First of all, several of the students had a difficult time working together
in a group, and several students decided they would rather work by themselves
on something else. For the most part this stemmed from one person moving
the other’s pieces and the desire to have ownership over one’s
work. Furthermore, since there was only one mouse, one child could only
really be in control at a time.
Secondly, one pair of students became distracted while completing the tasks. They discovered the buttons that changed the color of the tiles and proceeded to change each tile to another color. The color of each piece also became a topic of hot debate for the two children. Also, these students did not stay on task, and instead took to creating images out of the tangrams. (This distraction also came with the physical tangrams.)
In essence, I believe that these distractions and difficulties in the
groups came from the fact that the problems on the computer were much
more difficult than those on the worksheets. Students became frustrated,
then irritated and then distracted. This distraction came easily since
they were working in a pair. While I was concerned that the students might
have some trouble manipulating the pieces on the screen, this turned out
to be of no concern. A few students did not figure out that pieces immediately
that pieces could be turned or flipped. However, as soon as they were
shown this concept, they had no problem using the manipulative. Since
students who were working with tangrams by themselves were having an easier
time, I suggest that a combination of group dynamics and a difficult problem
made this manipulative less successful than it could have been.
Comparing the Web-Based Manipultaive to the Physical Manipulative:
Before I discuss changes I would make using an Internet-manipulative the next time around, I would like to reflect on the differences between the web-based and physical tangram sets. Initially, I was not exactly sure whether there would be some clear advantage to using either of these manipulatives over the other. After watching the students work with these two different manipulatives, I still do not believe that one is definitively better than the other. Rather, one of the manipulatives might work better for some students, and vice-versa.
Some students might benefit from the fact that they can actually touch pieces of the physical tangrams. For some students, this meant that they could push pieces into one another and twist them together until the puzzle worked. With the Internet-based manipulative this was more difficult since a student could only move one piece at a time. Also, some students were more apt to flip the pieces because they could actually pick them up and flip them. However, some students never lifted the physical tangram pieces off of their tables, and simply pushed them around with their hands. For these students, having a reminder that they could flip the pieces, or in other words, the "flip" button, might have been beneficial.
On the computer, there were outlines for students to fit the pieces into. For some students this might help them to conceptualize the puzzles. While outlines can be printed for the physical pieces, they may also be distorted in the copy machine.
On the computer, there was no way to lose pieces, or to have a set that was not complete or correct. The physcial sets of tangrams frequently got mixd together and students would drop a piece on the floor and then be frustrated that they couldn't get their puzzle to work. At the same time, the physical tangrams allowed all of the students to work at the same time and are extremely cheap, as each student could cut their own set off a photocopy.
Improvements to the Lesson and Suggestions for For Future Use
Even though the lesson did not go as planned, students definitely did get something out of working with the computer. In the future, I might facilitate a lesson using a web-based manipulative by structuring time so that students could work independently at the computers. Or I might go over strategies for group work or set guidelines before the lesson, so that students can work together peaceably. I would also be aware that students might need more time to work on an activity. Since the activity was quite difficult, I might consider having students work through simpler problems on the Internet and then moving to a physical manipulative.
Certainly, there are many interesting and useful web-based manipulatives, including many that have no physical counterpart. It seems to me that it is worthwhile to invest classroom time in using these manipulatives with the students as there are many things to gain from them. However, before using these activities it is important to think about classroom space and time, as well as the appropriateness of task.