Exploring Symmetry with First Graders

George Avrunin
Department of Mathematics
University of Massachusetts
Amherst, MA 01003-4515
avrunin@math.umass.edu
Jan Demers
Crocker Farm School
280 West St.
Amherst, MA 01002
jademers@k12.oil.umass.edu

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Mathematicians in Math Education
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Last spring, we did a small project on symmetry with a first grade class at the Crocker Farm School in Amherst, Massachusetts. The project was centered around a visit to the mathematics department at the University of Massachusetts where the first graders could see adult mathematicians at work and do some problem-solving of their own, but also included significant work in the classroom before and after the visit. In this document, we describe what we did and talk briefly about the changes we would make if we were to do the project again.

1. In the classroom

By spring, the first graders had had many experiences solving a variety of mathematical problems. They had had daily opportunities to talk about math and they were enthusiastic participants during math time. Needless to say, they loved the idea of taking a field trip to the university math department.

In order to make the experience both meaningful and productive, we planned specific activities for before, during, and after the trip. We decided to explore the topic of symmetry with the first graders since, at the time, they were working with two-dimensional geometric shapes.

First, the concept of congruence was introduced with a series of geoboard lessons. Given a simple definition of congruence, the children identified congruent figures. The formed congruent shapes with rubberbands on geoboard and looked a the same figures in different positions and orientations.

Next, they demonstrated slides, flips, and turns by moving a stuffed bear, their own bodies, and pictures of themselves drawn on 3 X 5 index cards. This activity is described more fully on pages 18--20 of the{Geometry and Spatial Sense} volume of the NCTM {Addenda} series.

2. The visit to the university

The class came to the math department by bus in the middle of the morning. The math department at UMass is located in a 16-story tower and has a large lounge and colloquium room at the top of the tower, where we assembled. After we let the children look out all the windows for a few minutes (which may have been the most memorable part of the trip for some of them), they were welcomed to the department by the department head. (The children had some questions about the "largest number", so the welcome actually turned into a short discussion of big numbers and infinity.) We then took them to a computer lab used for research in geometry.

In the lab, we showed them some surfaces and some of the ways the software allowed different views. Partly this was for the oohs and ahs, but we pointed out that this was really just a way of drawing pictures to help in thinking about something. We also noted some 3-dimensional models and people discussing a problem at a whiteboard, and pointed to the similarities with the way the children do their own mathematical work.

We then took them back to the lounge. Jan split them up into teams of two children each, and we gave each team a sheet of paper with a square outline on it and a square piece of paper that fit the outline. As shown below, the square had an off-center mark (on both sides), so changes in its position would be visible.

We asked them to find as many different ways of putting the square into the outline as they could, and to record the ways they found. They worked in the 2-person teams [view samples of student work]. After a few minutes, when things seemed to be slowing down a bit, we called them all together again and talked about how many ways they had found. (Most, but not all, of the teams had found all eight.) We then asked them to create a list at the blackboard, with the various teams adding to the list as we went. They had used several different methods for recording the ways of placing the square on the outline.

After some cookies and juice, George talked for a few minutes about the kind of mathematics represented by problems like this, about "applications" like mail-handling equipment and industrial robots, and about extensions to 3 dimensions and crystals. He suggested that they explore the same problem for additional shapes at school.

3. Back in the classroom

Back in the classroom, the first graders explored turns and flips using other shapes, including a rectangle, a trapezoid, a rhombus, an equilateral triangle, and a regular hexagon.

As they shared their results, they were asked if they noticed any patterns. Some of the children noticed that for the square, triangle and hexagon, the number of turns and flips was double the number of sides. One child asked about a circle, which prompted a lively discussion. They decided to write to George to report on their work with shapes other than the square, and to ask about the circle. (Their questions about the circle had to with counting the number of rotations.)

George wrote back to them with a suggestion on how they might go about investigating flips and turns with a circle. He also offered to visit with them during one of their math classes and do some more activities with symmetry.

George then visited the class about a month after the field trip. After answering some questions, we divided the children into the same teams they had worked in at the university and handed out the sheets with the square outline and the marked squares, along with the sheets on which the teams had recorded their answers. We then gave each team a small Mylar mirror, a little bigger than the square they were positioning, and showed the class how "flipping" the square around a horizontal axis could be accomplished "in the mirror" without lifting the square off the larger sheet. We asked them to see exactly which of the motions they had listed could be achieved in the mirror this way.

In a few minutes, the children discovered that all, and only, the reflections of the original position of the square could be obtained in the mirror. We then called the class together and discussed this connection between rotating the square through the third dimension and reflecting it in a mirror. George then talked about rotating a cube, and the fact that there are symmetries we can get in a mirror that we don't obtain by moving the cube around physically, since we can't rotate it through a fourth dimension. We tried to illustrate this with a large mirror and a Tinkertoy cube, showing how the reflection changed the orientation of edges and could only be accomplished by taking the cube apart and re-assembling it. This wasn't completely successful, at least partly because we turned out not to have exactly the right Tinkertoy parts (we should have known better and run through all the demonstrations in advance), but at least some of the children got the idea that 3-dimensional objects were different from 2-dimensional ones in this respect. George suggested that they explore reflections with the other 2-dimensional shapes they had investigated.

We then gave them "Mirror Game" sets that ask them to duplicate patterns using 2 marked cubes and a mirror, and let them explore various aspects of reflections.

4. Discussion

The least successful part of the activity was the discussion of 3-dimensional objects, a topic that the first graders had not had an opportunity to explore in the classroom beforehand. Although it's possible that the children were simply not ready for this particular sort of abstraction (and, as noted above, our presentation was handicapped by not having the right props on hand), we think it's more likely that the children simply needed more experience with congruence and orientation of 3-dimensional objects, like the geoboard activities and the exercises with flips and turns they had done with 2-dimensional ones. If we try this again this academic year, we will expand this aspect.

It was evident that the first graders enjoyed the field trip to the math department, and it was certainly valuable for them to see grown-ups doing mathematics, in ways that are not all that different from the way first graders do it. But the most valuable and exciting parts of this project depended on the whole sequence of activities that occurred in the classroom. We think that the most successful interactions between college faculty and elementary students will involve extended collaborations with the classroom teacher and the class.

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