A+Teacher's+Perspective

The background for the creation of this wiki is explained here. As I watched another person try to assemble //The Fifth Chair,// I realized it would support kids working with spatial visualization and composing and decomposing shapes and solids. Since an upcoming math unit was on ratios, I figured we could use the pieces of this puzzle to not only explore the solids and how they went together, but also tie in the ratio work as well.

We've had software for several years now that allows our students to work with shapes and nets and explore turning 2 dimensional shapes in to a net, which can then be printed and trimmed on digital fabrication printers. As they've worked with these tools, I have realized how crucial it is for students to have opportunities to construct and deconstruct figures to truly understand how they work and how faces, edges and vertices can be manipulated to change a figure. Many of these students (but not all) had experience with the digital fabricator. The students have had prior experiences with power blocks, exploring the patterns and relationships with in them. Many of these students (but, again, not all) had done an activity with me in third grade where they attempt to create the various nets that form a cube and then try to build them to check their work.

My goals for this activity were :
 * 1) to increase students' spatial visualization skills through manipulating nets in their mind
 * 2) to measure the various sizes of the brainteaser to find ratios and relationships between them
 * 3) to see if recognizing those patterns would help in solving the puzzle--if they knew the relationships, proportions and ratio of one piece to another, would they use that to help themselves solve the puzzle, or would they simply do trial and error as they usually did to solve a brainteaser?
 * 4) to use our digital fabricator to recreate the brainteaser with different measurements, but the same ratios, so that each child could take a set home

So we began this unit with an activity where we created generational pop-ups to introduce ratios and relationships between the line segments making up the pop up folds. (Instructions can be found here, and you can see my students working here.) The generational popup work was done with Willy Kjellstrom, a University of Virginia graduate student who has been supporting the digital fabrication work in my room for 2 years now. Willy is an amazing educator who created the materials and put together the videos for the generation popup lessons.

From there, we moved to the fifth chair work, where I first asked the kids to show me which nets on this sheet would turn into cubes and which wouldn't. I had kids using how many faces were on a cube, and the number of edges to eliminate some of the nets. They almost all began by first eliminating ones they knew would NOT. After they finished that sheet and gave it to me, then I asked them to complete this task. While I knew many of them had experience with manipulating 2-dimensional shapes into 3-D objects with our software, and while many of them had had experience with building things with our digital fabricator, I really didn't expect every single one of them to manipulate this shape mentally as quickly and accurately as they did. As they explained their thinking, I found the varying perspectives to show differences in both thinking and working.

Several of the kids were initially unsure of how to picture this task "in their minds."

Ty's Perspective shows his initial confusion and his burgeoning understanding-"Oh, I see..."

In Teryn's Perspective, Teryn said she "used her hands and flipped and folded and did some other stuff" so I asked her what she meant. It was fascinating to watch her show me how she first folded up her fingers to make a 90 degree fold, and explained she was thinking of that as the first two colors. She then talked about what color would be on her fingers or palm and turned her hand over to describe which colors would then be on the bottom of the cube she was creating in her mind with her hands.

Noa was initially unsure of how to do the task, but when she heard Teryn explaining her thinking, she immediately began folding and twisting her hands. Then, in her description, she described the placement of the colors accurately, using the first two colors as her anchor, or "base."

Several of the kids clearly described their "folding" of the net in their mind. Some of the more explicit ones include __McGee's Perspective __, __Aimee's Perspective __, __Jacob's Perspective __, and Vera's Perspective.

__Tyler's Perspective __ is very matter of fact and uses everyday language to describe the placement of the colors (left/right, top, etc.)

Noah's Perspective shows that his descriptions involve comparing a cube to familiar surroundings, as he equates the sides and top and bottom of the cube to the ceiling, floor and walls.

__Evan's perspective __ shows he clearly understands geometric language as he incorporates that into his description of his thinking.

Looking back, I think giving them the sheet of nets helped tremendously, as most of them were already trying to visualize the nets into solids from that work. This task is well worth doing again, as it gave me a nice pre-assessment as to language the kids used naturally, their ability to describe a process and their ability to envision folding a net into a cube.

As we moved into the next piece of the exploration of the fifth chair, I asked them NOT to try to put it together, but instead to measure the pieces of the puzzle. I gave them no tools, no units, curious to see what they would go for. Most immediately got rulers and attempted to use inches, which were NOT easy measurements of these particular objects. Some then tried centimeters, which were even and easy, and some got cubes and started trying figure out either a non-standard measure or the volume. Pretty quickly someone stated they were comparable through ratios, and they started sharing what they were finding, writing equivalencies of the heights, widths and depths as they explored.

We also discovered pretty quickly we had to set a standard position to be able to talk side or base or height or width and we had to decide how to name them so we could compare them and keep track of those comparisons. Once they had figured out many of the relationships, class was over and they left anxious to return the next day and put it together. We kept their work on the tables where they had been recording.

During the next math class, the goal was to assemble the fifth chair. Kids were in pairs working on it, and I began by sharing my reflection on watching another teacher try to put it together. I then asked them to be conscious of their thinking as they were watching each other. Not only did most kids put it together in less than 15-20 minutes, but the discussion of how they were looking at it was pretty amazing. As they watched another person fiddle with it, they saw it in different ways and many groups began by putting the two small ones together to make a matching piece before attempting to build the fifth chair--so they were clearly using the relationship of the pieces that they had discovered while measuring them.