Using 3-dimensional grids to present views of 3-dimensional shapes.
Problem Solving Steps and Problem Solving Talk
The challenges involve constructing a 3-dimensional grid "stage" to create or recreate 3-dimensional shapes. I tried it out on my PYP Coordinator and colleagues first. This gave me practice in being receptive and using questions to focus on process and thinking rather than on results and being directive. Then I presented the first challenge to my class - Space Map A. I found that to use effective types of questions which cause students to think I needed to hold back. I had to think quick to recall, formulate or use questions from the support poster. (See my blog Math IS How We Organise Ourselves.) The mind gymnastics on my part was well worth it. I began to see students thinking for themselves, reflecting, concluding and proving their solutions. I also had to allow for possibilities I hadn't thought of or reasonable but unique interpretations such as negative space seen as positive space. I needed to read facial expressions and body language rapidly to inform my action choices. I also had to insist that students respect the value of the problem solving process and what it provides them, rather than jump straight into Step 3 Act.
With my "Be More Receptive" hat on, I also noticed for the first time the signs that my students' beliefs about Math were being challenged. For example, one conversation went:
"Ms Hutchins but that isn't a 3-D shape!"
Instead of stating, "Yes it is." Followed by my explanation as to why. Instead I tried, "What to you believe a 3-D shape is?"
To which was replied with a circling hand motion, "A shape that goes all the way around like a cylinder , pyramid or cube."
"Ok, does that explanation connect to this situation?" I had shown myself that through questioning students can build on their own knowledge and apply it to unfamiliar contexts creating their own new and accurate knowledge.
The most important point I learned was to distinguish between the two reasons why I may reframe a Math question. Firstly, reframing a question I formulated with good intentions but one that needs clarity is necessary. I try my best but often my questioning is only better with age - at least one use. That is why I like trialling things on colleagues first. Secondly, and the more dangerous pitfall to avoid is, reframing a question that makes the pathway to solving a good question easier or quicker. In doing so, and unbeknownst to my students, this immediately robs them of independently solving worthwhile Math.
The best part was the genuine confusion in their faces turn into a quiet smirk of pride when they got it. Also it was great watching chairs being pushed aside so as to get down and see the actual view at eye-level. I knew they had it when, as if an optical allusion was revealed like a magic trick, I heard, "Wow! Look at it - it really works."
Spacial Visualisation Challenges
Where Could It Go From Here?
- Find the area of each view.
- Find the perimeter of each view.
- Find the square units of the 3-dimensional shape.
- Find the 3-dimensional shape and square units of each layer.
- Find grid positions / co-ordinates.
- Transformations: rotations, translations, reflections.
- Isometric drawings.