Yes! I think we basically agree also! It just caught me that your list began with kindergarten learning the names of shapes and sides... this is what a lot of kindergarten teachers think geometry with small children should be, and there is much much more than that! I see classrooms where teachers are proud that kids know the names of shapes and can point to the red rectangle and say it's a rectangle. Show them a red triangle and they also say it's a rectangle, or turn the triangle upside down and it is no longer a triangle it is an ice cream cone. A big part of my job right now is to help kindergarten teachers and helpers know the big ideas that can and should be developed in kindergarten math.

I'm not blaming teachers - saying the problem is with how math is taught does not mean I blame teachers. I taught preservice teachers for 10 years and for the past 6 I've worked with teachers at all levels in classrooms across the country. I have the greatest respect for teachers and all they do and accomplish. I work with eager and excited teachers who are thrilled when they see what mathematics can be. Many teachers, along with most of the population, have no idea what mathematics can be. It is part of a HUGE cultural problem with what people have experienced in their own school as to what mathematics is.

I agree with you 100% that we need TIME to explore these ideas. I see too the rush that because today is the second Tuesday in October we will learn to add two numbers that sum to greater than 10. I enjoy teaching math art so much because there are few expectations or press for time, and we can explore and think and share ideas and become inspired. Funny you mention moebius strips -- I did moebius strips last week with 6th graders. We went through a planned progression of constructions for about 45 minutes and accumulated a list of questions on the board as we went along. The kids were beside themselves to do their own exploring -- and they got the next 45 minutes to try things out and share. I saw results I've never seen before and these kids had so many ideas. One kid even had the idea that taping two moebius strips together could give different results by cutting them depending on if the handedness of the strips matched or differed. Kids took things home to try out themselves -- I'm excited to see them tomorrow and see how many kept exploring on their own.

The level of engagement and excitement was so high -- and yet this is not something you can put on a test so this kind of activity gets passed over and is generally seen as unimportant. I have many many such activities that I think are some of the most important things we can be doing in math class, but they can't be assessed well and thus they are not important in the system.

Here's some of their creations. The top right was 4 non-twisted rings taped together at right angles, then each ring cut in half. Top left is 6 rings. 3 rings didn't work well, this student conjectured that it only made a form that sat flat with an even number of rings, and she had a good idea what it would look like with 8 rings. She also has an idea how to modify the end rings so that all the lengths are the same after the cut.

Math is about thinking. Too often in the classroom it is just about memorizing.