[EDIT: Sorry for the long post, basically we agree] I feel that I deeply understand that the name of the object is not the object, and that naming an object has no bearing on what it is. However, I have seen kids come into classrooms with no idea what a circle is and that is the problem, especially in poor schools. In order to discuss the language of math, you need to have a basis to work with, and as such, at that level counting and shapes seems like a good starting point. Even if it's not mathematics. (I'm sure you know this, but I'd like to clear it up for someone else who might be passing by):
If you and I want to talk about birds, and one of us says chicken while the other says humming, while we're both looking at a robin, we're going to be confused. We need to both agree "This is a robin" then we can discuss the structure of the wings, the red breast, the blue eggs it leaves in the nest. So, of course I think kids should be able to explore all about the shapes! That is pretty much my whole point. At this time, the student has no choice but to move on to the next subject. They get no time to explore the model they just learned, they test and then move on. Also, perhaps you are thinking that my opinion is that geometry is not important, or that spatial reasoning would not have a place in my dream curriculum. Quite the opposite! Being able to build and break down shapes, I have found has helped many students in other areas. I would encourage more play with physical shapes. I would not however ask them to calculate mindlessly. I would twist a two dimensional circle, I would ask them if they could find a way that you can twist a band and always end up on the same side! We would explore these things. We would have TIME because we aren't rushing toward the next subject. In fact one of my favorite authors has a great deal to say about spatial reasoning. If we look at George Pólya's seminal work "How to Solve It" one of the first examples used is that of a parallelepiped wherein the room is used as an example for finding the diagonal. But it's true we are at a place where most elementary educators fear math, I've heard it come from teachers mouths. They've never been lost while solving something, only to find the answer after a brief night of sleep and a cup of coffee. Alleviating the math anxiety is important to me also. But I can't blame the teachers, I just can't. They already deal with so much, especially in elementary where they have multiple subjects and activities. Please do not misunderstand me, I am not dismissing current research into mathematics, I am dismissing the rush of administrators in schools to push kids to the "next level" and not give kids the chance to play with an idea. The kids never get to see the fun puzzles that can come because we are all so ready to test them and move on, and it saddens me greatly. I am not a real teacher, just someone who volunteered for a while with my wife who is a real teacher.
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.
It's interesting to hear you both talk about teaching math. I work on math with my three year old, not intensively but I try to make it creative and fun. She counts into the twenties well but she losses focus beyond that and I'm not being a drill sargent. I do things like put out a pile of M&M's and we count them. Then she has to make two piles with the same number of M&M's in them and count them. I did even numbered piles the first few times we played this game and then I started introducing piles with odd numbers. "There is one left over! What is going on here!" Talk a bit about even and odd number piles, hope to move to piles of three with the same number of objects soon. No idea if this is a very good game for teaching but we have a good time. Puzzles seem great for young kids and spacial relations. We have some picture puzzles with boarders some that are in the shape of the object being puzzled (stuff like fruits) and another free form geometric shape puzzle where you can try and copy form or just do what you want. We talk about how triangles can look different and still be triangles but also that it's strange that rectangles, rhombuses and squares are called different thing but are so much alike. Reading what you guys wrote I'm realizing that it's time to go three dimensional, I see a construction project in our future this week. Cones, cylinders and spheres will be interesting.
That sounds great cgod! Those are all really good things to do! After your daughter has counted like, 8, things, try adding one more and asking "how many now?" and see if she can answer without counting again. Try taking them away one at a time to practice one-fewer relationships as well. Only 20% of 1st graders can count backwards from 25. Counting by 2s and 3s and 5s and other multiples are really fun too. By the time my daughter was in 1st grade she could add and subtract 3 digit numbers mentally, including numbers that crossed over hundreds, and she could manage things like 135 minus 200. Her amazing number sense continued until she learned the standard algorithm -- and then she could no longer add mentally. She eventually came back to mental math, but it really made me mad that they started teaching the algorithm in her class when most of the kids had very little number sense. Algorithms are the beginning of the end for kids and math, they literally stop thinking at that point. I see lots of complaints from parents in the US about the "new" methods of teaching -- many schools are starting to focus on a variety of mental math methods which I think is absolutely brilliant. Parents are dead set against it because they don't understand that this is really really important. Parents see kids doing strange ways of subtracting 17 from 45 for example by thinking "17 + 3 is twenty, plus 20 more is 23, and 5 more makes 28" or thinking 45 - 17 is the same as 48 - 20 (add 3 to both parts) and so its 28. Parents insist that this is a waste of time when they learned to just write out the algorithm and manipulate digits and get the answer. But that is a rant for another day. Keep up the good work!
Teaching that way blows my mind. When I help someone with math I guess I've always used the standard algorithm as a shared language but I know that when I've asked people how they resolve a problem in their head almost everyone does it differently. I've learned better ways of doing problems in my head by hearing how other people go about solving problems differently from me. Feels like there is something useful in having a standardized way of expressing basic problems even if I know that most people will have a bunch of "tricks" they use to actually do math. It's kind of a strange paradigm shift for me to think about not teaching the way I was I was taught. Even with that I already planed on teaching my kid easier ways to get to their final answer. Even in college mathematics I remember being penalized on tests and if I didn't work backwards to express things I had already calculated another way in standard notation. I don't think I even had a way to express how I calculated some things in notation, just little rule of thumb stuff.Parents see kids doing strange ways of subtracting 17 from 45 for example by thinking "17 + 3 is twenty, plus 20 more is 23, and 5 more makes 28" or thinking 45 - 17 is the same as 48 - 20 (add 3 to both parts) and so its 28.