Wouldn't it be an amazing world if no one ever forgot how fun learning is? I can't remember who mentioned that about kids that already have the joy of discovery knocked out of them, but it sure strikes a chord in me. In a country that is so full of leisure time, imagine what armchair mathematicians could contribute to the bank of knowledge if they weren't just sitting there watching ESPN? They would have more fun, more self worth, and be a greater benefit to society. One of the people I have worked with on science education projects keeps emphasizing that when we teach a person science, we give them an alternate way to look at the world, more things to wonder about and be amazed by and have their life enriched with. How does it get turned around to the point that learning becomes an unwelcome burden? We are biological learners. Our brains are designed for it. I guess it's the temptation to live with the least effort....
As many people have pointed out, there is a need for some amount of repetitive practice to gain the ability to do harder and more interesting problems. Suppose we accept for the moment that over 50% of teaching is psychology. How do we motivate this practice? Can it be done through helping them see the big picture, or making "fun" math available after the practice has been finished?
What do you use to demonstrate that the skills they can develop and the things that they can learn/discover are worthwhile? Projects? Do they need to see some result (other than a good grade) for their work? Will it help or hinder to have competitions (which is a big motivator for practice in sports)?
I am about to spend a lot of time working on these types of questions. How would ye who teach and ye who teach about teaching suggest that I go about finding out the answers? If you could dictate the job description of a person that, say, was going to develop curriculum materials, what would you require of them? How should they conduct their research?
Suppose that a mathematician or mathematics teacher was promised funding for two to three years and they could spend it however they wanted, with the only requirement that at the end of the time period they had to demonstrate that they had produced materials, software, curriculum ideas or whatever that had a positive effect on mathematics education. What should they do?
Say that the evaluation would be guided by the ideas in the NCTM standards, so that there is a definite means of assessment.
What would you like to see in existance that doesn't yet exist? One thing people have mentioned would be an online resource center for good real world problems. I have saved most of those idea proposals. What else? Do you think that well written software could be useful? Do you have a "dream software package" that would combine useful elements from various existing ones or that would be different altogether?
Within a few months I should be able to have room on some website devoted to this type of information. If the consensus is that these questions will put too much traffic into this list, it would be fine if people just wanted to mail to me and have my interpretation of a general representation of your ideas on a web page for everyone on the list to have access to (not that I am wanting to filter and shape the ideas, just an acknowledgement that every presentation reflects some of the presenter).
My personal opinion is that it would be healthy and "on topic" for us to talk about what _should_ exist, (using the Standards as the, um, standard) since that would cause us to consider how the standards affect the way we look at education.
Sorry this is so long. You would think I had a lot of time on my hands.