Ladnor was curious about my expectation for students.
Quite simply, there is a limit. I don't let them get to the point of frustration (at least I try not to).
What I propose to them - particularly students who are re-entering the schooling process or have had a lifetime of bad math experiences - is that we work together to help them restructure how they think about math, and how they think about themselves doing math.
Reconstructing means to look at the problem in the sense of don't automatically start writing something; don't automatically start searching for rules; don't automatically anything. Rather, slow down and realize that math is first of all seeing what the problem is; am I reading the problem correctly?
There's nothing deep or mysterious about this, but again what I've learned from students is that they are in a hurry and that "good" mathematicians doing things fast, or come to the "aha" moment quickly, or do all the work in their head and spit out the answer. Whether it's true or not or whether you believe it or not, what I propose to them is that there are a number of quantitative things they do quite quickly because our brains are set up to make judgements, estimates, and comparisons quickly. The key is for us to slow these processes down so that we can see what's happening.
An example: I will open the class the first day by asking for someone who is athletic to stand up, someone preferably who plays baseball or softball.
I then toss them an eraser and ask them to toss it back. After we have done this I ask the class what just happened. I ask them to describe in detail how we were both able to throw and catch. Now, again, you may not agree but think about all the math involved in those acts; think about it at the micro-level.
What I hope to accomplish is a sense of their already being mathematical and helping them understand how to slow it down and get a true mastery of the processes we use to capture all the operations and relationships . later, mark