Hiya All - Maria's comment reminded me that I had worked on the last reply offline ... Indeed I found a copy. So for better or worse :) - here it is + links to my screencasts:
Okay - I am taking a deep breath and starting over. I worded my original message incredibly poorly and I apologize deeply. I did in fact generalize that all of us contribute to this phobia/fear/hatred or to use the politically correct term "math anxiety". However, I most certainly include myself in this group (and if you can hang on, my personal example from yesterday is at the bottom).
That math anxiety exists cannot be denied. (Thank-you Brian for the link – that was indeed interesting!) Certainly parents/TV/others contribute to this anxiety, but their anxiety must also have come from somewhere. The very nature of mathematics - where the ability to master skills in the current school year is dependent upon the supposed mastery of skills from the previous year - also contributes to this problem. Many other reasons can easily be cited and proven. But, we as teachers must be extremely careful not to add to this problem.
1. The Sieve Problem. Here is what I got from my brother a month ago. http://mathcasts.org/x/lfs/Sieve%20of%20Eratosthenes.pdf . He want me to explain this to her so that she can understand it – THIS IS THE KEY to MY ORIGINAL COMMENT. We must answer the question rigorously (no guessing) at the level at which it was asked.
I had never heard of SE (I even call him Erasthmus in my screencast: http://www.screencast.com/t/ZDk1NjE4YTg). Of course, I know the standard prime rule, but the question specifically says "key to SE". I look up SE on Wikipedia and there is no mention of square root in the algorithm. It says you start with 2. Circle it and then cross off all multiples of 2 starting with 2^2. Find the next number not crossed off – it is a prime p=3. Circle it and then cross of all multiples of 3 starting with 3^2. Find the next number not crossed off – it is p=5. Circle it and … . You are done when p^2 > n (the end number in your table) simply because there are no more numbers to cross off. I think – OMG, how I am supposed to relate square root to this and explain it to a 4th grader?
I put to you the following. (1) I don't care how it was explained in class. Maybe she was sick or not listening. I should be able to look it up and find the answer. This is 4th grade. (2) The California standards do not use the word "roots" until grade 7 (http://www.mathcasts.org/mtwiki/Standards/CaK-7) so the question is out of bounds period. (3) The wording of the question is too sophisticated for a 4th grader and too vague to be asked of ANYBODY. This is wishy-washy math that confuses and then angers students.
I want kids to write; I want them to think. I might ask of an older student. "For each new prime p, we start our new crossing-off with p^2. Why don't we have to check all of the multiples of p starting with 2*p? (And this is badly worded, but it is at least rigorous and makes them think.)
2. The sqrt problem. (My screencast: http://www.screencast.com/t/sxtZODAjkBpC) Let's assume that this question was asked "nicely", i.e. Show that sqrt(x+sqrt(1-2x)+sqrt(x-sqrt(1-2x))=sqrt(2) on [0.5,1]. Notice the reversal between sqrt(2x-1), which is now sqrt(1-2x) and further assume that we have recently been working on problems about simplification that involve squaring both sides of an equation. I too see no problem manipulating the numbers and getting the "answer".
I put to you the following. (1) There is no learning going on here by manipulating and this causes math anxiety. (2) This is in no way a proof – there is no rigor* and mathematics must consistently be rigorous. (3) I would bet REALLY good money that less than 5% of all students who "answered" this question would understand why the original question and this question are the same question (and presumably this is the KEY to why we might want to solve this problem at college level) and (4) "Seeing the way" and jumping in is not a consistent approach to problem solving (and it REALLY annoys students). We "see" because we have extensive experience. Saying "this is easy" is saying "if you don't see this, you are dumb". #1 This is not true. #2. We must teach them how to think, how to experiment, how to look for solutions. (5) We must stop thinking that if a student has learned a technique – say because he successfully solved problems like that within that context in a previous class - that this means he has learned it forever and will instantly recall it even within a different context. If I tell you "square both sides" whether directly with words or implicitly by asking this problem in a lesson on that technique, I have shown you the road. I always tell my students – if someone shows you the road and you are persistent, you can make it to the end. It is deciding what road to take that is the hard part.
==And if you think "domain checking" is excessive rigor - in the common core standards (which will most probably be implemented across the US), one of the KEY K-12 questions is: 5. Core Concept C. A student performs the following steps in solving an equation: (x + 3)/(2x + 6)= 1. x + 3 = 2x + 6. x = -3 Is the solution correct? If yes, explain why. If no, explain what was wrong with the student’s reasoning. ==
Finally – I give you an example for myself just yesterday. I found an applet that showed the harmonic series as a series of cards being slid over. Cool animation – I thought. However I did not see any math learning other than visualizing the sum. The sum was written generally – regardless of the current value of n. From the animation, I could not understand the divergence. But cool show&tell. I said to myself "I can fix this into a wonderful learning applet". I work on it for a couple of days. I say "This is great. Now all the math matches the animation. It is color-coded. It shows graphically the standard proof of divergence…. Everyone will love this applet." But, I am used to myself after all of these years. So I show it to my friend who tells me ever so gently – they are not going to see divergence here either and of course that is the goal of studying the harmonic series. Otherwise, why bother. We spend time discussing how we "realize" divergence. I rewrite the applet. Show it to him again and again he gives me feedback – this time on the wording. He is correct – it is vague. I rewrite it again. He thinks it good, but of course we must try it on real students in a real classroom. (See: http://geogebramath.org/lms/nav/activity.jsp?sid=__shared&cid=e... and the "next" applet is my first attempt. Huge difference.)
Then, he says we should connect this to the area under the integral of 1/x since it diverges too – I NEVER would have thought of that, but I think – ahh yes, I have a Riemann sums applet that I was just showing to my son's girlfriend… And the cycle continues. Process and Connections.
My point is that we must constantly be vigilant about what and how we teach it. Our goals and our wording must be clear and our math rigorous. (Wishy-washy is MUCH harder to learn than rigor.) We must consult with our colleagues and friends. (Ask the SE question of anyone not in math and watch their face screw up. The wording will drive them nuts.)
We must ask ourselves - what do we want our students to know at the end of the year and what must they know to succeed in the next year under this principle. Teach them the absolute essentials – no partial credit. Teach them what you love, get them talking/writing about ANYTHING in math – they will learn to love math and to think logically. Make them learn process and the particular skill in YOUR class. But don't assume that skill outside of your class; completing the square might be fun for you – but nobody in their right mind solves a quadratic equation by completing the square. The process of learning the skill with rigor - not the skill itself - is important. I don't like completing the square. I do it once to prove the quadratic formula without mentioning 'completing the square' and then just use the formula. But I do like inequalities with absolute values and think them great fun. So I teach them. Neither completing the square, nor solving inequalities with absolute values is an essential skill. But the ability to LEARN these skills is essential.
My many thanks to Maria Droujkova of mathfuture.wikispaces.com for making me think about essential skills in a different way. My thanks to Subhashree for asking this question. And – my thanks to all of you for contributing to this discussion. Philip was indeed right – we all do care and that makes all of our voices important.