What Were You Thinking
These problems ask students to say what they are thinking when they solve a simple math problem. Sounds easy, no? It is not. I suspect the problem is that this skill simply is not taught.
This skill fits together with the skills of expressing patterns and computer programming (a.k.a. being the machine master).
The Basic Problems
I have two basic problems. One is to ask students to say "ouch" 13 times. Then I ask them what they were thinking. The second problem is to first ask them the sum of the numbers from 1 to 3. Then 1 to 4, then 1 to 5. I do this up to about the sum of the numbers from 1 to 10. Then I ask them what they were doing when solving the problem. (I have no preferred form for this question.)
Both problems have a repeating loop. The question is really to say what happens each interation of the loop. This ideally should be an abstract description. So these problems fit in with the problems of expressing a pattern (abstractly) and can be a further exercise in abstract expression. The loop of course is a concept from computer programming,
No one has had any difficultly actually doing the math part of the problems. So far, no one has come up with a remotely good description of what they were thinking on their first try.
What Were They Thinking?
To say "ouch" exactly 13 times (without guessing), the student have to count by one each time he/she says ouch. That is done consciously, so saying that requires only paying attention to conscious thinking. There also has to be a way of exiting from the loop. This is probably done unconsciously, but it can be inferred -- there must be an unconscious program to recognize the number 13, and when it occurs, to stop. But this program must be running continuously, which in a sense is every iteration. In computer language, after they count the next ouch, they run the command "Stop if count equals 13".
The problem of adding up the numbers from 1 to 8 takes a while, but not if you have just been given the problem of adding up the numbers from 1 to 7. If you know that the the sum of the numbers from 1 to 7 is 28, then the sum of the numbers from 1 to 8 is 28 + 8 or 36. That's an example. Abstractly, if the number of the numbers from 1 to n is k, then the sum of the numbers from 1 to n + 1 is k + n + 1. (If you give the problem of adding the numbers from 1 to 10, the student needs to create the same loop, so the problem could have been given that way. Given as a whole problem, the student needs to have the exit to the loop)
I suppose any problem could be used for this exercise, though my preference is to use only problems with loops. One problem is if someone does 1 pushup on March 1st, and each day does twice as many pushups, how many pushups do they do on March 31st?
A very difficult problem is simply to ask someone to count something, with the exercise being to say what it means to count. The answer is that people have learned to say the numbers in order, so the counting loop is (a) isolating one of the things to be counted, then (b) saying the next number. This would be just one of those mathematical exercises in trying to say something that is so basic it is difficult to say. Except, many computer programs require the computer to count, so the student has to think about what counting is.
Doing Math From the Heart
The conscious part of your brain is not all of your brain, or even the most skilled part of your brain. Consciousness is important, but in a sense the clever and creative thoughts, and the very best thinking and calculating, occurs in the unconscious part of your mind.
So if you want to perform a sport or craft to your highest potential, you have to stop your consciousness from interferring, and essentially hand the task to your unconscious brain to perform. The experience is sometimes called "not thinking". Actually, it's a very intense not thinking, because your unconscious mind is thinking very hard, and your conscious mind has to follow along and play close attention to the task as a whole. But your conscious brain isn't doing any calculations or making any decisions, so "not thinking" works as a rough description of the conscious experience.
Now suppose you are doing math to your highest potential. No one would call that experience "not thinking". However, to do math to your highest potential, you cannot be using just the conscious part of your brain. Remember, the unconscious part is the more skilled, and it's the seat of cleverness and creativity. So you have to use it to do well.
Our culture sometimes calls the unconscious mind the "heart". Using this metaphor, you have to do math with your mind AND your heart. If you are just in your head, you are manipulating symbols and using what you have memorized, which can work okay but isn't a good recipe for math.
The "What are you thinking" exercises develop this attitude. First, and probably most important, they show the student that he/she is doing a lot of thinking. Second, they help the student learn to identify this thinking. That might sound easy, but in fact it is difficult, and even more importantly, it is an important skill and a learnable skill. Finally, when tied with computer programming (machine problems), the problems show how to use the heart to program, not conscious memory of programming facts. And fitting everything together, the programming commands are a way of expressing what is happening in their heads.