Miguel is working hard at his school work. This is not a walk in the park. He is trying to complete three years in just one year. His exercise books, that must be marked by the teacher, give just one example for a situation before posing questions related to the subject matter.
It's been a long time since I did anything with fractions, and I've certainly never taught anyone anything about fractions. The concept of coming up with a common denominator in order to do anything with fractions was a hard one to teach. I didn't remember all that stuff either until I saw the example in the book. Ah, it's all coming back to me now...
I created numerous examples for Miguel to work through just for addition of unlike fractions. He wasn't grasping that he had to use the same number on the numerator that he used on the denominator, he kept wanting to use different numbers. We went over it and over it until I wanted to hit him over the head with the book. (I probably should never change professions and become a teacher).
I found a website for fractions, but it was in English. I retaught myself a few things about fractions, but that didn't do much to help Miguel.
Finally I drew circles - one ended up with 24 segments. But it was a light-bulb moment for Miguel. Ah ha - he learns better with pictures than with words (words made out of Spanish and English, no less). Off he went to work on the exercises, drawing his own circles to come up with the answers.
He went to school and worked with the teacher. He is moving along in his book. He moved away from pure fractions into area, radius, diameter, and Pi. Good grief, the only Pi I've thought about since high school is cherry Pi.
But the book had an example, and the answer was in the back of the book. Knowing the answer I'm striving for helps me put the right pieces together. I figured it out, we did the exercise, and Miguel went back to the teacher for additional help. He returned with these departing words from the teacher - "Pi = 3.14, now, and for the rest of eternity". For ever and ever...
Yesterday he sat out on the front terrace all day long, doing his schoolwork. In the afternoon he asked for some help. Lord, if it wasn't a tunnel of 10 arches, 20 m in length. They wanted to know the 'area', what I would call 'circumference', of one arch. Somehow we worked it out, Miguel actually did as much as I did to come up with the right anwser. Yeah!
Last night he was not tired, so I went up to bed and left him studying. About midnight he came up and said he'd been stuck on one question. He wasn't settled in for more than 15 minutes, as I was drifting off to sleep, when he spoke out loud - "OH! It's thirds, not quarters! Yes!". Lying there, away from the books, his mind came up with the answer to the question that was puzzling him.
The mind is a strange thing - it continues to process after we've given up. At times I've been stumped on a programming problem, only to wake up at 3 am with the answer. But I will never have to wake up at 3 am with the answer to "what is Pi?". It's 3.14, from now until eternity.