We studied for only a short time. One math page and a few pages of exercises in logical thinking.

As for the math, Robert did two pages of the same type of exercises. I chose that hoping that Robert would do that almost automatically and independently. I needed time to finish cooking. He made a 7 mistakes out of 50 problems. Not bad. Moreover, those mistakes were caused by the problems I have already diagnosed but not addressed yet. With whole numbers up to 100 and denominators up to 10, Robert doesn’t have any problems with finding a fraction of the number. He quickly divides by denominators and multiplies by numerator. But when the numbers get larger and the answers don’t present themselves immediately, Robert hesitates and makes mistakes . Only twice while solving those problems, Robert used long division and followed with multiplication. Other times he divided twice.

I was surprised that he couldn’t come with a quick answer to division facts where 12 was either a divisor or a quotient.

Robert knows all multiplication facts involving 12 but is lost when he has to transform them into division facts. All division facts up to 100, Robert learned from using families of facts and changing a * b = c into c : b = a. So why couldn’t he use, the fact he knew well 7 * 12=84 to solve 84 : 7 = ?

Probably, because we did not practice doing that at all. Somehow we omitted dividing by 12, despite learning to multiply by it.

Today, we also continued solving logical puzzles by placing pictures in order described by verbal cues.

I felt concerned that Robert had difficulties with following cues:

1. W is last.

2. X and Y are next to each other.

3. Z is between X and W.

The problem was with *between.*

When I placed Z, X, and W in front of Robert asking him to place Z between X and W, Robert was confused.

I was sure he knew what it meant to place something between two other objects. He has done it in the past. Then I realized that there was a difference. In the past he had to put an object in the EMPTY space between two things that were already placed in unmovable places. Now, he had to arrange all THREE objects: Z , X , and W in the same way.

He has never done that before.

I helped him. I should not have. I should have given him time to figure this on his own. He can** learn to think** independently only if he has** a chance to think** independently. All too often, I am taking that chance from him by helping him all too soon.

There was a time when it was necessary to “help” before Robert made a mistake. Errorless teaching was a powerful tool in creating basic concepts . What I am writing here,however, is entirely different.

It is about trying, making mistakes, and trying to correct them INDEPENDENTLY. I don’t think you can learn to think without confronting errors independently.

When I was in seventh grade I was not very good in math. In previous years, my brother often had helped me to solve those word problems that required writing an equation or a set of two equations. When he went to the University, I was left with this horrid problem: *At 3:00 the hands of a clock make right angles. What is the NEXT time the hands would make another right angle?*

And no, the answer is not 9:00.

I spent three hours trying to solve this problem. I cried. I threw the book on the floor. I gave up, I started again. I drew the clock hundred times, I almost discovered calculus, before coming with a simple equation.

For the next 10 years, I did not have problems with math. I went to study mathematics. I did not have special talent, I certainly was not creative, but I knew that any problem in a textbook had a solution and thus I could find it.

This is why I want Robert to find a way to place three pictures in such a way that Z ends up between X and W.