Here is a description of an experiment in learning.
Today we had a student who came to the GED center for the first time. We started her on the self-learning pilot on MILESTONE A1: Numbers & Place Values.
Milestone A1 is written in very simple English. It consists of lots of pictures that explain the abstract concept of regrouping and the use of it to generate the numbering system. So it is very easy to understand. This is the most basic concept in math, so no other knowledge of math is required. Therefore, this pilot provides a measure of the learning ability of a student to grasp an abstract mathematical concept.
I started this student on the pilot. She seemed quite enthusiastic. She claimed that she had made it to the 12th grade but then failed the FCAT. So, I expected her to zip through Milestone A1.
Milestone A1 consists of 15 lessons. Usually I go over the first two lessons with the student to give an idea of how to proceed through rest of the milestone by oneself. I did the same with this student, then let her continue by herself, while keeping an eye on her.
She proceeded through the lessons fast. Then I found her stopped on lesson A1.7. She didn’t ask for help. She just sat there. So I went to help her. I checked her knowledge of lesson A1.5, because that is the key lesson on regrouping on which the understanding of all other lessons depend. She said that she understood it. But when I asked her to demonstrate regrouping on abacus, she couldn’t do it.
This is the common problem I am finding with school dropouts. They think that they know a concept but they really don’t. Asking them to demonstrate is a quick way to check their understanding. You cannot see what is going in their mind. But you can definitely see whether they can demonstrate a concept or not.
This student didn’t get the idea of regrouping on abacus despite the fact that the procedure of regrouping was clearly shown with pictures in Lesson A1.5. It took repeated demonstrations of regrouping on actual abacus for about 30 minutes before this student could get it. But this effort was worth it. After that, she was able to read and write numbers in millions and billions, and was quite happy about it.
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You can improve your abstract thinking by learning to demonstrate abstract concepts using concrete elements. You may also use thought experiments to assure yourself of the consistency of the concept.
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Comments
In Milestone A1, there seem to be a way to quickly assess a person’s capability for abstract thinking right upfront.
Mathematics requires a certain aptitude for the recognition of abstract patterns. One can see 3 cats, 3 dogs, 3 trees and 3 chairs easily. But to observe “3” as an abstract pattern underlying what one sees requires a special aptitude for abstract thinking.
Children are expected to develop that aptitude, but that seems to happen by luck. I am not sure if the normal educational curriculum ever addresses this aptitude. That may be part of special education. So many children are bound to slip through the cracks.
In the current self-learning project, where we are dealing with school drop-outs, we need to assess and handle the lack of this aptitude before the person once again feels frustrated with math.
I think we are being quite bold in attempting to pick up the pieces of a failed education system.
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