My daughter is just starting to learn subtraction. She was very frustrated by it, and if I verbally asked “What’s seven minus five?” she was about 50% likely to give the right answer. I asked her a sequence of simple subtraction problems and she consistently performed at about that level. In the course of our back and forth I switch my phrasing to the form “You have seven apples and you take away five, how many left?” and she immediately started answering the questions 100% correctly, very rapidly too. Experimentally I switched back to the prior form and she started getting them wrong again. It was apparent to me that simply phrasing the problem in terms of concrete objects was activating something like visualization which made the problems easy, and just phrasing it as abstract numbers was failing to activate this switch. So as you say, for more tricky arithmetic problems, it may be the case that what mental circuits are “activated automatically” determine the first answer you arrive at, and you can exploit that effect with edge cases like this.
One of my most eye-opening teaching experiences occurred when I was helping a six-year-old who was struggling with basic addition – or so it appeared. She was trying to work through a book that helped her to the concept of addition via various examples such as “If Nellie has three apples and is then given two more, how many apples does she have?” The poor little girl didn’t have a clue.
However, after spending a short time with her I discovered that she could do 3+2 with no problem whatsoever. In fact, she had no trouble with addition. She just couldn’t get her head around all these wretched apples, cakes, monkeys etc that were being used to “explain” the concept of addition to her. She needed to work through the book almost “backwards” – I had to help her understand that adding up apples was just an example of an abstract addition she could do perfectly well! Her problem was that all the books for six-year-olds went the other way round.
So perhaps the girl you were teaching was a member of the cognitive-decoupling elite you wrote about? The problems you are explaining in that article are reversed in the early years of school math. That is, a strong use of tangibles to explain concepts.
She’s encumbered with all the tangible objects preventing her from answering the conceptual question. In later years she’ll be freed from tangible objects obscuring the real problem. Math will likely be more intuitive for her and she’ll start to soar. And it’s at just that point where the rest of us start to flounder..
I wonder if this concept has been discussed from a learning and pedagogy perspective. Seeking to tailor technical subjects to different types of thinkers?
My daughter is just starting to learn subtraction. She was very frustrated by it, and if I verbally asked “What’s seven minus five?” she was about 50% likely to give the right answer. I asked her a sequence of simple subtraction problems and she consistently performed at about that level. In the course of our back and forth I switch my phrasing to the form “You have seven apples and you take away five, how many left?” and she immediately started answering the questions 100% correctly, very rapidly too. Experimentally I switched back to the prior form and she started getting them wrong again. It was apparent to me that simply phrasing the problem in terms of concrete objects was activating something like visualization which made the problems easy, and just phrasing it as abstract numbers was failing to activate this switch. So as you say, for more tricky arithmetic problems, it may be the case that what mental circuits are “activated automatically” determine the first answer you arrive at, and you can exploit that effect with edge cases like this.
Strangely, it can sometimes also go the other way!
I think this is unusual though.
So perhaps the girl you were teaching was a member of the cognitive-decoupling elite you wrote about?
The problems you are explaining in that article are reversed in the early years of school math. That is, a strong use of tangibles to explain concepts.
She’s encumbered with all the tangible objects preventing her from answering the conceptual question.
In later years she’ll be freed from tangible objects obscuring the real problem. Math will likely be more intuitive for her and she’ll start to soar. And it’s at just that point where the rest of us start to flounder..
I wonder if this concept has been discussed from a learning and pedagogy perspective. Seeking to tailor technical subjects to different types of thinkers?
See also?
Ooh, I’d forgotten about that test, and how the beer version was much easier—that would be another good one to read up on.