Abstraction is really hard. An economics professor of my acquaintance said he started requiring calculus for his game theory class so he could be sure everyone was competent with algebra. Every time you increase the level of abstraction you lose some people. Some of them can be taught it but some can’t, no matter how hard they or their teachers try.
Interesting. That’s probably the best reason I’ve heard for why calculus and computer science prerequisites should be similar, even when the latter doesn’t use the former: if you don’t have the abstraction capacity for calculus, you probably won’t have it for programming.
That's exactly how I started teaching my 2nd grader. They got some problems like that for their homework and on the scratch paper I said, "well for the blanks we can just stick an x in there" and she didn't even blink and went with it. Because well blanks look just odd. They might look like a "-" if it is written too high.
This is an example of them taking something that's basic and making it more complicated and cumbersome by trying to "simplify" it.
Montessori uses clear test tube learning devices with beads. Less abstract and more easy to visualize. This is taught in 2nd grade. In your above example, the blank or variable is an empty test tube, the 4 is a tube with 4 beads, and the 9 is a tube with 9 beads.
That works well for teaching how to solve problems like that, especially in an early age.
However, do you know what's the Montessori approach to learn abstraction as such, so that you can move from the visual/tangible approach (which works well for simpler problems) to solving problems with the abstract symbol manipulation that's needed for all the more complicated tasks including applied math at physics, economics, etc ?
In my experience as a math tutor, they’re untaught natural abilities via poorly constructed (and hence, understandably confusing) “math” lessons taught by humanities majors turned primary school teacher.
I often make the same point, and have occasionally found just rewriting it as “4 + [x] = 9” is enough to help span the conceptual bridge. Sticky notes in white board equations is another way to make this point.
