I can't help but think that the majority of students algebra problems in high school come from the fact algebra is taught extremely poorly at the college level. Which is where their teachers learned how to teach.
In an Ivy League PhD physics class more than half of the students thought that equality was something between a mechanical operation, e.g. 1 + 3 = 4 means you add 1 + 3 and the result is 4, or assignment, e.g. x = 3 means that for everything that follows you can replace x by 3.
The idea that it's a binary predicate was something that only people who had done logic or something adjacent at undergrad even understood. Viz. the type of = is (* x *) -> Bool, or in words an equality just returns true if there exists a way to transform one input of the equality to the other and false otherwise.
The confusing part is that for an equation like x + 2 = 3 you're given the predicate for the of set/tuple defined by abstraction using that predicate, e.g. the set {x | x + 2 = 3 and x in R} or the tuple (x | x^2 = 0) if we care about repeated answers. Something that is not at all obvious from just being given x + 2 = 3, which without a lot of implicit context is a well formed sentence of algebra with no intrinsic meaning.
If all algebra problems were reduced to something like "List all members of the set {x | x^2 - 3 = 9 and x is real}" then we have solved a huge part of the problem.
In an Ivy League PhD physics class more than half of the students thought that equality was something between a mechanical operation, e.g. 1 + 3 = 4 means you add 1 + 3 and the result is 4, or assignment, e.g. x = 3 means that for everything that follows you can replace x by 3.
The idea that it's a binary predicate was something that only people who had done logic or something adjacent at undergrad even understood. Viz. the type of = is (* x *) -> Bool, or in words an equality just returns true if there exists a way to transform one input of the equality to the other and false otherwise.
The confusing part is that for an equation like x + 2 = 3 you're given the predicate for the of set/tuple defined by abstraction using that predicate, e.g. the set {x | x + 2 = 3 and x in R} or the tuple (x | x^2 = 0) if we care about repeated answers. Something that is not at all obvious from just being given x + 2 = 3, which without a lot of implicit context is a well formed sentence of algebra with no intrinsic meaning.
If all algebra problems were reduced to something like "List all members of the set {x | x^2 - 3 = 9 and x is real}" then we have solved a huge part of the problem.