The problem of algebra is not just the teaching of it. It lies in the fact
that most students ~75% are not ready to understand it because they don't
use proportional reasoning, and they can't do 2 variable reasoning. Then
when asked to do problems, they are often given problems that can be done by
simple reasoning, sometimes by proportional reasoning. The is totally
backwards. Then they are noodled with methods to solve equations rather
than being asked to figure out answers first, with alogorithm development
second.
One solution to the problem is to use something akin to the Benezit
experiment. Thinking Science and Feuerstein Instrumental Enrichment also
can be used for great benefit.
I always point out to students that solving equations can be done by a $10
hand held calculator much more quickly than they can do it. But so far
computers can not really write equations. Therein lies another big problem.
They should be writing equations from graphs and from descriptions. I have
the students write the wave equation as a class exercise. Y = A Sin( 2 pi
X/(Lamda) - 2 pi t/(T)). I present with the picture of the wave with the
wavelength drawn, the amplitude drawn and we start from there. I do provide
prompts such as we want Y to go to the same value when we move over one
wavelength. The number of wrong suggestions should not surprise any of us.
It takes a long time, but it is worthwhile. Incidentally Modeling has
students always writing the equations from experiments before using then in
class. I use extremely long wait times before giving hints, never answers.
It is very helpful to have the students modeling physical situations in
algebra rather than just doing the usual word problems, which teachers
always skip.
Along the way students MUST interact with graphs with scales and unit. The
X-Y plane is NOT a graph, it is a map. Once they grasp the difference they
will not try to use the distance formula to find a quantity on a graph.
Math teachers claim to do graphs, but they are not. They are doing only
maps, so no wonder students have not grasp of graphs. Incidentally Benezit
had students interacting with both graphs and maps before learning
arithmetic. They did word problems before memorizing. Memorization of math
tables was not required until after the need for them was established, and
by then they had already memorized some of them naturally in context. By
the end of 5th grade they had caught up to their peers on the memorization
tests. Unlike them they could do problems that their peers could not, and
they also were not fooled by "trick" problems because they used reasoning
first then memorized "arithmetic facts". Their peers tended to solve
problems by just doing random math without thinking about what the problem
was asking.
If anyone doubts the efficacy of pushing thinking over memorized algorithms
look at the ADAPT program at U Nebraska. It halved the failure rate for
remedial students on a subsequent calculus course compared to similar
students who took remedial algebra. Of course like most programs that are
effective it was terminated. Benezit's experiment was terminated because of
parent objections, not because it didn't work. It was a success, but the
failure was one of the school system for not pursuing it.