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This is a very commonly asked question and, even once answered, students are not satisfied at the answer and continue to lament the need to "solve for x."
x is an unknown quantity and that is what algebra is all about. When we use letters of the alphabet to denote unknowns we are just trying to convert English (or whichever language the student learns in into maths language. And one of the amazing things about maths language is that it is largely UNIVERSAL!
So one of the biggest advantages of using algebra and converting complex problems into a "maths language" is that it can be understood by far more than the English version of a word problem can be.
As an adult, a parent perhaps, with 2 or 3 children, trying to work out how much pocket money to give each child based on age, uses algebra without even realizing. When shopping for groceries and the necessities are bought, the shopper uses algebra to figure out how much he or she can afford for sweets, chocolates or maybe a gift. When considering a bargain in a shop window with a 20% discount, calculations are frantically assessed in the head to allow the buyer to decide it he or she is getting a bargain.
So APPLICATION is the part of algebra that will be used long after "solve for x" has left your lips. I must admit that this is basically the problem with algebra. No two situations are ever the same and students do not often make connections between a recently solved problem and the one they are now faced with. This is the dilemma as algebra ,by necessity is
a generalization of arithmetic.
Teachers teach algebra is its purest form so that students can relate it to ANY given scenario but students do not make the connection. It would be most beneficial to teach and apply simultaneously but there is no time for that. As previously mentioned, there are many different applications of algebra so to try and teach application for all of them would be too much to ask.
Furthermore, students just want to get the answer to the question and do not retain the information to use later unless the question is almost the same.
There is also the question of understanding other - seemingly- more relevant aspects of Maths such as statistics. To do statistics you need a basic understanding of algebra so algebra becomes the foundation. As unlikely as it sounds to most students, algebra SIMPLIFIES otherwise complicated problems or situations in need of analysis.
So unless you are a mathematician or using maths in your career or study path, the most useful concepts would not necessarily be calculus, quadratic equations, expressions, etc but application and the capacity to be a universal language.
As for the least useful - I can't think of any because, it's like anything, years later you will come to a realization that algebra sets you up to understand and contribute to a society which relies so heavily on figures and where "money talks."
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