This idea has been in my head for a while. When in mathematics, or preferably in a practical situation, such as an occupation, do we use expressions? What is the point of an expression if it does not contain an answer - something we can derive and use for a purpose?
In case my ideas of an expression are not entirely correct, I mean a mathematical statement without an answer (or equal, not equal, etc. sign), such as
3x+8
Or
sqrt(5)-3
Just as arbitary examples.
On
In math textbooks and classes, lone expressions like this separate from equations are given as exercises in simplification or in substitution and evaluation. You might, for example, be told to factor 3x^2 -x -10, or to multiply (a + b + 2c)(a + b - 2c), or to evaluate sec(-2), etc.
These are practice exercises with the goal of helping you to learn to do various calculations quickly and accurately. The expressions involved do not have any real meaning outside the problem. It's just practice, like playing scales in music or shooting the puck at the net in hockey or writing various alphabets in calligraphy or conjugating verbs in French or Latin. Practice is necessary to perfect and refine a skill.
Use of practice exercises without any reference to real applied problems can be a good pedagogical tool, if used appropriately. The best organization for a math course, proven by more than a century of real-world teaching and several recent decades of experiments in schools, is this: [over a period of several days or even weeks] (1) Present a problem with real-world implications. (2) Explore various ideas for solution. See what works and what doesn't. Find problems of excess time and inefficiency that make us want a systematic approach. (3) Present the standard format for solution, now that the students are motivated to understand it. (4) Practice some simple problems. (5) As needed, practice some subskills separate from the full problem (for example, when doing quadratic and higher power equations, you need to develop skills in factoring; when solving area and volume and distance problems, you need to practice multiplication of polynomials, etc.) (6) Bring it all together and do some more advanced problems applying the system developed in (3) and the skills developed in (5). (7) Apply what has been learned to the real world and to more advanced work.
Weak schools and programs try to be more "efficient" (please define "efficiency" in teaching -- a real problem) and to "save time" by short-circuiting the process and doing only steps (3) and (5) and throwing step (6) at the students on homework and tests. This is why they are weak, that they try to raise their standards by leaving out everything that makes the subject meaningful.
Really bad schools and programs do almost nothing but step (5) and then berate the students because they don't know anything.
If you are in this situation then yes, your question is very reasonable. You should be getting a big-picture introduction to math, and if you aren't, that is a problem of misconceptions about the nature of math and of teaching on the part of the people running the place. Thse misconceptions are very common.
You can get out of this situation by finding good teach-yourself books and websites and working through exercises.
What's the point of a word, if it's not a sentence - something we can say and be grammatically correct?