Why Must We Show Our Work, Check Our Solution?
Reminiscence with me for a second about what life was like in your high school Algebra class. Can you recall yourself slouching, daydreaming, trying your best to appear attentive as your teacher jabbered on about some letter “X” or “Y”? Can you recall getting the correct answer only to be told to “show your work” and “plug the answer back in to check your solution”? Do you recall taking multiple choice math tests and getting tongue-lashed for using the plug-in-the-solutions strategy? Yet, did your slightly peevish math teacher ever put down that red pen long enough to offer some form of reasoning as to why you should/shouldn’t do these things? Alas, although belated, let me offer some perspective as to the significance of showing your work and checking your solution. Let me try to sell you on the reasoning.
In elementary Algebra we make many statements about the equality (or inequality) of two expressions. These expressions are made from blocks of known quantities and unknown quantities which we label variables. We join these blocks with operators such as +, -, *, / and grouping symbols such as ( ), { }. An example may look like the following,
Solve the Following Equation for X (please show all work and check your solution): 2x + 1 = 3
But what does it actually mean to solve for X? Furthermore, why are we asked to show our work throughout the solution process? Furthermore, why are we asked to plug our proposed solution back into the equation to see if it is correct? To answer these questions I want to introduce some rudimentary ideas in logic.
The equation 2x + 1 = 3 is a proposition. The use of the connective “=” joins the two expressions into a statement that makes a truth claim. It now reads as an assertion about the equality of the two given expressions and like any proposition there are two possibilities. It is either true or not true. To determine whether the proposition is true or not true we would need to know (in addition) the value of the unknown x. Yet, since the value of the unknown is not given in addition to the proposition 2x + 1 = 3 we must proceed to determine its value. This process is known as making an argument.
However, before we begin to argue toward a solution in order to gain a better understanding I would like you to view the above question as an “if…then” statement. That is, instead of viewing the question as “solve the equation 2x + 1 = 3 for x” I would like you to view it as “If 2x + 1 = 3, then x = ?”. The equation now reads as a conditional statement. What’s beautiful about this perspective is that we get to assume that 2x + 1 = 3 is true in order to solve for x.
Many individuals find themselves in a state of stupefaction whenever mathematical notation is present. Furthermore, many become duped into accepting unfounded results simply because they are veiled in mathematical rigor. In fact, with regards to what is true, mathematics finds itself more concerned more with the validity of a proposition than its soundness. It is for this reason that we need not labor to prove our axioms or definitions, nor the rules that are drawn from them. We ask only that they do not lead to contradictions with any previously held axioms or definitions. You see, Mathematics is in fact a science of validity. Any claim of truth is merely an implication…conditioned upon the acceptance of prior established rules. For this reason, what follows the “If…” in an “if…then” statement (normally referred to as the antecedent) is accepted in order to evaluate the validity of what follows “…then” (normally referred to as the consequent).
By assuming that 2x + 1 = 3 is a true statement we can proceed to determine the condition under which the proposition 2x + 1 = 3 is true (if such a condition exists). The process of showing your steps in the solution process is known as making an argument. An argument consists of using a set of accepted facts and assumptions, which we call premises, to support the conclusion that you reach. In our example above we use the inverse of each operation with the assumption that the properties of equality are still preserved.
2x + 1 = 3
(2x + 1) -1 = (3) -1 (inverse operation of addition)
2x = 2
(2x) / 2 = (2) / 2 (inverse operation of multiplication)
X = 1 (your conclusion)
At this stage you have concluded the given conditional statement, that is
“If 2x + 1 = 3, then x = 1”.
Therefore, by arguing toward the proposed solution of x = 1 you have now engaged in the act of developing your own truth-claim. Your proposition is that “If 2x + 1 = 3, then x = 1” and since you are the agent making the truth-claim the burden of proof falls on you to prove that your statement is true.
Yet, how do you determine if your proposed solution that x = 1 is actually true? You must test it. How do you test it? One way is to assume your conclusion x=1 is true and proceed to argue its validity toward the conclusion that 2x+1 = 3. That is, switch the if…then statement to read “If x = 1, then 2x + 1 = 3”. At first glance this might seem like a trivial process but it surely is not. Take for example the conditional statement “if you live in Miami, then you live in Florida”. This conditional statement is true. Now switch the antecedent with its consequent. “If you live in Florida, then you live in Miami”. This statement is not necessarily true. Thus, the converse of a true conditional statement need not itself be true.
Thus, checking your solution should not be thought of as an optional process of plugging back in your solution to catch silly mistakes (although it certainly is that) instead it should be thought of as justification for your proposition. You don’t get to just assume the converse is true, you must prove it by substituting “1” in for “x” and evaluating the expression using the correct properties. If the result of this process demonstrations a true statement, in this case 3 =3, then you have the following true bi-conditional statement.
If 2x + 1 = 3, then x = 1” and “If x = 1, then 2x + 1 = 3”
At last, you turn in your quiz and your Algebra teacher smiles and rewards you with a giant red check mark.