Knowing the answer is not the goal.

Many students think that “knowing how to do a math question” necessitates that you know every single step that you need to do to get to the answer, but this could not be further from the truth. I would like to convince you that you do not need to know exactly how to solve any math questions — and that not knowing is actually better.

When faced with a very easy math problem, most of us can either see the answer immediately or successfully plan out every action one needs to do to arrive at the answer.

\begin{align} 2x&=6 \\ \\ x&=3\\ \end{align}

Here, you can see that only two times three will be equal to six, or you can see that you only need to divide each side by two in order to make x the subject and that’s simple enough to do in your head.

What about a more complex example?

\begin{align*} \frac{x+4}{-3+7x}+1 &= \frac{3}{5}\\ \end{align*}

This equation is far too complex to tell what the answer must be and if you were to ask me whether the answer was x=12 I would tell you that I do not know! That doesn’t mean I don’t know how to find the answer, though. I know that there are certain moves I need to do in a certain order, and the answer will reveal itself!

For now, I just want to get rid of the fractions.

\begin{align*} \frac{x+4}{-3+7x}&= \frac{3}{5}-1\\ \\ \frac{x+4}{-3+7x}&= \frac{3}{5}-\frac{5}{5}\\ \\ \frac{x+4}{-3+7x}&= -\frac{2}{5}\\ \\ 5(x+4)&=-2(-3+7x)\\ \end{align*}

Oh! Now I just expand the brackets…

\begin{align*} 5(x+4)&=-2(-3+7x)\\ \\ 5x+20&=6-14x\\ \end{align*}

…group the ‘x’s…

\begin{align*} 5x+20&=6-14x\\ \\ 19x+20&=6\\ \\ 19x&=-14\\ \\ x&=\frac{-14}{19}\\ \end{align*}

And there it is! I never could have guessed the answer just by looking at the question, but it didn’t matter! At every step of the way I knew what I had to do next.

Now when you get to very hard math problems, even just knowing what to do next becomes a luxury. When the going gets really tough, all you need to know is something you can try next, and if it doesn’t work, try something else.

This strategy, just knowing what to do next, is a better outcome to aim for than being able to “know” the answer to certain problems. The usefulness of mathematics is all about letting the rules guide you to answers you wouldn’t be able to find otherwise. Moreover, the fun of math is in the delight you experience when you solve a problem with an unexpected answer!

Many students get in their heads about “not knowing how to solve” their math problems. That’s rubbish. Nobody expects you to know the answer, but you probably already know how to solve it.

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