개념을 '아는 것'과 '쓰는 것'은 다르다 — 1등급의 개념 학습Knowing a Concept Is Not Using It: How Top Students Learn Concepts

Opening Story: “But I Know All the Concepts”

One day Hyunwoo came to me looking thoroughly aggrieved. “Sir, I really do know all the concepts. But the moment I enter the exam room, I can’t apply any of them.” He could answer every “concept check” question at the front of a problem book, but his hand stopped as soon as he reached the application problems at the back.

I asked him, “Explain what the maximum and minimum of a quadratic function are.” Hyunwoo recited the definitions fluently. So I asked again, “Then do you know why you need that concept in this question?” Hyunwoo could not answer.

Hyunwoo “knew” the concept, but he did not know how to “use” it. In Chapter 1 we looked at rebuilding a concept in your own words and deriving it by hand. This chapter takes the next step. It is about the power to go beyond knowing a definition and call that concept up for use on an actual problem.

A Concept Is Not a Point but a Net

The first mark of a student who cannot use concepts is that they memorise each one separately: quadratic functions over here, quadratic equations over there, quadratic inequalities somewhere else. But in the mind of a top student, all three form one picture.

Consider the graph of the quadratic function $y=x^2-4x+3$.

• Ask where this graph meets the $x$-axis ($y=0$), and you are asking for the two roots of the quadratic equation $x^2-4x+3=0$: $x=1$, $x=3$.

• Ask for the intervals where the graph lies above the $x$-axis ($y>0$), and you are asking for the solution of the quadratic inequality $x^2-4x+3>0$: $x<1$ or $x>3$.

Three different names — function, equation, inequality — are in truth only three views of one graph. Weave concepts into a net like this and one problem opens several doors at once. A student who has memorised each concept separately, by contrast, cannot call up knowledge about equations when the problem arrives wearing the face of a function. There is no road between the concepts.

Why You Cannot Call Up a Concept You Know

What is missing for a student who knows a concept but cannot use it in a problem is a bridge of translation between definition and question. Knowing the concept alone is not enough. You must also know when it appears.

Top students memorise a signal alongside every concept. They pair a phrase in a question with the concept it should call up.

• “The graph is tangent to the $x$-axis” ⇒ call up the discriminant condition $D=0$.

• “Maximum or minimum” ⇒ call up the vertex of a quadratic function.

• “Two distinct real roots” ⇒ call up the discriminant condition $D>0$. (This is the condition we met in Chapter 1.)

This habit does not stop with grade-10 Common Mathematics. It becomes even more powerful in the courses that follow.

• In Calculus I, “the gradient of the tangent at a point on the curve” ⇒ differentiate at that point (the derivative at the point).

• In Probability and Statistics, “at least one” ⇒ translate it into the complementary event, subtracting the opposite case from the whole. (The “translation trap” from Chapter 1 was, in fact, a signal calling up a concept.)

This is the advanced form of circling conditions from Chapter 1. Do not stop at finding the condition; connect it to the concept it calls. When studying a concept, also write down an answer to “In what words will this concept appear in a question?” Then, in the exam room, the concept will walk to your fingertips of its own accord.

How Top Students Check Their Concepts

Here are three ways to test whether a concept has reached the state in which you can use it.

① Retrieval on a Blank Page: Call It Up Without Looking

Close the textbook and write everything you know about the unit’s concepts on a blank page. Read with your eyes and it all seems familiar; face the blank page and not even half comes out. The discomfort of this “retrieval” is one of the most powerful tools for moving a concept into long-term memory.

And here the path of top students and everyone else divides once more. Not all blank pages are equal. One may be scattered with formulae; another may have them woven together with arrows. On the second, functions, equations, and inequalities are connected, with words such as “therefore” and “in other words” laid between them. Your true ability lies not in what you have written, but in how tightly it is woven.

② Build a Counterexample: Test the Boundary of the Concept

Is the statement “a quadratic function always has a minimum” true? It is true when the graph opens upwards (when the coefficient of $x^2$ is positive). But if it opens downwards, it has a maximum and no minimum. Ask yourself, “Is there a case in which this statement fails?” and search for a counterexample. Then you no longer know the concept vaguely; you know its exact boundary. The examiner’s traps always hide along that boundary.

③ Draw a Concept Map: See the Net

After finishing a unit, spread all its concepts across one page and connect them with arrows: function → equation → inequality. This map is the “net” from Section 1. If you can draw the map, the unit is yours. Any part you cannot draw is the place to study next.

Let us draw it a little more concretely. A concept map for the grade-10 unit on quadratic functions might look like this. In the centre, write “$y=a{x}^2+bx+c$,” then send out three arrows. ▸ “set $y=0$ → quadratic equation (two roots)”; ▸ “set $y>0$ or $y<0$ → quadratic inequality (range of solutions)”; ▸ “view it as a graph → vertex and axis, intersections with the $x$-axis.”

Then connect “$D=b^2-4ac$” to the end of all three branches. The discriminant determines how many roots there are and how many times the graph meets the $x$-axis. (To complete the solution of a quadratic inequality, you must also read which way the graph opens, the direction of the inequality sign, and whether equality is included.) Spread it all across one page and you can see that equations, inequalities, and graphs, which seemed to be learned separately, are really branches from the single root called a quadratic function. The moment you can draw this picture yourself is the moment you have truly mastered the unit.

In Closing: Concepts Are Connected, Not Memorised

Many students “know” concepts; few can “use” them. The difference is not a head for math. It is whether the concepts are left as separate points or woven into one net. Rebuild a definition in your own words (Chapter 1), connect concepts to one another, and learn the signals that call each one up in a question. Only then does a concept become a living tool.

We now know how to approach a problem and how to use concepts. In the next chapter we place these principles onto the actual courses. From Common Mathematics 1 and 2 through Algebra, Calculus I, and Probability and Statistics, I will point out the particular weak point to target in each.