Students of all ages often fall into the trap of thinking that they understand material as soon as their teacher is through with explaining how to do a problem related to said concept. Really, a student can easily look at another’s work and automatically nod, “I understand what is going on” with each step, but do they truly understand? Would they be able to replicate the steps on a similar problem by themselves?
Perhaps. But one can never know this for sure until they’ve actually gotten a chance to do so. Even if they can replicate another’s steps a single time, there is a differentiation between knowing and understanding that must be made. If one knows a procedure, it merely means they can robotically replicate the procedure for a different problem as long as no wrenches are thrown in. But once the person that knows a procedure is presented with a problem where the verbatim procedure they learned does not work, they feel defeated, frustrated, and helpless. The person who knows material is inflexible in their approach to solving problems, and that is never a good thing. However, students may not realize this until they are in college if previous teachers don’t allow students to test their flexibility and adaptability when solving problems. It is suggested that you evaluate the formatting of assignments you give to your students, as you may be able to improve their intellectual profit for completing your assignments.
Here is an analogy for the above situation: you hand your student a rock and tell him that he can shatter glass with it. He proceeds to shatter ten pieces of glass with the rock. Then you take him to a site filled with lava. With no other option previously presented to him, the student throws his rock at the lava, hoping it will “break” the lava. He is shocked and disappointed to see that the lava just eats up the rock. If you don’t tell the student to try digging for a hose to extinguish the lava, then you may have missed giving your student one of the tools he needs.
One who understands material, on the other hand, can manipulate the underlying concepts to achieve so much more than the student who merely knows material. Then the question arises, “how do we get students to go from knowing material to understanding material?” The answer may surprise you because it is so simple: practice. And when I say practice, I advise that you do not present a worksheet with twenty problems that are essentially identical. There are two disadvantages to this practice: one, students will eventually get bored of rote repetition and will subconsciously “tune out,” defeating the purpose of giving the assignment in the first place. The second disadvantage is that such assignments make students think that completing math problems is a one-dimensional science, whereas in reality it is a multidimensional art.
Thus part of your job as a teacher is to prepare students to face this reality as they advance in their academic careers. Assignments should contain a wide variety of problems where the exact chain of steps required to solve a problem is seldom reused. This allows students to exercise their creativity and critical thinking skills from early on, and trains them to be flexible while problem-solving, rather than robotic.
One of the best ways to do this is by putting problems into context. Disguising math problems as real-world scenarios gives students practice performing computations like non-word problems do, but they also have to set up equations themselves, which serves as a check for students as to whether they understand the underlying concepts and have not merely memorized equations. Believe me, studying mathematics in real life is about a lot more than memorizing equations. Having participated in math competitions for five years, I know that problems require a vast array of applications of fundamental concepts to be solved, and this same idea extends to other disciplines as well. The ideal math curriculum should prepare students for this, the real world, by promoting understanding and ingenuity. Those are the qualities that have allowed many famous mathematicians and scientists have risen from the ashes; try not to make it harder for your students to be the next of these!