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The Importance of Conceptual Understanding in Math
The importance of conceptual understanding in math is particularly misunderstood today, so it is worth commenting on further.
Mathematics as a subject faces a unique problem: at younger ages, it is possible for students to do relatively well without understanding the meaning of what they are doing. By following memorized steps, elementary age students can regularly “get the right answer” in math, even though they do not properly grasp the material.
Later in life, a student that does not understand what he is doing will not be able to “get the right answer”. In high school, college and the workplace, it will not be enough to memorize and apply the steps of a pattern. Students must be able to understand the methods they have learned and apply them in non-routine ways. If they are to be successful at math, students must be able to think conceptually.
At LePort, we understand that mathematical creativity is best fostered at a young age. By high school, it can be too late. Students that have always learned to approach math mechanistically find it incredibly challenging to then shift to a more critical, inquisitive perspective. Students need to learn-right from the beginning of their education in math-that mathematics is a tool that is used to solve real life problems, and that learning math means nothing less than actually learning how to use that tool to solve actual problems.
Consider the following question, from our grade six word problem text: “Three metal cubes were 3 cm long, 4 cm long, and 5 cm long respectively. They were melted and recast into a new cube. Find the length of the new cube.”
In order to answer this question, a student has to work backwards. He starts with the fact that he needs to find the length of the new cube, and has to ask himself how he can get this information. If he has properly learned the relationship between the side of a cube and its volume, he will see that to find the length he first needs to find its volume. So he then has to ask himself: how can he find the volume of the new cube? If he has learned that the volume of an object is the amount of space it fills-if he truly grasps that fact-then it will be clear to him that the volume of the new cube will be equal to the sum of the volumes of the smaller three cubes. (Because it’s the same amount of metal!) He now needs to find the volume of the three smaller cubes, which he knows he can get by multiplying the length x length x length of each cube. After finding the volume of the new cube, the student simply needs to calculate its cube root; this is the length of the new cube.
Notice that in order to answer this type of problem, a student has to have a conceptual understanding of math. He/she has to understand what it means to “cube” a number; it would not be enough for him/her to have memorized the rule that “cubing means multiplying a number times itself twice”. He/she has to understand that volume refers to the number of cubes that fill up a space; it would not be enough for him/her to have memorized that volume is when you write “3” in superscript beside your units.
To become good at solving word problems, a student must also learn-and practice-a method of approaching such problems. The LePort word problems program teaches such a method. Students learn how to read through questions critically. They learn that “before they can figure out the answer, they must figure out the question”. They learn to draw diagrams that help them grasp the problems. They learn how to work in reverse order to identify what missing information they need. They learn to double-check not only their answers, but also their understanding of the question.
At first, students may struggle with their word problems. Even students that are otherwise successful at math initially find the problems difficult to process. The most common error is the temptation to immediately start computing numbers using whichever method first comes to mind. Over time, however, students learn to approach problems patiently, to take the time to think the situation through and form a mental picture, to carefully select a valid method of answering the question, and only then to proceed with the computation.
The value of creative mathematical thinking is revealed in the pride that the students take in their success. Implicitly, students know the difference between merely memorizing and applying a straightforward rule, and actually solving a complicated, multi-faceted problem. When our students achieve success on their word problems, they recognize-and we remind them-that they are not just learning math, they are learning the irreplaceable skill of analyzing and solving problems.
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