Have you encountered times when you are able to solve many mathematics questions relaxingly without fuss, but with a twist to the way the questions are asked, you are stuck?
Most maths learners do face this sort of situation while learning maths. It is a learning phase in education.
What is therefore the message behind this phase?
In this phase, you will discover that purely understanding the steps or procedures in "attacking" a mathematics problem may not cover all aspect of the topic involved. Extra (hidden) elements are also mixed into the learning to form complete comprehension of the mathematics topics.
What normally is the strategy maths teacher use to teach, especially in a classroom setting, is to "drill" students into mastering a topic, practising till the idea goes into the head. It is an effective way to transfer knowledge, but with limitation.
Students are left, sometimes, to figure out the underlying concepts themselves. This is so due to many factors, examples like time resources, ability of the class to understand details, teaching style, and assumptions on the part of students and teachers.
All these contributed to gray areas in the learning of maths, thus producing gaps in the knowledge acquiring process.
What is a better way to handle maths in a classroom learning environment?
Do not blame. Take learning to be a personal challenge to be guided by the teacher. There are always too many factors in class to handle, and some aspect of learning and teaching will definitely be missed out, especially when different students has different aptitude towards maths.
After all, you, the maths student, is the one who is learning and aiming to be better, not the teacher!
What you can do with in maths learning is the understanding of maths concept as opposed to ability to solve rigid questions step-by-step. (You can solve many similar types of maths problems but still do not know its concepts - this is not true learning).
Let me cite a few examples to illustrate the "concept" element.
Finding the area of a given triangle with known height and base.
Formula is Area = (1/2) x Height x Base length.
Yes, you can calculate the area from the formula and given information.
If we give you another triangle with the SAME information but slanted in angles. What is the approach?
For those who did not master the geometrical area concept, they will start off with the calculation as usual, and obtaining an answer that is correct in number.
For those who had mastered the underlying area concept, they will know that the area calculation is not necessary, and the answer will be the same as previous. This is so since the area formula spells out clearly that angles of the triangle has no relation with the area computation. (Even if they showed the workings, they reflect it to get marks for the steps).
The two styles of learning, therefore, clearly, showed the strength in the latter case as it allows you to grasp an overall understanding of area. You are not mechanically drilled into rigid step-by-step solving. You are elevated into "flexible" application of concept at a higher learning platform when you practise the latter technique. You are able to link with a bigger picture in mind.
Solve 8 + 8 + 8 + 8 + 8 + 8 + 8 + 8 = ?
If you understand the concept of "Multiply", you will solve the question efficiently.
(Solution: 8 x 8 = 64 , short and sweet)
You can, however, solve this the rigid way by adding up all the "8"s also, but, is this the proper method when you already know of a better technique?
Simplify log3 X + log3 Y into a single logarithmic term.
You can solve this by applying the logarithmic Product formula to get an answer of log3 XY.
But what if the question changes to 1 + log3 X ?
Without understanding the Product concept, you will be stuck!
Product concept in logarithm calls for terms to be expressed in the "log" form for proper combination later.
(Solution: 1 has to be changed to log3 3 to be able to continue).
Therefore ways has to be found or linked to previous knowledge to capture another "log" term for the "1" in the new maths question to simplify the problem.
The above three examples, hopefully, demonstrated the importance of comprehending concepts (the extra hidden element) to reap effective learning. This addresses the issue of solving ANY mathematics questions regardless of how they are asked and how they are twisted.
The implication of studying with mastery of concepts in mind has far-reaching advantages in your future endeavours as it allows you to have the "flexible" approach to problem-solving in an effectiveness manner.