I have a Bachelors of Science in Mathematics. I once struggled in math, and I know from experience how frustrating math can be. I also know, from a different experience, how useful and rewarding an understanding of math can be. I want to help those who struggle as I did.
I look at education, in general, like I look at an orchestra. In order to play a nice piece of music well, one must first master the tool; then, apply those tools to something useful. I believe, if students spend all their time on developing tools, that they lose sight of the reasons they are developing them. Students need to see the usefulness of the tools as they are developed. In any music class, a few interesting songs go a long way to keep students focused on learning their scales and arpeggios.
In life, as in music, there are a variety of applications for our “tools”. Students, likewise, have a variety of interests. A good teacher will provide a variety of opportunities for application of a student’s skill set, focusing, not just on what a group of esoteric professionals deems worthy, but more so on what the students themselves find engaging. The rest will follow.
For instance, while learning to play an instrument, many musicians, or educators, may wish for them to learn “the classics” (Mozart, Chopin, or Beethoven). The students, on the other hand, may not yet have an appreciation for such. So, more popular and fun pieces may be required to hold their interest and develop a sense of achievement and enjoyment in their newly acquired skills. The "classics", may take some time to appreciate.
Mathematics, for example, is a very useful tool in a student’s life. If all they do is learn and drill on the tools, many are bound to become bored, or discouraged. Like a student learning only fundamentals but never experiencing the joy of playing a masterful piece of work in symphony, a math student who never gets to experience the fruits of their labors will never see the majesty of their newly honed skills.
A math student in high school or early college may not be able to calculate the Heisenberg uncertainty principle or understand Newton's, Maxwell's, or Hamilton's equations—this does not mean that they should not see the relevance and power of the discipline, watered down for their level. I am not talking simply about "real world applications" given in story problems but in projects, in hands-on tests of their understanding, and in seeing the results of their calculations.
Learning anything, like learning music, should be applicable. But, most of all, it should be fun.
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