Arbitrary vs. Necessary

After reading this article, it makes me reflect on how I can improve on giving lessons to my high school Math students. A lot of times, maybe due to time limitations, we tend to just give too much information that we think are arbitrary to the students. But in fact they can be worked out using arbitrary concepts. This means we are basically giving too much information to the students. They are suppose to learn the topics themselves with our guidance. For example, when we are doing exponents, the arbitrary is the exponent notations and the definition: " a number to the power of 2 means the number multiplied by itself". Then from there, a lot of the exponent laws can be derived from this arbitrary definition of exponent. Of course most students require guidance from the teachers to arrive at all the exponent laws (unless your students are very smart). From this example we can see that learning a mathematical topics does not require teachers to lecture for the whole class. All we have to do is give hints and guidance so students arrive at the desired theorem or learning outcome. 

Another thing is some concepts were taught during previous grades and those topics become the "arbitrary" topics. Teachers do not have to go over how we derive those laws and theorems again. That is why you do not see high school Math teachers teaching students simple adding and subtracting of whole numbers. These are conventions that students have learnt and acquired when they were in elementary school. 

Overall, I feel that to teach a Math class effectively, we have to differentiate the arbitrary and the necessary. I think that we spend less time talking about arbitrary and instead focus on the necessary to foster a thinking class. As teachers, we have to come up with good guiding questions so students can make good use of the arbitrary to derive the necessary, ie. the learning outcome.