In reading Richard Skemp’s article Relational Understanding and Instrumental Understanding, I was reminded of my own high school experiences. Just as Kemp argues that there are “two effectively different subjects being taught under the same name, mathematics,” there seemed to be two different types of mathematics that my peers and I were learning. While I went to class with the aim of understanding the concept then applying my knowledge to various homework problems, my best friend would go to the same class and take essentially the exact opposite approach: do the homework problems and only if absolutely necessary would bother with learning the concepts.
This resulted in huge discrepancies in grades. I rarely got less than 95% on any given math exam while she struggled to pass the course. Later in life she avoided math classes in university like the plague - eventually pursuing a degree in English - and has a general distaste for the subject. I went on the major in mathematics at UBC and it should go without saying I love it. I read books about it for fun. I took extra math courses for electives. Looking back at high school, those early methods of learning provide a clear indication of which one of us would end up liking math and which one of us would end up with a negative attitude towards the subject which is “unhappily so common, even among otherwise highly-educated people”. Changing societal attitudes towards mathematics is very important to me and is something that is partially responsible for my decision to become a teacher. In the article Hermann Bondi is quoted as saying, “The negative attitude to mathematics...is surely the greatest measure of out failure and a real danger to our society.” I think that Bondi is not exaggerating in calling this attitude a “danger to society”. Math illiteracy is rampant among the population and the lack of embarrassment (sometimes even pride!) in being ignorant and uninterested is disheartening. It is likely that being taught math in an instrumental way results in students who “find it boring, and the rules to be memorized would be so numerous that problems...would be too difficult for most. They would give up [math] as soon as possible, and remember it with dislike.” Skemp’s description of disillusioned students is startlingly accurate from my experiences in school. Many of my friends cannot even do simple calculations such as split a check three ways, or calculate the interest they would pay on a loan. These are people who have graduated high school and obtained a A-B average even in the subject of math. However, they retained nothing and consider it a chore to remember any mathematical knowledge they learned previously, mostly because they didn’t actually learn anything at all. They got B’s in math simply by regurgitating formulas and applying rigid algorithms to yield correct (but meaningless) answers. Teachers have produced students whose primary accomplishment is following a set of instructions a computer could follow for them - yielding an accomplishment that feels trivial, even to the student.
Skemp does offer hope of a better alternative to this system. Learning relationally - that is, learning the logic behind the solutions - will take more time than the instrumental approach as there is more to learn, however, “the result, once learnt, is more lasting.” Students won’t leave high-school to forever forget their mathematical training as they will be able to remember how things work logically and if they remember a formula incorrectly they will be able to correct it themselves using logic and mathematical understanding. Similarly if they encounter a problem unfamiliar to them they will be able to apply their base of knowledge and (hopefully) come up with a solution. This leads to the final- and I think most important - advantage of relational learning: the desire to continue learning. “If people get satisfaction from relational understanding, they may not only try to understand relationally new material put before them, but also actively seek out new material and explore new areas.” To produce students who not only have a firm grasp of basic mathematics, but also have the skills and interest to remain students throughout their lives would be the ultimate achievement in math education. If I can influence even one student to actively seek out mathematical knowledge throughout his/her life, I will know I improved the quality of not only their individual life but the quality of life for everyone in society who will benefit from living amongst that individual.
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