What is the significance of simply memorizing something for which there is no understanding? And having memorized without understanding, what, then, is the importance of that knowledge? How does one catalog random or fragmented notions in one's stock of knowledge? And is not the ultimate aim of teaching to prompt understanding?
It is most unfortunate that formal mathematics is introduced to children almost exclusively as skill, with memorization being one of the greatest determinants of children's ability to do math. As a result, most students learn to sit passively in classrooms while their teachers assign them trivial tasks and inundate them with useless facts of mathematics – useless because these facts offer no understanding of the discipline of mathematics. With an intense focus on memorization, rarely do students develop the perspective of mathematics as a logical network of ideas; theories that can be discussed or argued in an effort to substantiate their worth. Accepting the role as consumers of their teachers' knowledge, students are not expected to reason mathematically. They are not knowledgeable of principles of mathematics; thus, they are unable to build upon or extend their knowledge of these understandings. For many students math is fixed and static, rather than an ever-growing body of knowledge to which they can contribute and appreciate.
So why are some high school students uninterested in learning school mathematics? School mathematics has done little to reflect the true nature of the discipline of mathematics. Thus, many students have grown weary with the endless, incoherent bits of knowledge that they are expected to store for the seemingly infinite series of tests, only to dispose of the floating information following the tests. This accounts for the inordinate amount of time teachers spend each school year reviewing the previous year's learning objectives. By high school, no longer are students easily enthused with the deluge of supposed real-life problems that are meant to convince them of the value of mathematics. Has mathematics no value of itself? Is it only significant in that it helps one to find one's earnings after a commissioned sale or some other “real life scenario”? Mathematics does not need real-world application to give it value. Mathematics is inherently valuable because of the relationships that it elucidates and the understandings on which the relationships are premised.
How might we teach mathematics so as not to fall into this fate? If children, for instance, were presented with multiplication as a concept – a truly important mathematical idea that helps to increase their understanding of number – they will be more likely to store the facts of multiplication and develop automaticity. The facts would have relevance, being part of a larger, more complex idea. They would have knowledge that is transferable, allowing them to access other mathematical ideas. Indeed, developing understanding is a process; it takes time. But the benefit of the time spent in the period of development is well worth the effort.