The lies we tell about our duty and our purposes, the meaningless words of science and philosophy, are walls that topple before a bewildered little "Why".
John Steinbeck
The Log from the Sea of CortezAt some point all Junior High School algebra students invariably ask:
"Why do we have to learn this stuff?"
One answer is:
So that you will pass the course and won't have to take it again next year.
You need to take Math 8 before you can do Math 9 which you need before you can do Math 10 which you need before you can take Math 11 which you need to take Math 12 which you will need to get into university. If you don't pass this year you will have to do it again next year. That is why you have to learn this stuff.
A pragmatic argument. In the absence of a better explanation it is very compelling and students often accept it. Here is another:
It's only after you have learned this stuff that you would be able to understand the explanation, so I'm not going to attempt to explain it. You wouldn't understand the explanation. When you have learned the mathematics we are studying you will realize how useful it can be.
This sounds facetious. It's actually a pretty good answer in the sense that it there is a lot of truth to it. Students find it condescending and interpret it to mean that teacher doesn't really know why either. Here are some other answers:
Mathematics is part of our cultural heritage
Mathematics is an intimate historical part of most cultures. Further, mathematical systems and concepts, like all knowledge, have an intrinsic beauty of their own which makes them worthy of study for their own sake. No further justification is required and no liberal education can be considered complete without it.
This is a widely recognized argument with undeniable validity. Whether it also explains the selection of those particular mathematics topics that constitute the school curriculum and the comparative amount of time devoted to them is another matter. The same argument can be made even more strongly for History, Music, Art, Literature, Religion, Ethics, Economics and Philosophy. In most educational jurisdictions these other subjects, combined, do not receive as much time and resources as mathematics does.
Mathematics teaches clear and logical thinking
The qualities that most people tend to associate with mathematics are things like precision and logic and therefore mathematics is closely linked with clear, 'correct' thinking. Because it teaches students to think, the study of mathematics is beneficial even in the absence of plausible practical uses of the mathematics topics selected.
This notion is based on the transfer of learning theory which for centuries was used to justify teaching Latin. Learning Latin was presumed to make it easier to learn other languages. The same argument can be used to justify the inclusion of chess in school curricula. There is no objective data to support the notion, and certainly only a small minority of mathematics teachers believe it to have merit.
In any event, the type of logic which is encouraged in school mathematics is characterized by being prescriptive, convergent and linear as opposed to descriptive and divergent and creative. Hardly the sort of thing we would normally admit to encouraging.
Mathematics teaches problem solving
In recent years a notion has arisen that there exists a set of generic problem solving skills that transcend subject matter. These skills can be developed in isolation from any application. Terms such as 'rubric' and 'heuristics' have become popular in this connection. Skills acquired as a result of doing mathematical problems are presumed to diffuse into other disciplines. Therefore 'doing math problems' is 'good for you'.
This is a currently popular variation of the 'transfer of learning' theory, and is supported only by anecdotal reports.
Mathematics is useful
An oft-cited reason for teaching mathematics. No one would deny that mathematics can be useful. Paul Ernst maintains that this aspect is greatly overemphasised.
The questions that needs to be answered, however, are "Has the mathematics we teach in school and the way in which it is taught been selected on the basis of this criterion?" and, if the answer to that is "Yes", then, "Has anyone recently re-examined the present curriculum according this criterion?" A scrutiny of the application problems - sometimes called 'word problems' - found at the end of most chapters of most high school mathematics text books - would not lend credence to the notion that curriculum topics have been selected with plausible usefulness in mind.
I came across an item in a book by Lewis Thomas (Et Cetera, Et Cetera) the other day. Dr. Thomas is, or was, scholar-in-residence at Cornell University. He observes:
Most of what is called mathematics in the curriculum (he is referring to post secondary as well public school math here) in the United States is essentially that subject as it existed at the end of the eighteenth century.
Implicit in western cultural heritage is the notion that the nature is governed by mathematical laws. The doctrine of Determinism states that if we would but know nature's (mathematical) laws, we only need to plug in the relevant measurements in order to predict the future. Cause and predictable effect. Also, natural shapes imitate the perfections of Euclidian figures such as triangles and circles. In the words of Galileo Galileo, "Mathematics is the language with which God wrote the Universe" Therefore an understanding of mathematics is a pre-requisite to understanding and appreciation of nature... indeed all scientific study
God may have written the universe in mathematics, but it's getting pretty evident that he didn't use the equations and formulas studied in school mathematics when he did it. These all imply that natural phenomena obey some mathematical differentiable function or another. Applied mathematicians have known for some time that mathematics does not dictate nature; but rather imperfectly attempts to describe it. Recent discoveries have buried classic determinism very deeply. Two supporting opinions:
So far as the laws of mathematics refer to reality, they are not certain. And so far as they are certain, they do not refer to reality.
Albert Einstein
Geometry and ExperienceWhat we observe is not nature itself, but nature exposed to our method of questioning.
Werner Heisenberg
Physics and PhilosophyA curriculum based on a descriptive, as opposed to prescriptive paradigm of mathematics has yet to be developed.
Preparation for citizenship
Preparation for citizenship requires becoming a useful person in the sense of having vocational skills. Many of these involve mathematics.
Certainly a more than sufficient justification for teaching mathematics. I would add that students should have skills and abilities related to responsible personal budgeting. In addition, participation in the political decision making processes of modern societies requires a level of understanding of economic and statistical concepts.
Mathematics that is designed as preparation for citizenship is not taught in the main stream school mathematics curriculum. Modified (read 'watered down') topics of this type are offered to non-academic students under the heading Consumer Mathematics.
The things learned in high school mathematics are pre-requisites for essential topics encountered in academic and vocational training. Students who want to take post secondary training whether at university or technology schools will find that good marks in Senior High School mathematics are an essential requisite for admission. Recently, competition for post secondary education has become so great that good marks are no longer enough - one has to have 'best' marks.
What is also true is that those who are involved in setting criteria for entry to post secondary educational institutions find that mathematics marks are a convenient filter for screening applicants. The justification is that anyone who has achieved good marks on school mathematics have demonstrated a capacity for learning and is therefore likely to do well at other things. This is probably true; but it is hard to argue that an objectivist mathematics curriculum is in any way unique for the purpose. It may well be that other criteria could be found that have a greater correlation with with success.
One of the things that makes mathematics examinations desirable as educational filters is that in mathematics it's easy to specify right and wrong answers. This gives rise to the notion that this is a highly precise measuring instrument.
In any event, because secondary school mathematics marks on examinations are used as a screening device, these examinations are the ultimate raison d'etre for school mathematics and are the determiners of school mathematics content. Cultural heritage, training for thinking, problem solving, usefulness in everyday life, understanding of natural laws, informed citizenry and all of those things really have nothing to do with it.
Despite the fact that only something like 15% of high school students actually complete post secondary courses, often in fields to which traditional school mathematics topics contribute little, the K - 12 curriculum is determined by that mathematics which post secondary institutions find it convenient to state are needed for admission. And that's the bottom line.
For an academic discussion of this topic by a well known educator, see Paul Ernst's
Why Teach Mathematics