How Math Should Work
One of my favorite moments in C.S. Lewis's The Chronicles of Narnia series occurs in book two, The Lion, the Witch, and the Wardrobe. Peter and Susan, the older Pevensie children, are confused and troubled by Lucy (the youngest sibling, though very honest)'s insistence of a seemingly impossible tale of entering another world, and the contrary account of malicious and mean-spirited Edmund.
Peter and Susan take their concerns to the eccentric Professor Kirk, in whose house they are staying. The exchange in the story is revealing. "How do you know," the Professor asked, "that your sister's story is not true?" They continue back and forth for a few moments, contrasting the general reliability of Lucy over Edmund.
The story notes the Professor's telling conclusion:
"Logic!" said the Professor half to himself. "Why don't they teach logic at these schools? There are only three possibilities. Either your sister is telling lies, or she is mad, or she is telling the truth. You know she doesn't tell lies and it is obvious that she is not mad. For the moment then and unless any further evidence turns up, we must assume that she is telling the truth."
Sounding much like the chair of the math education department of the University of Georgia when I was in their graduate program in the mid-1990s (I've forgotten his name), mathematician John Wesley Young declared, "It is clear that the chief end of mathematical study must be to make the students think." I would add, not only to "think," but to think logically -- both deductively and inductively.
"The study of mathematics cannot be replaced by any other activity that will train and develop man's purely logical faculties to the same level of rationality," said mathematician and textbook author C.O. Oakley. Einstein beautifully declared, "Pure mathematics is, in its way, the poetry of logical ideas."
In my 20 years in the high school classroom (in Georgia), I've often told my students that studying mathematics is not so much (or at least not only) about mastering some specific "objective" or "standard" (as we now call them), nor is it about making some future use of every little concept that they learn. I point out that these are good things, but the study of mathematics is more. It is also about growing and developing that logical part of their brains that mathematics, in particular, serves.
What's more, "mathematical training," as the math department of the University of Arizona puts it, "is training in general problem solving." Or, as Thomas Aquinas College declares, mathematics "prepares the mind to think clearly and cogently, expanding the ability to know."
It is a widely held belief that the most influential and successful textbook ever written was Euclid's Elements. Written by the ancient Greek mathematician Euclid of Alexandria around 300 B.C., Elements is actually a collection of 13 books. The work deals mainly with what today is typically deemed Euclidean geometry, along with the ancient Greek version of elementary number theory. Euclid's work was so complete and superior to anything before it that all Greek writings on mathematics prior to Elements virtually disappeared.
Also, as a modern translator has noted, Elements has been instrumental in the development of logic and modern science. According to Howard Eves's An Introduction to the History of Mathematics (one of my graduate texts), "[n]o work, save the Bible, has been more widely used, edited or studied, and probably no work has exercised a greater influence on scientific thinking."
The beauty of Elements lies in Euclid's axiomatic approach, which, according to Eves, is "the prototype of modern mathematical form." In this form of thinking, one must show (prove) that a particular conclusion is a necessary logical consequence of some previously established conclusion. These, in turn, must be established from some still more previously established conclusions, and so on.
Since one cannot continue in this way indefinitely, one must, initially, establish and accept some finite set of statements (axioms) without proof. All other conclusions are logically deduced from these initially accepted axioms (or postulates).
Sadly, this approach is almost completely abandoned with the integrated mathematics (previously Math I, II, III, and IV; now coordinate algebra, analytic geometry, and so on) curriculum adopted by the state of Georgia five years ago. Although Georgia recently gave systems the option of returning to a more traditional (Euclidean) approach to mathematics, most stayed with the integrated math.
As most in Georgia well know, this type of curriculum integrates several different topics (namely algebra, geometry, and statistics/probability) throughout each year of high school math.
For example, currently, most freshmen in Georgia take coordinate algebra. This course consists of six units. The first three units are algebraic. They involve things like writing and solving linear equations and inequalities, solving systems of linear equations, graphing linear and exponential functions, using function notation and language, calculating rate of change (slope), and working with arithmetic and geometric sequences.
Unit four is a statistic unit which involves representing data in a variety of ways. Unit 5 involves transforming (rotate, reflect, translate) polygons in the coordinate plane. In Unit 6, students "prove" (a vague, and as far as I'm concerned, inaccurately used term) geometric theorems algebraically.
Note the jump from algebra to stats and then to geometry. The course has little logical flow. For the most part, new topics are not arrived at from previous ones. In other words, the axiomatic approach is almost completely ignored.
I returned to the public schools from several years in a private school just as Georgia made the switch to this integrated approach. I knew there was going to be trouble when, during a professional development opportunity to help prepare us to teach the new math, a visiting college professor noted that, with this new curriculum, we were "not going to be able to do much axiomatic development."
Also, a good deal of the time is spent reviewing concepts "learned" in middle school. For example, at the beginning of the stats unit, many teachers have to review things like measures of central tendency (mean, median, and mode). In the geometry units, students often have to be reminded about angles with parallel lines and the sum of the angles of a triangle.
Much of the decision to go to an integrated math was based on the fact that most countries outside the U.S. use such an integrated approach. (This point is also made in the Appendix of the mathematics standards in Common Core.) Georgia modeled its integrated math after the Japanese mathematics curriculum.
It is noteworthy that, of the four mathematics "Pathways" that Common Core provides (see the Appendix), two are for the integrated math approach. This is the case even though Georgia is the only state in the U.S. that has an integrated mathematics curriculum. (New York has, for the most part, abandoned its integrated math.)
Why would Common Core provide curriculum direction for an approach to mathematics that is almost never used in the U.S.? I believe that it is because of the international appeal of the integrated approach and the desire by many on the left to make us more like other nations.
Don't get me wrong: I'm not saying that every mathematics course in Georgia (or other states) high schools should be replete with rigorous proofs, but I think it benefits all concerned when the curriculum is laid out in a manner such that, for the most part, one topic logically flows (and proof could be incorporated) from the previous topic. This should be the case at least for college prep courses.
And certainly all students do not belong in such courses -- which presents another flaw in Georgia's curriculum that many across the country, including President Obama, seems to have bought into: virtually every high school student must take college prep mathematics.
Lastly, if the integrated approach is best, why don't our highest institutions of learning use it? I know of not one college or university that does so. With few exceptions, and whether the course is online or traditional, it is time that Georgia's (or any other state considering such a change) secondary schools abandon integrated mathematics.
Trevor Grant Thomas: At the Intersection of Politics, Science, Faith, and Reason. www.trevorgrantthomas.com