Lesson 6: The Tried and True Way to Teach Math
-math allows learners to go from the process of concrete figures to the abstract forms
-from concrete thought of actual objects to lifting those thoughts out of the concrete and into the abstract and passing from one to the other
Purpose of the study of mathematics
-the purpose leads to how we go about the study of mathematics
-it is important to learn how the ancients approached the study of mathematics
-mathematics affected the shift in educational philosophy
-lastly, it is important to look at the current views of math education
C.S. Lewis, The Screwtape Letters
-It consists of a fictional correspondence between demons. The senior demon, Screwtape, writes to his nephew and junior tempter, Wormwood. Screwtape acts as a mentor to the young Wormwood in his job to lead the patient, that is a human, to the father below, Satan, and away from the enemy, God.
Screwtape writes, “But are you not being a trifle naive? It sounds as if you supposed that argument was the way to keep him out of the Enemy’s clutches. That might have been so if he had lived a few centuries earlier. At that time the humans still knew pretty well when a thing was proved and when it was not–and if it was proved they really believed it. They still connected thinking with doing and were prepared to alter their way of life as the result of chains of reasoning. But with the weekly press and other such weapons, we have largely altered that. Your man has been accustomed, ever since he was a boy, to have a dozen incompatible philosophies dancing about together inside his head. He doesn’t think of doctrines as primarily ‘true’ or ‘false,’ but as ‘academic’ or ‘practical,’ ‘outworn’ or ‘contemporary,’ ‘conventional’ or ‘ruthless.” Jargon, not argument, is your best ally in keeping him from the Church.”
Lewis:
-criticizes the weekly press, the changes in educational philosophy, in particular, mathematics education
-Lewis saw the problems with argument and reasoning skills of his fellow man
The place of mathematics in history in the last 100 years:
-the Greeks that brought reasoning into mathematics
-prior to the Greeks, Egyptians and Babylonians developed arithmetic and geometry, in which they relied heavily on empirical data based on their experience
-a heavy emphasis on practicality of mathematics, not answering questions beyond what was necessary to operate in daily life
-they built up algebra and geometry to deal with such issues as money
-algebra: simple and compound interest, computing wages, dividing inheritances
-geometry: computing volumes of graineries in areas of fields
-they also created calendars to predict natural occurrences for agricultural and religious ceremony
-advancements in mathematics relied on empirical data instead of abstract reasoning
Greeks
-they first lifted mathematics out of this experience and empirical data, and began studying mathematical objects in the abstract
-they relied on the nature and properties of the objects, rather than the objects themselves grounded in the physical world
-they were able to push mathematical knowledge beyond what was previously known
-set up in the abstract, reasoning can take over to form the basis of mathematical thought
-mathematical results became a chain of p propositions, arising out of preliminary assumptions and advanced by reasoning
Example of abstract reasoning that the Greeks brought into mathematics
-this example is in chapter 23 of book one of Aristotle’s Prior Analytics
-dates back to the earlier Pythagorean school
-the Pythagorean school was founded by a man named Pythagoras
-Pythagorean’s rested on the assumption that whole numbers ruled the universe
-whole numbers are the cause for various qualities, not just quantities, of man or matter
-they took up a study of whole numbers and their properties to better understand the qualities
-there was a mystical view of whole numbers, the Pythagoreans thought that all numbers should be made up as ratios of whole numbers (what we call fractions)
-abstract thinking: they were able to show that the square root of 2 could not be written as a ratio of whole numbers
-5th century B.C.: short argument (must remember what it means to be an even number)
-an even number is divisible by 2
-an even number can be rewritten as 2 times some other whole number
-the square root of 2 as a ratio of whole numbers, p divided by q
-p/q be written in reduced form (ex. ½ or ⅘ )
-the square root of 2 is written as a fraction in its reduced form
-the reduction is called reduction ad absurdum
-Supposition:
-it is impossible for 2 to be a square root of a whole number
-we can reject the statement if we do not agree with it
-we reject the statement because of faulty reasoning
-you need to test each step on sound reasoning
**We must understand that the arithmetic foundational skills of arithmetic in elementary school matter
-addition & subtraction
-times tables and division
-algebra and geometry is built up
Ex. Euclid’s Element, a set of 13 volumes covering geometry of the plane and space, as well as, many results in number theory
-300 years before Christ, Euclid brought together much of what was known in mathematics up to that point, and organized it in such a way, that beginning with a short list of abstract statements, assumed to be true and armed with reasoning, he pieced together a body of knowledge as an extended chain. He did so in such a way that Euclid’s Elements became the standard textbook in geometry for the next 2,000 years
-the elements have been discarded as required reading for educated people, has only occurred in the last 100 years
-the strengthening of the connection between mathematics and reasoning by Greeks, mathematics would next closely tied to philosophy and theology, and the natural sciences
Plato’s Republic Book 7, “We must endeavor that those who are to be the principal men of our State to go and learn arithmetic, not as amateurs, but they must carry on the study until they see the natural numbers with the mind only–arithmetic has a great and elevating effect, compelling the soul to reason about abstract number, and rebelling against the introduction of the visible and tangible objects into the argument.”
-the abstraction that Greeks brought into mathematics should never be discounted in importance
Morris Kline, Mathematics for the Nonmathematician
“The abstractions of mathematics possessed a special importance for the Greeks. The philosophers pointed out that, to pass from a knowledge of the world of matter to the world of ideas, man must train his mind to grasp ideas. These highest realities blind the person who is not prepared to contemplate them. He is, to use Plato’s famous simile, like one who lives continuously in the deep shadows of a cave and is suddenly brought out into the sunlight. The study of mathematics helps make the transition from darkness to light. Mathematics is in fact ideally suited to prepare the mind for higher forms of thought because it, on one hand, pertains to the world of visible things, and, on the other hand, it deals with abstract concepts. Hence through the study of mathematics man learns to pass from concrete figures to abstract forms– moreover, the study purifies the mind by drawing it away from the contemplation of the sensible imperishable and leading it to the eternal ideas.”
-an author summed up Plato’s position on mathematics
“While a little knowledge of geometry and calculation suffices for practical needs, the higher and more advanced portions tend to lift the mind of man above mundane considerations and enable it to apprehend the final aim of philosophy–the idea of the good. Mathematics, then, is the best preparation for philosophy.”
-emphasis on a deep and strong foundation, and reasoning to advance a subject, how did we get to Lewis’s criticism which he wrote in 1940 in The Screwtape Letters
20th Century:
-progressive movement began to exert an enormous influence over mathematics education in the United States through education professor, William Kilpatrick, who was a protege of John Dewey
-One of the results of this influence was the dumbing down of math education by rejecting the notion that the study of mathematics contributed to mental discipline, and subscribing to the idea that mathematics should be taught only to students based on direct practical value.
-this flies in the face of the Greek thought of mathematics, and is much more in line with the Babylonians and the Egyptians
Excerpt from A Brief History of American K-12 Mathematics Education in the 20th Century by mathematician, David Kline
“Reflecting mainstream views of progressive education, Kilpatrick rejected the notion that the study of mathematics contributed to mental discipline. His view was that the subjects should be taught to students based on their direct practical value, or if students independently wanted to learn those subjects. This point of view toward education comported well with the pedagogical methods endorsed by progressive education. Limiting education primarily to utilitarian skills sharply limited academic content, and this helped to justify the slow pace of student centered, discovery learning, the centerpiece of progressivism.”
-Kilpatrick proposed that the study of algebra and geometry in high school be discontinued ‘except as an intellectual luxury.’
-Kilpatrick: mathematics is ‘harmful rather than helpful to the kind of thinking necessary for ordinary living.’
-Kilpatrick lectured, “We have in the past taught algebra and geometry to too many, not too few.”
1st half of the 20th Century:
-algebra enrollment decreased by 57%
-geometry enrollment decreased by 63%
New York Times:
-an article, Is Algebra Necessary, Andrew Hacker, Professor Emeritus of Political Science at Queens college, City University of New York, writes:
“A typical American school day finds some six million high school students and two million college freshmen struggling with algebra. In both high school and college, too many students are expected to fail. Why do we subject American students to this ordeal? I found myself moving toward the strong view that we shouldn’t.”
“It would be far better to reduce, not expand, the mathematics we ask our young people to imbibe.”
-he blames the subject of algebra specifically for the many of the ills in public education, including dropout rates
-his words echo Kilpatrick's words nearly 100 years ago
-he suggests an alternative course of study, but can’t be accessed without a background in algebra
After progressive movement of the early 20th century
-the next movement known as the “new math” of the 1960s
-this movement was in response to the dumbing down of the math curriculum by the Progressives and the Soviets launching of Sputnik
-the counter movement arose to put rigor and reasoning back into the K-12 mathematics
-the new math never grounded itself in concrete mathematics at an early grade
-students never developed a solid foundation, and any reasoning they tried failed
***students had no firm foundation to build on
The Battle Today:
-Back to the basics vs. progressive ideas
Event: publishing of the NCTM standards in 1991
-NCTM: stands for National Council of Teachers of Mathematics
-the NCTM was founded in 1920 to fight against the progressive agenda of the math education community at that time
-standard algorithms, such as long division algorithm, are downplayed or completely ignored in the standards
-empowering students when they know they can answer a question every time
-standard algorithms are stand for a reason because they have stood the test of time to answer large groups of questions
-NCTM standards has deductive reasoning and geometry has been pushed aside as having little practical value
-the logic and reasoning which is the backbone for a math education is increasingly ignored at all levels in the curriculum
-this is relevant because with the decrease of mathematics so went the students ability to make an argument and to lay out a structure for an essay
-to some degree some of the lower-level college textbooks
-look at the textbooks published after 1990 has been influenced by the NCTM standards
****the so-called standards see mathematics as a tool for social change rather than a subject worthy of study
Current National Standards:
-Hillsdale teacher mathematics and deductive reasoning as the core course in mathematics
-students 1st study logic through both syllogisms and symbolically, to understand how it is that we can reach conclusions
-then we apply that logic to the study in depth of Euclid’s Elements, not for any particular geometric result, but to appreciate order and beauty in the subject
“”Logical structure of elementary school math or arithmetic-look for the 2013 Barney Charter School Initiative lecture
***We study math to understand what it is to be human