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Whither Mathematics Education in the 21 st Century? by James Nickel, B.A., B.Th., B.Miss., M.A. Copyright  2008

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Presentation on theme: "Whither Mathematics Education in the 21 st Century? by James Nickel, B.A., B.Th., B.Miss., M.A. Copyright  2008"— Presentation transcript:

1 Whither Mathematics Education in the 21 st Century? by James Nickel, B.A., B.Th., B.Miss., M.A. Copyright  2008

2 A Math Exam Name and define the Fundamental Rules of Arithmetic. District Number 33 has a valuation of $35,000. What is the necessary levy to carry on a school seven months at $50 per month and have $104 for incidentals? Find the interest of $ for 8 months and 18 days at 7 percent. What is the cost of 40 boards 12 inches wide and 16 feet long at $20 per meter? Copyright  2008

3 Gulp! These questions are part of an 8 th grade Final Exam, Salina, Kansas (1895). These questions are taken from the original document on file at the Smokey Valley Genealogical Society and Library in Salina, Kansas, and reprinted by the Salina Journal ( Remember when grandparents and great- grandparents stated that they only had an 8 th grade education? Copyright  2008

4 Whither Mathematics Education? Mathematics is a tool to teach: –Logic. –Analytical skills. –The structure and analysis of scientific law. Mathematics is not just a tool … it is a tool of wonder. Copyright  2008

5 Questions What perspectives should undergird mathematics curricula? Does God’s Word speak to mathematical issues? Copyright  2008

6 The Value of Logic “…studied and nearly mastered the six books of Euclid when he was a member of Congress. He began a course of rigid mental discipline with the intent to improve his facilities, especially his powers of logic and language. Hence his fondness for Euclid, which he carried with him on the circuit till he could demonstrate with ease all the theorems in the six books; often studying far into the night … while his fellow-lawyers, half a dozen in a room, filled the air with interminable snoring.” Abraham Lincoln ( ), 16 th president of the United States, Biographical sketch, 1860 Presidential campaign. Copyright  2008

7 The Value of Logic “Mathematics as an expression of the human mind reflects the active will, the contemplative reason, and the desire for aesthetic perfection. Its basic elements are logic and intuition, analysis and construction, generality and individuality … it is only the interplay [of these elements – JN]... that constitute the life, usefulness, and supreme value of mathematical science.” Richard Courant and Harold Robbins, What is Mathematics? p. xv. Copyright  2008

8 The Value of Logic “[Ratiocination] is the great principle of order in our thinking; it reduces a chaos into harmony; it catalogues the accumulation of knowledge; it maps out for us the relations of its separate departments; it puts us in the way to correct its own mistakes. It enables the independent intellects of many, acting and re-acting on each other, to bring their collective force to bear upon one and the same subject-matter, or same question.” Copyright  2008

9 Logic “If language is an inestimable gift to man, the logical faculty prepares it for our use. Though it does not go so far as to ascertain truth, still it teaches us the direction in which truth lies, and how propositions lie towards each other. Nor is it a slight benefit to know what is probable, and what is not so, what is needed for the proof of a point, what is wanting in theory, how a theory hangs together, and what will follow, if it be admitted. Though it does not itself discover the unknown, it is one principal way by which discoveries are made.” John Henry Newman, An Essay in Aid of a Grammar of Assent, p Copyright  2008

10 Logic: Two Aspects Symbolic logic. Justification of propositions (as in the logical flow of mathematical proofs). Copyright  2008

11 Symbolic Logic Hardware: computer circuitry (i.e., integrated circuits). Software: Branching statements (If this condition is true, then do that). Copyright  2008

12 Problem Solving The logical methods of mathematics train the mind in systematically approaching and solving, not only mathematical problems, but … … other problems as well; especially those germane to conflict resolution or ethical infractions. Copyright  2008

13 Proviso This does not mean that mathematical training guarantees ethical purity or that the ability to reason guarantees liberty (personally or culturally). The French Revolution posited the ability to reason independently is the sole guarantor of liberty. Truth: Reason is merely a tool of liberty, not the basis of it. The deification of reason resulted, not in liberty, but in heads rolling from the guillotine. Copyright  2008

14 Failsafe for Logic Loving God in Christ (the logos or the ultimate logic) does not consist of ratiocination only; it means loving Him with the totality of your being (Mark 12:29-31). Copyright  2008

15 Foundational Laws of Logic Law of the excluded middle. Law of contradiction. Copyright  2008

16 Law of the excluded middle For every formal statement B, either B is true or B is false (thesis/antithesis) Given B, either B or ~B (B is false). B: The infinite, personal and Triune God of Scripture exists and has spoken Truth in His word … or ~B. Copyright  2008

17 Law of contradiction A formal statement cannot be both true and false at the same time. You cannot say that B and ~B are both true. Copyright  2008

18 Thinking antithetically The presupposition of antithetical thinking … Thinking in terms of thesis and antithesis. A statement is true (thesis) or it is not true (antithesis) … subsumes these two laws. Copyright  2008

19 Transcendental Nature These laws are founded upon the transcendental nature of thesis/antithesis, not upon the Greek philosophy Aristotle ( BC). who formulated them. Copyright  2008

20 You cannot escape … To deny the validity of the law of the excluded middle assumes that it is either true or false. You must assume the law of the excluded middle in order to deny it. Copyright  2008

21 You cannot escape … To deny the law of contradiction means that you are saying that the law is false. By denying this law you are proving it. Copyright  2008

22 You cannot escape … Avicenna ( ), Islamic physician and philosopher from Persia, “Anyone who denies this law should be beaten and burned until he admits that to be beaten is not the same as not to be beaten, and to be burned is not the same as not to be burned.” Copyright  2008

23 You cannot escape … These two laws are prerequisite for coherence of the human experience. If not, then we have no basis for intelligible communication. We have the “forks coming forth from the tongue” scenario. Copyright  2008

24 More logic Informal logical fallacies: ad hominem (attack of an opponent’s character) and equivocation. Formal logical fallacies: –Asserting the consequent (converse error). –Denying the antecedent (inverse error). Copyright  2008

25 Conditional Statements P  Q Q  P (Converse) ~P  ~Q (Inverse) ~Q  ~P (Contrapositive). Copyright  2008

26 Logical Equivalence P  Q  ~Q  ~P (Contrapositive). Q  P (Converse)  ~P  ~Q (Inverse) Copyright  2008

27 More logic Disjunctive syllogisms. –P or Q. –~P. –Therefore Q. Reductio ad absurdum argumentation (reasoning to contradictions or law of contraposition). Copyright  2008

28 Resources Morris S. Engel, With Good Reason, 6 th ed. (New York: Bedford Books/St. Martin’s Press, 2000). Irving M. Copi, Introduction to Logic, 10 th ed. (Englewood Cliffs, NJ: Prentice-Hall, 1998). Copyright  2008

29 Analytical Skills Don’t believe a statement to be true simply because numbers are attached to it. Benjamin Disraeli ( ), British Prime Minister, “There are three kinds of lies: lies, damned lies, and statistics.” Rejoinder: “It is easy to lie with statistics, but it is easier to lie without them.” Copyright  2008

30 1936 Presidential “Polls” Franklin Delano Roosevelt vs. Alf Landon. 10 million voters were sampled by Literary Digest. Copyright  2008

31 10 million voters 1,293,669 said they would vote for Landon (53%). 972,897 said they would vote for Roosevelt (47%). On the basis of these results, the Literary Digest predicted that Landon would win the election. Result: Roosevelt won by a landslide 61% to 37%. Copyright  2008

32 What went wrong? Only 20% responded. The sample of voters relied heavily on lists of automobile and telephone owners. In the 1930s, these people would be the more affluent voters. Copyright  2008

33 Random Sample Bias Under coverage: some groups of the population are left out of the process of choosing the sample. Non response: sample individuals cannot be contacted or do not cooperate (give false responses). Copyright  2008

34 Buyer, Beware! Given a dataset, be very cautious when making extrapolations or interpolations. If not, sometime in the distant future, someone will run the mile in 0 seconds. Copyright  2008

35 Scientific Law: Structure and Analysis Algebra, as a language, teaches structured thinking; a thinking that resonates with the patterns revealed in the physical creation (scientific law). How many teachers teach Algebra in its historical context, a context that connects it directly to the physics of motion? Copyright  2008

36 Tool of Wonder The fact that a honeybee constructs a honeycomb to maximize packing space (for honey) given a minimum of packing material (beeswax) can be confirmed by the differential calculus. Copyright  2008

37 Wonder leads to … Wonder is the foundation of creativity. Creativity is the foundation of invention. Copyright  2008

38 Inventors Intense persistence. Optimism. Originality of approach combined with an almost mystical conviction that there are more effective, more elegant ways of doing things. Copyright  2008

39 Keys to Invention/Creativity Contemplation (includes analytical and logical comparison of one thing to another). Imagination. Wonder. There are more variables in the equation of invention and creativity than mere logic alone. Copyright  2008

40 Sir Isaac Newton ( ) Stanley L. Jaki (1924-) notes Newton’s reflection on the nature of God’s Works, “Newtonian science was the product of a truly inventive intellect pondering the witness of the senses.” The Road of Science and the Ways to God, p Copyright  2008

41 Big Question “Years ago I went into my laboratory and said, “Dear Mr. Creator, please tell me what the universe was made for?” The Great Creator answered, “You want to know too much for that little mind of yours. Ask for something more your size, little man.” Copyright  2008

42 Smaller Question Then I asked. “Please, Mr. Creator, tell me what man was made for?” Again the Great Creator replied, “You are still asking too much. Cut down on the extent and improve the intent.” Copyright  2008

43 Tiny Question So then I asked, “Please, Mr. Creator, will you tell me why the peanut was made?” “That’s better, but even then it’s infinite. What do you want to know about the peanut?” Copyright  2008

44 The Tiny Peanut “Mr. Creator, can I make milk out of the peanut?” “What kind of milk do you want? Good Jersey milk or just plain boarding house milk?” “Good Jersey milk.” Copyright  2008

45 The Peanut And then the Great Creator taught me to take the peanut apart and put it together again. And out of the process have come for all these products! Cited in Ethel Edwards, Carver of Tuskegee (Cincinnati: Ethel Edwards & James T. Hardwick, 1971), pp Copyright  2008

46 The Peanut George Washington Carver (ca ) discovered nearly 300 derivative products from his investigation of the peanut. Copyright  2008

47 Mathematics Curriculum What to look for? The how and the why. Unity in diversity. Induction and deduction. History. Operational Science. Philosophy. Remove isolation. Copyright  2008

48 The How and the Why A balance between the how (pure mechanics) and the why (reasoned justifications). For example, what is the logical justification for (-1)(-1) = +1? Copyright  2008

49 Unity in Diversity Show connections.  (pi), the ratio of the circumference to the diameter of a circle, and Probability. Breadsticks, tile floors, and the calculation of . Conic Sections, Quadratic equations, and the Physics of Motion. Copyright  2008

50 Induction and Deduction Balance between induction and deduction. Induction: search for patterns (engenders discovery and wonder). Deduction: justification of patterns induced (engenders logical analysis). Copyright  2008

51 Induction and Deduction Johannes Kepler ( ) discovered the three laws of planetary motion using induction. Isaac Newton, starting from his Universal Law of Gravitation, deduced Kepler’s three laws! Copyright  2008

52 Who flunked? Martin Gardner (1914-), in his introduction to the textbook Mathematics: A Human Endeavor (by Harold R. Jacobs), “An old anecdote tells of a meeting of the entire faculty of a major university. The president said his listeners would be amused to know the names of two new students who had flunked freshman classes. A lad named Cicero had failed Latin. Everybody laughed.” Copyright  2008

53 Who is Gauss? “Another student named Gauss had failed mathematics. Only the mathematicians and physicists laughed. This would not have been the case if the members of the liberal arts faculty had explored the world of mathematics with Harold Jacobs.” Copyright  2008

54 History We must connect mathematical propositions to the people who discovered them. We must know the critical place of mathematics in the development of Western Culture. Copyright  2008

55 Operational Science Mathematics is the handmaiden of science. Dr. Morris Kline ( ), in Mathematics and the Search for Knowledge, p. 144, “Radio waves and light waves operate in a physical darkness illuminated only for those who would carry the torch of mathematics.” Copyright  2008

56 Philosophy (the love of wisdom) Two primary questions. 1.Cosmological question: What is the origin, nature, and destiny of the cosmos (all things)? 2.Anthropological question: What is the origin, nature, and destiny of man? Copyright  2008

57 Four Subsidiary Questions 1.The ontological question: What is the nature of existence? 2.The epistemological question: What is the nature and limits of knowledge? 3.The axiological question: What is ultimate value? 4.The teleological question: Where are we going? Copyright  2008

58 Answers The answers to these four philosophical questions determine how one answers the two primary questions. The answers to the two primary questions determine how one structures the totality of life. The propositions of Scripture generate a worldview that provides enduring answers to the larger questions which are applicable to the totality of life. Copyright  2008

59 Truth and Consequences: The totality of life Civil/social. Law. Economics. International politics. Aesthetics. Ecclesiastical. Educational … Mathematics included! Copyright  2008

60 Resources Dr. Glenn R. Martin ( ), Prevailing Worldviews of Western Society Since 1500 (Marion: Triangle Publishing), 2006 ( The works of Francis Schaeffer ( ): The God Who is There, Escape from Reason, He is There and He is Not Silent. Copyright  2008

61 Remove isolation! Never teach mathematics in isolation; knowing mathematics should open the door to: Copyright  2008

62 Remove isolation! The wonders of Creation. The heritage (indeed, providential) of history. The appreciation of philosophical presuppositions governing the Judeo- Christian foundations of Western Culture. Copyright  2008

63 Review Mathematics and logic. Mathematics and wonder. Mathematics curricula. Mathematics and truth. Copyright  2008

64 Conclusion in Truth Proverbs 1:7, “The fear of the LORD is the beginning of knowledge; fools despise wisdom and instruction.” Copyright  2008

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