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MathematicsDecoding Theory of Knowledge pages 39-49Introduction.Mathematics is elegant, beautiful, simple, systematic and readily repeatable. Is mathematics the most perfect system of knowledge we have?Einstein said:“Insofar as the statements of mathematics are certain, they do not refer to reality; and insofar as they refer to reality, they are not certain”What does this mean? Do you agree?Numbers (page 39)Some ancient groups and some groups today such as the Piraha in Brazil only have words for one, two and many.Why do you think this happened?What are the advantages / disadvantages of this system?The decimal system.First used in 476CEWhy do we use the decimal system? Would it be better to use the hexadecimal system based on groups of 60?Should / could we change the way we tell the time to the decimal system?Real life situation.Aly has two small bars of chocolate and Risa has one large bar. We might say that Ali has twice the number or chocolate bars as Risa and there are three bars altogether. But, if they break up all the chocolate into equal sized squares, Risa has more pieces than Ali. Risa has fewer chocolate bars than Ali but more chocolate.To what extent does counting relate to quantity?How did you learn about numbers?Small children learn to count by applying numbers to objects.I can teach a child that 3 apples plus 2 apples equals 5 apples-easy. BUT…What if I have 3 apples and say I want to eat 4? How many apples are left now? Natural numbers can`t extend simply to negative numbers, so how can I make a child understand this?How would you teach a child negative numbers or algebra? (It is hard to do after we have built up an understanding of numbers and objects) Is there a reason to learn numbers that aren`t related to real life objects?Mathematics gets its power from working with abstractions.Explain what this means.`When looking at Geometry we use approximations. Perfect circles (in Euclidean geometry) don`t exist in the real world`.Do you accept this? Why can`t there be such a thing as perfect circles or straight lines in the real world?Think and discuss.Why do we need negative numbers?What would the world be like without zero?Knowledge framework.Historical development of mathematics.Is mathematical knowledge discovered or invented?How does the discovery of mathematical knowledge differ from other areas of knowledge?Applied mathematics (Page 41).How is mathematics used when designing a bridge?The Bay Bridge Disaster 1989.The bridge broke after an earthquake, causing 68 deaths.Did the bridge break because of the misuse of mathematics?Can all problems be fixed using mathematics?Read page 42 `Applied Mathematics` The book says that the bridge example shows that mathematics does not apply to the real world.Explain the reasons given in the book. Do you agree?How does the book justify the second half of Einstein`s statement `insofar as the statements of mathematics refer to reality they are not certain`?Think.Calculating the time you need to get to school each morning involves applying mathematics to a real issue. You need to leave a margin of error to avoid being late. Even then, an unexpected traffic jam, a delayed bus or a puncture in your bicycle tyre may still male you late if your margins are not big enough. What other calculations do you make that require a margin of error? Do you need to do this in science? Give an example.Knowledge framework.Personal Knowledge.How do individuals contribute to mathematics?What responsibilities does a mathematician have as a result of his or her mathematical knowledge?Mathematical proof (page 42).Mathematics is seen to be certain as we deal with mathematical proof, using deductive reasoning.Only when something in mathematics is proved do we say it is true.But what is proof?People think that proof establishes truth. “There, I have proved it, it must be true!”But proofs do no more than preserve truth, they do not create it or add to it.Certain assumptions (premises, axioms)PLUSLaws of logic (deductive reasoning) correctly applied. EQUALSThe conclusions we come to (theorems) will be as true as the original assumptions.THEREFOREProof only tells us what we already assumed.Give an example to show this thinking. THINK……If proofs don`t tell us anything new, why are they so important in mathematics?A maths riddleImagine you leave your house and walk 10km due south. You then turn 90 degrees left and walk 20km due east before turning 90 degrees left again and walking 10 km due north only to find a bear on your front doorstep. What colour is the bear?10. Maths logic (page 44)Logic is the study of valid forms of reasoning. Remember deductive reasoning?Premise 1: All rectangles have 4 sides.Premise 2: A square is a rectangleConclusion: Therefore a square has four sidesThis is an easy example and is true as long as both premises are true. It shows us certainty but doesn`t tell us anything we don`t know or assume to start with.Not all logics is so straightforward. Modal logics look at things like possibility where the logical connections between `certain`, `possible` and `likely`Premise 1: Some quadrilaterals are trapezoids.Premise 2: This shape is a quadrilateralConclusion: Therefore, this shape could be a trapezoid. This time there is no certainty, only a possibility, and we don`t learn anything new. These considerations are used for more complex mathematical problems.When following the rules of deductive reasoning we must start with the general and move to the particular and not the other way round. For example it would be wrong to sayPremise 1: A square is a rectangle.Premise 2: A square has four sides of equal length.Conclusion: Rectangles have four sides of equal length.Be careful with premises.Even if we use valid logic we are likely to make incorrect conclusions if out premises are not true. Altough most schools use deductive reasoning a good mathematician when seeing a new problem will imagine what the answer will look like before using deduction. Sometimes, however, logic can be counter-intuitive. It is possible to make statements that we know to be true that are not provable within a mathematical system. In 1931 Godel`s incompleteness theorems shocked the mathematical world by saying that the truth of something need not depend on the ability to7 prove it.Think…..Is imagination important in mathematics? Compare it to other areas of knowledge.Think…..Deductive reasoning is important in mathematics. Is it as important in other areas of knowledge?11.Mathematics and reality (page 47) Explain the mathematical problems that can arise due to the fact that some numbers such as go on forever (use the poem on page 47 to help) ................
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