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MAthematices and the SEA (and other things)

MAthematices and the SEA (and other things). Johan Deprez SEAMA-conference , Antwerp , 31/05/10 www.ua.ac.be / johan.deprez > Documenten. Overview. Introduction Example 1: Journey of the drilling rig Yatzy Some comments Example 2: The future of the Belgian population

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MAthematices and the SEA (and other things)

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  1. MAthematices and the SEA (and other things) Johan Deprez SEAMA-conference, Antwerp, 31/05/10 www.ua.ac.be/johan.deprez > Documenten

  2. Overview • Introduction • Example 1: Journey of the drillingrigYatzy • Somecomments • Example 2: The future of the Belgianpopulation • Somecomments

  3. Introduction secondexample comes frommy classes about myself: • mathematics teacher • for about 20 years • higher education at university level (but not university) • students in applied economics • basic mathematics course • …

  4. Introduction about myself: • mathematics teacher • … • mathematics educator • related to secondary mathematics education • informally: Uitwiskeling = magazine for secondary math teachers • formally: teacher education at university (KULeuven, Universiteit Antwerpen) for about 15 years • … firstexample comes fromteacherswithwhom I work

  5. Introduction about myself: • mathematics teacher • … • mathematics educator • … • researcher in mathematics education • for about 2 years • applications in mathematics education • concrete versus abstract in math education

  6. Journey of the drilling rig Yatzy • ownedby Diamond Offshore • operated in BrazilbyPetrobras (NOC) • built byshipyard Boelwerf Temse (1989)

  7. Journey of the drilling rig Yatzy

  8. Journey of the drilling rig Yatzy • Yatzy was transportedalong the river Scheldt fromTemse to Rotterdam on 11 January 1989 • problem: passing power line hanging over river • engineers’ problem is transformedinto a problemforsecondary school math classes (actually: forteachers …) • Dirk De Bock and Michel Roelens, The journey of the DrillingRigYatzy: TodayonTelevision, Tomorrow as a Large-scale Modelling Exercise. in: Jan de Lange et al. (1993) Innovations in MathsEducationby Modelling and Applications, EllisHorwood

  9. Introduction to the context depthchart of the river Scheldt at the crossing of the power line

  10. Introduction to the context height of the power line informationabout the tides

  11. First analysis • conclusions: • problem is at the top • pass nearleft bank (x=180) • pass at low water spring tide • studentsadd • riverprofileunder power line • (lowest point of) power line • water levels at high and low water spring tide • and then experiment usingscale model of Yatzy

  12. A mathematical model for the cable (engineer’s version) • power line follows catenary curve • height is given by a function of the form • using coordinates of lowest point: c = 422.92, d = 79.80 • using left end (numerical methods) a = 1491.09

  13. A mathematical model for the cable (student’s version) • power line approximately follows parabola • height is given by a function of the form • using coordinates of lowest point: c = 422.92, d = 79.80 • using left end a = 0.00033758

  14. A mathematical model for the cable comparison of the two models

  15. Can Yatzy pass? How much margin? • at x=180 Yatzy has a margin of 2m on top of safety margin • notrealistic to pass at exactly x=180! • Solvequadraticinequality • Mathematicalsolution: • In reality: horizontal margin of about 15 meters

  16. How much time is there? • notrealistic to arrive at exactly 1:24 PM (=forecasted low water)! • water must notrisetoomuch! • mathematical model water level: (a, b, c and d canbecalculatedfrom data below) • solveinequality: 10:57 AM  t  3:50 PM

  17. How much time is there? evaluation of the model (afterwards) using observed water levels • in reality there was much more time than predicted by the model • after all, sines are rather poor models for water level

  18. Yatzy on television

  19. Comments • In the 1970’s and 1980’s secondary math education in Flanders was strongly influenced by New Math • very strong emphasis on deduction and proof, formalism, classical mathematical structures, pure mathematics, … • very little attention for problem solving, geometrical insight, applications of mathematics and mathematical models, … • Example 1 marks the start of an evolution towards inclusion of more applications in secondary mathematics education.

  20. Comments • Example 1 was constructed before the integration of technology in secondary mathematics education. • Nowadays, technology is used in nearly all Flemish secondary school math classes: • mainly graphing calculators graphs, numerical calculations, matrix calculations, statistics, … • nearly no symbolic calculations factoring, symbolic differentiation and integration, … • Example 1 would be different when used in class now, but not too much.

  21. Comments • Twoevolutions in secondarymatheducation: • inclusion of more applications of mathematics • integration of technology • In general, there was nosimilarevolution in highereducation • sometimestechnology is notadmitted • purelymathematicalaspects are still more emphasized • notmuchattentionforrelationbetweenmathematics and main subject(s) of the students Question: How aboutyour country?

  22. Comments • Example 1 is anauthenticapplication. • Manyapplications in Flemishsecondary school math are non-authentic. • It is not easy to construct authenticapplications … • … and it is not evident to useauthenticapplications (time-consuming, nottoo easy forstudents, …) • … butit is important thatstudents meet anauthenticapplicationfrom time to time • … and it is possible to find/construct them • in books, magazines, on the internet, in-service training, … • i.e. start fromnewspaperarticles

  23. Comments Example 1 contains important aspects of mathematical modelling • translation of realityintomathematics i.e. from data to inequalities • interpretation of mathematicalresults in reality i.e. solution of the quadraticinequality • comparison of model and reality • mathematical models do not match realityperfectly i.e. sinefunction is poor model for water level • different models canbeusedfor the samephenomenon • comparison of different models i.e. parabolic versus cosh-model

  24. Comments Pure math and applications do not always match! • Example 1: • sine functions in mathematics versus physics: • minor differences: different letters used, no d in physics major differences: different form for the argument of sine and, hence, different interpretation of c vs.  • both math teachers and physicists have good reasons to prefer their form (i.e. nice interpretation for c) • My advice: Do the ‘translation’ twice: both in math and physics classes!

  25. Comments • Another example: functions • mathematics: f is the function y=x2 other subjects: s=t2 • first important difference: standard names in math for variables (independent: x, dependent:y) implication: • inverse of a function in pure mathematics: first, solve x in terms of y and next, interchange notations x and y • inverse of a function in applications: solve t in terms of s (and DO NOT interchange the notations for the variables) • similarly, composition of functions is different in pure mathematics compared to applications

  26. Comments • Another example: functions • mathematics: f is the function y=x2 other subjects: s=t2 • … • second important difference: use of variables (x and y or other names) versus functions (f) implication: different notations • derivative using function (f’) versus using variables (dy/dx) • composition of functions has a notation using functions (g◦f) but not using variables

  27. The future of the Belgian population we do nottakemigrationinto account! basedon data of BelgianStatistical Bureau, cfr. www.statbel.fgov.be after 20 years 98% of the individuals in age group I is still alive during a period of 20 years an individual in age group I is responsible for an average of 0.43 births

  28. The future of the Belgian population number of age 0-19 in 2023: number of age 20-39 in 2023: number of age 40-59 in 2023: number of age 60-79 in 2023: number of age 80-99 in 2023:

  29. A matrix model for the evolution of the Belgian population Leslie model fertilityrates Leslie matrix survival rates population on 1 Januari 2003

  30. A matrix model for the evolution of the Belgian population number of age 0-19 jaar in 2023: number of age 20-39 in 2023: number of age 40-59 in 2023: number of age 60-79 in 2023: number of age 80-99 in 2023: 2003 2023 from 2003 to 2023: L ... from 2023 to 2043: L ... from 2043 to 2063: L ...

  31. Leslie model for the internal growth of a population onlydeath and birth, nomigration! population is subdivided in age groups of equal width fertilityrates and survival rates initial population = column vector X(0) Leslie matrix = square matrix containing transition perunages between age groups over a period equal to the width of the age groups population after n periodes = X(n) recursive equation: explicit equation:

  32. The future of the Belgian population long term: graphs of all age groups show great and common regularity babyboomers babyboomers babyboomers ‘short’ term

  33. Long term: first observation -15,7% growth percentages: -16,1%

  34. Long term: first observation 0.84 is the long term growth factor • in the long run the number of individuals in eachagegroup • decreasesby 16% every 20 years • is multipliedby 0.84 every 20 years growth percentages:

  35. Long term: first observation if n is a verylargenumber, then in eachagegroup number of individuals at time n number of individuals at time n-1  0.84  in mathematical notation: X(n)  0.84·X(n-1) equivalent forms: X(n+1)  0.84·X(n) LX(n)  0.84·X(n)

  36. Long term: second observation in the long run the distribution over the agegroupsstabilizes long term age distribution percentages give the distribution of the population over the age classes

  37. Long term: second observation n-th line in table on previous slide = distribution of population over the age classes after n periods = X(n)/t(n), where t(n) is total population after n periods stabilization of age distribution means: if n is a very large number, then X(n)/t(n)X(n-1)/t(n-1) a limit age distribution is defined by

  38. LT growth factor and LT age distribution • LT growth factor and LT age distribution wereobserved in tables, foundbymassivecalculations • Can LT growth factor and LT age distribution bedetermined in a more elegant way? • method to determine LT age distribution if LT growth factor is alreadyknown • divide LX(n)0.84X(n) by t(n) and take limit: LX=0.84X • ifyou do notalready know X, youcanfind X bysolving the system LX=0.84X (and adding the conditionthatsum of components is 1)

  39. LT growth factor and LT age distribution • Can LT growth factor and LT age distribution bedetermined in a more elegant way? • … • method to determine LT growth factor • LX=0.84X has non-trivialsolutions • this is exceptional! • LT growth factor is the onlystrictlypositivenumberλforwhich LX= λX has non-trivialsolutions • i.e. … forwhichdet(L-λI)=0

  40. Decontextualising • long term growth factor is an eigenvalue of the matrix L • long term age distribution is an eigenvector of the matrix L Definitions A a square matrix (n  n) • A number  is an eigenvalue of A iff det (A-In)=0. • A column matrix X (≠ 0) is an eigenvector of A corresponding to the eigenvalue  iff AX = X.

  41. Comments Experiences • example 2 is not easy butfeasible • students report thatithelpsthem to seethatmathematicalconcepts are useful • studentsmaster the mathematics at same level as with traditional approach • decontextualising is necessary • LT growth factors are strictlypositive, but eigenvalues mayalsobenegativeor zero • LT agedistributions have sum of theircomponentsequal to 1, buteigenvectorsneednotsatisfythissupplementarycondition

  42. Comments • Ex. 1: application AFTER mathematics has been covered Ex. 2: application INTRODUCES mathematics • Rationale: • shows relevance of studiedmathematics right from the start! • you show abstractionprocess (instead of onlyresult of abstractionprocess) • I usethis at several occasions: • speed of growth -> derivative • multiplier in economics -> geometric series • discrete/continuousdynamicmarket model -> difference/differentialequations • …

  43. Conclusion Relationbetweenmathematics and the ‘rest of the world’ is • different fromstudying pure mathematics • different fromstudyingmathematics as a bag of tricks • worthstudying in highereducationmathematics

  44. Thank you!

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