Pomperipossa om Skolmatematik: Lite för Alla eller Mycket för Några?

Det moderna digitala samhället bygger på avancerad matematik och det är av vital betydelse att det finns samhällsmedborgare som kan bära och utveckla denna kunskap.

Skolans matematik har som mål att ge alla det minimum av matematisk kunskap, som anses nödvändigt för fungera som rationell samhällsmedborgare och inga resurser skall sparas för att nå detta mål. Genom massiva stödinsatser med start i förskolan skall varje elev garanteras att uppnå miniminivån, som en del av en samhällelig "läsa-skriva-räkna garanti".

Samhället har alltså behov av ett relativt fåtal som kan använda avancerad matematik, i likhet med avancerad medicin, medan skolan fokuserar på trivial matematik att användas av alla.

Detta går inte ihop. Det samhälleliga värdet av avancerad matematik i händerna på ett fåtal är stort, i likhet med avancerad medicin. Men denna kan inte ersättas av trivial matematik som alla kan, som att veta hur man sätter på ett plåster. Man kan inte ersätta hjärttransplantationer utförda av några specialister med en massa plåster applicerade av många.

För att skolmatematiken skall få mening och motsvara samhällets behov av matematik, fordras att nuvarande obligatorium i form av lite för alla, ersätts av valfrihet utan obligatorium och minimikrav, där de som vill ges möjlighet att lära sig så avancerad matematik som möjligt utan maximibegränsning.

Nuvarande mantra om mimimum för alla, måste alltså överges. Men det kommer att sitta hårt inne eftersom en hel skolbyråkrati bygger på detta mantra.

Vidare behöver lärarutbildningen reformeras. Med minimum för alla som mantra räcker det med lärare med ett minimum av kunskap. Med maximum för några, krävs lärare som kan mer än minimum.

Det finns många saker att jämföra med, som tex pianospel vilket i likhet med matematik är svårt att  lära sig och krävande att utöva. Vilket samhälle är då att föredra? Ett samhälle där alla hjälpligt kan traggla sig igenom en pekfingervals på vita tangenter som resultat av en massiv utbildningsinsats från tidiga skolår, men ingen kan spela Chopin, eller ett samhälle där några kan glädja många genom riktigt spel och kanske några inte ens kan pekfingervalsen eftersom det finns så mycket annat som kan vara mer givande, som att lära sig bemästra olika datorspel? Astrid Lindgren skulle säkert ha kunnat låta Pomperipossa ta sig an denna problematik, och då kanske något skulle kunna hända?

Programmering i Skolan: Trivium eller Quadrivium?

Skolverkets förslag till nya läroplaner för grundskolan med programmering som nytt inslag, utformat på uppdrag en Regering som registrerat vibrationer från omvärlden, verkar ha fastnat i systemet då ingen proposition till Riksdagen om programmering är i sikte.

Lika bra det, eftersom Skolverkets förslag har tillkommit efter principen att göra så lite som möjligt, dock med den stora fördelen någon vidareutbildning av lärare i det nya ämnet programmering inte behövs.

Men ute i kodstugor och på bibliotek växer en underground-rörelse fram, där barn får prova på att programmera i Scratch:

• Scratch helps young people learn to think creatively, reason systematically, and work collaboratively — essential skills for life in the 21st century.
• Scratch is a project of the Lifelong Kindergarten Group at the MIT Media Lab. 

Det informella samhället har här tagit ett initiativ till folkbildning, enligt gammal god svensk socialdemokratisk tradition, för att kompensera brister i skolans utbildning. 

Tanken med Scratch är alltså att hjälpa barn från tidig ålder att tänka kreativt och resonera systematiskt, eftersom detta anses vara väsentliga färdigheter i det vuxenliv som väntar barnen. 

Denna tanke ligger också till grund för den ökning av antalet undervisningstimmar i matematik på låg- och mellanstadium med 225 timmar, som Riksdagen beslutat: Med en ordentlig dos matematik under tidiga skolår för alla barn kommer både individ och samhälle att frodas, därför att matematik i likhet med programmering bygger på systematiskt resonerande, och det skall man lära sig i småskolan!

Men om nu systematiskt resonerande/matematik/programmering är så viktigt, och faktiskt inte så vidare enkelt ens för vuxna, vore det då inte bättre att vänta lite till dess barnen är mogna att ta emot mer än bara det enklaste? Och inte bränna det mesta krutet i förtid utan vidareutbildning av lärare, utan istället ge lärarna ordentlig vidareutbildning så att de kan förmedla något bortom det triviala? 

Klassik utbildning bestod av inledande trivium = grammatik, logik och retorik, följd av quadrivium = aritmetik, geometri, musik och astronomi. Vi ser att trivium (väsentligen språk) kom tidigt, medan quadrivium (väsentligen matematik) låg senare i utbildningen. Kanske något att tänka på även idag?

Mathematics as Magics 3: Towards a New School Mathematics

School mathematics with its 150 year history is based on an idea/fiction of unreasonable effectiveness of mathematics (according to Wigner), which when confronted with the reality of the unreasonable ineffectiveness of mathematics (according to Hamming), as any deep contradiction between ideal an reality, has resulted in a big mess and lots of frustration.

Both students and teachers are brain-washed to believe that mathematics is very powerful, while their experience is the opposite.

The idea of the unreasonable effectiveness of mathematics goes back to the declared success of analytical mathematics/Calculus of Newton's Principia Mathematica, which allowed Newton with pen-and-paper to play the role of God as monitor of the Universe.

School mathematics is supposed to bring some of this power to the people. But Principia Mathematica was very difficult to read, in fact written by Newton so as to make criticism from "little smatterers" impossible.  All efforts since then to make analytical mathematics/Calculus simple, have failed and the result is a an analytical pen-and-paper school mathematics which has resisted all efforts to be brought to the people.

This could mean the end of school mathematics, because teaching a subject experienced to be unreasonable ineffective in our society, like Latin, cannot be sustained over time.

But the computer has changed the game completely, and in fact has made mathematics fulfil the prophecy of almost god-like quality, as the basis and work horse of the digital society.

To convince young minds about the usefulness and power of mathematics + computer, it is sufficient to point at computer games such as Minecraft.

The digital society is an expression of the reasonable effectiveness of mathematics + computer and as such can be a meaningful new form of school mathematics, which can be brought to any young mind that can be trigged by a computer game.

PS Is it possible that the tremendous efforts which were made before the computer to develop school mathematics into its prominent position, were made in anticipation of the computer revolution, which would come sooner or later according the vision of Leibniz as a father of both Calculus and the computer? Yes, I guess it may well be that a collective unconscious awareness can motivate a change in society for which the true reason shows only later. Leibniz:

• It is unworthy of excellent men to lose hours like slaves in the labour of calculation which could safely be relegated to anyone else if machines were used.

Mathematics as Magics 2

The idea of the unreasonable effectiveness of mathematics in Wigner's formulation is based on the analytical solution of the two-body problem given in Newton's Principia Mathematica showing that a single planet subject to the inverse square law of gravitation from a fixed sun, will move in an elliptic (or parabolic or hyperbolic) orbit.

Newton could thus confirm Kepler's laws from a single hypothesis of the inverse square law, with  Newton as mathematician thereby convincingly playing the role God! This gave mathematics a tremendous boost into the queen of sciences with immense (seemingly magical) power, which is the basic argument behind extensive compulsory school mathematics: Learn math and play God!

But if Newton was playing with one planet, God is playing with many planets and thus solves the N-body problem of N bodies moving under mutual gravitational attraction with N any number. But already the 3-body problem has resisted analytical solution since Newton, which can be seen to signify the unreasonable ineffectiveness of analytical mathematics in Hamming's formulation.

But the N-body problem can be solved by computational mathematics for N very large, which expresses a reasonable effectiveness of mathematics + computer.

You can explore the N-body problem in the following apps for young minds:

• Dark Energy: NewMath by Claes Johnson
  https://appsto.re/se/5rkebb.i
• Mechanics1: NewMath by Claes Johnson
  https://appsto.re/se/mMzbbb.i

We learn that mathematics + computer (as NewMath) is powerful and should be taught as a subject of reasonable effectiveness, understanding that analytical mathematics alone may be unreasonably ineffective.

Mathematics as Magics 1

Why does the subject of mathematics have such a prominent position in basic school forcing children through lengthy math hours during 9 years, while the fact is that most adults have forgotten most of their own school mathematics and get by very well with a bare minimum of simple arithmetics?

As an expression of this prominent position the total math hours on basic school in Sweden (grades 1-9) has recently been expanded from 900, in two steps with 120 hours in 2013 and additional 105 hours in 2016 to a total of 1.125 math hours, out of a total of about 6.685 in all subjects, thus roughly 1/6 math or almost one full day a week of math for 9 years.

The logic of the expansion is presented to be (i) math is important for both individual and society and (ii) the result of all the math hours invested is close to zero for many students (not even simple arithmetics mastered), from which the conclusion is drawn that (iii) more hours are required.

Of course the logic is a bit weak: if 900 hours gives no result, why would 1125 give better result?
But maybe this is irrelevant, since anyway most children when adults will not miss whatever math they missed to learn in school. But if 900 gives no result you could as well argue that cutting down to a half would give the same result and that would save hours to something more meaningful.

But this is not the way the argument goes. It is instead: mathematics is very important for both individual and society and thus no effort should be spared for the purpose of mathematical enlightenment of the minds of all young people of a nation (like Sweden or China). Of course we can expect another expansion in 2019 and so on until the school day is filled with math!

But why is mathematics viewed to be so important, when most people have little use of more than a bare minimum of arithmetics? Who is selling this idea? How come that it is so uncritically embraced by just about everybody? Take any subject and claim that it should be expanded by 225 hours and see if you can succeed! Math is the unique subject for which this is possible.

Let us see what answer we can find in the book Is God a Mathematician? by Mario Livio:

• A few years ago, I was giving a talk at Cornell University. One of my PowerPoint slides read: “Is God a mathematician?” As soon as that slide appeared, I heard a student in the front row gasp: “Oh God, I hope not!”
• My rhetorical question was neither a attempt to define God for my audience nor a shrewd scheme to intimidate the math phobics. Rather, I was simply presenting a mystery with which some of the most original minds have struggled for centuries—the apparent omnipresence and omnipotent powers of mathematics.
• What is it that gives mathematics such incredible powers? Or, as Einstein once wondered: “How is it possible that mathematics, a product of human thought that is independent of experience [the emphasis is mine], fits so excellently the objects of physical reality?”
• This sense of utter bewilderment is not new. Some of the philosophers in ancient Greece, Pythagoras and Plato in particular, were already in awe of the apparent ability of mathematics to shape and guide the universe, while existing, as it seemed, above the powers of humans to alter, direct, or influence it. 
• The English political philosopher Thomas Hobbes (1588–1679) could not hide his admiration either. In Leviathan, Hobbes’s impressive exposition of what he regarded as the foundation of society and government, he singled out geometry as the paradigm of rational argument: 
• “Seeing then that truth consisteth in the right ordering of names in our affirmations, a man that seeketh precise truth had need to remember what every name he uses stands for, and to place it accordingly; or else he will find himself entangled in words, as a bird in lime twigs; the more he struggles, the more belimed. And therefore in geometry (which is the only science that it hath pleased God hitherto to bestow on mankind), men begin at settling the significations of their words; which settling of significations, they call definitions, and place them in the beginning of their reckoning”. 
• Millennia of impressive mathematical research and erudite philosophical speculation have done relatively little to shed light on the enigma of the power of mathematics. If anything, the mystery has in some sense even deepened.
• Physics Nobel laureate Eugene Wigner (1902–95) was equally dumbfounded: (a success that Wigner dubbed “the unreasonable effectiveness of mathematics”): “The miracle of the appropriateness of the language of mathematics to the formulation of the laws of physics is a wonderful gift which we neither understand nor deserve. We should be grateful for it and hope that it will remain valid in future research and that it will extend, for better or worse, to our pleasure, even though perhaps also to our bafflement, to wide branches of learning”. 
• The person who presents what may be the most extreme and most speculative version of the “mathematics as a part of the physical world” scenario is an astrophysicist colleague, Max Tegmark of MIT. Tegmark argues that “our universe is not just described by mathematics—it is mathematics”. 

We here find the idea of a mystery of an unreasonable effectiveness of mathematics as a positive answer to the question Is God a mathematician? We see this idea expressed by physicists (Wigner and Tegmark), while many mathematicians would take the (opposite) position of the British mathematician Godfrey Harold Hardy (1877–1947):

• Hardy was so proud of the fact that his work consisted of nothing but pure mathematics that he emphatically declared: “No discovery of mine has made, or is likely to make, directly or indirectly, for good or ill, the least difference to the amenity of the world.”  

Hardy's view of the uselessness of mathematics is supported by the mathematician and computer scientist Hamming:

• “There is only one thing which is more unreasonable than the unreasonable effectiveness of mathematics in physics, and this is the unreasonable ineffectiveness of mathematics in biology.” 

We thus find a range of views concerning the usefulness or effectiveness of mathematics from unbounded admiration to controlled skepticism. Most adults would say that they believe that math is very important, as something they learned in school, while they have themselves found very little use math beyond (if they can) helping their children through the frustrations of school math.

We see that the idea of math as something of god-like quality gets stronger as we move away from pure mathematicians (Hardy), over physicists (Wigner, Tegmark) to ordinary people. A similar pattern may be found in religion with a priest as covert non-believer and a community of strong believers.

The less you know of mathematics, the more powerful you tend to believe it is. To convey this idea is in fact an explicitly stated goal of Swedish school mathematics, and in this respect the education is very successful: When finishing school all students, independent of success in math, are fully convinced that math is very powerful and important and in addition very beautiful!

Mathematics is thus viewed as truly magical by many, which makes rational reasoning about school mathematics very difficult, or simply impossible.  How to be rational about magics?

I will continue with some examples of the magical character of math....and eventually I will land on a standpoint of reasonable effectiveness,  between Wigner's unlimited optimism of unreasonable effectiveness and Hamming's deep pessimism of unreasonable ineffectiveness.

Why Does Trump Not Play the Climate Card?

Trump is known to be skeptic to climate alarmism, according to The Guardian as the world's only national leader to express such an insight. It may come from a sound gut feeling that something is very fishy with that story.

Trump could tell the American people, and the people of the world, the truth that climate alarmism has no real scientific support and battling a non-existent climate enemy at a projected cost of 2-10% of GNP, thus is not needed. He could then say that these enormous resources could better be used for a meaningful purpose such as improving living conditions for the all the poor people of the world. That it would be more than enough to eliminate poverty.

But Trump is not playing such a strong climate card, which could tilt the game to his favour. Why?

Is it that this card would be harassed as being so much worse than any of the politically incorrect cards he has been playing, you name it, so bad that it simply cannot be played? Would freeing resources to eliminate poverty thus be so incredibly politically incorrect?

Or is Trump just waiting to play the card, to get maximal effect?

Hur kunde KI gå på Macchiarini?

Hur kunde KI's högsta vetenskapliga kompetens tro att ett plaströr bestruket med stamceller inopererad på en patient som ersättning för en defekt luftstrupe, skulle utvecklas till en normal fungerade luftstrupe i en helt ny helt oprövad och fullständigt spekulativ form av regenerativ bioteknik?

Jo, det gick till så här enligt Sten Heckschers utredning, där prefekten Felländer-Tsai ger svaret:

• Det är sällsynt med kirurger som inte tvekar inför att skära bort sjuka luftrör på folk och ersätta dessa med decellulariserade nekrograft som hottats upp med stamceller och tillväxtfaktorer. 
• Men det är banbrytande och platsar i Lancet och kan på sikt leda till nya behandlingsparadigm. 
• Jag känner så väl igen fenomenet, har erfarenhet av liknande personer och inom vår verksamhet gjordes ju Sveriges första njur- och levertransplantation (under intressanta former...), till stor nytta för mänskligheten och patienterna (även om vissa dog/dör på operationsbordet). 
• Det fordras en särskild personlighet för att eviscerera folk in vivo. 
• Gränsen mellan succé och fiasko är dock hårfin och succén hänger naturligtvis på en mängd randvillkor och också andra nyckelpersoner som kan balansera det hela.

En vild chansning alltså, ju vildare desto bättre, på decellulariserade nekrograft som hottats upp med stamceller och tillväxtfaktorer utförd av en läkare känd för att eviscerera folk in vivo, som skulle kunna vara banbrytande och göra KI inte bara till utdelare av Nobelpris, utan även till mottagare.  

En vild chansning alltså, som blev verklighet genom ett stark rekommendationsbrev från 14 av KI's mest framstående stamcellsforskare att anställa Macchiarini att utföra sin vivisektion på KI med dödlig utgång, forskare som på något sätt måste ha trott på Macchiarinis nya helt oprövade regenerativa teknik.

Ingen av dessa 14 undertecknare framträder i media, men deras världsledande stamcellsforskning fortsätter mot nya höjdpunkter med stora statliga anslag till utvalda "centers of excellency". Rektorer har fått avgå, men ingen av de forskare som låg bakom rektorernas fatala beslut. Rektor Harriet Wallberg säger i media ungefär att hon blev lurad att först anställa Macchiarini, och så var det nog. Rektor Hamsten kanske också tycker att han blev lurad att förlänga anställningen (2 ggr). Rektorer är ju oftast bara passiva reagenter på underliggande krafter från aktiva forskare eller finansiärer.

Det var inte inkorrekt formalia som ledde till katastrofen, vilket Heckscher menar, utan bristande omdöme hos ledande forskare. Formalia i all ära, men det är till slut omdöme med mer eller mindre korrekt formalia, som är avgörande.   

Om KI representerar svensk forsknings kronjuvel, vad säger detta om övrig svensk forskning?

  

Climate vs Chaos and Turbulence

Both climate alarmists and skeptics like to suggest deep understanding by expressing that global climate is a non-linear chaotic system and as such is unpredictable (as discussed by Kip Hansen in a recent sequence of posts):

• The climate system is a coupled non-linear chaotic system, and therefore the long-term prediction of future climate states is not possible. (IPCC TAR WG1, Working Group I: The Scientific Basis)
• It is my belief that most climate variability and even climate change could simply be the result of chaos in the climate system. (Roy Spencer)

But to simply say that a chaotic system is unpredictable is not the entire story. It is true that point values in space/time of a chaotic system are unpredictable, due to strong pointwise sensitivity to pointwise perturbations, but mean values of a chaotic system typically are predictable.

It is certainly impossible to predict the daily temperature of a specific city within one degree one year ahead, but meaningful monthly temperatures are routinely reported in tourist guides.

The book Turbulent Incompressible Fluid Flow presents the following analysis of turbulence as prime example of chaos:

1. Point values are unpredictable due to local exponential instability.
2. Mean values are predictable due to cancellation of instability effects.

It may thus well be possible (with a high degree of certainty) to predict that the global mean temperature will be the same 100 years from now, within a degree up or down.

For the Lorenz system, as a key example of a chaotic system, it is impossible to predict in which lobe a trajectory will be long ahead in time, but the total time spent in each lobe is observed to become nearly equal over long time. About the weather in Scandinavia, we know for sure that it will be variable with alternating low and high pressures, with sunshine following rain and vice versa as a result of the dynamics. 

Vilka Kunskaper i Matematik Kommer att Behövas i Yrkeslivet?

Huvudargumentet för att öka den redan omfattande undervisningstiden i matematik i den obligatoriska grundskolan, vilket Riksdagen beslutat enligt föregående bloggpost, formuleras på följande sätt:

1. För de enskilda eleverna är det av stor vikt att de får de kunskaper i matematik de kommer att behöva i yrkeslivet eller för fortsatta studier. 
2. Att de har sådana kunskaper är viktigt även för samhället i stort.

Eleverna skall alltså bibringas de kunskaper i matematik de kommer att behöva i yrkeslivet. Men detta  är ju en synnerligen kryptisk närmast logiskt cirkulär formulering.  

Men vilka kan då dessa kunskaper vara? Ett sätt att närma sig denna fråga är att kartlägga vilken matematik som idag de facto används inom olika yrkesgrupper, lämpligen genom en bred enkätundersökning hos dagens yrkesverksamma. Har en sådan undersökning gjorts, och vad var i så fall resultatet?