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Adjectives online worksheet for Grade 4. You can do the exercises online or download the worksheet as pdf.

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Adjectives ending in -ed and -ing interactive worksheet for grade 4. You can do the exercises online or download the worksheet as pdf.

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Participles adjectives ed or ing worksheet

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Adjectives online worksheet for B1. You can do the exercises online or download the worksheet as pdf.

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Liveworksheets transforms your traditional printable worksheets into self-correcting interactive exercises that the students can do online and send to the teacher.

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Reflexive pronoun online worksheet for 8. You can do the exercises online or download the worksheet as pdf.

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Reflexive pronoun online activity for 6th. You can do the exercises online or download the worksheet as pdf.

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Zero Conditional online worksheet for 5ºth Grade. You can do the exercises online or download the worksheet as pdf.

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Zero, first and second conditional interactive worksheet

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Liveworksheets transforms your traditional printable worksheets into self-correcting interactive exercises that the students can do online and send to the teacher.

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Liveworksheets transforms your traditional printable worksheets into self-correcting interactive exercises that the students can do online and send to the teacher.

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Liveworksheets transforms your traditional printable worksheets into self-correcting interactive exercises that the students can do online and send to the teacher.

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Liveworksheets transforms your traditional printable worksheets into self-correcting interactive exercises that the students can do online and send to the teacher.

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Are You Living an Insta Lie? Social Media Vs. Reality

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Deepfakes: Why you can't believe everything you see and hear | Joseph Palermo | TEDxToronto

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Tom Cruise recreated in uncanny deepfake video using AI technology

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This video looks like Tom Cruise talking to camera, but the whole thing is fake - created using a lookalike and computer technology.

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Follow-up questions:

In these kinetic sculptures, how would you describe the roles of the applied sciences and the arts, and their interaction?  What does each contribute?

How does the imagination act as a way of knowing in both the creation and the response to Jansen’s Sandbeasts, Anthony Howe’s wind sculptures or the airport’s intricate play of raindrops.

Which works do you like? Why?

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Kinetic Rain - World's largest kinetic art sculpture @ Changi Airport Terminal 1

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II

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Perfect Kinetic Art - OCTO by Anthony Howe

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What ways of knowing are involved in a response to the “sandbeasts”?

What areas of knowledge come together to create these splendid “beasts” that move with the wind?

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Theo Jansen's Strandbeest wind sculptures

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Theo Jansen's "Strandbeest" sculptures are more like a species of artificial animals than a work of art. Their movements are incredibly life-like. Meaning "beach animals" in Dutch, Jansen's creations move with the wind and are made with plastic tubes and lemonade bottles.

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http://www.johnedmark.com

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Edmark, who lectures in engineering at Stanford, stresses the intersection of mathematics and the world:

“If change is the only constant in nature, it is written in the language of geometry.

“Much of my work celebrates the patterns underlying space and growth. Through kinetic sculptures and transformable objects, I strive to give viewers access to the surprising structures hidden within apparently amorphous space.

“While art is often a vehicle for fantasy, my work is an invitation to plunge deeper into our own world and discover just how astonishing it can be. In experiencing a surprising behavior, one’s sense of wonder and delight is increased by the recognition that it is occurring within the context of actual physical constraints. The works can be thought of as instruments that amplify our awareness of the sometimes tenuous relationship between facts and perception.”

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Jean-Baptiste Michel: The mathematics of history

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What can mathematics say about history? According to TED Fellow Jean-Baptiste Michel, quite a lot. From changes to language to the deadliness of wars, he shows how digitized history is just starting to reveal deep underlying patterns.

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Finding the right mate is no cakewalk -- but is it even mathematically likely? In a charming talk, mathematician Hannah Fry shows patterns in how we look for love, and gives her top three tips (verified by math!) for finding that special someone.

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Prawdziwi geniusze - osoby pokroju Einsteina, Diraca czy Feynmana - pojawiają się na arenie dziejów stosunkowo rzadko. Jednak jeszcze rzadziej rodzi się ktoś o potencjale Ramanujana.

"Żaden matematyk, nie powinien nigdy zapomnieć, że matematyka, bardziej niż jakakolwiek inna dziedzina sztuki czy nauki, jest domeną ludzi młodych."
- G. H. Hardy

W filmie Buntownik z wyboru zawarto interesującą scenę. Próbując opisać tytułowego bohatera, prof. Sullivan porównuje go do pewnego młodzieńca żyjącego sto lat temu w Indiach. Chłopak nigdy nie chodził do szkoły, ale pewnego razu razu trafił na brytyjski podręcznik matematyki. Z tej jednej książki wyciągnął niesamowite wnioski, które rychło wprawiły w osłupienie najznamienitszych uczonych z Europy. Choć brzmi to fantastycznie, ów matematyczny samorodek żył naprawdę, a nazywał się Srinivasa Ramanujan.

Przyszedł na świat w Indiach, w okolicach Madrasu, jeszcze przed epoką Gandhiego. Życie w brytyjskiej kolonii zdecydowanie go nie rozpieszczało, lecz przynajmniej mógł odebrać elementarne wykształcenie, poznać język angielski, a co najważniejsze - zetknąć się ze zdobyczami zachodniej nauki. Punktem zwrotnym w życiu nastoletniego Hindusa było trafienie na zgubiony przez któregoś z wyspiarzy egzemplarz książki Synopsis of Elementary Results in Pure Mathematics, pióra George'a Shoobridge'a Carra. W momencie gdy przeciętny mieszkaniec subkontynentu miał ogromny problem ze zdobyciem kartki papieru, zakochany w liczbach Ramanujan wpadł akurat na publikację uznanego matematyka z Cambridge. Był to mokry całus od losu dla całego pokolenia matematyków.

Zauważyłem, że biografowie często przypisują wydaną w 1880 roku w wysłaniu jego niesamowitych prac aż do Wielkiej Brytanii. Wylądowały one na biurku Godfrey'a Harolda Hardy'ego - wybitnego znawcy teorii liczb i wykładowcy Cambridge. Podobno profesor zignorował trzy pierwsze listy. Postawa zrozumiała: gdy codziennie jesteś bombardowany wypocinami pseudonaukowców i szarlatanów, trudno ci traktować poważnie wieści o przełomowych badaniach jakiegoś chłopaka z Indii o trudnym do zapamiętania nazwisku. A jednak coś tchnęło Hardy'ego, aby w końcu pochylił się nad egzotycznymi obliczeniami. musiał przywyknąć do zachodniego rygoru dotyczącego publikacji naukowych i sposobu zapisu swoich nowatorskich formuł. Synopsis, do literatury popularnonaukowej, o dość podstawowym zakresie. Nie ulegnijcie jednak złudzeniu, jakoby Ramanujan wertował coś na kształt Krótkiej historii czasu czy podręcznika z gimnazjum, po czym wyciągnął zeń niestworzone teorie. Aż tak bajkowo nie było, bo choć dzieło Carra nie wykraczało ponad pewien poziom, to jednak stanowiło dość pokaźny przegląd istotnych zagadnień: począwszy od algebry, przez równania różniczkowe i całki, po rachunek wariacyjny. Podobno autor rekomendował swoją pracę kandydatom przygotowującym się do egzaminów wstępnych na Cambridge. Tymczasem Hindus, który nigdy nie miał szans na odebranie wykształcenia równie starannego co rówieśnicy z Anglii, przyswoił ten wykład w sposób błyskawiczny i bez niczyjej pomocy. Co ciekawsze, wiedza ta okazała się dla niego zbyt ciasna i dalszych odpowiedzi postanowił szukać na własną rękę. Z mocą superkomputera rozpoczął szkicowanie autorskich twierdzeń i wzorów. Rzecz jasna, chłopak nie mógł wiedzieć, które z rozpatrywanych przez niego problemów zostały rozwikłane przez Europejskich uczonych już wcześniej - toteż leciał za porządkiem, poznając królową nauk i jednocześnie poszerzając jej granice. Z rewelacyjnym skutkiem.
Dalej matematyczny wirtuoz podążał śladami niewiele starszego Alberta Einsteina. Podobnie do Szwajcara, Ramanujan znalazł pracę na stanowisku szeregowego urzędnika. I choć uprawianie tak ogłupiających i schematycznych zajęć powinno zostać zakazane, nudne posiedzenia w biurze dawały geniuszom czas na rozwój ich oryginalnych zainteresowań. Również podobnie, przełom w twórczości obu panów przypadł akurat na 26 rok życia. W tak młodym wieku Einstein przyszykował zręby szczególnej teorii względności, zaś nasz bohater otrzymał długo wyczekiwaną szansę wypłynięcia na szerokie wody. Dość dosłownie. Kierownik madraskiej placówki - również zapalony matematyk - poznał się na talencie swego podwładnego i pomógł mu Listy zawierały około 120 twierdzeń, w tym dotyczących hipotezy Goldbacha, problemu Waringa i innych zagadnień bliskich brytyjskiemu matematykowi. Jego szok był na tyle duży, że bez ceregieli postanowił wyciągnąć ręce po analityczny diament i ściągnąć go prosto do siebie.

"Nie posiadam wyższego wykształcenia, ale przebyłem kurs w szkole podstawowej. Po jej ukończeniu, wykorzystałem wolny czas i swoje predyspozycje do prac matematycznych. Nie zostałem spętany przez konwencjonalne metody wpajane podczas zajęć uniwersyteckich, lecz wytyczyłem dla siebie nową ścieżkę. Zająłem się badaniem szeregów rozbieżnych, a wyniki jakie otrzymuję są nazywane przez miejscowych matematyków "zaskakującymi"."
- Fragment listu Ramanujana do prof. Hardy'ego

W ten oto sposób, dwudziestoparoletni obcokrajowiec bez dyplomu, trafił do najbardziej prestiżowej uczelni świata. Sam ten fakt pozostaje ogromnym osiągnięciem, bowiem przybysz z Azji mijający się na korytarzach z Jamesem Chadwickiem, Ernestem Rutherfordem, Arthurem Eddingtonem czy Bertrandem Russellem, był w tym okresie ekstremalnie rzadkim widokiem. Oczywiście nie mógł on liczyć na zbyt wielu przyjaciół. W końcu, kto to widział aby jakiś Hindus bez szkoły, który w życiu nie przesiedział ani godziny na porządnym uniwersyteckim wykładzie, miał być nagle traktowany jako autorytet? Jednak Godfrey Hardy nie znosił sprzeciwu i bzdurnych konwenansów, zwłaszcza jeśli w grę wchodziło dobro nauki.
Dziwny to był duet. Wyobraźcie sobie uczonego o usposobieniu podobnym do Richarda Dawkinsa, którego podopieczny odprawia regularne modlitwy, wypełnia religijne nakazy żywieniowe i wreszcie stwierdza, że jego najlepsze pomysły są owocem natchnienia przez siłę wyższą. A tak właśnie było! Hardy, niespełna czterdziestoletni kawaler, zatwardziały racjonalista, ateista i pracoholik, musiał nauczyć się współpracy z nieobytym zarówno z europejskimi tradycjami jak i metodologią naukową, rodzinnym (żona została w ojczyźnie) i mocno uduchowionym młokosem. Jednak, jak przystało na ludzi, których łączne IQ przebijało 300, w końcu znaleźli wspólny język. Anglik ścierpiał opowieści o tym, jakoby równania spływały do umysłu Ramanujana podczas snów, za sprawą hinduskiej bogini Namagiri; zaś cudowny młodzieniec

Parafrazując słowa fizyka Freemana Dysona, nasiona posadzone przez Ramanujana i Hardy'ego zostały rozdmuchane po całym naukowym krajobrazie i kiełkują do dziś. Przez trzy urodzajne lata, tandem z Cambridge rozpracował właściwości liczb Bernoulliego, twierdzenia Bertranda-Czebyszewa i funkcji theta Jacobiego. W ten sposób do podręczników wkroczyły szeregi Ramanujana-Sato, liczby pierwsze Ramanujana, funkcja theta Ramanujana, funkcja tau Ramanujana, hipoteza Ramanujana-Peterssona i wiele, wiele innych. Nie trzeba posiadać doktoratu aby pojąć, z jak gigantycznym dorobkiem mamy do czynienia.

Dodatek: O co chodzi z kongruencją Ramanujana?

Niech p(n) oznacza liczbę możliwości przedstawienia liczby naturalnej n w postaci sumy liczb naturalnych (kolejność składników nie gra roli). Ramanujan odkrył, że jeśli reszta z dzielenia n przez 5 wynosi 4, to p(n) jest podzielne przez 5. Np. liczbę 4 można przedstawić w postaci sumy na 5 sposobów: 4, 1+3, 1+1+2, 2+2, 1+1+1+1, więc p(4) jest - jak widać - liczbą podzielną przez 5. Dla liczb 9, 14, 19, ... trudno już wyznaczyć wartości p(n). Wszystkie są jednak podzielne przez 5.

Tego rodzaju relacja nazwana została kongruencją Ramanujana. Inne przykłady takich kongruencji, również odkrytych przez Ramanujana (także inni badacze odkryli podobne kongruencje): jeśli n=7k+5, to p(n) jest podzielne przez 7, jeśli n=11k+6, to p(n) jest podzielne przez 11. Zagadnienia związane z liczbą sposobów sumowania pojawiają się w różnych działach matematyki, a także w fizyce.

Gościnnie: Szymon Wąsowicz z blogu Być matematykiem.

Godfrey Hardy lubił zwracać uwagę na napotykane na co dzień numery i dumać nad ich znaczeniem oraz właściwościami. Podobno odwiedzając pewnego razu swojego podopiecznego, poskarżył się, że jego taksówka nosiła wyjątkowo Do legendy przeszła również anegdota dotycząca powstania liczby "taksówkowej" Ramanujana-Hardy'ego. Wybitni matematycy miewają specyficzny humor i dziwaczne rozrywki. nieciekawy numer 1729. Ramanujan odparł, że wręcz przeciwnie. Tak się składa, iż 1729 jest najmniejszą liczbą naturalną, jaką można wyrazić jako sumę dwóch sześcianów na dwa sposoby. Trudno powiedzieć, czy rzeczona scena rzeczywiście miała miejsce, niemniej dobrze oddaje błyskotliwość analitycznego umysłu Ramanujana.

Niestety triumfalny pochód ku międzynarodowej sławie został brutalnie przerwany. Od wiosny 1917 roku stan zdrowia Ramanujana wyraźnie się pogorszył. Dręczyła go bezsenność, wychudł, miewał koszmarne nocne gorączki, problemy z oddychaniem, a narastającym atakom kaszlu zaczęła towarzyszyć krew. Na domiar złego, postęp choroby ułatwiała uboga dieta Hinduisty, dodatkowo uszczuplona niedoborami związanymi z trwającą I wojną światową. Pozbawiony swoich ulubionych warzyw Ramanujan potrafił pracować przez większość doby, bez odpoczynku i pożywienia. Tak dopadła naszego bohatera gruźlica. Dzisiaj rzadka i jak najbardziej uleczalna, w ówczesnych warunkach oznaczała niemal pewny wyrok śmierci. (Chociaż dziś niektórzy lekarze podejrzewają, że tak naprawdę zarzewie problemu leżało nie w płucach, lecz żołądku, gdzie grasowały pełzaki czerwonki). Mimo to, po niemal rocznej tułaczce po szpitalach i sanatoriach, trzydziestolatek niemal wrócił do zdrowia. Przynajmniej na tyle, że zdołał odbyć rejs powrotny do domu. Na miejscu dopadł go nawrót choroby, tym razem ostateczny.

Życiorys Ramanujana pozostaje najlepszą akademicką wersją historii typu "od pucybuta do milionera". Człowiek, który zaczął w nędznej wiosce opodal Madrasu, w niedługim czasie trafił do Trinity College w Cambridge i jako drugi Hindus w dziejach zasiadł w składzie Towarzystwa Królewskiego. A wszystko to w ciągu niesprawiedliwie krótkiego, zaledwie 33-letniego życia.

Literatura uzupełniająca:
M. Murty, V. Kumar, The Mathematical Legacy of Srinivasa Ramanujan, Londyn 2013;
G. S. Carr, A Synopsis of Elementary Results in Pure and Applied Mathematics, Londyn 1880;
A. Janardhanan, A passage to infinity: The untold story of Srinivasa Ramanujan, [online: http://indianexpress.com/article/lifestyle/life-style/a-passage-to-infinity-the-untold-story-of-srinivasa-ramanujan/];
I. Ellis, The Mystery of Srinivasa Ramanujan's Illness, [online: www.todayinsci.com/R/Ramanujan_Srinivasa/RamanujanSrinivasa-IllnessMystery.htm];
A jeśli kogoś interesuje nieco podkoloryzowana biografia Ramanujana, to polecam film Człowiek, który poznał nieskończoność, w reż. Matta Browna.

Naukowy totalitarysta. Jeśli nie chcesz aby wpadli do Ciebie naukowi bojówkarze, zostaw komentarz.

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Najgenialniejszy samouk w dziejach - Srinivasa Ramanujan

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Prawdziwi geniusze - osoby pokroju Einsteina, Diraca czy Feynmana - pojawiają się na arenie dziejów stosunkowo rzadko. Jednak jeszcze rzadziej rodzi się ktoś o potencjale Ramanujana.W filmie Bun­tow­nik z wyboru zawarto inte­re­su­jącą scenę. Próbując opisać tytu­ło­wego bohatera, prof. Sullivan porów­nuje go do pewnego mło­dzieńca żyjącego sto lat temu w Indiach.

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A few months ago the BBC conducted a survey to find the most beautiful mathematical equation in the world. There is of course a long history of the relationship between the idea of beauty and mathematics. The ancients certainly pondered that relationship as well as connecting it to the concepts of truth and of goodness. In Plato's mind certainly, as well as in the mind of others, the true, the good and the beautiful were objectively one in the transcendental realm, whilst they could only be perceived as pale reflections in physical reality. To what degree can beauty and mathematics be associated? What do we mean by beauty? Are beautiful equations always true, and in what sense? Is mathematics the best conceptual as well as concrete expression of beauty? These questions should be a typical part of a TOK exploration of Mathematics, let's see what answers to some of them might look like.

The first step for TOK of course would be an attempt to understand the concept of beauty in the first place. It has become commonplace to quote this tired old cliché that "beauty is in the eye of the beholder." This is to say that the range of things described as beautiful is so vast that one cannot possibly agree on a set of criteria or qualities (tone, symmetry or form, the presence of the Golden Ratio...) by which all things can then be judged. If this is so, then beauty is indeed an entirely subjective concept and as long as one prefaces any statement about individual things having some of the elements of beauty with "In my humble opinion...", then one is perfectly safe from the beauty-objectivists out there. However, Plato has found some support from modern philosopher Charles Sanders Peirce, who concluded that " Logic ( truth) follows ethics ( goodness) and both follow aesthetics ( beauty)" (my italics). In fact Peirce, a logician and mathematician came to the view that beauty should govern all human conduct and thought. Would all or most Mathematicians agree with this?

In an effort to answer this question, a few years ago, British researchers strapped a number of Mathematicians to a functional magnetic resonance imaging scanner ( fMRI), and showed them a number of famous equations. It turned out that many of these equations triggered activity in the brain usually associated with the experience of visual and musical beauty (in field A1 of the medial orbito-frontal cortex, apparently). This happened so consistently over the number of subjects that the researchers concluded that not only was beauty recognised in mathematical equations but, taking other research into consideration, they also triggered activity connected with the beauty of understanding and of moral goodness. Plato's view that the most perfect expression of beauty was in mathematics, and that it is intimately connected to truth and goodness, has therefore been vindicated in a way few people would have anticipated. Sadly, for the poor numerically illiterates out there, they'll have to be content with expressions of beauty in less elevated forms.

But what of the winning equation in the BBC survey? Out of twelve chosen equations, Paul Dirac's equation, which describes how particles behave when they travel close to the speed of light, came out as the clear winner. And what of the equation which so moved those Mathematicians' brains in the fMRI scanner? Well, Euler's Identity garnered 50% fewer votes than Dirac's, so it may well be that, after all, beauty is indeed in the eye of the mathematical beholder.

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A few months ago the BBC conducted a survey to find the most beautiful mathematical equation in the world. There is of course a long history of the relationship between the idea of beauty and mathematics. The ancients certainly pondered that relationship as well as connecting it to the concepts of truth and of goodness.

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Many students will be busy putting the finishing touches to their TOK presentations, here are a few last minute pieces of advice. The planning document is the only evidence the IBO will have of your presentation, make sure it is filled in to ensure the best possible mark. As a matter of good practice, initially try and choose two or three Real Life Situations (RLS) before settling on one of them, not all RLS will necessarily provide good TOK material. Make a note of all the questions raised by the RLS and try and connect the latter to TOK concepts and material (e.g. WoKs, AoKs, Personal/Shared Knowledge...). From those you should be able to create one or more Knowledge Questions (KQ), then choose the one you are most comfortable with or most interested in exploring.

Let's go through the form and highlight the dos and donts of the most important sections of the TOK Presentation Planning Document.

1. The title: It should not be the Knowledge Question but a brief, if possible, memorable one highlighting the main theme of the presentation (e.g. The Pitfalls of Social Media).

2. The RLS: This is the foundation of the whole presentation and therefore must be got right. An RLS can be any event rooted in time and place. Make sure it is described in sufficient detail including the key facts about its context (where, when, who, what, why...). Make sure any source used is acknowledged at some point.

3. The KQ: The KQ should clearly emerge from the RLS and be expressed in general terms, it should not refer to the RLS explicitly (e.g. To what extent does Reason provide more reliable knowledge than Emotion in Ethics?).

4. The Connections: Explain the steps taken from choosing the RLS to the creation of the KQ. The steps suggested in the opening paragraph could be used here. This is about making the connection between the RLS and the KQ as clear and explicit as possible.

5. The Development: This is where the core of the presentation is to be found. There is no need to repeat the RLS or KQ at this point. Make sure you identify the subsidiary KQs (SKQ) you are going to use to develop your ideas, and briefly outline your answer. Also make sure you refer back to your RLS to illustrate your points. This section should be clearly presented, making sure that bullet points/numbers/letters are used to indicate the different sections. Depending on the length of the presentation two SKQs by candidate should be enough.

6. The conclusions: This section should be in three parts. Firstly, outline your conclusion(s) and the reasons for them, make sure you have answered your main KQ. Secondly, explain why these are important or why they matter (e.g. their significance), if possible how they provide interesting insights into your chosen RLS. Lastly, show how your conclusions may be relevant to other RLSs (two should be sufficient). Be careful not to choose RLSs which are identical or too similar to your original one. A good way of doing this is to connect those to WoKs or AoKs not referred to in the core of your presentation.

7. Lastly, make sure your sources are clearly acknowledged.

The planning document is in many ways an outline not unlike an essay plan. Make sure you aim to cover all the key criteria set out in the presentation assessment instrument.

Namely, your outline will show clear connections between your chosen RLS and your KQ. The latter will be explored in detail through relevant subsidiary KQs and the original RLS will be effectively used to illustrate the arguments. An appreciation for the limitations (different perspectives) of your arguments will have been shown. Clear, relevant and pertinent conclusions will be stated with appropriate implications of the latter will be pointed out.

Good luck!

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The TOK Presentation Planning Document - OSC IB Blogs

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Many students will be busy putting the finishing touches to their TOK presentations, here are a few last minute pieces of advice. The planning document is the only evidence the IBO will have of your presentation, make sure it is filled in to ensure the best possible mark.

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"Those who cannot learn from history are doomed to repeat it." Ho hum? Heard this before? In the first place, this famous "quotation" is apparently a distortion-albeit a slight distortion-from George Santayana's actual assertion, "Those who cannot remember the past are condemned to repeat it. " More to the point, the statement raises important questions about the nature of historical knowledge. In fact, it doesn't really matter what a particular philosopher actually said (or even whether, as is often claimed, that it was Edmund Burke who said it first). What matters is that the widely quoted version has captured the imagination of those who accept the vital importance of historical knowledge, not as an end in itself, but for the way it can translate into understanding-and acting on- current social and political forces.

But this isn't enough. Of what is this knowledge supposed to consist? Surely not merely the sequence of incidents! The assertion makes most sense if we understand it to refer to causes and effects. As historian Charles Freeman says, "the excitement of history lies in trying to offer explanations for developments." (TOK Course Companion, p. 283) In this centennial of the start of World War I, the widespread scrutiny of the war's causes thus becomes not just discussion of Kaiser Wilhelm's personality, Serbian resentment of Austrian domination and so on, but more generalized principles of what kind of action leads to what kind of consequence. *
Counterfactual history and causality
A new book casts a provocative light on the nature of historical knowledge-and suggests all sorts of interesting exercises and discussions for a Theory of Knowledge class. In Altered Pasts: Counterfactuals in History, Richard J. Evans examines the phenomenon of so-called "counterfactual histories". For those who might be a little hazy on this phenomenon, a counterfactual history is a kind of "what if" speculation on how big chunks of history would have been completely different if a single key event or decision had been different from what it actually was.

At first blush, this kind of speculation might seem like the kind of hypothesis we expect of most academic disciplines. The "hard sciences", of course, lose much of their purpose if they stop with description of what has happened or is happening. On the contrary, the physical universe can be understood fully only as complex pattern of "if/then" links. (The same is true even for biology, with its complex web or nature/nurture causalities).

But history? In the first place, much is to be gained by simply knowing, descriptively, what has, in fact, happened. In this case, the chief knowledge questions involve knowledge sources, perspectives, and ways of knowing discussed in the Course Companion pp. 276-281. Second, though, when we try to understand cause and effect, though crucial to the most thoughtful approach to history, complexity can explode in our faces. We might think that by asking, "What if?" we gain deeper understanding into why. In an article published the month before the release of his book, (and using the particular example of the first world war) Evans says no:

...this kind of fantasising is now all the rage, and threatens to overwhelm our perceptions of what really happened in the past, pushing aside our attempts to explain it in favour of a futile and misguided attempt to decide whether the decisions taken in August 1914 were right or wrong. For that way, of course, leads not to historical understanding but to all kinds of wishful thinking, every hypothesis political in motivation....
Problems with counterfactual histories
This isn't the only charge that Evans brings against this kind of hunt for cause and effect through speculation:

"Counterfactuals", as such "what-if" speculations are generally termed by the aficionados, are often claimed to open up the past by demonstrating the myriad possibilities, thus freeing history from the straitjacket of determinism and restoring agency to the people. But in fact they imprison the past in an even tighter web: one tiny change in the timeline - Archduke Franz Ferdinand escapes assassination in Sarajevo, the British cabinet decides not to enter the war - leads inevitably to a whole series of much larger changes, sometimes stretching over decades almost up to the present day.

Although he doesn't say so directly, he seems to be suggesting that counterfactual histories are a kind of butterfly effect/chaos theory in reverse. Remove that fluttering butterfly with a snop of a hungry bird's beak, and what would happen to that hurricane on the other side of the world? "Calm as a millpond", would apparently be the answer for a counterfactual historian.

More to the point, Evans argues that counterfactual histories ignore the importance of other random events in history: "...this ignores, of course, an infinite number of chances that might have deflected the predicted course of events along the way -". If we apply Evans' point to our hapless butterfly, then it seems that even if it does become a tasty dish for a starling before it flutters its hurricane-inducing wings, there are other butterflies to take its place-or, even more appropriately, other sources of initially tiny air turbulence.

In addition, as Evans points out, historians actually attempt to look at a complex web of causes and effects: "every historian tries to balance out the elements of chance on the one hand, and larger historical forces (economic, cultural, social, international) on the other."

A third objection to counterfactual history is that it seems to be shackled to two narrow and outmoded kinds of historical thought. First is the "great man" view of history, long defunct, and second is the "kings-and-battles" summation of the past.

Even given Evans' objections, we might guess it is more in the practice of counterfactual history than in its essence that these previous two limitations are evident. Arguably, the same is true of his fourth objection, that it is mostly churned out in the form of "reactionary fantasy"-where 'what if' counterfactual history is really little more than 'if only'".

Or, more put more harshly, "Counterfactuals are the byproduct of a paranoid nostalgia specific to the right, where "greatness" is forever being sabotaged by leftists, liberals or whoever."
Using counterfactual history to illuminate history as an AOK
A discussion in TOK class could well involve not only these four objections, but also some others, and ones that beautifully combine WOKs and particular claims of history as an AOK.

After all, as sharp students will point out, "What if" is a question absolutely fundamental to the "hard sciences" and, perhaps to a lesser extent, the human sciences. Forming a hypothesis, they will point out, is what the sciences are all about. What they are also all about, though, of course, is testing the hypotheses.

And the very fact that historical hypotheses are largely untestable is the main reason history is separated from the human sciences as an AOK. That is, they are not testable by replicating the events (as closely as is possible with variable human beings) and making further observations! And yet - in history, interpretation of the past is still constrained by the evidence. We cannot re-run a specific battle, a particular plague, or the development of a farming technology in order to take better notes. However, the evidence that has come down to us in documents or artifacts provides the basis for any interpretation of the past; it must not be "imagined away".

But what might substitute for testing a hypothesis in a counterfactual history? When the real events of the past are indeed "imagined away" - with the substitution of events as they did not happen - then the hypothesis conflicts with the evidence. It is false. When a sequence of counterfactual consequences then follows as a result of a false account of the past, then we have left the realm of historical knowledge.

Nevertheless, counterfactual histories may usefully illuminate the way that history is constructed, and also illuminate the nature of specific historical events.
Counterfactual histories as a popular trend
Why-other than to push ideologies-are such counterfactual histories suddenly so popular? Evans speculates that much strikes to the heart of fundamental questions of shared knowledge:

Perhaps it's because we're living in a postmodern age where the idea of progress has largely disappeared, to be replaced by uncertainty and doubt, and where linear notions of time have become blurred; or because truth and fiction no longer seem such polar opposites as they once did; or because historians now have more licence to be subjective than they used to.

In terms of TOK what will probably stand out most are two issues that raise knowledge questions:

1. the contemporary blurring of truth and fiction-an idea that Evans expands upon in a podcast discussion of his book, where he singles out the spurious knowledge claims so common on the internet.

2. the trend towards increasing subjectivity in writing history. His comment becomes a reflection on this particular AOK and how, according to Evans, it increasingly combines researched shared knowledge with personal, ideological, or cultural perspectives of the historian. (See the Course Companion pp. 285-287)

In this anniversary year of the First World War, students and teachers alike are likely to come across lots of shows, articles, re-enactments, and speculations. In fact, given the current popularity of all counterfactuals, they are likely to come across many other areas of history making a "What if" claim on their credulity. Whether in the context of class-history class or Theory of Knowledge class-the fortunate I.B. student will be well equipped to react thoughtfully and analytically to such claims.

What if?
References http://www.theguardian.com/books/2014/mar/13/counterfactual-history-what-if-waste-of-time http://www.theguardian.com/books/2014/apr/17/altered-pasts-counterfactuals-in-history-review http://www.timeshighereducation.co.uk/comment/podcasts/ 28 March, 2014
Photo Credit: Robin Bain via Compfight cc

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TOK and Counterfactual History - OSC IB Blogs

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"Those who cannot learn from history are doomed to repeat it." Ho hum? Heard this before? In the first place, this famous "quotation" is apparently a distortion-albeit a slight distortion-from George Santayana's actual assertion, "Those who cannot remember the past are condemned to repeat it.

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Real life situations for history

Although the 'heart' of TOK is the concept of the knowledge question, you explore these questions - in both the essay and the presentation - via real life situations. These can be either experiences that you have had yourself (in DP lessons, during CAS projects, and events that have happened outside the school), or issues that you've read about, and that are going on in the world right now.

Good real life situations should be engaging, up-to-date, original, and lead on naturally to questions about knowledge. They should also relate to one or more ways of knowing and areas of knowledge. Finally, they should also help to answer one of our 8 Big Questions. This is our framework for the exploration of the TOK course, and acts as another lens (along with the WOKs and AOKs) through which we can view and focus on what's going on in the world.

Our newsletter for TOK provides teachers and students with real life situations taken from a massive range of sources from around the world. You can subscribe to the free version of the newsletter here, giving you access to 8 new RLSs every month. If you'd like to 'go pro', and receive the premium newsletter, giving you access to 15 RLSs in depth, and 5 further 'quick' RLSs, then choose one of our memberships. These also provide you with innovative and engaging lesson resources for TOK, that combine with the newsletter to create an incredible experience for students.

We've picked out a few examples of the sort of real life situations for history that have featured in the newsletter; note that in the premium newsletter, we also provide an outline of the RLS, more KQs, and a guide for how to analyse each story in more detail.

Links coming soon!

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Although the 'heart' of TOK is the concept of the knowledge question, you explore these questions - in both the essay and the presentation - via real life situations. These can be either experiences that you have had yourself (in DP lessons, during CAS projects, and events that have happened outside the school), or issues that you've read about, and that are going on in the world right now.

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Pictures That Reveal The Truth Behind Photography

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SENSE PERCEPTION

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Are You Living an Insta Lie? Social Media Vs. Reality

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HUMAN SCIENCE EXPERIMENT

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Brown eyes and blue eyes Racism experiment (Children Session) - Jane Elliott

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Jane Elliot Experiment with 3rd graders following the Assasination of Reverend Dr. Martin Luther King, Jr.

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Best Scene of "The Theory of Everything"

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ISLAMIC ART

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An introduction to mathematical theorems - Scott Kennedy

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View full lesson here: http://ed.ted.com/lessons/scott-kennedy-how-to-prove-a-mathematical-theory Euclid of Alexandria revolutionized the way that mathematics is written, presented or thought about, and introduced the concept of mathematical proofs. Discover what it takes to move from a loose theory or idea to a universally convincing proof. Lesson by Scott Kennedy, animation by Karrot Animation.

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5 Riddles Popular on Logic | To Test Your Brain

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Task 8.         ANSWER THESE QUESTIONS:                                                 To what extent do the explanations for the Monty Hall Problem, outlined in the videos, hold up as proof?                                                     In the Monty Hall "hands-on" experiment in the second video, there were twenty repetitions. Notwithstanding a fairly large sample size, the actual results were an exaggerated caricature of the expected results. What was going on?                                                                                        To what extent are probability and certainty in mathematics mutually exclusive?            

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Task 8.         ANSWER THESE QUESTIONS:                                                 To what extent do the explanations for the Monty Hall Problem, outlined in the videos, hold up as proof?                                                     In the Monty Hall "hands-on" experiment in the second video, there were twenty repetitions. Notwithstanding a fairly large sample size, the actual results were an exaggerated caricature of the expected results. What was going on?                                                                                        To what extent are probability and certainty in mathematics mutually exclusive?            

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As promised in a previous post, here's some math explanation behind the Monty Hall Problem.

Questions or thoughts? Shoot me an email via the contact page or comment below.
Share the knowledge! Tweet this!
This post may contain affiliate links. When you use them, you support us so we can continue to provide free content!

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The Math Behind the Monty Hall Problem - MathFour

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As promised in a previous post, here's some math explanation behind the Monty Hall Problem. Questions or thoughts? Shoot me an email via the contact page or comment below. Share the knowledge! Tweet this! This post may contain affiliate links. When you use them, you support us so we can continue to provide free content!

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To the untrained eye, these vivid images might appear to be random sets of colourful swirls and circles.

But they are in fact precise visual representations of mathematical theory known as dynamical systems.

Some of the images - created by mathematicians from across the world - have gone on display at the University of Liverpool.

Here, mathematician Lasse Rempe explains how they are made - and considers their artistic merits.

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BBC NEWS | Science & Environment | Audio slideshow: The art of mathematics

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To the untrained eye, these vivid images might appear to be random sets of colourful swirls and circles. But they are in fact precise visual representations of mathematical theory known as dynamical systems. Some of the images - created by mathematicians from across the world - have gone on display at the University of Liverpool.

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How can technology help us to extend mathematical knowledge?

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Conrad Wolfram: Teaching kids real math with computers

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KQ- the arts
What is the role of beauty in mathematics?

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Brain scans show a complex string of numbers and letters in mathematical formulae can evoke the same sense of beauty as artistic masterpieces and music from the greatest composers.

Mathematicians were shown "ugly" and "beautiful" equations while in a brain scanner at University College London.

The same emotional brain centres used to appreciate art were being activated by "beautiful" maths.

The researchers suggest there may be a neurobiological basis to beauty.

The likes of Euler's identity or the Pythagorean identity are rarely mentioned in the same breath as the best of Mozart, Shakespeare and Van Gogh.

The study in the journal Frontiers in Human Neuroscience gave 15 mathematicians 60 formula to rate.

One of the researchers, Prof Semir Zeki, told the BBC: "A large number of areas of the brain are involved when viewing equations, but when one looks at a formula rated as beautiful it activates the emotional brain - the medial orbito-frontal cortex - like looking at a great painting or listening to a piece of music."

The more beautiful they rated the formula, the greater the surge in activity detected during the fMRI (functional magnetic resonance imaging) scans.

"Neuroscience can't tell you what beauty is, but if you find it beautiful the medial orbito-frontal cortex is likely to be involved, you can find beauty in anything," he said.

A thing of great beauty

To the untrained eye there may not be much beauty in Euler's identity, but in the study it was the formula of choice for mathematicians.

At first you don't realise the implications it's a gradual impact, perhaps as you would with a piece of music and then suddenly it becomes amazing as you realise its full potential.
Prof David Percy, Institute of Mathematics and its Applications

It is a personal favourite of Prof David Percy from the Institute of Mathematics and its Applications.

He told the BBC: "It is a real classic and you can do no better than that.

"It is simple to look at and yet incredibly profound, it comprises the five most important mathematical constants - zero (additive identity), one (multiplicative identity), e and pi (the two most common transcendental numbers) and i (fundamental imaginary number).

"It also comprises the three most basic arithmetic operations - addition, multiplication and exponentiation.

"Given that e, pi and i are incredibly complicated and seemingly unrelated numbers, it is amazing that they are linked by this concise formula.

"At first you don't realise the implications it's a gradual impact, perhaps as you would with a piece of music and then suddenly it becomes amazing as you realise its full potential."

He said beauty was a source of "inspiration and gives you the enthusiasm to find out about things".

Mathematician and professor for the public understanding of science, Marcus du Sautoy, said he "absolutely" found beauty in maths and it "motivates every mathematician".

He said he loved a "small thing [mathematician Pierre de] Fermat did". He showed that any prime number that could be divided by four with a remainder of one was also the sum of two square numbers.

So 41 is a prime, can be divided by four with one left over and is 25 (five squared) plus 16 (four squared).

"So if it has remainder one it can always be written as two square numbers - there's something beautiful about that.

"It's unexpected why should the two things [primes and squares] have anything to do with each other, but as the proof develops you start to see the two ideas become interwoven like in a piece of music and you start to see they come together.

He said it was the journey not the final proof that was exciting "like in a piece of music it's not enough to play the final chord".

He said this beauty of maths was missing from schools and yet amazing things could be shown with even primary school mathematical ability.

In the study, mathematicians rated Srinivasa Ramanujan's infinite series and Riemann's functional equation as the ugliest of the formulae.

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Brain scans show a complex string of numbers and letters in mathematical formulae can evoke the same sense of beauty as artistic masterpieces and music from the greatest composers. Mathematicians were shown "ugly" and "beautiful" equations while in a brain scanner at University College London.

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Eugene Wigner said “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.” This quote pulls out several aspects of this created versus invented discussion. All phenomena observed in classical electricity and magnetism can be explained by means of just four mathematical equations (Livio). In some cases, Math seems to be so perfectly in-line with natural reality that it must be discovered. Whereas there are other examples of Math where there is no clear relation to reality, suggesting that math is invented. But Mario Livio explains that, "mathematicians [have] developed abstract branches of mathematics with absolutely no applications in mind; yet decades, or sometimes centuries later, physicists discovered that those theories provided necessary mathematical underpinnings for physical phenomena." This again suggests that mathematicians are discovering math.

Also, some theories (explanations of reality) can be mathematically correct, but not correct in the real world. Zeno's Paradox is an example of this.

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"The abstract science of number, quantity, and space, either as abstract concepts (pure mathematics), or as applied to other disciplines such as physics and engineering (applied mathematics)." (Oxford Dictionary)

"The science of numbers, quantities, and shapes and the relations between them." (Merriam Webster Dictionary)

"The systematic treatment of magnitude, relationships between figures and forms, and relations between quantities expressed symbolically." (Dictionary.com)

"The study of the measurement, properties, and relationships of quantities and sets, using numbers and symbols." (The Free Dictionary)

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"Mathematics is the abstract key which turns the lock of the universe" (John Polkinghorne).

"Everything that can be counted does not count. Everything that counts cannot be counted" (Albert Einstein).

"A mathematician is a machine for turning coffee into theorems" (Paul Erdos).

"When you have satisfied yourself that the theorem is true, you start proving it" (Arthur Koestler).

"If mathematics describes an objective world just like physics, there is no reason why inductive methods should not be applied in mathematics just the same as in physics" Kurt Godel).

"On each decision, the mathematical analysis only got me to the point where my intuition had to take over." (Robert Jensen)

"Mathematics is the most beautiful and most powerful creation of the human spirit." (Stefan Banach)

"The essence of mathematics lies in its freedom" (Georg Cantor).

"The mathematics is not there till we put it there" (Arthur Eddington).

"Logic and mathematics are nothing but specialised linguistic structures" (Jean Piaget).

"Film is one of the three universal languages, the other two: mathematics and music" (Frank Capra).