MISCONCEPTIONS

Familiarity has a lot of influence on our prior knowledge before learning something; we usually have more knowledge in fields we encounter on a regular basis. Often it is hard for us to accept the fact that while some things come as a no brainer for us, it might not like that for other people.

Want an example? When I was in high school I nearly got into a fight with my friend because she won’t accept the fact that people in some countries cannot go the beach because they do not have one.  She thought that every country must have beaches. Now for some of us, when confronted with questions like this it might come as automatic to simply summon a model of earth inside our mind and imagine the middle part of continents which leads to identifying Mongolia, all those countries with –tan in Central Asia, Zimbabwe, Sudan and such,  Austria, Switzerland, etc. They are called landlocked countries, by the way. It just seems like such a basic idea that we often get hopeless and even frustrated when people do not view it the same. But some people really cannot wrap their heads around it – either because they understand it differently or even never gave it a thought in a first place. This brings us to misconception.

(well because I learn about this today and this childhood memory somehow popped up into my mind)

 LEARNING FAILURES AND HOW IT HAPPENS

When knowledge is acquired, concepts in regards to that knowledge as well as relationship between said concepts are consequently changing. This process is called conceptual change and it happens continuously over the course of our lifetime.

Some theories have arisen to explain how conceptual change happens; most put a lot of emphasis on prior knowledge since it is the foundation the future knowledge is built upon. One of them is theory of conceptual change in learning physical world, by Vosniadou (1994). She distinguished two kinds of prior knowledge; naïve framework theory and specific theory. These are not concepts actually; they are more like structures the concepts are embedded in. Framework theory is believed to have been built in infancy, comprising entrenched presuppositions about the epistemological and ontological nature of the reality around us. Specific theory, in the other hand, is kind of self-explanatory; it is more specific and explains the nature of the conceptual domain within which the concepts are embedded. Hence to put it simply, framework theory stores concepts that are fundamental and serve as basic understanding of the world around us, while specific theory is much more specific (you don’t say) and narrow.

Vosniadou (1994) also differentiated two kinds of conceptual change; enrichment and revision. Enrichment happens when we simply put the new knowledge on top of the old ones, while revision occurs when we have to make adjustment, either to our presuppositions or relationship between concepts, in order to fit our new knowledge in. Both can happen at the level of specific theory or framework theory.  Enrichment usually does not cause any major riot in someone’s mind, as well as revision at the level specific theory. But if the revision happens at the level of framework theory, shit just got real. It is considered the most difficult since it involve changing someone’s fundamental understanding of the world, which is prone to misconception.

Misconception is one of the failures of learning which happens when we fail to assimilate our knowledge to our old ones. Truth to be told though, it is not the only form of learning failures. There are two others, inconsistency and inert knowledge (Vosniadou, 1994), although misconceptions turns out to be the most popular term.

Inconsistency happen when the conflicting pieces of information are simply added up on top of the existing knowledge. So even though a kid has learnt about earth rotation, for example, when being asked about day/night cycle they still use their prior knowledge which is “it goes behind the mountain”. Inert knowledge happens when someone learns something that has nothing to do with their prior knowledge and the information ends up not getting used. For example, you procrastinate until the day before test and end up cramming everything to your head without trying to make sense of everything. An hour after the test, you totally forgot everything (I so can relate to this). Misconceptions, on the other hand, are triggered when we try to reconcile the inconsistent piece of information, producing a mental model that stands in the middle; it is not quite right yet, but it is getting there.

One of the most common misconceptions is the Earth, especially the shape. Constructing correct mental model of earth is especially difficult for children because it violates the presuppositions of the naïve framework theory within which the concept of earth is embedded (Vosniadou & Brewer in Vosniadou, 1994). To be specific, children often think of Earth as a physical object, rather than astronomical, and apply to it all the prepositions that apply to physical objects in general. Two of these presuppositions are 1) space is organized in terms of the directions of up and down with respect to a flat ground, and 2) any unsupported objects fall to this ground. During the conceptual change, information about earth which is contradictory to these presuppositions is assimilated into the existing conceptual structure creating misconceptions.

Below are some graphic representations that depict children’s conceptual change regarding this topic, done in various studies across the world (Vosniadou, 1994). 

                Children’s earlies model of the Earth usually portrays it as a flat disc or rectangular object with the sky, stars and the sun floating above it. This model is heavily influenced by children’s own observation of the universe, which conceived in the presuppositions explained above. This is their presuppositions; it is not a misconception yet.

      This is what happens when children try to reconcile their deeply embedded presuppositions with the culturally accepted knowledge. The dual earth portrays 2 ‘earth’; one is flat where they live and do stuff, the other is up in the sky being round like other people said. This shows that the children are not ready to give up their previous knowledge and instead forcing the new ones at the end of it. The hollow sphere model commences when the children give up their presumption that Earth needs to be supported, but have not yet comprehend how people can walk on a ball and do not fall off the surface. The flattened sphere model happens when the children have embraced the concept of gravity, but still believe that the ground people walk on is flat. This is the misconceptions.

 

 This is the scientifically correct one. Children have to go a long way before they arrive at this point; some go through shorter journeys that others, other may will never be there. 

POPULAR OPINION = POPULAR MISCONCEPTION

The misconception of the shape of the earth is one of the most popular misconceptions in children – others are day/night cycle and seasonal chnge. I still don’t know how to categorize my friend though. We live in Indonesia, which has the longest shoreline in the world (I think), hence beaches are usually perceived as congenital. We got our first extensive geographical science in middle school, if I’m not mistaken; we discussed countries, their distinctive traits and drew maps, therefore the fact that some countries are located in the middle of their respective continent and therefore do not have shoreline had to be exposed to her at some point. Maybe she failed to connect the knowledge she learn to what she already know, which is a case of inconsistency, or maybe she had some sort of weird mental model of earth that somehow allows every countries to have beaches, which is a case of misconception. Either way, I’ll never find out.

Aside from that, misconceptions do happen a lot and this term is mostly assigned to an idea that is wrong yet widely believed by people. In a non-scientific way, misconception is defined to be ‘a view or opinion that is incorrect because based on faulty thinking or understanding.’ They range from small, daily facts to great ones that are strong enough to determine the storyline of the history. I think it is mostly because scientific ideas are hard to apprehend, and sticking with prior knowledge or constructing an alternative knowledge are much easier. Below is a list I made of misconceptions which are either too popular or fatal, or simply surprising to me.

• Tomatoes are not fruit, they are vegetables, Cinderella never wore glass slippers, Napoleon Bonaparte is not that short, and Marie Antoinette never said ‘Let them eat cake.’
• The Great Wall of China cannot be seen from the Moon; the air pollution can though. Shame on you, China.
• Hindu caste system. Oh God, this pisses me off so bad I cannot explain. I’ll have to write a separate post on this one.
• During the Black Death, Europe was still engulfed in superstitiousness and believed that black cats are the root of all evil, including the plague; therefore they had to be destroyed. What they did not know is the plague is caused by pes bacteria, which is carried by rat. The result? The population of rat exploded and the world witnessed the most devastating pandemics in human history during when Europe lost 30-60% of its total population. Whats with that, huh.
• The Evolution Theory by Charles Darwin is not about monkeys turning into human – this is the most common misconception about Darwin’s theory. Darwin never actually said this, nor will any respectable biologist. This myth was actually spread by religious zealots during the 19^th century in order to try and discredit Darwin and promote anti-evolutionism among the religious. Here, try reading this.
• Homosexuality is not a choice. First, I myself did not choose to like boys, so did other straight people. Secondly, why would people choose to be homos?
• Humans do not have 5 senses; we actually have 17.
• The term third world country does not refer to a country that is poor or underdeveloped. It is actually a term invented during WWII for countries that were neither capitalist nor communist.

... and many more. Interesting, isn't it. So never, ever, believe in something just because other people believe the same thing. Misconceptions can kill, people.

Sources:

Vosniadou, S. 1994. "Capturing and Modeling the Process of Conceptual Change."  Learning and 

                 Instruction, Vol. 4, pp. 45-69. Great Britain: Elsevier Science.

MATH JOKES YAY

Educating through the Game Fifteen

Nine cards are marked with digits 1 through 9 and placed face-up on a table between two players. The players take turn picking one card at a time. The winner is the first player to get the sum of exactly 15 from among any three of his or her cards.

  

  This game was proposed by our professor Marteen during our first meeting, as an early morning mind-exercise.  It was fun and quite a work out for brain, but he made it harder for us by demanding what he called as “the winning strategy.” Basically, he asked us whether or not there is a strategy that can ensure us to always win this game, and if such strategy exists, to find it. We worked in group of 5 people, and none of us could come up with that kind of strategy. I did note the following cases though:

1.    We made a list of all possible combinations of numbers that sum up as 15, in which 5 has the most frequent appearance. So we used that as strategy and always started with 5, but it did not ensure us to always win the game.

2.   There was this check-mate like situation in which whatever cards your opponent pick, you will always win. For example, I start first and I pick 8. You pick 4. Then I choose 6. You definitely have to pick 1 to prevent me from winning with 8+6+1=15. Then I choose 2. Now I will always win since I have 2 opportunities: 8+2+5=15 and 6+2+7=15, while you only have 4 and 1. The problem is to find out how to recreate such situation. 

3.    When our opponent picked a certain card, it was almost automatic for us to do quick calculation and pick a card that would prevent them for getting the sum of 15, therefore blocking them. It was only when Marteen brought it to our attention that we realized that it was not us being smart, but our opponent controlling us. 

It turned out that there is a winning strategy, only slightly went out of what it is defined to be.

MAGIC SQUARES AND TIC TAC TOE

First thing though, I am not digressing. This actually has something to do with what I was talking about. 

Magic square is an arrangement of numbers (usually integers) in a square grid, where the numbers in each row and in each column, and the numbers in the forward and backward main diagonals, all add up to the same number.  The size of a magic square is denoted by its number of columns and rows; magic square with 3 rows and 3 columns is said to be of size 3x3. There are 9 grids and the numbers in each row and in each column as well as the numbers in the diagonals, all add up to 15.

It is easy to see that playing this game here is like playing tic-tac-toe on a magic square.  Now we just have to ensure that we can always win in a tic-tac-toe.

DA WINNING STRATEGY: DOES IT EXIST?

Winning strategy in general means just like what Marteen said. But in tic-tac-toe, there are no sure-win strategies; a winning strategy is simply one that will maximize the chance of a player winning (Yeo, 2012). So you will either end up a winner or in a draw. Some winning strategies for tic-tac-toe follow. 

Imagine a tic-tac-toe game where player 1 uses X and player 2 uses O.

Case 1: player 1 starts in the center. The game will always end in a draw, assuming each player makes optimal moves.

Case 2: player 1 put the first x in a corner. If player 2 does not put the first O in the center, they will lose. For example, if player 2 put the first O in the corner opposite of the first X, player 1 can place second X in any of the remaining corners. There are two choices right? Player 2 will be forced to put an O in between two Xs to prevent player 1 from winning. Player 1 can win by placing an X in the last corner.

It is easy to figure out how to win if player 2 put the first O in any for the edge squares.

Case 3: player 1 put the first x in a corner. If player 2 put the first O in the center, there is still a way to win, but only if player 1 puts the second X at the corner opposite the first X and player 2 puts the second O in any of the remaining corners.  Then player 1 can win by placing the third X in the last corner.

Case 4: player 1 put the first X in an edge-square. If this is the case, then the game is wide open. Either player can win, although the game will most often end in a draw. Below are example of this case.

Therefore, the winning strategy for player 1 is to place the first X at the corner. For player 2, pray that player 1 will not start at the corner and if they do, put the first O in the center. 

Back to game we were talking about before, it is clear that any player can win the game by following this tic-tac-toe strategy on a magic square. That is why we never won when we started with 5, because 5 is located at the center. Although it will take extra effort since the players must visualize the magic square; otherwise, opponents will be aware of the strategy.

WHAT DOES IT HAVE TO DO WITH MATH EDUCATION?

Yeo (2012) introduced this game as Fifteen and stated that it can help students learn mathematics and develop heuristic skills such as examining all possible scenarios and systematic listing, spatial visualization, and thinking skills, including predicting, conjecturing, generalizing and checking. The question is, to children of what age should we introduce this game? 

This game is exciting because it requires the players to work his best and develop a strategy to win the game. But not every student will employ that particular mental action against the game. How children respond to games is apparently linked to their developmental levels. Kamii and Nagahiro (2008) summarize these developmental levels as follows. 

- Level 0: the child does not try to win. They are focused on doing what they are supposed to do, here by getting a sum of 15.
- Level 1: the child tries to win. They know they are supposed to get a sum of 15 and do it before the other player does.
- Level 2:  the child tries to block the opponent.
- Level 3: the child is at the most advanced level; they compare the consequences of winning the game now with blocking the opponent or foresee that the opponent has two possible ways of winning.

Most 4-year-old are at level 0, while 6 and 7-year-olds are generally at the highest level. Therefore, it will best to introduce this game to students in 6^th grade or higher. 

Aside from all those technical matters, we teachers know that most students love to play games and from the beginning I already though that this game will be a fun addition in the classroom. Fifteen is an interesting and thought-provoking game that helps students learn mathematics at the same time.

Sources:
Yeo, J.B.W. 2012. "Fifteen: Combining Magic Squares and Tic Tac Toe". Mathematics Teacher, vol 106, no. 1. USA: NCTM.

Kamii, et. al. 2008. "The Educational Value of Tic-Tac-Toe for Four to Six-Year-Olds". Teaching Children Mathematics. USA: NCTM.

Review: the Act of Killing (2012), From an Indonesian's Perspective

Director : Joshua Oppenheimer
Genre    : Documentary
Rating   : 95% (Rotten Tomatoes)

I have heard about this movie and the hype surrounding it months ago, but did not have the will to download and watch it, thanks to the endearing internet speed of Surabaya. But then Jack Harries recommended it and I knew I just have to watch it.

ermahgerd look at all the awards

 The Act of Killing is a documentary movie about the chaotic genocidal period following the coup d’etat against Indonesia’s first president and founding father, Soekarno, during the 60's. It is directed by Joshua Oppenheimer and nominated for last year Academic Award for Best Documentary.  Yep, the dark history of my motherland is now bared all for the whole world to see – well, the whole world who bother to see this kind of stuff. It follows Anwar Congo, one of the main executioners of PKI, as he describes and reenacts the memories of his past life as a ‘jagal’ (butcher) together with his old friends and affiliates. For your information, PKI is how people used to call the communist party back then, as well as marginalized people and Chinese ethnics. 

That’s it? Well, basically yes. Oppenheimer himself said that this a 'documentary of imagination' - we are going to see nothing that directly depicts all that murders and massacres, aside from Congo’s recalling, retelling and reenactment. 

The first thing that crossed my mind is now I know what German people feel. I know about G-30S/PKI (that’s how we call it – since it is started at September 30, 1965), we did learn about it in grade school, but like Pak Adi (one of the character in the movie) said, the history is told and written from the winner’s point of view. Back then, I only had the history book our teacher gave us. But then I grew up and started to have interest in history. I read a lot about World War 2, African Civil War, and other stuff, as well as movies, but never once it crossed my mind to open the history book of my own country. 

G-30S/PKI, there is no other way to say it, is a violation of human rights. It is cruel, sadistic, illogical, and unfair. It is basically mirror image of what has been happening throughout the world since the dawn of the humankind; one group of people think they are right, better and above the rest, therefore they have the right to judge, punish, kill and well, basically act as a god. I am talking about the people doing the killing here; in the grand scheme of things, greater reason may exists. But these people, regardless of what these massacres actually evolved from, definitely do not think that what they did is wrong. 

Now, years later, after the media are no longer doing their role to bias our opinion, we can look back with clean logic and know that it was wrong. But they can’t. They are standing in the winner box of the history and clearly do not feel like getting out anytime soon. Every sick thing they do is not only right; it is respectable, heroic and even honorable. It is just so disturbing the way they boast about what they did without even a slightest hint of regret.

Also, they are so stupid. Non-Indonesian might not know this since it is not recognizable with English subtitle, but when you are Indonesian and you have to listen to the way they talk and see the way they carry themselves … god. 

Especially the way they treat the Chinese ethnics. For all of you who don’t know, Chinese people have been the black sheep of our society since God knows when. Most Chinese people in our country totally fulfill their stereotype as hardworking, good with counting, and stingy people. They, of course, are rich as hell. Indonesians, of course, envy them. What did we do when we envy people for their money? Definitely not working as hard as or even more than them. We hate them, that’s what we do. And from the moment we hate people, all reason to  segregate, mistreat and banish them from the universe suddenly seems fair. 

Yay. Isn’t it awesome being a part of people like this.

You know what, I am Indonesian and I am fully aware of how outsiders perceive us as a nation. I used to be surprised and a little bit pissed off because well, I did not feel that way. Of course, we are not Netherlands-perfect, but we are doing well; I’m not hungry or being forced to quit school and marry someone 40 years older. But watching this movie, it hits me so hard that maybe my country is actually just as much of a third-world country as everyone think; I just do not see it because I have been living in a frickin buble.  

I thought my homeland, the place where I was born, is better than this piece of shit. 

But you know what, maybe I am wrong. Maybe my country IS better than this. Who knows right? I am feeling this way because I am watching this movie, and this movie is created by a director; a human. He took control on what he put in the movie and who knows if he told 100% truth. We all know the power of media. All that matter is everything comes back to the single individuals. You can’t just bunch people into one bundle of trash and throw it into the garbage. Nice people exist and even though they might never be able to fix a country, at the very least they won’t turn into one of those mindless criminals. 

The movie end with Congo being eaten by the nightmares of his past life and having a mental breakdown so severe he retches on the floor where he used to kill his victims. One thing I like about this movie is that the director is kind enough to put a human face on top of this unspeakable, shameful crime. Congo’s regret definitely healed me a bit from all those traumas.

Trivia: Joshua Oppenheimer is a foreigner but can speak Indonesian. He acts as interviewer throughout the movie. One of the co-director is credited to an anonymous Indonesian. In fact, most of the crew is credited as anonymous.The film is not granted release in Indonesia, for fear that it might provoke something in society. None of the people I know knows about the existence of this movie.

Spooky isn't it.

Chocolate Oatmeal Mug-Cake

When I first moved to Utrecht, I was pretty excited about our kitchen. It’s a standard middle-class European kitchen, but for me it’s awesome. It has microwave, oven, fridge, stove – I usually had to make do with just a stove. 

To be noted though, cooking is not really my hobby. I just like doing it. I also don’t like cooking food that takes a long time to make, that’s why I tend to pursue recipes that are fast and easy.

Now, meet my Chocolate Oatmeal Mug Cake.

It doesn’t look very appealing, I know, but it taste pretty good. This is especially for breakfast, because it is fulfilling and easy to make. Basically, it’s a regular oatmeal-and-milk version 2.0. The recipe is very flexible, you can make substitute and use everything according to your liking. Basically, try with equal parts of wet and dry stuffs, and then adjust accordingly. In this recipe I used half a mug of milk and half a mug of oatmeal. 

1.    Wet stuff: Get a nice big mug. Crack in an egg and half a mug of your preferred milk. Mix well with fork.

2.   Dry stuff: Get another cup. Combine half a mug of oat, a quarter teaspoon of baking powder, a pinch of salt, a teaspoon of cocoapowder and sugar, if you like your breakfast sweet. You can adjust the amount of sugar according to your taste, but use more than what you use when you make oatmeal, because the egg fade out the sweetness a lot.  

3.    Mix the stuff: Add half the dry mix to the wet mix, stir well, then add the rest and stir again. Think cake batter, not cookie mix – if it looks too thick, add another splash of milk. 

4.    Cook the stuff: Put it in the microwave. I put mine in my lunch box, and it takes 3 minutes to go from a cup of goopy ingredients, to a Cup-of-Awesome, slightly chewy on top, soft and cake-y through the middle, and soft and fudge-y on the bottom. You can also put some Nutella in the mix, so when its cooked you’ll get melted chocolate in the middle of your cake.