Saturday, March 1, 2014

The Use of Contextual Problem in PISA: Is It Unfair?

Dispute between the use of concrete and abstract in Mathematics has been going around for quite a long time. One group favors the old curriculum which focuses on computational and symbolic manipulation skill, others say that the old curriculum does not work and children have to learn Mathematics which is centered on real-world problems. It is obvious the majority of education society tends to favor the latter, with the development of mathematics learning approaches like RME, as well as terms like ‘constructivism’ and ‘contextual teaching learning’ being used widely and extensively. However it does not imply that no one wants to defend the use of abstract learning materials. Kaminski, Sloutsky, and Heckler’s (2008) publication, for instance, caused a major stir by stating that students may gain more from learning Mathematics through abstract, symbolic representation than from concrete examples.

Unaware of all the elitist arguments, research and publications done by people of influence though, Mathematics classrooms in most developing countries (which is basically most countries) for various reasons are inclined to believe the former standpoint rather than the latter. It is mainly because using concrete and contextual learning material tend to use up a lot of time and money, which is simply too problematic to carry out in an economically unaccommodating condition. Interestingly enough, this is claimed to be the reason behind these countries’ low rank in PISA. Indonesia, for example, only ranks as the 57th in 2009 and the 64th in 2012. Zulkardi et. al. (2013) claimed that it is due to students’ unfamiliarity with solving contextual problem. This leads to a suggestion that some countries are put on clear advantage due to their economic capability to fund Mathematics contextual teaching and learning. Is PISA assessment unfair? Does the type of the problems need to be changed to fit the participants’ background better?

The answer to this question lies in the nature of PISA itself. First, PISA is designed to find out whether or not students can use what they have learnt in school and apply that knowledge to real life situations and problems (OECD, 2009). Of course, the use of contextual problems comes as a no brainer. Second, PISA is often seen as the world cup of international education, but testing and ranking the students to see which one is better is not the main emphasis here. PISA uses the test results to compare the effectiveness of each country’s education system, determine the main characteristic of the higher ranking countries and use it as a foundation for future policy making (OECD, 2009). Therefore the use of contextual problem should not be a problem; not using it may kill the purpose of the test. 

The reason PISA is biased, if at all, might lies at other things. The context itself is one of them, along with the statistical method. Questions used in PISA have been culturally or linguistically biased toward certain countries, but when OECD tried to tackle this problem by ruling out questions suspected of bias, it can “smooth out” the key difference between countries (Stewart, 2013). The data is also problematic. Shanghai, for instance, scored number one in all three domains during the 2009 test, leading to China being crowned the country with best education system in the world. Shanghai province, whose population is 1.7% of the entire China’s and historically known to comprise the elite and the wealthy, in no way can be seen as representative of China. Some problem with sampling also occurs in countries like UK and Denmark (Stewart, 2003), which led to a conclusion that PISA “tend to say too much for what it can do and tends not to publicize the negative or the weaker aspect” (Stewart, 2003).

References:
Kaminski, J. A., Sloutsky, V.M, and Heckler, A.F. (2008). The advantage of abstract examples in learning math. Science, 320, 454-455.
OECD. (2009). PISA 2009 Assessment Framework Key Competencies in Reading, Mathematics and Science. OECD.
Stewart, W. (2013, July, 26).  Is PISA fundamentally flawed?. TES. Retrieved from http://www.tes.co.uk/article.aspx?storycode=6344672
Lutfianto, et. al. (2013). Unfinished Student Answer in PISA Mathematics Contextual Problem. IndoMS-JME, Volume 4, No. 2, July 2013, pp. 188 – 193.
 

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 19th 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.