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HYPERLINK "http://news.pamojaeducation.com/author/timknight/" \o "Tim Knight" ByTim Knight
23 April, 2013
Type into a Google search hardest IB subjects and you will quickly find blogs and forums with IB students discussing how Mathematics HL is the toughest higher level subject. Despite this perceived challenge, ten thousand students worldwide took IB Math HL in May 2011. So why is Math HL so tough? What then is the attraction for so many students to take it and is it really as difficult as the global chatter between students and schools implies?
So, starting at the beginning, what is it that makes Math HL seem to be so much more challenging than Math SL? In terms of content there is a difference but it does not seem that more substantial than additional topics in other HL and SL subjects. There are some key advanced topics in Math HL (such as Complex Numbers) that are not seen in SL and there is an Option Topic. However, it seems the big difference is the questions that are asked in Math HL exams compared to Math SL. Some teachers argue that the Math HL exams are too hard, however, I would counter that there is a fundamental difference between HL and SL Mathematics and this can be best described as a difference in the way of thinking. This is reflected in the types of problems asked in Math HL; in essence Math HL begins to cover topics and question types that require true mathematical thinking.
So what do we mean by mathematical thinking? What on earth were students doing for the ten years of schooling in mathematics before they took Math HL? Why develop thinking mathematically?
Mathematics in modern times has been described as the science of pattern (I strongly recommend that you read HYPERLINK "http://www.google.com/url?sa=t&rct=j&q=&esrc=s&source=web&cd=2&ved=0CD8QFjAB&url=http%3A%2F%2Fwww.ddtwo.org%2F~wdowd%2Ffiles%2Ftokdevlinwhatismathematics.doc&ei=YWFuUa6PBovIrQf7g4HwCA&usg=AFQjCNH9pSAlzllB6Lc7YdKgFyNPOcs0Dw&sig2=4Dglqnf9gYfIQ0ijj2_l9w&bvm=bv.45368065,d.bmk" this article on the issue of the nature of Mathematics by HYPERLINK "http://www.stanford.edu/~kdevlin/" Keith Devlin). Discovering new mathematics has involved looking for patterns in nature and within mathematics itself before showing that these patterns are generally true. Finding a pattern is to make a conjecture and then showing the pattern to be general is to prove a theorem. This is the key process when thinking mathematically. Once we have our theorem of course we can apply it to find new conjectures, prove new theorems, solve problems and model the world around us mathematically. This all sounds quite straight forward but actually requires extremely precise use of language and notation as well as a lot of hard work coming to grips with the essentials of rigorous proof. It does not come easier, even to the most talented and dedicated of young mathematicians.
Most of the time spent doing mathematics in school we are using known, learnt results and calculating. This is a useful and powerful skill that many of us go onto use in later studies and careers. However, it is not the type of thinking that is used when studying HYPERLINK "http://intjforum.com/showthread.php?t=35544" mathematics beyond High School. The difference can be seen in the type of math questions that are asked in school. Whilst proof is the very essence of mathematical thinking, it is extremely rare to see the word proof in a question asked in school mathematics. When proof is seen in school it somehow seen as a topic apart.
When we look at a Mathematics HL exams we will see that students are sometimes asked to prove results, or at least show that. This requires student to create correct mathematical arguments to justify a result. This is mathematical thinking. There is an expectation in Math HL that results are not just known but are well understood with their derivation and proofs a key part of the course. There are still a good proportion of typical school mathematics calculations in Math HL but there are enough mathematical thinking type problems that there is a qualitative difference between Math HL and Math SL. By extension there is a difference for students in the outcome of studying Math HL that is congruent with this difference in question type. The different outcome is that a student will be ready to think mathematically and so at university successfully pursue mathematics beyond High School.
Mathematics HL is where a student can make the transition from the somewhat HYPERLINK "http://www.math.tamu.edu/~dallen/masters/hist_frame.htm" antique mathematics of High School to the new, modern mathematics that is studied at university and will require mathematical thinking skills. This initial exposure to the topics and methods of university is what on the one hand makes Math HL such a challenge and on the other makes is such a key subject for students who may consider continuing their studies in a subject that requires rigorous, modern mathematics.
The argument made to this point may seem to suggest that only students interested in studying mathematics at university should take Math HL. Fortunately IB students are much smarter than this and have realised (at least intuitively) that developing mathematical thinking skills is both fun and a goal in itself. The real motivation to take ones own personal mathematical journey as far as it can reasonably go is for the pure pleasure of engaging in mathematical thought and the surprise and joy of discovering new and often counter intuitive results that we KNOW to be true.
In the spirit of paradox, surprise and beauty that is mathematics I would like to leave you with a question.
Imagine a square drawn ten centimetres by ten centimetres with perfect accuracy. Now draw a diagonal. What is the length of this diagonal? (Nice easy school math so far! Remember Pythagoras?)
Now it gets interesting. Explain why it is impossible to measure exactly the length of this diagonal regardless of how accurate your ruler is and regardless of how much you can zoom in to see the length more accurately.
If you want some clues about this problem consider what HYPERLINK "http://en.wikipedia.org/wiki/Square_root_of_2" irrational numbers are.
You might also want to work out the rectangles where you CAN measure exactly the length of the diagonal. Is this more or less common that those that we cannot?
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