The Trivial Group
It's been a pretty long time since I last updated this blog!
This is what happens when you replace your old computer by one that can actually run any game and then you just lose the habit of writing. My new laptop is still working just fine, but there's been a lull in strategy games I want to try, so I figured it was the perfect chance to revive this!
As you can see, victory on the Eastern front is at hand, so I'm not in pressing need of military advice. Instead, I'll just write about something I've been pondering about lately: The new math curriculum for the IB DP, and more specifically, the decision to get rid of the options and integrate some of their content into the new routes.
That being said, I don't intend to write about the new curriculum itself, it's just the starting point of this post. What I do want to talk about is one of the things that is being lost completely: the "Sets, Relations, and Groups" option (from here on out referred to as "Groups option").
Let's not beat around the bush, I don't think it's a good decision.
To preface my explanation for this opinion, I'll start by saying that I understand the decision based on the apparent intentions of the new curriculum. Getting the fluff out of the explanations the IB gives, this new curriculum's main driving force seems to be "usefulness".
Statistics, calculus, etc... they are all used in a very wide range of different fields of knowledge and irl. On the other hand, even intro abstract algebra topics like groups and isomorphisms are studied by a much narrower set of people; people who would have to study calculus, etc... anyways because of how university math courses are structured. Coupled with the fact that the Groups option is the most abstract one (and hence less directly applicable), it's not hard to understand why out of the four, it ended up like this:
I think that, while not being as widespread and immediately "useful" irl compared to the others, the Groups option has something special to offer, something that nothing else in the curriculum covers that well: being closer to how math is at looked by mathematicians.
I'm not saying this because of the "difficulty" of the topic, or because it is "higher level" math. Almost every topic in middle/high school can be both of those if looked from certain angles, just look at IMO level problems. The difference is that the Groups option doesn't follow the same spiral/tower structure the rest of the curriculum follows:
Take calculus for example. Its seeds are laid all throughout secondary education. Gradients of lines, areas, average speed problems, vertex of a parabola, etc... are introduced at different stages and they converge in calculus. This is not a problem per se, the issue is that it is all constructed to solve some exercise, to do some application. There's not much time to talk about what's behind it or why any of the precalc things work like they do. Even if there was, transforming a calculus class into an analysis one or being completely rigorous in middle school would be just nonsense.
This phenomenon occurs even in "proof" intensive topics like geometry. Each year students use the theorems of previous years to prove new ones, but the axioms of Euclidean geometry get buried by the years. I doubt the percentage of students who could identify an axiom from a theorem in geometry is very high.
This is where the Groups option comes in! Instead of progressing the same way as the other topics do, it yells "Stop!" and makes you take a different look at things that years of education have set in stone. Just by doing that, it is already valuable and incredibly refreshing.
Unlike calculus, the initial group theory topics are very easy to approach. However, having to question "axioms" they've had drilled into their heads since they were little makes students struggle quite a bit to wrap their heads around what's initially happening.
Because of this, the Groups option is the easiest and the hardest of them all! There's no need to use fancy techniques to solve a differential equations or accept on faith that the Gaussian function defines a distribution; but understanding just what equivalence classes are is arguably more difficult.
Returning to the word "rigorous" that I used before, the Groups option allows the introduction of a higher level of rigor and application of logic. You may think that, if need be, other math topics could be approached in such a way, but the reality is that it would most likely be counterproductive. Rigor shouldn't precede creativity and in most secondary education contexts a big amount of it isn't particularly necessary (it should be given in just enough doses!).
However, since intro abstract algebra is so basic and devoid of baggage, much higher level of rigor is not only more accesible to the student, but it is also clear to see why/when it's necessary. It's the ideal situation to do it! Most people's first encounter with something similarly rigorous is the $\epsilon - \delta$ definition of continuity, and I don't think I need to mention how that usually goes.
Stepping back and looking at things you think you know, being rigorous when needed, looking at the basis of your studies... these are the type of things that students don't usually get from their high school math enough, or even freshman/sophomore university courses for that matter. They are things that would be useful for everyone, not only "math people", in basically any area of knowledge.
Even for them in particular, regardless of how deep the analysis and approaches route goes into differential equations, a student would at most be able to skip the intro calc classes offered in college. They would still need to take a diff. eq. course and that course would still start at a "end of calc 2" place, the advantage wouldn't be too big.
On the other hand, abstract algebra courses cater to a much more select crowd, and thus they usually progress rather fast. The advantage of having experienced how sets, functions, and algebraic structures are treated in higher math would make a huge difference for anyone going into analysis, abstract algebra, topology, etc...
And so we come to the end of this post. What's done is done, so this is nothing more than a rant (as usual). I think that based on their objectives, the new curriculum will be successful enough in its "useful math"/modelling focus that any abstract algebra is gone for good. All that's left is to adapt and work with what is given once the time comes. Better take advantage of the last few years(?) left!
 |
| The type problems that have keep me up at night |
As you can see, victory on the Eastern front is at hand, so I'm not in pressing need of military advice. Instead, I'll just write about something I've been pondering about lately: The new math curriculum for the IB DP, and more specifically, the decision to get rid of the options and integrate some of their content into the new routes.
That being said, I don't intend to write about the new curriculum itself, it's just the starting point of this post. What I do want to talk about is one of the things that is being lost completely: the "Sets, Relations, and Groups" option (from here on out referred to as "Groups option").
Let's not beat around the bush, I don't think it's a good decision.
To preface my explanation for this opinion, I'll start by saying that I understand the decision based on the apparent intentions of the new curriculum. Getting the fluff out of the explanations the IB gives, this new curriculum's main driving force seems to be "usefulness".
Statistics, calculus, etc... they are all used in a very wide range of different fields of knowledge and irl. On the other hand, even intro abstract algebra topics like groups and isomorphisms are studied by a much narrower set of people; people who would have to study calculus, etc... anyways because of how university math courses are structured. Coupled with the fact that the Groups option is the most abstract one (and hence less directly applicable), it's not hard to understand why out of the four, it ended up like this:
 |
| Can you guess who is who? |
I think that, while not being as widespread and immediately "useful" irl compared to the others, the Groups option has something special to offer, something that nothing else in the curriculum covers that well: being closer to how math is at looked by mathematicians.
I'm not saying this because of the "difficulty" of the topic, or because it is "higher level" math. Almost every topic in middle/high school can be both of those if looked from certain angles, just look at IMO level problems. The difference is that the Groups option doesn't follow the same spiral/tower structure the rest of the curriculum follows:
Take calculus for example. Its seeds are laid all throughout secondary education. Gradients of lines, areas, average speed problems, vertex of a parabola, etc... are introduced at different stages and they converge in calculus. This is not a problem per se, the issue is that it is all constructed to solve some exercise, to do some application. There's not much time to talk about what's behind it or why any of the precalc things work like they do. Even if there was, transforming a calculus class into an analysis one or being completely rigorous in middle school would be just nonsense.
This phenomenon occurs even in "proof" intensive topics like geometry. Each year students use the theorems of previous years to prove new ones, but the axioms of Euclidean geometry get buried by the years. I doubt the percentage of students who could identify an axiom from a theorem in geometry is very high.
This is where the Groups option comes in! Instead of progressing the same way as the other topics do, it yells "Stop!" and makes you take a different look at things that years of education have set in stone. Just by doing that, it is already valuable and incredibly refreshing.
Unlike calculus, the initial group theory topics are very easy to approach. However, having to question "axioms" they've had drilled into their heads since they were little makes students struggle quite a bit to wrap their heads around what's initially happening.
Because of this, the Groups option is the easiest and the hardest of them all! There's no need to use fancy techniques to solve a differential equations or accept on faith that the Gaussian function defines a distribution; but understanding just what equivalence classes are is arguably more difficult.
Returning to the word "rigorous" that I used before, the Groups option allows the introduction of a higher level of rigor and application of logic. You may think that, if need be, other math topics could be approached in such a way, but the reality is that it would most likely be counterproductive. Rigor shouldn't precede creativity and in most secondary education contexts a big amount of it isn't particularly necessary (it should be given in just enough doses!).
However, since intro abstract algebra is so basic and devoid of baggage, much higher level of rigor is not only more accesible to the student, but it is also clear to see why/when it's necessary. It's the ideal situation to do it! Most people's first encounter with something similarly rigorous is the $\epsilon - \delta$ definition of continuity, and I don't think I need to mention how that usually goes.
 |
| Puzzle: What is commutative wine made of? |
Stepping back and looking at things you think you know, being rigorous when needed, looking at the basis of your studies... these are the type of things that students don't usually get from their high school math enough, or even freshman/sophomore university courses for that matter. They are things that would be useful for everyone, not only "math people", in basically any area of knowledge.
Even for them in particular, regardless of how deep the analysis and approaches route goes into differential equations, a student would at most be able to skip the intro calc classes offered in college. They would still need to take a diff. eq. course and that course would still start at a "end of calc 2" place, the advantage wouldn't be too big.
On the other hand, abstract algebra courses cater to a much more select crowd, and thus they usually progress rather fast. The advantage of having experienced how sets, functions, and algebraic structures are treated in higher math would make a huge difference for anyone going into analysis, abstract algebra, topology, etc...
---------------------------------------------------------------
And so we come to the end of this post. What's done is done, so this is nothing more than a rant (as usual). I think that based on their objectives, the new curriculum will be successful enough in its "useful math"/modelling focus that any abstract algebra is gone for good. All that's left is to adapt and work with what is given once the time comes. Better take advantage of the last few years(?) left!
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