I have a post prepared to upload in a bit that will announce a possible hiatus from this WordPress blog. The reason is just that I found a cool book I want to try to absorb, The Princeton Companion to Mathematics by Gowers, Barrow-Green and Leader. Doubtless I will not be able to absorb it all in one go, so I will likely return to blogging periodically. But there is also teaching and research to conduct, so this book will slow me down. The rest of this post is a light weight brain-dump of some things that have been floating around in my head.
Recently, while watching a lecture on topology I was reminded that a huge percentage of the writings of Archimedes were lost in the siege of Alexandria. The Archimedean solids were rediscovered by Johannes Kepler, and we all know what he was capable of! Inspiring Isaac Newton is not a bad epitaph to have for one’s life.
The general point about rediscovery is a beautiful thing. Mathematics, more than other sciences, has this quality whereby a young student can take time to investigate previously established mathematics but then take breaks from it to rediscover theorems for themselves. How many children have rediscovered Pythagoras’ theorem, or the Golden Ratio, or Euler’s Formula, or any number of other simple theorems in mathematics?
Most textbooks rely on this quality. It is also why most “Exercises” in science books are largely theoretical. Even in biology and sociology. They are basically all mathematical, because you cannot expect a child to go out and purchase a laboratory set-up to rediscover experimental results. So much textbook teaching is mathematical for this reason.
I am going to digress momentarily, but will get back to the education theme later in this article.
The entire cosmos itself has sometimes been likened to an eternal rediscovery. The theory of Eternal Inflation postulates that our universe is just one bubble in a near endless ocean of baby and grandparent and all manner of other universes. Although, recently, Alexander Vilenkin and Audrey Mithani found that a wide class of inflationary cosmological models are unstable, meaning that could not have arisen from a pre-existing seed. There had to be a concept of an initial seed. This kind of destroys the “eternal” in eternal inflation. Here’s a Discover magazine account: “What Came Before the Big Bang? — Cosmologist Alexander Vilenkin believes the Big Bang wasn’t a one-off event”. Or you can click this link to hear Vilenkin explain his ideas himself: FQXi: Did the Universe Have a Beginning? Vilenkin seems to be having a rather golden period of originality over the past decade or so, I regularly come across his work.
If you like the idea of inflationary cosmology you do not have to worry too much though. You still get the result that infinitely many worlds could bubble out of an initial inflationary seed.
Below is my cartoon rendition of eternal inflation in the realm of human thought:
Oh to be a bubble thoughtoverse of the Wittenesque variety.
Quantum Fluctuations — Nothing Cannot Fluctuate
One thing I really get a bee in my bonnet about are the endless recountings in the popular literature about the beginning of the universe is the naïve idea that no one needs to explain the origin of the Big Bang and inflatons because “vacuum quantum fluctuations can produce a universe out of nothing”. This sort of pseudo-scientific argument is so annoying. It is a cancerous argument that plagues modern cosmology. And even a smart person like Vilenkin suffers from this disease. Here I quote him from a quote in another article on the PBS NOVA website::
Vilenkin has no problem with the universe having a beginning. “I think it’s possible for the universe to spontaneously appear from nothing in a natural way,” he said. The key there lies again in quantum physics—even nothingness fluctuates, a fact seen with so-called virtual particles that scientists have seen pop in and out of existence, and the birth of the universe may have occurred in a similar manner.
At least you have to credit Vilenkin with the brains to have said it is only “possible”. But even that caveat is fairly weaselly. My contention is that out of nothing you cannot get anything, not even a quantum fluctuation. People seem to forget quantum field theory is a background-dependent theory, it requires a pre-existing spacetime. There is no “natural way” to get a quantum fluctuation out of nothing. I just wish people would stop insisting on this sort of non-explanation for the Big Bang. If you start with not even spacetime then you really cannot get anything, especially not something as loaded with stuff as an inflaton field. So one day in the future I hope we will live in a universe where such stupid arguments are nonexistent nothingness, or maybe only vacuum fluctuations inside the mouths of idiots.
There are other types of fundamental theories, background-free theories, where spacetime is an emergent phenomenon. And proponents of those theories can get kind of proud about having a model inside their theories for a type of eternal inflation. Since their spacetimes are not necessarily pre-existing, they can say they can get quantum fluctuations in the pre-spacetime stuff, which can seed a Big Bang. That would fit with Vilenkin’s ideas, but without the silly illogical need to postulate a fluctuation out of nothingness. But this sort of pseudo-science is even more insidious. Just because they do not start with a presumption of a spacetime does not mean they can posit quantum fluctuations in the structure they start with. I mean they can posit this, but it is still not an explanation for the origins of the universe. They still are using some kind of structure to get things started.
Probably still worse are folks who go around flippantly saying that the laws of physics (the correct ones, when or if we discover them) “will be so compelling they will assert their own existence”. This is basically an argument saying, “This thing here is so beautiful it would be a crime if it did not exist, in fact it must exist since it is so beautiful, if no one had created it then it would have created itself.” There really is nothing different about those two statements. It is so unscientific it makes me sick when I hear such statements touted as scientific philosophy. These ideas go beyond thought mutation and into a realm of lunacy.
I think the cause of these thought cancers is the immature fight in society between science and religion. These are tensions in society that need not exist, yet we all understand why they exist. Because people are idiots. People are idiots where their own beliefs are concerned, by in large, even myself. But you can train yourself to be less of an idiot by studying both sciences and religions and appreciating what each mode of human thought can bring to the benefit of society. These are not competing belief systems. They are compatible. But so many believers in religion are falsely following corrupted teachings, they veer into the domain of science blindly, thinking their beliefs are the trump cards. That is such a wrong and foolish view, because everyone with a fair and balanced mind knows the essence of spirituality is a subjective view-point about the world, one deals with one’s inner consciousness. And so there is no room in such a belief system for imposing one’s own beliefs onto others, and especially not imposing them on an entire domain of objective investigation like science. And, on the other hand, many scientists are irrationally anti-religious and go out of their way to try and show a “God” idea is not needed in philosophy. But in doing so they are also stepping outside their domain of expertise. If there is some kind of omnipotent creator of all things, It certainly could not be comprehended by finite minds. It is also probably not going to be amenable to empirical measurement and analysis. I do not know why so many scientists are so virulently anti-religious. Sure, I can understand why they oppose current religious institutions, we all should, they are mostly thoroughly corrupt. But the pure abstract idea of religion and ethics and spirituality is totally 100% compatible with a scientific worldview. Anyone who thinks otherwise is wrong! (Joke!)
Also, I do not favour inflationary theory for other reasons. There is no good theoretical justification for the inflaton field other than the theory of inflation prediction of the homogeneity and isotropy of the CMB. You’d like a good theory to have more than one trick! You know. Like how gravity explains both the orbits of planets and the way an apple falls to the Earth from a tree. With inflatons you have this quantum field that is theorised to exist for one and only one reason, to explain homogeneity and isotropy in the Big Bang. And don’t forget, the theory of inflation does not explain the reason the Big Bang happened, it does not explain its own existence. If the inflaton had observable consequences in other areas of physics I would be a lot more predisposed to taking it seriously. And to be fair, maybe the inflaton will show up in future experiments. Most fundamental particles and theoretical constructs began life as a one-trick sort of necessity. Most develop to be a touch more universal and will eventually arise in many aspects of physics. So I hope, for the sake of the fans of cosmic inflation, that the inflaton field does have other testable consequences in physics.
In case you think that is an unreasonable criticism, there are precedents for fundamental theories having a kind of mathematically built-in explanation. String theorists, for instance, often appeal to the internal consistency of string theory as a rationale for its claim as a fundamental theory of physics. I do not know if this really flies with mathematicians, but the string physicists seem convinced. In any case, to my knowledge the inflation does not have this sort of quality, it is not a necessary ingredient for explaining observed phenomena in our universe. It does have a massive head start on being a candidate sole explanation for the isotropy and homogeneity of the CMB, but so far that race has not yet been completely run. (Or if it has then I am writing out of ignorance, but … you know … you can forgive me for that.)
Anyway, back to mathematics and education.
You have to love the eternal rediscovery built-in to mathematics. It is what makes mathematics eternally interesting to each generation of students. But as a teacher you have to train the nerdy children to not bother reading everything. Apart from the fact there is too much to read, they should be given the opportunity to read a little then investigate a lot, and try to deduce old results for themselves as if they were fresh seeds and buds on a plant. Giving students a chance to catch old water as if it were fresh dewdrops of rain is a beautiful thing. The mind that sees a problem afresh is blessed, even if the problem has been solved centuries ago. The new mind encountering the ancient problem is potentially rediscovering grains of truth in the cosmos, and is connecting spiritually to past and future intellectual civilisations. And for students of science, the theoretical studies offer exactly the same eternal rediscovery opportunities. Do not deny them a chance to rediscover theory in your science classes. Do not teach them theory. Teach them some theoretical underpinnings, but then let them explore before giving the game away.
With so much emphasis these days on educational accountability and standardised tests there is a danger of not giving children these opportunities to learn and discover things for themselves. I recently heard an Intelligence2 “Intelligence Squared” debate on academic testing. One crazy women from the UK government was arguing that testing, testing, and more testing — “relentless testing” were her words — was vital and necessary and provably increased student achievement.
Yes, practising tests will improve test scores, but it is not the only way to improve test scores. And relentless testing will improve student gains in all manner of mindless jobs out there is society that are drill-like and amount to going through routine work, like tests. But there is less evidence that relentless testing improves imagination and creativity.
Let’s face it though. Some jobs and areas of life require mindlessly repetitive tasks. Even computer programming has modes where for hours the normally creative programmer will be doing repetitive but possibly intellectually demanding chores. So we should not agitate and jump up and down wildly proclaiming tests and exams are evil. (I have done that in the past.)
Yet I am far more inclined towards the educational philosophy of the likes of Sir Ken Robinson, Neil Postman, and Alfie Kohn.
My current attitude towards tests and exams is the following:
- Tests are incredibly useful for me with large class sizes (120+ students), because I get a good overview of how effective the course is for most students, as well as a good look at the tails. Here I am using the fact test scores (for well designed tests) do correlate well with student academic aptitudes.
- My use of tests is mostly formative, not summative. Tests give me a valuable way of improving the course resources and learning styles.
- Tests and exams suck as tools for assessing students because they do not assess everything there is to know about a student’s learning. Tests and exams correlate well with academic aptitudes, but not well with other soft skills.
- Grading in general is a bad practise. Students know when they have done well or not. They do not need to be told. At schools if parents want to know they should learn to ask their children how school is going, and students should be trained to be honest, since life tends to work out better that way.
- Relentless testing is deleterious to the less academically gifted students. There is a long tail in academic aptitude, and the students in this tail will often benefit from a kinder and more caring mode of learning. You do not have to be soft and woolly about this, it is a hard core educational psychology result: if you want the best for all students you need to treat them all as individuals. For some tests are great, terrific! For others tests and exams are positively harmful. You want to try and figure out who is who, at least if you are lucky to have small class sizes.
- For large class sizes, like at a university, do still treat all students individually. You can easily do this by offering a buffet of learning resources and modes. Do not, whatever you do, provide a single-mode style of lecture+homework+exam course. That is ancient technology, medieval. You have the Internet, use it! Gather vast numbers of resources of all different manners of approach to your subject you are teaching, then do not teach it! Let your students find their own way through all the material. This will slow down a lot of students — the ones who have been indoctrinated and trained to do only what they are told — but if you persist and insist they navigate your course themselves then they should learn deeper as a result.
Solving the “do what I am told” problem is in fact the very first job of an educator in my opinion. (For a long time I suffered from lack of a good teacher in this regard myself. I wanted to please, so I did what I was told, it seemed simple enough. But … Oh crap, … the day I found out this was holding me back, I was furious. I was about 18 at the time. Still hopelessly naïve and ill-informed about real learning.) If you achieve nothing else with a student, transitioning them from being an unquestioning sponge (or oily duck — take your pick) to being self-motivated and self-directed in their learning is the most valuable lesson you can ever give them. So give them it.
So I use a lot of tests. But not for grading. For grading I rely more on student journal portfolios. All the weekly homework sets are quizzes though, so you could criticise the fact I still use these for grading. As a percentage though, the Journals are more heavily weighted (usually 40% of the course grade). There are some downsides to all this.
- It is fairly well established in research that grading using journals or subjective criteria is prone to bias. So unless you anonymise student work, you have a bias you need to deal with somehow before handing out final grades.
- Grading weekly journals, even anonymously, takes a lot of time, about 15 to 20 times the hours that grading summative exams takes. So that’s a huge time commitment. So you have to use it wisely by giving very good quality early feedback to students on their journals.
- I still haven’t found out how to test the methods easily. I would like to know quantitatively how much more effective journal portfolios are compared to exam based assessments. I am not a specialist education researcher, and I research and write a about a lot of other things, so this is taking me time to get around to answering.
I have not solved the grading problem, for now it is required by the university, so legally I have to assign grades. One subversive thing I am following up on is to refuse to submit singular grades. As a person with a physicists world-view I believe strongly in the role of sound measurement practice, and we all know a single letter grade is not a fair reflection on a student’s attainment. At a minimum a spread of grades should be given to each student, or better, a three-point summary, LQ, Median, UQ. Numerical scaled grades can then be converted into a fairer letter grade range. And GPA scores can also be given as a central measure and a spread measure.
I can imagine many students will have a large to moderate assessment spread, and so it is important to give them this measure, one in a few hundred students might statistically get very low grades by pure chance, when their potential is a lot higher. I am currently looking into research on this.
OK, so in summary: even though institutions require a lot of tests you can go around the tests and still given students a fair grade while not sacrificing the true learning opportunities that come from the principle of eternal rediscovery. Eternal rediscovery is such an important idea that I want to write an academic paper about it and present at a few conferences to get people thinking about the idea. No one will disagree with it. Some may want to refine and adjust the ideas. Some may want concrete realizations and examples. The real question is, will they go away and truly inculcate it into their teaching practices?
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