Creative mathematical tasks contribute to deeper learning in mathematics

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Mathias Norqvist

Mathias Norqvist

Working with creative mathematical tasks is important for pupils both to reflect on mathematics as well as for their subsequent test results. Being faced with creative tasks during exercise has evident effects on all pupils, both on weak and high performers. This according to studies at Umeå University in Sweden.
“The results of my dissertation show the importance for pupils to work with creative reasoning and not always get methods and rules presented in advance. This is something both publishers and teachers could take into account more often when designing mathematical tasks,” says Mathias Norqvist, doctoral student at the Department of Mathematics and Mathematical Statistics at Umeå University.
The studies show that pupils at upper secondary school who work with exercises designed to encourage creative mathematical reasoning more easily remember what they have learnt and, as a result, perform better.
“Contrary to common belief, it seems to be the low performing pupils who benefit most from practicing with creative tasks, in comparison to more imitative tasks where focus lies on how to use the given solutions,” says Mathias Norqvist.
There is a great risk that pupils who are presented one method, will use it without further reflection. Although, there are of course certain methods in mathematics that should be automated to relieve the pressure on the working memory, but it should not come at a cost to the understanding of the underlying mathematics. Since well-designed creative exercises can focus on central mathematical properties, they are important for all pupils since they force pupils to reflect on the mathematics and to base their reasoning on what they already know.
A total of about 300 upper secondary school pupils participated in the studies that formed the basis of the dissertation.

A quantum theory for thrones fans

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Sydney University‘s delightful video in which academics predict who is going to win the Game of Thrones based on their disciplinary knowledge and understandings has had 62,500 Facebook likes, 900 YouTube hits and 10,000 Twitter impressions. The university has now uploaded, the full five-minute video of Michael Biercuk‘s quantum theory, which predicted a major event from the finale before it aired: ‘Tommin’s gotta die’. Biercuk has since been asked for further quantum physics theories, including how Bran can see into and interact with the past. The uni obviously harbours some hard core GoT fans. Back in 2014 it produced a video of Amy Johansen playing the GoT theme on the carillon, which was even watched by Davos Seaworth from the show.

via The Australian

Institutionalizing creationism

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by Michael Baltzley on Science 10 Jun 2016, Vol. 352, Issue 6291, pp. 1285-1286 DOI: 10.1126/science.aaf7386


Biology faculty who teach evolution at U.S. colleges and universities often worry about the efforts of creationists to include the teaching of “intelligent design” in publicly funded high school biology courses. Now we also have cause to worry about students at publicly funded colleges and universities earning science credits for learning creationism.
The Western Interstate Commission for Higher Education (WICHE) is developing an Interstate Passport Initiative, funded in part by the U.S. Department of Education, which would streamline the learning outcomes for courses across institutions to facilitate the transfer of credits(1). Unfortunately, with the Passport Initiative, WICHE proposes making the creationist “teach the controversy” strategy as a standard part of college biology courses. In their document “Faculty handbook: Constructing your institution’s Passport block,” WICHE suggests that to demonstrate scientific literacy, students should “watch the Ken Hamm [sic]–Bill Nye evolution-creation debate and evaluate the scientific evidence and arguments used by the participants”(2).
This suggestion validates creationism as science by stating explicitly that both participants have scientific evidence. Middle school, high school, and college instructors who support creationism can point to the WICHE Passport Initiative as evidence that there is a scientific debate that includes creationism. The Answers in Genesis website has already promoted the debate as a way to get creationism into science classrooms(3).
If the goal of the curriculum is to help students use scientific evidence to debunk myths, the suggested class activity should be rephrased to read, “Watch the Ken Ham–Bill Nye evolution-creation debate and evaluate the arguments used by the participants.” However, even with better wording, by including the debate in a science class, WICHE is promoting the use of the Ham-Nye debate as an example of a scientific controversy. There are hundreds of genuine biological debates, both current and historical, that good educators can make interesting. WICHE should choose real examples of scientific debates and avoid advocating for creationism in science classrooms.
A student who takes general education courses at a WICHE Passport institution will soon be able to transfer the credits to any other Passport institution. The receiving institution cannot reject individual courses from approved institutions. Currently, WICHE lists 24 public institutions representing more than 150 campuses in seven U.S. states as participants in developing the Passport Initiative. WICHE plans to expand the Passport Initiative to six more states. As the Initiative grows, more and more public postsecondary institutions will be awarding science credits for courses that include creationism. To prevent the insertion of religion into science classrooms, scientists must speak out against the Passport Initiative until WICHE removes creationism from their suggested curriculum.


(1) Western Interstate Commission for Higher Education, The Interstate Passport.
(2) Interstate Passport, “Faculty handbook: Constructing your institution’s Passport block” (2016); p. 43 (pdf).
(3) Answers in Genesis, “Public schools and the Bill Nye/Ken Ham debate” (2014).

Dawn of the quark ages

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by Michael Brooks from NewScientist 3024, 6 june 2015


Ask them to name their heart’s truest desire, and many a science nut might say the answer to life, the universe and everything – or, failing that, a fully functioning lightsaber.
Odd, then, that one field of scientific enquiry that could conceivably provide both gets so little press. After all the hoopla of the past few years, you could be forgiven for believing that understanding matter’s fundamentals is all about the Higgs boson – the “God particle” that explains where mass comes from.
The Higgs is undoubtedly important. But it is actually pretty insignificant for real stuff like you and me, accounting for just 1 or 2 per cent of normal matter’s mass. And the huge energy needed to make a Higgs means we’re unlikely to see technology exploiting it any time soon.
Two more familiar, though less glamorous, particles might offer more. Get to grips with their complexities, and we can begin to explain how the material universe came to exist and persist, and explore mind-boggling technologies: not just lightsabers, but new sorts of lasers and materials to store energy, too. That’s easier said than done, granted – but with a lot of computing muscle, it is what we are starting to do.
Chances are you know about protons and neutrons. Collectively known as nucleons, these two particles make up the nucleus, the meaty heart of the atom. (In terms of mass, the weedy electrons that orbit the nucleus are insignificant contributors to the atom.)
The headline difference between protons and neutrons is that protons have a positive electrical charge, whereas neutrons are neutral. But they also differ ever so slightly in mass: in the units that particle physicists use, the neutron weighs in at 939.6 megaelectronvolts (MeV) and the proton at 938.3 MeV.
That’s a difference of just 0.14 per cent, but boy does it matter. The neutrons’ extra mass means they decay into protons, not the other way around. Protons team up with negatively charged electrons to form robust, structured, electrically neutral atoms, rather than the world being a featureless neutron gloop.
“The whole universe would be very different if the proton were heavier than the neutron,” says particle theorist Chris Sachrajda of the University of Southampton in the UK. “The proton is stable, so atoms are stable and we’re stable.” Our current best guess is that the proton’s half-life, a measure of its stability over time, is at least 1032 years. Given that the universe only has 1010 or so years behind it, that is a convoluted way of saying no one has ever seen a proton decay.
The exact amount of the neutron’s excess baggage matters, too. The simplest atom is hydrogen, which is a single proton plus an orbiting electron. Hydrogen was made in the big bang, before becoming fuel for nuclear fusion in the first stars, which forged most of the other chemical elements. Had the protonneutron mass difference been just a little bigger, adding more neutrons to make more complex elements would have encountered energy barriers that were “difficult or impossible” to overcome, says Frank Wilczek of the Massachusetts Institute of Technology. The universe would be stuck at hydrogen.
But had the mass difference been subtly less, hydrogen would have spontaneously changed to the more inert, innocuous helium before stars could form – and the cosmos would have been an equally limp disappointment. Narrow the gap further, and hydrogen atoms would have transformed via a process called inverse beta decay into neutrons and another sort of neutral particle, the neutrino. Bingo, no atoms whatsoever.
All of that leads to an unavoidable conclusion about the proton and neutron masses. “Without these numbers, people wouldn’t exist,” says Zoltán Fodor of the University of Wuppertal, Germany.
But where do they come from?
The question is fiendishly difficult to answer. We’ve known for half a century that protons and neutrons are not fundamental particles, but made of smaller constituents called quarks. There are six types of quark: up, down, strange, charm, bottom and top. The proton has a composition of up-up-down, while the neutron is up-down-down.

A full explanation of where stuff gets its mass from is buried deep in the atomic nucleus

A full explanation of where stuff gets its mass from is buried deep in the atomic nucleus

Quark QuirksDown quarks are slightly heavier than up quarks, but don’t expect that to explain the neutron’s sliver of extra mass: both quark masses are tiny. It’s hard to tell exactly how tiny, because quarks are never seen singly (see “Quark quirks“, right), but the up quark has a mass of something like 2 or 3 MeV, and the down quark maybe double that – just a tiny fraction of the total proton or neutron mass.
Like all fundamental particles, quarks acquire these masses through interactions with the sticky, all-pervasive Higgs field, the thing that makes the Higgs boson. But explaining the mass of matter made of multiple quarks clearly needs something else.
The answer comes by scaling the sheer cliff face that is quantum chromodynamics, or QCD. Just as particles have an electrical charge that determines their response to the electromagnetic force, quarks carry one of three “colour charges” that explain their interactions via another fundamental force, the strong nuclear force. QCD is the theory behind the strong force, and it is devilishly complex.
Electrically charged particles can bind together by exchanging massless photons. Similarly, colour-charged quarks bind together to form matter such as protons and neutrons by exchanging particles known as gluons. Although gluons have no mass, they do have energy. What’s more, thanks to Einstein’s famous E = mc2, that energy can be converted into a froth of quarks (and their antimatter equivalents) beyond the three normally said to reside in a proton or neutron. According to the uncertainty principle of quantum physics, these extra particles are constantly popping up and disappearing again (see diagram).
To try and make sense of this quantum froth, over the past four decades particle theorists have invented and refined a technique known as lattice QCD. In much the same way that meteorologists and climate scientists attempt to simulate the swirling complexities of Earth’s atmosphere by reducing it to a three-dimensional grid of points spaced kilometres apart, lattice QCD reduces a nucleon’s interior to a lattice of points in a simulated space-time tens of femtometres across. Quarks sit at the vertices of this lattice, while gluons propagate along the edges. By summing up the interactions along all these edges, and seeing how they evolve step-wise in time, you begin to build up a picture of how the nucleon works as a whole.
Trouble is, even with a modest number of lattice points – say 100 by 100 by 100 separated by one-tenth of a femtometre – that’s an awful lot of interactions, and lattice QCD simulations require a screaming amount of computing power. Complicating things still further, because quantum physics offers no certain outcomes, these simulations must be run thousands of times to arrive at an “average” answer. To work out where the proton and neutron masses come from, Fodor and his colleagues had to harness two IBM Blue Gene supercomputers and two suites of cluster-computing processors.
The breakthrough came in 2008, when they finally arrived at a mass for both nucleons of 936 MeV, give or take 25 MeV – pretty much on the nose (Science, vol 322, p 1224). This confirmed that the interaction energies of quarks and gluons make up the lion’s share of the mass of stuff as we know it. You might feel solid, but in fact you’re 99 per cent energy.
But the calculations were nowhere near precise enough to pin down that all-important difference between the proton and neutron masses, which was still 40 times smaller than the uncertainty in the result. What’s more, the calculation suffered from a glaring omission: the effects of electrical charge, which is another source of energy, and therefore mass. All the transient quarks and antiquarks inside the nucleon are electrically charged, giving them a “self-energy” that makes an additional contribution to their mass. Without taking into account this effect, all bets about quark masses are off. Talk about one compound particle being more massive than another because of a difference in quark masses is a “crude caricature”, says Wilczek, who won a share of a Nobel prize in 2004 for his part in developing QCD.
The subtle roots of the proton-neutron mass difference lie in solving not just the equations of QCD, but those of quantum electrodynamics (QED), which governs electromagnetic interactions. And that is a theorist’s worst nightmare. “It’s awfully difficult to have QED and QCD in the same framework,” says Fodor. The electromagnetic self-energy can’t even be calculated directly. In a limited lattice simulation, its interactions create an infinity – a mathematical effect rather like a never-ending reverberation inside a cathedral.
Fodor and his colleagues’ new workaround involves solving the QED equations for various combinations of quarks inside different subatomic particles. The resulting subtle differences are used to replace the results of calculations that would invoke infinities, and so grind out a value for the proton-neutron mass difference (Science, vol 347, p 1452).
The figure the team came up with is in agreement with the measured value, although the error on it is still about 20 per cent. It is nonetheless “a milestone”, says Sachrajda. Wilczek feels similarly. “I think it’s exciting,” he says. “It’s a demonstration of strength.”
You might be forgiven for wondering what we gain by calculating from first principles numbers we already knew. But quite apart from this particular number’s existential interest, for Wilczek the excitement lies in our ability now to calculate very basic things about how the universe ticks that we couldn’t before.
Take the processes inside huge stars that go supernova – the events that first seeded the universe with elements heavier than hydrogen and helium. Our inability to marry QED and QCD meant we couldn’t do much more than wave our hands at questions such as the timescale over which heavy elements first formed – and we couldn’t make a star to test our ideas. “Conditions are so extreme we can’t reproduce them in the laboratory,” says Wilczek. “Now we will be able to calculate them with confidence.”
The advance might help clear up some of the funk surrounding fundamental physics. The Large Hadron Collider’s discovery in 2012 of the Higgs boson, and nothing else so far, leaves many open questions. Why did matter win out over antimatter just after the big bang (New Scientist, 23 May, p 28)? Why do the proton and electron charges mirror each other so perfectly when they are such different particles? “We need new physics, and simulations like ours can help,” says Kálmán Szabó, one of Fodor’s Wuppertal collaborators. “We can compare experiment and our precise theory and look for processes that tell us what lies beyond standard physics.”

An open road

For Sachrajda, this kind of computational capability comes at just the right time, as the LHC fires up again to explore particle interactions at even higher energies. “We all hope it will give an unambiguous signal of something new,” he says. “But you’re still going to have to understand what the underlying theory is, and for that you will need this kind of precision.”
If that still sounds a little highfalutin, it’s also worth considering how modern technologies have sprung from an ever deeper understanding of matter’s workings. A century or so ago, we were just getting to grips with the atom – an understanding on which innovations such as computers and lasers were built. Then came insights into the atomic nucleus, with all the technological positives and negatives – power stations, cancer therapies, nuclear bombs – those have brought.
Digging down into protons and neutrons means taking things to the next level, and a potentially rich seam to mine. Gluons are far more excitable in their interactions with colour charge than are photons in electromagnetic interactions, so it could be that manipulating colour-charged particles yields vastly more energy than fiddling with things on the atomic scale. “I think the possibility of powerful X-ray or gamma-ray sources exploiting sophisticated nuclear physics is speculative, but not outrageously so,” says Wilczek.
Star Wars' lightsaberGluons, unlike photons, also interact with themselves, and this could conceivably see them confining each other into a writhing pillar of energy – hence Wilczek’s tongue-incheek suggestion they might make a Star Wars-style lightsaber. More immediate, perhaps, is the prospect of better ways to harness and store energy. “Nuclei can pack a lot of energy into a small space,” says Wilczek. “If we can do really accurate nuclear chemistry by calculation as opposed to having hit-andmiss experiments, it could very well lead to dense energy storage.”
For Fodor, that’s still a long way off – but with the accuracy that calculations are now reaching, the road is at last open. “These are mostly dreams today, but now we can accommodate the dreams, at least,” he says. “You’ve reached a level where these technological ideas might be feasible.”
Welcome, indeed, to the quark ages.