Let’s face it, physics has an image problem. If we were to play a little word association game, what are the first words that comes to mind when I say “physics”?
boring
hard
math-y (i.e. see points 1 and 2!)
Or, if we play an image association game, what are the first things you picture when I say “physics”?
Photo by Markus Spiske temporausch.com on Pexels.com
Photo by toomuchforme on Creative Commons
But here’s the thing – physics is so much more than cosmology and quantum computers, and you don’t have to be ‘Sheldon-Cooper smart’ to appreciate it. Fundamental physics principles help us to understand everything from how the heart pumps blood through the circulation system to why soap films form spheres when you blow air into them.
This is why I write: I love that a surprisingly small set of fundamental physical principles can explain the ‘why’ of just about any everyday phenomenon. I want to share that with anyone who will listen. The world is full of incredible examples of physics in action and you don’t need a PhD in quantum mechanics to appreciate them! And, although I feel a little late to the party in starting this blog now, I’m taking inspiration from something I saw at #SciCommTO 2020:
What do giraffe spots, garlic bulbs, tree canopies, and ice formations all have in common? They are just a few gorgeous examples of Voronoi patterns, also known as Thiessen polygons or Dirichlet tessellations.
I’ve been thinking about Voronoi patterns a lot lately because they have inspired our architecture friends in their visualizations of the science centre we hope to build one day – a building that is both inspired by nature and powered by nature. It doesn’t take much for me to fall down a fascinating rabbit hole and now I’m finding these shapes everywhere it seems!
For example, last week I was in Kingston to visit with physics colleagues at Queen’s University and they put me up in a lovely hotel right beside Lake Ontario. When I woke up on that chilly Friday morning in early March and looked down from my balcony, a thin layer of ice had formed on the water in the harbour overnight, creating a lovely display of Thiessen polygons to greet me.
And I thought: “of course it creates a Voronoi pattern!”
Ice forms through a process called nucleation – the molecules in the cold liquid start to cluster together and the resulting ice expands outward at a steady pace from each starting cluster. If many nucleation sites are randomly scattered across the surface of the water, individual ice sheets will grow outward from each seed point until they reach the edges of neighbouring ice sheets. By definition, the resulting pattern consists of Theissen polygons as illustrated in this excellent gif from Wikipedia:
The ice fragments shown above aren’t perfect Thiessen polygons because there are wind and water currents jostling the pieces around, but you can definitely see the overall tendency. Each polygon surrounding a seed point encloses all the locations on the 2D plane that are closer to that seed point than any other in the region. The edges of each polygon are locations that are equidistant to two neighbouring seed points.
You can explore this phenomenon by drawing your own Voronoi pattern, with a little patience. First randomly scatter a few seed points on a piece of paper. (The more points you have, the longer it will take to draw.) Then draw lines connecting all dots to their neighbours – I used dashed lines here. The final step is to draw a (solid) line across each connecting/dashed line that is at the midway point between two dots – this is part of the boundary of the resulting polygon. Here’s what my shapes look like for three central dots in my random scatter:
For giraffe spots, their signature look comes from a random scattering of the cells that produce melanin, the dark skin pigment, in the skin of the giraffe embryo. As the giraffe grows, the dark spots expand outward from those random seed points to fill the available space with Thiessen polygons, and the resulting pattern is largely unique to the individual. This study does a lovely job of modeling giraffe spots with the mathematics of Voronoi patterns, resulting in some pretty realistic looking creatures! On the left is an image of real giraffe skin and on the right is an image that was computer generated by the authors.
Individual garlic cloves in a bulb are a great 3D version of the same process – each clove grows radially outward at a steady rate from the central random seed point, eventually pushing up against adjacent cloves to fill the space with Thiessen polygons.
Bubbles on a 2D surface, cells in a leaf, the outward spreading canopies of individual trees in a dense forest … they all tend to fill the available space with these closely packed, irregular-shaped, polygons.
The hexagonal shape of honeycombs (as discussed in a previous blog post) is actually just a special case of these patterns. With honeycombs, bees are creating a Voronoi pattern with regularly-spaced seed points rather than randomlydistributed starting points. When we start with a structured pattern of seed points, as shown in my sketch here, we end up with hexagons surrounding each point. Those honeybees do like their symmetry!
And, although my tendency is to focus on examples from nature, there are lots of other scenarios where we encounter Voronoi patterns too, such as this incredible map of the globe that has been divided up into Thiessen polygons based on a location’s proximity to a regional airport:
Or this map of the 1854 cholera outbreak in London, England, where the white “P” on blue circle indicates the location of a shared public water pump. Dr. John Snow, a local physician, created the original version of this map to dramatically show the connection between proximity to a specific contaminated water pump (the Broad Street pump) and the highest density of cholera deaths, which was contrary to the commonly held belief at the time that the disease was spread through the air. I’ve superimposed the Thiessen polygon (lightly shaded yellow) that you get when you enclose all the locations in London that are closer to the Broad Street pump than any other pump in the neighbourhood, and it clearly encompasses almost all of the cholera hotspots shown in pink/red.
Someday, I hope you’ll be able to wander through our exhibit space in the science centre that might look a little like this:
But, until then, keep your eyes open for Thiessen polygons all around you!
Student Works: Cellular Tessellation pavilion lights the way in Sydney – Dec 30, 2014 by Amelia Taylor-Hochberg in Archinet; Photos by Patrick Boland photography
I snapped this photo in my backyard while enjoying the last bit of the summer holidays – just look at how busy this little critter has been, flying from flower to flower, collecting pollen. I’m fascinated by the role that physics plays in pollination, and not just the unlikely aerodynamics of bumblebee flight. Electricity plays an important part too, with electric forces pulling pollen from the insect to the plant, as well as from the plant to the insect. The same way your hair stands on end when a charged balloon is nearby, so too are the pollen particles drawn away from the stigma as the insect comes closer.
When flying around, insects such as bees become charged through friction in a process called the triboelectric effect, like when you shuffle in your slippers across the carpet and discharge when you touch a metal door handle. Different materials have different tendencies to become charged through this process – some become negatively charged, some become positively charged. Some easily pick up and hold a lot of charge, others don’t.
Way back in 1962, an experiment was conducted to assess the tendency of bumblebees to accumulate charge, concluding that they were one of the most easily charged materials tested. They also found that bumblebees always become positively charged through friction, building up charges of +30 to +50 pC during flight. These tiny charges have been measured experimentally by training insects to fly into Faraday cups, the charge-measuring devices used by nuclear and particle physicists in accelerator beams – bee counting rather than beam counting, if you will.
A Faraday cup used to count charged particles (or charged bees)Particle accelerator used in experimental physics
While the insects become slightly positively charged in the air, the plants on the ground possess small amounts of negative charge. As the charged pollinator gets closer, that positive charge pulls more negative charge up from the ground into the plant where it accumulates at the tips in significant amounts. There is enough opposite charge on the two objects to pull pollen particles off the surface, with forces greater than their adhesion and greater than the force of gravity too. There is a lovely illustration of this process in a recent paper by Dominic Clarke and collaborators, showing the mathematical model they used to study the phenomenon:
Figure 7 from doi: 10.1007/s00359-017-1176-6, open source
The image on the left shows the accumulation of charge on the pointy parts of the two surfaces involved and the image on the right shows the resulting paths taken by pollen grains moving between the two objects. This model doesn’t include wind and humidity effects, but it’s a pretty compelling study showing the importance of electrostatics to the overall exchange of pollen between plants, facilitated by our charged friends, the bees. And, electricity doesn’t just play a passive role here, lazily moving pollen from one place to another as an unsuspecting bee buzzes around. Not only can bees sense the electric fields around flowers, but they can even learn to use them to find nectar.
A study reported in Science in 2013 demonstrated that bumblebees could be trained to preferentially fly to artificial flowers containing a sucrose reward versus identical artificial flowers containing a bitter solution made with quinine (like in tonic water), just by adding an electric field. The sucrose flowers were indistinguishable from the quinine ones except for the presence of this electric field and the bees chose correctly 81% (+/- 3%) of the time after training. When the electric field was then removed, the bees did no better than random guessing, hitting the sugar jackpot only 54% (+/- 4%) of the time.
The insects in this 2013 study could even distinguish between flowers with different electric field geometries, such as fields around flowers with longer stigma or flowers in which the petals are more open. A subsequent study by this group looked at the effect of applied electric fields on the tiny hairs on bumblebees, showing that these fields cause the hairs to vibrate and thereby identifying the physiological mechanism for electric field detection. The image here shows a closeup of the hairs examined and the measured velocity of vibrations that resulted from different frequencies of applied electric fields.
I think it’s pretty cool that electrostatics is the driving force behind the transfer of pollen between insects and plants. But I think it’s even cooler that bees aren’t just along for the ride and free nectar! Bees can detect electric fields, learn to find nectar based on these fields, and may even communicate to their hive-mates through an electrified waggle dance upon their return from foraging. Now that’s something to buzz about!
Waggle dance performed by a honey bee to communicate details of where to find food sources when returning to the hive
Image Citations
Opening image – insect on yellow flower – image taken by the author
Cartoon image of someone getting shocked when touching a doorknob – Shutterstock.com ID 2080565146
Mathematical model to understand the process of electrostatic pollen transfer – Figure 7 from The bee, the flower, and the electric field: electric ecology and aerial electroreception. Dominic Clarke, Erica Morley, and Daniel Robert, 2017 J Comp Physiol A 203:737-748 doi: 10.1007/s00359-017-1176-6, open source
Close up of bee to show the hairs being studied and the resulting data collected of vibrational speed as a function of frequency of applied electric field – Figure 1 from The bee, the flower, and the electric field: electric ecology and aerial electroreception. Dominic Clarke, Erica Morley, and Daniel Robert, 2017 J Comp Physiol A 203:737-748 doi: 10.1007/s00359-017-1176-6, open source
Mechanosensory hairs in bumblebees (Bombus terrestris) detect weak electric fields. GP Sutton, D Clarke, EL Morley, D Robert, 2016, Proc Natl Acad Sci doi: 10.1073/pnas.1601624113 The bee, the flower, and the electric field: electric ecology and aerial electroreception. Dominic Clarke, Erica Morley, and Daniel Robert, 2017 J Comp Physiol A 203:737-748 doi: 10.1007/s00359-017-1176-6
Our amazing crew at Royal City Science has been brainstorming ideas for travelling exhibits that we can take to schools, festivals, Farmers’ Markets, etc. to share our passion for all things STEAM (science, technology, engineering, arts, and mathematics), while we fundraise to build a permanent home in Guelph. A blog post from the NFB recently caught my eye and sparked a little bit of an obsession with the idea of reproducing our own replica iglu/igloo because, as I dug into the details, I was truly blown away by the incredible brilliance of the design. As June is National Indigenous History Month, it seems timely to share what I’ve learned so far.
Can’t you just picture it? We arrive at a school gymnasium and throw 70-100 perfectly shaped pre-cut plastic blocks on the floor for the students to puzzle over and work together to assemble just so. And then they take turns to climb inside, to marvel at the stable and aesthetically beautiful structure Inuit are masters at assembling with nothing more than snow and a knife. The Arctic is an unforgiving landscape, with no trees to provide timber for the typical structures built in warmer regions. Shelter is essential for survival and Inuit succeed in this harsh landscape through their ingenuity, patience, and understanding of the natural world around them, collaborating with nature rather than trying to conquer.
If you live in a region that regularly gets snow in the winter, then you have possibly tried to assemble your own iglu at some point. Naively you cut rectangular blocks and position them in a ring, but it probably isn’t long before you realize that this simple approach isn’t going to work. First of all, you need the walls to slope inwards to form that classic dome shape, so your blocks need to be shorter on the inside edge than on the outside to a precise degree, as shown in the sketch I’ve made below from the NFB summary of their project.
The next thing to notice is that, starting at Block #1 in this sketch, the first layer of blocks is actually cut to form a ramp or slope that is maintained all the way up to the top of the structure in a continuous spiral. Picture a screw thread superimposed on the dome. Why? When you place the next block up the ramp, part of its weight is supported by the block beside it that you just positioned, giving it more stability. This is particularly important as you near the top where the blocks are angled quite significantly inwards. I mean the genius of this design is truly breathtaking.
The dome, as well as the related arch, have been used in building structures for thousands of years. A domed roof is a fantastic way to enclose a relatively big area without any internal supports – the curved shape does a great job of spreading forces applied at the top all around to the whole structure, making it incredibly stable. This is also why an egg is so strong, even when mumma hen is sitting on it!
So maybe now you can see why I want to make this happen and I reached out to some brilliant colleagues in the Department of Physics at the University of Guelph to explore the idea further. My first stop was Steve Wilson, our fabulous machine shop supervisor. He was (understandably) a little worried about the NFB blog description that “it was too complicated to pre-plan all the angles”; they measured and cut each piece by hand to fit for their exhibit. Fair enough, so my next stop was Bernie Nickel, a retired faculty member in our department who loves a puzzle, and in no time at all he came back with a mathematical model that seems to work remarkably well.
The images below compare Bernie’s models with the shape of the iglu shown in the NFB documentary, traced out on the screen and assuming that camera and screen capture distortions aren’t a big issue here. The blue symbols are the tracing marks from the film. The red curve in the first image is a catenary arch; the black is a catenoid iglu. Bernie wasn’t satisfied with either of these, so the second image shows his refined model of a catenoid iglu at the base, with a spherical cap placed on top. He is now working on a scale model with cardboard to see how it works and I am so excited to see the end product! Stay tuned!
UPDATE! (July 27, 2022)
Bernie is working away on the smaller scale prototype and I got to see it today …. IT. IS. AWESOME!!
Images provided by Bernie Nickel, personal communication, comparing his mathematical models with the shape of the igloo featured in the NFB documentary cited here
Image taken by me in Bernie’s office – 39 of the full 80 blocks have been made by hand out of cardboard by Bernie and he is working away on completing the set. Then we talk to Steve about scaling it up! (The red string is to mark where the door is)
References
How to build an igloo by Laurence Desrosiers-Guité and Dan Thornhill, NFB Education.
How to build an igloo – Documentary film by Douglas Wilkinson, filmed in 1949 with support of the NFB. Please note this message from the NFB before viewing: this is an archival film that makes use of the word “Eskimo,” an outdated and offensive term. While the origin of the word is a matter of some contention, it is no longer used in Canada. The term was formally rejected by the Inuit Circumpolar Council in 1980 and has subsequently not been in use at the NFB for decades. This film is therefore a time-capsule of a bygone era, presented in its original version. The NFB apologizes for the offence caused.
Mathematics of Igloo Design – by Bernie Nickel, personal communication
The Igloo, by Charlotte and David Yue, Houghton Mifflin Company, Boston, USA, 1988
Last month, the world celebrated the 7th annual International Day of Women and Girls in Science (IDWGS). I was honoured to be featured by my institution for my work in fostering the natural curiosity of young girls through our fun video series, among other initiatives. I LOVE talking and doing science with kids, so getting the opportunity to share this passion more widely is fantastic. But today, on International Women’s Day (IWD), I find myself struggling to ‘celebrate’. I HATE that, in 2022, we still need an international day to shine a light on the lack of gender equality in so many fields, including science.
A little over 30 years ago, I graduated from high school in Toronto and headed off to university to study science. My strongest marks by far were in physics and math, but all my teachers ever focused on when giving career advice was medicine or nursing: no one suggested I pursue physics or engineering. Being a little contrary, I decided to give the applied physics program a go, telling myself I could always switch out if it proved too difficult. After all, girls didn’t do physics or engineering 30 years ago, it was almost exclusively a boys’ club back then.
Fast forward 30 years and here’s the incredibly depressing thing – the statistics haven’t changed. Physics, math, and engineering are still boys’ clubs.
How significant is this stagnant imbalance? Comparing StatsCan census data on post-secondary qualifications by field of study, we find that the percentage breakdowns by gender in every STEM (science, technology, engineering, and math) category have barely changed from 1996 to 2016:
More recent data from colleagues at the University of Guelph (2019) demonstrate that we lose women from physics and engineering at just about every step of their secondary schooling in Ontario:
Image from the “Closing the Gender Gap” report from the Ontario Network of Women in Engineering, 2019
It’s not like we’ve had our collective heads stuck in the sand for three decades. There are scores of non-profit organizations in Canada that have worked for 20 or 30 years to increase the number of students pursuing the STEM disciplines at the post-secondary level. These organizations all recognize the ongoing lack of women in certain disciplines, and countless programs have sprung up in response.
Some progress has been made, as seen in tracking what happens when we ask children to ‘draw a scientist’: the percentage of girls drawing a female scientist has increased substantially over a 50-year period. But we still have a lot of work to do, as both girls and boys are more likely to draw a male scientist as they get older:
Why does this matter? Because men don’t have a monopoly on curiosity and wonder. Spend some time in a kindergarten classroom and you’ll see that the girls are just as inquisitive about how the world works as the boys. Somewhere along the way, the boys get the message that physics and engineering are the fields in which to explore this curiosity and girls find themselves heading towards health-related fields. I’m furious that we still find ourselves here, 30 years later. The definition of insanity is doing the same thing over and over again and expecting a different result. Something must change.
And let’s deal with the elephant in the room right now: yes, girls are just as capable of understanding physics as boys are. Data from standardized exams in the UK as recently as 2018 show that 30% of the girls writing the A-level physics exam earned the highest two grades possible compared with 29.5% of the boys who wrote the same exam. But just like Canadian students, the numbers are hugely skewed: 8300 girls wrote that exam compared to 29,400 boys, i.e. 22% vs. 78%. Sound familiar? I cite the UK data for two reasons: 1) the standardized nation-wide exam gives us large n statistics to look at and 2) the Institute of Physics (IOP) seems to be lightyears ahead of other national professional organizations in championing this issue and aggressively seeking solutions.
The IOP is currently running a suite of gender balance programs, building upon decades of research into the issue. The main conclusion so far seems to be that tinkering around the edges by working with individual teachers or even whole science departments in specific schools doesn’t move the needle. However, effective change has been achieved in pilot projects that holistically tackle the culture of schools from early-learning centres through to secondary schools.
One pilot project worked on improving the confidence of 13 and 14-year-old girls directly and worked with physics teachers on best practices for inclusive teaching techniques, while simultaneously taking a whole-school approach in working with leaders, teachers, and students across all subjects on gender equity. This project saw the number of girls starting A-level physics (at age 16, comparable to our Grade 11) increase by a factor of three over a two-year period.
Change is possible if we are prepared to look in the mirror and recognize that we are failing our students by tinkering around the edges. STEM-themed summer camps and after-school programs for girls are great, as are targeted scholarships for women in postsecondary studies, but they tend to reach the minority who manage to maintain their passion for the discipline despite the structural biases working against them. Let’s follow the lead of our UK colleagues and recognize that systemic change, from kindergarten to graduate school and beyond, is the only way to truly address this decades-long issue, before another 30 years goes by and my grandchildren are facing the same obstacles. In the words of Marie Curie, progress is neither swift nor easy. But when it comes to gender equity in science, it is long overdue.
This adorable example of everyday physics took me entirely by surprise. We’ve all seen a sweet little floating parade in the springtime at our local river or pond, with a mumma duck or goose followed closely by a line of babies paddling quickly to keep up. But I never really noticed that the spacings of each duckling is quite consistent down the line. Or that the formation tends to be single file much more than other possible arrangements. Why? It’s all about the wave pattern generated by the duck in the lead position and how the wakes of the other ducklings in line interact with that pattern. Scientists have recently concluded that these sweet little fuzzballs find the exact right position in line every time so that they can surf along with a boost of energy to keep them moving with ease.
Two adult geese followed by three goslings in single file – a video I shot last May, completely unaware of the physics on display!
It honestly makes me smile to think that I have witnessed a fantastic example of wave interference countless times without realizing it. The only reason I now understand what’s happening is because I stumbled across a lovely little paper in the Journal of Fluid Mechanics a month or so ago by a collaboration from the University of Strathclyde (Glasgow, Scotland) & Jiangsu University of Science & Technology (Jiangsu, China). This team set about modelling the fluid dynamics both mathematically and numerically, discovering that these wee fluffs are both cute and smart: when they swim at the ‘sweet spots’ behind mumma, they instinctively choose positions that correspond to dips in the overall wave pattern. For ducklings in positions 1, 2, and 3, riding the crest of a wave (just under their little butts) gives enough of a boost that it is energetically favorable for them to swim there. The model found that ducklings in positions beyond #3 don’t gain in energy, but there is also no energy loss to drag in the water – a delicate dynamic equilibrium is established. Plus, ducklings have been observed to switch up their order in line, which might be an egalitarian approach to sharing the surfing boost among sibs.
Close up of the photo seen at the beginning of the article, with red lines added by me to show the crests of the wave pattern just under the butts of each of the three ducklings immediately behind mumma (left) and an image from the paper by Yuan et al (right) showing the optimal positions of ducklings based on modelling.
It honestly blows my mind a little that math and physics can explain why ducklings swim in single file and can even predict the positions they will choose behind mumma; this article by Yuan and team was a lovely discovery. Their paper also led me to an equally delightful experimental study in which biologist Frank E. Fish from West Chester University measured the mechanical effort exerted by ducklings swimming in formation, with and without the presence of a mumma generating a wake for them to swim in. Dr. Fish imprinted his subjects to a decoy and then had the babies swim behind the decoy in a tank in which the flow rate they swam against was controlled. The decoy was always visible in front of the ducklings, but it could be raised slightly above the water so that it didn’t create a wake.
Sketch of the clever experimental setup, figure 1 from the paper by Dr. Fish, copyright Wiley-Liss Inc.
Instinctively, the ducklings self-organized into single-file formation behind the decoy. Dr. Fish then analyzed video recordings from the side of the clear tank, calculating the arc length of the foot stroke taken by a duckling as a measure of the effort exerted. Mumma’s wake, when she was lowered into the tank, resulted in a reduction in effort for all the ducklings studied – more so for the younger babies (3 days old) than the older ones (14 days old). Consistent with the mathematical modelling from Yuan et al, by swimming in the sweet spots, the ducklings take advantage of wave interference to expend less energy to paddle along behind their mumma.
Even though I wasn’t actually wondering why in this instance, now I know the physics behind the formation! So next spring, when I see this amazing example of fluid dynamics in action, I’ll be smiling at both the spectacle and the science.
Ducklings swimming behind mumma – Shutterstock.com ID 1381730477
Video of Canada geese family – taken by the author, May 2021, Rockwood, ON
Mathematical model results of the wave pattern and the coefficient of drag (CDR) for mumma duck and 6 babies – Figure 4 from Yuan, Z., Chen, M., Jia, L., Ji, C., & Incecik, A. (2021). Wave-riding and wave-passing by ducklings in formation swimming. Journal of Fluid Mechanics,928, R2. doi:10.1017/jfm.2021.820 (Huge thank you to Cambridge University Press for believing in Open Access publishing!)
Close up of ducklings swimming behind mumma – same Shutterstock image as the opening file with red lines indicating wave crests in the overall pattern added by the author
Sketch of the experimental setup of Dr. Fish, Figure 1 from the Journal of Experimental Zoology (1995) 273: 1-11, Copyright Wiley-Liss Inc.
(Alt text – A cotton candy vendor walking under the forest canopy during a morning in Bardiya National Park in Nepal, light filtering through the canopy. By Gaurav Aryal, Shutterstock.com)
Like everyone else, the start of the new school year has been incredibly stressful for me. Going for walks in the arboretum on campus has been a huge coping mechanism, especially on the days when I just want to go home and curl up in a ball with the blankets over my head.
The arboretum is a spectacular sight this time of year, with maples, birches, and beeches showing off in russets, scarlets, and golds. But there’s a quiet little grove in the middle of this amazing space that also has a treat for your nose: if you’re standing in the right spot and inhale deeply, be ready for caramel, fresh-out-of-the-oven cake cooling on the counter, or, in my opinion, cotton candy. The source of this incredible scent is the katsara tree (Cercidiphyllum japonicum), a tree native to Japan and China that was imported to North America in the mid-1800s.
As the heart-shaped leaves begin to turn from green to yellow to brown every autumn, an aromatic molecule called maltol is released to the air, in concentrations high enough that it can smell like walking past the waffle station at your favourite brunch spot. The scent is so strong and so distinctive that katsuras are known as “kuchenbaum” (cake tree) in German or “arbre à caramel” (caramel tree) in French.
Maltol, chemical formula C6H6O3, occurs naturally in certain foods like coffee and cocoa, getting converted from sugars to this amazing smelling molecule in the baking or roasting process. Maltol is also found in pine needles, the bark of larch trees, and red ginseng. The white crystalline powder with the butterscotch fragrance is extensively used in the food industry as a flavouring agent as well as an additive in perfumes.
When I first noticed the smell a few years ago, I was honestly completely baffled. It was always in the same spot but it wasn’t always noticeable. Some days it would be incredibly strong but then months would go by without the slightest whiff. Looking around for the source of the scent, there are no flowers (or bakeries) in sight. But, with the help of the arboretum signage and Google, the mystery was revealed. And I was greatly relieved to find out that I wasn’t having some kind of stress-induced olfactory hallucination!
The map of the arboretum – the katsura collection is #6 here
The legend from the arboretum map
The sign at the foot of the katsura tree that helped me to figure out the source of this amazing smell
The path through the katsura grove in the Arboretum at the University of Guelph. Trees on either side of a dirt path, a light dusting of yellow/brown leaves on the ground.
With a little digging, I found myself reading about some pretty incredible properties of this deliciously scented molecule. Researchers are exploring its antioxidant capabilities and its potential application in treating a huge array of concerns, from liver disease to osteoarthritis, glaucoma to bacterial resistance to common antibiotics. Among other functions, maltol works to control the amount of metal in the human body, forming stable complexes with metallic ions like Al+3, Fe+3, Zn+2, etc. Some of these complexes have been investigated as possible treatments for anemia, Alzheimer’s, cancer, and diabetes. So it turns out that my favourite spot to destress in the arboretum is full of amazing science, right under my nose.
References
Maltol, a Food Flavoring Agent, Attenuates Acute Alcohol-Induced Oxidative Damage in Mice. Ye Han, Qi Xu, Jiang-ning Hu, Xin-yue Han, Wei Li and Li-chun Zhao Nutrients 2015, 7(1), 682-696
Novel Synthesis of Maltol Capped Copper Nanoparticles and Their Synergistic Antibacterial Activity with Antibiotics. Naqvi et al 2021 Plasmonicshttps://doi.org/10.1007/s11468-021-01452-3
Maltol prevents the progression of osteoarthritis by targeting PL3K/Akt/NF-kB pathway: in vitro and in vivo studies. Lu et al 2021 J Cell Mol Med25: 499-509 https://doi.org/10.111/jcmm.16104
(An ashy-headed goose demonstrating its exceptional balancing ability; image from Shutterstock.com)
The other day I spent some time exploring the “arm” at Guelph Lake, curious to see how far up the Speed River I could get in my kayak before bottoming out. (Turns out, not very far. Water shoes would have been a good idea. I definitely need to try this earlier in the season when the water is higher.)
(map of Guelph Lake with Speed River indicated)
It’s a very quiet stretch of water and, as I came around a bend with a magnificent willow holding court on the bank, I paused in my paddling to drift soundlessly past a large flock of Canada geese napping and preening in its shade. Some were standing, some were resting on the ground with their heads tucked around on their backs, and quite a few were balancing nonchalantly on one leg. Why is this a comfortable resting position? If I have a few quiet minutes on a summer afternoon, I don’t feel inclined to strike a tree pose on a riverbank …. what’s up with our feathered friends?
Canada goose balancing on one leg on a rocky beach; from Shutterstock.com
Woman in tree pose beside water at sunset; from Shutterstock.com
Canada geese are not the only bird to strike such a pose – flamingos and other long-legged aquatic birds such as storks and herons are perhaps more often pictured in this position. A recent report in Functional Ecology estimates that standing on one leg has been observed in approximately one third of the 852 avian species in their study.
In order to stay upright in this position, a two-footed creature (goose, heron, human, etc.) has to shift their weight slightly to the side to get their centre of mass over the single supporting foot, as is evident in the head-on photo of the ashy-headed goose at the top of this article, as well as in this video clip of my ever-helpful daughter, Mara, filmed for our AMASE video series: (sound on)
It takes effort for us to maintain this crucial vertical alignment of our centre of mass and the small area of support afforded us when only one foot is touching the ground. So what’s the advantage for birds to stand this way? There are two possible motivations commonly proposed by biologists: heat regulation and reducing muscle fatigue. However, it’s also possible, based on a recent study reported in Biology Letters, that it’s all about forces and torques (aka mechanics)!
First, let’s talk about the standard theories. When you see a long-legged bird standing on one foot, the retracted leg is often tucked up underneath the body, hidden inside the cozy plumage. This gives rise to the heat regulation theory: when the surrounding air is cold, a lot of body heat is lost through the legs since they are not covered with insulating feathers. Birds do adjust for this by reducing blood flow to their legs when its cold outside, as well as employing a neat heat-exchange trick between the cold blood coming back to the body from the feet and the warm blood heading down from the heart. Biologists have speculated that the one-legged pose is a means of further reducing heat loss during chilly weather by insulating one of the bare legs inside the chest feathers. But this couldn’t be the reason for my Guelph Lake geese friends, as the weather was a balmy 24°C that afternoon. And observations in the literature are mixed as to whether this pose is seen more often at lower ambient temperatures. Plus, the lifted leg is not always hidden inside the plumage; sometimes it’s only partially lifted and sometimes it’s extended back behind the bird. It seems that heat conservation has a role to play, but it can’t be the only reason.
Ducks regulating heat loss by standing on one leg on an icy surface, as well as shielding their bills in their feathers to prevent heat loss there. From Shutterstock.com
The other theory is something you have probably done – shift your weight from one leg to another when you stand for long periods of time. By shifting your weight, you are temporarily giving the muscles in one leg a little break. Seems reasonable that birds might do the same.
(Gif of the Brady Bunch, rapidly switching weight from one leg to the other, from Giphy)
However, there is a big difference here: we don’t lift the resting leg completely off the ground into mid-air when we give it a break. This takes energy and requires more muscle firing in the supporting leg to keep our balance over a smaller supporting area. According to a thorough summary by Dr. Reinhold Necker, professor emeritus from the University of Bochum in Germany, “there is so far no experimental support” that this one-legged bird pose is struck to reduce muscle fatigue. Which brings us to the 2017 paper in Biology Letters.
The study conducted by Young-Hui Chang and Lena H. Ting (Georgia Tech and Emory University, respectively) looked at both the anatomy and the behaviour of the one-legged stance in flamingos. With cadaver samples, the team manipulated the body of the flamingo while it was held in a one-legged pose by clamps, testing the passive stability of this position. The researchers also analyzed the patterns of the pressure exerted by the foot of live birds while they stood on one foot on a force plate, both awake and asleep.
The clever combination of experimental manipulations of cadavers and measurements of exerted pressures on force plates from living subjects led the team to conclude that it may be energetically favourable to stand on one leg due to a passive, gravity-driven body weight support mechanism in the lower joints of these birds.
From the cadaver studies, Chang and Ting observed a balance between the rotation about the knee and hip joints due to gravity and a countermeasure arising from anatomical limits on the range of motion of these joints – i.e. certain bony and cartilaginous structures may act to block the joints against the destabilizing rotations that would otherwise happen. Chang and Ting observed that this anatomical countermeasure only activates when the foot is positioned a little inside the hip joint, as it naturally is when a bird (or Mara, see above) tries to stand on one leg!
The force plate data from the live subjects further confirmed that flamingos are incredibly good at passively balancing on one foot, as the point of highest pressure remained static for long periods of time, even when the birds had their eyes closed. Try standing in tree pose for a few minutes and focus on how much the contact between your foot and the floor is shifting to maintain your balance – I predict that it moves around considerably more than the corresponding observations in flamingos by Chang and Ting.
(Image from Chang and Ting, 2017, Biology Letters13: 20160948)
As a physicist, even I realize that applying a study in flamingos to understanding an observation in Canada geese is a little suspect. But, as a physicist, I love that an everyday sight such as geese relaxing on a riverbank has led me to discover a free-body diagram in a paper in Biology Letters! Partially thermodynamics, partially mechanics, and wholly fascinating, I now have a slightly better understanding of why geese are yoga gurus. Can’t wait to share this with my first-year biological science students the next time I’m at the front of the room in PHYS*1300!
Chang Y-H and Ting L. H. 2017 Mechanical evidence that flamingos can support their body on one leg with little active muscular force Biology Letters13: 20160948 http://dx.doi.org/10.1098/rsbl.2016.0948
Pavlovic G., Weston M. A., Symonds M. R. E. 2019 Morphology and geography predict the use of heat conservation behaviours across birds Functional Ecology33(2) 286-296 https://doi.org/10.1111/1365-2435.13233
On an icy cold morning at the end of February, I tentatively headed down the slope behind our house, on my way out for some daily exercise. As I stepped off a retaining wall down to the walkway below, my foot went right out from under me and I crashed down, hard. The small of my back came down on the edge of the six-inch high concrete step as the rest of me landed on the lower level − I experienced the most intense pain I have ever felt. And that includes the drug-free labour and delivery of our first child.
There was no one in the house … no one to hear me scream. No matter how I tried to gingerly move, bolts of searing pain shot through me. I knew I was in trouble, so I sheepishly phoned a friend who raced over, took one look at me, and said: “I’m calling 911!” The combination of this fall[1] and that call quite possibly saved my life because, on our way to the local hospital, the incredibly wonderful paramedic said: “I know you’re in a lot of pain and quite stressed right now, but your blood pressure is really high. You need to follow up on that!”
How high? It was consistently reading over 180/100 (mmHg) that morning. This is WAY above the normal range of 120/80, and solidly in the danger zone of risk of heart attack or stroke. According to the FDA, readings in the 180/110 range or higher are considered “hypertensive crisis” and need immediate medical attention. And here’s the truly scary part in hindsight – I had absolutely no idea. It’s not called ‘the silent killer’ for nothing.
High blood pressure (hypertension[2]) affects almost 1 in 4 Canadian adults according to a 2015 study, with the prevalence increasing dramatically with age. We tend to think of this as a concern for our parents or grandparents, or at least I naively did, but the incidence rate in women in the 40 to 59 age range (i.e. me) is actually 21%. Of the hypertensive women in my age range, about 20% were not aware of their condition either.
Why is this an issue? Because hypertension is currently the leading risk factor for death in the world. High blood pressure means that your heart has to work harder to move blood around your body – like trying to pump air into a bike tire that is already fairly full – so the muscle of your heart thickens and is more prone to suffering a heart attack or failing. The vessels themselves can also narrow and get damaged, making blood flow more difficult. And it’s not just damage to your heart, as if that isn’t serious enough. The damage to blood vessels happens everywhere, including in the brain. Left unchecked, restricted blood flow can lead to blockages or bursts, i.e. strokes and their debilitating consequences. High blood pressure also raises the risk of developing vascular dementia, the second most common form of dementia. Reading that stopped me cold in my tracks.
For a middle-aged woman, I’m reasonably fit. Our family eats mostly home-cooked meals, avoiding the very high salt content typically found in prepared foods. I walk daily, play with my kids, row/kayak in the summers, snowshoe in the winters; 10,000 steps are pretty standard on a daily basis. But I do have a family history of concern and I should have been paying closer attention. It’s just so easy, when running a busy household and a demanding career, to de-emphasize your own health in the ever-fluctuating list of priorities. I also experienced no symptoms whatsoever, at least I didn’t think I did. It turns out that the symptoms can be pretty vague and easily chalked up to daily stresses. I mean ‘headaches’, ‘fatigue’, … who doesn’t suffer from these regularly, particularly in our current pandemic crisis?
I can guarantee you that I would not have scheduled a routine check-up with my family doctor for many months (years?) to come, given all that is going on in our lives. But my guardian-angel paramedic’s warning on my very first (and hopefully last) ambulance ride struck a chord, particularly since I lost my mother to a massive stroke when she was only 67 years old. Her premature dementia in the final years before her stroke, as well as the stroke itself, are fates I wish to avoid. Fast forward a few months and my family doc now has me on some low dose meds, I’ve amped up my exercise regime, and I have significantly reduced my alcohol intake (yes, during a pandemic, it can be done!).
To all my fellow middle-aged women, consider this a friendly nudge. It’s all too easy to dismiss little warnings as no big deal; I did, even though I should have known better, and it could have cost me my life.
[1] It turns out I was INCREDIBLY lucky in my fall – no broken bones, no organ damage, no slipped disks. Some deep muscle bruising and strain that took a couple of weeks of bedrest to begin to heal. More or less full mobility was back in about a month. HUGE thanks to my good friend, Cindy Johnson, for rushing to help!
[2] I’m using the definition of hypertension of a systolic pressure (the top number) greater than 140 mmHg OR a diastolic pressure (the bottom number) greater than 90 mmHg. This is technically Stage 2 hypertension, with Stage 1 usually considered at systolic greater than 130 mmHg or diastolic greater than 80 mmHg.
A few weeks ago during dinner there was a sudden burst of sunlight in the midst of a rain shower. Knowing that these were ideal rainbow-making conditions, we jumped up from the table and ran out to investigate. Sacrificing our meal was entirely worth it: this photo was taken just a few hundred metres down the road from our house. Once again, I am blown away by the beauty of physics in the world around me.
The bright feature in the photo labeled A is the primary bow, which is always observed with the red band on the outer edge of the arc and the violet band on the inner edge. Light coming from the Sun, behind you, scatters in all directions from the raindrops in front of you. The light doesn’t scatter equally in all directions though, as shown in the following image, modified from a 2016 paper by Alexander Haußmann. In this plot of intensity versus scatter angle, zero degrees represents light that is scattered straight back towards you, while light scattered directly away from you is at an angle of 180°.
(image from Haußmann’s 2016 paper)
We can see that the intensity is pretty steady at small angles, coming straight back towards you, then reaches a peak at 42°. This is the angle where you see the primary rainbow: 42° is the maximum angle of the cone of light that scatters straight back towards you from one bounce inside a raindrop.
At the very edge of this cone of scatter, we see the separation of the different colours of light that come from the Sun: the highest wavelength, red, scatters back at a slightly higher maximum angle (42.2°) than the lower wavelengths, like blue, which has a maximum angle of 41.2°, as we see in the sky with the red on the outside edge and violet on the inside of the primary bow.
Moving along the relative intensity graph from 42° to the next large spike at 51°, there is a region known as Alexander’s band, named after Alexander of Aphrodisias (circa 200 AD), which is marked B in my photo at the top. The band ends at 51°, which is the minimum scatter angle for light to bounce twice inside the drop and then reach our eyes as sketched here. The angle of 51° corresponds to the location of the secondary bow (C).
Alexander’s band is dark because it spans the ‘dead zone’ between the maximum angle for scattering with one bounce and the minimum angle for scattering with two bounces inside the droplet. Since light can’t bounce 1 ½ times, the intensity in this in-between range is quite low and the sky looks darker here than above the secondary bow or below the primary bow. It’s not midnight black though, because light is being scattered from clouds, other droplets, the front surface of droplets, etc.
The secondary bow, with its sharp spike at 51°, will always appear with the red band on the inside and the violet band on the outside, opposite to the order seen in the primary band. It’s not always easy to see this feature though; the intensity is less than that of the primary bow. With each bounce, some of the light continues through the droplet rather than reflecting, which means that the tertiary (and higher order) bows are rarely observed. The added challenge with seeing the tertiary bow is that it appears at about 140°, very much in the forward direction, away from the observer. To see the tertiary or quaternary bows, you have to be facing the Sun with scattering raindrops between you and the Sun, which, shining brightly, tends to wash out faint features like these weak bows. The angular spread of the colours increases with each order of bow, so this smearing out also makes them harder to see. The fifth order bow is predicted to appear just inside the secondary bow, as seen as the little spike in the intensity graph at around 50°.
We have marveled at, and tried to understand, rainbows for millennia. Way back in 1637, Descartes published a chapter in his Discours de la Méthode in which he correctly calculated the angles observed for the primary and secondary bows, although the colours were still a mystery in his day. Newton famously solved this mystery by deducing that white light is made up of all these colours. But it wasn’t until the early 1800s when Thomas Young linked the colour of light to its wavelength that physics was able to explain the observation of “supernumary arcs”, shown here in a cropped piece of my photo, marked as D.
Bright and dark bands inside the primary bow are called supernumerary arcs
These bands of light and dark regions just inside the primary bow are classic interference patterns caused by light scattered at the same angle but travelling different path lengths through a raindrop. Young’s theory correctly predicted the effect of drop size on this feature of the rainbow and provided further support for the wave theory of light.
Once James Clerk Maxwell figured out that light is transmitted in the form of electromagnetic (EM) waves (around 1864), a new theory emerged to understand light scattering from small objects like raindrops, known as Mie scatter after the German physicist Gustav Mie. By breaking the incoming EM wave into a series of partial waves, we then calculate how each scatters from a sphere of a particular size and electrical properties, and then add up the results. Intensive calculations with Mie theory were not possible until we had the help of computers in the latter part of the 20th century, but now millennia of wonder and delight has been explained, matching observations beautifully.
Refinements continue today, with tweaks to deal with non-spherical droplets (affectionately known as the ‘hamburger bun’ model), different sizes of droplets within the rain shower, effects of the intensity of rain in different regions of the shower, and more. Science marches on, so grab your rubber boots and let’s dash out, chasing rainbows. My little helper unicorn likes to refresh with a rainbow treat when we get back inside!
References:
Rainbows in nature: recent advances in observation and theory by Alexander Haußmann 2016 Eur. J. Phys.37 063001
As I strode across campus to meet my second-year physics students in our Electricity and Magnetism class a few years ago, it suddenly struck me that I had the makings of a fantastic opportunity tucked under my arm. My teaching assistant had returned their last assignment of the semester to me, so I quickly formulated my plan. As I entered the room and prepped for our session, I projected a stern and serious air, quite unlike my usual smiley/jokey self. As everyone settled down, I told them that I was very disappointed to hear from our TA that there had been a considerable amount of ‘sharing’ in the assignments that had been submitted, which is completely unacceptable. I told the class that I wanted them to take out a piece of paper immediately and work through the solution to one question again, individually, right then and there.
There was much muttering and sharing of furtive looks around the room, but everyone complied as I projected the question onto the screen with the document camera. One or two students looked at me suspiciously, but I maintained my grave look and told them to get started. After a few minutes, I said to the class: “Please make sure you write your name and the date on the page …. today is April 1st.” As my students slowly looked up at me, realization dawning on their faces, I shouted: “APRIL FOOLS!” It took a good 10 minutes before we were all settled down again and ready to discuss the physics of magnetic materials, but I maintain that it was 10 minutes well spent.
Incentivize Lecture Attendance
Study after study has demonstrated the value of a ‘flipped’ classroom, in which students engage with each other and the instructor in meaningful and deep learning activities (1,2). This little April Fools’ prank was certainly not such an activity, but it is a good example of the value of being a little silly with your students: it incentivizes lecture attendance. Step one for maximizing classroom engagement is to make your session an adventure, an event that students don’t want to miss because they never know what might happen. You care about their learning, you go to great lengths to fine-tune your pedagogy based on current physics-education research, but you need to get them out of bed and through the classroom door for all your hard work to pay off!
This is something I have struggled with during the pandemic with all my teaching done remotely. Being jokey and improvisational with my students has been incredibly difficult when only a tiny fraction of the class turns their camera and/or mic on – it is almost impossible to ‘read the room’ in Zoom. Even with these challenges, it seems more important than ever to find ways to inject some levity given the state of the world around us: I’m a firm believer in the notion that laughter truly is the best medicine. So, after grumbling about the countless ways in which technology foils my best-laid plans for an engaging virtual class, I decided to take advantage of the humour in the farcical: my students and I created a bingo game based on the myriad ways in which our sessions go sideways! (A spherical cow stress ball, pictured below, is the prize for completing a row or column.)
Break Down Barriers
In addition to incentivizing lecture attendance, injecting humour into our sessions also helps to break down barriers between my students and me. As evidenced by our bingo card, my humour tends towards the self-deprecating kind, which subtly communicates to my class that I really don’t take myself too seriously. I want my classroom to be a safe haven, where students can ask questions to each other and to me without worrying about being judged. No one, including me, has all the answers or is right all the time; mistakes are the best opportunities for learning.
So, rather than projecting an air of unassailable authority, one who, for example, would never drop a negative sign, I celebrate these moments with my students with Smarties. As we work through an example together, I want them to be trying to follow along with each step as we go, rather than just passively writing it all down to figure out later (i.e. the night before the next test). In our sessions, everyone is on high alert because the first person to catch a mistake, such as that inevitable loss of a pesky negative sign, is rewarded with a box of Smarties. Usually, in our face-to-face sessions, I have a couple of boxes in my laptop bag to hand out on the spot. With remote teaching I had to find a work-around and now I keep a running tally during the semester on my Smarties List, with payout at the end of term. This year, since I was actually keeping track, I added a personalized Smarties certificate to this special delivery, with one pictured below on proud display on the family fridge door! Students seemed to really appreciate the personalized touch so I’m keeping this new tradition going even after we are back in face-to-face sessions. (FYI, if you want to adopt this approach, definitely have extra boxes on hand if you need to teach after taking cold medication. Just saying.)
A COVID-inspired modification to my use of SmartiesTM in class to encourage careful scrutiny during our in-class examples as well as to normalize mistakes.
Improve Student Attitudes
I am not a stand-up comic in academic regalia. But you don’t have to be hysterical to make good use of humour in the classroom and it can have a profound effect on students’ attitudes. A recent study looked at the effect of a positive attitude towards math and the brain’s ability to learn and remember (3). The study concluded that even when they controlled for IQ, working memory, math anxiety, general anxiety and general attitude toward academics, children with poor attitudes toward math rarely performed well in the subject. And, let’s face it, in our big first-year service courses there are a lot of incoming students who do not have a positive attitude towards physics. In addition to teaching second-year Electricity & Magnetism to physics majors, a preacher to the converted, I frequently find myself in front of hundreds of first-year biology students, which is more like being an emissary to a hostile nation. In my very first lecture every year, I ask them to tell me how they are feeling about taking this course using their clickers to respond anonymously, and the breakdown invariably looks like this:
Typical responses from first-semester biology students who are required to take my introductory physics course.
There are times when there are identically zero responses in the “excited” category. In a room full of 400 students. *sigh* Making a concerted effort to help shift our students’ attitudes towards the subject is key to opening the door for them to do well, as that positive attitude results in enhanced memory and more efficient engagement of the brain’s problem-solving capacities according to the work of the team at Stanford (3).
A Little Goes a Long Way
Whenever I can I’m injecting little scenarios to make students smile. We work through a lot of example problems in class and I try to give them a little twist. For example, in free fall calculations, we work through a problem in which “Gord is spitting (sunflower seeds) from a bridge. Just to see how far down it really is.” This precise turn of phrase won’t mean much to non-Canadian readers, but this is a line from a Tragically Hip song, a hugely popular band that appeals across generations. The delivery is important when I read the question aloud and, although not everyone will get the reference, there are plenty of smiles around the room. We have also been known to analyze the position versus time graph of someone canoeing across a lake to an island where they suddenly encounter a mumma bear and her cub, as well as determining how far across an ice floe a sweet little penguin will slide before coming to a stop. This sometimes leads to a discussion on how one would experimentally determine the coefficient of kinetic friction between a penguin’s belly and an icy surface in terms of the factors one would have to control for, but I digress.
I am always looking for opportunities for student engagement and these often take the form of multiple-choice conceptual questions to use in the peer instruction mode of delivery, originally popularized by Eric Mazur at Harvard (4). I ask a question, students vote individually and we see the results. They then chat about their choices and vote again, after which we have a discussion about the correct answer (and, often, why the incorrect answers are wrong). Throwing a ridiculous choice or fun scenario into the mix often lightens the mood, like the following question that I pose as we start talking about conservation of momentum right around Halloween:
Two physicists decide to blow up a pumpkin. When it explodes, it splits into 3 pieces of equal mass, as shown from above in the illustration here. What is the direction of the velocity of the third piece?
(Yes, the image here is a still from a video that my colleague Chris and I made of blowing up a pumpkin with a bottle of liquid nitrogen!)
I like to couple our peer-instruction discussions with demonstrations, trying to include student helpers whenever possible. I always structure these as Predict-Observe-Explain, where students vote on what they think will happen, the demonstration/experiment is conducted, and then we discuss what happened and why. A popular one we do when talking about resistors in series and in parallel is the smoothie race – a head-to-head student battle with two straws in series vs. two straws in parallel to see which “battery” generates the greater current. This is a fun way to visualize currents/batteries/resistors as well as another good opportunity to discuss experimental design. But just as importantly, there is lots of cheering and laughter; a spoonful of sugar to help the physics go down!
The great smoothie race: an in-class demonstration of the difference between resistors in series and in parallel.
Every teacher develops their own style with practice and guidance, and I had the good fortune to be mentored by exceptional educators as I began to shape mine at the University of Guelph almost 20 years ago. I learned from some of the best in the business that humour is a powerful tool in the lecture hall for incentivizing class attendance, creating a welcoming environment, and improving student attitudes. To my mind, there is always room for a little tomfoolery in teaching physics!
References:
F Finkenberg and T Trefzger 2019 Flipped classroom in secondary school physics education J. Phys.: Conf. Ser.1286 012015
David C.D. van Alten, Chris Phielix, Jeroen Janssen, Liesbeth Kester 2019 Effects of flipping the classroom on learning outcomes and satisfaction: A meta-analysis Educational Research Review28 100281 https://doi.org/10.1016/j.edurev.2019.05.003
Chen L, Bae SR, Battista C, et al. Positive Attitude Toward Math Supports Early Academic Success: Behavioral Evidence and Neurocognitive Mechanisms. Psychological Science. 2018;29(3):390-402. doi:10.1177/0956797617735528
Wonder Why 
