Kids' real-world arithmetic skills don't transfer to the classroom
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Welcome back to The Nature Podcast. This week, how maths learnt in the classroom doesn't transfer to the real world. And modelling the movement of massive crowds. I'm Benjamin Thompson. And I'm Lizzie Gibney.

First up, a new paper in Nature looks at the differences in how children learn mathematics in the real world versus the classroom and why those math skills may not translate between the two. Here's reporter Anand Jagatia with the story. Imagine you're walking through a street market in a big city like Mumbai.

There are stalls here selling everything from mobile phones to shoes, but you want to buy some vegetables. So you find a stall and you ask for some carrots. They cost 60 rupees a kilo and you go for 400 grams. You pay with a 100 rupee note. How much change are you due back? Now my mental maths is pretty bad so it took me a while to work out the answer, 76 rupees.

But in India, it's not uncommon to find young children working on market stalls like this. And they can do calculations like that one in seconds. I grew up in India. I used to tag along with my grandfather to go shopping. So I had been to markets and I could see that there were kids there, eight-year-olds who could do the mathematics involved in selling. This is Abhijit Banerjee, an economist at MIT.

You know, you could buy five things and then give them some money and they have to give you the correct change back. You know, quite complicated calculations. And I would say, yes, they got it right. And this was true almost every time. He was impressed by the arithmetic agility of the kids he saw working at markets. But when he compared what he'd seen with the results from India's nationwide tests of children's maths ability, something didn't add up.

There's one very famous one in India called ASAR, Simple Subtractions and Multiplication Divisions kind of test. It's a nationwide 200,000 kids test, and it always finds that most kids just can't do grade-level math. So somebody who's in fourth grade is sort of more likely to be able to do second-grade math than fourth-grade math. And it made me wonder if these kids who are in markets, if they're somehow special.

Abhijit wondered if these children, who could rapidly perform complex sums in their heads at the market, had an advantage over their peers in the classroom.

So to formally test this, the researchers went out into markets in Kolkata posing as undercover shoppers and bought goods from 200 kids. And we made sure these quantities were unusual. Instead of 500 grams, it was 800 grams and 600 grams. So that, you know, it's not something you can completely do by rote. And we found what I expected, which is that the kids are very good at doing the math.

And then after we did that, we actually revealed ourselves and said, look, you know, can we actually give you the other test that what I mentioned?

And when we gave them the nationwide test, it was very clear that they were worse than the state average. And yet they were quite exceptional at the math that is required in selling things. The striking thing about these results, according to Abhijit, is that if anything, the market problems were harder.

They involved several operations, like the example with the carrots you heard earlier, and 95% of kids were correct on these problems by their second try. Most of them worked out the answers in their heads in just a few seconds. By contrast, only 32% of the children could solve a long division problem like 769 divided by 3.

So what might explain why they did so much worse on the abstract maths problems? Well, it didn't seem to be because they were written rather than verbal, as the kids did much better when written problems were phrased as market transactions.

You could also argue that giving someone the correct change when you're at work has much higher stakes than doing some exam questions. But the authors tested for that too. We told them that if you get it right, we'll give you a substantial amount of money, the amount of money they make in a day, to just get a few problems right. But that didn't have any effect. It wasn't because they weren't trying.

To explore this further, Abhijit wanted to compare the working market children to 200 non-working kids who attended school full-time. We found that the school kids could do the school math better.

But they absolutely could not do the market math. We created a fake market for them with plastic vegetables and fruits. And in all of these, they were just, you know, frozen. We could see that they didn't have the facility with the numbers that the market kids had, which

We took pictures of what they were doing, and they were writing out the math and computing and adding and then subtracting using at least the conventional steps. But it was just a very elaborate exercise. There was nothing instantaneous or effortless. Ten minutes was not long enough for them sometimes.

So while working children could rapidly perform multiple step calculations in their heads at the market, they couldn't leverage this ability to help them with school problems. And similarly, non-working school children, who, by the way, for their age, were on average a full school grade above the market group when it came to maths, weren't able to apply what they'd learned in school to concrete applied problems.

To understand what might be going on here, many different things need to be teased apart. We know from mathematical cognition research that a lot of factors influence arithmetic performance. This is Iro Xenadu-Dervu, reader in mathematical cognition at the Department of Mathematics Education at Loughborough University, who wasn't affiliated with the research.

It's influenced by age, by social class, by size of the numbers involved in the problems, the form of the task presentation. So is it purely symbolic only with Arabic digits or is it a word problem? You also have non-cognitive factors like children's anxious thoughts. So what I'm trying to say is that it's very hard to then only pick one dimension that will explain all of the differences you see between groups.

Those caveats aside, Eero says it's perhaps not surprising that the working children did so much better than the school children on transactional calculations.

They have been practicing these skills every day for a long time and actually probably under pressure, right? And loud environments, challenging circumstances. There's an additional factor to consider here, and that's motivation for learning, which is key, I think, for working children. Getting these calculations right daily is important for their and their family's livelihood. That is not a small matter, right?

The more puzzling thing is why didn't those amazing mental math skills translate to more abstract math problems, even when the same kinds of strategies could have been used to solve them? Abhijit has some observations. As soon as you say, here's a math problem for you, their entire instinctive system for manipulating these numbers seems to shut down. I think they really have disparate strategies

systems of processing. One kid once told me when I showed him the math problem, after he had done all the market math right, he said, if I could do math, I would be in school. So they think math is something else from what they do. And as soon as it's phrased as mathematics, they think they can't do it. That's our guess.

And Abhijit also points out that even though they did better than the market kids, the non-working school kids also struggled with the algorithms they'd been shown in class. In other words, the series of steps they'd been taught to follow to do arithmetic like long multiplication and division. The school kids, they learn the algorithm, but they don't learn it very well. They're actually very slow at implementing the algorithm. They're not very accurate.

And if you give them a complicated problem, they just can't do it. In general, it's not just a problem for market kids. It's a problem that cuts across all kids. Both Abhijit and Iroh stress that these results have real-world consequences. Maths attainment isn't just important for academic achievement. We know that poor mathematic skills are associated with unemployment, low income and poor quality of life.

We also know that good mathematics skills are important not just for an individual, but also for society as a whole because they predict civic engagement and a country's economic growth. In light of their results, the authors say maths education needs to do more to bridge the gap between the real world and the classroom. Iroh agrees. The findings of this paper resurface the importance of giving meaning to numbers and mathematics.

by building on children's experiences. For example, budgeting, exchange rates, sports league tables. And this can be achieved through play-based activities and hands-on activities where children get to manipulate objects or toys to better understand the mathematical ideas and their relationships.

Abhijit says that maths curricula should also try to cultivate a more intuitive relationship with numbers, like that displayed by the market children.

One way of doing that could be to devote more time to the skill of approximation. One of the ideas that we want to test is moving away from an exact answer to approximation, something that's much more given to intuitive reasoning. Here's a problem. Who can do it fastest? We don't need to know what's the exact product of these two numbers. We need to know which of these two products is larger.

That's a very different way of thinking about mathematics. Iroh agrees that our ability to estimate and have a good instinctive feel for numbers is important, and it's something she's looked at in her own research.

Imagine that you're going to a shop, right? While you're doing this approximate calculation in your head, it's like a checking mechanism to see, is the amount that I'm paying approximately correct? Because if it's completely off, that mechanism will immediately identify and say, hang on a minute, a mistake has been made here, right?

Both are important, both exact arithmetic, but also your ability to estimate is a very important ability and they're linked. In my own studies, I have found that our ability to approximate, so symbolic approximate arithmetic, as we called it, was the most important predictor of future mathematics achievement.

In the end, maths should be about giving children tools that can help them deal with situations inside and outside the classroom. But unfortunately, Abhijit says, that isn't always the case. What matters is kids can systematically solve problems, not...

that they use the right algorithm. That's something that I encountered as a school student in India, that, you know, you would do the same problem to a different method, the teachers would get upset with you. If you keep telling kids that, you know, there's one perfect way to do it and don't explain it very well, then they feel that, okay, I don't know mathematics. You know, there is what I know how to do, and then there is mathematics. And I think that distinction is extremely costly, actually.

That was Abhijit Banerjee from MIT in the US. You also heard from Iruksena Dudovo from Loughborough University here in the UK. For more on that story, check out the show notes for some links. Coming up, how studying a giant fiesta in Spain provided some unexpected insights into the movements of massive crowds. Right now, though, it's time for the Research Highlights with Dan Fox.

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They saw that wolverines are expanding their territory from high mountains along the border of Norway and Sweden into the neighboring forests, at times leading to increased conflicts with herders of domestic sheep and reindeer. The analysis shows that the recovery of large predators on a landscape altered to meet human interests is possible, but the authors say that long-term monitoring will be needed to understand how the wolverine population evolves.

Sink your teeth into that research in the proceedings of the National Academy of Sciences of the United States of America. Flexible arrows with a weighted tip can achieve super propulsion and launch faster than rigid ones.

Previous studies have shown that launching devices like bows impart more energy to flexible projectiles than rigid ones, an effect known as super propulsion. But this has only been experimentally validated for simple shapes. Now researchers have used a high-speed camera to chart the speed and position of arrow-like rods launched vertically.

They found that flexible rods, weighted at the top, buckled quickly before they were released from the device. This boosted the amount of energy transferred from the bow compared with rigid or unweighted counterparts and led to faster ejection speeds. The authors used simulations to show how to maximise this super propulsion effect and saw that at its peak, a wobbly arrow's kinetic energy could be boosted to 160% of its rigid counterpart.

Results, the authors say, that could lead to more efficient robot movements. Fly like an arrow to Physical Review E to see if that research hits the target. If you're anything like me, there have been a lot of occasions where you have been in an immensely crowded space with a lot of other people. Gigs, football matches, that sort of thing.

And these situations can be uncomfortable, to say the least. Potentially dangerous, even. But our second story this week concentrates on even more extreme situations.

Whenever I have a discussion about this project, people tell me, oh, I've been to this concert when I was young and the crowd was even dancer than the one you're talking about. Or I've been to a mall and it was Black Friday. It's very difficult to estimate the number of people around us. And I can tell you that the number of situations where the density is as high as nine people per square meters

This number is very low. It's very unlikely that you have experienced this situation in your life. This is Denis Bartolo. Denis and his team have been using principles from a particular branch of physics to help explain the behaviour of crowds, dense crowds, where up to nine people can be sharing one square metre. That's about the size of a shower cubicle. And they hope that their findings could help save lives.

So what is really dangerous is confinement. As soon as the motion of the people is blocked by a rigid structure, then all the power accumulated by these hundreds of people can be transmitted through

and crush your body. So it can be walls, but it can be also the ground. If you fall, you can be, of course, compressed. There have been countless tragic examples where intense overcrowding has led to deaths. But understanding why situations arise when people can get crushed or fall has been hard to figure out. A lot of research and modelling has looked at the dynamics of small and medium-sized crowds.

But studying huge, super-dense crowds has been difficult. You can't set one up in a lab for practical and ethical reasons. And in the real world, when they happen is unpredictable. And they're one-off events. Except when they aren't. Every July, the Spanish city of Pamplona celebrates the San Fermín Festival. It's a big deal.

The opening of the festival is called the Chupinazo. It's super famous in Spain. It's huge. And thousands of people gather on the main square of the old city. This square is about 50 meters long, 20 meters wide. And more than 5,000 people gather on this square to wait for the opening of the festival. So that's the massive crowd in a confined space. So in principle, you would expect this situation to be super dangerous.

And as it turns out, it's not. The event has been repeated like 200 times. No serious casualties have ever been reported there.

This joyous gathering happens in an area roughly the size of four tennis courts. And it's surrounded by tall buildings, which gave Dennis and his colleagues a precious chance to study the crowd from above, using a combination of camera footage and machine vision technology to identify the position and movement of each person in the crowd.

But Denis wasn't interested in individuals, rather the movement of the mass of people as a whole. And, sure enough, as the area filled up, the team noticed something weird. About 30 minutes before the festival opening,

the crowd becomes so dense that all of a sudden you can see these gigantic displacements of the crowd. And it's very clear that these displacements are completely uncontrolled. Past the critical density, all of a sudden, the crowd starts moving as a whole. With no warning, spontaneous movement occurs within the crowd. So we were very surprised to see

that this spontaneous dynamics is orbital. That is to say that large groups of people suddenly started moving in giant circular patterns. Denis calls them oscillations that each lasted about 18 seconds. So this spontaneous orbital motion is not always clockwise or counterclockwise. What's very clear is that you have groups of hundreds of people all moving in the clockwise directions.

Next to groups of people, all moving in the counterclockwise direction. And this rotation pattern was seen each year the team checked. And what's more, as the crowd became more confined, the oscillations increased. At noon, a band exits the city hall and divides space into two halves, restricting the space completely.

We found this divide does not suppress the oscillation. On the contrary, reducing space results in an increase of the oscillation frequency. So this told us that confinement is crucial to explain these oscillations.

It seems that there's a relationship between increasing crowd density and this characteristic movement. And it turned out that the team could model the crowd as if it were a fluid, using knowledge from a branch of physics called active matter to explain how this could happen, despite there being no external forces applied. But there was still an outstanding question.

Was this rotation phenomenon something that was unique to this event in this context? Could they find any other examples to corroborate their theory?

Well, it turns out they did, in a much less joyous situation. In 2010, in Germany, at the Love Parade, super dense crowds formed, leading to horrible accidents. This music festival took place in the city of Duisburg, where a crush in a crowded tunnel led to hundreds of people being injured and 21 losing their lives.

Security camera footage of this tragic event has been released and studied since the event, and Denis and his colleagues analysed it, looking for evidence of collective rotation. The same orbital oscillations, they happened, and we found that the signature were the same. Antoine Tordeau studies the dynamics of crowded situations involving people or vehicles.

He's written an analysis of the study for nature and says that identifying this phenomenon and laying out some of the physics to help explain why large groups can suddenly behave in this way is an important step forward to creating better models of crowd dynamics.

But there's still lots to learn. For example, Antoine wants to know if this really is a universal phenomenon.

N might equal 2, but will it equal 3, 4, 5 and so on? And what role does context play? The Spanish crowd is a young, energetic one, he says. Could the actions of individuals contribute to the movement of the whole? I'm sure this collective motion cannot be decided by the people. There's really some things that emerge and that's the main point.

whether it can be amplified because of the festive situation, this can be something to think about. Dennis also wants to understand the role of individual interactions, but more broadly, the exact relationship between density and space. He's shown that when a threshold of crowd density is hit, oscillations begin. But exactly what that threshold is and whether it varies is unknown.

Regardless, he hopes the findings could increase safety and highlight the need to look at crowds in a different way.

Recognising this spontaneous mass movement within a crowd might help avert life-threatening situations where people fall or are crushed. I would like to convince people to stop trying to measure the crowd density with increasing accuracy, but rather focus on the crowd dynamics. So by monitoring the dynamics of the crowd, we could tell, be careful, the crowd starts moving. And if you keep on accumulating people in this region,

then the motion is going to be huge and it's going to be dangerous. That was Denis Bartelot from INS de Lyon in France. You also heard from Antoine Tordeau, who's at the University of Wuppertal in Germany. To read Denis's paper and Antoine's news and views article, look out for links in the show notes. Finally on the show, it's time for the briefing chat, where we discuss a couple of articles that have been highlighted in the Nature Briefing. Ben, if you don't mind, I will crack on first this week. So,

So we're talking about an exclusive story in Nature this week that's on some of the turmoil that's going on in the United States in terms of federal funding. So you might remember that President Trump issued an executive order that was to halt funding for diversity, equity and inclusion efforts, as well as some other areas like foreign assistance and climate science. And these executive orders, they direct the US government's actions, but they can't override existing laws. So that's something we need to keep in mind for later. So...

What's been going on is that places like the National Science Foundation is they are trying to comply with this order. So to do that, they're doing a massive review of all their grants that would cease non-compliant grants ultimately. But for some context as to why there is quite as much chaos as there is...

Initially, there was a freeze on issuing grants as a result of this executive order. And that included cutting funding for postdocs who rely on NSF money. Some federal judges have now blocked that freeze and said that these are kind of temporary orders that say you cannot freeze these grants, you cannot terminate the grants. But

But nobody really knows if that is going to hold, if those temporary orders will last or if they might be appealed. And in any case, the executive order underlying it all is still in place. So ultimately, the NSF's review is still going ahead. So that's what's happening at the moment. And Nature spoke to six employees who remained anonymous, but telling us about what is actually going on inside NSF right now. Right. I mean, obviously, funding is a perennial topic of discussion on the Nature podcast. So important for science. But what are the employees saying then?

So they talked about how they're doing this review. So there are 10,000 grants that are currently flagged for review and they're looking out for very specific criteria. So anything to do with broadening participation, words like foreign assistance, climate science, DEI, they are all things that are being flagged as being potentially non-compliant with the executive order. Directors from one directorate reviewing the grants of another. So for instance, the geoscience director is reviewing grants in social science.

And what's happening is they have to indicate when a grant potentially violates the executive order, put that in a spreadsheet that had to be done by yesterday. We're saying yesterday, today, recording on the 4th of February. It's quite important probably to say that because this is changing very, very quickly. And so what's happening is from here on, these grants that are flagged could be cancelled. They could be archived. They could be modified. And

And a team at the NSF will review any revisions to the grants and ultimately that will all go off to the office of the director of the NSF for the final say. Wow. So, I mean, a great deal going on then. Of course, a lot of questions remain hanging in the air. Absolutely. One big question is, is this even legal?

So one staffer at the NSF that we quote said this executive order appears to be illegal. So in 1980, Congress mandated that the NSF should seek to broaden participation for underrepresented groups. And so the thing with the executive orders is they instruct the government's actions, but they can't override existing laws. So if there is an existing law saying that the NSF should be seeking to broaden participation, having an executive order that says that is not allowed, those things don't seem to be coherent anymore.

And there's also a lot of confusion because effectively all NSF grants include some element of broadening participation because, again, Congress mandated that, that when you apply for these grants, you have to describe how you're going to broaden the impacts on society. And that includes increasing participation of underrepresented groups. So there's effectively a chance that...

This executive order could affect every single NSF grant because they've been told for decades that they are supposed to be doing this. Which is, of course, a huge amount of money, I'm sure. It's a huge amount of money. To many, many people, it's hugely important and part of efforts to make science more equitable and better in so many ways that we've spoken about on this podcast many times before. And the immediate impacts are some people are just looking for alternative sources of funding. They don't feel like federal funding is secure anymore. Right.

Postdocs, as I mentioned at the start, some of them had their fellowships frozen. And there's just a huge amount of worry and uncertainty. Like, what if that happens again? Postdocs are not known for being particularly flush. Like, a week's even money being lost can have a huge impact on whether someone can pay their rent or not. And the communication itself coming out of the NSF is apparently rather controversial.

cryptic. So there's an enormous amount of uncertainty, worry and turmoil. Right. And there's so many different groups of people, as you say, on this continuum who are potentially going to be impacted in different ways. Absolutely. And the question I was dreading you asking me is what happens next? Because

It seems like we don't know. So the federal judges have issued these temporary orders. Will they be upheld? Are there enough of them to kind of give a broad legal opinion that this reach by the president is not allowed? Or will there be appeals? Will they go through on appeal?

Nobody really knows. All we know at the moment is that the review is happening or has happened and that some grants are very much in the firing line. Well, lots of questions, as you say there, Lizzie. And of course, there's lots of different funding streams in the US, and I'm sure a lot of that is going to be looked at. And while the answers may not be clear just yet, listeners, I would urge you to head over to nature.com slash news, where our colleagues will be covering all the developments in this space.

But let's move on to our second story this week. Lizzie, couldn't be more different. It's a story I read about in Nature, based on a paper in Science. And it's about having a bit of a scratch. OK. I thought this would be some light relief. But actually, I had a little look at the story and it's kind of making me shudder a little bit. Well, I don't know. Scratches can be a very satisfying thing to do. Right. I actually got an insect bite last week, just on my rib cage. And I'm very good at scratching.

at trying to avoid scratching them. But every now and again, just a little go, God, it feels good, right? Doesn't it? And researchers have been looking at what's going on when we do this kind of short-term scratching, which might help explain why it is so satisfying. And it all seems to be that it activates a very specific immune response.

That's so interesting because I have always thought that it's a bad idea to scratch when you have an itch, right? Because you might cause infection. But of course, we do scratch. So I've never thought that. Why do we have the urge if it is a bad thing? I guess you're right to tell me. Yes, I certainly hope so. Well, it turns out lots of animals do scratch, right? Not just humans. And a lot of the work I'm about to describe was in mice. Okay, now...

It was thought that scratching in general maybe was to remove parasites or irritants. But if you think about a mosquito bite, that mosquito is long gone by the time you sort of get the old fingernails in there and have a little scratch. So what they've done in this instance, they've taken some mice and painted a synthetic allergen on their ear. OK, now this induces a very specific sort of inflammation called contact dermatitis, a bit like if you touch poison ivy, that sort of thing. And when mice scratched this ear,

Itch. Their ears swelled up and became full of what are called neutrophils, this type of immune cell. And mice that couldn't scratch because of what they call in the article the cone of shame, you know, that animals have sometimes when they go to the vet or the Elizabethan collar. And this prevented the mice from scratching and they showed less inflammation and fewer neutrophils. And mice that didn't have itch-sensing neurons, they didn't do this either. They didn't do the scratch.

So it seems like the act of scratching increases inflammation. You might think that seems bad, right? But of course, inflammation is very important for... Your body's response. The immune response, right. So they look to see what was going on in this instance. And it seems like at the scratch sites, pain-sensing neurons released a potent nervous system messenger called substance P. Now, as I've joked on the podcast many times, the immune system is the most complex thing on earth.

I'll give you the kind of bullet point notes here. Okay, so substance P released at the site of the scratch. This activates a type of white blood cell involved in allergies called mast cells. Okay, these mast cells recruit neutrophils, the immune cells we talked about at the start, to the scratched site, driving inflammation. And these mast cells are known to directly respond to allergens, right? Like maybe pollen in the nose, that sort of thing, right? And kick off an allergic reaction. But what's interesting here is indirect reactions

activation of the mast cell. So the scratch ultimately ends with them being activated, which then leads to the neutrophils gathering at the site. So what's the benefit of having these cells there? There's an improved immune response. Yeah, this is where things take a bit of an unexpected twist for me. Okay, so the team behind it studied the skin microbiome. Ding, ding, ding. It turned out to be a microbiology story. Now, one day after an allergen

mice that could scratch were less likely to have potentially dangerous Staphylococcus aureus bacteria on their ears. So it seems like the act of scratching activates this pathway, which leads to inflammation and neutrophils gathering at this site more.

which may have an antibacterial benefit, which maybe explains why this can be pleasurable, right? There's obviously a gap between the two, but it might be a good thing to do. Now, it has to be said that this is acute scratching, just a quick itch. Chronic itching can lead to skin damage, and that in turn can give Staph aureus a foothold, right? So we need to make it clear that this is just a quick situation, but it could alter

ultimately help people who do experience chronic itching which can be caused by things like diabetes and eczema and so this study then shows that you know there's these two sets of nerves involved one says please itch this site and the other responds to the itch by increasing inflammation and in the article they say well if you could decouple these and block the needing to itch

but keep the inflammation maybe we could get some of the benefits without the you know need to scratch which can be unbearable for some people yeah I bet that'd be an enormous relief if you're able to do that

I mean, obviously, this is miles away. I guess understanding the underlying process is just that first step. Yeah, right. I mean, it's one of those things that I guess we all experience and knowing a bit more about it is obviously interesting, but could potentially one day be used to make therapeutics, that sort of thing. Well, let's keep looking out for that then. And listeners, for more on those stories and where you can sign up to The Nature Briefing to get more like them, check out the show notes for some links.

And that's all for this week. As always, you can keep in touch with us on email. We're podcast at nature.com. You'll also find us on Blue Sky and X. I'm Benjamin Thompson. And I'm Lizzie Gibney. Thanks for listening.

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