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Why America is bad at math
Research shows that the way math is taught in schools and how its conceptualized as a subject is severely impairing American student's ability to learn and understand the material.
- Americans continually score either in the mid- or bottom-tier when it comes to math and science compared to their international peers.
- Students have a fundamental misunderstanding of what math is and what it can do. By viewing it as a language, students and teachers can begin to conceptualize it in easier and more practical ways.
- A lot of mistakes come from worrying too much about rote memorization and speedy problem-solving and from students missing large gaps in a subject that is reliant on learning concepts sequentially.
It comes as no surprise to most people that Americans perform worse in math and sciences than many of their international peers on the world stage. The numbers don't lie: A recent national survey from the Organization for Economic Cooperation and Development found that 82% of adults couldn't determine the cost of carpeting when given its dimensions and price per square yard.
Unlike the more difficult and comprehensive math tests given to test students' comprehension, this test was for basic numeracy skills. The United States fell behind in 22nd place.
For a country that has boasted, or at least hosted, some of the smartest minds and most competitive research labs, companies, and universities in the world, there is a strange disconnect between our overall mathematical ability and our professional output. There is no doubt that a lot of Americans are bad at math and even suffer crippling math anxiety from a very young age.
But why? It has to do with a few reasons: how math is presented as a subject, how it's taught, and what's expected from American students.
Why the U.S. needs to change its collective view of mathematics
Mathematics has been taught a certain way for decades in U.S. schools. Maybe it's time for that to change.
One of the first things that comes to many people's minds when they think about math is rote memorization, impracticability, and the old slacker adage, "When am I ever going to need to use this?" The quadratic formula, sines, and cosines have gotten a bad rap and taken a verbal beating by an innumerable amount of high schoolers for probably more than a century.
The vast majority of people who haven't had to use an equation since their senior year or cram session in college just don't see the value in math. That's because they fundamentally misunderstand what mathematics is.
Neil deGrasse Tyson put it succinctly when he said, "Math is the language of the universe. So the more equations you know, the more you can converse with the cosmos."
Now, that's part of the equation, but not all of it. Math, in a sense, is a way to speak and manipulate the world in a logic- and reason-based system using a specialized written language.
It is the language of numbers, quantity, and space, and it's used in applications for engineering, physics, and so on. It's doubtful that math is presented this way to children or students at an early age. But that's just one part of the problem with how we approach math.
Why it's easy to fall behind in math
Professor Po-Shen Loh at Carnegie Mellon believes that everyone is a math person; all they lack is proper instruction. In an interview with Big Think, he went on to say that math is a language that builds upon itself, and not understanding the foundations of math is like not understanding the roots and structure of a language.
Essentially if a student doesn't catch on in their first years of instruction, it's going to be very difficult for them to reverse course and excel later on down the line. He believes it is essential to catch this early on and address it before a student's issues with math reach a point where they feel "they're just not good at math."
Professor Loh goes on to say that "Mathematics is the principles of reasoning. There are ways to show you how these basic building blocks of reasoning can be used to deduce surprising and difficult things."
One major reason that mathematics is difficult to understand is because it is a network of prerequisites. Everything, all of the concepts, are chained in sequences of dependencies.
If you miss an important concept earlier on—say, not being able to understand how to chart a simple algebraic equation on a line graph, you'll have no idea how to go on to charting even more complicated equations.
Loh goes on to say that this is much more prevalent in mathematics than history, for example. If you didn't fully understand the War of 1812, it's not going to impact how you learn about the Civil War—aside from the occasional historical patterns you may or may not recognize, of course.
The way to address this is to provide a learning environment for everyone that moves at their own pace, to make sure to fill in the gaps, and to catch those lapses in understanding before they get out of control.
And if you're already in too deep, say, as a college grad or just an adult who wants to learn… well, it's time to start from square one.
A faulty learning and teaching methodology
A few years back, the Programme for International Student Assessment (PISA) dug a little deeper into how math is taught. A 2012 assessment questioned how students approach the subject. Their responses were categorized in three learning styles: some students relied mostly on memorization, others tried to relate new concepts to ones they've already learned, and finally, some used a self-monitoring approach in which they evaluated their understanding and focused on concepts they had yet to learn.
Without much of a surprise, it turned out that the memorizers were the least likely to achieve high scores and understanding. The United States ranked in the top three for this learning method. A more in-depth look showed that memorizers were about a half year behind students who used either relational or self-monitoring strategies.
Research has shown and most likely loads of anecdotal evidence shows that most math classrooms in the United States equate comprehension and skill with speed. Students who are the fastest on their time tables race against the clock to see how fast they can write down their memorized lines. This is not learning, this is not comprehension.
Studies show that stress interferes with the part of our brain we use to manipulate mathematical facts.
Studies have shown that children manipulate math facts with their working memory, an area of the brain that will go offline when they experience stress.
Now put together 45-minute timed tests in a condensed school year or semester combined with math anxiety, faulty instruction and expectations, poor learning methods, potential lapses in the fundamentals, and the problems start to pile up. As a result, the part of the brain responsible for mathematical thinking literally shuts off, and you start to see why Americans are so bad at math.
Leading mathematician Laurent Schwartz wrote in his autobiography that he was a slow thinker in math and even believed that he was stupid. That is until he realized that "What is important is to deeply understand things and their relations to each other. This is where intelligence lies. The fact of being quick or slow isn't really relevant."
The problem has been diagnosed, and a few pieces of the solution have been put together, but something is still missing.
Why new methods of teaching math aren’t working
We've tried different methods of teaching math over the years, but have any of them worked?
Many potentially great minds have probably been turned off by the fast-paced timed tests and wonky teaching methods presented through the years. The language of math needs to be presented in a way that shows how it connects to the world and demonstrates its great capacity for understanding and manipulating reality.
If more people could tap into this infinite matrix of power, they'd be able to engage in the wondrous world of math and unlock unknown potentials. It's not for a lack of trying that we've failed; it comes down to instruction yet again.
Despite being today's newest fad and the subject of ire from many on both sides of the political spectrum, Common Core is our latest panacea for our math woes. Yet we still suffer from what math professor and author John Allen Paulos calls innumeracy—a mathematical illiteracy akin to not being able to read or write.
What is needed is a fundamental shift in how we view mathematics as a subject so we can learn to imagine how it can benefit and help us in different fields. In addition, we need to make sure that it's taught in a way that no student skips past the fundamentals. Instructors and teachers at all levels must make a systemic change if we're to see any progress. Could this change be Common Core or a different teaching philosophy? We'll find out in the years to come.
Andy Samberg and Cristin Milioti get stuck in an infinite wedding time loop.
- Two wedding guests discover they're trapped in an infinite time loop, waking up in Palm Springs over and over and over.
- As the reality of their situation sets in, Nyles and Sarah decide to enjoy the repetitive awakenings.
- The film is perfectly timed for a world sheltering at home during a pandemic.
Richard Feynman once asked a silly question. Two MIT students just answered it.
Here's a fun experiment to try. Go to your pantry and see if you have a box of spaghetti. If you do, take out a noodle. Grab both ends of it and bend it until it breaks in half. How many pieces did it break into? If you got two large pieces and at least one small piece you're not alone.
But science loves a good challenge<p>The mystery remained unsolved until 2005, when French scientists <a href="http://www.lmm.jussieu.fr/~audoly/" target="_blank">Basile Audoly</a> and <a href="http://www.lmm.jussieu.fr/~neukirch/" target="_blank">Sebastien Neukirch </a>won an <a href="https://www.improbable.com/ig/" target="_blank">Ig Nobel Prize</a>, an award given to scientists for real work which is of a less serious nature than the discoveries that win Nobel prizes, for finally determining why this happens. <a href="http://www.lmm.jussieu.fr/spaghetti/audoly_neukirch_fragmentation.pdf" target="_blank">Their paper describing the effect is wonderfully funny to read</a>, as it takes such a banal issue so seriously. </p><p>They demonstrated that when a rod is bent past a certain point, such as when spaghetti is snapped in half by bending it at the ends, a "snapback effect" is created. This causes energy to reverberate from the initial break to other parts of the rod, often leading to a second break elsewhere.</p><p>While this settled the issue of <em>why </em>spaghetti noodles break into three or more pieces, it didn't establish if they always had to break this way. The question of if the snapback could be regulated remained unsettled.</p>
Physicists, being themselves, immediately wanted to try and break pasta into two pieces using this info<p><a href="https://roheiss.wordpress.com/fun/" target="_blank">Ronald Heisser</a> and <a href="https://math.mit.edu/directory/profile.php?pid=1787" target="_blank">Vishal Patil</a>, two graduate students currently at Cornell and MIT respectively, read about Feynman's night of noodle snapping in class and were inspired to try and find what could be done to make sure the pasta always broke in two.</p><p><a href="http://news.mit.edu/2018/mit-mathematicians-solve-age-old-spaghetti-mystery-0813" target="_blank">By placing the noodles in a special machine</a> built for the task and recording the bending with a high-powered camera, the young scientists were able to observe in extreme detail exactly what each change in their snapping method did to the pasta. After breaking more than 500 noodles, they found the solution.</p>
The apparatus the MIT researchers built specifically for the task of snapping hundreds of spaghetti sticks.
(Courtesy of the researchers)
What possible application could this have?<p>The snapback effect is not limited to uncooked pasta noodles and can be applied to rods of all sorts. The discovery of how to cleanly break them in two could be applied to future engineering projects.</p><p>Likewise, knowing how things fragment and fail is always handy to know when you're trying to build things. Carbon Nanotubes, <a href="https://bigthink.com/ideafeed/carbon-nanotube-space-elevator" target="_self">super strong cylinders often hailed as the building material of the future</a>, are also rods which can be better understood thanks to this odd experiment.</p><p>Sometimes big discoveries can be inspired by silly questions. If it hadn't been for Richard Feynman bending noodles seventy years ago, we wouldn't know what we know now about how energy is dispersed through rods and how to control their fracturing. While not all silly questions will lead to such a significant discovery, they can all help us learn.</p>
The multifaceted cerebellum is large — it's just tightly folded.
- A powerful MRI combined with modeling software results in a totally new view of the human cerebellum.
- The so-called 'little brain' is nearly 80% the size of the cerebral cortex when it's unfolded.
- This part of the brain is associated with a lot of things, and a new virtual map is suitably chaotic and complex.
Just under our brain's cortex and close to our brain stem sits the cerebellum, also known as the "little brain." It's an organ many animals have, and we're still learning what it does in humans. It's long been thought to be involved in sensory input and motor control, but recent studies suggests it also plays a role in a lot of other things, including emotion, thought, and pain. After all, about half of the brain's neurons reside there. But it's so small. Except it's not, according to a new study from San Diego State University (SDSU) published in PNAS (Proceedings of the National Academy of Sciences).
A neural crêpe
A new imaging study led by psychology professor and cognitive neuroscientist Martin Sereno of the SDSU MRI Imaging Center reveals that the cerebellum is actually an intricately folded organ that has a surface area equal in size to 78 percent of the cerebral cortex. Sereno, a pioneer in MRI brain imaging, collaborated with other experts from the U.K., Canada, and the Netherlands.
So what does it look like? Unfolded, the cerebellum is reminiscent of a crêpe, according to Sereno, about four inches wide and three feet long.
The team didn't physically unfold a cerebellum in their research. Instead, they worked with brain scans from a 9.4 Tesla MRI machine, and virtually unfolded and mapped the organ. Custom software was developed for the project, based on the open-source FreeSurfer app developed by Sereno and others. Their model allowed the scientists to unpack the virtual cerebellum down to each individual fold, or "folia."
Study's cross-sections of a folded cerebellum
Image source: Sereno, et al.
A complicated map
Sereno tells SDSU NewsCenter that "Until now we only had crude models of what it looked like. We now have a complete map or surface representation of the cerebellum, much like cities, counties, and states."
That map is a bit surprising, too, in that regions associated with different functions are scattered across the organ in peculiar ways, unlike the cortex where it's all pretty orderly. "You get a little chunk of the lip, next to a chunk of the shoulder or face, like jumbled puzzle pieces," says Sereno. This may have to do with the fact that when the cerebellum is folded, its elements line up differently than they do when the organ is unfolded.
It seems the folded structure of the cerebellum is a configuration that facilitates access to information coming from places all over the body. Sereno says, "Now that we have the first high resolution base map of the human cerebellum, there are many possibilities for researchers to start filling in what is certain to be a complex quilt of inputs, from many different parts of the cerebral cortex in more detail than ever before."
This makes sense if the cerebellum is involved in highly complex, advanced cognitive functions, such as handling language or performing abstract reasoning as scientists suspect. "When you think of the cognition required to write a scientific paper or explain a concept," says Sereno, "you have to pull in information from many different sources. And that's just how the cerebellum is set up."
Bigger and bigger
The study also suggests that the large size of their virtual human cerebellum is likely to be related to the sheer number of tasks with which the organ is involved in the complex human brain. The macaque cerebellum that the team analyzed, for example, amounts to just 30 percent the size of the animal's cortex.
"The fact that [the cerebellum] has such a large surface area speaks to the evolution of distinctively human behaviors and cognition," says Sereno. "It has expanded so much that the folding patterns are very complex."
As the study says, "Rather than coordinating sensory signals to execute expert physical movements, parts of the cerebellum may have been extended in humans to help coordinate fictive 'conceptual movements,' such as rapidly mentally rearranging a movement plan — or, in the fullness of time, perhaps even a mathematical equation."
Sereno concludes, "The 'little brain' is quite the jack of all trades. Mapping the cerebellum will be an interesting new frontier for the next decade."
What happens if we consider welfare programs as investments?
- A recently published study suggests that some welfare programs more than pay for themselves.
- It is one of the first major reviews of welfare programs to measure so many by a single metric.
- The findings will likely inform future welfare reform and encourage debate on how to grade success.
Welfare as an investment<p>The <a href="https://scholar.harvard.edu/files/hendren/files/welfare_vnber.pdf" target="_blank">study</a>, carried out by Nathaniel Hendren and Ben Sprung-Keyser of Harvard University, reviews 133 welfare programs through a single lens. The authors measured these programs' "Marginal Value of Public Funds" (MVPF), which is defined as the ratio of the recipients' willingness to pay for a program over its cost.</p><p>A program with an MVPF of one provides precisely as much in net benefits as it costs to deliver those benefits. For an illustration, imagine a program that hands someone a dollar. If getting that dollar doesn't alter their behavior, then the MVPF of that program is one. If it discourages them from working, then the program's cost goes up, as the program causes government tax revenues to fall in addition to costing money upfront. The MVPF goes below one in this case. <br> <br> Lastly, it is possible that getting the dollar causes the recipient to further their education and get a job that pays more taxes in the future, lowering the cost of the program in the long run and raising the MVPF. The value ratio can even hit infinity when a program fully "pays for itself."</p><p> While these are only a few examples, many others exist, and they do work to show you that a high MVPF means that a program "pays for itself," a value of one indicates a program "breaks even," and a value below one shows a program costs more money than the direct cost of the benefits would suggest.</p> After determining the programs' costs using existing literature and the willingness to pay through statistical analysis, 133 programs focusing on social insurance, education and job training, tax and cash transfers, and in-kind transfers were analyzed. The results show that some programs turn a "profit" for the government, mainly when they are focused on children:
This figure shows the MVPF for a variety of polices alongside the typical age of the beneficiaries. Clearly, programs targeted at children have a higher payoff.
Nathaniel Hendren and Ben Sprung-Keyser<p>Programs like child health services and K-12 education spending have infinite MVPF values. The authors argue this is because the programs allow children to live healthier, more productive lives and earn more money, which enables them to pay more taxes later. Programs like the preschool initiatives examined don't manage to do this as well and have a lower "profit" rate despite having decent MVPF ratios.</p><p>On the other hand, things like tuition deductions for older adults don't make back the money they cost. This is likely for several reasons, not the least of which is that there is less time for the benefactor to pay the government back in taxes. Disability insurance was likewise "unprofitable," as those collecting it have a reduced need to work and pay less back in taxes. </p>