Anyone Can Be a Math Person Once They Know the Best Learning Techniques
So you think you're "not a math person"? International Mathematical Olympiad coach Po-Shen Loh strongly disagrees.
Po-Shen Loh, PhD, is associate professor of mathematics at Carnegie Mellon University, which he joined, in 2010, as an assistant professor in the Department of Mathematical Sciences. As a Hertz Fellow, Professor Loh received his PhD in combinatorics of the Pure Math Department at Princeton University. His thesis discussed several original results that he discovered during his graduate study in joint projects with his advisor and other collaborators. Professor Loh studies questions that lie at the intersection of two branches of mathematics: combinatorics (the study of discrete systems) and probability theory.
Prior to his work at Princeton, Loh received the equivalent of a master's degree in mathematics from the University of Cambridge (United Kingdom) in 2005, where he was supported by a Winston Churchill Foundation Scholarship. He received his undergraduate degree in mathematics from Caltech in 2004, graduating first in his class, and his undergraduate thesis later received the Honorable Mention for the 2004 AMS-MAA-SIAM Morgan Prize.
In his spare time, Loh has maintained his involvement with the United States Mathematical Olympiad program. He is now the head coach of the national delegation, as well as a lead fundraiser for the organization. As a high school student, he won a silver medal at the 1999 International Mathematical Olympiad (IMO), and following his win continued to be active in the training of high school students at the U.S. national Math Olympiad Summer Program. In 2004, he served as the deputy leader for the U.S. team at the IMO in Athens, Greece, where our national team placed second. After completing his PhD, Loh again, served as deputy team leader for the United States at the International Mathematical Olympiad from 2010 to 2013. Afterwards Professor Loh was promoted to national head coach of the U.S.A. IMO team, and on his second attempt, Team U.S.A. won first place, in a competition with teams from over 100 countries represented.
Earlier this year, Loh received an NSF CAREER award, the most prestigious NSF award for junior faculty, which honors outstanding research combined with a commitment to teaching. Professor Loh is the founder of the educational technology startup expii.com, a crowd-sourced platform for the world to share interactive lessons in math and science.
Po-Shen Lo: I think that everyone in the world could be a math person if they wanted to. The keyword though, I want to say, is if they wanted to. That said, I do think that everyone in America could benefit from having that mathematical background in reasoning just to help everyone make very good decisions. And here I'm distinguishing already between math as people usually conceive of it, and decision making and analysis, which is actually what I think math is.
So, for example, I don't think that being a math person means that you can recite the formulas between the sines, cosines, tangents and to use logarithms and exponentials interchangeably. That's not necessarily what I think everyone should try to concentrate to understand. The main things to concentrate to understand are the mathematical principles of reasoning.
But let me go back to these sines, cosines and logarithms. Well actually they do have value. What they are is that they are ways to show you how these basic building blocks of reasoning can be used to deduce surprising things or difficult things. In some sense they're like the historical coverages of the triumphs of mathematics, so one cannot just talk abstractly about “yes let's talk about mathematical logic”, it's actually quite useful to have case studies or stories, which are these famous theorems.
Now, I actually think that these are accessible to everyone. I think that actually one reason mathematics is difficult to understand is actually because of that network of prerequisites. You see, math is one of these strange subjects for which the concepts are chained in sequences of dependencies.
When you have long chains there are very few starting points—very few things I need to memorize. I don't need to memorize, for example, all these things in history such as “when was the war of 1812?” Well actually I know that one, because that's a math fact—it was 1812—but I can't tell you a lot of other facts, which are just purely memorized. In mathematics you have very few that you memorize and the rest you deduce as you go through, and this chain of deductions is actually what's critical.
Now, let me contrast that with other subjects like say history. History doesn't have this long chain, in fact if you fully understand the war of 1812 that's great, and it is true that that will influence perhaps your understanding later of the women's movement, but it won't to be as absolutely prerequisite. In the sense that if you think about the concepts I actually think that history has more concepts than mathematics; it's just that they're spread out broader and they don't depend on each other as strongly. So, for example, if you miss a week you will miss the understanding of one unit, but that won't stop you from understanding all of the rest of the components.
So that's actually the difference between math and other subjects in my head. Math has fewer concepts but they're chained deeper. And because of the way that we usually learn when you had deep chains it's very fragile because you lose any one link—meaning if you miss a few concepts along the chain you can actually be completely lost. If, for example, you're sick for a week, or if your mind is somewhere else for a week, you might make a hole in your prerequisites. And the way that education often works where it's almost like riding a train from a beginning to an end, well it's such that if you have a hole somewhere in your track the train is not going to pass that hole.
Now, I think that the way to help to address this is to provide a way for everyone to learn at their own pace and in fact to fill in the holes whenever they are sensed. And I actually feel like if everyone was able to pick up every one of those prerequisites as necessary, filling in any gap they have, mathematics would change from being the hardest subject to the easiest subject.
I think everyone is a math person, and all that one has to do is to go through the chain and fill in all the gaps, and you will understand it better than all the other subjects actually.
Po-Shen Loh is a Hertz Foundation Fellow and Carnegie Mellon mathematics professor who thinks that history is a much harder subject than math. Do you agree? Well, your position on that might change before and after this video. Loh illuminates the invisible ladders within the world of math, and shows that it isn't about memorizing formulas—it's about processing reason and logic. With the support of the Fannie and John Hertz Foundation, Po-Shen Loh pursued a PhD in combinatorics at the Pure Math Department at Princeton University.
The Hertz Foundationmission is to provide unique financial and fellowship support to the nation's most remarkable PhD students in the hard sciences. Hertz Fellowships are among the most prestigious in the world, and the foundation has invested over $200 million in Hertz Fellows since 1963 (present value) and supported over 1,100 brilliant and creative young scientists, who have gone on to become Nobel laureates, high-ranking military personnel, astronauts, inventors, Silicon Valley leaders, and tenured university professors. For more information, visit hertzfoundation.org.
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China has reached a new record for nuclear fusion at 120 million degrees Celsius.
This article was originally published on our sister site, Freethink.
China wants to build a mini-star on Earth and house it in a reactor. Many teams across the globe have this same bold goal --- which would create unlimited clean energy via nuclear fusion.
But according to Chinese state media, New Atlas reports, the team at the Experimental Advanced Superconducting Tokamak (EAST) has set a new world record: temperatures of 120 million degrees Celsius for 101 seconds.
Yeah, that's hot. So what? Nuclear fusion reactions require an insane amount of heat and pressure --- a temperature environment similar to the sun, which is approximately 150 million degrees C.
If scientists can essentially build a sun on Earth, they can create endless energy by mimicking how the sun does it.
If scientists can essentially build a sun on Earth, they can create endless energy by mimicking how the sun does it. In nuclear fusion, the extreme heat and pressure create a plasma. Then, within that plasma, two or more hydrogen nuclei crash together, merge into a heavier atom, and release a ton of energy in the process.
Nuclear fusion milestones: The team at EAST built a giant metal torus (similar in shape to a giant donut) with a series of magnetic coils. The coils hold hot plasma where the reactions occur. They've reached many milestones along the way.
According to New Atlas, in 2016, the scientists at EAST could heat hydrogen plasma to roughly 50 million degrees C for 102 seconds. Two years later, they reached 100 million degrees for 10 seconds.
The temperatures are impressive, but the short reaction times, and lack of pressure are another obstacle. Fusion is simple for the sun, because stars are massive and gravity provides even pressure all over the surface. The pressure squeezes hydrogen gas in the sun's core so immensely that several nuclei combine to form one atom, releasing energy.
But on Earth, we have to supply all of the pressure to keep the reaction going, and it has to be perfectly even. It's hard to do this for any length of time, and it uses a ton of energy. So the reactions usually fizzle out in minutes or seconds.
Still, the latest record of 120 million degrees and 101 seconds is one more step toward sustaining longer and hotter reactions.
Why does this matter? No one denies that humankind needs a clean, unlimited source of energy.
We all recognize that oil and gas are limited resources. But even wind and solar power --- renewable energies --- are fundamentally limited. They are dependent upon a breezy day or a cloudless sky, which we can't always count on.
Nuclear fusion is clean, safe, and environmentally sustainable --- its fuel is a nearly limitless resource since it is simply hydrogen (which can be easily made from water).
With each new milestone, we are creeping closer and closer to a breakthrough for unlimited, clean energy.
The symbol for love is the heart, but the brain may be more accurate.
- How love makes us feel can only be defined on an individual basis, but what it does to the body, specifically the brain, is now less abstract thanks to science.
- One of the problems with early-stage attraction, according to anthropologist Helen Fisher, is that it activates parts of the brain that are linked to drive, craving, obsession, and motivation, while other regions that deal with decision-making shut down.
- Dr. Fisher, professor Ted Fischer, and psychiatrist Gail Saltz explain the different types of love, explore the neuroscience of love and attraction, and share tips for sustaining relationships that are healthy and mutually beneficial.
Inventions with revolutionary potential made by a mysterious aerospace engineer for the U.S. Navy come to light.
- U.S. Navy holds patents for enigmatic inventions by aerospace engineer Dr. Salvatore Pais.
- Pais came up with technology that can "engineer" reality, devising an ultrafast craft, a fusion reactor, and more.
- While mostly theoretical at this point, the inventions could transform energy, space, and military sectors.
The U.S. Navy controls patents for some futuristic and outlandish technologies, some of which, dubbed "the UFO patents," came to light recently. Of particular note are inventions by the somewhat mysterious Dr. Salvatore Cezar Pais, whose tech claims to be able to "engineer reality." His slate of highly-ambitious, borderline sci-fi designs meant for use by the U.S. government range from gravitational wave generators and compact fusion reactors to next-gen hybrid aerospace-underwater crafts with revolutionary propulsion systems, and beyond.
Of course, the existence of patents does not mean these technologies have actually been created, but there is evidence that some demonstrations of operability have been successfully carried out. As investigated and reported by The War Zone, a possible reason why some of the patents may have been taken on by the Navy is that the Chinese military may also be developing similar advanced gadgets.
Among Dr. Pais's patents are designs, approved in 2018, for an aerospace-underwater craft of incredible speed and maneuverability. This cone-shaped vehicle can potentially fly just as well anywhere it may be, whether air, water or space, without leaving any heat signatures. It can achieve this by creating a quantum vacuum around itself with a very dense polarized energy field. This vacuum would allow it to repel any molecule the craft comes in contact with, no matter the medium. Manipulating "quantum field fluctuations in the local vacuum energy state," would help reduce the craft's inertia. The polarized vacuum would dramatically decrease any elemental resistance and lead to "extreme speeds," claims the paper.
Not only that, if the vacuum-creating technology can be engineered, we'd also be able to "engineer the fabric of our reality at the most fundamental level," states the patent. This would lead to major advancements in aerospace propulsion and generating power. Not to mention other reality-changing outcomes that come to mind.
Among Pais's other patents are inventions that stem from similar thinking, outlining pieces of technology necessary to make his creations come to fruition. His paper presented in 2019, titled "Room Temperature Superconducting System for Use on a Hybrid Aerospace Undersea Craft," proposes a system that can achieve superconductivity at room temperatures. This would become "a highly disruptive technology, capable of a total paradigm change in Science and Technology," conveys Pais.
High frequency gravitational wave generator.
Credit: Dr. Salvatore Pais
Another invention devised by Pais is an electromagnetic field generator that could generate "an impenetrable defensive shield to sea and land as well as space-based military and civilian assets." This shield could protect from threats like anti-ship ballistic missiles, cruise missiles that evade radar, coronal mass ejections, military satellites, and even asteroids.
Dr. Pais's ideas center around the phenomenon he dubbed "The Pais Effect". He referred to it in his writings as the "controlled motion of electrically charged matter (from solid to plasma) via accelerated spin and/or accelerated vibration under rapid (yet smooth) acceleration-deceleration-acceleration transients." In less jargon-heavy terms, Pais claims to have figured out how to spin electromagnetic fields in order to contain a fusion reaction – an accomplishment that would lead to a tremendous change in power consumption and an abundance of energy.
According to his bio in a recently published paper on a new Plasma Compression Fusion Device, which could transform energy production, Dr. Pais is a mechanical and aerospace engineer working at the Naval Air Warfare Center Aircraft Division (NAWCAD), which is headquartered in Patuxent River, Maryland. Holding a Ph.D. from Case Western Reserve University in Cleveland, Ohio, Pais was a NASA Research Fellow and worked with Northrop Grumman Aerospace Systems. His current Department of Defense work involves his "advanced knowledge of theory, analysis, and modern experimental and computational methods in aerodynamics, along with an understanding of air-vehicle and missile design, especially in the domain of hypersonic power plant and vehicle design." He also has expert knowledge of electrooptics, emerging quantum technologies (laser power generation in particular), high-energy electromagnetic field generation, and the "breakthrough field of room temperature superconductivity, as related to advanced field propulsion."
Suffice it to say, with such a list of research credentials that would make Nikola Tesla proud, Dr. Pais seems well-positioned to carry out groundbreaking work.
A craft using an inertial mass reduction device.
Credit: Salvatore Pais
The patents won't necessarily lead to these technologies ever seeing the light of day. The research has its share of detractors and nonbelievers among other scientists, who think the amount of energy required for the fields described by Pais and his ideas on electromagnetic propulsions are well beyond the scope of current tech and are nearly impossible. Yet investigators at The War Zone found comments from Navy officials that indicate the inventions are being looked at seriously enough, and some tests are taking place.
If you'd like to read through Pais's patents yourself, check them out here.
Laser Augmented Turbojet Propulsion System
Credit: Dr. Salvatore Pais
A new study suggests that reports of the impending infertility of the human male are greatly exaggerated.
- A new review of a famous study on declining sperm counts finds several flaws.
- The old report makes unfounded assumptions, has faulty data, and tends toward panic.
- The new report does not rule out that sperm counts are going down, only that this could be quite normal.
Several years ago, a meta-analysis of studies on human fertility came out warning us about the declining sperm counts of Western men. It was widely shared, and its findings were featured on the covers of popular magazines. Indeed, its findings were alarming: a nearly 60 percent decline in sperm per milliliter since 1973 with no end in sight. It was only a matter of time, the authors argued, until men were firing blanks, literally.
Well… never mind.
It turns out that the impending demise of humanity was greatly exaggerated. As the predicted infertility wave crashed upon us, there was neither a great rush of men to fertility clinics nor a sudden dearth of new babies. The only discussions about population decline focus on urbanization and the fact that people choose not to have kids rather than not being able to have them.
Now, a new analysis of the 2017 study says that lower sperm counts is nothing to be surprised by. Published in Human Fertility, its authors point to flaws in the original paper's data and interpretation. They suggest a better and smarter reanalysis.
Counting tiny things is difficult
The original 2017 report analyzed 185 studies on 43,000 men and their reproductive health. Its findings were clear: "a significant decline in sperm counts… between 1973 and 2011, driven by a 50-60 percent decline among men unselected by fertility from North America, Europe, Australia and New Zealand."
However, the new analysis points out flaws in the data. As many as a third of the men in the studies were of unknown age, an important factor in reproductive health. In 45 percent of cases, the year of the sample collection was unknown- a big detail to miss in a study measuring change over time. The quality controls and conditions for sample collection and analysis vary widely from study to study, which likely influenced the measured sperm counts in the samples.
Another study from 2013 also points out that the methods for determining sperm count were only standardized in the 1980s, which occurred after some of the data points were collected for the original study. It is entirely possible that the early studies gave inaccurately high sperm counts.
This is not to say that the 2017 paper is entirely useless; it had a much more rigorous methodology than previous studies on the subject, which also claimed to identify a decline in sperm counts. However, the original study had more problems.
Garbage in, garbage out
Predictable as always, the media went crazy. Discussions of the decline of masculinity took off, both in mainstream and less-than-reputable forums; concerns about the imagined feminizing traits of soy products continued to increase; and the authors of the original study were called upon to discuss the findings themselves in a number of articles.
However, as this new review points out, some of the findings of that meta-analysis are debatable at best. For example, the 2017 report suggests that "declining mean [sperm count] implies that an increasing proportion of men have sperm counts below any given threshold for sub-fertility or infertility," despite little empirical evidence that this is the case.
The WHO offers a large range for what it considers to be a healthy sperm count, from 15 to 250 million sperm per milliliter. The benefits to fertility above a count of 40 million are seen as minimal, and the original study found a mean sperm concentration of 47 million sperm per milliliter.
Healthy sperm, healthy man?
The claim that sperm count is evidence of larger health problems is also scrutinized in this new article. While it is true that many major health problems can impact reproductive health, there is little evidence that it is the "canary in the coal mine" for overall well-being. A number of studies suggest that any relation between lifestyle choices and this part of reproductive health is limited at best.
Lastly, ideas that environmental factors could be at play have been debunked since 2017. While the original paper considered the idea that pollutants, especially from plastics, could be at fault, it is now known that this kind of pollution is worse in the parts of the world that the original paper observed higher sperm counts in (i.e., non-Western nations).
There never was a male fertility crisis
The authors of the new review do not deny that some measurements are showing lower sperm counts, but they do question the claim that this is catastrophic or part of a larger pathological issue. They propose a new interpretation of the data. Dubbed the "Sperm Count Biovariability hypothesis," it is summarized as:
"Sperm count varies within a wide range, much of which can be considered non-pathological and species-typical. Above a critical threshold, more is not necessarily an indicator of better health or higher probability of fertility relative to less. Sperm count varies across bodies, ecologies, and time periods. Knowledge about the relationship between individual and population sperm count and life-historical and ecological factors is critical to interpreting trends in average sperm counts and their relationships to human health and fertility."
Still, the authors note that lower sperm counts "could decline due to negative environmental exposures, or that this may carry implications for men's health and fertility."
However, they disagree that the decline in absolute sperm count is necessarily a bad sign for men's health and fertility. We aren't at civilization ending catastrophe just yet.