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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.
If machines develop consciousness, or if we manage to give it to them, the human-robot dynamic will forever be different.
- Does AI—and, more specifically, conscious AI—deserve moral rights? In this thought exploration, evolutionary biologist Richard Dawkins, ethics and tech professor Joanna Bryson, philosopher and cognitive scientist Susan Schneider, physicist Max Tegmark, philosopher Peter Singer, and bioethicist Glenn Cohen all weigh in on the question of AI rights.
- Given the grave tragedy of slavery throughout human history, philosophers and technologists must answer this question ahead of technological development to avoid humanity creating a slave class of conscious beings.
- One potential safeguard against that? Regulation. Once we define the context in which AI requires rights, the simplest solution may be to not build that thing.
Duke University researchers might have solved a half-century old problem.
- Duke University researchers created a hydrogel that appears to be as strong and flexible as human cartilage.
- The blend of three polymers provides enough flexibility and durability to mimic the knee.
- The next step is to test this hydrogel in sheep; human use can take at least three years.
Duke researchers have developed the first gel-based synthetic cartilage with the strength of the real thing. A quarter-sized disc of the material can withstand the weight of a 100-pound kettlebell without tearing or losing its shape.
Photo: Feichen Yang.<p>That's the word from a team in the Department of Chemistry and Department of Mechanical Engineering and Materials Science at Duke University. Their <a href="https://onlinelibrary.wiley.com/doi/abs/10.1002/adfm.202003451" target="_blank">new paper</a>, published in the journal,<em> Advanced Functional Materials</em>, details this exciting evolution of this frustrating joint.<br></p><p>Researchers have sought materials strong and versatile enough to repair a knee since at least the seventies. This new hydrogel, comprised of three polymers, might be it. When two of the polymers are stretched, a third keeps the entire structure intact. When pulled 100,000 times, the cartilage held up as well as materials used in bone implants. The team also rubbed the hydrogel against natural cartilage a million times and found it to be as wear-resistant as the real thing. </p><p>The hydrogel has the appearance of Jell-O and is comprised of 60 percent water. Co-author, Feichen Yang, <a href="https://today.duke.edu/2020/06/lab-first-cartilage-mimicking-gel-strong-enough-knees" target="_blank">says</a> this network of polymers is particularly durable: "Only this combination of all three components is both flexible and stiff and therefore strong." </p><p> As with any new material, a lot of testing must be conducted. They don't foresee this hydrogel being implanted into human bodies for at least three years. The next step is to test it out in sheep. </p><p>Still, this is an exciting step forward in the rehabilitation of one of our trickiest joints. Given the potential reward, the wait is worth it. </p><p><span></span>--</p><p><em>Stay in touch with Derek on <a href="http://www.twitter.com/derekberes" target="_blank">Twitter</a>, <a href="https://www.facebook.com/DerekBeresdotcom" target="_blank">Facebook</a> and <a href="https://derekberes.substack.com/" target="_blank">Substack</a>. His next book is</em> "<em>Hero's Dose: The Case For Psychedelics in Ritual and Therapy."</em></p>
What would it be like to experience the 4th dimension?
Physicists have understood at least theoretically, that there may be higher dimensions, besides our normal three. The first clue came in 1905 when Einstein developed his theory of special relativity. Of course, by dimensions we’re talking about length, width, and height. Generally speaking, when we talk about a fourth dimension, it’s considered space-time. But here, physicists mean a spatial dimension beyond the normal three, not a parallel universe, as such dimensions are mistaken for in popular sci-fi shows.
An algorithm may allow doctors to assess PTSD candidates for early intervention after traumatic ER visits.
- 10-15% of people visiting emergency rooms eventually develop symptoms of long-lasting PTSD.
- Early treatment is available but there's been no way to tell who needs it.
- Using clinical data already being collected, machine learning can identify who's at risk.
The psychological scars a traumatic experience can leave behind may have a more profound effect on a person than the original traumatic experience. Long after an acute emergency is resolved, victims of post-traumatic stress disorder (PTSD) continue to suffer its consequences.
In the U.S. some 30 million patients are annually treated in emergency departments (EDs) for a range of traumatic injuries. Add to that urgent admissions to the ED with the onset of COVID-19 symptoms. Health experts predict that some 10 percent to 15 percent of these people will develop long-lasting PTSD within a year of the initial incident. While there are interventions that can help individuals avoid PTSD, there's been no reliable way to identify those most likely to need it.
That may now have changed. A multi-disciplinary team of researchers has developed a method for predicting who is most likely to develop PTSD after a traumatic emergency-room experience. Their study is published in the journal Nature Medicine.
70 data points and machine learning
Image source: Creators Collective/Unsplash
Study lead author Katharina Schultebraucks of Columbia University's Department Vagelos College of Physicians and Surgeons says:
"For many trauma patients, the ED visit is often their sole contact with the health care system. The time immediately after a traumatic injury is a critical window for identifying people at risk for PTSD and arranging appropriate follow-up treatment. The earlier we can treat those at risk, the better the likely outcomes."
The new PTSD test uses machine learning and 70 clinical data points plus a clinical stress-level assessment to develop a PTSD score for an individual that identifies their risk of acquiring the condition.
Among the 70 data points are stress hormone levels, inflammatory signals, high blood pressure, and an anxiety-level assessment. Says Schultebraucks, "We selected measures that are routinely collected in the ED and logged in the electronic medical record, plus answers to a few short questions about the psychological stress response. The idea was to create a tool that would be universally available and would add little burden to ED personnel."
Researchers used data from adult trauma survivors in Atlanta, Georgia (377 individuals) and New York City (221 individuals) to test their system.
Of this cohort, 90 percent of those predicted to be at high risk developed long-lasting PTSD symptoms within a year of the initial traumatic event — just 5 percent of people who never developed PTSD symptoms had been erroneously identified as being at risk.
On the other side of the coin, 29 percent of individuals were 'false negatives," tagged by the algorithm as not being at risk of PTSD, but then developing symptoms.
Image source: Külli Kittus/Unsplash
Schultebraucks looks forward to more testing as the researchers continue to refine their algorithm and to instill confidence in the approach among ED clinicians: "Because previous models for predicting PTSD risk have not been validated in independent samples like our model, they haven't been adopted in clinical practice." She expects that, "Testing and validation of our model in larger samples will be necessary for the algorithm to be ready-to-use in the general population."
"Currently only 7% of level-1 trauma centers routinely screen for PTSD," notes Schultebraucks. "We hope that the algorithm will provide ED clinicians with a rapid, automatic readout that they could use for discharge planning and the prevention of PTSD." She envisions the algorithm being implemented in the future as a feature of electronic medical records.
The researchers also plan to test their algorithm at predicting PTSD in people whose traumatic experiences come in the form of health events such as heart attacks and strokes, as opposed to visits to the emergency department.