46 Matching Annotations
  1. Apr 2019
    1. Andrews, P., & Hatch, G. (2001). Hungary and its characteristic pedagogical flow. Proceedings of the British Congress of Mathematics Education, 21(2). 26-40. Stockton, J. C. (2010). Education of Mathematically Talented Students in Hungary. Journal of Mathematics Education at Teachers College, 1(2), 1-6.

      references to read

    2. Moreover, the teacher repeatedly asks, “Did anyone get a different answer?” or “Did anyone use a different method?” to elicit multiple solutions strategies. This highlights the connections between different problems, concepts, and areas of mathematics and helps develop students’ mathematical creativity. Creativity is further fostered through acknowledging “good mistakes.” Students who make an error are often commended for the progress they made and how their work contributed to the discussion and to the collective understanding of the class.
  2. Feb 2019
    1. And so it makes most sense to regard epoch 280 as the point beyond which overfitting is dominating learning in our neural network.

      I do not get this. Epoch 15 indicates that we are already over-fitting to the training data set, on? Assuming both training and test set come from the same population that we are trying to learn from.

    2. If we see that the accuracy on the test data is no longer improving, then we should stop training

      This contradicts the earlier statement about epoch 280 being the point where there is over-training.

    3. It might be that accuracy on the test data and the training data both stop improving at the same time

      Can this happen? Can the accuracy on the training data set ever increase with the training epoch?

    4. What is the limiting value for the output activations aLj

      When c is large, small differences in z_j^L are magnified and the function jumps between 0 and 1, depending on the sign of the differences. On the other hand, when c is very small, all activation values will be close to 1/N; where N is the number of neurons in layer L.

  3. Dec 2018
    1. Unfortunately, many classrooms focus on math facts in isolation, giving students the impression that math facts are the essence of mathematics, and, even worse, that mastering the fast recall of math facts is what it means to be a strong mathematics student. Both of these ideas are wrong, and it is critical that we remove them from classrooms, as they play a key role in creating math-anxious and disaffected students.

      This article uses the word "unfortunately quite a lot.

    2. If you think mathematics is difficult, tough, or you're scared of it, this article will indicate why and potentially show you a way forward for yourself and your children.

    3. The hippocampus, like other brain regions, is not fixed and can grow at any time,15 but it will always be the case that some students are faster or slower when memorizing, and this has nothing to do with mathematics potential.
    4. Notably, the brain can only compress concepts; it cannot compress rules and methods.
    5. Unfortunately for low achievers, they are often identified as struggling with math and therefore given more drill and practice—cementing their beliefs that math success means memorizing methods, not understanding and making sense of situations. They are sent down a damaging pathway that makes them cling to formal procedures, and as a result, they often face a lifetime of difficulty with mathematics.
    6. The low achievers did not know less, they just did not use numbers flexibly—probably because they had been set on the wrong pathway, from an early age, of trying to memorize methods and number facts instead of interacting with numbers flexibly.4
  4. Nov 2018
    1. Stochastic Gradient Descent Optimizes Over-parameterized Deep ReLU Networks

      这个哥们的文章和这个月内好几篇的立意基本一致(1811.03804/1811.03962/1811.04918) [抓狂] ,估计作者正写的时候,内心是崩溃的~[笑cry] 赶快强调自己有着不同的 assumption~

    2. Accelerating Natural Gradient with Higher-Order Invariance

      每次看到研究梯度优化理论的 paper,都感觉到无比的神奇!厉害到爆表。。。。

    3. Learning and Generalization in Overparameterized Neural Networks, Going Beyond Two Layers

      全篇的数学理论推导,意在回答2/3层过参的网络可以足够充分地学习和有良好的泛化表现,即使在简单的优化策略(类SGD)等假定下。(FYI: 文章可谓行云流水,直截了当,标准规范,阅读有种赏心悦目的感觉~)

    4. Gradient Descent Finds Global Minima of Deep Neural Networks

      全篇的数学理论证明:深度过参网络可以训练到0。(仅 train loss,非 test loss)+(GD,非 SGD)


    5. A Convergence Theory for Deep Learning via Over-Parameterization

      又一个全篇的数学理论证明,但是没找到 conclusion 到底是啥,唯一接近的是 remark 的信息,但内容也都并不惊奇。不过倒是一个不错的材料,若作为熟悉DNN背后的数学描述的话。

  5. Apr 2018
    1. ConvexHull

      In mathematics, the convex hull or convex envelope of a set X of points in the Euclidean plane or in a Euclidean space (or, more generally, in an affine space over the reals) is the smallest convex set that contains X. For instance, when X is a bounded subset of the plane, the convex hull may be visualized as the shape enclosed by a rubber band stretched around X. -Wikipedia

  6. Nov 2017
    1. As Uri Treisman said, “The most common use of algebra in the adult world is helping their kids with algebra.”

      CUNY found that more students went on to graduate when they were allowed to take statistics instead of remedial algebra. And the results were the same regardless of race or ethnicity.

  7. Jul 2017
    1. there are two different measurements for the length of a foot in the United States: the International Foot (also commonly called the foot) and the U.S. Survey Foot. The International Foot (which we were all taught in school) is defined as 0.3048 meters, whereas the U.S. Survey Foot is defined as 0.3048006096 meters. The difference of the two equates to 2 parts per million.

      For example, in a measurement of 10,000 feet, the difference would be 0.02 feet (just less than one-quarter of an inch). In a measurement of 1 million feet, the difference is 2 feet.

  8. Apr 2017
    1. Two people can have one conversation. Three people have four unique conversation groups (three different two-person conversations and a fourth conversation between all three as a group). Five people have 26. Twenty people have 1,048,554.

      what's the equation for that?

  9. Jan 2017
    1. Whether you're a student, parent, or teacher, this book is your key to unlocking the aha! moments that make math click -- and learning enjoyable.

      You had me already at the Coffee Cup picture over the equations! :)

  10. Nov 2016
    1. If you accept this, thenit seems fair to say that untilPversusNPis solved, the story of Hilbert's Entscheidungsproblem|itsrise, its fall, and the consequences for philosophy|is not yet over.

      If you accept this bizarre interpretation, then you can suspend the belief in the fact, known to everybody, that \(P \ne NP\), because it hasn't been mathematically proved, and say the question isn't solved yet. Wow, how interesting!

  11. Oct 2016
    1. Sunil Singh asks us to stop promoting mathematics based on its current applications in business and science. Math is an art that should be enjoyed for its own sake.

      This reminded me of A Mathematician's Lament by Paul Lockhart. This is a 25-page essay which was later worked into a 140-page book. (And Sunil Singh has read at least one of them. He credits Lockhart in one of the replies.)

      It also reminds me of this article on the history of Gaussian elimination and the birth of matrix algebra. Newton's algebra text included instructions for solving systems of equations -- but it didn't have much practical use until later. (Silly word problems are as old as mathematics.)

    1. We should let people learn at their own pace. We should neither rush them, nor hold them back. If they show a talent, then encouraging them to push themselves is fine.

    1. Math isn't for everyone, and that's fine. The same is true of any other subject. We should help people learn what they are interested in learning.

  12. Jul 2016
    1. I always found it incredible. He would start with some problem, and fill up pages with calculations. And at the end of it, he would actually get the right answer! But he usually wasn’t satisfied with that. Once he’d gotten the answer, he’d go back and try to figure out why it was obvious. And often he’d come up with one of those classic Feynman straightforward-sounding explanations. And he’d never tell people about all the calculations behind it. Sometimes it was kind of a game for him: having people be flabbergasted by his seemingly instant physical intuition, not knowing that really it was based on some long, hard calculation he’d done.

      Straightforward intuition isn't just intuition.

  13. Jun 2016
    1. The Hardy-Littlewood Rule

      "The rule states that anyone that joins collaboration in good faith will be listed equally as an author, regardless of the relative contributions they end up making.

  14. Apr 2016
  15. Mar 2016
    1. New property of prime numbers discovered. Primes greater than 5 can end with 1, 3, 7, or 9. The next prime is less likely to end with the same digit, and biased toward one of the remaining three. For instance, a prime ending in 3 is most likely to be followed by a prime ending in 9. The bias evens out as the primes get larger, but only very slowly.


  16. Feb 2016
    1. Borges “Like works of literature, mathematical ideas help expand our circle of empathy, liberating us from the tyranny of a single, parochial point of view.”
    1. Design-Oriented Analysis Rules and Tools, Dr. R. David Middlebrook

      Real problems usually have more variables than equations. So approximations and inequalities are essential.

    1. Free courses and tutorials from the Santa Fe Institute on subjects related to complex systems science.

  17. Jan 2016
    1. Category Theory for the Sciences by David I. Spivak<br> Creative Commons Attribution-NonCommercial-ShareAlike 4.0<br> MIT Press.

  18. Dec 2015
    1. All this time, however, category theory was consistently seen by much of the mathe-matical community as ridiculously abstract. But in the 21st century it has finally cometo find healthy respect within the larger community of pure mathematics. It is the lan-guage of choice for graduate-level algebra and topology courses, and in my opinion willcontinue to establish itself as the basic framework in which mathematics is done
  19. May 2015
    1. an infamous number conundrum

      One of the central problems in these articles is the glaring lack of math. These personal choices and the environment they have set up for themselves tells us rather only the minuscule part of the story!

    2. eccentricities of Grigori Perelman

      What eccentricities? He just lives with his parents and did not accept either the Fields' Medal or the Claymath's Millennium Prize.