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Articles 1 - 13 of 13
Full-Text Articles in Mathematics
Education In The Era Of Google, Wikipedia, And Deep Learning: Are We Humans Still Needed And If Yes For What?, Miroslav Svitek, Olga Kosheleva, Vladik Kreinovich
Education In The Era Of Google, Wikipedia, And Deep Learning: Are We Humans Still Needed And If Yes For What?, Miroslav Svitek, Olga Kosheleva, Vladik Kreinovich
Departmental Technical Reports (CS)
One of the main purposes of education is to teach skills needed in future life and future jobs. What is important and what is useful changes with time. Before the industrial revolution, routine mechanical work was an important part of human activity – now machines can do it (and do it better). Before printing, copying was an important activity – now copy machines do it. Before computers, humans were needed for computing – now computer do it better. With Wikipedia and Google, there is not much need for scholars being erudite. Even extracting dependencies from data – one of the …
When Is Deep Learning Better And When Is Shallow Learning Better: Qualitative Analysis, Salvador Robles Herrera, Martine Ceberio, Vladik Kreinovich
When Is Deep Learning Better And When Is Shallow Learning Better: Qualitative Analysis, Salvador Robles Herrera, Martine Ceberio, Vladik Kreinovich
Departmental Technical Reports (CS)
In many practical situations, deep neural networks work better than the traditional "shallow" ones, however, in some cases, the shallow neural networks lead to better results. At present, deciding which type of neural networks will work better is mostly done by trial and error. It is therefore desirable to come up with some criterion of when deep learning is better and when shallow is better. In this paper, we argue that this depends on whether the corresponding situation has natural symmetries: if it does, we expect deep learning to work better, otherwise we expect shallow learning to be more effective. …
A Natural Causality-Motivated Description Of Learning, Olga Kosheleva, Vladik Kreinovich
A Natural Causality-Motivated Description Of Learning, Olga Kosheleva, Vladik Kreinovich
Departmental Technical Reports (CS)
Teaching is not easy. One of the main reasons why it is not easy is that the existing descriptions of the teaching process are not very precise -- and thus, we cannot use the usual optimization techniques, techniques which require a precise model of the corresponding phenomenon. It is therefore desirable to come up with a precise description of the learning process. To come up with such a description, we notice that on the set of all possible states of learning, there is a natural order s ≤ s' meaning that we can bring the student from the state s …
Video Or Text? Bullets Or No Bullets? Why Not Both?, Olga Kosheleva, Vladik Kreinovich, Christian Servin
Video Or Text? Bullets Or No Bullets? Why Not Both?, Olga Kosheleva, Vladik Kreinovich, Christian Servin
Departmental Technical Reports (CS)
Some students – which are, in terms of pop-psychology – more left-brain – prefer linear exposition, others – more right-brain ones – prefer 2-D images and texts with visual emphasis (e.g., with bullets). At present, instructors try to find a middle grounds between these two audiences, but why not prepare each material in two ways, aimed at both audiences?
Shall We Ignore All Intermediate Grades?, Christian Servin, Olga Kosheleva, Vladik Kreinovich
Shall We Ignore All Intermediate Grades?, Christian Servin, Olga Kosheleva, Vladik Kreinovich
Departmental Technical Reports (CS)
In most European universities, the overall student's grade for a course is determined exclusively by this student's performance on the final exam. All intermediate grades -- on homework, quizzes, and previous texts -- are, in effect, ignored. This arrangement helps gauge the student's performance by the knowledge that the student shows at the end of the course. The main drawback of this approach is that some students do not start studying until later, thinking that they can catch up and even get an excellent grade -- and this hurts their performance. To motivate students to study hard throughout the semester, …
How To Teach Advanced Highly Motivated Students: Teaching Strategy Of Iosif Yakovlevich Verebeichik, Olga Kosheleva, Vladik Kreinovich
How To Teach Advanced Highly Motivated Students: Teaching Strategy Of Iosif Yakovlevich Verebeichik, Olga Kosheleva, Vladik Kreinovich
Departmental Technical Reports (CS)
The paper describes and explains the teaching strategy of Iosif Yakovlevich Verebeichik, a successful mathematics teacher at special mathematical high schools -- schools for students interested in and skilled in mathematics. The resulting strategy seems counterintuitive and contrary to all the pedagogical advice. Our explanation is not complete: it worked well for this teacher, but others who tried to follow seemingly the same strategy did not succeed. How he made it work, how can others make it work -- this is still not clear. In the words of Verebeichik himself, while mathematics itself is a science, teaching mathematics is an …
Rosenzweig, Equality, And Assignment, Olga Kosheleva, Vladik Kreinovich
Rosenzweig, Equality, And Assignment, Olga Kosheleva, Vladik Kreinovich
Departmental Technical Reports (CS)
In his seminal book "The Star of Redemption", the renowned philosopher Franz Rosenzweig illustrated his ideas by the intuitive difference between mathematical statements A=B and B=A. Of course, from the purely mathematical viewpoint, these two statements are always equivalent, so to a person trained in mathematics -- even in simple school mathematics -- this illustration is puzzling. What we show is that from the viewpoint of common folks, there is indeed a subtle difference between how people understand these two equalities. To us, the understanding of this difference helped us better understand Rosenzweig's ideas. But we believe that this difference …
A Recent Result About Random Metrics Explains Why All Of Us Have Similar Learning Potential, Christian Servin, Olga Kosheleva, Vladik Kreinovich
A Recent Result About Random Metrics Explains Why All Of Us Have Similar Learning Potential, Christian Servin, Olga Kosheleva, Vladik Kreinovich
Departmental Technical Reports (CS)
In the same class, after the same lesson, the amount of learned material often differs drastically, by a factor of ten. Does this mean that people have that different learning abilities? Not really: experiments show that among different students, learning abilities differ by no more than a factor of two. This fact have been successfully used in designing innovative teaching techniques, techniques that help students realize their full learning potential. In this paper, we deal with a different question: how to explain the above experimental result. It turns out that this result about learning abilities -- which are, due to …
Egyptian Fractions Re-Revisited, Olga Kosheleva, Vladik Kreinovich, Francisco Zapata
Egyptian Fractions Re-Revisited, Olga Kosheleva, Vladik Kreinovich, Francisco Zapata
Departmental Technical Reports (CS)
Ancient Egyptians represented each fraction as a sum of unit fractions, i.e., fractions of the type 1/n. In our previous papers, we explained that this representation makes perfect sense: e.g., it leads to an efficient way of dividing loaves of bread between people. However, one thing remained unclear: why, when representing fractions of the type 2/(2k+1), Egyptians did not use a natural representation 1/(2k+1) + 1/(2k+1), but used a much more complicated representation instead. In this paper, we show that the need for such a complicated representation can be explained if we take into account that instead of cutting a …
In Education, Delayed Feedback Is Often More Efficient Than Immediate Feedback: A Geometric Explanation, Francisco Zapata, Olga Kosheleva, Vladik Kreinovich
In Education, Delayed Feedback Is Often More Efficient Than Immediate Feedback: A Geometric Explanation, Francisco Zapata, Olga Kosheleva, Vladik Kreinovich
Departmental Technical Reports (CS)
Feedback is important in education. It is commonly believed that immediate feedback is very important. That is why instructors stay often late at night grading students' assignments -- to make sure that the students get their feedback as early as possible. However, surprisingly, experiments show that in many cases, delayed feedback is more efficient that the immediate one. In this paper, we provide a simple geometric explanation of this seemingly counter-intuitive empirical phenomenon.
Maybe The Usual Students' Practice Of Cramming For A Test Makes Sense: A Mathematical Analysis, Francisco Zapata, Olga Kosheleva, Vladik Kreinovich
Maybe The Usual Students' Practice Of Cramming For A Test Makes Sense: A Mathematical Analysis, Francisco Zapata, Olga Kosheleva, Vladik Kreinovich
Departmental Technical Reports (CS)
We always teach students that cramming for a test is a bad idea, that they should study at the same speed throughout the semester – but many still cram. We ourselves are not that different: when we prepare papers for a conference, we often “cram” in the last days before the deadline instead of working with a regular speed for the whole time before the conference. The ubiquity of cramming makes us think that maybe it is not necessarily always a bad idea. And indeed, a simple model of a study process shows that an optimal solution often involve some …
How To Teach Implication, Martha Osegueda Escobar, Olga Kosheleva, Vladik Kreinovich
How To Teach Implication, Martha Osegueda Escobar, Olga Kosheleva, Vladik Kreinovich
Departmental Technical Reports (CS)
Logical implication is a somewhat counter-intuitive notion. For students, it is difficult to understand why a false statement implies everything. In this paper, we present a simple pedagogical way to make logical implication more intuitive.
Why Rozenzweig-Style Midrashic Approach Makes Rational Sense: A Logical (Spinoza-Like) Explanation Of A Seemingly Non-Logical Approach, Olga Kosheleva, Vladik Kreinovich
Why Rozenzweig-Style Midrashic Approach Makes Rational Sense: A Logical (Spinoza-Like) Explanation Of A Seemingly Non-Logical Approach, Olga Kosheleva, Vladik Kreinovich
Departmental Technical Reports (CS)
A 20 century German Jewish philosopher Franz Rosenzweig promoted a new approach to knowledge, an approach in which in addition to logical reasoning, coming up with stories with imagined additional details is also important. This approach is known as midrashic since it is similar to the use of similar stories -- known as midrashes -- in Judaism. While stories can make the material interesting, traditionally, such stories are not viewed as a serious part of scientific discovery. In this paper, we show that this seemingly non-logical approach can actually be explained in logical terms and thus, makes perfect rational sense.