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PROBLEMS AND DIFFICULTIES ENCOUNTERED BY STUDENTS TOWARDS MASTERING LEARNING COMPETENCIES IN MATHEMATICS
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PhD dissertations that solve an established open problem
I search for a big list of open problems which have been solved in a PhD thesis by the Author of the thesis (or with collaboration of her/his supervisor).
In my question I search for every possible open problem but I prefer (but not limited) to receive answers about those open problems which had been unsolved for at least (about) 25 years and before the appearance of the ultimate solution, there had been significant attentions and efforts for solving it. I mean that the problem was not a forgotten problem.
If the Gauss proof of the fundamental theorem of algebra did not had a gap, then his proof could be an important example of such dissertations.
I ask the moderators to consider this question as a wiki question.
- soft-question
- open-problems
- mathematics-education
- 17 $\begingroup$ Godel's thesis comes to mind. $\endgroup$ – user2277550 Apr 24, 2018 at 7:39
- 81 $\begingroup$ Don't all PhD theses solve open problems...? $\endgroup$ – Najib Idrissi Apr 24, 2018 at 7:42
- 20 $\begingroup$ @NajibIdrissi: I understood this question to be about open problems in the narrow sense of a precise mathematical question that at least some researcher(s) have previously articulated and tried to answer , e.g. “what is the dimension of such-and-such space?”. Most theses I know only solve open problems only in a much broader sense: open-ended questions that researchers may have wondered about, like “what can we say about the homology of such-and-such space?”, or “can we develop a useful theory of homotopy-coherent diagrams in such-and-such setting?”. $\endgroup$ – Peter LeFanu Lumsdaine Apr 24, 2018 at 9:04
- 9 $\begingroup$ @NajibIdrissi: Not always: sometimes a thesis provides a new insight into an already solved problem, or a better way to solve it. Tate's thesis is the first to come to mind. $\endgroup$ – Alex M. Apr 24, 2018 at 11:27
- 6 $\begingroup$ Eric Larson proved the Maximal Rank Conjecture (a significant problem in the algebraic geometry of curves) in his MIT PhD thesis this year: math.mit.edu/research/graduate/thesis-defenses-2018.php $\endgroup$ – Sam Hopkins Apr 24, 2018 at 13:32
24 Answers 24
I find George Dantzig's story particularly impressive and inspiring.
While he was a graduate student at UC Berkeley, near the beginning of a class for which Dantzig was late, professor Jerzy Neyman wrote two examples of famously unsolved statistics problems on the blackboard. When Dantzig arrived, he assumed that the two problems were a homework assignment and wrote them down. According to Dantzig, the problems "seemed to be a little harder than usual", but a few days later he handed in completed solutions for the two problems, still believing that they were an assignment that was overdue. Six weeks later, Dantzig received a visit from an excited professor Neyman, who was eager to tell him that the homework problems he had solved were two of the most famous unsolved problems in statistics. Neyman told Dantzig to wrap the two problems in a binder and he would accept them as a Ph.D. thesis.
The two problems that Dantzig solved were eventually published in: On the Non-Existence of Tests of "Student's" Hypothesis Having Power Functions Independent of σ (1940) and in On the Fundamental Lemma of Neyman and Pearson (1951).
- 2 $\begingroup$ Thank you for this very interesting answer on George Dantzig's story. $\endgroup$ – Ali Taghavi Apr 24, 2018 at 7:54
- 9 $\begingroup$ For those who like the story, a Snopes article contains some interesting excerpts from an interview with George Dantzig: snopes.com/fact-check/the-unsolvable-math-problem $\endgroup$ – JohnEye Apr 24, 2018 at 10:20
- 10 $\begingroup$ That's freaking crazy. $\endgroup$ – iammax Apr 24, 2018 at 18:45
- 5 $\begingroup$ If the problems were solved concurrently, why were they published 11 years apart? $\endgroup$ – Michael Sep 25, 2018 at 20:39
- 3 $\begingroup$ @Michael --- Wikipedia explains that "Years later another researcher, Abraham Wald, was preparing to publish an article that arrived at a conclusion for the second problem, and included Dantzig as its co-author when he learned of the earlier solution." $\endgroup$ – Carlo Beenakker Sep 26, 2018 at 0:31
I am quite surprised that nobody has mentioned Grothendieck's thesis. Apparently Laurent Schwartz gave Grothendieck a recent paper listing a number of open problems in functional analysis at one of their initial meetings. (Schwartz had just won the Fields at the time.) Grothendieck went away for a few weeks/ months and then returned with solutions to many (or all?) of the questions. In the course of the next few years Grothendieck became one of the world's leading functional analysts, before turning his attention to algebraic geometry.
This is the story I heard as part of mathematical gossip many, many years ago. Maybe someone who is more knowledgeable can chime in.
Another utterly spectacular thesis was Noam Elkies'. Among other things he settled a 200 year old problem posed by Euler!
- 7 $\begingroup$ I’d upvote this a dozen times if I could. As I understand it, we owe the modern theory of tensor products of topological vector spaces—and hence, in particular, the theory of nuclear topological vector spaces—entirely to Grothendieck’s PhD thesis. A quick sketch can be found, for instance, in this survey chapter by Fernando Bombal. $\endgroup$ – Branimir Ćaćić Sep 30, 2020 at 0:00
- $\begingroup$ I just skimmed the Ph.D. thesis of Noam Elkies. I don't think that the result you mentioned is in there. He did solve that problem of Euler's around the same time that he wrote his Ph.D. thesis, but the solution doesn't appear in his Ph.D. thesis as far as I can tell. $\endgroup$ – Timothy Chow Nov 26, 2022 at 18:27
Godel's Completeness Theorem, was part of his PHD thesis.
It was definitely an active field of research, but I don't know to what degree the problem was an open one, in the way we understand it today.
.. when Kurt Gödel joined the University of Vienna in 1924, he took up theoretical physics as his major. Sometime before this, he had read Goethe’s theory of colors and became interest in the subject. At the same time, he attended classes on mathematics and philosophy as well. Soon he came in contact with great mathematicians and in 1926, influenced by number theorist Philipp Furtwängler, he decided to change his subject and take up mathematics. Besides that, he was highly influenced by Karl Menger’s course in dimension theory and attended Heinrich Gomperz’s course in the history of philosophy. Also in 1926, he entered the Vienna Circle, a group of positivist philosophers formed around Moritz Schlick, and until 1928, attended their meetings regularly. After graduation, he started working for his doctoral degree under Hans Hahn. His dissertation was on the problem of completeness. In the summer of 1929, Gödel submitted his dissertation, titled ‘Über die Vollständigkeit des Logikkalküls’ (On the Completeness of the Calculus of Logic). Subsequently in February 1930, he received his doctorate in mathematics from the University of Vienna. Sometime now, he also became an Austrian citizen.
- $\begingroup$ I don't know whether this specific question was open, but it is certainly in the family of open problems highlighted by Hilbert's 2nd Problem and the Entscheidungsproblem. $\endgroup$ – Joshua Grochow Sep 26, 2018 at 2:37
- 2 $\begingroup$ It was an open problem (posed by Hilbert and Ackermann) but only a few years earlier. It also falls out of earlier results by Skolem. $\endgroup$ – none Sep 26, 2018 at 7:11
The thesis of Martin Hertweck answered the at that time 60-years-old isomorphism problem for integral group rings in the negative, by constructing a counterexample. That is, a pair of non-isomorphic finite groups $G$ and $H$ such that the group rings $\mathbb{Z}G$ and $\mathbb{Z}H$ are isomorphic. This result has been published afterwards in the Annals of Mathematics .
- 1 $\begingroup$ I never heard about this very great thesis. Thank you for this answer. $\endgroup$ – Ali Taghavi May 20, 2018 at 19:12
- $\begingroup$ @AliTaghavi: You are welcome. -- And thank YOU! $\endgroup$ – Stefan Kohl ♦ Sep 11, 2022 at 18:52
Scott Aaronson's thesis , Limits on Efficient Computation in the Physical World , refuted some popular wisdom .
In the first part of the thesis, I attack the common belief that quantum computing resembles classical exponential parallelism, by showing that quantum computers would face serious limitations on a wider range of problems than was previously known. In particular, any quantum algorithm that solves the collision problem -- that of deciding whether a sequence of $n$ integers is one-to-one or two-to-one -- must query the sequence $\Omega(n^{1/5})$ times. This resolves a question that was open for years; previously no lower bound better than constant was known. A corollary is that there is no "black-box" quantum algorithm to break cryptographic hash functions or solve the Graph Isomorphism problem in polynomial time.
There was even a second part to that thesis...
...Next I ask what happens to the quantum computing model if we take into account that the speed of light is finite -- and in particular, whether Grover's algorithm still yields a quadratic speedup for searching a database. Refuting a claim by Benioff, I show that the surprising answer is yes.
Lisa Piccirillo, who recently obtained her PhD from the University of Texas, Austin, showed that the Conway knot is not slice, answering a relatively famous open problem in topology. You can read a popular account of her work in Quanta here: https://www.quantamagazine.org/graduate-student-solves-decades-old-conway-knot-problem-20200519/ . Her paper proving this result was published in the Annals of Math; but I'm pretty sure it also constituted her dissertation (see https://gradschool.utexas.edu/news/studying-knots-and-four-dimensional-spaces ).
John von Neumann's dissertation seems to be an example with just the right timing.
But at the beginning of the 20th century [ in 1901, to be precise ], efforts to base mathematics on naive set theory suffered a setback due to Russell's paradox (on the set of all sets that do not belong to themselves). The problem of an adequate axiomatization of set theory was resolved implicitly about twenty years later by Ernst Zermelo and Abraham Fraenkel [i.e. there was active research on the question]. Zermelo–Fraenkel set theory provided a series of principles that allowed for the construction of the sets used in the everyday practice of mathematics, but they did not explicitly exclude the possibility of the existence of a set that belongs to itself. In his doctoral thesis of 1925, von Neumann demonstrated two techniques to exclude such sets—the axiom of foundation and the notion of class. The axiom of foundation proposed that every set can be constructed from the bottom up in an ordered succession of steps by way of the principles of Zermelo and Fraenkel. If one set belongs to another then the first must necessarily come before the second in the succession. This excludes the possibility of a set belonging to itself. To demonstrate that the addition of this new axiom to the others did not produce contradictions, von Neumann introduced a method of demonstration, called the method of inner models, which later became an essential instrument in set theory. The second approach to the problem of sets belonging to themselves took as its base the notion of class, and defines a set as a class which belongs to other classes, while a proper class is defined as a class which does not belong to other classes. Under the Zermelo–Fraenkel approach, the axioms impede the construction of a set of all sets which do not belong to themselves. In contrast, under the von Neumann approach, the class of all sets which do not belong to themselves can be constructed, but it is a proper class and not a set. Van Heijenoort, Jean (1967). From Frege to Gödel: a Source Book in Mathematical Logic, 1879–1931. Cambridge, Massachusetts: Harvard University Press. ISBN 978-0-674-32450-3. OCLC 523838.
A question, a book, and a couple of dissertations; the most relevant, I think, is the thesis by Petkovšek. Hopefully this is an acceptable MO answer. First, the question comes from Knuth in The Art of Computer Programming :
[50] Develop computer programs for simplifying sums that involve binomial coefficients. Exercise 1.2.6.63 in The Art of Computer Programming, Volume 1: Fundamental Algorithms by Donald E. Knuth, Addison Wesley, Reading, Massachusetts, 1968.
(For those unfamiliar, there is a pseudo log scale to rate each problem such that [50], as above, is the most difficult exercise, expected to take some years to answer).
A solution to this exercise is given by the book A = B by Marko Petkovsek, Herbert Wilf, and Doron Zeilberger (fully availble from the linked page).
On page 29 of the book, the authors mention a Ph.D. dissertation, and one of the author's Ph.D. thesis (among a few other works) that provide the main content of the book:
[Fase45] is the Ph.D. dissertation of Sister Mary Celine Fasenmyer, in 1945. It showed how recurrences for certain polynomial sequences could be found algorithmically. (See Chapter 4.) ... [Petk91] is the Ph.D. thesis of Marko Petkovšek, in 1991. In it he discovered the algorithm for deciding if a given recurrence with polynomial coefficients has a "simple" solution, which, together with the algorithms above, enables the automated discovery of the simple evaluation of a given definite sum, if one exists, or a proof of nonexistence, if none exists (see Chapter 8). A definite hypergeometric sum is one of the form $f(n) = \sum^{\infty}_{k=-\infty} F(n, k)$, where $F$ is hypergeometric.
Sources are
[Fase45] Fasenmyer, Sister Mary Celine, Some generalized hypergeometric polynomials , Ph.D. dissertation, University of Michigan, November, 1945.
[Petk91] Petkovšek, M., Finding closed-form solutions of difference equations by symbolic methods , Ph.D. thesis, Carnegie-Mellon University, CMU-CS-91-103, 1991.
- $\begingroup$ Thank you very much for your very interesting answer and your attention to my question. $\endgroup$ – Ali Taghavi Sep 10, 2022 at 19:16
June Huh's recent proof of Rota's conjecture (stated by Read in 1968 for graphical matroids and Rota in 1971 for all matroids) formed his 2014 Ph. D. thesis . For matroids over $\mathbb{C}$ , this appeared first in Huh's 2010 preprint ; for matroids over any field, this appeared in his work with Eric Katz (2011) ; for arbitrary matroids, see Adiprasito, Huh and Katz (2015) . As the dates would suggest, the thesis covers matroids over any field, but not the result on general matroids.
- 7 $\begingroup$ Just to clarify: this is not the Rota's conjecture , but a conjecture by Rota; which apparently also is known as Rota-Heron-Welsh conjecture. $\endgroup$ – Max Horn Sep 29, 2020 at 22:08
- 1 $\begingroup$ Is this the work for which Huh was awarded a Fields medal? $\endgroup$ – Gerry Myerson Aug 28, 2022 at 1:41
- 1 $\begingroup$ @GerryMyerson It was indeed one of the things for which he was awarded a Fields medal, according to the short citation . $\endgroup$ – Timothy Chow Nov 26, 2022 at 18:31
Vladimir Arnold's thesis was about his solution to Hilbert's 13th problem, which he had done a few years earlier. This info is missing from Wikipedia but some details are in Mactutor: https://mathshistory.st-andrews.ac.uk/Biographies/Arnold/
Does Serre's (Jean-Pierre) thesis qualifies ? He computed there a lot of homotopy groups of spheres. But I don't know how old was this problem in 1951.
- 2 $\begingroup$ +1 (!!!). The problem of computing the homotopy groups of spheres must be as old as Witold Hurewicz's definition of homotopy groups. Every single computation was a huge success, there were very few of them. Then Serre's revolution came. $\endgroup$ – Wlod AA Nov 5, 2020 at 20:04
Stephen Bigelow showed that braid groups are linear in his thesis at Berkeley in 2000 (the paper had already appeared in 1999 in JAMS, but he included it in his thesis).
John Thompson's thesis solved the famous and long-standing conjecture that a Frobenius kernel is nilpotent in the late 1950s. Not only was this noteworthy enough to be reported in the New York Times, but many of the techniques developed in the thesis played a major role in shaping finite group theory for decades to come.
Richard Laver's dissertation proved a long-standing conjecture of Fraïssé, that the scattered order types are well-quasi-ordered. But maybe that was not quite 25 years old at the time.
A very recent example is Eric Larson's 2018 dissertation The maximal rank conjecture [Lar1], which proves the following old conjecture:
Conjecture. (Maximal rank conjecture) Let $C \subseteq \mathbb P^r$ be a general Brill-Noether¹ curve. Then the restriction map $$H^0(\mathbb P^r, \mathcal O_{\mathbb P^r}(k)) \to H^0(C, \mathcal O_C(k))$$ has maximal rank, i.e. is injective if $h^0(\mathbb P^r, \mathcal O(k)) \leq h^0(C, \mathcal O(k))$ and surjective otherwise.
Historical remarks. Although I have been unable to find a definite place where this conjecture was stated, it is attributed to M. Noether by Arbarello and Ciliberto [AC83, p. 4]. Cases of the problem have been studied by M. Noether [Noe82, §8], Castelnuovo [Cas93, §7], and Severi [Sev15, §10].
In modern days, the conjecture had regained attention by 1982 [Har82, p. 79]. Partial results from around that time are mentioned in the introduction to [Lar2].
Larson's work culminates a lot of activity, including many papers by himself with other authors. An overview of the proof and how the different papers fit together is given in [Lar3].
References.
[AC83] E. Arbarello and C. Ciliberto, Adjoint hypersurfaces to curves in $\mathbb P^n$ following Petri . In: Commutative algebra (Trento, 1981) . Lect. Notes Pure Appl. Math. 84 (1983), p. 1-21. ZBL0516.14024 .
[Cas93] G. Castelnuovo, Sui multipli di una serie lineare di gruppi di punti appartenente ad una curva algebrica . Palermo Rend. VII (1893), p. 89-110. ZBL25.1035.02 .
[Har82] J. D. Harris, Curves in projective space . Séminaire de mathématiques supérieures, 85 (1982). Les Presses de l’Université de Montréal. ZBL0511.14014 .
[Lar1] E. K. Larson, The maximal rank conjecture . PhD dissertation, 2018.
[Lar2] E. K. Larson, The maximal rank conjecture . Preprint, arXiv:1711.04906 .
[Lar3] E. K. Larson, Degenerations of Curves in Projective Space and the Maximal Rank Conjecture . Preprint, arXiv:1809.05980 .
[Noe82] M. Nöther, Zur Grundlegung der Theorie der algebraischen Raumcurven . Abh. d. K. Akad. d. Wissensch. Berlin (1882). ZBL15.0684.01 .
[Sev15] F. Severi, Sulla classificazione delle curve algebriche e sul teorema d’esistenza di Riemann . Rom. Acc. L. Rend. 24 .5 (1915), p. 877-888, 1011-1020. ZBL45.1375.02 .
¹ Brill-Noether curves form a suitable component of the Kontsevich moduli space $\overline M_g(\mathbb P^r, d)$ of stable maps $\phi \colon C \to \mathbb P^r$ from a genus $g$ curve whose image has degree $d$ .
- 2 $\begingroup$ Unfortunately, the references given in [Larson2] (which I blindly copied) are not accurate. I have been unable to find the conjecture in either [Harris] or the Severi reference given therein. I spent a few hours trying to sort this out, which got me a bit closer, but I still didn't really find the original sources. I would be grateful if someone could clear this up. $\endgroup$ – R. van Dobben de Bruyn Sep 26, 2018 at 1:21
- 2 $\begingroup$ In arxiv.org/abs/1505.05460 Jensen and Payne cite "F. Severi. Sulla classificazione delle curve algebriche e sul teorema di esistenza di Riemann. Rend. R. Acc. Naz. Lincei, 24(5):887–888, 1915." and " J. Harris. Curves in projective space, volume 85 of Séminaire de Mathématiques Supérieures. Presses de l’Université de Montréal, Montreal, Que., 1982. With the collaboration of David Eisenbud" as sources for the Maximal Rank Conjecture. (By the way, I posted this example as a comment earlier- glad to see it upgraded into an answer.) $\endgroup$ – Sam Hopkins Sep 26, 2018 at 15:13
- $\begingroup$ @SamHopkins: Thanks, I had found those as well. But they do not settle the question of the origin of the conjecture. If I understand the paper correctly, Severi does not state the general case as a conjecture. $\endgroup$ – R. van Dobben de Bruyn Sep 26, 2018 at 23:02
Does Scholze's PhD thesis count, as if I remember rightly he applied perfectoid spaces to prove some important special case of Deligne's weight-monodromy conjecture? (Not an expert, correct me if I'm wrong).
Also I believe Mirzakhani gave a proof of the Witten conjecture when she was still a student, so I don't know if that proof is incorporated into her PhD thesis. Coincidentally, Kontsevich's proof of the Witten conjecture was also given in his PhD dissertation and then published as a journal article Intersection Theory on the Moduli Space of Curves and the Matrix Airy Function .
Edit: I have not read Mirzakhani's thesis, but it does indeed seem to be the case that she gave a new proof of Witten's conjecture in that thesis. Taken from the following article :
In 2002 I received a rather apologetic letter from Maryam, then a student at Harvard, together with a rough draft thesis and a request for comments. After reading only a few pages, I was transfixed. Starting with a formula discovered by Greg McShane in his 1991 Warwick PhD, she had developed some amazingly original and beautiful machinery which culminated in a completely new proof of Witten’s conjecture, a relation between integrable systems of Hamiltonian PDEs and the geometry of certain families of 2D topological field theories.
A recent and rather spectacular quasi-example is the Ph.D. thesis of María Pe (advisor Javier Fernández de Bobadilla), entitled “On the Nash Problem for Quotient Surface Singularities” (2011).
While it was not was the full solution of the Nash problem (dated back to 1968) it included great steps towards the full solution, eventually presented by María and Javier themselves in Annals of Math. in 2012.
Peter Weinberger's Ph.D. thesis is a superb example:
Proof of a Conjecture of Gauss on Class Number Two
See: https://en.wikipedia.org/wiki/Peter_J._Weinberger
- $\begingroup$ I couln't find this PhD thesis online. How is his strategy of proof? $\endgroup$ – user19475 Sep 27, 2018 at 3:06
- $\begingroup$ @TKe, sorry, you need to ask a specialist, I am not (too bad). $\endgroup$ – Wlod AA Sep 27, 2018 at 7:29
Since the OP mentions Gauss, this entry could be an appropriate addition to the list:
Manjul Bhargava's PhD thesis, Higher composition laws (2001), concerns a problem going back to Gauss. In the nineteenth century, Gauss had discovered a fundamental composition law for binary quadratic forms which are homogeneous polynomial functions of degree two in two variables. No formula or law of the Gauss type was known for cubic or higher degree forms. Bhargava broke the impasse of 200 years by producing a composition law for cubic and higher degree forms.
What makes Bhargava’s work especially remarkable is that he was able to explain all his revolutionary ideas using only elementary mathematics. In commenting on Bhargava’s results Andrew Wiles, his Ph.D. advisor said “He did it in a way that Gauss himself could have understood and appreciated.”
[ source1 and source2 ]
I found this interview from 2014, after Bhargava won the Fields medal, an inspiring read: "Somehow, he extracts ideas that are completely new or are retwisted in a way that changes everything. But it all feels very natural and unforced — it’s as if he found the right way to think.”
- $\begingroup$ I just now noticed that M. Khan suggested in a 2018 comment to the OP that Bhargava should be on this list. So here he is. $\endgroup$ – Carlo Beenakker Aug 30, 2022 at 21:28
A bit surprised that Leslie F. Greengard has not been mentioned. His PhD thesis from Yale was supervised by Vladimir Rokhlin, and is often cited for the development of the fast multipole method (FMM) which reduces the computation of the electrostatic or gravitational potential field/force for an N-particle system from $O(N^2)$ to $O(N)$ . Together with FFT, the Monte Carlo method, the simplex method for linear programming, Quicksort, QR algorithm, etc., FMM is regarded as one of the top 10 algorithms of the 20th century . To quote from the link,
This algorithm overcomes one of the biggest headaches of $N$ -body simulations: the fact that accurate calculations of the motions of $N$ particles interacting via gravitational or electrostatic forces (think stars in a galaxy, or atoms in a protein) would seem to require $O(N^2)$ computations—one for each pair of particles. The fast multipole algorithm gets by with $O(N)$ computations. It does so by using multipole expansions (net charge or mass, dipole moment, quadrupole, and so forth) to approximate the effects of a distant group of particles on a local group. A hierarchical decomposition of space is used to define ever-larger groups as distances increase. One of the distinct advantages of the fast multipole algorithm is that it comes equipped with rigorous error estimates, a feature that many methods lack.
Maria Chudnovsky's PhD thesis gives a proof of the Strong Perfect Graph Conjecture . This is a conjecture of Claude Berge from 1961 (hence meets the 25 year criterion), and was considered one of the hardest and most important open problems in graph theory at the time. It is joint work with Neil Robertson, Paul Seymour, and Robin Thomas and was published in the Annals in 2006. See The strong perfect graph theorem .
I suppose my own thesis fulfils this. Let $M$ be a monoid generated by a finite set $A$ and defined by a single defining relation $w=1$ . The language of all words $v \in A^\ast$ representing the identity element in $M$ is called the congruential language for $M$ . If the congruential language for $M$ is a context-free language, then Louxin Zhang [1] proved that the group of all units (two-sided invertible elements) $U(M)$ of $M$ is a virtually free group.
Zhang then asked in 1992 (conference held in 1990) the following questions:
- If $U(M)$ is virtually free, is the congruential language of $M$ a context-free language?
- Is it decidable, taking $w$ as input, whether the congruential language of $M$ is context-free?
In Chapter 3 of my thesis, I give an affirmative answer to both questions. In fact, I generalise the answer to Question 1 to to all monoids defined by arbitrarily many relations of the form $w_i = 1$ .
My thesis can be found on my website .
[1] Zhang, Louxin , Congruential languages specified by special string-rewriting systems, Ito, Masami (ed.), Words, languages and combinatorics, Kyoto, Japan, August 28–31, 1990. Singapore: World Scientific. 551-563 (1992). ZBL0875.68589 .
Robin Moser in his PhD thesis found a constructive proof for the Lovasz Local Lemma , a problem that was essentially open for decades. This earned him a Godel Prize.
Bart Plumstead's 1979 thesis at Chicago included (among other things) proofs of all three of the Eisenbud-Evans conjectures, which were widely considered to be among the most significant open questions in commutative algebra at the time.
- $\begingroup$ That is very interesting thank you but it does not satisfy the condition of at least 25 years of openness. I give the bounty to your answer and I will give another bounty to other recent answer which satisfy (i think) this c9nditiin $\endgroup$ – Ali Taghavi Sep 3, 2022 at 20:13
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The case for 'math-ish' thinking
by Stanford University
For everyone whose relationship with mathematics is distant or broken, Jo Boaler, a professor at Stanford Graduate School of Education (GSE), has ideas for repairing it. She particularly wants young people to feel comfortable with numbers from the start—to approach the subject with playfulness and curiosity, not anxiety or dread.
"Most people have only ever experienced what I call narrow mathematics—a set of procedures they need to follow, at speed," Boaler says. "Mathematics should be flexible, conceptual, a place where we play with ideas and make connections. If we open it up and invite more creativity, more diverse thinking, we can completely transform the experience."
Boaler, the Nomellini and Olivier Professor of Education at the GSE, is the co-founder and faculty director of Youcubed, a Stanford research center that provides resources for math learning that has reached more than 230 million students in over 140 countries. In 2013 Boaler, a former high school math teacher, produced How to Learn Math, the first massive open online course (MOOC) on mathematics education . She leads workshops and leadership summits for teachers and administrators, and her online courses have been taken by over a million users.
In her new book, " Math-ish: Finding Creativity, Diversity, and Meaning in Mathematics ," Boaler argues for a broad, inclusive approach to math education, offering strategies and activities for learners at any age. We spoke with her about why creativity is an important part of mathematics, the impact of representing numbers visually and physically, and how what she calls "ishing" a math problem can help students make better sense of the answer.
What do you mean by 'math-ish' thinking?
It's a way of thinking about numbers in the real world, which are usually imprecise estimates. If someone asks how old you are, how warm it is outside, how long it takes to drive to the airport—these are generally answered with what I call "ish" numbers, and that's very different from the way we use and learn numbers in school.
In the book I share an example of a multiple-choice question from a nationwide exam where students are asked to estimate the sum of two fractions: 12/13 + 7/8. They're given four choices for the closest answer: 1, 2, 19, or 21. Each of the fractions in the question is very close to 1, so the answer would be 2—but the most common answer 13-year-olds gave was 19. The second most common was 21.
I'm not surprised, because when students learn fractions, they often don't learn to think conceptually or to consider the relationship between the numerator or denominator. They learn rules about creating common denominators and adding or subtracting the numerators, without making sense of the fraction as a whole. But stepping back and judging whether a calculation is reasonable might be the most valuable mathematical skill a person can develop.
But don't you also risk sending the message that mathematical precision isn't important?
I'm not saying precision isn't important. What I'm suggesting is that we ask students to estimate before they calculate, so when they come up with a precise answer, they'll have a real sense for whether it makes sense. This also helps students learn how to move between big-picture and focused thinking, which are two different but equally important modes of reasoning.
Some people ask me, "Isn't 'ishing' just estimating?" It is, but when we ask students to estimate, they often groan, thinking it's yet another mathematical method. But when we ask them to "ish" a number, they're more willing to offer their thinking.
Ishing helps students develop a sense for numbers and shapes. It can help soften the sharp edges in mathematics, making it easier for kids to jump in and engage. It can buffer students against the dangers of perfectionism, which we know can be a damaging mindset. I think we all need a little more ish in our lives.
You also argue that mathematics should be taught in more visual ways. What do you mean by that?
For most people, mathematics is an almost entirely symbolic, numerical experience. Any visuals are usually sterile images in a textbook, showing bisecting angles, or circles divided into slices. But the way we function in life is by developing models of things in our minds. Take a stapler: Knowing what it looks like, what it feels and sounds like, how to interact with it, how it changes things—all of that contributes to our understanding of how it works.
There's an activity we do with middle-school students where we show them an image of a 4 x 4 x 4 cm cube made up of smaller 1 cm cubes, like a Rubik's Cube. The larger cube is dipped into a can of blue paint, and we ask the students, if they could take apart the little cubes, how many sides would be painted blue? Sometimes we give the students sugar cubes and have them physically build a larger 4 x 4 x 4 cube. This is an activity that leads into algebraic thinking.
Some years back we were interviewing students a year after they'd done that activity in our summer camp and asked what had stayed with them. One student said, "I'm in geometry class now, and I still remember that sugar cube, what it looked like and felt like." His class had been asked to estimate the volume of their shoes, and he said he'd imagined his shoes filled with 1 cm sugar cubes in order to solve that question. He had built a mental model of a cube.
When we learn about cubes, most of us don't get to see and manipulate them. When we learn about square roots, we don't take squares and look at their diagonals. We just manipulate numbers.
I wonder if people consider the physical representations more appropriate for younger kids.
That's the thing—elementary school teachers are amazing at giving kids those experiences, but it dies out in middle school, and by high school it's all symbolic. There's a myth that there's a hierarchy of sophistication where you start out with visual and physical representations and then build up to the symbolic. But so much of high-level mathematical work now is visual. Here in Silicon Valley, if you look at Tesla engineers, they're drawing, they're sketching, they're building models, and nobody says that's elementary mathematics.
There's an example in the book where you've asked students how they would calculate 38 x 5 in their heads, and they come up with several different ways of arriving at the same answer. The creativity is fascinating, but wouldn't it be easier to teach students one standard method?
That narrow, rigid version of mathematics where there's only one right approach is what most students experience, and it's a big part of why people have such math trauma. It keeps them from realizing the full range and power of mathematics. When you only have students blindly memorizing math facts, they're not developing number sense.
They don't learn how to use numbers flexibly in different situations. It also makes students who think differently believe there's something wrong with them.
When we open mathematics to acknowledge the different ways a concept or problem can be viewed, we also open the subject to many more students. Mathematical diversity, to me, is a concept that includes both the value of diversity in people and the diverse ways we can see and learn mathematics.
When we bring those forms of diversity together, it's powerful. If we want to value different ways of thinking and problem-solving in the world, we need to embrace mathematical diversity.
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10 Hard Math Problems That Continue to Stump Even the Brightest Minds
Maybe you’ll have better luck.
For now, you can take a crack at the hardest math problems known to man, woman, and machine. For more puzzles and brainteasers, check out Puzzmo . ✅ More from Popular Mechanics :
- To Create His Geometric Artwork, M.C. Escher Had to Learn Math the Hard Way
- Fourier Transforms: The Math That Made Color TV Possible
- The Game of Trees is a Mad Math Theory That Is Impossible to Prove
The Collatz Conjecture
In September 2019, news broke regarding progress on this 82-year-old question, thanks to prolific mathematician Terence Tao. And while the story of Tao’s breakthrough is promising, the problem isn’t fully solved yet.
A refresher on the Collatz Conjecture : It’s all about that function f(n), shown above, which takes even numbers and cuts them in half, while odd numbers get tripled and then added to 1. Take any natural number, apply f, then apply f again and again. You eventually land on 1, for every number we’ve ever checked. The Conjecture is that this is true for all natural numbers (positive integers from 1 through infinity).
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Tao’s recent work is a near-solution to the Collatz Conjecture in some subtle ways. But he most likely can’t adapt his methods to yield a complete solution to the problem, as Tao subsequently explained. So, we might be working on it for decades longer.
The Conjecture lives in the math discipline known as Dynamical Systems , or the study of situations that change over time in semi-predictable ways. It looks like a simple, innocuous question, but that’s what makes it special. Why is such a basic question so hard to answer? It serves as a benchmark for our understanding; once we solve it, then we can proceed onto much more complicated matters.
The study of dynamical systems could become more robust than anyone today could imagine. But we’ll need to solve the Collatz Conjecture for the subject to flourish.
Goldbach’s Conjecture
One of the greatest unsolved mysteries in math is also very easy to write. Goldbach’s Conjecture is, “Every even number (greater than two) is the sum of two primes.” You check this in your head for small numbers: 18 is 13+5, and 42 is 23+19. Computers have checked the Conjecture for numbers up to some magnitude. But we need proof for all natural numbers.
Goldbach’s Conjecture precipitated from letters in 1742 between German mathematician Christian Goldbach and legendary Swiss mathematician Leonhard Euler , considered one of the greatest in math history. As Euler put it, “I regard [it] as a completely certain theorem, although I cannot prove it.”
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Euler may have sensed what makes this problem counterintuitively hard to solve. When you look at larger numbers, they have more ways of being written as sums of primes, not less. Like how 3+5 is the only way to break 8 into two primes, but 42 can broken into 5+37, 11+31, 13+29, and 19+23. So it feels like Goldbach’s Conjecture is an understatement for very large numbers.
Still, a proof of the conjecture for all numbers eludes mathematicians to this day. It stands as one of the oldest open questions in all of math.
The Twin Prime Conjecture
Together with Goldbach’s, the Twin Prime Conjecture is the most famous in Number Theory—or the study of natural numbers and their properties, frequently involving prime numbers. Since you've known these numbers since grade school, stating the conjectures is easy.
When two primes have a difference of 2, they’re called twin primes. So 11 and 13 are twin primes, as are 599 and 601. Now, it's a Day 1 Number Theory fact that there are infinitely many prime numbers. So, are there infinitely many twin primes? The Twin Prime Conjecture says yes.
Let’s go a bit deeper. The first in a pair of twin primes is, with one exception, always 1 less than a multiple of 6. And so the second twin prime is always 1 more than a multiple of 6. You can understand why, if you’re ready to follow a bit of heady Number Theory.
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All primes after 2 are odd. Even numbers are always 0, 2, or 4 more than a multiple of 6, while odd numbers are always 1, 3, or 5 more than a multiple of 6. Well, one of those three possibilities for odd numbers causes an issue. If a number is 3 more than a multiple of 6, then it has a factor of 3. Having a factor of 3 means a number isn’t prime (with the sole exception of 3 itself). And that's why every third odd number can't be prime.
How’s your head after that paragraph? Now imagine the headaches of everyone who has tried to solve this problem in the last 170 years.
The good news is that we’ve made some promising progress in the last decade. Mathematicians have managed to tackle closer and closer versions of the Twin Prime Conjecture. This was their idea: Trouble proving there are infinitely many primes with a difference of 2? How about proving there are infinitely many primes with a difference of 70,000,000? That was cleverly proven in 2013 by Yitang Zhang at the University of New Hampshire.
For the last six years, mathematicians have been improving that number in Zhang’s proof, from millions down to hundreds. Taking it down all the way to 2 will be the solution to the Twin Prime Conjecture. The closest we’ve come —given some subtle technical assumptions—is 6. Time will tell if the last step from 6 to 2 is right around the corner, or if that last part will challenge mathematicians for decades longer.
The Riemann Hypothesis
Today’s mathematicians would probably agree that the Riemann Hypothesis is the most significant open problem in all of math. It’s one of the seven Millennium Prize Problems , with $1 million reward for its solution. It has implications deep into various branches of math, but it’s also simple enough that we can explain the basic idea right here.
There is a function, called the Riemann zeta function, written in the image above.
For each s, this function gives an infinite sum, which takes some basic calculus to approach for even the simplest values of s. For example, if s=2, then 𝜁(s) is the well-known series 1 + 1/4 + 1/9 + 1/16 + …, which strangely adds up to exactly 𝜋²/6. When s is a complex number—one that looks like a+b𝑖, using the imaginary number 𝑖—finding 𝜁(s) gets tricky.
So tricky, in fact, that it’s become the ultimate math question. Specifically, the Riemann Hypothesis is about when 𝜁(s)=0; the official statement is, “Every nontrivial zero of the Riemann zeta function has real part 1/2.” On the plane of complex numbers, this means the function has a certain behavior along a special vertical line. The hypothesis is that the behavior continues along that line infinitely.
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The Hypothesis and the zeta function come from German mathematician Bernhard Riemann, who described them in 1859. Riemann developed them while studying prime numbers and their distribution. Our understanding of prime numbers has flourished in the 160 years since, and Riemann would never have imagined the power of supercomputers. But lacking a solution to the Riemann Hypothesis is a major setback.
If the Riemann Hypothesis were solved tomorrow, it would unlock an avalanche of further progress. It would be huge news throughout the subjects of Number Theory and Analysis. Until then, the Riemann Hypothesis remains one of the largest dams to the river of math research.
The Birch and Swinnerton-Dyer Conjecture
The Birch and Swinnerton-Dyer Conjecture is another of the six unsolved Millennium Prize Problems, and it’s the only other one we can remotely describe in plain English. This Conjecture involves the math topic known as Elliptic Curves.
When we recently wrote about the toughest math problems that have been solved , we mentioned one of the greatest achievements in 20th-century math: the solution to Fermat’s Last Theorem. Sir Andrew Wiles solved it using Elliptic Curves. So, you could call this a very powerful new branch of math.
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In a nutshell, an elliptic curve is a special kind of function. They take the unthreatening-looking form y²=x³+ax+b. It turns out functions like this have certain properties that cast insight into math topics like Algebra and Number Theory.
British mathematicians Bryan Birch and Peter Swinnerton-Dyer developed their conjecture in the 1960s. Its exact statement is very technical, and has evolved over the years. One of the main stewards of this evolution has been none other than Wiles. To see its current status and complexity, check out this famous update by Wells in 2006.
The Kissing Number Problem
A broad category of problems in math are called the Sphere Packing Problems. They range from pure math to practical applications, generally putting math terminology to the idea of stacking many spheres in a given space, like fruit at the grocery store. Some questions in this study have full solutions, while some simple ones leave us stumped, like the Kissing Number Problem.
When a bunch of spheres are packed in some region, each sphere has a Kissing Number, which is the number of other spheres it’s touching; if you’re touching 6 neighboring spheres, then your kissing number is 6. Nothing tricky. A packed bunch of spheres will have an average kissing number, which helps mathematically describe the situation. But a basic question about the kissing number stands unanswered.
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First, a note on dimensions. Dimensions have a specific meaning in math: they’re independent coordinate axes. The x-axis and y-axis show the two dimensions of a coordinate plane. When a character in a sci-fi show says they’re going to a different dimension, that doesn’t make mathematical sense. You can’t go to the x-axis.
A 1-dimensional thing is a line, and 2-dimensional thing is a plane. For these low numbers, mathematicians have proven the maximum possible kissing number for spheres of that many dimensions. It’s 2 when you’re on a 1-D line—one sphere to your left and the other to your right. There’s proof of an exact number for 3 dimensions, although that took until the 1950s.
Beyond 3 dimensions, the Kissing Problem is mostly unsolved. Mathematicians have slowly whittled the possibilities to fairly narrow ranges for up to 24 dimensions, with a few exactly known, as you can see on this chart . For larger numbers, or a general form, the problem is wide open. There are several hurdles to a full solution, including computational limitations. So expect incremental progress on this problem for years to come.
The Unknotting Problem
The simplest version of the Unknotting Problem has been solved, so there’s already some success with this story. Solving the full version of the problem will be an even bigger triumph.
You probably haven’t heard of the math subject Knot Theory . It ’s taught in virtually no high schools, and few colleges. The idea is to try and apply formal math ideas, like proofs, to knots, like … well, what you tie your shoes with.
For example, you might know how to tie a “square knot” and a “granny knot.” They have the same steps except that one twist is reversed from the square knot to the granny knot. But can you prove that those knots are different? Well, knot theorists can.
✅ Up Next: The Amazing Math Inside the Rubik’s Cube
Knot theorists’ holy grail problem was an algorithm to identify if some tangled mess is truly knotted, or if it can be disentangled to nothing. The cool news is that this has been accomplished! Several computer algorithms for this have been written in the last 20 years, and some of them even animate the process .
But the Unknotting Problem remains computational. In technical terms, it’s known that the Unknotting Problem is in NP, while we don ’ t know if it’s in P. That roughly means that we know our algorithms are capable of unknotting knots of any complexity, but that as they get more complicated, it starts to take an impossibly long time. For now.
If someone comes up with an algorithm that can unknot any knot in what’s called polynomial time, that will put the Unknotting Problem fully to rest. On the flip side, someone could prove that isn’t possible, and that the Unknotting Problem’s computational intensity is unavoidably profound. Eventually, we’ll find out.
The Large Cardinal Project
If you’ve never heard of Large Cardinals , get ready to learn. In the late 19th century, a German mathematician named Georg Cantor figured out that infinity comes in different sizes. Some infinite sets truly have more elements than others in a deep mathematical way, and Cantor proved it.
There is the first infinite size, the smallest infinity , which gets denoted ℵ₀. That’s a Hebrew letter aleph; it reads as “aleph-zero.” It’s the size of the set of natural numbers, so that gets written |ℕ|=ℵ₀.
Next, some common sets are larger than size ℵ₀. The major example Cantor proved is that the set of real numbers is bigger, written |ℝ|>ℵ₀. But the reals aren’t that big; we’re just getting started on the infinite sizes.
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For the really big stuff, mathematicians keep discovering larger and larger sizes, or what we call Large Cardinals. It’s a process of pure math that goes like this: Someone says, “I thought of a definition for a cardinal, and I can prove this cardinal is bigger than all the known cardinals.” Then, if their proof is good, that’s the new largest known cardinal. Until someone else comes up with a larger one.
Throughout the 20th century, the frontier of known large cardinals was steadily pushed forward. There’s now even a beautiful wiki of known large cardinals , named in honor of Cantor. So, will this ever end? The answer is broadly yes, although it gets very complicated.
In some senses, the top of the large cardinal hierarchy is in sight. Some theorems have been proven, which impose a sort of ceiling on the possibilities for large cardinals. But many open questions remain, and new cardinals have been nailed down as recently as 2019. It’s very possible we will be discovering more for decades to come. Hopefully we’ll eventually have a comprehensive list of all large cardinals.
What’s the Deal with 𝜋+e?
Given everything we know about two of math’s most famous constants, 𝜋 and e , it’s a bit surprising how lost we are when they’re added together.
This mystery is all about algebraic real numbers . The definition: A real number is algebraic if it’s the root of some polynomial with integer coefficients. For example, x²-6 is a polynomial with integer coefficients, since 1 and -6 are integers. The roots of x²-6=0 are x=√6 and x=-√6, so that means √6 and -√6 are algebraic numbers.
✅ Try It Yourself: Can You Solve This Viral Brain Teaser From TikTok?
All rational numbers, and roots of rational numbers, are algebraic. So it might feel like “most” real numbers are algebraic. Turns out, it’s actually the opposite. The antonym to algebraic is transcendental, and it turns out almost all real numbers are transcendental—for certain mathematical meanings of “almost all.” So who’s algebraic , and who’s transcendental?
The real number 𝜋 goes back to ancient math, while the number e has been around since the 17th century. You’ve probably heard of both, and you’d think we know the answer to every basic question to be asked about them, right?
Well, we do know that both 𝜋 and e are transcendental. But somehow it’s unknown whether 𝜋+e is algebraic or transcendental. Similarly, we don’t know about 𝜋e, 𝜋/e, and other simple combinations of them. So there are incredibly basic questions about numbers we’ve known for millennia that still remain mysterious.
Is 𝛾 Rational?
Here’s another problem that’s very easy to write, but hard to solve. All you need to recall is the definition of rational numbers.
Rational numbers can be written in the form p/q, where p and q are integers. So, 42 and -11/3 are rational, while 𝜋 and √2 are not. It’s a very basic property, so you’d think we can easily tell when a number is rational or not, right?
Meet the Euler-Mascheroni constant 𝛾, which is a lowercase Greek gamma. It’s a real number, approximately 0.5772, with a closed form that’s not terribly ugly; it looks like the image above.
✅ One More Thing: Teens Have Proven the Pythagorean Theorem With Trigonometry. That Should Be Impossible
The sleek way of putting words to those symbols is “gamma is the limit of the difference of the harmonic series and the natural log.” So, it’s a combination of two very well-understood mathematical objects. It has other neat closed forms, and appears in hundreds of formulas.
But somehow, we don’t even know if 𝛾 is rational. We’ve calculated it to half a trillion digits, yet nobody can prove if it’s rational or not. The popular prediction is that 𝛾 is irrational. Along with our previous example 𝜋+e, we have another question of a simple property for a well-known number, and we can’t even answer it.
Dave Linkletter is a Ph.D. candidate in Pure Mathematics at the University of Nevada, Las Vegas. His research is in Large Cardinal Set Theory. He also teaches undergrad classes, and enjoys breaking down popular math topics for wide audiences.
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(e.g., OECD, 2010). Word problems constitute one of the most common types of problem solving (Jonassen, 2003) and serve as "a vehicle for developing students' general problem-solving skills" (Verschaffel, Greer, & De Corte, 2007, p.583). Word problem solving is a complex process, which requires students to integrate
Guzman Gurat M. (2018) "Mathematical problem-solving strategies among student teachers", Journal on Efficiency and Responsibility in Education and Science, Vol. 11, No. 3, pp. 53-64, online ISSN 1803-1617, printed ISSN 2336-2375, doi: 10.7160/ eriesj.2018.110302. MATHEMATICAL PROBLEM-SOLVING STRATEGIES AMONG STUDENT TEACHERS Abstract
The aim of the present paper is to present and discuss the recent progress of the. problem solving process in mathematics education. 1. Introduction. Problem - Solving (P-S) is a principal ...
A number of research projects in educational assessment reveal that students struggle when it comes to accomplishing problem-solving tasks in Mathematics. Such a struggle is primarily due to the ...
iii The following papers are included in this thesis: Paper I: Fülöp, E. (2015). Teaching problem-solving strategies in mathematics.LUMAT, 3(1), 37-54. Paper II: Fülöp, E. (under review) Developing problem-solving
mathematical problem-solving skills, and effect/role of assessments on students' mathematical problem-solving skills' as keywords, 63 studies were obtained. With a deep analysis of the collected data, 32 studies were related to teaching strategies in enhancing problem- solving skills, and these studies were filtered out, while ...
Microsoft Word - 445434-convertdoc.input.433044.2qsjx.docx. The Effects of Constructs Related to Mathematical Persistence. on Student Performance during Problem Solving. A thesis submitted to the Department of Education and Human Development of the College at.
mathematicians conceptualize word problems as part of a larger problem-solving component of the mathematics curriculum in which students must overcome barriers in order to obtain and explain a solution to a mathematical problem that is not directly apparent (Heddens & Speer, 2001). Based on this conceptualization of solving word
TEACHERS AND STUDENTS MATHEMATICAL PROBLEM-SOLVING BELIEFS AND SKILLS WITH A FOCUS ON PISA PROBLEMS by Seyma Pekgoz Dissertation Committee: Professor J. Philip Smith, Sponsor Professor Erica Walker Approved by the Committee on the Degree of Doctor of Education Date 12 February 2020
Problems, difficulties and pressures abound everywhere. In Mathematics, much has been said and heard of students struggling with problem solving. This study therefore primarily aimed to develop models that could address the problem solving difficulties of students through their coping strategies. Specifically, it aimed to determine the students' strategies in coping with their difficulties ...
1.1. Problem-Solving Strategies and their Significance . There are a number of distinct categorizations regarding problem-solving strategies, which are crucial to the problem-solving process (Altun, Memnun, & Yazgan, 2007; Ministry of National Education [MoNE], 2018; Posamentier & Krulik, 1998). These categorizations, however, basically
require problem-solving and reasoning; elaboration; and real-world applications (Erickson, 1999; Gavin et al., 2009; Rotigel & Fello, 2004; VanTassel-Baska, 2013). One such pedagogical technique that combines many of these attributes is problem-based learning (PBL). Project Gifted Education in Math and Science (Project GEMS) was established in
THE EFFECTS OF PROBLEM-BASED LEARNING ON INTEREST IN MATHEMATICS FOR ELEMENTARY STUDENTS ACROSS TIME. A Thesis Presented to The Faculty of the Department of Psychology Western Kentucky University Bowling Green, Kentucky. In Partial Fulfillment Of the Requirements for the Degree Master of Arts. By Kerry Douglas Duck.
PROBLEM SOLVING IN MATHEMATICS suggested, the relationship between problem-solving ability and its under-lying skills, particularly higher-order verbal skills, is probably more com-plex than has been supposed. Two factor analytic studies of problem solving in mathematics were synthesized by Werdelin (1966), who rotated the two factor matrices to
Mathematical problem solving is a complex cognitive activity, as a process to overcome a problem encountered and to solve it requires a number of strategies. View full-text. Article.
Taking into consideration the aforementioned different theories of leaning embedded on mathematical modeling, this study attempted to determine the effects of integrating mathematical modeling to the problem solving performance and math anxiety of Grade 9 students. In this study, much is given importance to the process rather than the product.
the following definition of mathematical problem solving in this dissertation: Mathematical problem solving is a nonsequential process that involves creativity and the application of mathematical knowledge—resources, strategies, and so on—to solve a nonroutine task for which . 4
Assessing the Mathematics Achievement of College Freshmen using Piaget's Logical Operations. Competencies in Mathematics. Unpublished Master's Thesis. Bataan Polytechnic State College, Bataan, Philippines. [14] Limjap, A. A. (2002). Issues on problem solving: Drawing implications for a Technomathematics Curriculum at the Collegiate Level.
This thesis is dedicated to my mother, Selma Nghipundaka and my late father, Kleopas Naivela ... mathematical problem solving skills in Grade 3 at rural farm schools in the Kunene region. In the introductory chapter, I briefly discuss the background context of the study, the statement of the ...
This study was conducted to determine the effect of creative problem solving strategy on the performance of students in mathematics problem solving. The study sample comprised of 30-paired teacher education students of Aklan Catholic College. The study adapted the experimental pretest posttest control group. The experimental group was exposed to creative problem solving while the control group ...
37. Godel's Completeness Theorem, was part of his PHD thesis. It was definitely an active field of research, but I don't know to what degree the problem was an open one, in the way we understand it today. .. when Kurt Gödel joined the University of Vienna in 1924, he took up theoretical physics as his major.
In the book I share an example of a multiple-choice question from a nationwide exam where students are asked to estimate the sum of two fractions: 12/13 + 7/8. They're given four choices for the ...
The research utilized a quasi-experimental design to find out and compare the mathematics performance of the students and problem-solving skills when they were exposed to Experiential Learning ...
One of the greatest unsolved mysteries in math is also very easy to write. Goldbach's Conjecture is, "Every even number (greater than two) is the sum of two primes.". You check this in your ...
Picture math puzzles Do the math. BrainSnack. Enter numbers in each row and column to arrive at the end totals. Only numbers 1 through 9 are used, and each only once. Answer: BrainSnack. Tricky ...
Online math solver with free step by step solutions to algebra, calculus, and other math problems. ... Try Math Solver. Type a math problem. Quadratic equation { x } ^ { 2 } - 4 x - 5 = 0. Trigonometry. 4 \sin \theta \cos \theta = 2 \sin \theta . Linear equation. y = 3x + 4 ... See how to solve problems and show your work—plus get definitions ...
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Online math solver with free step by step solutions to algebra, calculus, and other math problems. Get help on the web or with our math app. ... Type a math problem. Examples. Quadratic equation { x } ^ { 2 } - 4 x - 5 = 0. Trigonometry. 4 \sin \theta \cos \theta = 2 \sin \theta. Linear equation ...
Mathematics, as we know, " Mathematics is the Queen and servant of Sciences. Confluent learning, as reemphasized by Loon and Nichol (2015), is holistic, it aims to activate and engage. all of ...
Related Concepts. Trigonometry is a branch of mathematics concerned with relationships between angles and side lengths of triangles. In particular, the trigonometric functions relate the angles of a right triangle with ratios of its side lengths. The field emerged in the Hellenistic world during the 3rd century BC from applications of geometry ...