https://mklab.gr/ Sat, 12 Sep 2026 08:26:58 +0000 MyBB https://mklab.gr/showthread.php?tid=1774 Tue, 01 Sep 2026 18:38:56 +0300 mklabgr]]> https://mklab.gr/showthread.php?tid=1774 Book:Mathematical Surprises
Author: Mordechai Ben-Ari
Publication date: 2022
Publisher: Springer Nature Switzerland AG
Subject: Recreational/elementary mathematics, geometry, algebra, combinatorics
Level: Advanced secondary-school mathematics and above
Open Access: Yes — Creative Commons Attribution 4.0 (CC BY 4.0).

Summary

Mathematical Surprises is a collection of 16 largely independent chapters devoted to mathematical results that are unexpected, elegant, or surprisingly accessible. Ben-Ari deliberately chooses topics that normally do not appear in school or introductory university textbooks but whose proofs can still be understood with a strong secondary-school background. The book is not intended as a conventional textbook; rather, it is designed for mathematical enrichment, particularly for secondary-school students, college seminars, teachers, and interested readers willing to work through sometimes lengthy proofs.
A major theme is that apparently simple mathematical questions can hide deep structures. The book begins with classical straightedge-and-compass constructions, explaining why Euclid's collapsing compass is just as powerful as a modern fixed compass and emphasizing the danger of trusting geometric diagrams without proof. It then examines the famous impossible classical problems of trisecting an arbitrary angle and squaring the circle, while also showing alternative instruments that can accomplish these constructions. Ramanujan's remarkably accurate geometric approximations to $\pi$ are another example of the sort of result Ben-Ari considers mathematically surprising.

The scope then expands beyond classical geometry. The reader encounters the five- and six-color theorems, graph theory, the art-gallery or museum-guarding problem, unusual applications of mathematical induction, Fibonacci and Fermat numbers, the Josephus problem, Po-Shen Loh's approach to quadratic equations, Ramsey theory, Pythagorean triples, SAT solving and Langford's problem. One particularly attractive example is the museum theorem: what initially looks like a geometry problem is solved elegantly by translating it into a graph-coloring problem.
Several chapters explore the unexpectedly rich mathematics of origami. Ben-Ari presents seven axioms of mathematical origami, Lill's method and the Beloch fold, and demonstrates that origami can perform constructions impossible with an ordinary straightedge and compass—including angle trisection, doubling the cube and constructing a regular nonagon. This is one of the subjects that most surprised the author himself; he explains that he had originally doubted whether origami contained serious mathematics at all.

Perhaps the book's most striking results concern the apparent limitations of classical geometric instruments. The Mohr–Mascheroni theorem says that every construction possible with straightedge and compass can actually be performed with a compass alone. Mohr discovered the result in 1672 and Mascheroni independently proved it in 1797. Even more surprisingly, the Poncelet–Steiner theorem shows that a straightedge alone is sufficient if just one fixed circle is already provided—the circle may have essentially arbitrary position and radius.
Another unexpected chapter asks whether two triangles having the same perimeter and the same area must be congruent. They need not be: for example, triangles with sides $(17,25,28)$ and $(20,21,29)$ both have perimeter $70$ and area $210$. The analysis eventually connects an elementary-looking geometry problem to elliptic curves, illustrating the book's recurring theme that simple questions can lead to unexpectedly sophisticated mathematics.
The culmination is Gauss's celebrated proof that a regular heptadecagon (17-gon) can be constructed with straightedge and compass. Gauss expresses the necessary quantities using arithmetic operations and repeated square roots, linking geometric constructibility to algebra and the roots of polynomials. The book supplements Gauss's algebraic argument with an explicit geometric construction and related constructions of the regular pentagon.
What makes the book interesting

The unifying idea is not a single branch of mathematics but the experience of mathematical surprise: familiar assumptions turn out to be false, apparently impossible constructions become possible when the rules change slightly, elementary questions connect to unexpected areas of mathematics, and difficult-looking theorems sometimes possess elegant proofs. The author deliberately selects results that are rarely encountered in normal curricula while remaining accessible through algebra, Euclidean and analytic geometry, and trigonometry.

Key takeaways

• Elementary mathematics can produce genuinely deep and surprising results without requiring advanced university machinery.
  
• Geometric construction problems reveal profound connections between geometry and algebra.
  
• Origami is mathematically more powerful than straightedge-and-compass geometry for certain constructions.
  
• The Mohr–Mascheroni, Poncelet–Steiner and Gauss heptadecagon results are among the book's standout surprises.
  
• The book is particularly suitable for mathematics teachers, strong secondary-school students, competition-oriented students and mathematically curious readers rather than someone looking for a standard course textbook.
  

BOOK DOWNLOAD]]> Book:Mathematical Surprises
Author: Mordechai Ben-Ari
Publication date: 2022
Publisher: Springer Nature Switzerland AG
Subject: Recreational/elementary mathematics, geometry, algebra, combinatorics
Level: Advanced secondary-school mathematics and above
Open Access: Yes — Creative Commons Attribution 4.0 (CC BY 4.0).

Summary

Mathematical Surprises is a collection of 16 largely independent chapters devoted to mathematical results that are unexpected, elegant, or surprisingly accessible. Ben-Ari deliberately chooses topics that normally do not appear in school or introductory university textbooks but whose proofs can still be understood with a strong secondary-school background. The book is not intended as a conventional textbook; rather, it is designed for mathematical enrichment, particularly for secondary-school students, college seminars, teachers, and interested readers willing to work through sometimes lengthy proofs.
A major theme is that apparently simple mathematical questions can hide deep structures. The book begins with classical straightedge-and-compass constructions, explaining why Euclid's collapsing compass is just as powerful as a modern fixed compass and emphasizing the danger of trusting geometric diagrams without proof. It then examines the famous impossible classical problems of trisecting an arbitrary angle and squaring the circle, while also showing alternative instruments that can accomplish these constructions. Ramanujan's remarkably accurate geometric approximations to $\pi$ are another example of the sort of result Ben-Ari considers mathematically surprising.

The scope then expands beyond classical geometry. The reader encounters the five- and six-color theorems, graph theory, the art-gallery or museum-guarding problem, unusual applications of mathematical induction, Fibonacci and Fermat numbers, the Josephus problem, Po-Shen Loh's approach to quadratic equations, Ramsey theory, Pythagorean triples, SAT solving and Langford's problem. One particularly attractive example is the museum theorem: what initially looks like a geometry problem is solved elegantly by translating it into a graph-coloring problem.
Several chapters explore the unexpectedly rich mathematics of origami. Ben-Ari presents seven axioms of mathematical origami, Lill's method and the Beloch fold, and demonstrates that origami can perform constructions impossible with an ordinary straightedge and compass—including angle trisection, doubling the cube and constructing a regular nonagon. This is one of the subjects that most surprised the author himself; he explains that he had originally doubted whether origami contained serious mathematics at all.

Perhaps the book's most striking results concern the apparent limitations of classical geometric instruments. The Mohr–Mascheroni theorem says that every construction possible with straightedge and compass can actually be performed with a compass alone. Mohr discovered the result in 1672 and Mascheroni independently proved it in 1797. Even more surprisingly, the Poncelet–Steiner theorem shows that a straightedge alone is sufficient if just one fixed circle is already provided—the circle may have essentially arbitrary position and radius.
Another unexpected chapter asks whether two triangles having the same perimeter and the same area must be congruent. They need not be: for example, triangles with sides $(17,25,28)$ and $(20,21,29)$ both have perimeter $70$ and area $210$. The analysis eventually connects an elementary-looking geometry problem to elliptic curves, illustrating the book's recurring theme that simple questions can lead to unexpectedly sophisticated mathematics.
The culmination is Gauss's celebrated proof that a regular heptadecagon (17-gon) can be constructed with straightedge and compass. Gauss expresses the necessary quantities using arithmetic operations and repeated square roots, linking geometric constructibility to algebra and the roots of polynomials. The book supplements Gauss's algebraic argument with an explicit geometric construction and related constructions of the regular pentagon.
What makes the book interesting

The unifying idea is not a single branch of mathematics but the experience of mathematical surprise: familiar assumptions turn out to be false, apparently impossible constructions become possible when the rules change slightly, elementary questions connect to unexpected areas of mathematics, and difficult-looking theorems sometimes possess elegant proofs. The author deliberately selects results that are rarely encountered in normal curricula while remaining accessible through algebra, Euclidean and analytic geometry, and trigonometry.

Key takeaways

• Elementary mathematics can produce genuinely deep and surprising results without requiring advanced university machinery.
  
• Geometric construction problems reveal profound connections between geometry and algebra.
  
• Origami is mathematically more powerful than straightedge-and-compass geometry for certain constructions.
  
• The Mohr–Mascheroni, Poncelet–Steiner and Gauss heptadecagon results are among the book's standout surprises.
  
• The book is particularly suitable for mathematics teachers, strong secondary-school students, competition-oriented students and mathematically curious readers rather than someone looking for a standard course textbook.
  

BOOK DOWNLOAD]]> https://mklab.gr/showthread.php?tid=843 Sat, 04 Jul 2026 16:14:12 +0300 mklabgr]]> https://mklab.gr/showthread.php?tid=843

Learning Mathematics in the 21st Century
EDITORS : Elena Arias Ortiz Julian Cristia Santiago Cueto

Summary

The report argues that learning mathematics in the 21st century should shift away from rote procedures toward deeper conceptual understanding, supported by modern technology and an awareness of how students actually think. It emphasizes that technology is not just an add-on but can reshape what counts as mathematical competence by enabling visualization, experimentation, feedback, and personalized learning paths. 

A key idea is that students develop mathematical understanding in diverse ways, so teaching should connect and build on their intuitive approaches rather than forcing a single rigid method. The document also stresses the importance of teacher development, showing how educators need support to move toward more interactive, discussion-based “math talk” classrooms where reasoning is made visible. Visual models and digital tools are highlighted as essential bridges between informal thinking and formal mathematics. 

Overall, the message is that improving mathematics education requires combining pedagogy, cognition, and technology into a more flexible and student-centered system aimed at genuine understanding rather than memorization.


BOOK]]>

Learning Mathematics in the 21st Century
EDITORS : Elena Arias Ortiz Julian Cristia Santiago Cueto

Summary

The report argues that learning mathematics in the 21st century should shift away from rote procedures toward deeper conceptual understanding, supported by modern technology and an awareness of how students actually think. It emphasizes that technology is not just an add-on but can reshape what counts as mathematical competence by enabling visualization, experimentation, feedback, and personalized learning paths. 

A key idea is that students develop mathematical understanding in diverse ways, so teaching should connect and build on their intuitive approaches rather than forcing a single rigid method. The document also stresses the importance of teacher development, showing how educators need support to move toward more interactive, discussion-based “math talk” classrooms where reasoning is made visible. Visual models and digital tools are highlighted as essential bridges between informal thinking and formal mathematics. 

Overall, the message is that improving mathematics education requires combining pedagogy, cognition, and technology into a more flexible and student-centered system aimed at genuine understanding rather than memorization.


BOOK]]> https://mklab.gr/showthread.php?tid=347 Sun, 14 Jun 2026 15:58:58 +0300 mklabgr]]> https://mklab.gr/showthread.php?tid=347 Algebra 
by Gelfand and Shen

Summary

Algebra by I.M. Gelfand and Alexander Shen is a uniquely engaging math textbook that focuses on understanding "why" rather than just memorizing formulas. Aimed at motivated middle and high school students, the book teaches through carefully curated problems, their solutions, and deep mathematical concepts, rather than traditional rote exercises.

BOOK PAGE]]> Algebra 
by Gelfand and Shen

Summary

Algebra by I.M. Gelfand and Alexander Shen is a uniquely engaging math textbook that focuses on understanding "why" rather than just memorizing formulas. Aimed at motivated middle and high school students, the book teaches through carefully curated problems, their solutions, and deep mathematical concepts, rather than traditional rote exercises.

BOOK PAGE]]> https://mklab.gr/showthread.php?tid=346 Sun, 14 Jun 2026 15:57:10 +0300 mklabgr]]> https://mklab.gr/showthread.php?tid=346 Trigonometry 
by Gelfand and Saul

Summary  Trigonometry by Israel M. Gelfand and Mark Saul is an unconventional, highly acclaimed textbook that teaches trigonometry as a unified, conceptually rich branch of mathematics rather than a collection of formulas to memorize

BOOK PAGE]]> Trigonometry 
by Gelfand and Saul

Summary  Trigonometry by Israel M. Gelfand and Mark Saul is an unconventional, highly acclaimed textbook that teaches trigonometry as a unified, conceptually rich branch of mathematics rather than a collection of formulas to memorize

BOOK PAGE]]> https://mklab.gr/showthread.php?tid=267 Wed, 10 Jun 2026 06:54:37 +0300 mklabgr]]> https://mklab.gr/showthread.php?tid=267 Uses of Technology in Upper Secondary Mathematics Education
by Stephen Hegedus

Summary This survey addresses the use of technology in upper secondary mathematics education from four points of view: theoretical analysis of epistemological and cognitive aspects of activity in new technology mediated learning environments, the changes brought by technology in the interactions between environment, students and teachers, the interrelations between mathematical activities and technology, skills and competencies that must be developed in teacher education. Research shows that the use of some technologies may deeply change the solving processes and contribute to impact the learning processes. The questions are which technologies to choose for which purposes, and how to integrate them, so as to maximize all students’ agency.  In particular the role of the teacher in classrooms and the content of teacher education programs are critical for taking full advantage of technology in teaching practice.

BOOK PAGE
]]> Uses of Technology in Upper Secondary Mathematics Education
by Stephen Hegedus

Summary This survey addresses the use of technology in upper secondary mathematics education from four points of view: theoretical analysis of epistemological and cognitive aspects of activity in new technology mediated learning environments, the changes brought by technology in the interactions between environment, students and teachers, the interrelations between mathematical activities and technology, skills and competencies that must be developed in teacher education. Research shows that the use of some technologies may deeply change the solving processes and contribute to impact the learning processes. The questions are which technologies to choose for which purposes, and how to integrate them, so as to maximize all students’ agency.  In particular the role of the teacher in classrooms and the content of teacher education programs are critical for taking full advantage of technology in teaching practice.

BOOK PAGE
]]> https://mklab.gr/showthread.php?tid=265 Wed, 10 Jun 2026 06:47:08 +0300 mklabgr]]> https://mklab.gr/showthread.php?tid=265 Summary 

Teaching Mathematics is nothing less than a mathematical manifesto. Arising in response to a limited National Curriculum, and engaged with secondary schooling for those aged 11 ̶ 14 (Key Stage 3) in particular, this handbook for teachers will help them broaden and enrich their students’ mathematical education. It avoids specifying how to teach, and focuses instead on the central principles and concepts that need to be borne in mind by all teachers and textbook authors—but which are little appreciated in the UK at present.

This study is aimed at anyone who would like to think more deeply about the discipline of ‘elementary mathematics’, in England and Wales and anywhere else. By analysing and supplementing the current curriculum, Teaching Mathematics provides food for thought for all those involved in school mathematics, whether as aspiring teachers or as experienced professionals. It challenges us all to reflect upon what it is that makes secondary school mathematics educationally, culturally, and socially important.

BOOK PAGE

]]> Summary 

Teaching Mathematics is nothing less than a mathematical manifesto. Arising in response to a limited National Curriculum, and engaged with secondary schooling for those aged 11 ̶ 14 (Key Stage 3) in particular, this handbook for teachers will help them broaden and enrich their students’ mathematical education. It avoids specifying how to teach, and focuses instead on the central principles and concepts that need to be borne in mind by all teachers and textbook authors—but which are little appreciated in the UK at present.

This study is aimed at anyone who would like to think more deeply about the discipline of ‘elementary mathematics’, in England and Wales and anywhere else. By analysing and supplementing the current curriculum, Teaching Mathematics provides food for thought for all those involved in school mathematics, whether as aspiring teachers or as experienced professionals. It challenges us all to reflect upon what it is that makes secondary school mathematics educationally, culturally, and socially important.

BOOK PAGE

]]> https://mklab.gr/showthread.php?tid=229 Wed, 10 Jun 2026 02:33:16 +0300 mklabgr]]> https://mklab.gr/showthread.php?tid=229 Elementary Algebra 

Authors : Ellis and Burzynski

Summary  Elementary Algebra is a work text that covers the traditional topics studied in a modern elementary algebra course. Use of this book will help the student develop the insight and intuition necessary to master algebraic techniques and manipulative skills.Elementary Algebra is a work text that covers the traditional topics studied in a modern elementary algebra course. It is intended for students who (1) have no exposure to elementary algebra, (2) have previously had an unpleasant experience with elementary algebra, or (3) need to review algebraic concepts and techniques.

BOOK PAGE

]]> Elementary Algebra 

Authors : Ellis and Burzynski

Summary  Elementary Algebra is a work text that covers the traditional topics studied in a modern elementary algebra course. Use of this book will help the student develop the insight and intuition necessary to master algebraic techniques and manipulative skills.Elementary Algebra is a work text that covers the traditional topics studied in a modern elementary algebra course. It is intended for students who (1) have no exposure to elementary algebra, (2) have previously had an unpleasant experience with elementary algebra, or (3) need to review algebraic concepts and techniques.

BOOK PAGE

]]> https://mklab.gr/showthread.php?tid=227 Wed, 10 Jun 2026 02:26:12 +0300 mklabgr]]> https://mklab.gr/showthread.php?tid=227 Real Numbers and Fascinating Fractions
by N. M. Beskin

Summary  The booklet is an elegant, highly accessible introduction to the theory and application of continued fractions, aimed at high school students, teachers, and anyone interested in mathematical olympiads or number theory. It bridges the gap between basic arithmetic and advanced approximation methods.

BOOK PAGE]]> Real Numbers and Fascinating Fractions
by N. M. Beskin

Summary  The booklet is an elegant, highly accessible introduction to the theory and application of continued fractions, aimed at high school students, teachers, and anyone interested in mathematical olympiads or number theory. It bridges the gap between basic arithmetic and advanced approximation methods.

BOOK PAGE]]> https://mklab.gr/showthread.php?tid=208 Wed, 10 Jun 2026 01:13:11 +0300 mklabgr]]> https://mklab.gr/showthread.php?tid=208 Just the Maths
by A. J. Hobson

Summary  Just the Maths is a collection of separate units, in chronological topic-order, intended to service foundation level and first year degree level courses in higher education, especially those delivered in a modular style. It concentrates on the core mathematical techniques required by any scientist or engineer.

BOOK PAGE]]> Just the Maths
by A. J. Hobson

Summary  Just the Maths is a collection of separate units, in chronological topic-order, intended to service foundation level and first year degree level courses in higher education, especially those delivered in a modular style. It concentrates on the core mathematical techniques required by any scientist or engineer.

BOOK PAGE]]> https://mklab.gr/showthread.php?tid=207 Wed, 10 Jun 2026 01:09:12 +0300 mklabgr]]> https://mklab.gr/showthread.php?tid=207 Algebraic Problems and Exercises for High School
by I. Goian

Summary  In this book, you will find algebra exercises and problems, grouped by chapters, intended for higher grades in high schools or middle schools of general education. Its purpose is to facilitate training in mathematics for students in all high school categories, but can be equally helpful in a standalone workout.

BOOK PAGE]]> Algebraic Problems and Exercises for High School
by I. Goian

Summary  In this book, you will find algebra exercises and problems, grouped by chapters, intended for higher grades in high schools or middle schools of general education. Its purpose is to facilitate training in mathematics for students in all high school categories, but can be equally helpful in a standalone workout.

BOOK PAGE]]> https://mklab.gr/showthread.php?tid=198 Wed, 10 Jun 2026 00:19:21 +0300 mklabgr]]> https://mklab.gr/showthread.php?tid=198 Beginning and Intermediate Algebra
by Tyler Wallace

Summary  Topics covered include: pre-algebra review, solving linear equations, graphing linear equations, inequalities, systems of linear equations, polynomials, factoring, rational expressions and equations, radicals, quadratics, and functions including exponential, logarithmic and trigonometric.

BOOK PAGE]]> Beginning and Intermediate Algebra
by Tyler Wallace

Summary  Topics covered include: pre-algebra review, solving linear equations, graphing linear equations, inequalities, systems of linear equations, polynomials, factoring, rational expressions and equations, radicals, quadratics, and functions including exponential, logarithmic and trigonometric.

BOOK PAGE]]>