Mathematics is the universal language

Mathematics is...

a universal language that describes the natural world, allows communication

between languages and cultures, and teaches the ability to think sequentially

ASSESMENT:

Assembling evidence from a variety of assessment strategies

is a routine part of the mathematics classroom

and assists in forming an accurate picture of

an individual student’s progress toward

learning goals. Data gathered from

assessments is used to inform

instruction.

  Communication

Mathematics instruction pro-vides opportunities for students

to effectively communicate their mathematical thinking, orally and

in written form, to peers, teachers, and others by using the precise

language of mathematics. Through communication, students evaluate their

own mathematical thinking as well as analyze the strategies of others

Conceptual Understanding

Teachers provide sufficient time and experiences which enable

students to actively build new knowledge from prior knowledge

so that understanding deepens and the ability to apply math-

ematics in new situations expands. Conceptual understanding

supports retention and prevents common errors. 

Connections

Mathematics represents a network of interconnected concepts

and procedures. Connections are made within the same mathematical structure,

 between mathematical strands, and to other  disciplines and daily living.

Cooperative Learning

Cooperative learning offers students opportunities to explore and discuss

challenging problems which would normally be beyond the capacity of an individual.

This creates opportunities for students to discover and test conjectures based

upon mathematical principles. Through  working together, students increase

self-confidence, deepen mathematical understanding, and utilize social skills.

Multiple Representations

Students should create, select, and apply various representations to organize,

record, communicate, and prove mathematical ideas. These representations may include

diagrams, tables, graphs, symbolic expressions, and physical models. Representations should

 support students’ understanding of mathematical concepts and relationships.

Experiences with concrete models followed by pictorial representations assist in developing abstract thinking.

Problem-Solving

Knowledge is built through the solving of problems that arise

in mathematics and in real world contexts. Students use

and adapt a variety of strategies that enable them to monitor

and reflect on mathematical processes. The teacher’s role

is crucial in choosing appropriate problems that encourage

students to explore, take risks, share failures and successes,

and question one another.

Technology

Technology plays an increasingly important role, helping the

current generation of students visualize and learn mathemat-

ics. While technology should not be used as a replacement

for basic understanding and computational proficiency, mul-

tiple forms of technology used properly in the mathematics

classroom can deepen students’ understanding of mathematical concepts

 When technology is placed in the hands of students, attitudes toward

mathematics are improved, allowing focus on decision making

 reflection, reasoning and problem-solving, while also preparing students for 21st

century life and careers.

Parent Involvement

Parents play a critical role in a child’s math achievement.

Parents need to support students in developing an attitude

that effort, more than natural talent, leads to increased stu-

dent achievement. Changing children’s beliefs from a focus

on ability to a focus on effort increases their engagement in

mathematics learning.

Prime numbers

•       Remember how the early scientists visited Africa and some other parts of the world and ‘discovered’ naturally existing things like mountains, rivers and other places, something close to that has happened again.

•      This time round, though, it is not in either of the above areas.

•      Two mathematicians have uncovered a simple, yet previously unnoticed quality of prime numbers. So, apparently, the prime number sequence isn’t as random as earlier thought.

• Prime numbers as you will hopefully recall from your early formation years in primary school, are whole numbers that are only divisible by 1 and themselves. They include 2,3,5,7,11and so on.

•       In a paper by researchers studying the primes at Stanford University, the prime numbers, it seems, are trying very hard not to be similar.

•       Kannan Soundararajan and Robert Lemke Oliver, present new evidence that prime numbers ward off other would-be primes that end in the same digits.

via GIPHY

•      From the initial set (numbers less than ten) 2 and 5 were part of the primes, but when they appear next as in 12, 15, 22, 25, they are no longer primes as they are divisible by other numbers other than 1, and themselves. Thus, all other prime numbers can only end in one of four digits: 1, 3, 7, or 9.

•       Armed with the fact that prime numbers appear to be random, or so it was thought, the pair of mathematicians decided to test this ‘randomness’ assumption on primes. They discovered a strange phenomenon entrenched among the integers that prove there’s certainly an un-random pattern at play.

•             If they were truly random as earlier thought, primes ending in 1 would be followed by a prime ending in 1 about 25% of the time. However, the mathematicians found this order occurred in only 18.5 percent of cases.

• It was discovered that a prime number was followed by a prime ending in 3 or 7 was about 30 percent of the time and about 22 percent for 9.

•            Although their study is yet to checked and approved by experts before it can be officially published in a peer-reviewed journal, the study has received a lot of attention from other mathematicians.

•           As fascinating as the new study appears, George Dvorsky in an article argues that it likely won’t help with other prime-related challenges including the twin-prime conjecture or the Riemann hypothesis. He adds that the new discovery may not have any practical implications or use to math and number theory.

 

Homotopy type theory

       

Homotopy Type Theory: Univalent Foundations of Mathematics.

•      Homotopy type theory is a new branch of mathematics that combines aspects of several different fields in a surprising way. It is based on a recently discovered connection between homotopy theory and type theory. It touches on topics as seemingly distant as the homotopy groups of spheres, the algorithms for type checking, and the definition of weak ∞-groupoids.

•        Homotopy type theory brings new ideas into the very foundation of mathematics. On the one hand, there is Voevodsky’s subtle and beautiful univalence axiom. The univalence axiom implies, in particular, that isomorphic structures can be identified, a principle that mathematicians have been happily using on workdays, despite its incompatibility with the “official” doctrines of conventional foundations. On the other hand, we have higher inductive types, which provide direct, logical descriptions of some of the basic spaces and constructions of homotopy theory: spheres, cylinders, truncations, localizations, etc. Both ideas are impossible to capture directly in classical set-theoretic foundations, but when combined in homotopy type theory, they permit an entirely new kind of “logic of homotopy types”.

•       This suggests a new conception of foundations of mathematics, with intrinsic homotopical content, an “invariant” conception of the objects of mathematics — and convenient machine implementations, which can serve as a practical aid to the working mathematician. This is the Univalent Foundations program.

• The present book is intended as a first systematic exposition of the basics of univalent foundations, and a collection of examples of this new style of reasoning — but without requiring the reader to know or learn any formal logic, or to use any computer proof assistant. We believe that univalent foundations will eventually become a viable alternative to set theory as the “implicit foundation” for the unformalized mathematics done by most mathematicians.

•      In brief, this is another step in the quest to unify all mathematics on consistent foundations. You could say that that project began with the axiomatisation of geometry leading to Euclid’s Elements, and really got going at the start of the 20th century with Russell and Whitehead’s Principia Mathematica, which tried to state everything in terms of sets. Set theory had a huge paradox in the middle of it, so mathematicians have spent the time since Principia trying to come up with something that puts it all on a firm footing. Bourbaki restates all of maths in terms of Zermelo-Fraenkel set theory with the axiom of choice, but some people have serious objections to that too. Meanwhile, category theory and model theory have been providing as much abstraction as anyone can usefully apply, and the advent of computer proof has made constructivism, where you can only accept something exists once you’ve explicitly constructed it, much more popular.

•       Homotopy type theory unifies type theory, arising from logic and computer science, with homotopy theory, which talks about topological spaces. In classical set theory, the base objects are sets, and propositions about those sets exist elsewhere. In homotopy type theory, objects and propositions about objects are instances of the same kind of thing – types – and you can think of types as being spaces of points, which is where the homotopy theory comes in. The philosophy of the book is generously constructivist – they avoid non-constructive concepts like the axiom of choice or law of the excluded middle right up until they need them for things like defining the real numbers.

•      Like Bourbaki, the presentation of these ideas is just as important as the ideas themselves, and the book delivers on that too. The authors have learnt the lesson of Russell and Whitehead, and use notation deftly to avoid losing the reader in a morass of symbols. The introductory sections are short and to the point, giving definitions and justification concisely and clearly. The language is approachable and not at all high-falutin’.

latest dicoveries in the field

What are the latest discoveries in the field of mathematics?

     People make new mathematical “discoveries” everyday. Well, I suppose it really depends on what you mean by the term “discoveries.” If proving results and building theories counts, then we are on the same page.

Below are some notable, recent mathematical discoveries, which constitute a far-from-exhaustive list, rife with personal bias. That is the nature of such lists. Mathematics is vast, and most discoveries come from niche fields with very little exposure to even the general mathematical public. Nonetheless, some of the most famous discoveries are included.

The abc Conjecture (Mochizuki)

This conjecture, despite its simple statement, is incredibly important in number theory. The proof itself, as usual, is the real story. Not only is Mochizuki's proof is truly massive, fleshing out a mathematical theory its author has called "Inter-Universal Teichmüller Theory" (IUTT or IUT or IU-Tech or, using Fesenko’s terminology, “arithmetic deformation theory”). Unfortunately, Mochizuki’s proof has yet to be confirmed, so it could be flawed. Few, if any, fully understand it.

Here is a one particular statement of the conjecture: For each ϵ>0

, there are only finitely many triples (a,b,c) of coprime positive integers for which a+b=c

that satisfy the equation

c>d1+ϵ,

where d

is the product of the distinct prime factors of the product abc

.

The Twin Prime Conjecture (Zhang, Tao, et al.)

This conjecture is somewhat simpler to state but no less remarkable or important for number theory. Terry Tao, the mathematician heading some of the recent development on this, is also better known than Mochizuki, who was unheard of by most until a few years ago due to the obscurity of his preferred field of anabelian geometry, among other reasons. Tao is a Fields medalist who is commonly thought to be one of the greatest mathematicians of this century.

Anyway, about the conjecture and the progress. Here is the statement of the conjecture itself: There are infinitely many twin primes (i.e., numbers with no factors other than one and themselves in pairs where they are separated by only two such as 3 and 5).

The progress on this is thanks to a previously unknown mathematician named Yitang Zhang, who proved that there were infinitely many primes with prime-gap approximately 70 million (the first finite gap ever to be established). This was a big deal. More recently, the hugely collaborative Polymath Project, which is partly maintained on Tao's blog, has substantially tightened the bound. Thus far it has been found that there are infinitely many primes of separation of 6, assuming some other conditions.

The Navier-Stokes Problem (Tao – Again)

There's a reason I said Tao might be one of the greatest mathematicians around. Here we see he is involved in another substantial development.

This problem belongs to the field of mathematical physics and has to do with fluid mechanics. The Navier-Stokes equation is an extraordinary complex differential equation for which the smoothness and existence problem for the three-dimensional case is unsolved. This is also a Millennium Prize Problem, meaning that the Clay Mathematics Institute offers one million dollars to whoever solves it completely in the form they present it.

The partial result that Tao published handles an averaged version of the problem. This not only gives some insight into the full problem by solving a simpler version, but also provides a methodology of proof that could assist in solving the full problem.

The Cobordism Hypothesis (Introduced: Baez & Dolan; Proved: Lurie)

This is an older result than the other three, and it is much more difficult to describe even vaguely, but it is no less important.

The result has to do with an insightful classification of topological quantum field theories (yes, physics again) using, among other things, the abstract nonsense known as category theory. The proof comes from a very well-known Harvard mathematician and MacArthur fellow.