You do math in your head most of the time, but you can also teach your body how to do it. Researchers investigated how our brain processes and understands numbers and number size. They show that movements and sensory perception help us understand numbers.
Math

Most Topular Stories

Doing math with your body
Mathematics News  ScienceDaily2 Oct 2014  7:09 am 
Adding uncertainty to improve mathematical models
Mathematics News  ScienceDaily29 Sep 2014  3:05 pmMathematicians have introduced a new element of uncertainty into an equation used to describe the behavior of fluid flows. While being as certain as possible is generally the stock and trade of mathematics, the researchers hope this new formulation might ultimately lead to mathematical models that better reflect the inherent uncertainties of the natural world. 
At the interface of math and science: Using mathematics to advance problems in the sciences
Mathematics News  ScienceDaily29 Sep 2014  3:03 pmIn popular culture, mathematics is often deemed inaccessible or esoteric. Yet in the modern world, it plays an ever more important role in our daily lives and a decisive role in the discovery and development of new ideas  often behind the scenes. In new research, scientists have developed new mathematical approaches to gain insights into how proteins move around within lipid bilayer membranes. 
Garrett Lisi Explains His Grand Unified Theory
Scientific American  Math15 Oct 2014  8:00 amDeep down, the particles and forces of the universe are a manifestation of exquisite geometry  Read more on ScientificAmerican.com 
New theorem determines age distribution of populations from fruit flies to humans
Mathematics News  ScienceDaily6 Oct 2014  8:41 amThe initial motivation of a new study was to estimate the age structure of a fruit fly population, the result a fundamental theorem that can help determine the age distribution of essentially any group. This emerging theorem on stationary populations shows that you can determine the age distribution of a population by looking at how long they still have to live.

Search for "math OR mathematics"

Mount Washington College Names Dr. Donald G. Knezek as Board of Trustees Member
21 Oct 2014  8:51 amMount Washington College today announced that Dr. Donald G. Knezek, former Chief Executive Officer of International Society for Technology in Education , has joined the College's Board of Trustees. "Don brings more than 40 years of experience and leadership in education technology," said Francis Mulgrew, President of Mount Washington College. 
Street Spirits force has trustee hopes
21 Oct 2014  8:50 amAndrew Burton, the founder and artistic director of the Street Spirits Theatre Company, wants to bring his experience working with youth to the School District 57 board of trustees. Burton, who holds a master's degree in education, was a nominee for Prince George's citizen of the year this year. 
Getting into elite colleges; Cylvia Hayes is no Eleanor Roosevelt: Opinion roundup
21 Oct 2014  8:45 amWant to increase your chances of getting into the college of your dreams ? Become an elite athlete, writes guest columnist Ed Harris in The Seattle Times. Harris, who founded a college counseling company, says "you don't need a math degree from MIT to realize an allstate linebacker with a 700 SAT in math might edge out a less athletically gifted candidate with a higher score." 
Common Core test: Can Smarter Balanced scores be compared to state test results? Surprising answer is yes
21 Oct 2014  8:44 amAt Alberta Rider Elementary School in Tigard, Evan Gerhard works on a reading lesson geared to the rigorous Common Core State Standards. Oregon students will be tested using the Common Corealigned Smarter Balanced tests for the first time this spring. 
Chevron's $20M to improve Pennsylvania workforce
21 Oct 2014  8:43 amChevron Corp. is spending $20 million to launch its Appalachia Partnership Initiative, which is meant to improve schools and workforce development in 27 counties in southwestern Pennsylvania, northern West Virginia and eastern Ohio. Chevron is working with the Allegheny Conference on Community Development, the Claude Worthington Benedum Foundation and RAND Corporation to improve Science, Technology, Engineering and Math education and other measures to produce skilled workers for the energy and manufacturing industries spurred by Marcellus Shale drilling.

Mathematics News  ScienceDaily

New theorem determines age distribution of populations from fruit flies to humans
6 Oct 2014  8:41 amThe initial motivation of a new study was to estimate the age structure of a fruit fly population, the result a fundamental theorem that can help determine the age distribution of essentially any group. This emerging theorem on stationary populations shows that you can determine the age distribution of a population by looking at how long they still have to live. 
Doing math with your body
2 Oct 2014  7:09 amYou do math in your head most of the time, but you can also teach your body how to do it. Researchers investigated how our brain processes and understands numbers and number size. They show that movements and sensory perception help us understand numbers. 
Adding uncertainty to improve mathematical models
29 Sep 2014  3:05 pmMathematicians have introduced a new element of uncertainty into an equation used to describe the behavior of fluid flows. While being as certain as possible is generally the stock and trade of mathematics, the researchers hope this new formulation might ultimately lead to mathematical models that better reflect the inherent uncertainties of the natural world. 
At the interface of math and science: Using mathematics to advance problems in the sciences
29 Sep 2014  3:03 pmIn popular culture, mathematics is often deemed inaccessible or esoteric. Yet in the modern world, it plays an ever more important role in our daily lives and a decisive role in the discovery and development of new ideas  often behind the scenes. In new research, scientists have developed new mathematical approaches to gain insights into how proteins move around within lipid bilayer membranes. 
Taking advantage of graphene defects: Security screening
24 Sep 2014  5:51 amScientists have discovered a potential application for graphene in security screening. A new theoretical model estimates electric current rectification in graphene. Electronic transport in graphene contributes to its characteristics. Now, a Russian scientist proposes a new theoretical approach to describe graphene with defectsin the form of artificial triangular holesresulting in the rectification of the electric current within the material. Specifically, the study provides an analytical and numerical theory of the socalled ratchet effect.

Scientific American  Math

Garrett Lisi Explains His Grand Unified Theory
15 Oct 2014  8:00 amDeep down, the particles and forces of the universe are a manifestation of exquisite geometry  Read more on ScientificAmerican.com 
Beyond Emmy and Sophie: Resources for Learning about Women in Math
14 Oct 2014  12:00 pmToday is Ada Lovelace Day, an international celebration of women in science, technology, engineering, and math. If you’d like to read about women in math for the occasion, you're in serious...  Read more on ScientificAmerican.com 
Atomic Reporters Curbs “Egregious” Coverage of Nuclear Perils
13 Oct 2014  4:32 amYes, the Cold War ended long ago, but we still live in a nucleararmed world, in which the possibility of nuclear war, terrorism and accidents is all too real.  Read more on ScientificAmerican.com 
How Well Do You Know Martin Gardner? [Quiz]
10 Oct 2014  11:00 amDo you think of yourself as a true fan of Martin Gardner’s? Test your knowledge of him here  Read more on ScientificAmerican.com 
Make Your Own Hexaflexagons…and Snap Pictures of Them
10 Oct 2014  6:00 amUse the templates here or design your own flexagons. Share the results with us  Read more on ScientificAmerican.com

NYT > Mathematics

Preparing Your Child for Common Core Tests
6 Oct 2014  9:00 pmThe when and what kind of tests depend on the state, but make sure your children take the tutorial about using the computer. 
The Math of Whips, Chains and Ropes
5 Oct 2014  9:00 pmSome people in applied mathematics are looking at the physics behind socalled viscous threads and elastic rods. 
The Odds, Continually Updated
29 Sep 2014  9:00 pmIn Bayesian statistics, new data is used to shape assumptions, the opposite of the frequentist (classical) approach. 
MacArthur Awards Go to 21 Diverse Fellows
16 Sep 2014  9:00 pmStipends will be awarded to a psychologist studying racial perceptions in crimes, the maker of the film “The Act of Killing” and a law professor seeking to help American Indian women, among others. 
Incentives Aid A.P. Science and Math Scores
1 Sep 2014  9:00 pmA Pittsburgh high school took part in an initiative to raise its performance in math and science, but did its results match the high expectations?

Ars Mathematica

Arguesian Lattices
23 Sep 2014  12:46 pmAs is wellknown, the lattice of submodules of a module is modular. What I did not know is that the converse is not true, and that lattices of submodules must satisfy a stronger property, the arguesian law. The Arguesian law is a latticetheoretic analogue of Desargues’ theorem in projective geometry. I read the statement of the theorem several times and I have no intuition about what it means. There is a kind of converse to this result: a complemented lattice can be embedded into the lattice of submodules of a module if and only if it is arguesian. (I found the result in… 
K2, not the mountain
20 Mar 2014  2:18 pmChandan Singh Dalawat has a nice survey article about K2. It just gives the highlights of the theory, without proofs, so it’s closer to a teaser trailer than it is to fulllength movie. But sometimes you just want a teaser trailer to tell you if you want to invest the time in the movie. 
Cayley Bacharach Theorem through History
10 Feb 2014  3:04 pmI came across this terrific article that describes a sequence of results beginning with Pappas’ theorem through the CayleyBacharach theorem to modern formulations in terms of the Gorenstein (!) condition. The connection between classical topics in algebraic geometry and modern techniques is fascinating. 
Nonassociative Algebras
30 Dec 2013  2:00 pmI periodically feel like I should learn more about nonassociative algebra. (I’ve studied Lie algebras, and technically Lie algebras are nonassociative, but they’re pretty atypical of nonassociative algebras.) There’s a mysterious circle of “exceptional” examples that are all related — the octonions, the five exceptional Lie algebras, the exceptional Jordan algebra — that I would like to understand better. John Baez has an article about the direct connection that I post about before, but what I don’t understand about the general theory is how… 
Determinacy
30 Nov 2013  1:27 pmOne of my ambitions in life is to understand projective determinacy. Fortunately, Tim Gowers has written a series of posts to explain Martin’s proof that Borel sets are determined. The main source of interest in determinacy is that results suggest that it is the strongest regularity property that a set can have, in that it it tends to imply other nice properties such as Lebesgue measurability. Here is a short proof by Martin that determinacy implies Lebesgue measurability. Justin Palumbo has a nice set of lecture notes that relate determinacy to other regularity properties. (One nuance…

Loren on the Art of MATLAB

Taking the Pulse of MOOCs
16 Oct 2014  5:45 amCoursera is a technology platform that kickstarted the current MOOCs boom. Even though there are more MOOCs players now, it still remains one of the leading companies in this space. But how are they doing these days for delivering higher education to the masses online?Today's guest blogger, Toshi Takeuchi, would like to share an analysis using Courera's data.I am a big fan of MOOCs and I benefited a lot from free online courses on Coursera, such as Stanford's Machine Learning course. Like many websites these days, Coursera offers its data through REST APIs. Coursera offers a number of APIs,… 
MATLAB R2014b Graphics – Part 2: Using Graphics Objects
14 Oct 2014  7:55 amToday, David Garrison, our guest blogger, will continue his series on the new graphics system in R2014b.Part 1: Features of the New Graphics SystemPart 2: Using Graphics ObjectsPart 3: Compatibility Considerations in the New Graphics SystemHere is Part 2 of the series.ContentsWhat have we learned so far?The MATLAB Graphics SystemPreR2014b Numeric HandlesR2014b Graphics ObjectsGetting and Setting Object PropertiesUsing set and getHave you starting using graphics objects in R2014b?Next up  Part 3: Compatibility Considerations in the New Graphics SystemWhat have we learned so far?In Part 1 of… 
Reversal of Sorts – New in Release R2014b
9 Oct 2014  5:36 amI wanted to show you a glimpse of some of the new math functionality available in R2014b.ContentsThe QuestionMy Original AnswerSolution with R2014bWhat New Math Have You Enjoyed in R2014b?The QuestionRecently on the MATLAB newsgroup, Christoph asked this question:I have a vector A shown below, which has 6 elements. the elements are already sorted in descending order. now i want to create vector C by deleting elements from A, starting with element a1, until the sum of the vector equals or is smaller the value BA= 26 23 20 19 15 14 B=70So, the output should beC= 20 19 15 14Any idea how to do… 
MATLAB R2014b Graphics – Part 1: Features of the New Graphics System
3 Oct 2014  1:00 amToday I’d like to introduce a guest blogger, David Garrison, who is a MATLAB Product Manager here at MathWorks. This is the first in a series of blogs over the next few weeks describing the new graphics system in R2014b and how some of the changes will affect you.Part 1: Features of the New Graphics SystemPart 2: Using Graphics ObjectsPart 3: Compatibility Considerations in the New Graphics SystemHere is Part 1 of the series.ContentsBig Changes in R2014bThe New MATLAB Graphics SystemThe New Look of MATLAB GraphicsRotatable Tick LabelsAutomated Updating of datetime Tick LabelsAnimated… 
Symbolic Math Solves a Linear Algebra Conundrum
24 Sep 2014  5:49 amI'd like to introduce this week's guest blogger Alan Weiss. Alan writes documentation for mathematical toolboxes here at MathWorks.An economist I know (well, he's my son) asked me a question the other day. I was able to answer him quickly by using Symbolic Math Toolbox™ calculations. You might find the steps to be an interesting case of a somewhat nonstandard use of symbolic mathematicsContentsA Linear Algebra QuestionA Symbolic ApproachFeedbackA Linear Algebra QuestionMy son was working through a paper that described a computational algorithm. He got stuck on one step of the…

Homeschool Math Blog

Math Mammoth South African version, grades 4 and 5
15 Oct 2014  7:05 amMath Mammoth South African version is now available for grades 4 and 5! Math Mammoth South African version, Grade 4Math Mammoth South African version, Grade 5You can read detailed descriptions and download free samples at the above links. The South African version has been customized to South Africa in the following manners:The names used are South African names (instead of Jack and Jill, there are Ansie and Mampho).The currency used in word problems is rand. The money chapter teaches both rand and cents, of course.The material is "all metric". In other words, the US customary measuring… 
Math Mammoth giveaway!
14 Oct 2014  7:33 amIt is time for a giveaway of my Math Mammoth products! There will be 17 prizes! The "grand prize":ONE (1) winner will get Math Mammoth All Inclusive bundle  either a download or a CDTWO (2) winners will get Math Mammoth Light Blue Series bundle  either a download or a CDFOUR (4) winners will get Math Mammoth Blue Series bundle  either a download or a CDTEN (10) winners will get one grade level of Math Mammoth Light Blue series  download  the grade level is chosen by each winner.The winners will be chosen by a random number generator. Something SPECIAL This giveaway is COUPLED with… 
Lesson on Fibonacci numbers
12 Oct 2014  7:09 pmHave you ever heard about the Fibonacci numbers and the Golden ratio?A tiling that uses squares whose side lengths are successive Fibonacci numbers. Ask students to continue it! Image from Wikipedia. Here's a lesson I just wrote about them:Fibonacci numbers and the golden section – lesson for middle and high school studentsIf you like it, share it!You might ask, "Should our children or students even learn about Fibonacci numbers or the golden ratio?"True, they aren't any standard fare in math books. However, I feel that yes, students should know about them. I think it's important that our… 
Worksheets in PDF form
11 Oct 2014  7:56 pmYou can now make worksheets for these topics in both html and PDF formats at my site: Basic division facts (grades 34)Long division (grades 46) Division with remainders (grades 35)Equivalent fractionsClockMeasurement units, grade 3Measurement units, grade 4Measurement units, grade 5Measurement units, grade 6The pages also include lots of readymade PDF worksheets. Enjoy! 
Timed tests
22 Sep 2014  4:15 pmI wrote a new piece outlining some of the dangers of using timed tests in math, such as how they promote math anxiety and cause children to be afraid of making mistakes in math classes.Should you use timed tests for math facts?Go check it out!

Let's Play Math!

Horseshoes: A Place Value Game
20 Oct 2014  7:00 am[Feature photo above by Johnmack161 via Wikimedia Commons (CC BY 2.5).] I first saw place value games on the old PBS Square One TV show (video below). Many teachers have posted versions of the game online, but Snugglenumber by Anna Weltman is by far the cutest variation. Anna kindly gave me permission to use the game in my upcoming Math You Can Play book series, and I added the following variation: Horseshoes Math Concepts: place value, strategic thinking. Players: two or more. Equipment: one deck of playing cards, or a double deck for more than three players. Separate out the cards numbered… 
Math Teachers and Homeschool Bloggers: We Want You!
15 Oct 2014  2:47 pm[Photo by Olga Berrios via flickr.] Do you have a favorite blog post about math activities, games, lessons, or handson fun? The Math Teachers at Play (MTaP) math education blog carnival would love to feature your article! We welcome math topics from preschool through the first year of calculus. Old posts are welcome, as long as they haven’t been published in past editions of this carnival. Click here to submit your blog post. [Note: Sometimes the automated Google form refuses to load in my browser. If you have trouble, leave a link to your entry in the comments below.] Browse all the past… 
Math Storytelling Day: The Hospital Floor
24 Sep 2014  6:14 am[Feature photo above by Christiaan Triebert via flickr (CC BY 2.0).] Have you ever heard of Math Storytelling Day? On September 25, people around the world celebrate mathematics by telling stories together. The stories can be real — like my story below — or fictional like the tale of Wizard Mathys from Fantasia and his crystal ball communication system. Check out these posts for more information: Happy Math Storytelling Day Math Storytelling Day resources Moebius Noodles: Math Storytelling Day archive My Math Story My story begins with an unexpected adventure in pain. Appendicitis… 
Math Teachers at Play #78 via 1001 Math Problems
23 Sep 2014  12:35 pmMath Teachers at Play is a traveling collection of math tidbits — games, lesson ideas, and more — from around the Internet. It moves around from month to month, and the September edition is now posted at 1001 Math Problems blog. What a fun list of math posts to browse! Special Blog Carnival Edition of 1001 Math Problems Welcome to the 78th edition of the Math Teachers At Play math education blog carnival, which I am thrilled to be hosting this month in celebration of my soontobereleased book, Camp Logic. What is the blog carnival? It is a monthly snapshot of some interesting… 
Calling All Math Teacher Bloggers and Homeschoolers: Carnival Time!
15 Sep 2014  3:01 pmby Bob Jagendorf via flickr The monthly Math Teachers at Play (MTaP) math education blog carnival is almost here. If you’ve written a blog post about math, we’d love to have you join us! Each of us can help others learn, so in a sense we are all teachers. Posts must be relevant to students or teachers of schoollevel mathematics (that is, anything from preschool up to firstyear calculus). Old posts are welcome, as long as they haven’t been published in past editions of this carnival. Click here to submit your blog post. Browse all the past editions of the Math Teachers at Play…

Basic mathematics blog

Online Matrix Calculator
16 Oct 2014  4:31 amThis online matrix calculator will do addition, subtraction, multiplication, determinant, and inverse of matrices 
Introduction to Matrices
15 Oct 2014  8:06 pmEasy to follow introduction to matrices learn how to add, subtract and multiply matrices. Learn how to find the determinant and the inverse of matrices 
Union of Sets Calculator
11 Oct 2014  4:14 pmunion of sets calculator: Easily find the union of sets 
How to calculate the weighted mean
10 Oct 2014  10:22 amThis lesson will show you how to calculate the weighted mean using a simple formula 
Standard Deviation Problems
10 Oct 2014  7:44 amThese standard deviation problems will help you understand better the meaning of standard deviation and how it is related to the mean

ChapterZero

Mirror descent is, in a precise sense, a second order algorithm
1 Oct 2014  10:47 pmFor one of our projects at eBay, I’ve been attempting to do a Poisson MLE fit on a large enough dataset that Fisher scoring is not feasible. The problem is that the data also has such large variance in the scales of the observation that stochastic gradient descent does not work, period — because of the exponentiation involved, you need to take a very tiny step size to avoid overflow errors, but this step size is shared by all the parameters, so you can’t make progress in this way. An alternative is adagrad, which maintains separate stepsizes for each parameter, but that… 
Algebra: it matters
15 Sep 2014  4:42 pmI’m looking at two different models for learning polynomial functions, and trying to determine if they are equivalent. After a couple days of thinking, I’ve reduced the question to the following: Can every symmetric polynomial of degree \(r\) in \(d\) variables that has no constant term be written as a sum of the \(r\)th powers of linear polynomials in \(d\) degrees and a homogeneous polynomial of degree \(r\) each of whose monomials involves at most \(d1\) variables? 
Julia, once more
9 Aug 2014  12:43 pmJulia + PyCall + CCall + Gadfly or PyPlot (+ Julia Studio ?) looks delicious. The only feature that absolutely needs to be added is shared memory parallelism (why wasn’t this an initial core feature of the language?), but I’m extremely excited by the current awesomeness of the Julia ecosystem. I recommend you get into it now, if you’re a scientific computation person. Update: Julia has experimental support for sharedmemory arrays on Unix, which is really all that I need at this point. Great! 
a bit on word embeddings
25 Jul 2014  9:31 pmLately I’ve been working almost exclusively on continuous word representations, with the goal of finding vectorial representations of words which expose semantic and/or syntactic relationships between words. As is typical for any interesting machine learning problem, there are a glut of clever models based on various assumptions (sparsity, hierarchical sparsity, lowrankedness, etc.) that yield respectable embeddings. Arguably, however, the most well known of these representations are the word2vec models due to Mikolov et al., which are part of a larger class of neural networkbased… 
Installing Hadoop on Ubuntu (works for Ubuntu 12.04 and Hadoop 2.4.1)
18 Jul 2014  4:11 pmI’m trying to use LDA on a large amount of data. A quick recap: Tried vowpal wabbit … it’s fast, I’ll give it that, but it’s also useless: the output is dubious (what I think are the topics look like they haven’t changed very much from the prior) *and* I have no idea how it maps onto topics and documents (the documentation is AWFUL, and the dimensions of the output files are WONKY). Tried two implementations of SCVB0, a stochastic collapsed variational bayes LDA algorithm: one doesn’t work at all (as in, it stalls on any amount of data — so…

Computational Complexity

Martin Gardner Centennial
21 Oct 2014  8:09 amMartin Gardner was born on October 21, 1914, so today is his Centennial (he died on May 22, 2010, at the age of 95). We've mentioned him in the blog before: The Life of Martin Gardner Contribute to the Gardner Centennial Another Post on Martin Gardner I used the anagram Tim Andrer Gran in both my review of the LiptonRegan book (see here) and my Applications of Ramsey Theory to History paper (see here) So what can I add on his centennial? He was not the first person to write on recreational mathematics, but he was certainly early and did it for a long time. I suspect he… 
The Curious Case of NP and NEXP
16 Oct 2014  9:34 amNP (nondeterministic polynomial time) and NEXP (nondeterministic exponential time) are provably different classes by the nondeterministic time hierarchy. No surprise, given exponentially more time we expect to solve more problems. But the proof requires collapses at many input lengths and odd things happen when we look at the infinitelyoften question. We say a language L is in i.o.C for a complexity class C if there is an A in C such that for infinitely many n, A and L agree on strings of length n (for all x of length n, x is in A if and only if x is in L). Straightforward diagonalization… 
Luddite or not?
13 Oct 2014  9:33 amMy first ever guest post for Lance was on Are you a luddite. I certainly am to some extent a luddite, but there are some things where it not clear if they are ludditeish or not. I prefer reading books to blogs. This came up when I reviewed both Lipton and LiptonRegan blogbooks, and I am now reading some of Terry Tao's Blog book. l look forward to reading Scott's Blog book. At first I thought that preferring books was ludditeish. But some high tech people and some young people who I've asked AGREE with me. Why is this? when reading a blog (or doing anything on line) its so easy… 
2014 Fall Jobs Post
9 Oct 2014  5:45 amTis the season for the fall jobs post. Please list any jobs, academic or industrial, in theoretical computer science broadly construed in the comments to this post. If you are a job seeker check this page often as new jobs get added over time. As always the best places to look for academic CS positions are the job sites at the CRA and the ACM. Also check out postdoc and other opportunities on Theory Announcements. It never hurts to check out the webpages of departments or to contact people to see if positions are available. I expect the computer science market to be quite… 
The Complexity of NIM. Open?
6 Oct 2014  11:21 amRecall 1pile NIM: Let A be a finite set of Naturals. NIM(A) is the following game: There are n stones on the board. Players I and II alternate removing a\in A stones. The first player who can't win loses. Note that if 1\in A then `can't move' means that the other player took the last stone. If (say) 2 is the min elt of A then its possible there is 1 stone on the board and a player can't move. The following are known and easy to prove: If A={1,L} and L is even then II wins iff n\equiv 0,2,4,...,L2 mod L+1 If A={1,L,L+1} and L is odd then II wins iff n\equiv 0,2,4,...L1 mod 2L+1…

Mathematics and Computation

TEDx “Zeroes”
16 Oct 2014  12:01 amI spoke at TEDx University of Ljubljana. The topic was how programming influences various aspects of life. I showed the audence how a bit of simple programming can reveal the beauty of mathematics. Taking John Baez’s The Bauty of Roots as an inspiration, I drew a very large image (20000 by 17500 pixels) of all roots of all polynomials of degree at most 26 whose coefficients are $1$ or $1$. That’s 268.435.452 polynomials and 6.979.321.752 roots. It is two degrees more than Sam Derbyshire’s image, so consider the race to be on! Who can give me 30 degrees? The code… 
Reductions in computability theory from a constructive point of view
19 Jul 2014  5:50 amHere are the slides from my Logic Coloquium 2014 talk in Vienna. This is joint work with Kazuto Yoshimura from Japan Advanced Institute for Science and Technology. Abstract: In constructive mathematics we often consider implications between nonconstructive reasoning principles. For instance, it is well known that the Limited principle of omniscience implies that equality of real numbers is decidable. Most such reductions proceed by reducing an instance of the consequent to an instance of the antecedent. We may therefore define a notion of instance reducibility, which turns out to have a… 
Seemingly impossible constructive proofs
8 May 2014  7:15 amIn the post Seemingly impossible functional programs, I wrote increasingly efficient Haskell programs to realize the mathematical statement $\forall p : X \to 2. (\exists x:X.p(x)=0) \vee (\forall x:X.p(x)=1)$ for $X=2^\mathbb{N}$, the Cantor set of infinite binary sequences, where $2$ is the set of binary digits. Then in the post A Haskell monad for infinite search in finite time I looked at ways of systematically constructing such sets $X$ with corresponding Haskell realizers of the above omniscience principle. In this post I give examples of infinite sets $X$ and corresponding constructive… 
Brazilian type checking
6 May 2014  2:59 amI just gave a talk at “Semantics of proofs and certified mathematics”. I spoke about a new proof checker Chris Stone and I are working on. The interesting feature is that it has both kinds of equality, the “paths” and the “strict” ones. It is based on a homotopy type system proposed by Vladimir Voevodsky. The slides contain talk notes and explain why it is “Brazilian”. Download slides: braziliantypechecking.pdf GitHub repository: https://github.com/andrejbauer/tt Abstract: Proof assistants verify that inputs are correct up to judgmental… 
Intuitionistic Mathematics and Realizability in the Physical World
4 Mar 2014  7:57 amThis is a draft version of my contribution to “A Computable Universe: Understanding and Exploring Nature as Computation”, edited by Hector Zenil. Consider it a teaser for the rest of the book, which contains papers by an impressive list of authors. Abstract: Intuitionistic mathematics perceives subtle variations in meaning where classical mathematics asserts equivalence, and permits geometrically and computationally motivated axioms that classical mathematics prohibits. It is therefore wellsuited as a logical foundation on which questions about computability in the real world…

WordPress Tag: Mathematics

Partitions, perforations and tilings
19 Oct 2014  12:22 amWe are “geometric” in our thinking – perhaps, we are hence a little more Greek or the old type Arya than the later Hindu (who is more algebraic) in mentality. Long back in college we were fascinated by implicit trigonometric relations but were utterly defeated by the difficulty in visualizing. Without visualization, of what use are these relations to one who is inclined towards visible geometry? Then we wrote a bit of code that allowed us to visualize the same. Today far more efficient programs exist to do the same and we tend to revisit these relations, perhaps as a reenactment of our… 
Maths is actually somehow my favourite subject now (Maths Reflection)
18 Oct 2014  2:18 pmFirst things first, I am a straight male. Now that’s out of the way, I am actually in love with my maths tutor. Seriously, the guy is enthusiastic, intellectual, commanding, engaging, humorous and has an ability to transmit knowledge with ease. I want to be just like him as a teacher, only…less old and more muscular. He has singlehandedly smashed English down a notch and established Maths as the subject to be, the subject to love, the subject to want to do. I feel more confident at the subject every seminar and no longer dread a subject with somewhat of a reputation as a… 
Engineering Education and Skinning Cats
18 Oct 2014  1:55 pmAmong all the courses I took in engineering school, I recall only one as being a “true” engineering course…that was a course in servomechanisms and feedback control systems. But it wasn’t the course content that made it “true.” I entered Northwestern University’s Technological Institute in the fall of 1957. I was on my way to becoming an electrical engineer. At least that was the plan. I had chosen engineering because one of my high school math teachers thought I had the right skills to be an engineer. I chose electrical engineering because I had always been interested in… 
Saved by Pythagoras: defeating a recurring nightmare
18 Oct 2014  1:36 pmDreaming of waking up in a strange place. Modern civilization, but language resembling nothing I’ve encountered. Can’t communicate or read signs. More than foreign — alien almost, although the people are definitely human. Just doesn’t feel like my world at all. And to crown everything — I’m at large in my pajamas! Eventually, I wake up in a nighttime scene. There are three moons in the sky, confirming my initial assessment of an alien world. I panic — really panic. Earth suddenly seems out of reach and I believe I will not be returned home this time. 
Fighting Back Against "I'm Terrible at Math"
18 Oct 2014  12:26 pmMany teachers, and many people in general feel terrible at math. In the article, The Myth of “I’m Bad at Math”, by Miles Campbell and Noah Smith analyzes this phenomenon and makes a very compelling case for why this is so dangerous in our society. They say “We hear it all the time. And we’ve had enough. Because we believe that the idea of “math people” is the most selfdestructive idea in America today. The truth is, you probably are a math person, and by thinking otherwise, you are possibly hamstringing your own career.” I grew up thinking I was bad…

Mr. L's Math

Programming with GeoGebra
14 Oct 2014  1:58 pmGeoGebra logo Here’s a nice post by Riley EynonLynch from the Point of Inflection website – PROGRAMMING WITH GEOGEBRA Some of his main points: This post is about some of the virtues of programming computers in math class. I include a long anecdote and a quick geogebra tutorial. The punchline: teaching kids to program introduces them to an environment that gives instantaneous, continuous, 100% correct, 0% helpful feedback without judgement. The computer doesn’t say, “you’ve made a mistake here,” it just shows you a result, and it’s up to you to interpret it, decide if… 
GeoGebra Loved by StudentsTeachersSchools
5 Oct 2014  6:42 amGeoGebraa Powerful Tool for Students GeoGebra.org has had a facelift and is worth visiting/revisiting. The new interface suits computers and mobile devices well, and has something for everyone. Its strength is ease of use paired with great power to visualize mathematics. The following was taken directly from the newly configured site: GeoGebra is a multiplatform mathematics software that gives everyone the chance to experience the extraordinary insights that math makes possible. Students love it because… it makes math tangible – GeoGebra makes a link between Geometry and Algebra in… 
Visual Pattern Site by Fawn Nguyen
26 Sep 2014  2:06 pmpattern no. 155 from Fawn Nguyen’s site visualpatterns.org When I taught 7th grade for six years visual patterns were used to start the school year because they did so many great things for students. They were engaging to the students, visually stimulating, allowed all students easy entry to the math involved, worked great for student projects, and addressed many math standards. Here’s a site with a lot of patterns you can use in your classroom, along with commentary for teacher use. As of the date of this post (Sept 2014) there are 145 patterns, along with the Equation… 
Malin ChristerssonDigital Math for GeoGebra Enthusiasts
25 Sep 2014  5:46 pmMalin Christersson’s site, Digital Mathematics, is a great place to spend some time for GeoGebra enthusiasts. It has some of the best tutorials on the web, organized into seven clusters. Malin provides clear and detailed explanations, some with embedded videos, that help the new as well as experienced user to get more out of GeoGebra. Malin also has provided further work in the areas of NonEuclidean Geometry, Latex/LyX, Geometry, Functions, Trigonometry, Calculus, Statistics, Linear Algebra, and Fractals. From the site: “This is a collection of material that I have used when… 
Circle Dissection PuzzleFour Equal Curved Areas
24 Sep 2014  4:30 pmDissect the circle into four parts of equal area by drawing three curved lines of equal length. – idea from Arithmetrics, by Jerome S. Meyer, pg 88 Move slider halfway to reveal a hint if you’re stuck. Questions for you or your students: 1 Why are the 3 curved lines of equal length? 2 How do you show that the 3 curved lines make equal areas? The downloadable file can be found here. My other GeoGebraTube apps can be found here. here.

MathNotations

Round your answer to nearest cent: $1.29 or $1.30?
20 Oct 2014  10:27 amTweeted (@dmarain) the above a couple of days ago. Moderate reaction so far which I find fascinating since I've done my own "random" survey... SCENARIO 6th gr student calculates an *exact* answer of $1.29. Directions read "round ans to nearest cent." Student writes $1.30 in the answer box on the test. Teacher notes $1.29 was correct but the answer in box was wrong. No credit for problem... COREFLECTIONS Making too big a deal of this? After all "rounded to nearest cent" means "round to nearest hundredth". So $1.29 is already rounded to the nearest… 
Implement The Core  Opposite Corners of a Square
18 Oct 2014  12:12 pmTwitter Problem 101814 If (a,b),(a,b) are opposite vertices of a square, show that its area=2(a^2+b^2) EXTENSION: What if (a,b),(a,b) are adjacent? COREFLECTIONS (1) What do you believe will challenge your geometry students here? The abstraction? "Show that"? (2) Predict how many of your students would "complete the rectangle" by incorrectly drawing sides  to the axes? (3) Even if not an assessment question, is it a good strategy to "plug in" values for a&b? This is worthy if more dialog IMO... (4) How many of your students would question the lack… 
Implement The Core: Arithmetic Patterns & Generalizations in Middle School Math
16 Oct 2014  9:54 amAs tweeted on 101614... Pattern #1 Explore on calculator... 352×11=3872 527×11=5797 365×11=4015 Keep going! Discuss! Explain! Generalize! Pattern #2 18=9×2,81=9×9 27=9×3,72=9×8 36=9×4,63=9×7 Keep going! Describe, extend,generalize! Is 407×9=3663 unrelated? COREFLECTIONS... (1) But these are just math curiosities, Dave. They don't really tie into the Common Core, do they? Well, doesn't multiplying by 11 connect nicely to the Distributive Property: 352×11=352×(10+1)=3520+352 etc. How about 9? (2) My goal has always been to expose our students to engaging and meaningful… 
Implement The Core: 'Dates' and 'Figs'  Middle School Investigation
13 Oct 2014  4:59 amTwitter Problem @dmarain... Yesterday's date here in the US was 101214: an arithmetic sequence. (a) List the other 5 such dates this year (b) List them for 2015 & 2016 (c) Observations & Explanations In your group make at least 5 observations and/or conjectures. Explain/prove or show they are false. Examples... (1) Observation: There are fewer such dates in 2016 than in 2015. Possible Explanation: In 2015, the months are the 7 odd numbers from 1 through 13; in 2016, the months are the 6 evens from 212 . There is no 141516. Note:Would the same be true for all even years from… 
Implement The Core: f(3)=5,f(5)=5 and much more
11 Oct 2014  3:59 amTwitter Problem 101114f is a linear function with f(3)=5 and f(5)=3. f(0)=?COREFLECTIONS...(1) The title has an error and omits the critical linear condition. Note that f(5)=3 not f(5)=5.(2) The Mathematics Practice Standards ask us to extend student thinking, make connections and go beyond the superficial qualities of a problem. My hope is that you will see the Twitter problem as a door marked ENTER not EXIT...How do we do this?One possibility is to ask our students to generalize. Note that the responsibility is shifted from us to them. We can guide this by prompting with: "Suppose…

mathrecreation

GSP and LOGO (for MITx: 11.132x)
20 Oct 2014  7:51 pmNote: This post is an assignment for the Edx MOOC MITx: 11.132x Design and Development of Educational Technology. The assignment had to be posted online, and since it relates somewhat to the themes of this blog, I put it here.Educational Technology Then and Now: Geometer's Sketchpad and LOGOGeomter's Sketchpad (GSP) is an example of current educational technology that is based on design and educational principles that can generally be described as constructionist. Widely used in contemporary classrooms, GSP is based on ideas about computerhuman interaction that date back to the 1960s, and… 
circles, hexagons, flowers
2 Oct 2014  7:02 pmWas playing around with intersecting circles, as in this post. 
A year of tinkering
29 Sep 2014  6:37 pmYou really should take advantage of the free until August 2015 license that is currently being offered with a fresh download TinkerPlots. Would that it was freely available in perpetuity without condition, but a year of tinkering is nice.If you are a middle school teacher, then this is designed for you and yours. If, like me, you are not, you may find it fun to play with anyway. Here is something I was playing with recently:An elementary school number sense activityIn the JUMP math curriculum for grades 3 and 4, there are lessons where students investigate the patterns formed when… 
modular tables
17 Sep 2014  7:13 pmNo, not a post about IKEA furniture. A while ago I put up a post on colouring multiplication tables by assigning ranges of numbers a colour value. You end up with something that looks like a rainbow.This image was made in Tinkerplots, so it was easy to go from a 10 x 10 table to a 50 x 50 table (removing the numbers and just keeping the colours, and shrinking each cell down a bit):Inspired by the "Zn Multiplication visualizer" found here and mentioned here, and thinking about modular arithmetic from the last post, I decided to make a few more images.If you take the values in this… 
squashing multiples
15 Sep 2014  7:21 pmAn elementary school exercise leads to writing a simple program, a little proof by contradiction, and learning about some mostlyforgotten calculation tricks: just some of the fun that can be had when playing with simple math. Sound good? It all starts with squashing numbers...No doubt you've noticed some patterns in the nonzero multiples of 9: 9, 18, 27, 36, 45,... One thing to notice is that if you (repeatedly) add up all the digits of a multiple of 9, you always get 9 as your answer.This works immediately for many multiples of 9, like 9*14 = 126 (1 + 2 + 6 = 9), for others you need to…

MathFour

Using the Hundreds Chart to Teach Beginning Multiplication
21 Oct 2014  5:50 amWe typically think of beginning multiplication as memorizing your math facts for 2’s, 3’s, 4’s and 5’s. But in my new job, things aren’t so typical. So I’m trying something new. Sieve of Eratosthenes It started with a brilliant idea to teaching the Sieve of Eratosthenes. I began by asking the students to color in (or cross out) each of the multiples of 2, 3, 5 and 7. Alas, the plan went awry quickly. Students got confused when they needed to cross out a number that was previously crossed out (like 6). So they moved to the next number and crossed it out:… 
Really Big Numbers: The 100 Dots Project
13 Oct 2014  10:26 amNote: this post has affiliate links. If you use these, you support my work in giving you free content. The AMS sent me a copy (for free, yay me!) of their first publication ever – Really Big Numbers by Evan Schwartz. I loved reading the first few pages with K8, and it gave me an idea. What if I used this in class to introduce numbers? I started teaching at a private, special education school recently. I quickly figured out that everything I know about math education is wrong. I thought this book (and the activity that spawned from it) might work well. And for once, with these kids, I… 
Halloween Geometric Bat – FREE DOWNLOAD
2 Oct 2014  2:14 amCheck out this little Halloween bat made of rectangles, squares and triangles. There’s even an opportunity to talk about trapezoids! Both of the downloads have the same bat template, they just have different discussion questions. Supplies One of the free downloadable templates: Geometric Bat for Older Kids or Geometric Bat for Younger Kids Black and orange construction paper Scissors Glue or glue stick Googly eyes Instructions Use the template shapes to cut out the geometric pieces from black construction paper. Follow the design on the download to glue the pieces onto the orange piece… 
How to Tell the Difference between 0.3 and 0.33 – Visually
26 Sep 2014  4:39 amA student was working on this problem the other day: Find the volume of a cone with radius 2.5 units and height 5 units. Of course we all remember the formula as So she plugged in her numbers and got 29.45 cubic units. She checked the back of the book and was disappointed to see the answer listed as 32.72 cubic units. Turns out she was using 0.3 instead of 1/3! What’s wrong with the answer? I asked her to calculate with 0.33 instead. She got 32.39 – far closer to the answer from the book. Then I asked her to use 0.333. This yielded 32.69. She was amazed at how adding another 3 got… 
Teaching Math to Special Needs Children
16 Sep 2014  7:38 pmI’m out of my depth. Like 3 bazillion leagues out of my depth. I took a math teaching position at a school for kids with neurological differences. I knew it would be hard. But I didn’t think it would be this hard. Lesson 1: Everything you know is wrong. It’s a Weird Al song, but it also applies to teaching kids with special needs. I gave this great math artwork activity that I thought would be perfect. My students are all 1219 years old. They can communicate, be polite and follow instructions. And they are all listed as over 2nd grade in abilities. So this should have be a…

CSE Blog  quant, math, computer science puzzles

Diminishing Differences Puzzle
18 Oct 2014  3:05 amSource: Australian Mathematical Society Gazette Puzzle Corner 34 Problem: Begin with n integers x1, . . . , xn around a circle. At each turn, simultaneously replace all of them by the absolute differences Repeat this process until every number is 0, then stop. Prove that this process always terminates if and only if n is a power of 2. Shameless plug: Follow CSE Blog on CSE Blog  Twitter and CSE Blog on Quora. :) 
Balancing Act Puzzle
19 Sep 2014  1:21 amSource: Australian Mathematical Society Gazette Puzzle Corner 35 Problem:There are some weights on the two sides of a balance scale. The mass of each weight is an integer number of grams, but no two weights on the same side of the scale share the same mass. At the moment, the scale is perfectly balanced, with each side weighing a totalof W grams. Suppose W is less than the number of weights on the left multiplied by the number of weights on the right.Is it always true that we can remove some, but not all, of the weights from each side and still keep the two sides balanced? 
"Flawless Harmony" Puzzle
26 Aug 2014  2:40 amSource: AUSTMS Puzzle Corner 35 Problem: Call a ninedigit number flawless if it has all the digits from 1 to 9 in some order. An unordered pair of flawless numbers is called harmonious if they sum to 987654321. Note that (a, b) and (b, a) are considered to be the same unordered pair.Without resorting to an exhaustive search, prove that the number of harmonious pairs is odd. 
Minimum sum of numbers in an array
10 Aug 2014  10:21 pmSource: Asked to me on quora ( cseblog.quora.com ) Problem: Given an array of n positive numbers (n ~ 100000), what is the algorithmic approach to find the minimum possible sum (>=0) by using all the numbers in an array? Example 1: 1 2 2 3 4 Answer : 0 (1+223+4) Example 2: 2 3 4 7 13 Answer: 1 (+2347+13) 
Caterer's Problem
5 Aug 2014  10:07 pmSource: Puzzle Toad CMU Problem: You are organizing a conference, with a festive dinner on the first day. The catering service has 1024 different dinner choices they know how to make, out of which you need to choose 10 to be in the dinner menu (each participant will choose one of these during the dinner). You send an email to the 6875 participants of the conference, with the list of all 1024 choices, asking them to rank the choices in linear order from their favorite to their unfavorite. You want to find a list L of 10 choices, such that for any dinner choice d not in the list L, if we run a…

Math for all Grade blog

Oct 20, Complementaryangles
20 Oct 2014  1:12 amTwo angles are called complementary angles, if the sum of their degree measures is 90. Each angle is called complement of the other 
Oct 20, Compoundinterestformula
20 Oct 2014  12:14 amcompound interest formula to calculate the amount for annual compounding and other investment time periods. 
Oct 20, Areaofcircle
19 Oct 2014  11:47 pmArea of circle is Π times the radius of the circle squared or Π times onefourth the diameter of circle squared. 
Oct 17, Alternateinteriorangles
17 Oct 2014  12:32 amAlternate interior angles formed by parallel lines cut by a transversal are equal in measure. 
Oct 17, Area of a Trapezoid
17 Oct 2014  12:22 am: Area of a trapezoid is half the product of the sum of the parallel sides and distance between the sides.