The New York Times recently (Sep 16) published an interesting article – “Gut Instinct’s surprising role in Math”. The article shares an interesting view about human mind’s ability to instinctively solve math problems. It shows how the mind is wired to perform some seemingly complex math tasks with ease.
To quote it from the article “Whenever we choose a shorter grocery line over a longer one, or a bustling restaurant over an unpopular one, we rally our approximate number system, an ancient and intuitive sense that we are born with and that we share with many other animals. Rats, pigeons, monkeys, babies — all can tell more from fewer, abundant from stingy. An approximate number sense is essential to brute survival: how else can a bird find the best patch of berries, or two baboons know better than to pick a fight with a gang of six?” It’s interesting to know that not only the human brain works this way but even baboons’ and rodents’ work in the same manner.
In fact, research has shown that new born babies if found to be good at deciphering approximations with ease, will carry on the same trait when they grow up and have to deal with abstruse mathematical problems. NYT states: “One research team has found that how readily people rally their approximate number sense is linked over time to success in even the most advanced and abstruse mathematics courses. Other scientists have shown that preschool children are remarkably good at approximating the impact of adding to or subtracting from large groups of items but are poor at translating the approximate into the specific.” In fact, it goes further and suggests that Math teachers should emphasize on the power of approximation early on in the child’s education so he can hone this natural skills to an advantage. To quote “Taken together, the new research suggests that math teachers might do well to emphasize the power of the ballpark figure, to focus less on arithmetic precision and more on general reckoning.”
At LazyMaths.com, we precisely try to encourage our students to leverage these skills of approximation. We have found that with practice, the line between approximation and precision can be diluted if not fully erased. In the section of Don’t Solve, LazyMaths offers a large number of examples of a variety of Math problems attempted using the techniques of approximation.
Learning math on Twitter? Is this just a thought, a dream or mere wishful thinking? Not anymore!
LazyMaths.com has been sharing Free Speed Math shortcuts with its Twitter followers since the beginning of this year. Yes, One Free Shortcut a Day. Could it get better than this?
This has been so popular with our followers that they have been wanting for more! So, we plan to now offer something else for free too! More Math on Twitter!!
Starting June 15, we plan to offer one free Smart Math technique for our Twitter followers. If you are taking the GMAT, GRE, SAT, CAT, CET or any multiple choice based Math tests, this is an absolute must for you. All Smart Math techniques are in the Don’t Solve section of LazyMaths. The techniques are based on concepts of approximation, elimination and reverse substitution. Not only does using these techniques, save a ton of time to get to the answer, but it also helps in avoiding silly mistakes.
The Don’t Solve section contains problems in areas of Algebra, Percentages, Average, Ratio Proportion, Time, Speed & Distance, etc. Check out free samples here.
Eager to learn a new kind of Math right now? Go ahead, sign up with LazyMaths.com and select your choice of membership. We guarantee that you would not regret.
While we publish free speed math shortcuts and free smart math techniques, we also publish math puzzles, downloadable math learning resources like Number classification. All of these are available to our Twitter followers as well as on our blog – Zzzlog.
In short, learn math the web 2.0 way – right on Twitter! Simply follow LazyMaths on Twitter and become a pro in Twitter Math!!
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We have been using numbers in everyday life. Everything from 0 to 22/7 might sound the same to most, but numbers differ from one another. Based on their characteristics, they are classified in groups. Based on all the different types of numbers Mathematician’s have named, we have built the ultimate guide to Number classification.
The associated chart (at the bottom of the article) shows how the number groups are related to each other. Read below for more details on each number group.
Real Numbers – All kinds of numbers that you usually think of – from bus route numbers, to your weight, to pi and even the square root of pi! In short everything!! Everything? Really? Well…almost
Imaginary numbers – Have you ever tried finding the square root of -1? If you haven’t, try it on your calculator. It might show an error (if it is a dumb calc) or it might show an ‘i’. That little ‘i’ is called an imaginary number. In short square roots of negative numbers make imaginary numbers.
Complex Numbers – It’s rather simple! Make a combination of Real and Imaginary numbers and voila! You get a Complex number. Stuff like 3+2i or 3/4i make up complex numbers. Just think of it when you mix a real number with an imaginary one, things do get a bit complex!
Rational Numbers – Any number that can be written as a fraction is a rational number. So numbers like ½, ¾, even 22/7 and all integers are also rational numbers.
Irrational Numbers – Simply the opposite of rational numbers i.e. numbers that cannot be written as fraction, like square roots of prime numbers, the golden ratio, the real value of pi (22/7 is a mere approximation not the real value of pi) are irrational numbers.
Integers – Any number that is not a fraction and does not have a tail after the decimal point is an integer. This includes both negative as well as positive numbers as well as zero.
Fractions – Numbers that are expressed in a ratio are called fractions. This classification is based on the number arrangement and not the number value. Remember that even integers can be expressed as fractions – 3 = 6/2 so 6/2 is a fraction but 3 is not.
Proper Fractions – Whenever the value of the numerator in a fraction is less than the value of the denominator, it is called a proper fraction. i.e. it’s bottom heavy.
Improper Fractions – Whenever the value of the denominator in a fraction is less than the value of the numerator, it is called a proper fraction. i.e. it’s top heavy.
Mixed Fractions – All improper fractions can be converted into an integer with a proper fraction. This combination of an integer with a proper fraction is called a mixed fraction.
Natural Numbers – All positive integers (not including the zero) are Natural numbers. Simply put, whatever you can count in Nature uses a natural number.
Whole Numbers – All positive integers inclusive of the zero are Whole numbers. Not a big deal different from Natural numbers.
Even Numbers – All integers that end with a 0, 2, 4, 6, or 8 (including the numbers 0, 2, 4, 6 & 8 themselves) are even numbers. Note that ‘0’ itself is an even number. Also note that negative numbers can also be even so long as they can be integrally divided by 2.
Odd Numbers – All integers that are not even numbers are odd number
Prime Numbers – A natural number, more than one, which has exactly two distinct natural number divisors: 1 and itself – is called a Prime number. There can be infinite prime numbers.
Composite Numbers – A positive integer which has a positive divisor other than one or itself is a composite number. In other words, all numbers that are not prime are composite
Did you notice something unusual about today’s date? 3/3/09 – See something?
3 x 3 = 9 in other words, the square root of 9 is 3. Today’s date has all that unique combination that shows the relationship between a perfect square and its square root.
You may ask, so what? Well, the importance of this (rather such) date(s) really lies in their rarity. Let’s see when was the last such date? 2/2/04 – This was back in 2004 (almost 5 years back). So, when’s the next one? 4/4/16 – That’s in 2016! Over 7 years later. Now you get it?
So, mark out 4/4/16 in that calendar and don’t miss it for you would have to wait until 5/5/25 – 9 long years!!
This is like asking why plural and not singular. I mean, why use plural and not just singular – always. Well, anyone (pretty much) who knows a bit of language grammar will use plural where the reference is to more than one and singular to refer a single item. No brainer here!
So, why do we use the term Math and not Maths to refer to Mathematics. Remember we don’t say or use Mathematic!!
We all know that Mathematics is a really a body of knowledge related to quantities, figures and forms. It includes a variety of areas ranging from simple Arithmetic and Algebra to complex Calculus and Combinatorics. (BTW, here’s a nice list of all Math(s) areas for those of you who are interested: http://en.wikipedia.org/wiki/Lists_of_mathematics_topics) With a list as large as this, doesn’t it make sense to use a plural term instead of singular? It definitely is more that one!
Well, for my non American readers, all this may sound a bit nonsensical, that because in most other English speaking countries – the UK, India, etc., Maths is the term being used. It is one of the oddities of the American English that allows the term Math to be used to refer something plural – much like the term ‘hair’ or the term ‘fish’ which stay the same for both singular and plural.
In all fairness to the American English, one could also argue that just as a tree has many branches, like Math(s) has / have and yet a tree is singular, so can the word Math! Because Mathematics is a body of knowledge and not many bodies of knowledge, the term should be Math and not Maths!
Having looked at both viewpoints, what’s you take? Which one would you agree?
Well, to sum it up, we shouldn’t really care too much whether it is Math or Maths as long as we are understood. Isn’t that really the purpose of all languages of the world – to make oneself understood?
The topic of whether men or women are better at math has almost always evoked a debate. Some even call it sexist to ask such a question – or should we call just a social scientist? Why are there fewer women in math oriented careers than men? Is it because of math skills or just social causes? (http://discovermagazine.com/2003/sep/breakgirls)
Outcomes of experiments are also influenced by who is present in the experiment. A study by Brown University suggests that women actually perform 12% better when men are not around them! (http://www.sciencedaily.com/releases/2000/09/000913083409.htm)
Another interesting study conducted in one of the world’s largest math experiment suggests that women are actually better than men not in general math but instant Math!
Regardless of what all the studies say, the fact of the matter is by practicing the tricks and techniques shown on LazyMaths.com, all can ace at all kinds of math – whether man or woman!
This generated some really interesting debate. here are some selected excerpts from what members of the community say…
Amish Bhavsar says -
Should calculators be allowed in schools for Math problems?
As a Math lover and as well as a Math teacher, I am convinced that excessive use of calculators for simple calculations cause over dependence on them. This undermines the mind’s ability to do very simple additions. This causes a lot of discomfort for the students who eventually start to hate math as a subject.
In India, school students do not use any machine based computational device (not even log tables) upto the 10th grade. The standard basic arithmetice calculators are introduced in 11th grade along with log tables. This forces Indian kids to keep their brains active and actually helps them understand the fundamentals of math. This is one reason you see Asian kids excelling in Math compared to American kids.
Just as nature took away the tail as humans evolved from a monkey, as it was not used (even joints of the body atrophy when not used / exercised), so would the brain’s ability to do simple math atrophy if we become overly dependent on calculators.
Encourage your kids and students to compute mentally. Let’s get back our brains!
Howard Wright says -
I remember a study that was done in the late 70’s using adding machines. What they did was rigged the machines to deliberly to give wrong answers and see how many subjects will question it. The results were rather discouraging for only 11% actually caught on and questioned the results. What made it really amazing? One of the questions was 2+2 (the machines were rigged to answer 5) and 87% wrote the wrong answer. I remember to this day what the article was call: Electronic Bullies.
Having done a year of teaching math I have seen with my own eyes how much kids today relay on machines and it’s not just math. Their social life and entertainment are completely depended on computers. Just try to get them to function a day without their cell phones, Ipods, PSP’s, etc. They would go insane. Other than the secrete code to WOW, they can’t think or critically analyze for their life. Given how our education system caters to the lowest common denominator, it’s comes as no suprise we allow kids to become addicted to calculators since they’re just specilized computers to them. But we should not fret. In a few years, nobody will know what the lowest common denominator is anymore anyways. I guess they could look it up on the internet.
John Purvis says –
That is sad, dismal, pessimistic and downright insulting to kids.
Unfortunately, it is absolutely true.
Julie Kaletsch says –
I teach 8th/9th grade algebra. Our school district has a very advanced pace of math. All students by the grade of 6th use mainly algebra based books. I teach Algebra 1 and 2 level.
Using calculators is a way for students to see how graphing, intersections, and parabolas actually work and look. When I start teaching graphs they are allowed to play on the calculators changing y and x to see the effects on the graph. It is a visual that many learners need. They can then do it on their own.
I have had the pleasure of working closely with Claran Einfeldt who is one of the writers of the Illinois ISAT. We have had this conversation many times. She says that many students start to hate math in junior high because some schools continually teach the same things which have failed at before. Then the students shut down. To continually go over facts once they have reached 6th grade fails them. And with all that must be taught in one year, they would fall even farther below grade level. The US schools are very aware that China and India are ahead of us mathematically. We are making tremendous gains though.
My view is some people are simply not good at arithmetic….but math is more than just math facts and arithmetic. It is really about problem solving. I love to witness a child who has never been good at basic math concepts use a calculator and be able to understand and complete higher level concepts. Some of these kids are great problem solvers….they just can’t get the arithmetic. This is not reason to not allow them to see they can be successful in the most difficult of math reasoning.
Melissa Mausolf says –
I’m hoping to be alive and to retain the information I know today in a few years, so to say that “In a few years, nobody will know what the lowest common denominator is anymore anyways” is both pessimistic and unrealistic. This kind of thinking is why older adults seem to have lost respect for the younger generations. I think it is unfortunate that people who could have the greatest impact on today’s youth by being an active part of the educational process are instead complaining about them in a group the students do not visit (and, therefore, cannot defend against).
I wonder, Howard, why is it that you only taught math for one year?
Amish, some school teachers and administrators are taking an active role in lessening our students’ dependence on calculators. For instance, in our school, students are not allowed to use calculators in the classroom except in special circumstances. Those circumstances include exploring iterated sequences, finding the irrational roots of cubic polynomials, and some higher level mathematical operations. Even then, students are encouraged to do as much as they can without a calculator (e.g.: find the exact expression for the value of x before plugging the information into the calculator and rounding it off).
I see students everyday who still hate math… perhaps more because they have to spend 10 minutes calculating (1.04)^10 instead of being able to round it off and move on to the more interesting application of 4% compounded interest and how much money they should invest to have $10,000 in 10 years. Students will always find a reason to hate something you hate teaching to them. I think a more immediate problem that needs to be addressed is that of the teachers who dislike what they do being allowed to slowly seep that hatred into their students.
Bonnie Crowder says –
Indeed, calculators are the robber of skills. What is 8*36? 8*(30+6) it is. If we give kiddos the evil button box they will never learn to use the distributive property. Why do we think then they should miraculously accept x(y+z) = xy + xz?
We hand them calculators, destroy their confidence in themselves, take away their skills in numbers and then throw them into fields of math where you need confidence and skills.
As a Math lover and as well as a Math teacher, I am convinced that excessive use of calculators for simple calculations cause over dependence on them. This undermines the mind’s ability to do very simple additions. This causes a lot of discomfort for the students who eventually start to hate math as a subject.
In India, school students do not use any machine based computational device (not even log tables) upto the 10th grade. The standard basic arithmetice calculators are introduced in 11th grade along with log tables. This forces Indian kids to keep their brains active and actually helps them understand the fundamentals of math. This is one reason you see Asian kids excelling in Math compared to American kids.
Just as nature took away the tail as humans evolved from a monkey, as it was not used (even joints of the body atrophy when not used / exercised), so would the brain’s ability to do simple math atrophy if we become overly dependent on calculators.
Encourage your kids and students to compute mentally. Let’s get back our brains!
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