Tool or Crutch?

Parents fret today over their children's use of digital devices. They fear that iPods, Xbox 360, or SmartPhones are taking over their collective brains and lives. Well think again. The most harmful device your child uses today may very well be their handheld calculator.

Many classrooms, middle school and up, allow calculators to be used. Calculators are particularly helpful with upper-level math problems such as advanced calculus and such.  But, in many instances, they become overused.  

Calculators are sometimes being used to do basic arithmetic operations when a child lacks the underlying math skill.  Calculator over-reliance is responsible for atrophied mental math skills.  These waning skills, if left uncorrected, will lead to problems not only in their next grade level, but beyond that in executing life’s daily challenges.

Mathematicians have always searched for efficient ways to perform mundane calculations to free their minds to focus on more complex problems.   The abacus and sliderule are two such examples.  Thus, the real question is not “if” to use a calculator, but “when”.  

There a many opinions as to when it is OK to allow a child to use a calculator. With all the divergent points of view, what’s a parent to do? Here's a simple test.

If your child is currently using a calculator in class, give them this 10-question test below.  They should be able to answer each in their head within about 2 seconds.

1.      9 times 7   [63]

2.      17 minus 9   [8]

3.      68 plus 15  [83]

4.      72 divided by 8  [9]

5.      80 times 3   [240]

6.      51 minus 14  [37]

7.      7 times 8 [56]

8.      24 divided by 3  [8]

9.      8 plus 7  [15]

10.  150 divided by 25  [6]

If your child can do the above exercises in their head, you can feel pretty confident they are not using a calculator in class as a crutch.  If not, don’t worry so much about their FaceBook, SmartPhone and texting usage ...worry instead about their over-reliance on calculators.  

Math to Ponder:  The Birthday Paradox

There is a 50% chance that in any group of 23 people, 2 of them will have a birthday on the same day.   Once you have a group of 57 people the chance of any 2 of them having the same birthday jumps to 99%!

To understand this probability problem, let’s work backwards using the converse probability – i.e. that no two people have the same birthday.

Ignoring leap year the first person has a probability of 365/365 (their birthday is on 1 day in the year) and the second has a probability of 364/365 i.e. there are 364 days on which their birthday can fall without being the same as the first person.   Thus the probability that these 2 people do NOT have the same birthday is 365/365 x 364/365 = 99.72%; for a third person the probability becomes 363/365 (there are only 2 days when there could be a common birthday)and so the probability that none of the 3 share a birthday becomes 365/365 x 364/365 x 363/365 = 99.17%.

As we add people the series continues so for the 23rd person it becomes:

365/365 x 364/365 x 363/365 x 362/365 x … x 344/365 x 343/365 = 49.27% or less than 50% chance that 2 people do NOT have the same birthday, hence there is greater than 50% chance that there are 2 people who share a birthday.   If we continue the above pattern we will find that with 57 people there is a 99% chance that 2 will share a birthday.   Next time you are in a large group amaze people by predicting that at least 2 of them will share a birthday.   It really is true.

Math Humor: Old mathematicians never die; they just lose some of their functions.