Why Multiplication Tables?

Although BatBoy has taken it upon himself to know exponents of 2 and SpiderGirl has been accumlating multiplication facts for some time, I can't imagine why anyone without foreknowledge of the power of memorized multiplication tables would bother sitting down to memorize 145 multiplication facts.  That's right:  the power of memorized multiplication tables.  Having multiplication facts in your head, retrievable at a moment's notice, is incredibly powerful, mathematically speaking.



Multiplication tables are unlike addition tables in important ways.  Addition has its power in the concrete.  One can visualize what it means to add two or more numbers together.  Concretely understanding simple addition supports the understanding of adding multiple digit numbers, "carrying," subtracting, and adding numbers that are not Whole.  Moreover, there are no special numbers when it comes to addition.  Every number can be the sum of countless other numbers.

Multiplication, on the other hand, is the gateway to the world of the abstract.  Certainly, one needs to understand that multiplication is repeated addition (or subtraction, if multiplying by a negative number).  One can make connections with skip counting, area, and volume.  However, to access the power of multiplication, one needs to have some facts memorized. 

With multiplication tables in one's pocket, one can

• Understand division as a reversal of multiplication.  This becomes important in dividing fractions.  Try as we might force a visualization of dividing one fraction by another, it simply takes more mental acrobatics to rationalize flipping one fraction (and the correct one!) "upside down" and multiplying, than to deal with the division as a reverse of multiplication.  It's still not intuitive -- it takes thinking, and understanding of the fraction as a division itself -- but it generally makes much more sense.
• More easily find prime numbers.  Unlike in addition, from a multiplication point of view, some numbers are special.  There exist numbers that can no longer be divided without leaving the realm of Whole Numbers.  Prime factorization is not only important in basic arithmetic such as finding common denominators for fractions, but also is the foundation for much of the coding and computer protection algorithms in use today.
• Recognize significant numbers.  We can see quickly when a number from the multiplication tables are divisible and by what numbers.  This helps us from getting bogged down trying to think through multiplication (or division) during
• multiplying multiple-digit numbers,
• long division,
• short division,
• factoring,
• finding common denominators,
• simplifying radicals,
• simplifying fractions, 
• manipulating exponents and logarithms, and
  
• anything in algebra, geometry, statistics, or calculus requiring these operations.

In a way, when I work with numbers, I feel like I'm working magic.  I don't mean 'magic' in the sense that I'm doing something inexplicable, but in the sense that I am working with unseeable forces.  I sense my fingers move into the air to pull down tools to apply to the numbers, to pull them apart, to move them around, to fuse one to another, or to change their forms. 

Will children see multiplication tables in their course of daily life?  In this society, I'd say it's doubtful.  People here and now do not share math with others; for most, math exists internally or on paper.  We show young children the alphabet, which exists as a list of letters purely for the sake of young children.  Would an unschooler keep the alphabet away from their babies and toddlers because it isn't "real"?  Are we unschoolers yet if we show our children what worlds might be opened should they choose to attain this set of facts?