...in the first two weeks of school her student's math scores had risen. Tutoring had not started; the only change was beginning to work with flashcards.

It's not surprising.  Why?  Because the time and mental energy it takes to do simple calculations when you do not have them memorized are a distraction from the thinking work of doing math. 

Consider a simple two-digit by two-digit multiplication problem.  Let's take 47 x 98.  Here's what's involved:

  47

x98

8 x 7 = 56, 8 x 4 = 32, 32 + 5 = 37, 9 x 7 = 63, 9 x 4 = 36, 36 + 6 = 42, 6 + 0 = 6, 7 + 3 = 10, 3 + 2 = 5, 0 + 4 = 4... so  47 x 98 = 4,600

The human brain can retrieve a memorized fact in less than one second.  That means that all these calculations can be performed in about 10 seconds or less.  (Adding a few fractions of a second to account for the mental math strategy involved in 36 + 6 = 36 + 4 = 40, 40 + 2 equals 42). 

And learning long multiplication also involves memorizing the "pencil dance" of how to arrange these number on the paper.  That alone takes a lot of practice to memorize.

Imagine the frustration of having to stop that "dance" several times to painfully do the calculation by counting up on your fingers.

As for using a calculator, there is time involved in nerve impulses traveling from the brain to the fingers, and then in reading the answer and recording it.  Doubling the time it takes to do one long multiplication problem may seem unimportant -- as long as we're talking about one long multiplication problem.  Doubling the time it takes to do a homework assignment that was designed to take twenty minutes, so that it takes forty minutes... that starts to make a difference to a kid's day.  Or to a family's day, for that matter. 

And if calculators are not allowed, math facts that are not memorized can take three minutes.  When does the real learning of mathematical concepts happen if so much time is going to simple calculations? 

And it's exhausting.  Parents tell us that their children "space out" while doing math, that they "lose focus."  It's not that they're not working hard enough.  It's that they are working much too hard.

The solution is to help them memorize their math facts.  And although we've looked for computer programs that do that, we've never found one that works as effectively as daily practice with flash cards.  And then the parent-child stress goes away.  Math fact memorization simply becomes part of the daily routine, and while it may take two years for a child to memorize more than 600 addition, subtraction, multiplication and division facts, once it's done, it's done forever.  And the good effects can be seen quite quickly.