What does a 700 student look like?

Yesterday, a student who last scored in the 500's on the math section of the SAT asked me what it would take to score above 700 by fall.  She asked me during a class change at school, so there wasn't much time to say more than, "a lot of work." However, once she had gone on to class, I asked myself:

What is the difference between this young lady, and that hypothetical person who scores above 700?

Both students have taken all of the math courses listed as prerequisites and then some.  (This is true of all of my tutorees.)  Both have good grades in math (A's and B's.)  What is true of that 700 kid that isn't true of everyone else?

The short answer:  all of the kids can tell me that a certain math fact is true, but the 700 kid behaves as if the math fact is true.  For example, all of the kids can give me a definition of an even number.  They can all recognize one when they see it (if it is written as a number and not an expression.)  The 700 kid can glance at a problem, see that 2 will have to be a factor of the answer, and eliminate the answer choices that are not even.

A 700 student can promptly tell me that 1÷ (1/16) is 16.  A 500 student will either labor through the algorithm for dividing by a fraction or, more often, sit stymied because s/he doesn't recognize that the dividing-by-a-fraction algorithm is relevant. (Note:  this most often happens when the above exercise is expressed as a compound fraction in the first place.)

A 700 student understands additive inverses and therefore doesn't sit gaping in horror if I ask him or her to add all of the integers from -25 to 26 inclusive without a calculator.

A 700 student can tell me that the square of the square root of 2 is 2 without laboring through the algorithm for multiplying square roots.  (She or he also remembers from one day to the next what fractional and negative exponents represent.)

A 500 student may or may not be able to recite the commutative properties of addition and multiplication, although he or she will confirm that numbers can be added or multiplied in any order.  A 700 student may or may not be able to recite the commutative properties of addition and multiplication  depending on whether or not she or he remembers which property goes by the name "commutative."  However, she or he will not hesitate to rearrange addends or factors to find the most efficient way to compute the answer.

A 700 student recognizes that every integer has a unique prime factorization and understands that any factor of that integer must be the product of some combination of those prime factors. If asked, the 500 student can find the prime factorization of a positive integer, but he or she will not recognize those occasions when finding the prime factorization of an integer would be useful.  If the student has found the prime factorization of an integer (with or without prompting) to be 11 x 17, and if you ask the student if the original integer is divisible by 6, the 700 student will say, "no."  The 500 student will whip out a calculator and do the computation.

In short, the 700 student has a characteristic called "number sense".  The 500 student does not.  These two students might be in the same math class, at the same school, earning the same grade, but the 700 student is only working half as hard.  Furthermore, this will have been true for years.  I am currently working with two elementary school students - brothers.  One has number sense, the other doesn't.  I can already predict what their first SAT scores will be (or perhaps would have been if the test weren't changing.)

So now you're a junior and you don't have number sense, but you want that 700.  AND you're willing to do the work.  What should you do?  Check in tomorrow for instructions.