You want the 700, but you don't have number sense: try this!

This post is a follow up to yesterday's post which describes the difference between a student who scores in the 700's on the math section of the SAT and a student who scores in the 500's.

So I've told my 500's student that getting a 700 on the SAT will be "a lot of work."  But what should that work consist of?  I have some activities we could do and some problems we could work, but we need to do the activities and work the problems over a long period of time so that she can actually internalize the lessons.  Unfortunately, working one-on-one with me for that period of time would be prohibitively expensive for a lot of people.  Is there a cheaper alternative?  Can someone do it on her own?

I have been using materials from Art of Problem Solving with a handful of elementary school students, and I am struck with the fact that they place more emphasis on the number sense than they do the algorithms.  They walk students through steps toward comprehension that focus on the meaning and assume that the algorithm will come on its own.  That process works best for the advanced kids, but that is their target market.  Still, I thought they might have something useful for the high school student who has learned the algorithms, but would like to retro-actively work on the number sense.

Introducing Alcumus.  Alcumus is a free online problem bank.  Once you register, it will give you a math problem.  Get that right, and you will get a more difficult problem. Complete enough problems in a row, and you will move on to the next topic. It is somewhat similar to the practice modules on Khan academy with a couple of notable differences:  First, you will be given an explained solution even if your answer was correct!  In fact, to get anything out of this exercise, you need to carefully read every explanation to see if there was a different, more intuitive (as opposed to algorithmic) method of solving the problem.  There are a few videos to watch for more instruction, but Art of Problem Solving believes in a problem-first approach.  There are also references to chapters in Art of Problem Solving math text books. (The books can be a bit pricey.  If you have the means, buy a couple of copies and donate one to your school library.)

Try to see if these methods lead to being able to solve complicated-looking problems in your head.  (It goes without saying that you should NOT be using a calculator.)  You will be led through addition, subtraction, the distributive property - simple stuff, but there are lessons here for how to think about these problems differently.  How to use your head instead of that hand-held machine you have been using as a crutch.

Try it!  I'd love to hear how it works out!

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.