What is Number Sense and How Can You Improve It?

Date: 08/29/2007 at 11:55:56
From: Sarah
Subject: What does the term "number sense" refer to

What does the term "number sense" refer to on standardized math tests?
The other categories are:  Patterns, algebraic thinking,  data
analysis/prediction, measurement, geometric concepts.  Our daughter
scores pretty highly in all categories except "number sense"--in which 
her score is average.

Date: 08/29/2007 at 14:39:40
From: Doctor Ian
Subject: Re: What does the term 

Hi Sarah,

You can think of "number sense" as being short for "common sense
regarding numbers".  

For example, it's knowing that if I multiply 5.8 by 6.9, I shouldn't
get an answer greater than 100, or less than 1.  (Note that either of
those could happen if I'm using a calculator, and type in one or both
of the numbers incorrectly.)

It's knowing things like:  1/2 is greater than 1/3, 2/5 is greater
than 2/7, 2.6 is greater than 2.59, and so on.  Or that 2/5 and 4/10
and 0.4 all refer to the same number. 

Does this help? 

- Doctor Ian, The Math Forum
  http://mathforum.org/dr.math/ 

Date: 08/29/2007 at 14:46:41
From: Sarah
Subject: What does the term "number sense" refer to

Thanks, this does help!  Any suggestions on helping someone improve
their "math sense"?  If it really is like common sense, I don't know
if you can teach that.

Date: 08/29/2007 at 15:58:58
From: Doctor Ian
Subject: Re: What does the term 

Hi Sarah,

The way you develop common sense about anything is through experience.  

A concept that is closely related to number sense is estimation.  A
good habit to get into is always predicting what kind of answer you're
going to get, using estimation.  For example, if I'm going to multiply
75 by 259, I might think:  that's on the order of 100 times 200, or
20,000.  So if I get something a lot different than that, I have a
problem.  (The actual result is 19,425.)

A second thing you can do to develop number sense is to get in the
habit of always looking for simpler ways to do calculations, often by
changing representations.  For example, suppose I need to do this
division:
        _____
  0.02 ) 44.8

One way to approach this is to scale both numbers by the same amount,
to get
     _____
  2 ) 4480

Now I have a much simpler problem.  I could also arrive at it this way:
        _____    44.8
  0.02 ) 44.8  = ----
                 0.02

                 448/10
               = -------
                   2/100

                 448   100
               = --- * ---
                  10    2   

                 448   100
               = --- * ---
                  2     10  

A good measure of your number sense is probably the number of
different ways that you can think of to do any particular calculation.
A nice way to motivate this is to look for ways to do calculations in
your head. 

Here are a few more examples:

  1) 20 * 17 = (10 * 2) * 17

             = 10 * (2 * 17)

             = 10 * 34

             = 340

  2)   243 - 87

     = 246 - 90         <- Add 3 to each term

     = 256 - 100        <- Add 10 to each term

     = 156      
       
            _____
  3)    52 ) 2340

            _____
     = 104 ) 4680             <- Double each of them

                  _____
     = about 100 ) 4600       <- Make each a little smaller
                                 to get nicer numbers

     = about 46

     (Real answer is 45)

I hope this helps.  Write back if you'd like to talk more about this,
or anything else. 

- Doctor Ian, The Math Forum
  http://mathforum.org/dr.math/ 

Date: 08/29/2007 at 19:54:06
From: Sarah
Subject: Thank you (What does the term "number sense" refer to)

Thanks so much!  This makes "sense"!  I really appreciate you taking
the time to answer my questions in such depth and breadth.  This is a
super web site!!

Gratefully,

Sarah