Marsbarn Designs...

Evolution of arrays, or why they aren't just one trick ponies

I have been a slow (okay, glacial) convert to arrays.
They appear early in the Common Core curriculum. 
This is what they look like in third grade, where they
are used for basic multiplication skills.

Then, they move onto this in fourth grade:

They start out on grid paper, and, in the beginning,
students are encouraged to manually count squares.
The numbers being multiplied get larger, and are 
broken down into tens and ones.  Students are shown
that larger numbers can be broken down into components,
which can be multiplied and then added together.
Often, these components can be easier to work with
mentally. It's easier to think about 10x8 and 4x8 and 
adding them (80+32) than it is to mentally multiply
14x8.

Then they move away from arrays done to scale, 
where you can count grid squares, to skeleton arrays
that require multiplication work.  Again, larger numbers
are broken down into units of tens and ones, and those 
pieces are added together. Is this shorter than doing 
an algorithm?  No. BUT, this does demonstrate the 
varied way that we can pull apart and reassemble 
multiplication problems, which is HUGE for algebra.

Then, we start to move away from the visual, to
the number driven math:

(I wrote in little number number sentences
to show where the product in each cell came from.)

Here's a purely algebraic version, to illustrate where
the values in each square come from.

Oh, look, distributive property and algebra.
Hey, we're in middle school now!  8th grade, perhaps?

We've got our factored quadratic, and we're going
into standard form.  And, voila, it's work we've been
doing since 4th grade.  Isn't that nifty??
This is where we would foil in old school math.

If foiling is your jam, then you don't need this, but
if you are a student who looses track of what's been
multiplied and what hasn't, this is an easier to follow
method. (And yes, it works in reverse. You can use this 
method to factor a quadratic, but that's a different post.)

I'm not quite done.  This method continues to 
evolve.  Take the following:

Oh yes, imaginary numbers!  This is big kid, high school
math, with things I can't visualize! Three factors being 
multiplied by three factors! But, our array can handle 
it. And look at how pretty it is! I can see my pairs 
that cancel so clearly!

The symmetry and cleanness of this method makes
my little dyslexic heart SO happy.  SO HAPPY.
Now, look below at the method I learned in school.

You tell me this foiling method is better.  That mess 
above.. that is how mistakes are made. That is how
elements are lost and math students cry.  This is where
the new method is faster, neater, kinder, and sweeter.

Do I always love arrays? No.  Sometimes they are slow,
clunky square wheels that make me say, "Put more 
math in your math!" to my students, BUT, it has a 
purpose and it is a great, powerful tool to have. 

Arrays, or one of those mysteries parents don't understand

My main, number one, burning hot issue with common
core, is the alienation of parents in the curriculum.  The
"you can't possibly understand the intricacy of what we're
doing with your child" tone of the math roll out. But here's
the thing, if my fourth grader can understand this and
embrace it, why can't I?  

Take the array. An array is simply a grid which is 
colored in to represent mathematical situations.

 

The array is used in many different configurations in 
this "new" math world. It can be used for multiplication,
division,fractions, square numbers, square roots, decimals,
and factors. Oh, there are more ways to use it than that,
but these are just from the top of my  head.  In short,
arrays are a simple but versatile visual math tool.

They can be expanded to larger multiplication problems to 
demonstrate the commutative property of multiplication.

The purpose of the array above it to try and demonstrate
how more difficult multiplication problems can be broken
down to easier products, which are added together.
Is it successful?  Well, that depends on how you learn. 
No one-sized method fits all.

Here is an example of arrays being used to show
equivalent fractions. Again, a visual method to teach
what, in the past, was taught with numbers.

I primarily see arrays in elementary school, but they do 
pop up in middle school.  They can be used to demonstrate
the concept of square numbers and square roots, for 
instance.

(There is a rather beautiful introduction to the Pythagorean
theorem which uses them.  That particular example needs
a post of its own.)

Is it really the bomb? Meh. I have personal issues with it.  
I like the visual element of teaching math, but I think the
curriculum spends too long on it, and not enough time
transitioning to  number based math.  I see kiddos in 6th
grade who want to draw arrays for fractions or
multiplication, instead of using numbers, and honestly
there is a time when the array holds you back.  It is also
slow, and that can be discouraging to kids who are ready
to move forward.

Arrays are a tool.  I'm a woman who likes tools.  They
don't have to be fix-alls, they just have to have a purpose,
and they shouldn't be mysterious.