Understanding what it means to understand the structure of mathematics (use the language of the mathematical practices)

Implementing the recently adopted Common Core State Standards requires looking at the vertical progressions of important concepts across the grades. One of these concepts is how arithmetic is generalized in algebra. The algorithms we emphasize in arithmetic may ease the transition to algebraic understanding, or thwart that transition. The Common Core talks about the "traditional" algorithm but not until students have opportunities and experiences to deeply understand the foundational concepts.

Think, for instance, about the addition of the following two-digit numbers:
27 + 33

The way in which the computation is presented may make a difference to students. Some students will see this computation and round 27 up to 30 and 33 down to 30 to get the sum of 60. Others will see this computation written horizontally and need to rewrite it vertically before they can compute:

27

+ 33

If they understand place value they may show partial addends illustrating the following:

27

+ 33

10

+ 50

60

Or show the regrouping method:

1
2	7
+ 3	3
6	0

Of these methods, which do you think best relates to addition in algebra?

Let's think about using place value representation to compute:
 2 tens + 7 + 3 tens + 3.

Written this way it is clear that tens will be combined with tens, and units with units, for a sum of 5 tens + 10. Notice there is no temptation to mix the tens and the units.

Written vertically we get

2 tens	+ 7
+ 3 tens	+ 3
5 tens	+ 10

It makes perfect sense to combine like terms.  This can also be written as
2t + 7 + 3t + 3, which is a standard algebraic representation and would total 5t + 10.

The partial sum best models the foundational method for addition in algebra. Now, this is not to say students should always use the partial sums, but they do help solidify what values are being combined. Once this understanding is mastered then we expect our students to use a more efficient method, especially if we are adding three or four columns and rows.

The same rationalization can be used with multiplication which may also include additional links or connections to the structure of algebra. Consider the same two values but this time think of them as factors:

27 x 33.

Using partial products we get

	2	7		20	+ 7
x	3	3		x 30	+ 3
	2	1		7 x 3 =	21
	6	0		20 x 3 =	60
2	1	0		7 x 30 =	210
6	0	0		20 x 30 =	600
8	9	1			891

Or we could think about decomposing 27 into 30 – 3, and 33 into 30 + 3, which would give partial products of 900 + 90 - 90 – 9, which equals 891.

30	- 3
x 30	+ 3
-3 x 3 =	-9
30 x 3 =	90
-3 x 30 =	-90
30 x 30 =	900
	891

The mathematical structure for this computation is the difference of two squares, a² – b². This is an arithmetic way of showing 27 x 33 = (30 – 3) x (30 + 3) = 30² – 3², which equals 891.

I propose that when we are teaching arithmetic we keep in mind the fact that algebra is the generalization of arithmetic.  What do think?

I look forward to your comments,

Anne