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

Problem of the Week

A number of children are standing equally spaced around the edge of a large circle and are numbered consecutively.  Abby is 14^th and Emma is 32^nd and are standing directly across from each other.  How many children are there?

In a week, we will post solutions for this problem.  At that time we will also choose one commenter who gave the correct answer to get a copy of Zeroing in on Number and Operations (the commenter who wins can choose from PreK-K, 1-2, 3-4, 5-6, or 7-8).  If you want to be entered into the drawing please comment with your solution, and leave your first name last initial.

We look forward to seeing your responses!

Katie

 

 

Student Misconceptions...

I often wonder how effective we can be as teachers if we don’t first get an indication of what students misunderstand. I did a lot of work with Dynamic Classroom Assessment, George Bright and Jeane Joyner’s work and they talk about four ways to think about students’ written responses. They differentiate among “Communicating an Understanding”, “Communicating a Misunderstanding”, “Miscommunicating an Understanding”, and “Miscommunicating a Misunderstanding”. If we think about student work in terms of these categories then I believe it also means we need to have multiple representations for which we can determine whether there is understanding or not, and if it is computational or procedural, conceptual or representational.

Think about the students who submitted the following work (I am recreating it here but have not changed any of the students’ notations):

Which of the following procedures results in a larger numerical answer?

1 ½ ÷ ¾   or 1 ½ x ¾

Student A: They are the same because division is just the opposite of multiplication.

Student B: 1 ½ x ¾ because multiplication always gives a bigger answer.

Student C: They are the same because in division you just change the sign and multiply.

What was interesting is the fact that none of the students actually computed the quotient or product to allow them to check their thinking.

In talking with these students I asked them what it means to divide. None of the three students was able to articulate that division is used to determine how many equivalent groups into which a dividend can be arranged, nor given a number of groups what makes a fair share for each of the groups.

I am convinced that conceptual understanding must be developed rather than giving students “cutesy” sayings like “keep, change, flip”.  How might we, as teachers, develop a stronger conceptual model to deepen student understanding of division and then division by a fraction?  Please post your thoughts and I will be reading them and posting a follow-up to this in the next couple of weeks.

Anne