Throughout mathematics, the idea that objects and numbers can be decomposed and recomposed can be found almost everywhere. I plan on writing a few articles in the next while to discuss a few of these areas. In this post, I’d like to help us think about how and why we use visual representations and contexts to help our students make sense of the numbers they are using.
Decomposing and Recomposing
Foundational to almost every aspect of mathematics is the idea that things can be broken down into pieces or units in a variety of ways, and be then recomposed again. For example, the number 10 can be thought of as 2 groups of 5, or 5 groups of 2, or a 7 and a 3, or two-and-one-half and seven-and-one-half…
Understanding how numbers are decomposed and recomposed can help us make sense of subtraction when we consider 52-19 as being 52-10-9 or 52-20+1 or (40-10)+(12-9) or 49-19+3 (or many other possibilities)… Let’s take a look at how each of these might be used:
The traditional algorithm suggests that we decompose 52-19 based on the value of each column, making sure that each column can be subtracted 1 digit at a time… In this case, the question would be recomposed into (40-10)+(12-9). Take a look:
While this above strategy makes sense when calculating via paper-and-pencil, it might not be helpful for our students to develop number sense, or in this case, maintain magnitude. That is, students might be getting the correct answer, but completely unaware that they have actually decomposed and recomposed the numbers they are using at all.
Other strategies for decomposing and recomposing the same question could look like:
The first problem at the beginning was aimed at helping students see how to “regroup” or decompose/recompose via a standardized method. However, the second and third examples were far more likely used strategies for students/adults to use if using mental math. The last example pictured above, illustrates the notion of “constant difference” which is a key strategy to help students see subtraction as more than just removal (but as the difference). Constant difference could have been thought of as 52-19 = 53-20 or as 52-19 = 50-17, a similar problem that maintains the same difference between the larger and smaller values. Others still, could have shown a counting-on strategy (not shown above) to represent the relationship between addition and subtraction (19+____=53).
Why “Decompose” and “Recompose”?
The language we use along with the representations we want from our students matters a lot. Using terms like “borrowing” for subtraction does not share what is actually happening (we aren’t lending things expecting to receive something back later), nor does it help students maintain a sense of the numbers being used. Liping Ma’s research, shared in her book Knowing and Teaching Elementary Mathematics, shows a comparison between US and Chinese teachers in how they teach subtraction. Below you can see that the idea of regrouping, or as I am calling decomposing and recomposing, is not the norm in the US.
Visualizing the Math
There seems to be conflicting ideas about how visuals might be helpful for our students. To some, worksheets are handed out where students are expected to draw out base 10 blocks or number lines the way their teacher has required. To others, number talks are used to discuss strategies kids have used to answer the same question, with steps written out by their teachers.
In both of these situations, visuals might not be used effectively. For teachers who are expecting every student to follow a set of procedures to visually represent each question, I think they might be missing an important reason behind using visuals. Visuals are meant to help our students see others’ ideas to learn new strategies! The visuals help us see What is being discussed, Why it works, and How to use the strategy in the future.
Teachers who might be sharing number talks without visuals might also be missing this point. The number talk below is a great example of explaining each of the types of strategies, but it is missing a visual component that would help others see how the numbers are actually being decomposed and recomposed spatially.
If we were to think developmentally for a moment (see Dr. Alex Lawson’s continuum below), we should notice that the specific strategies we are aiming for, might actually be promoted with specific visuals. Those in the “Working with the Numbers” phase, should be spending more time with visuals that help us SEE the strategies listed.
Aiming for Fluency
While we all want our students to be fluent when using mathematics, I think it might be helpful to look specifically at what the term “procedural fluency” means here. Below is NCTM’s definition of “procedural fluency” (verbs highlighted by Tracy Zager):
Which of the above verbs might relate to our students being able to “decompose” and “recompose”?
Some things to think about:
- How well do your students understand how numbers can be decomposed and recomposed? Can they see that 134 can be thought of as 1 group of 100, 3 groups of 10, and 4 ones AS WELL AS 13 groups of 10, and 4 ones, OR 1 group of 100, 2 groups of 10, and 14 ones…….? To decompose and recompose requires more than an understanding of digit values!!!
- How do the contexts you choose and the visual representations you and your students use help your students make connections? Are they calculating subtraction questions, or are they thinking about which strategy is best based on the numbers given?
- What developmental continuum do you use to help you know what to listen for?
- How much time do your students spend calculating by hand? Mentally figuring out an answer? Using technology (a calculator)? What is your balance?
- How might the ideas of decomposing and recomposing relate to other topics your students have learned and will learn in the future?
- Are you teaching your students how to get an answer, or how to think?
If you are interested in learning more, I would recommend:
- The Importance of Contexts and Visuals
- How to make math visual
- The difference between Strategies and Models
I’d love to continue the conversation about assessment in mathematics. Leave a comment here or on Twitter @MarkChubb3