Spatial Puzzles: Cuisenaire Cover-ups

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 then be 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…

Earlier this year I shared a post discussing how we might decompose and recompose numbers to do an operations (subtraction). But, I would like us to consider why some students are more comfortable decomposing and recomposing, and how we might be aiming to help our students early with experiences that might promote the kinds of thinking needed.

Doug Clements and Julie Sarama have looked at the relationship between students’ work with space and shapes with students understanding of numbers.

“The ability to describe, use, and visualize the effects of putting together and taking apart shapes is important because the creating, composing, and decomposing units and higher-order units are fundamental mathematics. Further, there is transfer: Composition of shapes supports children’s ability to compose and decompose numbers”

Contemporary Perspectives on Mathematics in Early Childhood Education p.82, Clements and Sarama

The connection between composing and decomposing shapes and numbers is quite exciting to me. However, I am also very interested in the meeting place between Spatial tasks (composing/decomposing shapes) and Number tasks that involve composing and decomposing.

A few years ago I found a neat little puzzle in a resource called The Super Source called “Cover the Giraffe”. The idea was to cover an image of a giraffe outline using exactly 1 of each size of cuisenaire rods. The task, simple enough, was actually quite difficult for students (and even for us as adults). After using the puzzle with a few different classes, I decided to make a few of my own.

After watching a few classrooms of students complete these puzzles, I noticed an interesting intersection between spatial reasoning, and algebraic reasoning happening…. First, let me share the puzzles with you:

Objective:

To complete a Cuisenaire Cover-Up puzzle, you need exactly 1 of each colour cuisenaire rod. Use each colour rod once each to completely fill in the image.

Below are the 5 puzzles:

Assessment Opportunities

Knowing what to look for, helps us know how to interact with our students.

  • Which block are students placing first? The largest blocks or the smallest?
  • Which students are using spatial cues (placing rods to see which fits) and which students are using numerical cues (counting units on the grid)? How might we help students who are only using one of these cueing systems without over-scaffolding or showing how WE would complete the puzzle?
  • How do our students react when confronted with a challenging puzzle?
  • Who is able to swap out 1 rod for 2 rods of equivalent length (1 orange rod is the same length as a brown and red rod together)?
  • Which of the following strands of proficiency might you be noting as you observe students:
Adding It Up, 2001

Questions to Reflect on:

  • Why might you use a task like this? What would be your goal?
  • How will you interact with students who struggle to get started, or struggle to move passed a specific hurdle?
  • How might these puzzles relate to algebraic reasoning? (try to complete one with this question in mind)
  • How are you making the connections between spatial reasoning and algebraic reasoning clear for your students to see? How can these puzzles help?
  • How might puzzles allow different students to be successful in your class?

I’d love to continue the conversation about how we can use these puzzles to further our students’ spatial/algebraic reasoning.  Leave a comment here or on Twitter @MarkChubb3

If interested in these puzzles, you might be interested in trying:

Decomposing & Recomposing – How we subtract

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:

52 is decomposed into 40+10+2
19 is decomposed into 10+9
The problem is recomposed into (40-10) + (12-9)

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:

Maintain 52
Decompose 19 into 10+9
Subtract 52-10 (landing on 42), then 42-9
Some students will further decompose 9 as 2+7 and recompose the problem as 42-2-7
Maintain 52
Decompose 19 as 20-1
Recompose the problem as 52-20+1
Decompose 52 as 49-3
Recompose the problem as 49-19+3

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:

I’d love to continue the conversation about assessment in mathematics.  Leave a comment here or on Twitter @MarkChubb3