“Number Boxes”

A few weeks ago I was introduced to Jenna Laib‘s game “Number Boxes” and was very interested in using it as a dynamic game to help students learn a variety of new content — Jenna’s blog explaining the game can be found here: “One of My Favorite Games: Number Boxes“.

Basically the game involves students rolling dice (or spinning a spinner / drawing a card) to generate a random number and placing that number in one of their empty number boxes one-at-a-time. The game can progress in a variety of ways:

Rolling 1 number at a time, create the largest number you can.

Rolling one number at a time, create 2 numbers that will add to the largest number.

Rolling one number at a time, create an expression that is as close to 2000 as possible.

As you can see, the game is quite adaptive to the sizes of numbers and concepts your students are comfortable with. As students roll/spin/draw a number, they have to place it on the board. What makes this tricky is not knowing what future numbers will be. In the board above, you can see that there is also a “Throwaway” box that students can use if they do not like one of the numbers rolled/spun/drawn. This game is an excellent example of a “Dynamic Game” or “Dynamic Practice” as students are following the ideals on the right side of the chart below:

Originally published here

Blow is a gallery of some possible adaptations of this game or linked here is a slideshow

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Metric Conversions

I however, wanted to use Jenna’s game to help students practice a concept they often have difficulty with – Metric Conversions. Once students have had many opportunities to estimate and measure various distances, capacities, and masses, they should be able to start making connections between all of the units. I suggest a good balance between using problems that help students make sense of the relationships between the units, and opportunities to practice conversions on their own. However, instead of randomly generated worksheets or other rote practice, I think Jenna’s game could work perfectly. Take a look at some examples:

Rolling one number at a time, find the largest total distance possible

Rolling one number at a time, find the largest total mass possible

Rolling one number at a time, find the largest possible distance

Rolling one number at a time, how close to 5km can you get?

Reflection

It is important to offer tasks that allow students to make choices and decisions like the ones offered in this game. Learning needs to be more than handing out assignments, and collecting work… Learning takes time! Students need more time to explore, see what works, have peers challenge each others’ thinking, make important connections… Hopefully you can see these opportunities in this task.

Final Thoughts:

• If you play one of these games, or your own version, will you first offer a simplified version so your students get familiar with the game, or will you dive into the content you want to teach?
• Would you prefer your students to play this game as a class or with a group, a partner, or independently?
• How will you build in conversations with students so they discuss which numbers they think should be the highest / lowest numbers? How will you offer time for these strategic discussions?
• Should we adapt these to continually offer more challenge and deeper learning, or offer more opportunities to play the same game board? How will we know when to adapt and change?
• What does “practice” look like in your classroom?  Does it involve thinking or decisions?  Would it be more engaging for your students to make practice involve more thinking?
• How does this game relate to the topic of “engagement”?  Is engagement about making tasks more fun or about making tasks require more thought?  Which view of engagement do you and your students subscribe to?
• How have your students experienced measurement concepts like these? Are they learning procedural rules or are they thinking about the actual sizes of numbers / sizes of the units involved?
• As the teacher, what will you be doing when students are playing? How might listening to student thinking help you learn more about your students? (See: Noticing and Wondering: A powerful tool for assessment)

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

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:

• 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