One-Hole Punch Puzzle Templates

Recently Lowrie et al. published an article in the Journal of Experimental Education where they looked at the effects of a spatial intervention program for grade 8 students. This study followed the 876 grade 8 students across 9 schools as they received 20 hours of spatial interventions (as well as other grade 8 students in schools that conducted “business as usual” mathematics classes). Their findings were quite interesting. They found students who received spatial intervention programs achieved:

• significantly better on spatial tasks (13% higher)
• significantly better on Geometry – Measurement problems that included material not discussed in either class
• significantly better on Number – Algebra problems
• equally well on Statistic – Probability problems

While this study supports many studies showing a link between spatial abilities and mathematics performance (Mix and Cheng, 2012), and others showing how students’ spatial abilities are malleable at any age (can be learned with the right experiences) (Uttal et al., 2013), there seems to be a definite need for teachers to have at their fingertips good examples of experiences that will help our students develop spatial reasoning. At the end of this post are possible examples for us to try.

A recent meta-analysis of 217 studies, representing more than two decades of research on spatial training, found that a variety of activities improve spatial reasoning across all age groups. Not only did the authors find that spatial training led to improvements on spatial tasks closely related to the training task, but improvements were also seen on other types of tasks that were not part of the training.

Taking Shape, 2016

A focus on spatial reasoning, from my experience, has helped the students in my schools make sense of connections between concepts, it has been the underpinning of new learning, and has been the vehicle for so much of OUR learning as educators.

One-Hole Punch puzzles

You might be familiar with various cognitive tests that ask students to think through mental rotations, 2D/3D visualization, paper folding or other tasks where students’ abilities to visualize are measured. Spatial tasks like these are very predictive of a students’ math success, however, few resources are aimed to specifically help our students develop their spatial reasoning (see Taking Shape as an ideal K-2 example).

Below is an example of one of these cognitive test questions. As you can see, students here are asked to mentally fold a piece of paper and then punch 1 hole through the folds, then imagine what the paper would look like once opened.

Templates

Instead of trying to measure our students’ abilities by giving sample problems like this, it is far more productive to offer experiences where our students can learn to think spatially. This is why I have created a few sample experiences. Below are sample templates that you can use with your students. They will need squares of paper (linked below), as well as the puzzles they would be aiming to solve. Take a look:

• One-Hole Punch puzzle templates (use this link if first is restricted)
• 2 by 2 squares with cut lines
• 3 by 3 squares with cut lines

How to Solve a One-Hole Punch puzzle:

Directions for solving the puzzles are only written on the first page:

“These are One-Hole Punch puzzles.  To complete a puzzle, take a square of paper, fold it using as many folds as needed so that if you punch one hole and unfold it, you will match one of the puzzles.  Complete the following puzzles in any order.  Be prepared to discuss your strategies with others.”

Students will need several blank squares of paper and a single hole puncher. Student will need to visualize how to fold their paper and where to punch their one hole to match the puzzle they are working on. These puzzles are tricky because only 1 hole can be punched to create several holes.

Pro Tip: Make sure you have a lot of squares of paper for every student.

A few thoughts about using these:

• How will you introduce these puzzles to your students?  How much information about strategies and tips will you provide?  Will this allow for productive struggle, or will you attempt to remove as much of the struggle as possible?
• Would you use these as an activity you give all students, or something you provide to just some.  Why? (Something for students who finish early or something for everyone to try?)
• How would giving a page of puzzles to a pair of students be different than if you gave it to individuals?  Which were you assuming to do here?  What if you tried the other option?
• How will you orchestrate a conversation for your class to help consolidate the learning here? How might this be helpful before you give the next set of puzzles?
• What will you do if students give up quickly?  What questions / prompts will you provide?
• Who is successful at solving these puzzles? Sometimes, those who are often left out of mathematical discussions can become more involved when given spatial tasks. How might you capitalize on increasing your students’ interest in mathematics? Do you see tasks like this helping improve your students’ agency, identity and authority in mathematics?
• How might you see intentional decisions to focus on spatial reasoning as helping your students with mathematics in other areas? This post might be helpful.

A belief I have: Teaching mathematics is much more than providing neat things for our students, it involves countless decisions on our part about how to effectively make the best use of the problem / activity.  Hopefully, this post has helped you consider your own decision making processes!

As always, I’d love to hear your thoughts. Feel free to write a response here or send me a message on Twitter ( @markchubb3 ).

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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:

• Cover the Dog
• Cover the Elephant
• Cover the Giraffe (edited from the SuperSource)
• Cover the Rocket
• Cover the Sailboat

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:

• Zukei Puzzles
• Skyscraper Templates for Relational Rods
• Creating Voronoi
• Skyscraper Templates
• Making Math Visual

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

“Making Math Visual”

A few days ago I had the privilege of presenting at OAME in Ottawa on the topic of “Making Math Visual”.   If interested, here are some of my talking points for you to reflect on:

To get us started, we discussed an image created by Christopher Danielson and asked the group what they noticed:

We noticed quite a lot in the image… and did a “how many” activity sharing various numbers we saw in the image.  After our discussions I explained that I had shared the same picture with a group of parents at a school’s parent night followed by the next picture.

The picture above was more difficult for us as teachers to see the mathematics. While we, as math teachers, saw patterns in the placements of utensils, shapes and angles around the room, quantities of countable items, multiplicative relationships between utensils and place settings, volume of wine glasses, differences in heights of chairs, perimeter around the table…..  the group correctly guessed that many parents do not typically notice the mathematics around them.

So, my suggestion for the teachers in the room was to help change this:

While I think it is important that we tackle the idea of seeing the world around us as being mathematical, a focus on making math visual needs to by MUCH more than this. To illustrate the kinds of visuals our students need to be experiencing, we completed a simple task independently:

After a few minutes of thinking, we discussed research of the different ways we use fractions, along with the various visuals that are necessary for our students to explore in order for them to develop as fractional thinkers:

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When we looked at the ways we typically use fractions, it’s easy to notice that WE, as teachers, might need to consider how a focus on representations might help us notice if we are providing our students with a robust (let’s call this a “relational“) view of the concepts our students are learning about.

Data taken from 1 school’s teachers:

Above you see the 6 ways of visualizing fractions, but if you zoom in, you will likely notice that much of the “quotient” understanding doesn’t include a visual at all… Really, the vast majority of fractional representations here from this school were “Part – Whole relationships (continuous) models”. If, our goal is to “make math visual” then I believe we really need to spend more time considering WHICH visuals are going to be the most helpful and how those models progress over time!

We continued to talk about Liping Ma’s work where she asked teachers to answer and represent the following problem:

As you can see, being able to share a story or visual model for certain mathematics concepts seems to be a relative need. My suggestion was to really consider how a focus on visual models might be a place we can ALL learn from.

We then followed by a quick story of when a student told me that the following statement is true (click here for the full story) and my learning that came from it!

So, why should we focus on making math visual?

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We then explored a statement that Jo Boaler shared in her Norms document:

…and I asked the group to consider if there is something we learn in elementary school that can’t be represented visually?

If you have an idea to the previous question, I’d love to hear it, because none of us could think of a concept that can’t be represented visually.

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I then shared a quick problem that grade 7 students in one of my schools had done (see here for the description):

Along with a few different responses that students had completed:

Most of the students in the class had responded much like the image above.  Most students in the class had confused linear metric relationships (1 meter = 100 cm) with metric units of area (1 meter squared is NOT the same as 100cm2).

In fact, only two students had figured out the correct answer… which makes sense, since the students in the class didn’t learn about converting units of area through visuals.

If you are wanting to help think about HOW to “make math visual”, below is some of the suggestions we shared:

And finally some advice about what we DON’T mean when talking about making mathematics visual:

You might recognize the image above from Graham Fletcher’s post/video where he removed all of the fractional numbers off each face in an attempt to make sure that the tools were used to help students learn mathematics, instead of just using them to get answers.

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I want to leave you with a few reflective questions:

• Can all mathematics concepts in elementary be represented visually?
• Why might a visual representation be helpful?
• If a student can get a correct answer, but can’t represent what is going on, do they really “understand” the concept?
• Are some representations more helpful than others?
• How important is it that our students notice the mathematics around them?
• How might a focus on visual representations help both us and our students deepen our understanding of the mathematics we are teaching/learning?
• Where do you turn to help you learn more about or get specific examples of how to effectively use visuals?

I’d love to continue the conversation.  Feel free to write a response, or send me a message on Twitter ( @markchubb3 ).

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If you are interested in all of the slides, you can take a look here

Skyscraper Templates – for Relational Rods

Many math educators have come to realize how important it is for students to play in math class. Whether for finding patterns, building curiosity, experiencing math as a beautiful endeavour, or as a source of meaningful practice… games and puzzles are excellent ways for your students to experience mathematics.

Last year I published a number of templates to play a game/puzzle called Skyscrapers (see here for templates) that involved towers of connected cubes. This year, I decided to make an adjustment to this game by changing the manipulative to Relational Rods (Cuisenaire Rods) because I wanted to make sure that more students are becoming more familiar with them.

Skyscraper puzzles are a great way to help our students think about perspective while thinking strategically through each puzzle.  Plus, since they require us to consider a variety of vantage points of a small city block, the puzzles can be used to help our students develop their Spatial Reasoning!

How to play a 4 by 4 Skyscraper Puzzle:

• Build towers in each of the squares provided sized 1 through 4 tall
• Each row has skyscrapers of different heights (1 through 4), no duplicate sizes
• Each column has skyscrapers of different heights (1 through 4), no duplicate sizes
• The rules on the outside (in grey) tell you how many skyscrapers you can see from that direction
• The rules on the inside tell you which colour rod to use (W=White, R=Red, G=Green, P=Purple, Y=Yellow)
• Taller skyscrapers block your view of shorter ones

Below is an overhead shot of a completed 4 by 4 city block.  To help illustrate the different sizes. As you can see, since each relational rod is coloured based on its size, we can tell the sizes quite easily.  Notice that each row has exactly 1 of each size, and that each column has one of each size as well.

To understand how to complete each puzzle, take a look at each view so we can see how to arrange the rods:

If you are new to completing one of these puzzles, please take a look here for clearer instructions: Skyscraper Puzzles

Relational Rod Templates

Here are some templates for you to try these puzzles yourself and with your students:

4 x 4 Skyscraper Puzzles – for Relational Rods

5 x 5 Skyscraper Puzzles – for Relational Rods

A few thoughts about using these:

• How will you introduce these puzzles to your students?  Would you use the cube version I shared first? How much information about strategies and tips will you provide?  Will this allow for productive struggle, or will you attempt to remove as much of the struggle as possible?
• Would you use these as an activity you give all students, or something you provide to just some.  Why? (Something for students who finish early or something for everyone to try?)
• How would giving a puzzle to a pair of students be different than if you gave it to individuals?  Which were you assuming to do here?  What if you tried the other option?
• How might using Relational rods be different than paper-and-pencil or electronic versions? How might it be different than the version with cubes?
• How will you orchestrate a conversation for your class to help consolidate the learning here?
• What will you do if students give up quickly?  What questions / prompts will you provide?

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A belief I have: Teaching mathematics is much more than providing neat things for our students, it involves countless decisions on our part about how to effectively make the best use of the problem / activity.  Hopefully, this post has helped you consider your own decision making processes!

I’d love to hear how you and/or your students do!