An area of mathematics I wish more students had opportunities to explore is spatial/visualization. There are many studies that show just how important spatial/visual reasoning is for mathematical success (I discuss in more depth here), but often, we as teachers aren’t sure where to turn to help our students develop spatial reasoning, or now to make the mathematics our students are learning more spatial.
One such activity I’ve suggested before is Skyscraper Puzzles. I’ve shared these puzzles before (Skyscraper Puzzles and Skyscraper Templates – for relational rods). With the help of my own children, I decided to make new templates. The package includes a page dedicated to explain how to solve the puzzles, as well as instructions on each page.
For details about how to solve a Skyscraper Puzzle, please click here
*The above files are open to view / print. If you experience difficulties accessing, you might need to use a non-educational account as your school board might be restricting your access.
You’ll notice in the package above that some of the puzzles are missing information like the puzzle below:
Puzzles like these might include information within the puzzle. In the puzzle above, the 1 in the middle of the block refers to the height of that tower (a tower with a height of 1 goes where the 1 is placed).
You might also be interested in watching a few students discussing how to play:
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 physical blocks be different than paper-and-pencil or electronic versions?
If students are working on these away from school, what resources might you use instead of school manipulatives like snap cubes? Lego bricks? Checkers? Other household items?
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.
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:
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.
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?
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 ).
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.
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:
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 42-2-7Maintain 52 Decompose 19 as 20-1 Recompose the problem as 52-20+1Decompose 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:
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:
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?
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.
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.
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 ).
If you are interested in all of the slides, you can take a look here
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:
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?
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?
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!
Throughout this year I have been thinking a lot about perspective. It’s why I wrote a post about Spatial Reasoning here, and shared templates for skyscraper puzzles here and why I shared my presentation at this year’s OAME that included perspective here. It’s also why I shared videos like this or this or this.
So why is perspective so important? Mathematically, perspective taking involves us being able to mentally rotate objects in our mind, it includes us composing and decomposing shapes and figures into other shapes and figures… Really, perspective taking is one small piece of what Spatial Reasoning (Read this monograph to learn more) is all about:
Another look at perspective:
Have you ever been in a situation before where somebody is discussing with you their students – the ones you will be getting next year? Was it enlightening or awkward, helpful or fear-inducing? In my buildings, this is the time of year when lots of decisions start to happen for next year. Administrators moving to new schools and new ones coming, new teachers being hired, experienced teachers being given new assignments within the school or in a new school, and of course, students about to be promoted to the next grade. And with change comes uncertainty and anxiety.
In all of this uncertainty we often have opportunities to discuss with others some of our personal thoughts and feelings about our year, those who we work with, those who we teach…… So I thought I would challenge you to think about what your typical conversations look like.
A few things to think about when discussing others in your building:
Do you tend to see the good in others and describe things in a positive light or aim to help others see the potential issues and risks?
Do you tend to talk in generalities or specifics?
How will the other teacher/administrator perceive the messages you have given?
What information would you like to hear from others? How specific and detailed of a description would you like?
Most importantly, where is your line? You know, the line where it’s too much information and is either starting to cause you more stress or where you realize that the information is too negative.
A challenge for you:
What conversations have you had lately to describe to others about any of these changes? I think it can be powerful to hear both positive and negative experiences so we can think about how to navigate through changes.
If you are changing roles, or schools, or even getting new students, what conversations have you had? What might you have done differently?
For the last few months, a team of kindergarten teachers and myself have been working together to deepen our understanding of early years mathematics, spatial reasoning, and the importance of guided play as a vehicle to engage our students to think mathematically. Below is a copy of our slideshow presentation we shared at OAME 2017, and some of the documents we have created over the past few months.
A quick synopsis of our work first:
While our research led us toward Doug Clement’s work about trajectories, and research about spatial reasoning and early mathematics, much of the tasks we actually did with students directly came from the book shown above (Taking Shape) which I can’t recommend enough – if you can, get yourself a copy!
We discussed the quote above to help us realize what actually underpins mathematics success. More details about how the quote ends here.
We shared research showing just how important early mathematics is, and specifically what the kinds of instruction could / should look like to accomplish this learning. Duncan et. al., is a widely quoted piece of research that has led many to realize that early math learning needs to be a focus in schools – even more so than early reading!
We played a few games that helped us stretch our spatial reasoning abilities. The image above was part of our “See It, Build It, Check It” activity (found in Taking Shape). Everyone saw the image for a minute, then was asked to build it once the image was removed. What we noticed is just how difficult spatial tasks are for us!
After we had the opportunity to play for a bit, we dug back into the research about spatial reasoning and the jobs typically chosen based on (high school) spatial ability. Hopefully you noticed something interesting on the graph above!
So, we know how important spatial reasoning is, but the 3 pieces above (taken from Paying Attention to Spatial Reasoning document) might help us realize how important a focus on spatial reasoning is for both our students and us.
In our time together, we learned a lot about the importance of observing our students as they were engaged in the learning. From the initial choices they made, to how they overcame obstacles, to understanding the mental actions that were happening… Observing students in the moment is far more powerful than collecting correct answers!
See the link at the bottom of the page for our connections to Doug Clements’ work.
We also discussed the specific connections between the mathematics behaviours and the learning that happened beyond. In our Kindergarten program document, our students’ expectations fall under 4 frames (see above) so we linked the learning we saw to the program document in a way that helps us see the depth and breadth of the kindergarten program (see document linked below).
We then ended our presentation with a synopsis of what we learned throughout our work together. While the slideshow might be helpful here (I’d love for someone to comment on those slides at the end), the conversations that Sue, Kristi and Kristen had with others have shown me just how valuable it is to spend time learning together. I couldn’t be prouder to be able to work with such reflective and dedicated teachers!
I’ve been spending a lot of time lately observing students who struggle with mathematics, talking with teachers about their students who struggle, and thinking about how to help. There are several students in my schools who experience difficulties beyond what we might typically do to help. And part of my role is trying to think about how to help these students. It seems that at the heart of this problem, we need to figure out where our students are in their understanding, then think about the experiences they need next.
However, first of all I want to point out just how difficult it is for us to even know where to begin! If I give a fractions quiz, I might assume that my students’ issues are with fractions and reteach fractions… If I give them a timed multiplication test, I might assume that their issues are with their recall of facts and think continued practice is what they need… If I give them word problems to solve, I might assume that their issue is with reading the problem, or translating a skill, or with the skill itself……. Whatever assessment I give, if I’m looking for gaps, I’ll find them!
So where do we start? What are the foundations on which the concepts and skills you are doing in your class rest on? This is an honest question I have.
Take a look at the following quote. How would you fill in the blank here?
Really, take a minute to think about this. Write down your thoughts. In your opinion, what are the foundational skills of math? Why do you believe this? This is something I’ve really been reflecting on and need to continue doing so.
I’ve asked a few groups of teachers to fill in the blank here in an effort to help us consider our own beliefs about what is important. To be honest, many of the responses have been very thoughtful and showcased many of the important concepts and skills we learn in mathematics. However, most were surprised to read the full paragraph (taken from an explanation of dyscalculia).
Is this what you would have thought? For many of us, probably not. In fact, as I read this, I was very curious what they meant by visual perception and visual memory. What does this look like? When does this begin? How do we help if these are missing pieces later?
Think about it for a minute. How might you see these as the building blocks for later math learning? What specifically do these look like? Here are two excerpts from Taking Shape that might help:
Excerpt from Taking Shape
Excerpt from Taking Shape
Visual perception and visual memory are used when we are:
Perceptual Subitizing (seeing a small amount of objects and knowing how many without need of counting)
Conceptual subitizing (the ability to organize or reorder objects in your mind. To take perceptual subitizing to make sense of visuals we don’t know)
Comparing objects’ sizes, distances, quantities…
Composing & decomposing shape (both 2D or 3D)
Recognizing, building, copying symmetry designs (line or rotational)
Recognizing & performing rotations & reflections.
Constructing & recognizing objects from different perspectives
Orienting ourselves, giving & following directions from various perspectives.
Visualizing 3D figures given 2D nets
While much of this can be quite complicated (if you want, you can take a quick test here), for some of our students the visual perception and visual memory is not yet developed and need time and experiences to help them mature. Hopefully we can see how important these as for future learning. If we want our students to be able to compose and decompose numbers effectively, we need lots of opportunities to compose and decompose shapes first! If we want our students to understand quantity, we need to make sure those quantities make sense visually and conceptually. If we want students to be flexible thinkers, we need to start with spatial topics that allow for flexibility to be experienced visually. If we want students to be successful we can’t ignore just how important developing their spatial reasoning is!
In our schools we have been taking some of the research on Spatial Reasoning and specifically from the wonderful resource Taking Shape and putting it into action. I will be happy to share our findings and action research soon. For now, take a look at some of the work we have done to help our students develop their visual perception and visual memory:
Symmetry games:
Composing and decomposing shapes:
Relating nets to 3D figures:
Constructing unique pentominoes:
And the work in various grades continues to help support all of our students!
So I leave you with a few questions:
What do you do with students who are really struggling with their mathematics? Have you considered dyscalculia and the research behind it?
How might you incorporate spatial reasoning tasks / problems for all students more regularly?
Where in your curriculum / standards are students expected to be able to make sense of things visually? (There might be much more here than we see at first glance)
How does this work relate to our use of manipulatives, visual models and other representations?
What do we do when we notice students who have visual perception and visual memory issues beyond what is typical?
I spent some time with a wonderful kindergarten teacher a while back. The two of us were chatting when a little boy, probably 4 or 5 years old walked up to us with a inquisitive look and asked if it was home time yet. Since it was only about 10 in the morning I smiled at the question. But the teacher in a very calm and candid manner spread his arms out wider than shoulder width and said, “There’s still THIS much time left.“
The child, very satisfied with the response, happily headed over to another activity and continued to play. I, however, was quite stunned at what just transpired. The little boy wasn’t given an answer… or was he?
What was it about the gesture given that helped this child understand that there was still lots of time left in the day? I found myself quite interested in the above conversation for some time until I started thinking about the types of experiences our students have with time as young children.
Clocks like those below don’t have any meaning for most 4 or 5 year-olds. There are far too many numbers or symbols… different ways to read… for many to understand here!
But they have likely had many experiences looking at things like these:
Let’s think about these items for a minute. Children by age 4 or 5 have likely been exposed to a cell phone, a tablet or some type of electronic device that has a battery meter. Without any understanding of percent or without even being able to recognize large numbers, children instinctively know when the device is charged or when it is nearly “dead”. When they watch movies, they can determine if we are currently near the beginning or end based on the progression meter (when they hit pause of fast-forward)… or when downloading, they can tell by how much space is left before they can access the app or game… Even when waiting in a line-up, children can tell if it will be a short time or not without having to count the people in front of them.
Each of the pictures above show our students’ natural ability to determine information based on the relative space provided. This is a great example of how we use Spatial Reasoning to make sense of things.
There is a lot of research about Spatial Reasoning and it’s relationship with mathematics. Much of this idea was first brought to my attention through the article: Paying Attention to Spatial Reasoning. While you probably haven’t read this, I think it is really important to read through the whole article. Really… it will explain a lot better than I can.
If our students come into our math classes with an intuitive understanding of the spatial relationships behind concepts, shouldn’t we provide experiences that allows our students to access this form of reasoning BEFORE we start with formal rules, procedures, notations…?
For instance, how might a simple graph like this bring about a conversation of subtraction?
How might a number line be used so our students can visually see how subtraction works? And how subtraction is related to addition?
It is blurry, but the 132-94 was moved to show 138-100. How might this visual help us with future problems?
Or, if we know Mr. Stadel’s height is 193cm. How might that help us think about estimating Mrs. Stadel’s height?
How did you think about this? What spatial cues did you use? Can you see how others might see this as a subtraction or addition problem?
Spatial reasoning isn’t just for subtraction though. If we wanted to know how many chocolates were here what might students do that are just starting to learn about multiplication?
How many different ways can you think of solving this question?
An example that really illustrates the need to focus on Spatial Reasoning:
At the end of the year, the grade 7 students throughout our school board were given a year-end test. One of the questions our students did poorest on was this problem:
While the vast majority of students got this problem wrong because of issues with converting units, several students instead took a path that allowed them to make sense of the problem. Below is a diagram we saw a few students draw that represented what was going on in the problem quite nicely. What did they do here? What do you notice?
I notice dimensions that make sense. I notice a piece of cloth being cut up, much like what we would actually do if this were a real-life problem. I notice the student thinking about the space required to cut the bandanas.
So I am left wondering why did only a few students attempt to make a drawing? Why did so many make errors with converting between units? Why did so many students make errors that were not even close to an acceptable answer? What experiences did they have… and didn’t they have that led them to attempt this problem the way they did?
While many teachers might tell students that there are 100 x 100 cm in a square meter, and expect students to understand and remember… or show conversion charts… or offer a page of conversion questions… I think we might be missing a big piece of the learning process. We haven’t tapped into our students’ spatial reasoning at all!
After seeing so many questions like this attempted by students in such procedural ways, I think next year we will start to think more about the reasoning our students already come into our classes with. And continue to think about how we can support the development of our students Spatial Reasoning is in helping them make sense of things!
The National Research Council describes the current situation as a “major blind spot”in education and maintains that, without explicit attention to spatial thinking, the concepts, tools and processes that underpin it “will remain locked in a curious educational twilight zone: extensively relied on across the K–12 curriculum but not explicitly and systematically instructed in any part of the curriculum” (p. 7).
Thinking Mathematically 