Many teachers are comfortable allowing their students to read for pleasure at school and encourage reading at home for pleasure too. Writing is often seen as a creative activity. Our society appreciates Literacy as having both creative and purposeful aspects. Yet mathematics as a source of enjoyment or creativity is often not considered by many.
I want you to reflect on your own thinking here. How important do you see creativity in mathematics? What does creativity in mathematics even mean to you?
Marian Small might explain the notion of creativity in mathematics best. Take a look:
Several years ago, Marian Small tried to help us as math teachers see what it means to think and be creative in mathematics by sharing 2 different ways for our students to experience the same content. She called them “type 1” and “type 2” questions.
Type 1 problems typically ask students to give us the answer. There might be several different strategies used… There might be many steps or parts to the problem. Pretty much every Textbook problem would fit under Type 1. Every standardized test question would fit here. Many “problem solving” type questions might fit here too.
Type 2 problems are a little tricky to define here. They aren’t necessarily more difficult, they don’t need a context, nor do they need to have more steps. A Type 2 problem asks students to get to relationships about the concepts involved. Essentially, Type 2 problems are about asking something where students could have plenty of possible answers (open ended). Again, here is Marian Small describing some examples:
Examples of Type 1 & 2 Questions
Notice that a type 2 problem is more than just open, it encourages you to keep thinking and try other possibilities! The constraints are part of what makes this a “type 2” problem! The creativity and interest comes from trying to reach your goal!
Where do you look for “Type 2” Problems?
If you haven’t seen it before, the website called OpenMiddle.com is a great source of Type 2 problems. Each involve students being creative to solve a potential problem AND start to notice mathematical relationships.
Remember, mathematically interesting problems (Type 2 problems) are interesting because of the mathematical connections, the relationships involved, the deepening of learning that occurs, not just a fancy context.
Questions to Reflect on:
When do you include creativity in your math class? All the time? Daily? Toward the beginning of a unit? The end? What does this say about your program? (See A Few Simple Beliefs)
If you find it difficult to create these types of questions, where do you look? Marian Small is a great start, but there are many places!
How might “Type 2” problems like these offer your students practice for the skills they have been learning? (See purposeful practice)
What is the current balance of q]Type 1 and Type 2 problems in your class? Are your students spending more time calculating, or deciding on which calculations are important? What balance would you like?
If students start to understand how to solve type 2 problems, would you consider asking your students to make up their own problems? (Ideas for making your own problems here).
How do these problems help your students build their mathematical intuitions? (See ideas here)
Would you want students to work alone, in pairs, in groups? Why?
If you have struggled with developing rich discussions in your class, how might these types of problems help you bring a need for discussions? How might this change class conversations afterward?
Data management is becoming an increasingly important topic as our students try to make sense of news, social media posts, advertisements… Especially as more and more of these sources aim to try to convince you to believe something (intentionally or not).
Part of our job as math teachers needs to include helping our students THINK as they are collecting / organizing / analyzing data. For example, when looking at data we want our students to:
Notice the writer’s choice of scale(s)
Notice the decisions made for categories
Notice which data is NOT included
Notice the shape of the data and spatial / proportional connections (twice as much/many)
Notice the choice of type of graph chosen
Notice irregularities in the data
Notice similarities among or between data
Consider ways to describe the data as a whole (i.e., central tendency) or the story it is telling over time (i.e., trends)
While each of these points are important, I’d like to offer a way we can help our students explore the last piece from above – central tendencies.
Central Tendency Puzzle Templates
To complete each puzzle, you will need to make decisions about where to start, which numbers are most likely and then adjust based on what makes sense or not. I’d love to have some feedback on the puzzles.
How will your students be learning about central tendencies before doing these puzzles? What kinds of experience might lead up to these puzzles? (See A Few Simple Beliefs)
How might puzzles like these offer your students practice for the skills they have been learning? (See purposeful practice)
What is the current balance of questions / problems in your class? Are your students spending more time calculating, or deciding on which calculations are important? What balance would you like?
If students start to understand how to solve one of these, would you consider asking your students to make up their own puzzles? (Ideas for making your own problems here).
How do these puzzles help your students build their mathematical intuitions? (See ideas here)
Would you want students to work alone, in pairs, in groups? Why?
Would you prefer all of your students doing the same puzzle / game / problem, or have many puzzles / games / problems to choose from? How might this change class conversations afterward?
TRU Math (Teaching for Robust Understanding) a few years ago shared their thoughts about what makes for a “Powerful Classroom”. Here are their 5 dimensions:
Looking through the dimensions here, it is obvious that some of these dimensions are discussed in detail in professional development sessions and in teacher resources. However, the dimension of Agency, Authority and Identity is often overlooked – maybe because it is much more complicated to discuss. Take a look at what this includes:
This dimension helps us as teachers consider our students’ perspectives. How are they experiencing each day? We should be reflecting on:
Who has a voice? Who doesn’t?
How are ideas shared between and among students?
Who feels like they have contributed? Who doesn’t?
Who is actively contributing? Who isn’t?
Reflecting on our students’ experiences makes us better teachers! So, I’ve been wondering:
Who created today’s problem / game / puzzle?
For most students, math class follows the same pattern:
This pattern of a lesson leaves many students disinterested, because they are not actively involved in the learning – which might lead to typical comments like, “When will we ever need this?”. This lesson format is TEACHER centered because it centers the teachers’ ideas (the teacher provides the problem, the teacher helps students, the teacher tells you if you are correct). In this example, students’ mathematical identities are not fostered. There is NO agency afforded to students. Authority solely belongs to the teachers. But there are ways to make identity / agency / authority a focus!
STUDENT driven ideas
Today as a quick warm-up, I had students solve a little pentomino puzzle. After they finished, I asked students to create their own puzzles that others will solve. Here is one of the student created puzzles:
Here you can see a simple puzzle. The pieces are shown that you must use, and the board is included (with a hole in the middle). Now, as a class, we have a bank of puzzles we can attempt any day (as a warm-up or if work is finished).
You can read about WHY we would do puzzles like this in math class along with some examples (Spatial Reasoning).
What’s more important here is for us to reflect on how we are involving our own students in the creation of problems, games and puzzles in our class. This is a low-risk way to allow everyone in class do more than just participate, they are taking ownership in their learning, and building a community of learners that value learning WITH and FROM each other!
How to involve our students?
The example above shows us a simple way to engage our students, to expand what we consider mathematics and help our students form positive mathematical identities. However, there are lots of ways to do this:
Play a math game for a day or 2, then ask students to alter one or a few of the rules.
Have students submit questions you might want to consider for an assessment opportunity.
Have students look through a bank or questions / problems and ask which one(s) would be the most important ones to do.
Give students a sheet of many questions. Ask them to only do the 3 easiest, and the 3 hardest (then lead a discussion about what makes those ones the hardest).
Lead 3-part math lessons where students start by noticing / wondering.
Have students design their own SolveMe mobile puzzles, visual patterns, Which One Doesn’t Belong…
Questions to Reflect on:
Who is not contributing in your class, or doesn’t feel like they are a “math student”? Whose mathematical identities would you like to foster? How might something simple like this make a world of difference for those children?
Fostering student identities, paying attention to who has authority in your class and allowing students to take ownership is essential to build mathematicians. The feeling of belonging in this space is crucial. How are you paying attention to this? (See Matthew Effect)
If you do have your students create their own puzzles, will you first offer a simplified version so your students get familiar with the pieces, or will you dive into having them make their own first?
Would you prefer all of your students doing the same puzzle / game / problem, or have many puzzles / games / problems to choose from? How might this change class conversations afterward?
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:
Blow is a gallery of some possible adaptations of this game or linked here is a slideshow
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 possibleRolling one number at a time, find the largest total mass possibleRolling one number at a time, find the largest possible distanceRolling 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?
In 2001, the National Research Council, in their report Adding it up: Helping children learn mathematics, sought to address a concern expressed by many Americans: that too few students in our schools are successfully acquiring the mathematical knowledge, skill, and confidence they need to use the mathematics they have learned.
Developing Mathematical Proficiency The potential of different types of tasks for student learning, 2017
As we start a new school year, I expect many teachers, schools and districts to begin conversations surrounding assessment and wondering how to start learning given students who might be “behind”. I’ve shared my thoughts about how we should NOT start a school year, but I wanted to offer some alternatives in this post surrounding a piece often overlooked — our students’ confidence (including student agency, ownership and identity). If we are truly interested in starting a year off successfully, then we need to spend time allowing our students to see themselves in the math they are doing… and to see their strengths, not their deficits.
[The] goal is to support all students — especially those who have not been academically successful in the past — to develop a sense of agency and ownership over their own learning. We want students to come to see themselves as intellectually capable and competent — not by giving them easy successes, but by engaging them as sense-makers, problem solvers, and creators of meaningful and important ideas.
When we hear ideals like the above quote, what many of us see is as missing are specific examples. How DO we help our students gain confidence becomes a question most of us are left with. Adding It Up suggests that mathematical proficiency includes an intertwined mix of procedural fluency, conceptual understanding, strategic competence, adaptive reasoning and productive disposition. Which again sounds nice in theory, but in reality, these 5 pieces are not balanced in classroom materials nor in our assessment data. Not even close!
Adding It Up: helping children learn mathematics, 2001
So, again we are left with a specific need for us to build confidence in our students. There is a growing body of evidence to support the use of strategy games in math class as a purposeful way to build confidence (including student agency, authority and identity).
To be helpful, I’d like to share some examples of possible strategy games that are appropriate for all ages. Each game is a traditional game from various places around the world.
*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.
How to Play:
Each link above includes a full set of rules, but you might also be interested in watching a preview of these games (thanks to WhatDoWeDoAllDay.com)
A few things to reflect on:
Some students have missed a lot of school / learning. Our students might be entering a new grade worried about the difficulty level of the content. Beyond content, what other aspects of learning math might be a struggle for our students? How might introducing games periodically help with these struggles?
How do you see equity playing a role in all of this? Pinpointing and focusing on student gaps often leads to inequities in experiences and outcomes. So, how can the ideas above help reduce these inequities?
One of the best ways to tackle equity issues is to expand WHAT we consider mathematics and expand WHO is considered a math person. How might you see using games periodically as a way for us to improve in these two areas?
If you are distance learning, how might games be an integral part of your program? How do you see including games that are not related to content helpful for our students that might struggle to learn mathematics? (building confidence, social-emotional learning skills, community, students’ identities…)
If you are learning in person this year, but can not have students working together, how might you adapt some of these strategy games?
What might you notice as students are playing games that you might not be able to notice otherwise?
How might we see a link between gaining confidence through playing strategy games and improvement in mathematical reasoning?
Why do you think I choose the games above (I searched through many)? Hopefully you can see a benefit from seeing mathematics learning from various cultures.
If interested in more games and puzzles? Take a look at some of the following posts:
A few years ago Tracy Zager wrote a wonderful article called “How Not to Start Math Class in the Fall” where she shared the pitfalls of starting the year with diagnostic tests and instead gave a more positive and productive path which included setting a positive tone for learning mathematics and gathering useful formative data. While the article was a powerful reminder about what we should value and how we can help start the year off on a positive note, the article might be more important this year than most for us to consider.
The ending of the 2020 school year was (is) not ideal for many students (as we all know). Many students did not participate in learning from home platforms, and for those who did, many did not participate regularly. And even for those who did participate regularly – with no fault at all placed on teachers or schools – the ability to give students experience to learn new materials, to observe students’ thinking, to ask timely guiding questions, to monitor student progress, to know how/when/what to consolidate….. were not ideal or equitable (or possible in many cases) making learning mathematics difficult.
From conversations I have had with various teachers, I think we can all agree on a few things here:
Learning over the past few months has not been ideal for many students;
Learning about our students’ thinking has been difficult, at best, for us, making it difficult to sequence learning, consolidate big ideas, and use various students’ thinking to drive conversations; and
There will be a huge discrepancy between how much / what students have learned over the past few months
Because of these three points, when students finally get back into classrooms we will likely have many eagre to attempt to make the best of things. However, what first moves we make when school returns matters more this year than ever. This leads me to wonder, will our decisions be driven by thoughts of how to fill gaps or how to build a community of learners?
Whether or not things will go back to normal in the fall, even if we are back in schools, what we value and what we believe is important will have huge effects on the experiences our students have in our classrooms. For those who might be pushing a “Gaps Driven” message, I would like us to recognize the multitude of equity issues that surround this approach in normal circumstances. NCTM’s new resource Catalyzing Change in Elementary and Early Childhood Mathematics offers some advice:
At the early childhood and elementary school levels, the use of pre-assessment data at the start of a unit or mathematics workshop to create flexible ability groups might seem harmless on the surface and even helpful. Proponents say that this practice allows teachers to figure out children’s learning needs, then tailor the content and pace of instruction to children’s varying levels of performance. However, flexible groups often lead to differentiated learning expectations and experiences and thus, differentiated learning outcomes. Students are perceptive and soon realize they are usually put in the same groups with the same other students. Any ability grouping in mathematics education is an inequitable structure that perpetuates privilege for a few and marginality for others.
Catalyzing Change in Elementary and Early Childhood Mathematics, 2020
The idea that many of our students will be in different places academically will be at the front of our thinking, however, there are many issues that we need to be thoughtful about. Families that have been able to support children from home this spring are at a direct advantage in the fall. Students from economically disadvantaged homes, or are from families that have limited access to technology or have mental health concerns, or students that have struggled with motivation or self-monitoring…. are at a particular disadvantage right now, and potentially in the fall.
So, how could we start the fall productively? Somehow, the first few weeks need to be a time to build community, engage in rich learning experiences where we can notice student thinking and create opportunities for collaboration and discussion norms. Dr. Yeap Ban Har might have said it best:
We have no idea what next year will look like. So, whatever time we do have in classrooms, we need to build the kinds of relationships and norms that will help us in case we are expected to once again learn from home.
How TO Start?
If we really are worried about gaps in prior learning, thinking about how to start all new learning with experiences that will help bridge current understandings with what your students will be learning will need to be a focus. Instead of starting with a test that quantifies learning or sorts kids, how about you:
Anything that gets kids thinking, talking, sharing, testing ideas, playing with concepts, making conjectures, noticing patterns, building, representing…..
Content will come. Focusing on our kids as thinkers and doers of mathematics needs to come first. Doing so in ways that builds relationships and learning norms is where I would start!
A few things to reflect on:
Some students have missed a lot of school / learning. Beyond content, what other aspects of learning math might be a struggle in the fall?
How do you see equity playing a role in all of this? Pinpointing and focusing on student gaps often leads to inequities in experiences and outcomes. So, how can the ideas above help reduce these inequities?
What you do the first few days/weeks will show your students what you value. What will your first days/weeks say about you as a teacher and the subject of mathematics to your students?
If you noticed a lack of engagement this Spring, how can we better prepare for future disruptions by building the right kinds of relationships, norms and routines? What will you do in your first few days/weeks to start down this path?
Maybe if you can see that some of the above strategies can really help you get to know your kids personally and mathematically, you might realize that a test might not be as valuable as you had thought.
As always, I’d love to hear your thoughts. Leave a reply here on Twitter (@MarkChubb3)
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?
Ask any teacher anywhere what some of the most pressing challenges are that they face as a teacher and likely you will hear examples of how difficult it can be to meet the various needs within a classroom. When conversations on the topic arise, there are often discussions from one of two extremes:
One one side you might hear about reasons why a teacher might believe that it is best to make sure that every student be expected to learn the same things. These beliefs often lead to practices where everyone receives the same instruction, followed by individual assistance for students who were not able to follow classroom instructions. Equity here is believed to be related to equal outcomes.
On the other hand, some teachers might believe that it isn’t possible to expect every student to learn the same things at the same time. Their beliefs lead them to focus more attention on determining readiness and offering different groups of students different learning opportunities. Equity here is viewed as meeting each child’s unique need.
While I understand each of these points of view, part of the issue between these two views is the overall view of what mathematics is. If mathematics is seen as a set of linearly learned skills, where each skill is boiled down to a list of subskills, each learned in a specific sequence, it is difficult to imagine anything else. However, when mathematics is seen through the lens of rich connections, we might start to see students’ development of these connections as what can drive our decisions.
One way to think of a person’s understanding of mathematics is that it exists along a continuum. At one end is a rich set of connections. At the other end of the continuum, ideas are isolated or viewed as disconnected bits of information. A sound understanding of mathematics is one that sees the connections within mathematics and between mathematics and the world.
TIPS4RM: Developing Mathematical Literacy, 2005
The two views mentioned above do not account for a view of mathematics where connections between concepts is a focus. Instead of seeing the issue as simply whether or not we want a classroom of students to be doing the same things or if we should be providing some students with different things, we should also consider what is actually being learned by the students. Below you can see a matrix showing four different examples of how we could tackle the same concept in a classroom:
Same / Different Learning? Same / Different Tasks?
Same Tasks, Same Learning: The teacher offers everyone the same task, expects everyone to be able to follow the same procedures and might offer explicit help to specific students that aren’t following accordingly.
Different Tasks, Same Learning: While everyone is learning the same thing, the teacher offers some groups easier work and other groups more advanced work based on readiness.
Different Tasks, Different Learning: Based on diagnostic assessments, students are placed into groups based on what they need to continue learning. Some groups might be learning different materials within the same class.
Same Tasks, Different Learning: Every student is provided the same task, but there is variance in how and what is being learned.
For the readers here, I encourage you to think about which of the above models might you have experienced as a student, and which you might think would be best for your students.
Taking an Equity Stance
So, what does any of this have to do with equity? In my experience, a lot! Taking an equity stance means that we both believe that every student can achieve, and understand that every student might need different things from us. Keeping equity in mind requires us to analyze who has access to rich mathematical experiences and whose experiences are narrowed or reduced to lower-level skills (Access), whose ideas contribute to the group’s development of mathematical understanding and whose are not heard (Agency and Authority), who identifies with mathematics and who does not (Identity)… Without considering our beliefs and practices, we will never be able to notice which students are being underserviced, which structures promote some groups over others, or see which practices lead to the “Matthew Effect“.
How do we aim for Equity?
When thinking about how we aim for equity in mathematics, there seems to be 2 key tenets that help point us in the right direction:
Expand WHO is considered a math student
Expand WHAT is accepted as mathematics
The question is not whether all students can succeed in mathematics but whether the adults organizing mathematics learning opportunities can alter traditional beliefs and practices to promote success for all.
Principles to Action – NCTM (p.61)
WHO is considered a math person?
Teachers who come to recognize that some students identify with mathematics (and others do not) aim to promote tasks that allow more students to engage in mathematical reasoning via problems/tasks that are easily accessible (low-floor, high-ceiling tasks). If our students are going to see themselves as budding mathematicians, then we need to allow more opportunities for students to share their emerging ideas first!
Dr. Christine Suurtamm does a great job of articulating what this could look like in practice:
Dr. Christine Suurtamm
Students need to see themselves in the work they are doing. This includes knowing that mathematics is not created for and used by only some people (race/gender…), nor is it only useful for potential futures of some of our students, but is actually used by all of us RIGHT NOW. If we want to make sure our students see themselves as mathematicians, OUR STUDENTS need to be doing more of the thinking, they need to be part of the process of learning, not simply showing that they have mastered procedures.
Reflecting on WHO believes they are a math person might help us reflect on what messages our students have received over the years. If you haven’t already read about the “Matthew Effect“, I recommend that this might be a great place to help you reflect.
WHAT Counts as “Mathematics”?
To some, mathematics is a very narrow subject. Calculating (adding, subtracting, multiplying, dividing), converting (equivalent fractions), and carrying out other procedures accurately by using the requisite steps… Procedures dominate some textbooks and online practice sites and for some, this narrow vision of mathematics is where some students begin to struggle. However, if we are aiming for equity then we need to allow more opportunities for our students to show us what ARE good at.
One way to expand what counts as mathematics is for us to reflect on how much time we spend on each strand of mathematics (Patterning, Number Sense, Geometry, Measurement, Data Management). Analyzing how much time we spend on each of these strands, and specifically when in the year we might teach these concepts might help us reflect on what messages our students hear when they consider what counts as mathematics. For example, schools in my area typically start with several weeks of patterning because it can be experienced physically (manipulatives) and visually (visual patterns, graphing…), followed by several weeks of Geometry. These moves were strategic, because it allows our students more opportunities to talk, more opportunities to solve problems, more opportunities for our students to use visual/ spatial reasoning and more students to start their year with successes!
Expanding what mathematics means is much more than strands or concepts though. A focus on concrete and visual representations (not solely abstract symbolic representations) can be a path to expand what counts as mathematics. Allowing students to show their strategies, and accepting student strategies as part of the learning process means that preformal representations and strategies can be compared and learned from.
Spatial puzzles and games allow students to think mathematically in ways that differ from typical assignments. A story I often tell is of this young student who had never liked mathematics, and often struggled with mathematics. Here you can see her attempting to solve a difficult puzzle that one of her classmates created. Every child deserves to experience what this student experienced – productive struggle and success. Take a look:
Considerations
If we are aiming for equity in our own personal practices, we need to be aware of our own biases, our own patterns. This isn’t easy! It might mean noticing how we talk about race or gender or socio-economic groups and what our expectations are for each. It might mean reflecting on words we use to discuss students who might currently be struggling to learn mathematics or who are identified learners and what our expectations are of these students. Again, learning more about the Matthew Effect is where I would recommend you start. Planning with providing greater access for students to learn mathematics (same tasks/different learning – spatializing mathematics) is likely a first concrete step we can take.
I want to leave you with a few reflective questions:
How do you see the Same/Different Learning – Same/Different Tasks chart relating to equity? Which quadrant would you like provide for your students to be engaged with more frequently? What barriers are standing in the way?
We need to be aware that when schools group students by ability or assign different tasks to different students, those that are relegated to lower groups/tasks often receive lower level of cognitive demand tasks, which often puts them at a further disadvantage than their peers. How do you combat these inequities in your classroom?
Providing students with rich tasks and access to rich problems isn’t enough. We also need to be noticing our students’ thinking so we know how to respond to our students individually and as a group. This isn’t easy! How do you pay attention to their thinking? What structures do you have in place to listen to students and respond accordingly?
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 ).
Many math resources attempt to share the difference between teaching FOR problem solving and teaching THROUGH problem solving. Cathy Seeley refers to teaching THROUGH problem solving as “Upside-Down Teaching” which is the opposite of a “gradual release of responsibility” model:
And instead calls for us to flip how our students learn to a more active model:
So, instead of starting a unit on Geometry with naming shapes or developing definitions together, we decided to start with a little problem:
Create as many polygons as possible using exactly 2 pattern block pieces. Sort your polygons by how many sides they have.
As students started placing pattern block pieces together, all kinds of questions started emerging (questions we took note of to bring to the whole group in a few minutes):
What is this shape called? It has 6 sides, so is it a hexagon?
This shape has 12 sides. Am I allowed to do this?
Are these the same shapes or different? Do I have to line up the sides or can I place a shape in the middle like I did here (on the right)?
By the end of a period, students had worked through the definitions of what a polygon is (and isn’t), the difference between concave and convex polygons, defined the term “regular polygon” (which was not what they had been calling “regular” before), and were able to name and create triangles, quadrilaterals, pentagons, hexagons, heptagons, octagons, nonagons, decagons and undecagons. Recognizing a variety of possible ways a shape can look was very helpful for our students who might have experienced shapes more traditionally in the past.
One group compiled their polygons together (with one minor error):
Instead of starting with experiences where students accumulate knowledge (writing out definitions, taking a note, direct instruction), an upside down approach aims to start with students’ ideas. This way we would know which conversations to have with our students, and so our students are actively engaged in the process of learning.
I want to leave you with a few reflective questions:
Why might it benefit students to start with a problem instead of starting with the teachers’ ideas?
Why might it benefit teachers to listen to students’ thinking before instruction has occurred?
What does it mean to effectively monitor students as they are thinking / working? (See This POST for examples)
Can all mathematics topics begin with tasks that help our students make connections between what they already know, and what they are learning? Can you think of a topic that can not be experienced this way?
The final stage in the You-We-I model is where the teacher helps make specific learning explicit for their students. How do you find time to consolidate a task like this? How do you know what to share? (See This POST for an example)
How might this form of teaching relate to how we view assessment? (See This POST)
How might this form of teaching relate to how we view differentiated instruction? (See This POST)
How do you find problems that ask students to actively think before any instruction has occurred? (See This POST for examples)
I’d love to continue the conversation. Feel free to write a response, 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:
Many mathematicians are good at searching for patterns in numbers, however, an area that I think we all need to continue to explore is Visualizing.
Instead of just looking for procedural rules, or numeric patterns I encourage you to take one of the following and actually VISUALIZE what is going on.
Pick one of the above that interests you. Answer some of these questions:
What relationships do you notice here?
What are you curious about?
What visual might be helpful to represent this/these relationships?
Will these relationships work in other instances? When will it work/ when won’t it work?
How might a visual help others see the relationships you’ve noticed?
I’d love to hear some answers. You can respond here below, or via Twitter @MarkChubb3
Thinking Mathematically 