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?
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:
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 ).
Thinking Mathematically 