How Not to Start Math Class in the Fall – 2020

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:

• Start with a diagnostic Task for each new concept
• Choose a specific notice and wonder image as a shared experience where you can build important discussions about key concepts
• Use an open problem that is highly accessible. Then share specific examples with the group that lead to relationships between prior and new learning
• Choose a spatial task to help students learn to persevere when challenged
• Ask students to share what they know on a frayer model which can be updated throughout upcoming days
• Play a game that uses the concept you want to address so you can watch students’ in action, then bring up what you have noticed with the class
• Anything to get your students DOING so you can NOTICE their current thinking and WONDER about what to do next.
• 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)

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Taking an Equity Stance in Math Class

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?
• How do you monitor your students’ interests and thoughts about mathematics in general, or about specific concepts?
• How are you aiming to minimize the Matthew Effect and reduce inequities in your room?

As always, I’d love to hear your thoughts.  Leave a reply here on Twitter (@MarkChubb3)

Rushing for Interventions

I see students working in groups all the time…  Students working collaboratively in pairs or small groups having rich discussions as they sort shapes by specific properties, students identifying and extending their partner’s visual patterns, students playing games aimed at improving their procedural fluency, students cooperating to make sense of a low-floor/high-ceiling problem…..

When we see students actively engaged in rich mathematics activities, working collaboratively, it provides opportunities for teachers to effectively monitor student learning (notice students’ thinking, provide opportunities for rich questioning, and lead to important feedback and next steps…) and prepare the teacher for the lesson close.  Classrooms that engage in these types of cooperative learning opportunities see students actively engaged in their learning.  And more specifically, we see students who show Agency, Ownership and Identity in their mathematics learning (See TruMath‘s description on page 10).

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On the other hand, some classrooms might be pushing for a different vision of what groups can look like in a mathematics classroom.  One where a teachers’ role is to continually diagnose students’ weaknesses, then place students into ability groups based on their deficits, then provide specific learning for each of these groups.  To be honest, I understand the concept of small groups that are formed for this purpose, but I think that many teachers might be rushing for these interventions too quickly.

First, let’s understand that small group interventions have come from the RTI (Response to Intervention) model.  Below is a graphic created by Karen Karp shared in Van de Walle’s Teaching Student Centered Mathematics to help explain RTI:

Response to Intervention – Teaching Student Centered Mathematics

As you can see, given a high quality mathematics program, 80-90% of students can learn successfully given the same learning experiences as everyone.  However, 5-10% of students (which likely are not always the same students) might struggle with a given topic and might need additional small-group interventions.  And an additional 1-5% might need might need even more specialized interventions at the individual level.

The RTI model assumes that we, as a group, have had several different learning experiences over several days before Tier 2 (or Tier 3) approaches are used.  This sounds much healthier than a model of instruction where students are tested on day one, and placed into fix-up groups based on their deficits, or a classroom where students are placed into homogeneous groupings that persist for extended periods of time.

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Principles to Action (NCTM) suggests that what I’m talking about here is actually an equity issue!

Principles to Action

We know that students who are placed into ability groups for extended periods of time come to have their mathematical identity fixed because of how they were placed.  That is, in an attempt to help our students learn, we might be damaging their self perceptions, and therefore, their long-term educational outcomes.

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Tier 1 Instruction

While I completely agree that we need to be giving attention to students who might be struggling with mathematics, I believe the first thing we need to consider is what Tier 1 instruction looks like that is aimed at making learning accessible to everyone.  Tier 1 instruction can’t simply be direct instruction lessons and whole group learning.  To make learning mathematics more accessible to a wider range of students, we need to include more low-floor/high-ceiling tasks, continue to help our students spatalize the concepts they are learning, as well as have a better understanding of developmental progressions so we are able to effectively monitor student learning so we can both know the experiences our students will need to be successful and how we should be responding to their thinking.  Let’s not underestimate how many of our students suffer from an “experience gap”, not an “achievement gap”!

If you are interested in learning more about what Tier 1 instruction can look like as a way to support a wider range of students, please take a look at one of the following:

•  How do we meet the needs of so many unique students in a mixed-ability classroom?
• Differentiated Instruction: comparing 2 subjects
• Differentiating Mathematics Instruction

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Tier 2 Instruction

Tier 2 instruction is important.  It allows us to give additional opportunities for students to learn the things they have been learning over the past few days/weeks in a small group.  Learning in a small group with students who are currently struggling with the content they are learning can give us opportunities to better know our students’ thinking.  However, I believe some might be jumping past Tier 1 instruction (in part or completely) in an attempt to make sure that we are intervening. To be honest, this doesn’t make instructional sense to me! If we care about our content, and care about our students’ relationship with mathematics, this might be the wrong first move.

So, let’s make sure that Tier 2 instruction is:

• Provided after several learning experiences for our students
• Flexibly created, and easily changed based on the content being learned at the time
• Focused on student strengths and areas of need, not just weaknesses
• Aimed at honoring students’ agency, ownership and identity as mathematicians
• Temporary!

If you are interested in learning more about what Tier 2 interventions can look like take a look at one of the following:

• RTI for Adults
• Targeted Instruction
• Continuing to get better together

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Instead of seeing mathematics as being learned every day as an approach to intervene, let’s continue to learn more about what Tier 1 instruction can look like!  Or maybe you need to hear it from John Hattie:

Or from Jo Boaler:

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Final Thoughts

If you are currently in a school that uses small group instruction in mathematics, I would suggest that you reflect on a few things:

• How do your students see themselves as mathematicians?  How might the topics of Agency, Authority and Identity relate to small group instruction?
• What fixed mindset messaging do teachers in your building share “high kids”, “level 2 students”, “she’s one of my low students”….?  What fixed mindset messages might your students be hearing?
• When in a learning cycle do you employ small groups?  Every day?  After several days of learning a concept?
• How flexible are your groups?  Are they based on a wholistic leveling of your students, or based specifically on the concept they are learning this week?
• How much time do these small groups receive?  Is it beyond regular instructional timelines, or do these groups form your Tier 1 instructional time?
• If Karp/Van de Walle suggests that 80-90% of students can be successful in Tier 1, how does this match what you are seeing?  Is there a need to learn more about how Tier 1 approaches can meet the needs of this many students?
• What are the rest of your students doing when you are working with a small group?  Is it as mathematically rich as the few you’re working with in front of you?
• Do you believe that all of your students are capable to learn mathematics and to think mathematically?

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