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)

Advertisement

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).

---

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.

---

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.

---

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

---

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

---

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:

---

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 ).

Starting where our students are….. with THEIR thoughts

A common trend in education is to give students a diagnostic in order for us to know where to start. While I agree we should be starting where our students are, I think this can look very different in each classroom.  Does starting where our students are mean we give a test to determine ability levels, then program based on these differences?  Personally, I don’t think so.

Giving out a test or quiz at the beginning of instruction isn’t the ideal way of learning about our students.  Seeing the product of someone’s thinking often isn’t helpful in seeing HOW that child thinks (Read, What does “assessment drive instruction mean to you” for more on this). Instead, I offer an alternative- starting with a diagnostic task!  Here is an example of a diagnostic task given this week:

Taken from Van de Walle’s Teaching Student Centered Mathematics

This lesson is broken down into 4 parts.  Below are summaries of each:

---

Part 1 – Tell 1 or 2 interesting things about your shape

Start off in groups of 4.  One student picks up a shape and says something (or 2) interesting about that shape.

Here you will notice how students think about shapes. Will they describe the shape as “looking like a mountain” or “it’s an hourglass” (visualization is level 1 on Van Hiele’s levels of Geometric thought)… or will they describe attributes of that shape (this is level 2 according to Van Hiele)?

As the teacher, we listen to the things our students talk about so we will know how to organize the conversation later.

---

Part 2 – Pick 2 shapes.  Tell something similar or different about the 2 shapes.

Students randomly pick 2 shapes and either tell the group one thing similar or different about the two shapes. Each person offers their thoughts before 2 new shapes are picked.

Students who might have offered level 1 comments a minute ago will now need to consider thinking about attributes. Again, as the teacher, we listen for the attributes our students understand (i.e., number of sides, right angles, symmetry, number of vertices, number of pairs of parallel sides, angles….), and which attributes our students might be informally describing (i.e., using phrases like “corners”, or using gestures when attempting to describe something they haven’t learned yet).  See chart below for a better description of Van Hiele’s levels:

Van Hiele’s chart shared by NCTM

At this time, it is ideal to hold conversations with the whole group about any disagreements that might exist.  For example, the pairs of shapes above created disagreements about number of sides and number of vertices.  When we have disagreements, we need to bring these forward to the group so we can learn together.

---

Part 3 – Sorting using a “Target Shape”

Pick a “Target Shape”. Think about one of its attributes.  Sort the rest of the shapes based on the target shape.

The 2 groups above sorted their shapes based on different attributes. Can you figure out what their thinking is?  Were there any shapes that they might have disagreed upon?

---

Part 4 – Secret sort

Here, we want students to be able to think about shapes that share similar attributes (this can potentially lead our students into level 2 type thinking depending on our sort).  I suggest we provide shapes already sorted for our students, but sorted in a way that no group had just sorted the shapes. Ideally, this sort is something both in your standards and something you believe your students are ready to think about (based on the observations so far in this lesson).

---

In this lesson, we have noticed how our students think.  We could assess the level of Geometric thought they are currently using, or the attributes they are comfortable describing, or misconceptions that need to be addressed.  But, this lesson isn’t just about us gathering information, it is also about our students being actively engaged in the learning process!  We are intentionally helping our students make connections, reason and prove, learn/ revisit vocabulary, think deeper about specific attributes…

---

I’ve shared my thoughts about what I think day 1 should look like before for any given topic, and how we can use assessment to drive instruction, however, I wanted to write this blog about the specific topic of diagnostics.

In the above example, we listened to our students and used our understanding of our standards and developmental research to know where to start our conversations. As Van de Walle explains the purpose of formative assessment, we need to make our formative more like a streaming video, not just a test at the beginning!

If its formative, it needs to be ongoing… part of instruction… based on our observations, conversations, and the things students create…  This requires us to start with rich tasks that are open enough to allow everyone an entry point and for us to have a plan to move forward!

I’m reminded of Phil Daro’s quote:

For us to make these shifts, we need to consider our mindsets that also need to shift.  Statements like the following stand in the way of allowing our students to be actively engaged in the learning process starting with where they currently are:

• My students aren’t ready for…
• I need to start with the basics…
• My students have gaps in their…
• They don’t know the vocabulary yet…

These thoughts are counterproductive and lead to the Pygmalion effect (teacher beliefs about ability become students’ self-fulfilling prophecies).  When WE decide which students are ready for what tasks, I worry that we might be holding many of our students back!

If we want to know where to start our instruction, start where your students are in their understanding…with their own thoughts!!!!!  When we listen and observe our students first, we will know how to push their thinking!