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:

rti2
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!

P2A
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

intervention

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:


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:


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 ).

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How do you give feedback?

There seems to be a lot of research telling us how important feedback is to student performance, however, there’s little discussion about how we give this feedback and what the feedback actually looks like in mathematics. To start with, here are a few important points research says about feedback:

  • The timing of feedback is really important
  • The recipient of the feedback needs to do more work than the person giving the feedback
  • Students need opportunities to do something with the feedback
  • Feedback is not the same thing as giving advice

I will talk about each of these toward the end of this post.  First, I want to explain a piece about feedback that isn’t mentioned enough…  Providing students with feedback positions us and our students as learners.  Think about it for a second, when we “mark” things our attention starts with what students get right, but our attention moves quickly to trying to spot errors. Basically, when marking, we are looking for deficits. On the other hand, when we are giving feedback, we instead look for our students’ actual thinking.  We notice things as almost right, we notice misconceptions or overgeneralization…then think about how to help our students move forward.  When giving feedback, we are looking for our students strengths and readiness.  Asset thinking is FAR more productive, FAR more healthy, FAR more meaningful than grades!


Feedback Doesn’t Just Happen at the End!

Let’s take an example of a lesson involving creating, identifying, and extending linear growing patterns.  This is the 4th day in a series of lessons from a wonderful resource called From Patterns to Algebra.  Today, the students here were asked to create their own design that follows the pattern given to them on their card.

IMG_1395
Their pattern card read: Output number = Input number x3+2
IMG_1350
Their pattern card read:  Output number = Input number x7
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Their pattern card read:  Output number = Input number x4

 

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Their pattern card read:  Output number = Input number x3+1
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Their pattern card read:  Output number = Input number x8+2
IMG_1361
Their pattern card read: Output number = Input number x5+2

Once students made their designs, they were instructed to place their card upside down on their desk, and to circulate around the room quietly looking at others’ patterns.  Once they believed they knew the “pattern rule” they were allowed to check to see if they were correct by flipping over the card.

After several minutes of quiet thinking, and rotating around the room, the teacher stopped everyone and led the class in a lesson close that involved rich discussions about specific samples around the room.  Here is a brief explanation of this close:

Teacher:  Everyone think of a pattern that was really easy to tell what the pattern rule was.  Everyone point to one.  (Class walks over to the last picture above – picture 6).  What makes this pattern easy for others to recognize the pattern rule?  (Students respond and engage in dialogue about the shapes, colours, orientation, groupings…).

Teacher:  Can anyone tell the class what the 10th position would look like?  Turn to your partner and describe what you would see.  (Students share with neighbor, then with the class)

Teacher:  Think of one of the patterns around the room that might have been more difficult for you to figure out.  Point to one you want to talk about with the class.  (Students point to many different ones around the room.  The class visits several and engages in discussions about each.  Students notice some patterns are harder to count… some patterns follow the right number of tiles – but don’t follow a geometric pattern, some patterns don’t reflect the pattern listed on the card.  Each of these noticings are given time to discuss, in an environment that is about learning… not producing.  Everyone understands that mistakes are part of the learning process here and are eager to take their new knowledge and apply it.

The teacher then asks students to go back to their desks and gives each student a new card.  The instructions are similar, except, now she asks students to make it in a way that will help others recognize the patterns easily.

The process of creating, walking around the room silently, then discussing happens a second time.

To end the class, the teacher hands out an exit card asking students to articulate why some patterns are easier than others to recognize.  Examples were expected from students.


At the beginning of this post I shared 4 points from research about feedback.  I want to briefly talk about each:

The timing of feedback is really important

Feedback is best when it happens during the learning.  While I can see when it would be appropriate for us to collect items and write feedback for students, having the feedback happen in-the-moment is ideal!   Dan Meyer reminds us that instant feedback isn’t ideal.  Students need enough time to think about what they did right/wrong… what needs to be corrected.  On the other hand, having students submit items, then us giving them back a week later isn’t ideal either!  Having this time to think and receive feedback DURING the learning experience is ideal.  In the example above, feedback happened several times:

  1. As students walked around looking at patterns.  After they thought they knew the pattern, they peeked at the card.
  2. As students discuss several samples they are given time to give each other feedback about which patterns make sense… which ones visually represented the numeric value… which patterns could help us predict future visuals/values
  3. Afterward once the teacher collected the exit cards.

The recipient of the feedback needs to do more work than the person giving the feedback

Often we as teachers spend too much time writing detailed notes offering pieces of wisdom.  While this is often helpful, it isn’t a feasible thing to do on a daily basis. In fact, us doing all of the thinking doesn’t equate to students improving!  In the example above, students were expected to notice patterns that made sense to them, they engaged in conversations about the patterns.  Each student had to recognize how to make their pattern better because of the conversations.  The work of the feedback belonged, for the most part, within each student.

Students need opportunities to do something with the feedback

Once students receive feedback, they need to use that feedback to continue to improve.  In the above example, the students had an opportunity to create new patterns after the discussions.  After viewing the 2nd creations and seeing the exit cards, verbal or written feedback could be given to those that would benefit from it.


Feedback is not the same thing as giving advice

This last piece is an interesting one.  Feedback, by definition, is about seeing how well you have come to achieving your goal.  It is about what you did, not about what you need to do next.  “I noticed that you have switched the multiplicative and additive pieces in each of your patterns” is feedback.  “I am not sure what the next position would look like because I don’t see a pattern here” is feedback.  “The additive parts need to remain constant in each position” is not feedback… it is advice (or feedforward).

In the example above, the discussions allowed for ample time for feedback to happen.  If students were still struggling, it is appropriate to give direct advice.  But I’m not sure students would have understood any advice, or retained WHY they needed to take advice if we offered it too soon.


So I leave you with some final questions for you:

  • When do your students receive feedback?  How often?
  • Who gives your students their feedback?
  • Is it written?  Or verbal?
  • Which of these do you see as the most practical?  Meaningful for your students?  Productive?
  • How do you make time for feedback?
  • Who is doing the majority of the work… the person giving or the person receiving the feedback?
  • Do your students engage in tasks that allow for multiple opportunities for feedback to happen naturally?

PS.   Did you notice which of the students’ examples above had made an error.  What feedback would you give?  How would they receive this feedback?