Targeted Instruction

The other day I was asked about my opinion about entrance slips. Curious about their thoughts first, I asked a few question that helped me understand what they meant by entrance slips, what they would be used for, and how they might believe they would be helpful. The response made me a little worried. Basically, the idea was to give something to students at the beginning of class to determine gaps, then place students into groups based on student “needs”.  I’ll share my issues with this in a moment…  Once I had figured out how they planned on using them, I asked what the different groups would look like.  Specifically, I asked what students in each group would be learning. They explained that the plan was to give an entrance slip at the beginning of a Geometry unit. The first few questions on this entrance slip would involve naming shapes and the next few about identifying isolated properties of shapes. Those who couldn’t name shapes were to be placed into a group that learns about naming shapes, those who could name shapes but didn’t know all of the properties were to go into a second group, and those who did well on both sections would be ready to do activities involving sorting shapes.  In our discussion I continually heard the phrase “Differentiated Instruction”, however, their description of Differentiated Instruction definitely did not match my understanding (I’ve written about that here). What was being discussed here with regards to using entrance slips I would call “Individualized Instruction”.  The difference between the two terms is more than a semantic issue, it gets to the heart of how we believe learning happens, what our roles are in planning and assessing, and ultimately who will be successful.  To be clear, Differentiated Instruction involves students achieving the same expectations/standards via different processes, content and/or product, while individualized or targeted instruction is about expecting different things from different students.


Issues with Individualized / Targeted Instruction

Individualized or targeted instruction makes sense in a lot of ways.  The idea is to figure out what a student’s needs are, then provide opportunities for them to get better in this area.  In practice, however, what often happens is that we end up setting different learning paths for different students which actually creates more inequities than it helps close gaps.  In my experience, having different students learning different things might be helpful to those who are being challenged, but does a significant disservice to those who are deemed “not ready” to learn what others are learning.  For example, in the 3 pathways shared above, it was suggested that the class be split into 3 groups; one working on defining terms, one learning about properties of shapes and the last group would spend time sorting shapes in various ways.  If we thought of this in terms of development, each group of students would be set on a completely different path.  Those working on developing “recognition” tasks (See Van Hiele’s Model below) would be working on low-level tasks.  Instead of providing experiences that might help them make sense of Geometric relationships, they would be stuck working on tasks that focus on memory without meaning.

Figure-1-Examples-of-interview-items-aligned-with-van-Hiele-levels

When we aim to find specific tasks for specific students, we assume that students are not capable of learning things others are learning.  This creates low expectations for our students!  Van de Walle says it best in his book Student Centered Mathematics:

Determining how to place students in groups is an important decision.  Avoid continually grouping by ability.  This kind of grouping, although well-intentioned, perpetuates low levels of learning and actually increases the gap between more and less dependent students.  

Targeted instruction might make sense on paper, but there are several potential flaws:

  • Students enter into tracks that do not actually reflect their ability.  There is plenty of research showing that significant percentages of students are placed in the wrong grouping by their teachers.  Whether they have used some kind of test or not, groupings are regularly flawed in predicting what students are potentially ready for.
  • Pre-determining who is ready for what learning typically results in ability grouping, which is probably the strongest fixed mindset message a school can send students.  Giving an entrance ticket that determines certain students can’t engage in the learning others are doing tells students who is good at math, and who isn’t.  Our students are exquisitely keen at noticing who we believe can be successful, which shapes their own beliefs about themselves.
  • The work given to those in lower groups is typically less cognitively demanding and results in minimal learning.  The intent to “fill gaps” or “catch kids up” ironically increases the gap between struggling students and more independent learners.  Numerous studies have confirmed what Hoffer (1992) found: “Comparing the achievement growth of non-grouped students and high- and low-group students shows that high-group placement generally has a weak positive effect while low-group placement has a stronger negative effect. Ability grouping thus appears to benefit advanced students, to harm slower students.

 

The original conversation I had about Entrance Tickets illustrated a common issue we have.  We notice that there are students in our rooms who come into class in very different places in their understanding of a given topic.  We want to make sure that we provide things that our students will be successful with… However, this individualization of instruction does the exact opposite of what differentiated instruction intends to do.  Differentiated instruction in a mathematics class is realized when we provide experiences for our students where everyone is learning what they need to learn and can demonstrate this learning in different ways.  The assumption, however, is that WE are the ones that should be determining who is learning what and how much.  This just doesn’t make sense to me!  Instead of using entrance tickets, we ended up deciding to use this problem from Van de Walle so we could reach students no matter where they were in their understanding.  Instead of a test to determine who is allowed to learn what, we allowed every student to learn!  This needs to be a focus!

If we are ever going to help all of our students learn mathematics and believe that they are capable of thinking mathematically, then we need to provide learning experiences that ALL of our students can participate in.  These experiences need to:

  • Have multiple entry points for students to access the mathematics
  • Provide challenge for all students (be Problem-Based)
  • Allow students to actively make sense of the mathematics through mathematical reasoning
  • Allow students opportunities to students to express their understanding in different ways or reach an understanding via different strategies

Let’s avoid doing things that narrow our students’ learning like using entrance tickets to target instruction!  Let’s commit to a view of differentiated instruction where our students are the ones who are differentiating themselves (because the tasks allowed for opportunities to do things differently)!  Let’s continue to get better at leveraging students’ thinking in our classrooms to help those who are struggling!  Let’s believe that all of our students can learn!  


I want to leave you with a few reflective questions:

  • Why might conversations about entrance tickets and other ways to determine students ability be more common today?  We need to use our students’ thinking to guide our instruction, but other than entrance cards, how can we do this in ways that actually help those who are struggling?
  • Is a push for data-driven instruction fueling this type of decision making?  If so, who is asking for the data?  Are there other sources of data that you can be gathering that are healthier for you and your students?
  • If you’ve ever used entrance tickets or diagnostics, followed by ability groups, how did those on the bottom group feel?  Do you see the same students regularly in the bottom group?  Do you see a widening gap between those dependent on you and those who are more independent?
  • Where do you look for learning experiences that offer this kind of differentiated instruction?  Is it working for the students in your class that are struggling?

I encourage you to continue to think about what it means to Differentiate your Instruction.  Here are a few pieces that might help:

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

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Pick a Quote

Seems to me that many schools and districts are asking questions about assessment in mathematics.  So, I thought I would share a few quotes that might get you to think and reflect on your views about what it means to assess, why there might be a focus on assessment, and what our goals and ideals might look like.  I want you to take a look at the following quotes.  Pick 1 or 2 that stands out to you:

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A few things to reflect on as you think about the quotes above:

  • Which quotes caught your eye?  Did you pick one(s) that confirm things you already believe or perhaps ones that you hadn’t spent much time thinking about before?
  • Some of the above quotes speak to “assessment” while others speak to evaluation practices.  Do you know the difference?
  • Take a look again at the list of quotes and find one that challenges your thinking.  I’ve probably written about the topic somewhere.  Take a look in the Links to read more about that topic.
  • Why do you think so many discuss assessment as a focus in mathematics?  Maybe Linda Gojak’s article Are We Obsessed with Assessment? might provide some ideas.
  • Instead of talking in generalities about topics like assessment, maybe we need to start thinking about better questions to ask, or thinking deeper about what is mathematically important, or understanding how mathematics develops!

Please pick a quote that stands out for you and share your thoughts about it.

Leave a reply here or on Twitter (@MarkChubb3)

 

Which one has a bigger area?

Many grade 3 teachers in my district, after taking part in some professional development recently (provided by @teatherboard), have tried the same task relating to area.  I’d like to share the task with you and discuss some generalities we can consider for any topic in any grade.


The task:

As an introductory activity to area, students were provided with two images and asked which of the two shapes had the largest area.

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A variety of tools and manipulatives were handy, as always, for students to use to help them make sense of the problem.


Student ideas

Given very little direction and lots of time to think about how to solve this problem, we saw a wide range of student thinking.  Take a look at a few:

Some students used circles to help them find area.  What does this say about what they understand?  What issues do you see with this approach though?

Some students used shapes to cover the outline of each shape (perimeter).  Will they be able to find the shape with the greater area?  Is this strategy always / sometimes / never going to work?  What does this strategy say about what they understand?

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Example 3

Some students used identical shapes to cover the inside of each figure.

And some students used different shapes to cover the figures.

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Example 7
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Example 8
C-rods, difference
Example 9

Notice that example 9 here includes different units in both figures, but has reorganized them underneath to show the difference (can you tell which line represents which figure?).


Building Meaningful Conversations

Each of the samples above show the thinking, reasoning and understanding that the students brought to our math class.  They were given a very difficult task and were asked to use their reasoning skills to find an answer and prove it.  In the end, students were split between which figure had the greater area (some believing they were equal, many believing that one of the two was larger).  In the end, students had very different numerical answers as to how much larger or smaller the figures were from each other.  These discrepancies set the stage for a powerful learning opportunity!

For example, asking questions that get at the big ideas of measurement are now possible because of this problem:

“How is it possible some of us believe the left figure has a larger area and some of us believe that the right figure is larger?”

“Has example 8 (scroll up to take a closer look) proven that they both have the same area?”

“Why did example 9 use two pictures?  It looks like many of the cuisenaire rods are missing in the second picture?  What did you think they did here?”

In the end, the conversations should bring about important information for us to understand:

  • We need comparable units if we are to compare 2 or more figures together.  This could mean using same-sized units (like examples 1, 4, 5 & 6 above), or corresponding units (like example 8 above), or units that can be reorganized and appropriately compared (like example 9).
  • If we want to determine the area numerically, we need to use the same-sized piece exclusively.
  • The smaller the unit we use, the more of them we will need to use.
  • It is difficult to find the exact area of figures with rounded parts using the tools we have.  So, our measurements are not precise.

Some generalizations we can make here to help us with any topic in any grade

When our students are being introduced to a new topic, it is always beneficial to start with their ideas first.  This way we can see the ideas they come to us with and engage in rich discussions during the lesson close that helps our students build understanding together.  It is here in the discussions that we can bridge the thinking our students currently have with the thinking needed to understand the concepts you want them to leave with.  In the example above, the students entered this year with many experiences using non-standard measurements, and this year, most of their experiences will be using standard measurements.  However, instead of starting to teach this year’s standards, we need to help our students make some connections, and see the need to learn something new.  Considering what the first few days look like in any unit is essential to make sure our students are adequately prepared to learn something new!  (More on this here: What does day one look like?)

To me, this is what formative assessment should look like in mathematics!  Setting up experiences that will challenge our students, listening and observing our students as they work and think… all to build conversations that will help our students make sense of the “big ideas” or key understandings we will need to learn in the upcoming lessons.  When we view formative assessment as a way to learn more about our students’ thinking, and as a way to bridge their thinking with where we are going, we tend to see our students through an asset lens (what they DO understand) instead of their through the deficit lens (i.e., gaps in understanding… “they can’t”…, “didn’t they learn this last year…?).  When we see our students through an asset lens, we tend to believe they are capable, and our students see themselves and the subject in a much more positive light!

Let’s take a closer look at the features of this lesson:

  • Little to no instruction was given – we wanted to learn about our students’ thinking, not see if they can follow directions
  • The problem was open enough to have multiple possible strategies and offer multiple possible entry points (low floor – high ceiling)
  • Asking students to prove something opens up many possibilities for rich discussions
  • Students needed to begin by using their reasoning skills, not procedural knowledge…
  • Coming up with a response involved students doing and thinking… but the real learning happened afterward – during the consolidation phase

A belief I have is that the deeper we understand the big ideas behind the math our students are learning, the more likely we will know what experiences our students need first!


A few things to reflect on:

  • How often do you give tasks hoping students will solve it a specific way?  And how often you give tasks that allow your students to show you their current thinking?  Which of these approaches do you value?
  • What do your students expect math class to be like on the first few days of a new topic/concept?  Do they expect marks and quizzes?  Or explanations, notes and lessons?  Or problems where students think and share, and eventually come to understand the mathematics deeply through rich discussions?  Is there a disconnect between what you believe is best, and what your students expect?
  • I’ve painted the picture here of formative assessment as a way to help us learn about how our students think – and not about gathering marks, grouping students, filling gaps.  What does formative assessment look like in your classroom?  Are there expectations put on you from others as to what formative assessment should look like?  How might the ideas here agree with or challenge your beliefs or the expectations put upon you?
  • Time is always a concern.  Is there value in building/constructing the learning together as a class, or is covering the curriculum standards good enough?  How might these two differ?  How would you like your students to experience mathematics?

As always, I’d love to hear your thoughts.  Leave a reply here 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!van-de-walle-streaming-video

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:

daro-starting-point

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!