Differentiated Instruction: comparing 2 subjects

I’ve been thinking a lot about how to meet the various needs of students in our classrooms lately.  If we think about it, we are REALLY good at differentiated instruction in subjects like writing, yet, we struggle to do differentiated instruction well in subjects like math.  Why is this???

In writing class, everyone seems to have an entry point.  The teacher puts a prompt up on the board and everyone writes.  Because the prompt is open, every student has something to write about, yet the writing of every student looks completely different. That is, the product, the process and/or the content differs for each student to some degree.

Teachers who are comfortable teaching students how to write know that they start with having students write something, then they provide feedback or other opportunities for them to improve upon.  From noticing what students do in their writing, teachers can either ask students to fix or improve upon pieces of their work, or they can ask the class to work on specific skills, or ask students to write something new the next day because of what they have learned.  Either way the teacher uses what they noticed from the writing sample and asks students to use what they learned and improve upon it!


In Math class, however, many teachers don’t take the same approach to learning.  Some tell every student exactly what to do, how to do it, and share exactly what the finished product should look like.  OR… in the name of differentiated instruction, some teachers split their class into different groups, those that are excelling, those that are on track, and those that need remediation.  To them, differentiated instruction is about ability grouping – giving everyone different things.  The two teachers’ thinking above are very different aren’t they!

Imagine these practices in writing class again.  Teacher 1 (everyone does the exact same thing, the exact same way) would show students how to write a journal (let’s say), explain about the topic sentence, state the number of sentences needed per paragraph, walk every student through every step.  The end products Teacher 1 would get, would be lifeless replications of the teacher’s thinking!  While this might build some competence, it would not be supporting young creative writers.

Teacher 2 (giving different things to different groups) on the other hand would split the class into 3 or 4 groups and give everyone a different prompt.  “Some of you aren’t ready for this journal writing topic!!!”  Students in the high group would be allowed to be creative… students in the middle group wouldn’t be expected to be creative, but would have to do most of what is expected… and those in the “fix-up” group would be told exactly what to do and how to do it.  While this strategy might seem like targeted instruction, sadly those who might need the most help would be missing out on many of the important pieces of developing writers – including allowing them to be engaged and interested in the creative processes.


Teacher 1 might be helpful for some in the class because they are telling specific things that might be helpful for some.

Teacher 2 might be helpful for some of the students in the class too… especially those that might feel like they are the top group.

But something tells me, that neither are allowing their students to reach their potential!!!


Think again to the writing teacher I described at the beginning.  They weren’t overly prescriptive at first, but became more focused after they knew more about their students.  They provided EVERYONE opportunity to be creative and do the SAME task!

In math, the most effective strategy for differentiating instruction, in my opinion, is using open problems.  When a task is open, it allows all students to access the material, and allows all students to share what they currently understand.  However, this isn’t enough.  We then need to have some students share their thinking in a lesson close (this can include the timely and descriptive feedback everyone in the group needs).  Building the knowledge together is how we learn.  This also means that future problems / tasks should be built on what was just learned.

ambuigity


We know that to differentiate instruction is to allow for differences in the products, content and/or processes of learning… However, I think what might differ between teacher’s ability to use differentiated instruction strategies is if they are Teacher-Centered… or Student-Centered!

Differentiated Instruction.jpg

When we are teacher-centered we believe that it is our job to tell which students should be working on which things or aim to control which strategies each student will be learning.  However, I’m not convinced that we would ever be able to accurately know which strategies students are ready for (and therefore which ones we wouldn’t want them to hear), nor am I convinced that giving students different things regularly is healthy for our students.

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.  Instead, consider using flexible grouping in which the size and makeup of small groups vary in a purposeful and strategic manner.  When coupled with the use of differentiation strategies, flexible grouping gives all students the chance to work successfully in groups. Van de Walle – Teaching Student Centered Mathematics

If we were to have students work on a problem in pairs, we need to be aware that grouping by ability as a regular practice can actually lead students to develop fixed mindsets – that is they start to recognize who is and who isn’t a math student.


Obviously there are times when some students need remediation, however, I think we are too quick to jump to remediation of skills instead of attempting to find ways to allow students to make sense of things in their own way followed by bringing the learning / thinking together to learn WITH and FROM each other.


To make these changes, however, I think we need to spend more time thinking about what a good problem or rich task should look like!  Maybe something for a future post?


As always, I want to leave you with a few reflective questions:

  • I chose to compare what differentiated instruction looks like in mathematics to what it looks like in writing class.  However, I often hear more comparisons between reading and mathematics.  Do you see learning mathematics as an expressive subject like writing or a receptive subject like reading?
  • Where do you find tasks / problems that offer all of your students both access and challenge (just like a good writing prompt)?  How do these offer opportunities for your students to vary their process, product and/or content?
  • Once we provide open problems for our students, how do you leverage the reasoning and representations from some in the room to help others learn and grow?
  • Math is very different than Literacy.  Reading and writing, for the most part, are skills, while mathematics is content heavy.  So how do you balance the need to continually learn new things with the need to continually make connections and build on previous understanding?
  • What barriers are there to viewing differentiated instruction like this?  How can we help as an online community?

For more on this topic I encourage you to read How do we meet the needs of so many unique students in a mixed-ability classroom?  or take a look at our Ontario Ministry’s vision for Differentiated Instruction in math: Differentiating Mathematics Instruction

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

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The Same… or Different?

For many math teachers, the single most difficult issue they face on a daily basis is how to meet the needs of so many students that vary greatly in terms of what they currently know, what they can do, their motivation, their personalities…  While there are many strategies to help here, most of the strategies used seem to lean in one of two directions:

  1. Build knowledge together as a group; or
  2. Provide individualized instruction based on where students currently are

Let’s take a closer look at each of these beliefs:

Those that believe the answer is providing all students with same tasks and experiences often do so because of their focus on their curriculum standards.  They believe the teacher’s role is to provide their students with tasks and experiences that will help all of their students learn the material.  There are a few potential issues with this approach though (i.e., what to do with students who are struggling, timing the lesson when some students might take much more time than others…).

On the other hand, others believe that the best answer is individualized instruction.  They believe that students are in different places in their understand and because of this, the teacher’s role is to continually evaluate students and provide them with opportunities to learn that are “just right” based on those evaluations.  It is quite possible that students in these classrooms are doing very different tasks or possibly the same piece of learning, but completely different versions depending on each student’s ability.  There are a few difficulties with this approach though (i.e., making sure all students are doing the right tasks, constantly figuring out various tasks each day, the teacher dividing their time between various different groups…).

 


There are two seemingly opposite educational ideals that some might see as competing when we consider the two approaches above:  Differentiated Instruction and Complexity Science.  However, I’m actually not sure they are that different at all!

 

For instance, the term “differentiated instruction” in relation to mathematics can look like different things in different rooms.  Rooms that are more traditional or “teacher centered” (let’s call it a “Skills Approach” to teaching) will likely sort students by ability and give different things to different students.

teaching approaches touched up.png
From Prime Leadership kit
If the focus is on mastery of basic skills, and memorization of facts/procedures… it only makes sense to do Differentiated Instruction this way.  DI becomes more like “modifications” in these classrooms (giving different students different work).  The problem is, that everyone in the class not on an IEP needs to be doing the current grade’s curriculum.  Really though, this isn’t differentiated instruction at all… it is “individualized instruction”.  Take a look again at the Monograph: Differentiating Mathematics Instruction.

Differentiated instruction is different than this.  Instead of US giving different things to different students, a student-centered way of making this make sense is to provide our students with tasks that will allow ALL of our students have success.  By understanding Trajectories/Continuum/Landscapes of learning (See Cathy Fosnot for a fractions Landscape), and by providing OPEN problems and Parallel Tasks, we can move to a more conceptual/Constructivist model of learning!

Think about Writing for a moment.  We are really good at providing Differentiated Instruction in Writing.  We start by giving a prompt that allows everyone to be interested in the topic, students then write, we then provide feedback, and students continue to improve!  This is how math class can be when we start with problems and investigations that allow students to construct their own understanding with others!


The other theory at play here is Complexity Science.  This theory suggest that the best way for us to manage the needs of individual students is to focus on the learning of the class.

Complexity1.jpg
What Complexity Science Tells us about Teaching and Learning

The whole article is linked here if you are interested.  But basically, it outlines a few principles to help us see how being less prescriptive in our teaching, and being more purposeful in our awareness of the learning that is actually happening in our classrooms  will help us improve the learning in our classrooms.  Complexity Science tells us to think about how to build SHARED UNDERSTANDING as a group through SHARED EXPERIENCES.  Ideally we should start any new concept with problem solving opportunities so we can have the entire group learn WITH and FROM each other.  Then we should continue to provide more experiences for the group that will build on these experiences.


Helping all of the students in a mixed ability classroom thrive isn’t about students having choice to do DIFFERENT THINGS all of the time, nor is it about US choosing the learning for them… it should often be about students all doing the SAME THING in DIFFERENT WAYS.  When we share our differences, we learn FROM and WITH each other.  Learning in the math classroom should be about providing rich learning experiences, where the students are doing the thinking/problem solving.  Of course there are opportunities for students to consolidate and practice their learning independently, but that isn’t where we start.  We need to start with the ideas from our students.  We need to have SHARED EXPERIENCES (rich problems) for us to all learn from.


 

As always, I leave you with a few questions for you to consider:

  • How do you make sure all of your students are learning?
  • Who makes the decisions about the difficulty or complexity of the work students are doing?
  • Are your students learning from each other?  How can you capitalize on various students’ strengths and ideas so your students can learn WITH and FROM each other?
  • How can we continue to help our students make choices about what they learn and how they demonstrate their understanding?
  • Do you see the relationship between Differentiated Instruction and the development of mathematical reasoning / creative thinking?  How can we help our students see mathematics as a subject where reasoning is the primary goal?
  • How can we foster playful experience for our students to learn important mathematics and effectively help all of our students develop at the same time?
  • What is the same for your students?  What’s different?

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

How do we meet the needs of so many unique students in a mixed-ability classroom?

Explaining what something is can be really hard to do without that person actually experiencing the same thing as you.  One strategy that we often use to explain difficult concepts in math is to discuss non-examples.  Consider how the frayer model below could be used with any difficult concept you are discussing in class.

frayer-model

If we discussed fractions in class, many students might believe that they understand the concept, however, they might be over-generalizing.  Seeing non-examples would help all gain a much clearer idea of what fractions are.  Of the shapes below, which ones have 1/2 or 1/4 of the area shaded blue?  Which ones do not represent 1/2 or 1/4 of the shapes’ area?

Having students complete activities like this would be an excellent way for you to originally see students’ understanding, and then see students refine and develop ideas throughout the unit (they can continually add different models and correct misconceptions)


The purpose of this post, however, isn’t about fractions or even a Frayer Model.  I am actually writing about the often used phrase “Differentiated Instruction” (DI).  Hopefully we can think more about what DI looks like in our math classes by thinking about what DI is and isn’t.


How would you define Differentiated Instruction?

Take a look at the following graphic created by ASCD helping us think about what DI is and isn’t.

di

In many places, DI is looked at as grouping students by ability, or providing individualized instruction.  However, if you look at the graphic above, these are in the non-example section.  These views are probably more common in highly teacher-centered classrooms, where the teacher feels like they need to be in control of every student’s process, products and content.  For me, I don’t see how it is possible to have every student doing the same thing at the same time, or how productive it would be if we assigned different students to do different work (sounds like an access and equity issue here).


So how do we help all of our students in a mixed ability classroom???

First of all, a few words from Van de Walle’s Teaching Student Centered Mathematics p.43:

All [students} do not learn the same thing in the same way at the same rate.  In fact, every classroom at every grade level contains a range of students with varying abilities and backgrounds.  Perhaps the most important work of teachers today is to be able to plan (and teach) lessons that support and challenge ALL students to learn important mathematics.

Teachers have for some time embraced the notion that students vary in reading ability, but the idea that students can and do vary in mathematical development may be new.  Mathematics education research reveals a great deal of evidence demonstrating that students vary in their understanding of specific mathematical ideas.  Attending to these differences in students’ mathematical development is key to differentiating mathematics instruction for your students.

Interestingly, the problem-based approach to teaching is the best way to teach mathematics while attending to the range of students in your classroom.  In a traditional, highly directed lesson, it is often assumed that all students will understand and use the same approach and the same ideas as determined by the teacher.  Students not ready to understand the ideas presented by the teacher must focus their attention on following rules or directions without developing a conceptual or relational understanding (Skemp, 1978).  This, of course, leads to endless difficulties and can leave students with misunderstandings or in need of significant remediation.  In contrast, in a problem-based classroom, students are expected to approach problems in a variety of ways that make sense to them, bringing to each problem the skills and ideas that they own.  So, with a problem-based approach to teaching mathematics, differentiation is already built in to some degree.


When we take a student centered view of Differentiated Instruction, we start to see that all students can be given the SAME work, yet each individual student will be able to adjust the process, product and/or content naturally.  However, this requires us to start with things where students are going to make sense of them.  It requires us to move our instruction from a teaching FOR problem solving approach to a teaching THROUGH problem solving approach.  It requires us to offer things that are actually problems, not just practicing skills in contexts.


3 Strategies for Differentiating Instruction:

There seem to be 3 different ways we can help all of our students access to the same curriculum expectations, while attending to the various differences in our students:

  1. Open-Middle Problems
  2. Open-Ended Problems
  3. Parallel Problems

Open-middle problems, or open routed problems as they are sometimes called, typically have 1 possible solution, however, there are several different strategies or pathways to reach the solution.  These are a great way for us to offer something that everyone will have access to.  Ideally, these problems need to have an entry point that all students enter into the problem with, yet offer extensions for all.  The benefit of this type of problem is that we can listen to and learn from our students about the strategies they use.  We can then use the 5 Practices as a way to move instruction forward based on our assessments.

Open-ended problems, on the other hand, typically have multiple plausible answers.   These problems, in contrast, offer a much wider range of content.  Again, the benefit of this type of problem is that we can listen to and learn from our students about the types of thinking they are currently using, and from there, consider what they are ready for next.  Again, we can then use the 5 Practices as a way to move instruction forward based on our assessments.

Parallel Problems differ from the other two types in that we are actually offering different things for our students to work on.  Hopefully, our students are given choice here as to which path they are taking, so we don’t run into the issue earlier posted about DI not being about ability grouping.  Parallel problems are aptly named because while some of the pathways are easier than others, all pathways are designed to meet the same curricular expectations (content standards).  Again though, we should be using the 5 Practices as a way to share student thinking with each other so our students can learn WITH and FROM each other and so we can move instruction forward based on our assessments.


For an in depth understanding of how these help us, please read the article entitled Differentiating Mathematics Instruction.


As always, I want to leave you with a few reflective questions:

  • How do you balance the need to teach 1 set of standards with the responsibility of meeting your students where they are?
  • Do you hear student centered messages like the ones I’ve posted here about DI, or more teacher centered messages?  Which set of messages do you believe is easier for you to attempt as a teacher?  Which set of messages would you believe would make the learning in your classroom richer?
  • Who tends to participate in your classrooms?  Who tends to not participate?  How can the DI strategies above help change this dynamic so that the voices being heard in your classroom are more distributed?
  • What issues do you see being a barrier to DI looking like this?  How can the online community help?

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