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.

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.

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:

- Open-Middle Problems
- Open-Ended Problems
- 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).

Over the past 2 years I have been working with teachers in my school on using the 5 practices to plan and teach their instruction. It has transformed our teachers ability to design and guide tasks have a low floor and high ceiling. It has allowed teachers to let students to do most of the talking and most of the doing…2 key components of a differentiated classroom. This transformation has and continues to be hard work. First and foremost, we have worked on developing the deeper relational understandings that are necessary to plan tasks that allow students to discover the math and allow the teachers to weave the web of student work that fosters a comprehensive understanding of the ideas, practices and skills that exist within their work. Doing the math and spending time on the anticipation stage is key to being able to do this. Having a plan for what questions to ask, how to make thinking visible and how to sequence the work enables to to balance differentiation and the structure of the standards. It allows us to have freedom within the framework. While still a work in progress, teachers are seeing the value in this approach and are excited to see the changes they see in all students.

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I can’t tell you how awesome it is to hear how the hard work of so many teachers is paying off. Teaching is hard work! Learning to teach in ways that we never experienced as students takes lots of time and much support.

Thanks for sharing!

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It’s so great that technology has allowed us the capability of connecting and sharing with other educators and having meaningful discussions through professional learning communities. Feeling like so have more support through generous teachers who take the time to create sites and blogs to share their ideas and resources has really given me the confidence to try something new in the classroom. Thank you!

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Thanks for this post! I am a new(ish) Math coach in my district in northern BC and we have a large number of classrooms with very challenging compositions (complex academic, behaviour and social-emotional needs)… The most common question I am asked by overwhelmed teachers is “how do I differentiate in math?” – many of our schools are looking to find a streaming model that will make teaching easier. One of my plans this year is to create a collection of low-floor/high-ceiling resources that will support teachers to teach math with a DI model. I need an accessible alternative to offer teachers who are convinced that streaming will solve their composition challenges. I always appreciate the great resources and articles you share and your thoughtful posts! Thanks for helping me to move my own learning forward 🙂

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Hi Tamdeford. When our own experiences with mathematics has been showing students how to do a specific skill, followed by a handing out a bunch of closed questions, we think DI is about giving students different things. However, this creates many issues (fixed mindsets for teachers and students, more management issues for teachers now that many different things are being done at the same time, difficulty to achieve curriculum expectations, equity and access issues…..

Some of my first experiences with opening up tasks came from using the SuperSource materials, Cathy Fosnot’s contexts for learning resources and Marian Small’s work. Having many identified students in my classroom I needed things that allowed all of us to work on the same curricular standards, while allowing us to have shared experiences where we could learn WITH and FROM each other.

Today, there are so many resources available to us, but knowing which ones offer developmentally appropriate entry points, and then knowing where to go from there is difficult. I wonder if you have ever used any of Cathy Fosnot units? or explored Van de Walle before? or CGI?

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Thanks for the tips – I have used Marian Small and Van de Walle resources before, but not Cathy Fosnot or CGI, so I will look into them. I know that a big barrier for many teachers is a lack of time to look for or develop just the right problem or context on a daily basis. Especially for those teachers who don’t feel knowledgeable enough in math themselves, this is a huge undertaking. Thank-you!

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What are your thoughts regarding the best ways to teach a split grade classroom, where some of the standards are very different between the grades? In all of my reading (published books and Twitter) over the past few years, I haven’t seen this subject discussed. Thanks.

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This hasn’t been easy topic. Really it’s all about decision-making and knowing why you’ve made those decisions. Some people aim to teach every concept to everybody as a whole group (possibly “extending” for the older group), while others tend to lead to very separate and distinct lessons (one group works while one group listens). Both of these examples lack the nuance needed to successfully navigate two curricula at the same time. Really what we need to think about is the balance between the content we want our students leaving with, which are stated in our standards or expectations, and the mathematical practises standards or processes. If we focus solely on the content we will likely miss out on the rich conversations needed to build a community that learns together. If we Soley focus on the standards of practice, our students might not be prepared to learn a new concept next year.

So, I would suggest when possible, We aim to have everybody participate in the same learning so that we can hold rich conversations together about what we are learning. But when as you said the expectations from two grades are very different, this strategy won’t make sense. Instead we can have students working on different things. But ideally, one group would be doing something practice-ish that requires little to no teacher guidance, while the other group is learning something new (engaging in a problem that will lead to important discussions).

I would also suggest that in a split class the teacher should be doing far less talking because they will need to be navigating their students’ thinking more. You are definitely right that many administrators or educational gurus or researchers discount many of the things that are real struggles in classrooms. Probably because it has more to do with very specific instructional decisions than simple routines to follow. One of the best ways to really discuss this is to actually plan for a specific concept between 2 grades, then actually try to Do the learning together. The real learning would come when we analyze which decisions we made and why. Let me know if you have a specific topic in mind I’d love to continue chatting.

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Thanks so much for your quick and honest reply. What you’ve suggested sounds a lot like what I’ve been doing, but I know there is more that I can tweak to make sure that the students who are working without me are working on the best tasks possible. It’s just hard to find a lot of ideas for those best tasks, or even suggestions on what that could look like.

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Just noticed a typo in my email address as I posted. This comment is linked to the correct email address (sorry!)

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