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).
Thinking Mathematically 