## 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.

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:

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).

## Aiming for Mastery?

The other day, a good friend of mine shared this picture with me asking me what my thoughts were:

On first glance, my thoughts were mixed.  On one hand, I really like that students are receiving messages about mistakes being part of the process and that there is more than one opportunity for students to show that they understand something.

However, there is something about this chart that seems missing to me…  If you look at the title, it says “How to Learn Math” and the first step is “learn a new skill”.  Hmmm…… am I missing something here?  If the bulletin board is all about how we learn math, the first step can’t be “learn a new skill”.  For me, I’m curious about HOW the students learn their math?  While this is probably the most important piece (at least to me), it seems to be completely brushed aside here.

I want you to take a look at the chart below.  Which teaching approach do you think is implied with this bulletin board?  What do you notice in the “Goals” row?  What do you notice in the “Roles” row?  What do you notice in the “Process” row?

If “Mastery” is the goal, and most of the time is spent on practicing and being quizzed on “skills” then I assume that the process of learning isn’t even considered here.  Students are likely passively listening and watching someone else show them how… and the process of learning will likely be drills and practice activities where students aim at getting everything right.

I’d like to offer another view…

Learning is an active process.  To learn math means to be actively involved in this process:

• It requires us to think and reason…
• To pose problems and make conjectures…
• To use manipulatives and visuals to represent our thinking…
• To communicate in a variety of ways to others our thinking and our questions…
• To solve new problems using what we already know
• To listen to others’ solutions and consider how their solutions are similar or different than our own…
• To reflect on our learning and make connections between concepts…

It’s this process of learning that is often neglected, and often brushed aside.

While I do understand that there are many skills in math, seeing mathematics as a bunch of isolated skills that need to be mastered is probably neither a positive way to experience mathematics, nor will it help us get through all of the material our standards expects.  What is needed are deeper connections, not isolating each individual piece… more time spent on reasoning, not memorizing each skill… richer tasks and learning opportunities, not more quizzes…

“One way to think of a person’s understanding of mathematics is that it exists along a continuum. At one end is a rich set of connections. At the other end of the continuum, ideas are isolated or viewed as disconnected bits of information. A sound understanding of mathematics is one that sees the connections within mathematics and between mathematics and the world”
TIPS4RM: Developing Mathematical Literacy, 2005

So, while I don’t think putting up a bulletin board (or not) is really going to do much, I really do hope we are spending more of our time thinking about HOW our students learn and WHAT our goals are for our students (see the chart above again).

I still wonder what it means to be good at math?  I wrote about my questions here, but I am still looking for ways to show others how important mathematical reasoning is for students to develop.  Skills without reasoning won’t get you very far.  Maybe more about this another time…

Open problems are probably my most used strategy to help meet each student where they are.  Problems that offer a low floor and high ceiling are great because all students engage in the learning, then can participate and learn from each other.  However, some teachers also like to offer Parallel Tasks as a way to differentiate instruction.  The idea here is that students can be given a choice of a task/problem, some being more difficult than others, yet all of the tasks/problems deal with the same standard (curriculum expectation).  Let’s take a look at an example of how a quality parallel task can work:

Take a look at the problem below.  What is it asking us to do?

I’d love some actual responses here.  Build it using actual manipulatives (ideally) or using virtual manipulatives (This Illuminations Applet might help).

Notice that each choice allows students to do the same expectation related to proportional thinking, however, students are given choice about what numbers they want to think about.

Think about what the answers would look like?  When we discuss designs afterward, we should be able to discuss the solutions to each problem and compare the similarities and differences.

Here are some designs students made.  Can you tell which option each student chose?

Actually try to match the designs to each of the tasks/problems.  Take a moment to think this through.  What do you notice about the 4 images above?

This task was designed very cleverly to help make a point… to help us bring ALL of our students together to have a conversation  (Even when we ask our students to do different things from each other, we still need to make sure we come together and have shared experiences).

Did you notice anything about the 3 options?  Did you try decimals or fractions to solve any of the students’ designs?  If you did, you would notice that all 3 options used the same proportions.

A great parallel task helps us to learn things together…  It helps us see others’ thinking…  It allows every student to start to think where they are comfortable, yet be able to learn and grow from the ideas of others.

If you were to offer a parallel problem/task for your students would you:

• Choose which students get each choice, or allow students to pick themselves? (Does this matter?)
• Expect all students to create the design using blocks or digitally?  (Does this matter?)
• Ask students to work independently or in a small group?  (Does this matter?)
• Offer calculators or not?  (Does this matter?)
• Engage in 3 different conversations – 1 per group – or 1 conversation all together?  (Does this matter?)

The small decisions we make tell a lot about what we value!    Personally, IF I want to offer Parallel tasks/problems, I want to make sure that all of my students feel successful, that everyone realize their ideas are valued, that there isn’t a hierarchy of ability in the room… and of course, that the mathematics we are engaged is important.