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.


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 ).

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?

Taken from PRIME Leadership

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 is 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 curriculum expects.  What is needed is 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 we are 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…

Seeking Challenges in Math

I was working with a grade 7 teacher and his students a while back.  The teacher came to me with an interesting problem, his students were doing quite well in math (in general) but only wanted to do work out of textbooks, only wanted to work independently, and were very mark-driven. The teacher wanted his students to start being able to solve non-routine problems, not just be able to follow the directions from the textbook, and he wanted his students to see the value in working collaboratively and to listen to each other’s thoughts.

Our conversations quickly moved to the topic of mindsets. It sounded like many of his students had fixed mindsets, and didn’t want to take any risks.

For those of you who are not familiar with growth and fixed mindsets, students with fixed mindsets believe that their ability (in math for example) is an inborn trait.  They believe how smart they are in math is either a gift or a curse they are born with.  Those with growth mindsets, however, believe that their ability improves over time with the right experiences, attitude and effort.

When confronted with challenges, those with growth mindsets are willing to struggle, willing to make mistakes, knowing that they will continue to learn and grow throughout the learning process.  On the other hand, those that have fixed mindsets tend to avoid challenges.  They believe that struggle, making mistakes, and being challenged are signs of weakness.  Psychologically, they will avoid the feeling of discomfort in not knowing, as this threatens their belief about how smart they are.

Knowing this, we devised a plan to see whether or not his students were able to take on challenges.  We started the class by giving each student their own unique 24 card (see below).

24 card.jpeg

We explained that each card had 4 numbers that could be manipulated to equal 24.  For instance, the card above could be solved by doing 5 x 4 x 1 + 4 = 24.

We then explained that we would give them time to solve their own card (which had a front and a back), and that we would give them additional cards if they completed both problems.  We also explained the little white dots in the center of the card, 1 dot being an easy card, 2 dots being more complicated, and 3 dots being the most difficult.

As students continued to work, we noticed some students eagerly trying to solve the cards, and others starting to become frustrated by others’ successes.  After a few minutes, the first few students had completed both problems and asked for their next card.  We asked, “Would you like another easy card, or would you like to challenge yourself?” to which the vast majority asked for another easy card.  In fact, some students completed many cards, front and back, all at the easy level, never accepting a more challenging card (even bragging to others about how many they had completed).  Others, after giving up pretty quickly, asked if they could work with a classmate to make a pair.  While we were happy at first with this, none of the pairs had students working cooperatively together for most of the time.

Take a look at some of the challenging cards.  What do you do when confronted with something challenging?  Do you skip it and move on, or do you keep trying?


As soon as we were finished, we showed the class this video:

Watch the 3 minute video above as it ties in perfectly with the 24 problem from above.  We had a quick discussion about the video and why some of the students wanted to choose the easier puzzles.  The class quickly saw the parallels between the problems we had just done and the video.

While we had a great discussion about fixed and growth mindsets, it took most of the year to be able to get this group to see the value in collaboration, to focus on their learning instead of their marks, to be able to take on challenges and not get frustrated when they didn’t have immediate success.

Changing our mindset takes time and the right experiences!

I am really interested in why students who believe themselves to be “smart” at math would opt out of challenging themsleves.

Do any of your students exhibit any of the same signs as these students:

  • Not comfortable with tasks that require thinking
  • Eager for formulas and procedures
  • Competitive with others to show they are “smart”
  • Preference to work alone
  • Preference to work out of textbooks/ worksheets instead of on rich problems/tasks
  • See math as about being fast / right, not about thinking / creativity
  • Eager to do easy work that is repetative


So I leave you with some reflective questions:

What previous experiences must these students have had to create such fixed mindsets?

What would you do if your students avoided challenges?

What would you do if your students groaned each time you asked them to work with a partner?

How are you helping your students gain a growth mindset in math?

Can you recognize those in your class that have fixed mindsets?  Are you noticing those from different achievement levels, or just those who are struggling?

If our students find everything we do “easy” what will happen to them when they get to a math course that actually does offer them some challenge???



P.S.  Did you solve any of the 24 cards above?  Did you skip over them?  What do you typically do when confronted with challenges?

Who makes the biggest impact?

A few years ago I had the opportunity to listen to Damian Cooper (expert on assessment and evaluation here in Ontario). He shared with us an analogy talking to us about the Olympic athletes that had just competed in Sochi.  He asked us to think specifically about the Olympic Ice Skaters…

He asked us, who we thought made the biggest difference in the skaters’ careers:  The scoring judges or their coaches?

Think about this for a second…  An ice skater trying to become the best at their sport has many influences on their life…  But who makes the biggest difference?  The scoring judges along the way, or their coaches?  Or is it a mix of both???

Damian told us something like this:

The scoring judge tells the skater how well they did… However, the skater already knows if they did well or not.  The scoring judge just CONFIRMS if they did well or not.  In fact, many skaters might be turned off of skating because of low scores!  The scoring judge is about COMPETITION.  Being accurate about the right score is their goal.

On the other hand, the coach’s role is only to help the skater improve. They watch, give feedback, ask them to repeat necessary steps… The coach knows exactly what you are good at, and where you need help. They know what to say when you do well, and how to get you to pick yourself up. Their goal is for you to become the very best you can be!  They want you to succeed!

In the everyday busyness of teaching, I think we often confuse the terms “assessment” with “evaluation”   Evaluating is about marking, levelling, grading… While the word assessment comes from the Latin “Assidere” which means “to sit beside”.  Assessment is kind of like learning about our students’ thinking processes, seeing how deeply they understand something…   These two things, while related, are very different processes!


I have shared this analogy with a number of teachers.   While most agree with the premise, many of us recognize that our job requires us to be the scoring judges… and while I understand the reality of our roles and responsibilities as teachers, I believe that if we want to make a difference, we need to be focusing on the right things.  Take a look at Marian Small’s explanation of this below.  I wonder if the focus in our schools is on the “big” stuff, or the “little” stuff?  Take a look:


Marian Small – It’s About Learning from LearnTeachLead on Vimeo.

Thinking again to Damian’s analogy of the ice skaters, I can’t help but think about one issue that wasn’t discussed.  We talked about what made the best skaters, even better, but I often spend much of my thoughts with those who struggle.  Most of our classrooms have a mix of students who are motivated to do well, and those who either don’t believe they can be successful, or don’t care if they are achieving.

If we focus our attention on scoring, rating, judging… basically providing tasks and then marking them… I believe we will likely be sending our struggling students messages that math isn’t for them.  On the other hand, if we focus on providing experiences where our students can learn, and we can observe them as they learn, then use our assessments to provide feedback or know which experiences we need to do next, we will send messages to our students that we will all improve.

Hopefully this sounds a lot like the Growth Mindset messages you have been hearing about!

Take a quick look at the video above where Jo Boaler shows us the results of a study comparing marks vs feedback vs marks & feedback.

So, how do you provide your students with the feedback they need to learn and grow?

How do you provide opportunities for your students to try things, to explore, make sense of things in an environment that is about learning, not performing?

What does it mean for you to provide feedback?  Is it only written?

How do you use these learning opportunities to provide feedback on your own teaching?

As  always, I try to ask a few questions to help us reflect on our own beliefs.  Hopefully we can continue the conversation here or on Twitter.


So you want your students to have a Growth Mindset?

There has been much talk about Growth Mindset the past few years.  Many teachers recognize that there are a lot of students who exhibit a fixed mindset.  I often wonder when we introduce another educational term into the mainstream how many different ways the term can be misinterpreted.  Ask someone what they think about the terms growth or fixed mindsets and really listen to what they say.  I bet you would be surprised!

When others talk to me about mindsets, I often hear about students who struggle, needing to change their mindset… Makes sense doesn’t it.  We see a group of students in our school who don’t apply themselves, and we wish that they would just realize they could do much better if they just put forth more effort.  But is this what the whole growth/fixed mindset conversation is all about?

Those who see growth and fixed mindsets being about effort often think that encouraging their students with phrases or posters is what they need to succeed.  When we believe this, we might start to see a neat looking poster or bulletin board like these:


However, a poster alone might send a message to students that THEY are doing something wrong, THEY need to do something different… and that if they only tried harder, changed their words, became better people… that they would do better in math.

Personally, I don’t think this is all what the issue is with our students.  And I don’t believe this is what the research behind mindsets is pointing to either!  If we are looking to change our students’ mindsets, we need to do two things:

  1. Learn what it means to have a growth / fixed mindset.  We might be misinterpreting the whole idea here!
  2. Change OUR actions to promote growth mindsets.  Our students will only improve when WE change!

1.  Learning about Growth and Fixed Mindsets

While there are a lot of places you can go to learn more about a growth mindset, I think we need to make sure we are hearing from the experts. Here are a few I think you need to take a look at:

While I don’t want to spend much time restating what is already in these articles/videos/books, I do really hope you will check out at least one of the new ones you haven’t already seen.

I will give a few key points about the whole mindset movement though:

  • Students with fixed mindsets believe that they either have or do not have a math brain.  They see their intelligence as static or hereditary (a gift or a curse)
  • Students and adults across the achievement spectrum can exhibit a growth or fixed mindset (high achieving students are just as likely to have a fixed mindset).
  • Students with fixed mindsets are likely to avoid challenges and likely give up easily.
  • It is more common for a student to have a fixed mindset about mathematics than other subjects.
  • Our actions and words will either promote a fixed or growth mindset in others.

Do these 5 points fit your current understanding of Mindsets?  If any of these do not fit your thinking, please make an effort to read/watch some of the articles, videos linked above.

2.  Change OUR Actions to Promote Growth Mindsets

I think this is the part of the conversation that is completely missed!  We tend to focus on what everyone else needs to do instead of how we need to change.

Growth Mindset - My quote.jpg

Think again about the message above.  What are the common practices that happen in many math classrooms that help our students believe they are smart at math, or that they are not smart at math?

Practices that promote fixed mindsets:

  • Providing students with a large number of closed questions
  • Math fluency is often conducted via timed tests
  • Evaluating each assignment, especially early in the learning process
  • Grouping / Sorting students by ability
  • Providing different activities for different students
  • Asking struggling students to speak first, then more advanced students next
  • Competitive mathematics tasks (often based on speed)

Think about how each of these practices helps our students identify with mathematics.  Their mathematical identity is based on how WE present their successes.  Each of the practices above treats mathematics as a performance subject.  Students in classrooms like these are far more likely to believe that they are either a math person or not because the tasks ask them to recognize if they are better or worse than others in the room.

For example, students who are routinely given a page of closed questions to complete come to believe their role is to get them right.  Students who get them all right are likely to see that they are naturally good at math, while those who struggle through the questions are likely to see their “ability” in more negatively.  There is little room for students to see that they are growing!

On the other hand, teachers who view mathematics as a learning subject are more likely to have practices that allow students learn and grow.

Practices that promote growth mindsets:

  • Provide students with open questions (open-ended problems with more than 1 possible answer, or open-middle problems with more than 1 potential strategy to achieve the answer)
  • Math fluency/flexibility is based on reasoning and development of strategies
  • Assessment of students is non-evaluative and instead focuses on feedback
  • Grouping of students is flexible in nature
  • Open problems (low floor, high ceiling problems) provide natural differentiated instruction as they offer challenge for every student
  • All students are expected to contribute.  Students sharing in an important part of the learning, and all students recognize that all members can contribute
  • Cooperative tasks where students learn WITH and FROM each other


Providing experiences like the items above will more likely allow your students to see that they are capable of growing and learning since the focus is on allowing your students time to develop.  These experiences help move the focus away from whether or not students are already good at math, toward a focus on the the learning today.  Allowing our students room for growth promotes a growth mindset.

Think about the lists above.  There are possibly a few that might challenge your thinking! Many hear some of the messages about the types of practices that promote fixed mindsets and struggle to know what to do instead.  Moving toward classrooms that promote growth mindsets is not about lowering the bar, in fact, classrooms that are successfully moving in this direction are expecting much more of their students… and are especially helping our students who begin the year struggling.

If you or your school or your district are interested in taking a focus on growth mindset, I encourage you to not just consider the pedagogy involved, but to actually focus on the practices that will help us to understand the math deeper ourselves!

For example, to focus on growth mindset, we as teachers need to be able to understand the math deeper in order to be able to teach mathematics in way that allows our students to see mathematics as a learning subject.  Here are a few things we as teachers, as schools and/or districts that we can focus on to help us make these shifts:

  1. Consider focusing on tasks that help us see that we can develop number sense through the use of strategies, visual representations and the big ideas behind the numbers.  Number Talks or Strings are wonderful practices that both allow our students to develop appropriate number sense, think flexibly about operations, yet allow for creativity and reasoning to develop.  Our learning here needs to include understanding the developmental models, strategies and big ideas appropriate for our students to develop number sense.
  2. Consider focusing on opening up problems.  Accessing resources like Marian Small’s Open Questions for the 3-Part Lesson/Eyes on Math/Good Questions… or Cathy Fosnot’s Contexts for Learning units… or other sources that might help us see how we can use open problems as part of a sequence of learning.  These, along with structures like the 5 Practices for Orchestrating productive mathematics discussions, can help us recognize how we can learn THROUGH problem solving.
  3. Consider focusing on mini-lessons that focus on student reasoning, conceptual understanding, the use of visuals and spatial reasoning.  This can include using tasks like Estimation 180, Which One Doesn’t Belong, Fraction Talks, Visual Patterns, 101Qs Problems, SolveMe Mobiles… in a way that promotes student thinking, students sharing, and the development of reasoning together.  This can also include the development of and use of instructional routines like Notice and Wonder or Contemplate then Calculate.  Whichever of these we explore, we need to learn how to give our students the opportunity to think and reason… share with each other… learn FROM and WITH each other.  And to do these well, WE need to continue to learn the important mathematics behind the tasks, not just use them because they are fun or neat.
  4. Consider a focus on assessment.  If we want classrooms that promote a growth mindset, we need to reconsider what information we send our students about what is important.  Learning about developmental progressions, how to give feedback effectively, tasks that allow us to make observations, listen to students as they are thinking, have conversations in-the-moment as students are problem solving.  When we deepen our understanding of the mathematics our students are learning, we will become better at using assessment effectively.  While many believe they are focusing on growth mindset by looking at spreadsheets, this is exactly the opposite of what needs to be focused on.


Whatever your focus, keep in mind the unintended messages we send our students about who is a math student… and continue OUR learning about the mathematics itself so we know how to help our students as they struggle (developmentally appropriate tasks/problems).

Teaching for a growth mindset will require us to make changes in our beliefs, but also in our practices!  Let’s keep learning my friends!!!


If interested, here are a few more posts that might help us see how to make these ideals a reality: