…a child first has to learn the foundational skills of math, like______?

I’ve been spending a lot of time lately observing students who struggle with mathematics, talking with teachers about their students who struggle, and thinking about how to help.  There are several students in my schools who experience difficulties beyond what we might typically do to help.  And part of my role is trying to think about how to help these students.  It seems that at the heart of this problem, we need to figure out where our students are in their understanding, then think about the experiences they need next.

However, first of all I want to point out just how difficult it is for us to even know where to begin!  If I give a fractions quiz, I might assume that my students’ issues are with fractions and reteach fractions… If I give them a timed multiplication test, I might assume that their issues are with their recall of facts and think continued practice is what they need… If I give them word problems to solve, I might assume that their issue is with reading the problem, or translating a skill, or with the skill itself…….  Whatever assessment I give, if I’m looking for gaps, I’ll find them!

So where do we start?  What are the foundations on which the concepts and skills you are doing in your class rest on?  This is an honest question I have.

Take a look at the following quote.  How would you fill in the blank here?



Really, take a minute to think about this.  Write down your thoughts.  In your opinion, what are the foundational skills of math?  Why do you believe this?  This is something I’ve really been reflecting on and need to continue doing so.

I’ve asked a few groups of teachers to fill in the blank here in an effort to help us consider our own beliefs about what is important.  To be honest, many of the responses have been very thoughtful and showcased many of the important concepts and skills we learn in mathematics.  However, most were surprised to read the full paragraph (taken from an explanation of dyscalculia).

Here is the complete quote:

Taken from Dyscalculia Headlines


Is this what you would have thought?  For many of us, probably not. In fact, as I read this, I was very curious what they meant by visual perception and visual memory.  What does this look like?  When does this begin?  How do we help if these are missing pieces later?

Think about it for a minute.  How might you see these as the building blocks for later math learning?  What specifically do these look like?  Here are two excerpts from Taking Shape that might help:

Excerpt from Taking Shape
Excerpt from Taking Shape

Visual perception and visual memory are used when we are:

  • Perceptual Subitizing (seeing a small amount of objects and knowing how many without need of counting)
  • Conceptual subitizing (the ability to organize or reorder objects in your mind. To take perceptual subitizing to make sense of visuals we don’t know)
  • Comparing objects’ sizes, distances, quantities…
  • Composing & decomposing shape (both 2D or 3D)
  • Recognizing, building, copying symmetry designs (line or rotational)
  • Recognizing & performing rotations & reflections.
  • Constructing & recognizing objects from different perspectives
  • Orienting ourselves, giving & following directions from various perspectives.
  • Visualizing 3D figures given 2D nets

While much of this can be quite complicated (if you want, you can take a quick test here), for some of our students the visual perception and visual memory is not yet developed and need time and experiences to help them mature. Hopefully we can see how important these as for future learning.  If we want our students to be able to compose and decompose numbers effectively, we need lots of opportunities to compose and decompose shapes first!  If we want our students to understand quantity, we need to make sure those quantities make sense visually and conceptually. If we want students to be flexible thinkers, we need to start with spatial topics that allow for flexibility to be experienced visually.  If we want students to be successful we can’t ignore just how important developing their spatial reasoning is!

In our schools we have been taking some of the research on Spatial Reasoning and specifically from the wonderful resource Taking Shape and putting it into action.  I will be happy to share our findings and action research soon.  For now, take a look at some of the work we have done to help our students develop their visual perception and visual memory:

Symmetry games:

Composing and decomposing shapes:

Relating nets to 3D figures:

Constructing unique pentominoes:

And the work in various grades continues to help support all of our students!

So I leave you with a few questions:

  • What do you do with students who are really struggling with their mathematics?  Have you considered dyscalculia and the research behind it?
  • How might you incorporate spatial reasoning tasks / problems for all students more regularly?
  • Where in your curriculum / standards are students expected to be able to make sense of things visually?  (There might be much more here than we see at first glance)
  • How does this work relate to our use of manipulatives, visual models and other representations?
  • What do we do when we notice students who have visual perception and visual memory issues beyond what is typical?
  • How can Doug Clements’ trajectories help us here?
  • If we spend more time early with students developing their visual memory and visual perception, will fewer students struggle later???

I’d love to hear your thoughts.  Leave a comment here or on Twitter (@MarkChubb3).


Reflecting on 2016

Last June (2016) I started writing this blog.  I’m not exactly sure what got me started to be honest, probably because I have been inspired by so many others’ blogs, possibly  thanks to @MaryBourassa’s encouragement!  Whatever helped me get started, I am still not sure WHY I am blogging.

Some things I DO know:

  • I started writing last June
  • I wrote 32 blog posts last year
  • People from 123 countries have been reading
  • I try to include pedagogical decisions and mathematical content in every post
  • I tend to write more when I should be working on other things:)


My 10 most popular posts were:

  1. So you want your students to have a Growth Mindset?
  2. Concept vs Procedure: An anecdote about what it means to be good at math
  3. Questioning the pattern of our questions
  4. Focus on Relational Understanding
  5. Never Skip the Closing of the Lesson
  6. What Does Day 1 Look Like?
  7. Exit Cards – What do your’s look like?
  8. Is This “Real World”?
  9. How do you give feedback?
  10. How do we meet the needs of so many unique students in a mixed-ability classroom?

My least popular posts were:

  1. Purposeful Practice: Happy Numbers
  2. Aiming for Mastery?
  3. “I like math because it’s objective…”
  4. How to change everything and nothing at the same time!
  5. Is That Even A Problem???

So, I’m left wondering, why are some posts more popular and others less so?  Are my least popular posts less read because they are more confrontational?  Do they offer less for others to relate to?

And why do some posts get retweeted or commented on more?  Is it because they offer more chance for reflection, or is it the topic…?

And more importantly, is this what I’m aiming for?  Is the purpose of this blog to share with others and hope it will be read, or is it for me to continue to write so I can reflect on my own thinking/decisions???

Is it about making connections with others?  Or about my own learning?  Or about helping others reflect???

I am left wondering what about blogging is different than reading others’ blogs?  How is this helpful and to whom?

While I don’t think I have the answers to my questions, I do know that I am continuing to learn and that my thoughts are getting others to consider their own teaching.  Hopefully as I continue writing, I will start to find the answers to WHY I do what I do.

Hopefully this blog will continue to be an important aspect to my work in 2017 as well!  Thanks for reading.

I’d love to know why you read math blogs.  Or what it would take to get you started writing your own!

Leave a comment here, or on Twitter (@MarkChubb3)

Subtracting Integers – Do you see it as removal or difference???

If you are thinking about how we should develop an understanding of adding and subtracting integers, it is very important to first consider how primary students should be developing adding and subtracting natural numbers (positive integers).

Of the two operations, subtraction seems to be the one that many of us struggle to know the types of experiences our students need to be successful.  So let’s take a look at subtraction for a moment.

Subtraction can be thought of as removal…

We had 43 apples in a basket. The group ate 7.  How many are left?  (43-7 =___)

Or subtraction can be thought of as difference…  

I had 43 apples in a basket this morning.  Now I only have 38.  How may were eaten?  (43-___=38  or 38+____=43)

Each of these situations requires different thinking.  Removal is the most common method of teaching subtraction (it is simpler for many of us because it follows the standard algorithm – but it is not more common nor more conceptual as difference).  Typically, those who have an internal number line (can think based on understanding of closeness of numbers…) can pick which strategy they use based on the question.

For instance, removing the contexts completely, think about how you would solve these 2 problems:

25-4 =


Think for a moment like a primary student.  The first problem is much easier for many!  If the only strategy a student has is counting backwards, the second method is quite complicated!  In the end, we want all of our students to be able to make sense of subtraction as both the removal and difference!  (We want our students to gain a relational understanding of subtraction).

Take a minute to watch Dr. Alex Lawson discuss the power of numberlines:


Alex Lawson: The Power of the Number Line from LearnTeachLead on Vimeo.

How does this relate to Integers?

Subtraction being thought of as removal is often taught using integer chips (making zero pairs…).  Take a look at the examples below.  Can you figure out what is happening here?  What do the boxes mean?

integers subtraction.png
Enter a caption
Number lines are often used with Integer operations too, but the method of using them is typically removal as well.  Think about this problem for a minute:

Enter a caption
Again, students view of subtraction is removal here (or with the case of subtracting negative numbers here, students will be adding).

However, now let’s think about how a number line could help us understand integers more deeply if we thought of subtraction as difference:

Hopefully for many students, they might be able to see how far apart the numbers (+5) and (-2) are.  Without going through a bunch of procedures, many might already understand the difference between these numbers.  Many students come to understand something potentially difficult as (+5) – (-2) = 7 quite easily!


I encourage you to try to create 2 different number line representations of the following question, one using removal and the other using difference:  

(-4) – (-7) = 


Some final thoughts:

  • When is it appropriate for us to use difference?  When is it appropriate for us to use removal? 
  • Should students explore 1 first?  Which one?
  • Which is easier for you?  Are you sure it is also the easiest strategy for all of your students?
  • The questions above have no context of any kind.  I wonder if this makes this concept more or less difficult for our students?
  • How do you provide experiences for your students to make sense of things, not just follow rules and procedures?
  • How can we avoid the typical tricks used during this unit?

As always, I’d love to continue the conversation. Send me your number lines or share your thoughts / questions below or on Twitter (@markchubb3)

Questioning the pattern of our questions

I find myself spending more and more time trying to get better at two things.  Listening and asking the right kinds of questions that will push thinking.  While I find that resources have helped me get better at asking the right questions, I have learned that listening is actually quite difficult.  The quote below is something that made me really think and reflect on my own listening skills:listening

More about this in a minute…

A while ago I had the pleasure to work with a second grade teacher as we were learning how to do String mini-lessons (similar to Number Talks) to help her students reason about subtraction.  After a few weeks of getting comfortable with the routine, and her students getting comfortable with mental subtraction, I walked into the class and saw a student write this:


What would you have asked?

What would you have done?

Did she get the right answer?

My initial instincts told me to correct her thinking and show her how to correctly subtract, however, I instead decided to ask a few questions and listen to her reasoning.  When asked how she knew the answer was 13 she quickly started explaining by drawing a number line.  Take a look at her second representation:

She explained that 58 and 78 were 20 away from each other, but 58 and 71 weren’t quite 20 away, so she needed to subtract.

I asked her a few questions to push her thinking with different numbers to see if her reasoning would always work.

Is her reasoning sound?  Will this always work?  Try a few yourself to see!

Typically, we look at subtraction as REMOVAL (taking something away from something else), however, this student saw this subtraction question as DIFFERENCE (the space between two numbers).

I wonder what would have happened if I “corrected” her mathematics?  I wonder what would have happened if I neglected to listen to her thinking?  Would she have attempted to figure things out on her own next time, or would she have waited until she was shown the “correct” way first?

I also wonder, how often we do this as teachers?  All it takes is a few times for a student’s thinking to be dismissed before they realize their role isn’t to think… but to copy the teacher’s thinking.

Funneling vs. Focusing Questions

As part of my own learning, I have really started to notice the types of questions I ask.  There is a really big difference here between funneling and focusing questions:


Think about this from the students’ perspective.  What happens when we start to question them?

Screen Shot 2013-11-07 at 1.49.12 PM.png
Summarized by Annie Forest in her Blog

Please make sure you continue to read more about we can get better at paying attention to the pattern of our questions:

Questioning Our Patterns of Questioning by Herbel-Eisenmann and Breyfogle

Starting where our students are….. with THEIR thoughts

So I leave you with some final thoughts:

  • Do you tend to ask funneling questions or focusing questions?
  • How do we get better at asking questions and listening to our students’ thinking?
  • What barriers are there to getting better at asking questions and listening?  How can we remove these barriers?
  • Is there a time for asking funneling questions?  Or is this to be avoided?
  • What unintended messages are we sending our students when we funnel their thinking?  … or when we help them focus their thinking?
  • What if our students’ reasoning makes sense, but WE don’t understand?


I’d love to continue the conversation about the subtraction question above, or about questioning and listening in general.  Leave a comment here or on Twitter @MarkChubb3

What are your thoughts?

How our district improved…

Earlier this week news was made public about the Province’s and our district’s grade 3 and grade 6 math results.  For the past several years, there has been a negative trend both here and across the Province in grade 6 math, and that trend continued again this year across the Province… with our district being the outlier… we made significant gains.  In fact, our school board jumped 12% since the last testing (I’m not sure, but I don’t believe there has been such a spike for a school board our size before).  So I think it might be worth discussing WHY this might have happened.

If you are reading this article, you are probably a teacher, or administrator, or have some other role in education.  Before you continue, I want you to think for a minute about why you became an educator?  It was likely because you cared about students, saw value in providing children with opportunities that would positively affect their lives… probably it wasn’t to try to get scores of some test to go up.  I want you to keep this in mind as you continue to read.

Our district has put a concerted effort into mathematics teaching and learning over the past few years.  While we might want to look at what the changes were last year that made the difference, I want to reach back a little further to give you the bigger picture.


Our school board engaged in intensive training for several teachers in a program called SUM (Supporting Understanding in Mathematics).  This training involved willing teachers who wanted to learn more about the process of learning and teaching of mathematics.  These teachers delved into well-researched resources, co-taught lessons together, viewed others’ classrooms, and deepened their own understanding of mathematics in the process.  The goal for these sessions was to help build teachers’ capacity, help develop their math knowledge for teaching.  The drawback, was that few teachers across the board could participate, but the teachers who did participate, quickly became leaders in their buildings and within the system.  For many, these experiences changed their view of mathematics education significantly.


Our school board invested heavily in Cathy Fosnot’s Contexts for Learning.  Cathy herself trained many teachers.  Teachers had many opportunities to learn together in co-planning, co-teaching, co-debriefing sessions as they learned through using these resources.  The benefit from these sessions is that we learned what teaching THROUGH problem solving looks like, how we can assess using developmental landscapes, how we can build procedural fluency from conceptual development.  The teachers who participated learned mathematics in ways they had never experienced as students.  Personally, I learned to think mathematically because of these experiences.  The drawback, was that these units only covered 1 of the 5 strands in our curriculum, but it was a great way to help all of us see that mathematics could look different than it did when we were in school.  During this time, our board also invested heavily by purchasing a copy of Van de Walle’s Teaching Student Centered Mathematics for every teacher.

Discussions from this resource helped start conversations about other strands (lots of content learning and ideas), and helped foster changes in how we viewed the subject (relational understanding, assessment, differentiated instruction…).


Over the 2014-2015 years, our board increased the number of instructional coaches in the system, we implemented a flexible scope-and-sequence to allow conversations between teachers to happen about the same topic, we released math newsletters helping us deepen our understanding of math concepts, we gave every student a Dreambox account, and we started offering free Additional Qualifications (AQ) courses to any teacher who was willing to take one.  While each of these are important, I want to address the last point.  Between 2014 and 2016 we have provided over 550 AQ courses to the teachers in our school board (our board has approximately 1500-2000 teachers).  These courses provided experiences for teachers to deepen their understanding in mathematics THROUGH problem solving opportunities.  I believe that the reason why so many teachers in our board have invested their time (125+ hours after school per course) and energy in these courses is because of the initial investment our board put into its teachers.  Our teachers have seen just how important OUR learning is, and because of this they are willing to continue their learning.

As a system, over the past several years, the goals of our school board have been very clear.  As a system, we have continued to work towards:


These 3 goals are aimed at helping us reach our board goal for students:


I want you to notice which words/phrases you like above?  Which ones catch your attention?  Are each of these important to you?

So, I assume that there are going to be a lot of questions about why scores improved so much, and I hope 2 stories become front and center:

  1. We have invested in our teachers!  From SUM groups, to site-based instructional coaches, to providing AQ courses, we have put OUR learning as the focus.  If we want to provide a better education for our students, we need to understand the mathematics deeply, understand how mathematics concepts develop over time, we need to understand our curriculum deeply, we need to understand pedagogical moves that will help our students learn…  Changing the culture to help us become learners has made a huge difference!
  2. We are using researched-based resources.  From Cathy Fosnot’s Context for Learning units, to Beaty/Bruce’s From Patterns to Algebra resources, to Jo Boaler’s research, to Marian Small’s resources, to Cathy Bruce’s Fractions research, to Fosnot’s Dreambox and String mini-lessons, to our Province’s Guide to Effective Instruction work, to Van de Walle’s Teaching Student Centered Mathematics……..  We have delved into a lot of resources, and it is paying off.  While resources typically aren’t the answer, much of these resources have helped us understand mathematics relationally, they have helped us see and understand mathematics in ways that we never experienced as students.  They have helped us visualize, and conceptualize in ways that help us help our students.  More than just a resource to follow, these have become platforms for which we have been learning.

These two pieces, in my mind, are the big reasons we have started to make gains.  Yes, I am sure there are countless other factors, but none of them could possibly help without making sure we have invested in our teachers, and have provided appropriate resources that will help us learn.

I wanted to write this post, not as a “how-to…” for districts, but as a reminder for what is important.  When we focus on deepening OUR understanding, our students benefit… when WE learn through problem solving, we will likely see how we can help our students do the same… when WE see how to provide experiences for our students that are powerful learning opportunities, we won’t rely on gimmicks and fads…

Yes, our scores went up, but that’s not what is important.  What IS important is that our students learn mathematics in ways that allow them to gain a relational understanding.  Our students deserve lessons that are interactive and experiential… they deserve to learn mathematics through thinking and doing and building and exploring, not listening and copying… they deserve to have teachers who understand the mathematics they are teaching and are passionate about the subject… they deserve to have classrooms that are vibrant and full of rich mathematical discussions… they deserve to learn in safe classrooms that promote growth mindset messages… they deserve teachers that view games and puzzles as potential sources of learning or practice… they deserve to see mathematics in ways that makes sense, not through “answer getting” tricks… they deserve teachers who care about mathematics and see mathematics as valuable and fun and creative and beautiful… and they deserve a system that cares about their teachers, because a system that cares about their teachers has teachers who care about their students!

Yes, our scores have gone up, but what really matters is that our students are liking mathematics better… they see themselves as mathematicians… they believe that can succeed… they are starting to see mathematics as something that makes sense!  My hope is this doesn’t mean we are done learning.  We have a lot of work still to do with the list above!  Real change takes time!

So I leave you with a few thoughts:

  • How is your district supporting you?  Are the initiatives similar to the ones I’ve written about above?
  • Is your district interested in providing the best mathematics education for your students, or trying to get scores up?  Is this the same thing?
  • How are you deepening your own understanding of the concepts you teach?
  • What ways have you /could you collaborate with others to deepen your own understanding of the concepts you teach?
  • What opportunities are out there for you to continue to learn?
  • What resources have you used that have pushed your own thinking?
  • If we focus on scores will teaching and learning improve?  If we focus on teaching and learning will scores improve?  Are we mixing up the goal and the evidence of that goal?


As always, I’d love to continue the conversation here, or 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.


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).