Estimating – Making sense of things

I remember as a student being asked to estimate in mathematics class on a few occasions.  It was either an afterthought from my teachers telling the group to estimate before we do our work to help us with the “reasonableness” of our answers, or during measurement activities where we had to estimate then measure items around the classroom.  As a student though, when asked to actually estimate, I always did the calculations or measuring first, then wrote down a number that was near the actual answer as my “estimate”.  Was I confusing rounding with estimating?  Or was I avoiding thinking???

It might come as a surprise to consider just how much estimating we do outside of school.  Within the same day we might determine how much milk to pour on our cereal so it is covered but won’t get soggy, think about how early we need to leave for work to make sure we aren’t late, figure out if we can safely squeeze our car into a parking space, consider how loud to speak to someone across the room, determine an appropriate amount to tip the waitress at dinner, think about if there is enough time during the commercial break to use the restroom so you don’t miss any of your favourite show…  Whether we know it or not, nearly every minute of the day, we are estimating about physical spaces and numbers.

In school, however, the practice of estimating is often neglected.  Many of our students are estimating all the time without realizing it, but others might not be aware of the mental actions others are doing and don’t engage in the same active thinking processes!  Because of this, I believe we should be estimating more than we probably realize.  The skill of estimation is directly related to our Number Sense and our Spatial Reasoning, so we need to make estimating a priority!

Kinds of estimating:

Situations in which we estimate involve: computational estimations, measurement estimations, numerosity estimates (how many) and number line estimates.  Computational and numerosity estimations are directly related to students’ Number Sense (i.e., size of numbers, doubling, how much more or less…) and often involve students approximating numbers.  While estimates involving measurement and number lines involve our students’ Spatial Reasoning (i.e., considering the size and space of objects).  However, if we really delve deeply into any of the 4 kinds of estimating, they probably each deal with our Number Sense and each deal with our ability to think Spatially.

Questions that ask students to estimate distance or length:

Take a look at the following 5 questions.  Which type of question do you think is most common in school?  Which type of question is less common?


For many students, estimating a measurement is about “guessing”, then actually measuring.  To a student, the act of estimating becomes useless in this scenario.  If they are going to measure anyway, why did they estimate anything?  Many of the questions above ask students to go beyond guessing and ask them to develop benchmarks then think about subdividing or iterating those benchmarks.  While some of our students will naturally develop these benchmarks and strategies to subdivide/iterate, many others will not without rich tasks and discussions.  For these reasons, we likely need to spend more time than we might realize estimating and discussing our strategies / thinking.

Questions that ask students to estimate area:

Take a look at the 5 questions below.  Which ones might help your students better understand the concept of area?  Which ones might help them consider the attributes of area you want them to notice?

To many students, concepts like area are easy… plug in the numbers to a formula and you get your answer.  Estimating, however, requires far more thinking and understanding of the attributes than simple calculations.  For this reason, it is probably best if we start by asking our students to estimate well before they are ever given any formulas, and continually as they learn more complicated shapes.

Each of the above problems ask our students to actually consider the size and shape of the things they are thinking about.  Hopefully this happens ALL the time as our students learn measurement concepts!

Questions that ask students to estimate angles:

As a student I remember learning types of angles and how to read a protractor – very knowledge based and procedural in nature.  However, as a teacher I regularly see students who can easily tell me if an angle is greater or less than 90 degrees, but make seemingly careless mistakes when actually measuring an angle.  Personally, I believe that the issue isn’t about students being careless, it is more about a student’s experiences with angles.  Specifically, too many students are asked to DO something instead of them being asked to THINK about something when it comes to topics like angles.  Above are 5 problems / tasks that ask students to think first by estimating.  When the task is about estimating, it adds motivation for students to actually measure to see how close they might be!

Questions that ask students to estimate number or computations:

Look at the 6 problems/tasks below.  Which ones do you think are more common in classrooms?  Why do you think they are more common?  Which ones require your students to consider the space numbers take up?  Which ones help our students develop and use their number sense?

When the problems / tasks we give are about estimating, our students think about what they already know and use this as a basis for learning new things.  While the aim for many of us is to help our students determine reasonableness for our students’ answers, in real life, estimates are likely good enough most of the time!

We need to estimate more!

Estimating needs to be integrated into more of what we teach, instead of it being an isolated lesson/ concept.  Whether teaching probability or time or Geometry or patterns… we need to ask our students to think more before they start doing any calculation!

Special Thanks

Hopefully you have heard of Andrew Stadel’s Estimation 180.  This is an easy to use routine that can help us with some of the types of estimation I have talked about here!

Estimating slides-24

One of the best parts about these routines is that it helps students build benchmarks, use number sense, think spatially and consider the importance of a range of reasonable answers instead of just “guessing”.

Jamie Duncan added to the conversation when she shared snapshots of how she helped her students refine their ranges:

Estimating slides-25

After noticing her students’ ranges were quite large, she started asking her students to indicate their “brave” too low and “brave” too high estimates.  Ideally we want our students focused on their range of values, not their actual estimate.  Focusing on the actual estimate promotes guessing, while focusing on student ranges helps us think more about reasonableness.  Brilliant idea!

And of course, for me, two of the most powerful images on the topic have been shared by Tracy Zager (If you haven’t purchased her book: Becoming the Math Teacher You Wish You’d Had you need to!).  The image on the left shows the process we go through when solving a problem.  So much of what we want our students to do involves them making sense of things and considering their initial thoughts.  The image on the right to me is even more powerful though.  It shows just how important our intuition is, and how building our students’ intuition is key for them to build their logic!  These two go hand-in-hand!

Estimating slides-26

A few things to reflect on:

  • What does estimation look like in your class?  Is this a routine you do?
  • Many of the ideas shared above might be more specific to the content you are learning.  How do you help your students see the importance of estimation when you are learning new topics?
  • When your class is estimating, how do you promote the range?  Are some of your students still “guessing”?  If so, how can you use the ideas of others in the room to help?  How else can we improve here?
  • Many of the ideas I shared above involved estimating without giving a number.  These tasks often directly involve helping our students use their Spatial Reasoning.  How are you helping your students develop their Spatial Reasoning?
  • The students in our classrooms that are estimating all the time (even without them realizing it) do well in mathematics.  Those that struggle often aren’t using their intuition.  Why do some use their intuition more than others?  What do WE have to do to help everyone use their intuition more often?
  • Did you notice any relationships between the coloured images above?  All the yellow ones involve… all the red ones are…  I wonder which ones you gravitated toward?

As always, I encourage you to leave a message here or on Twitter (@markchubb3)!

Skyscraper Templates

A while ago I was introduced to Skyscraper Puzzles (I believe they were invented by BrainBashers).  I’ll explain below about the specifics of how to play, but basically they are a great way to help our students think about perspective while thinking strategically through each puzzle.  Plus, since they require us to consider a variety of vantage points of a small city block, the puzzles can be used to help our students develop their Spatial Reasoning!

I’ve written before about how to help your students persevere more in math class and I still think that one of the best ways to do this involves physically and visually thinking about tasks that involve Spatial Reasoning.

While I loved the idea of doing these puzzles the first time I saw them, I was less enthusiastic about having these puzzles as a paper-and-pencil or computer generated activity because it is difficult to help develop perspective without actually building the skyscrapers.  So, I created several templates that can easily be printed, where standard link-cubes can be placed on the grid structures.

Below are the instructions for playing and templates you are welcome to use.  Enjoy!

How to play a 4 by 4 Skyscraper Puzzle:

  • Build towers in each of the squares provided sized 1 through 4 tall
  • Each row has skyscrapers of different heights (1 through 4), no duplicate sizes
  • Each column has skyscrapers of different heights (1 through 4), no duplicate sizes
  • The rules on the outside (in grey) tell you how many skyscrapers you can see from that direction
  • Taller skyscrapers block your view of shorter ones

Below is an overhead shot of a completed 4 by 4 city block.  To help illustrate the different sizes, I’ve coloured each size of skyscraper a different colour.  Notice that each row has exactly 1 of each size, and that each column has one of each size as well.

skyscrapers 1
Top View

Below is the front view.  You might notice that many of the skyscrapers are not visible from this vantage point.  For instance, the left column has only 3 skyscrapers visible.  We can see two in the second column, one in the third column, and two in the far right column.

skyscrapers 5
Front View

Below is the view of the same city block if we looked at it from the left side. From left to right we can see 4, 1, 2, 2 skyscrapers.

skyscrapers 2
Left View

Below is the view from the back of the block.  From left to right you can see 1, 2, 3, 2 skyscrapers.

skyscrapers 3
Back View

Below is the view from the right side of the block.  From here we can see 2, 2, 4, 1 skyscrapers (taken from left to right).

skyscrapers 4
Right View

When playing a beginner board you will be given the information around the outside of your city block.  Each number represents the number of skyscrapers you could see if you were to look from that vantage point.  For example, the one on the front view (at the bottom) would indicate that you could only see 1 skyscraper and so on…  The white squares in the middle of the block have been sized so you can actually make the skyscrapers with standard link cubes.

blank 4 by 4

Templates for you to Download:

Beginner 4 by 4 Puzzles

Advanced 4 by 4 Puzzles

Beginner 5 by 5 Puzzles

Advanced 5 by 5 Puzzles

A few thoughts about how you might use these:

A belief I have: Teaching mathematics is much more than providing neat things for our students, it involves countless decisions on our part about how to effectively make the best use of the problem / activity.  Hopefully, this post has helped you consider your own decision making processes!

I’d love to hear how you and/or your students do!

The Same… or Different?

For many math teachers, the single most difficult issue they face on a daily basis is how to meet the needs of so many students that vary greatly in terms of what they currently know, what they can do, their motivation, their personalities…  While there are many strategies to help here, most of the strategies used seem to lean in one of two directions:

  1. Build knowledge together as a group; or
  2. Provide individualized instruction based on where students currently are

Let’s take a closer look at each of these beliefs:

Those that believe the answer is providing all students with same tasks and experiences often do so because of their focus on their curriculum standards.  They believe the teacher’s role is to provide their students with tasks and experiences that will help all of their students learn the material.  There are a few potential issues with this approach though (i.e., what to do with students who are struggling, timing the lesson when some students might take much more time than others…).

On the other hand, others believe that the best answer is individualized instruction.  They believe that students are in different places in their understand and because of this, the teacher’s role is to continually evaluate students and provide them with opportunities to learn that are “just right” based on those evaluations.  It is quite possible that students in these classrooms are doing very different tasks or possibly the same piece of learning, but completely different versions depending on each student’s ability.  There are a few difficulties with this approach though (i.e., making sure all students are doing the right tasks, constantly figuring out various tasks each day, the teacher dividing their time between various different groups…).


There are two seemingly opposite educational ideals that some might see as competing when we consider the two approaches above:  Differentiated Instruction and Complexity Science.  However, I’m actually not sure they are that different at all!


For instance, the term “differentiated instruction” in relation to mathematics can look like different things in different rooms.  Rooms that are more traditional or “teacher centered” (let’s call it a “Skills Approach” to teaching) will likely sort students by ability and give different things to different students.

teaching approaches touched up.png
From Prime Leadership kit
If the focus is on mastery of basic skills, and memorization of facts/procedures… it only makes sense to do Differentiated Instruction this way.  DI becomes more like “modifications” in these classrooms (giving different students different work).  The problem is, that everyone in the class not on an IEP needs to be doing the current grade’s curriculum.  Really though, this isn’t differentiated instruction at all… it is “individualized instruction”.  Take a look again at the Monograph: Differentiating Mathematics Instruction.

Differentiated instruction is different than this.  Instead of US giving different things to different students, a student-centered way of making this make sense is to provide our students with tasks that will allow ALL of our students have success.  By understanding Trajectories/Continuum/Landscapes of learning (See Cathy Fosnot for a fractions Landscape), and by providing OPEN problems and Parallel Tasks, we can move to a more conceptual/Constructivist model of learning!

Think about Writing for a moment.  We are really good at providing Differentiated Instruction in Writing.  We start by giving a prompt that allows everyone to be interested in the topic, students then write, we then provide feedback, and students continue to improve!  This is how math class can be when we start with problems and investigations that allow students to construct their own understanding with others!

The other theory at play here is Complexity Science.  This theory suggest that the best way for us to manage the needs of individual students is to focus on the learning of the class.

What Complexity Science Tells us about Teaching and Learning

The whole article is linked here if you are interested.  But basically, it outlines a few principles to help us see how being less prescriptive in our teaching, and being more purposeful in our awareness of the learning that is actually happening in our classrooms  will help us improve the learning in our classrooms.  Complexity Science tells us to think about how to build SHARED UNDERSTANDING as a group through SHARED EXPERIENCES.  Ideally we should start any new concept with problem solving opportunities so we can have the entire group learn WITH and FROM each other.  Then we should continue to provide more experiences for the group that will build on these experiences.

Helping all of the students in a mixed ability classroom thrive isn’t about students having choice to do DIFFERENT THINGS all of the time, nor is it about US choosing the learning for them… it should often be about students all doing the SAME THING in DIFFERENT WAYS.  When we share our differences, we learn FROM and WITH each other.  Learning in the math classroom should be about providing rich learning experiences, where the students are doing the thinking/problem solving.  Of course there are opportunities for students to consolidate and practice their learning independently, but that isn’t where we start.  We need to start with the ideas from our students.  We need to have SHARED EXPERIENCES (rich problems) for us to all learn from.


As always, I leave you with a few questions for you to consider:

  • How do you make sure all of your students are learning?
  • Who makes the decisions about the difficulty or complexity of the work students are doing?
  • Are your students learning from each other?  How can you capitalize on various students’ strengths and ideas so your students can learn WITH and FROM each other?
  • How can we continue to help our students make choices about what they learn and how they demonstrate their understanding?
  • Do you see the relationship between Differentiated Instruction and the development of mathematical reasoning / creative thinking?  How can we help our students see mathematics as a subject where reasoning is the primary goal?
  • How can we foster playful experience for our students to learn important mathematics and effectively help all of our students develop at the same time?
  • What is the same for your students?  What’s different?

I’d love to continue the conversation.  Write a response, or send me a message on Twitter (@markchubb3).

Who Makes the Biggest Difference?

A few years ago I had the opportunity to listen to Damian Cooper (expert on assessment and evaluation here in Ontario) give a talk at OAME.  He shared an analogy with us that I found particularly interesting.  He talked about the Olympic athletes that had just competed in Sochi, specifically ice skaters.

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

Think about this for a second, an ice skater trying to become the best at their sport has many influences in their life.  But who makes the biggest difference?  Who helps them become the top athletes in their field?  The scoring judges along the way, or their coaches?  Or is it a mix of both???Damian gave us some time to think and then told us something like this (I’m paraphrasing here):

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 their 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 when you fall. Their goal is for you to become the very best you can be!  They want you to succeed!

Thinking about this analogy, I can’t help but wonder how we might reflect on our own practices.  In the everyday busyness of teaching, I think we often confuse the terms “assessment” with “evaluation”.   Evaluating is about marking, levelling, grading.  Any time we put a check on a question, a mark on the top of a page, or a grade or level on an assignment we are evaluating our students’ work.  

On the other hand, assessing is actually something quite different.  The term “assess” comes from the Latin “assidere” which translates as “to sit beside”.  Assessment is kind of like learning about our students’ thinking processes, seeing how deeply they understand something.  It is a process where we observe our students in action to understand their thought process, consider what a student was thinking when looking at an assignment, listen carefully to students’ reasoning as they share with us and others.  Assessment, while related to evaluation, are very different processes!


I have shared this analogy with a lot of teachers in a number of settings because I think it is helpful for us to consider our role in the classroom.  Are we handing out assignments and marking them (like a scoring judge), or are we providing opportunities for our students to learn, then observing them and really considering what we need to do next?  While most agree with the premise of the coach having a bigger impact, many of us recognize that our job requires us to be the scoring judges too.  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


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 aren’t as motivated to achieve.

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 can all improve.

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

Jo Boaler – Assessments for Learning Encourage a Growth Mindset


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.  

As usual, I leave you with a few questions to reflect on:

  • Are you assessing or evaluating most of the time?  Can you see the difference?
  • Do your students see you as their coach or as a scoring judge?  
  • 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 can we capitalize on using various students’ thinking to help each other improve?
  • How do you use these learning opportunities to provide feedback on your own teaching?

Let’s continue the conversation here or on Twitter!

Quick fixes and silver bullets…

I find myself reflecting on what I believe is best for my students and best for my students’ beliefs about what mathematics is often.  When I get the opportunity to take a look at my students’ work and time to determine next steps, I can’t help but reflect on how my beliefs inform what next steps I would take.   However, I wonder, given the same students and the same results, if we would all give the same next steps?  Let’s take a look at a few common beliefs about what our students need to be successful and discuss each.

My kids need to know their facts:

Often we see students who make careless mistakes and wonder why they could have gone wrong with something so simple. To some, the belief here is that if we could just memorize more facts, they would be able to transfer those facts to the problems in the assessment. While I agree that we want our students being comfortable with the numbers they are working with, I’m not convinced that memorizing is the answer here. Our Provincial test includes a few computation questions (for grade 3 only) and none of these are timed. Most of our questions involve students making sense of things (across 5 strands), some with contexts and some without.

Instead of spending more time worrying about memorizing facts, I wonder if other strategies have been thought of to help our students as well?  For example, other than the 4 questions in grade 3, all other questions allow students to use manipulatives or calculators, and all questions have space for students to write in the margins any rough work or visual models might be using.  The question below is one of the few computational questions a grade 3 student is expected to do.  Many who might use the traditional algorithm might accidently pick 41.  How might a number line help our students visualize the space between the two numbers here?


My kid aren’t reading the questions:

Often we notice that our students understand a concept, but the question itself requires several steps and students don’t end up answering what is asked. For many the solution is having students do some kind of strategy (whether they need to or not) like highlighting key words.

I wonder what answer students might get to the above question?  Will they get the right answer here?  What would you have liked them to do instead?

Highlighting specific words or filling out a standard graphic organizer isn’t the answer for all kids, nor for all questions!

Personally, I think the issue isn’t that our students can’t read the questions, it is that they are jumping to a solution strategy too quickly.  Instead of believing the solution is to have our students highlight or fill out graphic organizers, what might be more appropriate is to help our students slow down and think deeper about the questions they are being asked.  I wonder, however, about how our students’ prior experiences might be a big part of why they jump to solution strategies too quickly?  If students typically receive questions that are simple and closed, and typically follow a lesson directly telling us how to answer those questions, then I wonder if the issue is that our students can’t read or if their experiences have actually been counter-productive? If students don’t experience mathematics in ways that help them make sense of a situation, and instead see math as answering a bunch of questions, then it is no wonder why they aren’t reading the whole question!  They have been trained to believe math is about getting answers quickly, and that we get rewarded (less homework, better grades…) when we are fast.

Instead of more time practicing reading and highlighting questions, or filling out graphic organizers, we might want to spend more time building questions together, asking students to pose their own problems, or providing experiences for students to notice and wonder. What if we started by showing this:


What do you notice here?  What do you wonder?

What might our students see?  They might notice things we didn’t realize they were not even aware of (e.g., each row has 7 boxes, some rows are missing numbers, the numbers go in order, there are 2 Ss and 2 Ts at the top, the letters at the top probably mean days of the week…..).

What might our students be curious about?  They might wonder why April 1st is on a Monday and not a Sunday.  Or wonder about the “Chapter 1” part.

Then we could continue to show more of the question and again ask what students notice and wonder.


No matter the grade level or content, my students need to realize that mathematics is about the development of mathematical reasoning, not just quickly jumping to a solution strategy (especially not one that my teacher told me to use all the time).  Taking the time to think deeply about our mathematics is what I want from my students!  Numberless word problems, notice and wonder strategy, contemplate and calculate… any strategy that helps my students slow down, pay attention to visuals, and start to think about the situation more will probably help many of our students given enough opportunities.

They need more practice with questions like these:

Many believe that if students are doing poorly on something, that the best course of action is to continue practicing that thing.  For example, if our students are doing poorly on Provincial testing questions, then giving students more questions like these will answer all of the issues.

To some, the answer to the problem is to give sample questions every week in a package or even more frequently.  While there are times when practice is helpful, if our students are struggling with the content, giving more questions will not be helpful!  Take a look at Daro’s quote:

More often than not, the quick fix solutions like this (noticing our students struggle with something, then providing the same instruction or same types of practice again) will not be successful.  Developing our math knowledge for teaching is probably the most difficult aspect of teaching mathematics, and is definitely NOT a quick fix, but it is probably the answer here!  If our students are struggling with concepts we believe they should be able to do, it is likely that they haven’t had the right experiences to help them learn!  Have we provided experiences for our students to deeply explore a variety of representations?  Have we provided experiences where our students are able to develop reasoning skills?  Have we provided ample opportunities for our students to consolidate their learning?

Remember, questions that are designed to show evidence OF student learning is not necessarily the WAY students learn!  Handing out these questions toward the end of the learning is far more reasonable.

My students need more stamina:

Often when giving young students extended time to sit and focus independently on an assessment task we have many that struggle to remain focussed.  For some, the solution is to help students build stamina through quiet seat work regularly.  While I do agree that we should have our students work independently at times, I’m not sure this is the answer to the stamina problem.

To me, I think the issue has more to do with how our students experience mathematics.  Do they get lots of short closed questions where the right answer is apparent quickly?  Or do they experience rich problems where they reason through and figure out their own way of making the question make sense?  Do they learn math through independent think time and cooperative problem solving, or are they told material then asked to remember all of the steps and terms.  Is their mathematics class structured in a way where students come to rely on themselves (individually or within their group) to make sense of challenging problems, or do they feel the need to access their teacher every time they don’t know what to do?  When our students are working, are we monitoring all of our students’ thinking, or are we spending a lot of time guiding our students’ thinking?

If stamina is the issue, I wonder if we are allowing our students to productively struggle enough?  If we see a bunch of hands raised around the classroom all wanting us to help, this is a huge red flag moment.  Our students are asking US to think for them!  If we find ourselves sitting beside our students helping out a small group all of the time, this might be another red flag.  Our students are learning that they always have access to us right beside them when they learn, but the unintended problem is that we aren’t allowing our students to struggle enough!

Providing our students with a variety of manipulatives to learn and puzzle through their mathematics on a daily basis might be a big step in the direction of allowing our students to gain the confidence and stamina they need to do well every day.  Notice how these students are using manipulatives to help them make sense of their work:



When our students learn their mathematics using manipulatives and have access to any manipulative at any time to solve new problems, we start to notice that our students come to realize their role is to slow down and make sense of things.  When our students have had various experiences with manipulatives and can see their role as “thinking tools”, we start to notice fewer hands asking for help, less need to have to sit down with a group, and more time for us to really notice our students’ thinking going on in our classroom.  When this starts to happen, we no longer see stamina as a big issue.

I want you to consider for a moment the differences between the beliefs I’ve mentioned. What messages are we sending to our students about what is important in mathematics?  Strategies that get us to do better on the test, or strategies that help us slow down and think more?  Is math about memorizing or figuring things out?  Is math about removing the context to mathematize a situation, or about using the context to make sense of things?

Sure I want my students to do well on any assessment they are given, but I want them to do well every day!  Quick fixes and silver bullets often don’t help our students in the long run though!!!

So I leave you with a few things to reflect on:

  • What are some of the quick fixes you’ve heard about?  Did you try any of them?  Did they work?
  • Is there a strategy that you see working for all of your students?  Was it actually helpful for everyone, or just some?  Do you expect everyone to use this strategy?
  • Have you been asked or told to use specific strategies?  Do you see it being successful for everyone?  Do you have the autonomy to choose here based on the students you have in front of you?  Do your students have any autonomy over the strategies they use?
  • When looking at your student work, are you determining next steps for your students, or for yourself?

It is far easier to determine what your students can and can’t do well, than it is to figure out what to do next.  While we absolutely need to help our students notice the things they could do to improve, we also need to do the hard work of reflecting on our own practices.  Our beliefs about what is important and how we learn mathematics have a direct effect on how our students will do in our classrooms!

Lessons learned from 3 Mistakes

I make mistakes all the time.  We all do.  As you’ve probably heard, mistakes are an important part of our learning.  However, I don’t think it is always as easy to learn from our mistakes as any catchy statement might make it sound.  So, I thought I’d share three quick stories and then reflect on them.

Story 1

The other day I was leading a string mini-lesson (similar to a number talk) and was modelling on a number line what the students were saying.  The string involved the question 78-29 and I took answers from a few students.

Student 1 explained that they started at 78, went back 30, then forward 1.  However, I modelled it on the numberline incorrectly.  I took what they mentioned, started at 78 and landed at 38 and 39 instead of 48 and 49.

Story 2

A few weeks ago I was co-planning / co-teaching with two teachers starting a new Contexts for Learning unit (Cathy Fosnot’s The T-Shirt Factory).  In our planning we looked at her landscapes of learning, the progression of lessons, did some of the questions together, anticipated student responses, and started gathering needed materials.  I joined the classrooms on day 4 of the unit and came in to hear some issues they had with some group members not doing much of the work.  When asked, they had told me they split the class into groups of 4 students because the resource told them to.  They put each group in charge of figuring out 1 size of T-shirts (as is the context in the unit).  They placed struggling students together to figure out the easier sizes, and “stronger” students together to work on the larger numbers.

After the first few lessons both teachers noted that none of the groups had all students working.  I suggested that it would be far easier for us to work with pairs instead of groups of 4.  I co-taught the same lesson with each teacher that day having students in pairs, but something was still not right.  We read through the unit again and realized that we had grouped students incorrectly and had been assigning problems incorrectly!  We should have placed 4 random students together and given EACH student their own size T-Shirt.  That way each student could work as part of the group, been given their own problem, and we could assign numbers that were appropriate for each student.

We ended up having to redo the previous 2 lessons over again so each student would have proper groups and their own responsibilities!

Story 3

In my third year of teaching I went to a workshop where a teacher shared their practices with the group.  The workshop was about having students create their own math dictionaries where definitions (with examples and pictures) were kept, and lessons and worked examples from each school day would be stored.  I quickly started working on making this a reality.  Over the summer I created the template for these books, I wrote out the notes I wanted every student to copy for each day of the year leaving blanks where their samples would go, and put together a package for each student that would be continually added upon throughout the year.

I was excited because these resources would house all of the definitions, all of the examples, all of the thinking from the entire year, and they would be able to use this resource for studying, for homework, and keep it for future years as a reference!

June quickly came and every student had their very own personal math dictionaries we created from countless hours of work.  Each was neat and complete, full of all of our thinking from the year!  On the last day of school, I watched as students packed up their belongings (forgotten sweaters, old shoes, pencil cases…), handed in their textbooks, and threw out their their unwanted worksheets and duo-tangs.  However, I wasn’t counting on just how happy many of my students were to throw out their math dictionaries.  I tried to pull out of the garbage and recycling bins each book to save anyway.

Some thoughts about the kinds of mistakes we make

We tend to make mistakes at different levels.  Story 1 showed one of the many simple mistakes I make.  They are quick little things that are often due to carelessness, forgetting things, accidents…  When we make mistakes like this, what matters is how we show our students how WE handle mistakes.  If we make a calculation error, do we get embarrassed?  Do we pretend it didn’t happen?  Do we turn it into a teachable moment?  This really matters!  If we show our students that we aren’t comfortable with us making mistakes, what unintended messages will this send?

In my second story I shared a mistake I made by not reading carefully and not being fully prepared for a lesson.  Mistakes that cause us to redo something because we noticed that something isn’t working are great mistakes to make.  When we realize that we have taken a wrong path, we come to see just how important the bigger things are.  We learn to be more intentional because of our mistakes!  In this situation, we realized that we could differentiate the learning in the classroom without ability grouping, and that if we have larger groups (4 students) we still need to give everyone something that they are responsible for.

The third story is a story I don’t share much.  Probably because I had spent an entire year getting my students to memorize, follow procedures, copy out worked examples…  My students eagerly throwing out their books was great feedback for me (I didn’t think it was great at the time though).  Over time, I came to see that I was teaching my students instrumentally!  Not only did my mistake help me to notice just how unhappy my students were in my math class, it helped me realize why my students had done so poorly on their provincial testing (they were the lowest results I’ve ever had).  I was also able to reflect on what it means to learn mathematics and what it means to be engaged in thinking mathematically.  Recognizing we are taking a path that isn’t beneficial for students requires us to see other approaches, understand the research, experience learning mathematics ourselves in ways that help us understand the concepts deeper.  Mistakes like this are not only difficult to recognize, they are difficult to change.  It has been a long road for me to continue to develop my own math knowledge for teaching, but I know that it started when I started realizing math is more than memorizing, more than rules and procedures, more than a collection of unrelated topics.

Some things to reflect on:

  • When you make little errors in front of your students, how do you react?  What do your students think about making mistakes in front of others?  Is there a relationship?
  • Can you think of a time you made a mistake with a lesson, or in teaching a unit?  These types of mistakes are easy to learn from, but we need to take advantage of these opportunities.  How have you learned from your mistakes?
  • Recognizing and reflecting on our practices that reflect our beliefs is probably the most difficult for us to do.  So, how can we find out which teaching practices are helping our students learn?  How can we find opportunities for our students to give US feedback?  What experiences help us reflect on our own beliefs about what is important for our students to do?  What experiences help us reflect on our own beliefs about how students learn mathematics?
  • How do the beliefs I have shared here in this post relate to yours?


As always, I encourage you to continue the conversation here or on Twitter (@MarkChubb3)

…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).


The smallest decisions have the biggest impact!

In my role, I have the advantage of seeing many great teachers honing and refining their craft, all to provide the best possible experiences for their students. The dedication and professionalism that the teachers I work with continue to demonstrate is what keeps me going in my role!

One particularly interesting benefit I have is when I can be part of the same lesson multiple times with different teachers.  When I am part of the same lesson several times I have come to notice the differences in the small decisions we make.  It is here in these small decisions that have the biggest impact on the learning in our classrooms. For instance, in any given lesson:

There are so many little decisions we make (linked above are posts discussing several of the decisions).  However, I want to discuss a topic today that isn’t often thought about: Scaffolding.

For the past few months, the teachers / instructional coaches taking my Primary/Junior Mathematics additional qualifications course have been leading lessons. Each of the lessons follow the 3-part lesson format, are designed to help us “spatialize” the curriculum (allow all of us to experience the content in our curriculum via visuals / representations / manipulatives), and have a specific focus on the consolidation phase of the lesson (closing). After each lesson is completed I often lead the group in a discussion either about the content that we experienced together, or the decisions that the leader choose. Below is a brief description of the discussion we had after one particular lesson.

First of all, however, let me share with you a brief overview of how the lesson progressed:

  1. As a warm up we were asked to figure out how many unique ways you can arrange 4 cubes.
  2. We did a quick gallery walk around the room to see others’ constructed figures.
  3. We shared and discussed the possible unique ways and debated objects that might be rotations of other figures, and those that are reflections (take a look at the 8 figures below).
  4. The 3 pages of problems were given to all (see below).  Everyone had time to work independently, but sharing happened naturally at our tables.
  5. The lesson close included discussions about how we tackled the problems.  Strategies, frustrations, what we noticed about the images… were shared.

Here are the worksheets we were using so you can follow along with the learning (also available online Guide to Effective Instruction: Geometry 4-6, pages 191-212):





While the teacher leader made the decision to hand out all 9 problems (3 per sheet) at the same time, I think some teachers might make a different decision. Some might decide to take a more scaffolded approach. Think about it, which would you likely do:

  1. Hand out all 9 problems, move around the room and observe, offer focusing questions as needed, end in a lesson close; or
  2. Ask students to do problem 1, help those that need it, take up problem 1, ask students to do problem 2, help those that need it, take up problem 2…

This decision, while seemingly simple, tells our students a lot about your beliefs about how learning happens, and what you value.

So as a group of teachers we discussed the benefits and drawbacks of both approaches. Here are our thoughts:

The more scaffolded approach (option 2) is likely easier for us. We can control the class easier and make sure that all students are following along. Some felt like it might be easier for us to make sure that we didn’t miss any of our struggling students. However, many worried that this approach might inhibit those ready to move on, and frustrate those that can’t solve it quickly. Some felt like having everyone work at the same pace wasn’t respectful of the differences we have in our rooms.

On the other hand, some felt that handing out all 9 puzzles might be intimidating for a few students at first. However, others believed that observing and questioning students might be easier because there would be no time pressure. They felt like we could spend more time with students watching how they tackle the problems.

Personally, I think our discussion deals with some key pieces of our beliefs:

  • Do we value struggle?  Are we comfortable letting students productively keep trying?
  • Are we considering what is best for us to manage things, or best for our students to learn (teacher-centered vs student-centered)?
  • What is most helpful for those that struggle with a task?  Lots of scaffolding, telling and showing?  Or lots of time to think, then offer assistance if needed?

In reality, neither of these ways will likely actually happen though. Those who start off doing one problem at a time, will likely see disengagement and more behaviour problems because so many will be waiting. When this happens, the teacher will likely let everyone go at their own pace anyway.

Similarly, if the teacher starts off letting everyone go ahead at their own pace, they might come across several of the same issues and feel like they need to stop the class to discuss something.

While both groups will likely converge, the initial decision still matters a lot.  Assuming the amount and types of scaffolding seems like the wrong move because there is no way to know how much scaffolding might be needed. So many teachers default by making sure they provide as much scaffolding as possible  however, when we over-scaffold, we purposely attempt to remove any sense of struggle from our students, and when we do this, we remove our students’ need to think!  When we start by allowing our students to think and explore, we are telling our students that their thoughts matter, that we believe they can think, that mathematics is about making sense of things, not following along!

So I leave you with a few thoughts:

  • Do your students expect you to scaffold everything?  Do they give up easily?  How can we change this?
  • When given an assignment do you quickly see a number of hands raise looking for help?  Why is this?  How can we change this?
  • At what point do you offer any help?  What does this “help” look like?  Does it still allow your students opportunities to think and make sense of things?

When we scaffold everything, we might be helping them with today’s work, but we are robbing them of the opportunity of thinking. When we do this, we rob them of the enjoyment and beauty of mathematics itself!

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)

How Big is “Big”?

In the last few weeks I have asked several groups of teachers to indicate where 1 billion would go on this number line:

It has been really interesting to me that many have placed the 1 billion mark in a variety of areas and have had a variety of reasons why.  Many have attempted to use their understanding of place value digits  (there are 12 zeros in 1 trillion and only 9 zeros in 1 billion, so 1 billion should be 3/4 of the way toward a trillion) or their knowledge of prefixes to help (million, billion, trillion… so it must be 2/3 the way along the line). Others thought about how many billions are in a trillion asking themselves, “Is their one-hundred or one-thousand billions in a trillion?” Using this strategy, everyone picked a spot toward the left, but some much closer to zero than others.

Others did something interesting though. They started placing other numbers on the number line to help them make sense of the question. Often placing 500 billion in the middle, then 250 billion at the 1/4 mark and so on until they realized just how close to 0 a billion is when we are considering 1 trillion.

What’s the point?

Really big numbers, and really small numbers (decimal numbers), are difficult to conceptualize!  They are hard to imagine their size!  Think about this:

How long is 1 million seconds?  Without doing ANY calculations would you guess the answer is several minutes, hours, days, weeks, months, years, decades, centuries…?  Can you even imagine a million seconds without calculating anything?

How about 1 billion seconds?  Or 1 trillion seconds?

I bet you’ve started trying to calculate right!  That’s because these numbers are so abstract for us that we can’t imagine them.

Because of this little experiment, I am left wondering three things:

  1. What numbers can/can’t the students in our classrooms conceptualize?
  2. What practices do we do that gets kids to think about digits more than magnitude?
  3. What practices could/should we be including that helps our students make these connections?

What numbers can/can’t the students in our classrooms conceptualize?

Before we start working with operations of any given size, I think we need to spend time making sure our students can visualize and estimate the size of the number.  Working with numbers we can’t imagine doesn’t seem productive for our young students!  In our rush to move our kids into more “complicated” mathematics, we often move too quickly through numbers to include numbers that are too abstract for our students!  We think that if a student can accurately carry out a procedure that they understand the numbers they are working with. However, I’m sure we have all seen many students who produce answers that are completely unreasonable without them noticing. Is this carelessness, or is it a lack of understanding of the magnitude of the numbers involved?  Or possibly that our students aren’t visualizing the size of and relationship between the numbers???

What practices do we do that gets kids to think about digits more than magnitude?

The other day, Jamie Garner shared her frustration on Twitter:

Think about the question from the textbook for a second. Students trying to think about 342 pencils (not sure why they would want that many) should be considering a strategy that makes sense. For example, if you had 342 pencils how many boxes of 10 would that be?  Thinking this way, student should answer 34 or 34.2, or maybe 35 boxes (if you wanted to purchase enough boxes).  However, the teacher’s edition tells us that none of these are the right answer. Take a look:

If our students attempt to make sense of the problem, they will be completely wrong!  In fact, many students will likely answer 4 because they’ve been trained not to think at all about the mathematics, and instead focus their attention on what they think the text wants them to do.

This is one of MANY cases where elementary mathematics focuses on digits over understanding magnitude or relative size.  Here are a few others:

These, along with pretty much any standard algorithm (see Christopher Danielson’s post: Standard Algorithms Unteach Place Value) tell our kids to stop thinking about what makes sense, and instead focus on steps that help kids get an answer without understanding.

What practices could/should we be including that helps our students make these connections?

If we want our students to understand numbers, and their relative size… if we want to help our students develop a conceptual understanding of operations… if we want our students make sense of the math they are learning… then we need to:

  • Use contexts that make sense to our students (not pseudo-contexts like the pencil question above).
  • Provide plenty of experiences where students are making sense of numbers visually.  When we allow our students to access their Spatial Reasoning we are allowing them to see the relationship between numbers and help them make connections between concepts.
  • Provide plenty of experiences estimating with numbers

Below are 2 activities taken from Van de Walle’s Student Centered Mathematics.  Think about how you could adapt these to work with numbers your students are starting to explore (really big or really small numbers).


A few questions for you to reflect on:

  • How might you see how well your students understand the numbers that are really large or really small?
  • How are you helping your students develop reasonableness when working with numbers?
  • What visuals are you using in your class that help your students visualize the numbers you are working with?
  • What practices do you use regularly that help with any of the 3 above?

P.S. Here are the answers to the seconds problem I posted earlier: