Professional Development: What should it look like?

A few weeks ago Michael Fenton asked on his blog this question:

Suppose a teacher gets to divide 100% between two categories: teaching ability and content knowledge. What’s the ideal breakdown?

The question sparked many different answers that showed a very wide range of thinking, from 85%/15% favouring content, to 100% favouring pedagogy.  In general though, it seems that more leaned toward the pedagogy side than the content side.  While each response articulated some of their own experiences or beliefs, I wonder if we are aware of just how much content and pedagogical knowledge we come into this conversation with, and whether or not we have really thought about what each entails?  Let’s consider for a moment what these two things are:

What is Content Knowledge?

To many, the idea of content knowledge is simple.  It involves understanding the concept or skill yourself.  However, I don’t believe it is that simple!  Liping Ma has attempted to define what content knowledge is in her book:  Knowing and Teaching Elementary Mathematics: Teachers’ Understanding of Fundamental Mathematics in China & the United States.  In her book she describes teachers that have a Profound Understanding of Fundamental Mathematics (or PUFM) and describes those teachers as having the following characteristics:

Connectedness – the teacher feels that it is necessary to emphasize and make explicit connections among concepts and procedures that students are learning. To prevent a fragmented experience of isolated topics in mathematics, these teachers seek to present a “unified body of knowledge” (Ma, 1999, p.122).

Multiple Perspectives – PUFM teachers will stress the idea that multiple solutions are possible, but also stress the advantages and disadvantages of using certain methods in certain situations. The aim is to give the students a flexible understanding of the content.

Basic Ideas – PUFM teachers stress basic ideas and attitudes in mathematics. For example, these include the idea of an equation, and the attitude that single examples cannot be used as proof.

Longitudinal Coherence – PUFM teachers are fundamentally aware of the entire elementary curriculum (and not just the grades that they are teaching or have taught). These teachers know where their students are coming from and where they are headed in the mathematics curriculum. Thus, they will take opportunities to review what they feel are “key pieces” in knowledge packages, or lay appropriate foundation for something that will be learned in the future.

Taken from this Yorku wiki

As you can see, having content knowledge means far more than making sure that you understand the concept yourself.  To have rich content knowledge means that you have a deep understanding of the content.  It means that you understand which models and visuals might help our students bridge their understanding of concepts and procedures, and when we should use them.  Content knowledge involves a strong developmental and interconnected grasp of the content, and beliefs about what is important for students to understand.  Those that have strong content knowledge, have what is known as a “relational understanding” of the material and strong beliefs about making sure their students are developing a relational understanding as well!

What is Pedagogical Knowledge?

Pedagogical knowledge, on the other hand, involves the countless intentional decisions we make when we are in the act of teaching.  While many often point out essential components like classroom management, positive interactions with students… let’s consider for a moment the specific to mathematics pedagogical knowledge that is helpful.  Mathematical pedagogical knowledge includes:

Which is More Important:  Pedagogy or Content Knowledge?

Tracy Zager during last year’s Twitter Math Camp (Watch her TMC16 keynote address here) showed us Ben Orlin’s graphic explaining his interpretation of the importance of Pedagogy and Content Knowledge.  Take a look:


In Ben’s post, he explains that teachers who teach younger students need very little content knowledge, and a strong understanding of pedagogy.  However, in the video, Tracy explains quite articulately how this is not at all the case (if you didn’t watch her video linked above, stop reading and watch her speak – If pressed for time, make sure you watch minutes 9-14).

In the end, we need to recognize just how important both pedagogy and content are no matter what grade we teach.  Kindergarten – grade 2 teachers need to continually deepen their content knowledge too!  Concepts that are easy for us to do like counting, simple operations, composing/decomposing shapes, equality… are not necessarily easy for us to teach.  That is, just because we can do them, doesn’t mean that we have a Profound Understanding of Foundational Mathematics.  Developing our Content Knowledge requires us to understand the concepts in a variety of ways, to deepen our understanding of how those concepts develop over time so we don’t leave experiential gaps with our students!  

However, I’m not quite sure that the original question from Michael or that presented by Ben accurately explain just how complicated it is to explain what we need to know and be able to do as math teachers.  Debra Ball explains this better than anyone I can think of.  Take a look:

Research Gate: Figure 1.  Mathematical Knowledge for Teaching (Ball et., al, 2008)

Above is a visual that explains the various types of knowledge, according to Deb Ball, that teachers need in order to effectively teach mathematics.  Think about how your own knowledge fits in above for a minute.  Which sections would you say you are stronger in?  Which ones would you like to continue to develop?

Hopefully you are starting to see just how interrelated Content and Pedagogy are, and how ALL teachers need to be constantly improving in each of the areas above if we are to provide better learning opportunities for our students!

 What should Professional Development Look Like?

The purpose of this article is actually about professional development, but I felt it necessary to start by providing the necessary groundwork before tackling a difficult topic like professional development.

Considering just how important and interrelated Content Knowledge and Pedagogy are for teaching, I wonder what the balance is in the professional development currently in your area?  What would you like it to look like?  What would you like to learn?

While the theme here is about effective PD, I’m actually not sure I can answer my own question.  First of all, I don’t think PD should always look the same way for all teachers, in all districts, for all concepts.  And secondly, I’m not sure that even if we found the best form of PD it would be enough because we will constantly need new/different experiences to grow and learn.  I would like to offer, however, some of my current thoughts on PD and how we learn.

Some personal beliefs:

  • We don’t know what we don’t know.  That is, given any simple concept or skill, there is likely an abundance of research, best practices, connections to other concepts/skills, visual approaches… that you are unaware of.  Professional development can help us learn about what we weren’t even aware we didn’t know about.
  • Districts and schools tend to focus on pedagogy far more than content.  Topics like grading, how to assess, differentiated instruction… tend to be hot topics and are easier to monitor growth than the importance of helping our students develop a relational understanding.  However, each of those pedagogical decisions relies on our understanding of the content (i.e., if we are trying to get better at assessment, we need to have a better understanding of developmental trajectories/continuums so we know better what to look for).  The best way to understand any pedagogy is to learn about the content in new ways, then consider the pedagogy involved.
  • Quality resources are essential, but handing out a resource is not the same as professional development.  Telling others to use a resource is not the same as professional development, no matter how rich the resource is!  Using a resource as a platform to learn things is better than explaining how to use a resource.  The goal needs to be our learning and hoping that we will act on our learning, not us doing things and hoping we will learn from it.
  • The knowledge might not be in the room.  An old adage tells us that when we are confronted with a problem, that the knowledge is in the room.  However, I am not sure this is always the case.  If we are to continue to learn, we need experts helping us to learn!  Otherwise we will continually recycle old ideas and never learn anything new as a school/district.  If we want professional development, we need new ideas.  This can come in the form of an expert or more likely in the form of a resource that can help us consider content & pedagogy.
  • Learning complicated things can’t be transmitted.  Having someone tell you about something is very different than experiencing it yourself.  Learning happens best when WE are challenged to think of things in ways we hadn’t before.  Professional development needs to be experiential for it to be effective!
  • Experiencing learning in a new way is not enough.  Seeing a demonstration of a great teaching strategy, or being asked to do mental math and visually represent it, or experiencing manipulatives in ways you hadn’t previously experienced is a great start to learning, but it isn’t enough.  Once we have experienced something, we need to discuss it, think about the pedagogy and content involved, and come to a shared understanding of what was just learned.
  • Professional learning can happen in a lot of different places and look like a lot of different things.  While many might assume that professional development is something that we sign up for in a building far from your own school, hopefully we can recognize that we can learn from others in a variety of contexts.  This happens when we show another teacher in our school what we are doing or what our students did on an assignment that we hadn’t expected.  It happens when we disagree on twitter or see something we would never have considered before.  We are engaging in professional development when we read a professional book or blog post that helps us to consider new things and especially when those things make us reconsider previous thoughts.  And professional learning especially happens when we get to watch others’ teach and then discuss what we noticed in both the teaching and learning of the material.
  • Some of the best professional development happens when we are inquiring about teaching / learning together with others, and trying things out together!  When we learn WITH others, especially when one of the others is an expert, then take the time to debrief afterward, the learning can be transformational.
  • Beliefs about how students learn mathematics best is true for adults too.  This includes beliefs about constructivist approaches, learning through problem solving, use of visual/spatial reasoning, importance of consolidating the learning, role of discourse…
  • Not everyone gets the same things out of the same experiences.  Some people are reflecting much more than others during any professional learning experience.  Personally, I always try to make connections, think about the leaders’ motivations/beliefs, reflect on my own practices… even if the topic is something I feel like I already understand.  There is always room for learning when we make room for learning!

Some things to reflect on:

As always, I like to ask a few questions to help us reflect:

  • Think of a time you came to make a change in your beliefs about what is important in teaching mathematics.  What led to that change?
  • Think of a time you tried something new.  What helped you get started?
  • Where do you get your professional learning?  Is your board / school providing the kind of learning you want/need?  If so, how do you take advantage of this more often?  If not, how could this become a reality?
  • Think about your answers to any of the above questions.  Were you considering learning about pedagogy or content knowledge?  What does this say about your personal beliefs about professional development?
  • Take a look again at any of the points I made under “Some Personal Beliefs”.  Is there one you have issue with?  I’d love some push-back or questions… that’s how we learn:)

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


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).