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:

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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:

Figure-1-Mathematical-Knowledge-for-Teaching-Ball-Thames-and-Phelps-2008
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)!

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Paying Attention to OUR Understanding!

I believe one of the most important things we can do as math teachers is to constantly deepen our understanding of the concepts our students are learning.


In my 2nd year of teaching I was assigned to teach a grade 6/7 split class with 35 students.  Being a new teacher, I taught my students the way I was taught… through direct instruction… and in a very procedural way… with the belief that learning happens by me telling, and students following the rules/steps that I shared.

Now, I could look back at my teaching and cringe with disbelief at my actions, however, I was doing the best I knew how!  Instead of looking back and thinking about how poor of a job I was doing, or even how far I have come, I think it is far more valuable to recognize the moments that helped me grow as an educator.  Here is one such incident:

So, my second year of teaching was well underway when the topic of adding fractions came along.  I knew how to add fractions, and so I taught my students the only way I knew how.  At the end of the unit I gave a test and was surprised at how poorly many of my students did.

I decided to offer those that did poorly a second chance, but assumed that first I needed to offer some help.  I called a student over to my desk and showed him a few of the errors he had made on his test.

His first incorrect answer looked something like this:

IMG_2509

Pretty common error right!  In fact, for each of the questions I had given where the denominators were different, this was his strategy.

So, slowly I “re-taught” him the steps of how to add fractions… I asked little mini-questions to walk him through each small step in finding the solution.  It looked like he understood until he said:

 “I get that you are telling me that 3/5 + 2/7 is 31/35, but look, on my test I got 3 of the 5 right on the front and 2 of the 7 right on the back.  Why did you put 5/12 on the top of my test?”

I, of course, wasn’t expecting this, and completely didn’t know what to do… so, I again showed him my method of adding fractions, completely dismissing his question.


While I think the example above illustrates an interesting moment where I recognized that I didn’t know enough about the math to understand how to react, it is equally interesting to point out what I did when confronted with something I didn’t understand.  Take a look at Phil Daro’s quote below:

Daro - Gaps

When we don’t know the math deeply, we jump to “answer getting” strategies… we tell the students procedures to remember… provide them with tricks… we focus on notations… we provide closed questions that are easily marked as right or wrong…..  and our attempts to help students that struggle includes doing the same things over and over again!

This brings us to where we started:

I believe one of the most important things we can do as math teachers is to constantly deepen our understanding of the concepts our students are learning.

If we want to get better at being math teachers, we need to learn more about the concepts our students explore!  We need to learn from knowledgeable others about how the concepts develop over time, and the experiences our students need to make sense of the mathematics!

For me, this has included learning from wonderful influential math leaders like Cathy Fosnot, Cathy Bruce, Van de Walle, and Marian Small (check out the links).

When the Ontario Ministry published Paying Attention to Fractions K-12 I finally saw the connection and differences between my understanding of 3/5 + 2/7 = 31/35 and my student’s comment showing 3/5 + 2/7 = 5/12.  Take a look.  Can you see how my student was seeing fractions?


While a blog might not be the easiest place to deepen our content knowledge, it can be a platform to encourage you to consider how we are challenging our understanding of the math your students are learning.

Do you have a knowledgeable other learning with you?  Do you read resources that challenge your current understanding of the concepts you teach?  How are you making connections between different representations, or between concepts?  How are you learning more about what is developmentally appropriate?

I think we owe it to our students to be continually learning!  This learning is often referred to as “Math Knowledge for Teaching” or MK4T.  Take a look:

Math Knowledge for Teaching

In the beginning of this blog I explained a time when I recognized that while I understood the content myself, I did not know what to do when my students struggled with the math.  This is where we need to spend our time learning!  This is what our focus needs to be as professionals!