## Reasoning & Proving

This week I had the pleasure to see Dan Meyer, Cathy Fosnot and Graham Fletcher at OAME’s Leadership conference.

Each of the sessions were inspiring and informative… but halfway through the conference I noticed a common message that the first 2 keynote speakers were suggesting:

Dan Meyer showed us several examples of what mathematical surprise looks like in mathematics class (so students will be interested in making sense of what they are learning, and to get our students really thinking), while Cathy Fosnot shared with us how important it is for students to be puzzled in the process of developing as young mathematicians.  Both messages revolved around what I would consider the most important Process Expectation in the Ontario curriculum – Reasoning and Proving.

###### Reasoning and Proving

While some see Reasoning and Proving as being about how well an answer is constructed for a given problem – how well communicated/justified a solution is – this is not at all how I see it.  Reasoning is about sense-making… it’s about generalizing why things work… it’s about knowing if something will always, sometimes or never be true…it is about the “that’s why it works” kinds of experiences we want our students engaged in.  Reasoning is really what mathematics is all about.  It’s the pursuit of trying to help our students think mathematically (hence the name of my blog site).

###### A Non-Example of Reasoning and Proving

In the Ontario curriculum, students in grade 7 are expected to be able to:

• identify, through investigation, the minimum side and angle information (i.e.,side-side-side; side-angle-side; angle-side-angle) needed to describe a unique triangle

Many textbooks take an expectation like this and remove the need for reasoning.  Take a look:

As you can see, the textbook here shares that there are 3 “conditions for congruence”.  It shares the objective at the top of the page.  Really there is nothing left to figure out, just a few questions to complete.  You might also notice, that the phrase “explain your reasoning” is used here… but isn’t used in the sense-making way suggested earlier… it is used as a synonym for “show your work”.  This isn’t reasoning!  And there is no “identifying through investigation” here at all – as the verbs in our expectation indicate!

A Example of Reasoning and Proving

Instead of starting with a description of which sets of information are possible minimal information for triangle congruence, we started with this prompt:

Given a few minutes, each student created their own triangles, measured the side lengths and angles, then thought of which 3 pieces of information (out of the 6 measurements they measured) they would share.  We noticed that each successful student either shared 2 angles, with a side length in between the angles (ASA), or 2 side lengths with the angle in between the sides (SAS).  We could have let the lesson end there, but we decided to ask if any of the other possible sets of 3 pieces of information could work:

While most textbooks share that there are 3 possible sets of minimal information, 2 of which our students easily figured out, we wondered if any of the other sets listed above will be enough information to create a unique triangle.  Asking the original question didn’t offer puzzlement or surprise because everyone answered the problem without much struggle.  As math teachers we might be sure about ASA, SAS and SSS, but I want you to try the other possible pieces of information yourself:

Create triangle ABC where AB=8cm, BC=6cm, ∠BCA=60°

Create triangle FGH where ∠FGH=45°, ∠GHF=100°, HF=12cm

Create triangle JKL where ∠JKL=30°, ∠KLJ=70°, ∠LJK=80°

If you were given the information above, could you guarantee that everyone would create the exact same triangles?  What if I suggested that if you were to provide ANY 4 pieces of information, you would definitely be able to create a unique triangle… would that be true?  Is it possible to supply only 2 pieces of information and have someone create a unique triangle?  You might be surprised here… but that requires you to do the math yourself:)

###### Final Thoughts

Graham Fletcher in his closing remarks asked us a few important questions:

• Are you the kind or teacher who teaches the content, then offers problems (like the textbook page in the beginning)?  Or are you the kind of teacher who uses a problem to help your students learn?
• How are you using surprise or puzzlement in your classroom?  Where do you look for ideas?
• If you find yourself covering information, instead of helping your students learn to think mathematically, you might want to take a look at resources that aim to help you teach THROUGH problem solving (I got the problem used here in Marian Small’s new Open Questions resource).  Where else might you look?
• What does Day 1 look like when learning a new concept?
• Do you see Reasoning and Proving as a way to have students to show their work (like the textbook might suggest) or do you see Reasoning and Proving as a process of sense-making (as Marian Small shares)?
• Do your students experience moments of cognitive disequilibrium… followed by time for them to struggle independently or with a partner?  Are they regularly engaged in sense-making opportunities, sharing their thinking, debating…?
• The example I shared here isn’t the most flashy example of surprise, but I used it purposefully because I wanted to illustrate that any topic can be turned into an opportunity for students to do the thinking.  I would love to discuss a topic that you feel students can’t reason through… Let’s think together about if it’s possible to create an experience where students can experience mathematical surprise… or puzzlement… or be engaged in sense-making…  Let’s think together about how we can make Reasoning and Proving a focus for you and your students!

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

## Reasoning and Proving

This week I had the pleasure to see Dan Meyer, Cathy Fosnot and Graham Fletcher at OAME’s Leadership conference.

Each of the sessions were inspiring and informative… but halfway through the conference I noticed a common message that the first 2 keynote speakers were suggesting:

Dan Meyer showed us several examples of what mathematical surprise looks like in mathematics class (so students will be interested in making sense of what they are learning), while Cathy Fosnot shared with us how important it is for students to be puzzled in the process of developing as young mathematicians.  Both messages revolved around what I would consider the most important Process Expectation in the Ontario curriculum – Reasoning and Proving.

###### Reasoning and Proving

While some see Reasoning and Proving as being about how well an answer is constructed for a given problem – how well communicated/justified a solution is – this is not at all how I see it.  Reasoning is about sense-making… it’s about generalizing why things work… it’s about knowing if something will always, sometimes or never be true…it is about the “that’s why it works” kinds of experiences we want our students engaged in.  Reasoning is really what mathematics is all about.  It’s the pursuit of trying to help our students think mathematically (hence the name of my blog site).

###### A Non-Example of Reasoning and Proving

In the Ontario curriculum, students in grade 7 are expected to be able to:

• identify, through investigation, the minimum side and angle information (i.e.,side-side-side; side-angle-side; angle-side-angle) needed to describe a unique triangle

Many textbooks take an expectation like this and remove the need for reasoning.  Take a look:

As you can see, the textbook here shares that there are 3 “conditions for congruence”.  It shares the objective at the top of the page.  Really there is nothing left to figure out, just a few questions to complete.  You might also notice, that the phrase “explain your reasoning” is used here… but isn’t used in the sense-making way suggested earlier… it is used as a synonym for “show your work”.  This isn’t reasoning!  And there is no “identifying through investigation” here at all – as the verbs in our expectation indicate!

A Example of Reasoning and Proving

Instead of starting with a description of which sets of information are possible minimal information for triangle congruence, we started with this prompt:

Given a few minutes, each student created their own triangles, measured the side lengths and angles, then thought of which 3 pieces of information (out of the 6 measurements they measured) they would share.  We noticed that each successful student either shared 2 angles, with a side length in between the angles (ASA), or 2 side lengths with the angle in between the sides (SAS).  We could have let the lesson end there, but we decided to ask if any of the other possible sets of 3 pieces of information could work:

While most textbooks share that there are 3 possible sets of minimal information, 2 of which our students easily figured out, we wondered if any of the other sets listed above will be enough information to create a unique triangle.  Asking the original question didn’t offer puzzlement or surprise because everyone answered the problem without much struggle.  As math teachers we might be sure about ASA, SAS and SSS, but I want you to try the other possible pieces of information yourself:

Create triangle ABC where AB=8cm, BC=6cm, ∠BCA=60°

Create triangle FGH where ∠FGH=45°, ∠GHF=100°, HF=12cm

Create triangle JKL where ∠JKL=30°, ∠KLJ=70°, ∠LJK=80°

If you were given the information above, could you guarantee that everyone would create the exact same triangles?  What if I suggested that if you were to provide ANY 4 pieces of information, you would definitely be able to create a unique triangle… would that be true?  Is it possible to supply only 2 pieces of information and have someone create a unique triangle?  You might be surprised here… but that requires you to do the math yourself:)

###### Final Thoughts

Graham Fletcher in his closing remarks asked us a few important questions:

• Are you the kind or teacher who teaches the content, then offers problems (like the textbook page in the beginning)?  Or are you the kind of teacher who uses a problem to help your students learn?
• How are you using surprise or puzzlement in your classroom?  Where do you look for ideas?
• If you find yourself covering information, instead of helping your students learn to think mathematically, you might want to take a look at resources that aim to help you teach THROUGH problem solving (I got the problem used here in Marian Small’s new Open Questions resource).  Where else might you look?
• What does Day 1 look like when learning a new concept?
• Do you see Reasoning and Proving as a way to have students to show their work (like the textbook might suggest) or do you see Reasoning and Proving as a process of sense-making (as Marian Small shares)?
• Do your students experience moments of cognitive disequilibrium… followed by time for them to struggle independently or with a partner?  Are they regularly engaged in sense-making opportunities, sharing their thinking, debating…?
• The example I shared here isn’t the most flashy example of surprise, but I used it purposefully because I wanted to illustrate that any topic can be turned into an opportunity for students to do the thinking.  I would love to discuss a topic that you feel students can’t reason through… Let’s think together about if it’s possible to create an experience where students can experience mathematical surprise… or puzzlement… or be engaged in sense-making…  Let’s think together about how we can make Reasoning and Proving a focus for you and your students!

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

## Rushing for Interventions

I see students working in groups all the time…  Students working collaboratively in pairs or small groups having rich discussions as they sort shapes by specific properties, students identifying and extending their partner’s visual patterns, students playing games aimed at improving their procedural fluency, students cooperating to make sense of a low-floor/high-ceiling problem…..

When we see students actively engaged in rich mathematics activities, working collaboratively, it provides opportunities for teachers to effectively monitor student learning (notice students’ thinking, provide opportunities for rich questioning, and lead to important feedback and next steps…) and prepare the teacher for the lesson close.  Classrooms that engage in these types of cooperative learning opportunities see students actively engaged in their learning.  And more specifically, we see students who show Agency, Ownership and Identity in their mathematics learning (See TruMath‘s description on page 10).

On the other hand, some classrooms might be pushing for a different vision of what groups can look like in a mathematics classroom.  One where a teachers’ role is to continually diagnose students’ weaknesses, then place students into ability groups based on their deficits, then provide specific learning for each of these groups.  To be honest, I understand the concept of small groups that are formed for this purpose, but I think that many teachers might be rushing for these interventions too quickly.

First, let’s understand that small group interventions have come from the RTI (Response to Intervention) model.  Below is a graphic created by Karen Karp shared in Van de Walle’s Teaching Student Centered Mathematics to help explain RTI:

As you can see, given a high quality mathematics program, 80-90% of students can learn successfully given the same learning experiences as everyone.  However, 5-10% of students (which likely are not always the same students) might struggle with a given topic and might need additional small-group interventions.  And an additional 1-5% might need might need even more specialized interventions at the individual level.

The RTI model assumes that we, as a group, have had several different learning experiences over several days before Tier 2 (or Tier 3) approaches are used.  This sounds much healthier than a model of instruction where students are tested on day one, and placed into fix-up groups based on their deficits, or a classroom where students are placed into homogeneous groupings that persist for extended periods of time.

Principles to Action (NCTM) suggests that what I’m talking about here is actually an equity issue!

We know that students who are placed into ability groups for extended periods of time come to have their mathematical identity fixed because of how they were placed.  That is, in an attempt to help our students learn, we might be damaging their self perceptions, and therefore, their long-term educational outcomes.

###### Tier 1 Instruction

While I completely agree that we need to be giving attention to students who might be struggling with mathematics, I believe the first thing we need to consider is what Tier 1 instruction looks like that is aimed at making learning accessible to everyone.  Tier 1 instruction can’t simply be direct instruction lessons and whole group learning.  To make learning mathematics more accessible to a wider range of students, we need to include more low-floor/high-ceiling tasks, continue to help our students spatalize the concepts they are learning, as well as have a better understanding of developmental progressions so we are able to effectively monitor student learning so we can both know the experiences our students will need to be successful and how we should be responding to their thinking.  Let’s not underestimate how many of our students suffer from an “experience gap”, not an “achievement gap”!

If you are interested in learning more about what Tier 1 instruction can look like as a way to support a wider range of students, please take a look at one of the following:

###### Tier 2 Instruction

Tier 2 instruction is important.  It allows us to give additional opportunities for students to learn the things they have been learning over the past few days/weeks in a small group.  Learning in a small group with students who are currently struggling with the content they are learning can give us opportunities to better know our students’ thinking.  However, I believe some might be jumping past Tier 1 instruction (in part or completely) in an attempt to make sure that we are intervening. To be honest, this doesn’t make instructional sense to me! If we care about our content, and care about our students’ relationship with mathematics, this might be the wrong first move.

So, let’s make sure that Tier 2 instruction is:

• Provided after several learning experiences for our students
• Flexibly created, and easily changed based on the content being learned at the time
• Focused on student strengths and areas of need, not just weaknesses
• Aimed at honoring students’ agency, ownership and identity as mathematicians
• Temporary!

If you are interested in learning more about what Tier 2 interventions can look like take a look at one of the following:

Instead of seeing mathematics as being learned every day as an approach to intervene, let’s continue to learn more about what Tier 1 instruction can look like!  Or maybe you need to hear it from John Hattie:

Or from Jo Boaler:

###### Final Thoughts

If you are currently in a school that uses small group instruction in mathematics, I would suggest that you reflect on a few things:

• How do your students see themselves as mathematicians?  How might the topics of Agency, Authority and Identity relate to small group instruction?
• What fixed mindset messaging do teachers in your building share “high kids”, “level 2 students”, “she’s one of my low students”….?  What fixed mindset messages might your students be hearing?
• When in a learning cycle do you employ small groups?  Every day?  After several days of learning a concept?
• How flexible are your groups?  Are they based on a wholistic leveling of your students, or based specifically on the concept they are learning this week?
• How much time do these small groups receive?  Is it beyond regular instructional timelines, or do these groups form your Tier 1 instructional time?
• If Karp/Van de Walle suggests that 80-90% of students can be successful in Tier 1, how does this match what you are seeing?  Is there a need to learn more about how Tier 1 approaches can meet the needs of this many students?
• What are the rest of your students doing when you are working with a small group?  Is it as mathematically rich as the few you’re working with in front of you?
• Do you believe that all of your students are capable to learn mathematics and to think mathematically?

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

## Which one has a bigger area?

Many grade 3 teachers in my district, after taking part in some professional development recently (provided by @teatherboard), have tried the same task relating to area.  I’d like to share the task with you and discuss some generalities we can consider for any topic in any grade.

As an introductory activity to area, students were provided with two images and asked which of the two shapes had the largest area.

A variety of tools and manipulatives were handy, as always, for students to use to help them make sense of the problem.

###### Student ideas

Given very little direction and lots of time to think about how to solve this problem, we saw a wide range of student thinking.  Take a look at a few:

Some students used circles to help them find area.  What does this say about what they understand?  What issues do you see with this approach though?

Some students used shapes to cover the outline of each shape (perimeter).  Will they be able to find the shape with the greater area?  Is this strategy always / sometimes / never going to work?  What does this strategy say about what they understand?

Some students used identical shapes to cover the inside of each figure.

And some students used different shapes to cover the figures.

Notice that example 9 here includes different units in both figures, but has reorganized them underneath to show the difference (can you tell which line represents which figure?).

###### Building Meaningful Conversations

Each of the samples above show the thinking, reasoning and understanding that the students brought to our math class.  They were given a very difficult task and were asked to use their reasoning skills to find an answer and prove it.  In the end, students were split between which figure had the greater area (some believing they were equal, many believing that one of the two was larger).  In the end, students had very different numerical answers as to how much larger or smaller the figures were from each other.  These discrepancies set the stage for a powerful learning opportunity!

For example, asking questions that get at the big ideas of measurement are now possible because of this problem:

“How is it possible some of us believe the left figure has a larger area and some of us believe that the right figure is larger?”

“Has example 8 (scroll up to take a closer look) proven that they both have the same area?”

“Why did example 9 use two pictures?  It looks like many of the cuisenaire rods are missing in the second picture?  What did you think they did here?”

In the end, the conversations should bring about important information for us to understand:

• We need comparable units if we are to compare 2 or more figures together.  This could mean using same-sized units (like examples 1, 4, 5 & 6 above), or corresponding units (like example 8 above), or units that can be reorganized and appropriately compared (like example 9).
• If we want to determine the area numerically, we need to use the same-sized piece exclusively.
• The smaller the unit we use, the more of them we will need to use.
• It is difficult to find the exact area of figures with rounded parts using the tools we have.  So, our measurements are not precise.

###### Some generalizations we can make here to help us with any topic in any grade

When our students are being introduced to a new topic, it is always beneficial to start with their ideas first.  This way we can see the ideas they come to us with and engage in rich discussions during the lesson close that helps our students build understanding together.  It is here in the discussions that we can bridge the thinking our students currently have with the thinking needed to understand the concepts you want them to leave with.  In the example above, the students entered this year with many experiences using non-standard measurements, and this year, most of their experiences will be using standard measurements.  However, instead of starting to teach this year’s standards, we need to help our students make some connections, and see the need to learn something new.  Considering what the first few days look like in any unit is essential to make sure our students are adequately prepared to learn something new!  (More on this here: What does day one look like?)

To me, this is what formative assessment should look like in mathematics!  Setting up experiences that will challenge our students, listening and observing our students as they work and think… all to build conversations that will help our students make sense of the “big ideas” or key understandings we will need to learn in the upcoming lessons.  When we view formative assessment as a way to learn more about our students’ thinking, and as a way to bridge their thinking with where we are going, we tend to see our students through an asset lens (what they DO understand) instead of their through the deficit lens (i.e., gaps in understanding… “they can’t”…, “didn’t they learn this last year…?).  When we see our students through an asset lens, we tend to believe they are capable, and our students see themselves and the subject in a much more positive light!

Let’s take a closer look at the features of this lesson:

• Little to no instruction was given – we wanted to learn about our students’ thinking, not see if they can follow directions
• The problem was open enough to have multiple possible strategies and offer multiple possible entry points (low floor – high ceiling)
• Asking students to prove something opens up many possibilities for rich discussions
• Students needed to begin by using their reasoning skills, not procedural knowledge…
• Coming up with a response involved students doing and thinking… but the real learning happened afterward – during the consolidation phase

A belief I have is that the deeper we understand the big ideas behind the math our students are learning, the more likely we will know what experiences our students need first!

A few things to reflect on:

• How often do you give tasks hoping students will solve it a specific way?  And how often you give tasks that allow your students to show you their current thinking?  Which of these approaches do you value?
• What do your students expect math class to be like on the first few days of a new topic/concept?  Do they expect marks and quizzes?  Or explanations, notes and lessons?  Or problems where students think and share, and eventually come to understand the mathematics deeply through rich discussions?  Is there a disconnect between what you believe is best, and what your students expect?
• I’ve painted the picture here of formative assessment as a way to help us learn about how our students think – and not about gathering marks, grouping students, filling gaps.  What does formative assessment look like in your classroom?  Are there expectations put on you from others as to what formative assessment should look like?  How might the ideas here agree with or challenge your beliefs or the expectations put upon you?
• Time is always a concern.  Is there value in building/constructing the learning together as a class, or is covering the curriculum standards good enough?  How might these two differ?  How would you like your students to experience mathematics?

## The Zone of Optimal Confusion

In the Ontario curriculum we have many expectations (standards) that tell us students are expected to, “Determine through investigation…” or at least contain the phrase “…through investigation…”.  In fact, in every grade there are many expectations with these phrases. While these expectations are weaved throughout our curriculum, and are particularly noticeable throughout concepts that are new for students, the reality is that many teachers might not be familiar with what it looks like for students to determine something on their own.  Probably in part because this was not how we experienced mathematics as students ourselves!

First of all, I believe the reason behind why investigating is included in our curriculum is an important conversation!   I’ve shared this before, but maybe it will help explain why we want our students to investigate:

The chart shows 3 different teaching approaches and details for each (for more thoughts about the chart you might be interested in What does Day 1 Look Like). Hopefully you have made the connection between the Constructivist approach and the act of “determining through investigation.”  Having our students construct their understanding can’t be overstated. For those students you have in your classroom that typically aren’t engaged, or who give up easily, or who typically struggle… this process of determining through investigation is the missing ingredient in their development. Skip this step and start with you explaining procedures, and you lose several students!

Traditionally, however, many teachers’ goal was to scaffold the learning.  They believed that a gradual release of responsibilities would be the most helpful.  Cathy Seeley in her book Making Sense of Math: How to Help Every Student Become a Mathematical Thinker and Problem Solver explains the issue clearly:

In the two pieces above Cathy explains the “upside-down teaching” approach.  This is exactly the approach we believe our curriculum is suggesting when it says “determine through investigation,”  and exactly the approach suggested here:

At the heart of this is the idea of “productive struggle”, we want our students actively constructing their own thinking.  However, I wonder if we could ever explain what “productive struggle” looks / feels like without ever experiencing it ourselves?  How might the following graphic help us reflect on our own understanding of “productive struggle” and “engagement”?

I think it would be a wonderful opportunity for us to share problems and tasks that allow for productive struggle, that have student reasoning as its goal, problems / tasks that fit into this “zone of optimal confusion”.

In the end, we know that these tasks, facilitated well, have the potential for deep learning because the act of being confused, working through this confusion, then consolidating the learning effectively is how lasting learning happens!

Let’s commit to sharing a sample, send a link to a problem / task that offers students to be confused and work through that confusion to deepen understanding.  Let’s continue sharing so that we know what these ideals look like for ourselves, so we can experience them with our own students!

## 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:

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?

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?
• 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:)

## 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.

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.

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).

## 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!