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 we 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.  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.  Telling someone 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, 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.
  • 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 you 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)!

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.

Complexity1.jpg
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).

Starting where our students are….. with THEIR thoughts

A common trend in education is to give students a diagnostic in order for us to know where to start. While I agree we should be starting where our students are, I think this can look very different in each classroom.  Does starting where our students are mean we give a test to determine ability levels, then program based on these differences?  Personally, I don’t think so.

Giving out a test or quiz at the beginning of instruction isn’t the ideal way of learning about our students.  Seeing the product of someone’s thinking often isn’t helpful in seeing HOW that child thinks (Read, What does “assessment drive instruction mean to you” for more on this). Instead, I offer an alternative- starting with a diagnostic task!  Here is an example of a diagnostic task given this week:

Taken from Van de Walle’s Teaching Student Centered Mathematics

This lesson is broken down into 4 parts.  Below are summaries of each:


Part 1 – Tell 1 or 2 interesting things about your shape

Start off in groups of 4.  One student picks up a shape and says something (or 2) interesting about that shape.


Here you will notice how students think about shapes. Will they describe the shape as “looking like a mountain” or “it’s an hourglass” (visualization is level 1 on Van Hiele’s levels of Geometric thought)… or will they describe attributes of that shape (this is level 2 according to Van Hiele)?

As the teacher, we listen to the things our students talk about so we will know how to organize the conversation later.


Part 2 – Pick 2 shapes.  Tell something similar or different about the 2 shapes.

Students randomly pick 2 shapes and either tell the group one thing similar or different about the two shapes. Each person offers their thoughts before 2 new shapes are picked.

Students who might have offered level 1 comments a minute ago will now need to consider thinking about attributes. Again, as the teacher, we listen for the attributes our students understand (i.e., number of sides, right angles, symmetry, number of vertices, number of pairs of parallel sides, angles….), and which attributes our students might be informally describing (i.e., using phrases like “corners”, or using gestures when attempting to describe something they haven’t learned yet).  See chart below for a better description of Van Hiele’s levels:

Van Hiele’s chart shared by NCTM

At this time, it is ideal to hold conversations with the whole group about any disagreements that might exist.  For example, the pairs of shapes above created disagreements about number of sides and number of vertices.  When we have disagreements, we need to bring these forward to the group so we can learn together.


Part 3 – Sorting using a “Target Shape”

Pick a “Target Shape”. Think about one of its attributes.  Sort the rest of the shapes based on the target shape.


The 2 groups above sorted their shapes based on different attributes. Can you figure out what their thinking is?  Were there any shapes that they might have disagreed upon?


Part 4 – Secret sort

Here, we want students to be able to think about shapes that share similar attributes (this can potentially lead our students into level 2 type thinking depending on our sort).  I suggest we provide shapes already sorted for our students, but sorted in a way that no group had just sorted the shapes. Ideally, this sort is something both in your standards and something you believe your students are ready to think about (based on the observations so far in this lesson).


In this lesson, we have noticed how our students think.  We could assess the level of Geometric thought they are currently using, or the attributes they are comfortable describing, or misconceptions that need to be addressed.  But, this lesson isn’t just about us gathering information, it is also about our students being actively engaged in the learning process!  We are intentionally helping our students make connections, reason and prove, learn/ revisit vocabulary, think deeper about specific attributes…


I’ve shared my thoughts about what I think day 1 should look like before for any given topic, and how we can use assessment to drive instruction, however, I wanted to write this blog about the specific topic of diagnostics.

In the above example, we listened to our students and used our understanding of our standards and developmental research to know where to start our conversations. As Van de Walle explains the purpose of formative assessment, we need to make our formative more like a streaming video, not just a test at the beginning!van-de-walle-streaming-video

If its formative, it needs to be ongoing… part of instruction… based on our observations, conversations, and the things students create…  This requires us to start with rich tasks that are open enough to allow everyone an entry point and for us to have a plan to move forward!

I’m reminded of Phil Daro’s quote:

daro-starting-point

For us to make these shifts, we need to consider our mindsets that also need to shift.  Statements like the following stand in the way of allowing our students to be actively engaged in the learning process starting with where they currently are:

  • My students aren’t ready for…
  • I need to start with the basics…
  • My students have gaps in their…
  • They don’t know the vocabulary yet…

These thoughts are counterproductive and lead to the Pygmalion effect (teacher beliefs about ability become students’ self-fulfilling prophecies).  When WE decide which students are ready for what tasks, I worry that we might be holding many of our students back!

If we want to know where to start our instruction, start where your students are in their understanding…with their own thoughts!!!!!  When we listen and observe our students first, we will know how to push their thinking!

An Unsolved Problem your Students Should Attempt

There are several great unsolved math problems that are perfect for elementary students to explore.  One of my favourites is the palindrome sums problem.
In case you aren’t familiar, a palindrome is a word, phrase, sentence or number that reads the same forward and backward.


The problem itself comes out of this conjecture:

If you take any number and add it to its inverse (numbers reversed), it will eventually become a palindrome.

Let’s take a look at the numbers 12, 46 and 95:

Palindrome adding1


Palindrome adding2


Palindrome adding3


As you can see with the above examples, some numbers can become palindromes with 1 simple addition, like the number 12.  12 + 21 = 33.  Can you think of others that should take 1 step?  How did you know?

Other numbers when added to their inverse will not immediately become a palindrome, but by continuing the process, will eventually, like the numbers 46 and 95.  Which numbers do you think will take more than 1 step?


After going through a few examples with students about the process of creating palindromes, ask your students to attempt to see if the conjecture is true (If you take any number and add it to its inverse (numbers reversed), it will eventually become a palindrome).  Have them find out if each number from 0-99 will eventually become a palindrome. Ask students if they need to find the answer for each number?  Encourage them to make their own conjectures so they don’t need to do all of the calculations for each number.


Mathematically proficient students look closely to discern a pattern or structure. They notice if calculations are repeated, and look both for general methods and for shortcuts. (SMP 7)

Some students will notice:

  • Some numbers will already be a palindrome
  • If they have figured out 12 + 21… they will know 21 + 12.  So they don’t have to do all of the calculations.  Nearly half of the work will have already been completed.
  • A pattern emerging in their answers … 44, 55, 66, 77, 88, 99, 121… and see this pattern regularly (well, almost)
  • A pattern about when numbers will have 99 as a palindrome, or 121… 18+81 = 27+72 = 36+63…

Thinking about our decisions:
  • What is the goal of this lesson?  (Practice with addition?  Looking for patterns?  Perseverance?  Making conjectures? …)
  • Do we share some of the conjectures students are making during the time when students are working, or later in the closing of the lesson?  
  • How will my students record their work?  Keep track of their answers?  Do I provide a 0-99 chart or ask them to keep track somehow?
  • Will students work independently / in pairs / in small groups?  Why?
  • Do I allow calculators?  Why or why not?  (think back to your goal)
  • How will I share the conjectures or patterns noticed with the class?
  • Are my students gaining practice DOING (calculations) or THINKING (noticing patterns and making conjectures)?  Which do you value?

The smallest of decisions can make the biggest of impacts for our students!  


So, at the beginning of this post I shared with you that this problem is currently unsolved.  While it is true that the vast majority of numbers have been proven to easily become palindromes, there are some numbers that require many steps (89 and 98 require 24 steps), and others that have never been proven to either work or not work (198 is the smallest number never proven either way).


Some final thoughts about this problem…

After using this problem with many different students I have noticed that many start to see that mathematics can be a much more intellectually interesting subject than they had previously experienced.  This problem asks students to notice things, make conjectures, try to prove their conjectures and be able to communicate their conjectures with others…  The problem provides students with the opportunity to both think and do.  It offers students from various ability levels access to the problem (low floor), and many different avenues to challenge those ready for it (high ceiling).  It tells students that math is still a living, growing subject… that all of the problems have not yet been figured out!  And probably the most important for me, it sends students messages about what it means to really do mathematics!

 

palindrome2
Taken from Marilyn Burns’ 50 Problem Solving Lessons resource

 

Focus on Relational Understanding

Forty years ago, Richard Skemp wrote one of the most important articles, in my opinion, about mathematics, and the teaching and learning of mathematics called Relational Understanding and Instrumental Understanding.  If you haven’t already read the article, I think you need to add this to your summer reading (It’s linked above).

Skemp quite nicely illustrates the fact that many of us have completely different, even contradictory definitions, of the term “understanding”.   Here are the 2 opposing definitions of the word “understanding”:

“Instrumental understanding” can be thought of as knowing the rules and procedures without understanding why those rules or procedures work. Students who have been taught instrumentally can perform calculations, apply procedures… but do not necessarily understand the mathematics behind the rules or procedures.

Instrumental2.png

“Relational understanding”, on the other hand, can be thought of as understanding how and why the rules and procedures work.  Students who are taught relationally are more likely to remember the procedures because they have truly understood why they work, they are more likely to retain their understanding longer, more likely to connect new learning with previous learning, and they are less likely to make careless mistakes.

Relational2.png

Think of the two types of understanding like this:

 

Shared by David Wees

Students who are taught instrumentally come to see mathematics as isolated pieces of knowledge. They are expected to remember procedures for each and every concept/skill.  Each new skill requires a new set of procedures.  However, those who are taught relationally make connections between and within concepts and skills.  Those with a relational understanding can learn new concepts easier, retain previous concepts, and are able to deviate from formulas/rules given different problems easier because of the connections they have made.

 

While it might seem obvious that relational understanding is best, it requires us to understand the mathematics in ways that we were never taught in order for us to provide the best experiences for our students. It also means that we need to start with our students’ current understandings instead of starting with the rules and procedures.

Skemp articulates how much of an issue this really is in our educational system when he explains the different types of mismatches that can occur between how students are taught, and how students learn.  Take a look:

Instrumental vs Relational

Notice the top right quadrant for a second.  If a child wants to learn instrumentally (they only want to know the steps/rules to solve today’s problem) and the teacher instead offers tasks/problems that asks the child to think or reason mathematically, the student will likely be frustrated for the short term.  You might see students that lack perseverance, or are eager for assistance because they are not used to thinking for themselves.  However, as their learning progresses, they will come to make sense of their mathematics and their initial frustration will fade.

On the other hand, if a teacher teaches instrumentally but a child wants to learn relationally (they want/need to understand why procedures work) a more serious mismatch will exist.  Students who want to make sense of the concepts they are learning, but are not given the time and conditions to experience mathematics in this way will come to believe that they are not good at mathematics.  These students soon disassociate with mathematics and will stop taking math classes as soon as they can.  These students view themselves as “not a math person” because their experiences have not helped them make sense of the mathematics they were learning.

While the first mismatch might seem frustrating for us as teachers, the frustration is short lived. On the other hand, the second mismatch has life-long consequences!


I’ve been thinking about the various initiatives/ professional development opportunities… that I have been part of, or have been available online or through print that might help us think about how to move from an instrumental understanding to a relational understanding of mathematics.  Here are a few I want to share with you:

Phil Daro’s Against Answer Getting video highlights a few instrumental practices that might be common in some schools.  Below is the “Butterfly” method of adding fractions he shares as an “answer getting” strategy.  While following these simple steps might help our students get the answer to this question, Phil points out that these students will be unable to solve an addition problem with 3 fractions.  These students “understand” how to get the answer, but in no way understand how the answer relates to addends.

Daro - Butterfly.jpg

On the other hand, teachers who teach relationally provide their students with contexts, models (i.e, number lines, arrays…), manipulatives (i.e., cuisenaire rods, pattern blocks…) and visuals to help their students develop a relational understanding.  If you are interested in learning more about helping your students develop a relational understanding of fractions, take a look at a few resources that will help:


Tina Cardone and a group of math teachers across Twitter (part of the #MTBoS) created a document called Nix The Tricks that points out several instrumental “tricks” that do not lead to relational understanding.  For example, “turtle multiplication” is an instrumental strategy that will not help our students understand the mathematics that is happening.  Students can draw a collar and place an egg below, but in no way will this help with future concepts!Turtle mult.jpg

Teachers focused on relational understanding again use representations that allow their students to visualize what is happening.  Connections between representations, strategies and the big ideas behind multiplication are developed over time.

Take a look at some wonderful resources that promote a relational understanding of multiplication:

Each of the above are developmental in nature, they focus on representations and connections.


So how do we make these shifts?  Here are a few of my thoughts:

  1. Notice instrumental teaching practices.
  2. Learn more about how to move from instrumental to relational teaching.
  3. Align assessment practices to expect relational understanding.

Goal 1 – Notice instrumental teaching practices.

Many of these are easy to spot.  Here is a small sample from Pinterest:

 

The rules/procedures shared here ask students to DO without understanding.  The issue is that there are actually countless instrumental practices out there, so my goal is actually much harder than it seems.  Think about something you teach that involves rules or procedures.  How can you help your students develop a relational understanding of this concept?


Goal 2 – Learn more about how to move from instrumental to relational teaching.

I don’t think this is something we can do on our own.  We need the help of professional resources (Marian Small, Van de Walle, Fosnot, and countless others have helped produce resources that are classroom ready, yet help us to see mathematics in ways that we probably didn’t experience as students), mathematics coaches, and the insights of teachers across the world (there is a wonderful community on Twitter waiting to share and learn with you).

I strongly encourage you to look at chapter 1 of Van de Walle’s Teaching Student Centered Mathematics where it will give a clearer view of relational understanding and how to teach so our students can learn relationally.

However we are learning, we need to be able to make new connections, see the concepts in different ways in order for us to know how to provide relational teaching for our students.


Goal 3 – Align assessment practices to expect relational understanding.

This is something I hope to continue to blog about.  If we want our students to have a relational understanding, we need to be clear about what we expect our students to be able to do and understand.  Looking at developmental landscapes, continuums and trajectories will help here.  Below is Cathy Fosnot’s landscape of learning for multiplication and division.  While this might look complicated, there are many different representations, strategies and big ideas that our students need to experience to gain a relational understanding.

Fosnot landscape2
Investigating Multiplication and Division Grades 3-5

Asking questions or problems that expect relational understanding is key as well.  Take a look at one of Marian Small’s slideshows below.  Toward the end of each she shares the difference between questions that focus on knowledge and questions that focus on understanding.


I hope whatever your professional learning looks like this year (at school, on Twitter, professional reading…) there is a focus on helping build your relational understanding of the concepts you teach, and a better understanding of how to build a relational understanding for your students.  This will continue to be my priority this year!