Monthly Archives: April 2013

Learning Trajectories: A research-based tool you can use right now


In an earlier post, I talked about the gap between the work being done by researchers and the work being done by teachers in classrooms.  Today, I’d like to talk about a resource that I think is a good example of a tool that connects research and practice.

The Common Core Standards have received a lot of attention lately.   I often struggle when trying to unpack the Standards in order to develop a lesson.  A research team from NC State has created an interactive tool, available at, which I have found very useful in designing lessons.  The researchers developed 18 learning trajectories for the K-8 Common Core Standards for Mathematics.   These learning trajectories describe how students develop an understanding of a particular concept, or set of concepts, over time.

The hexagon map allows you to click on one of the trajectories or on an individual standard and get detailed information about how students might move from prior knowledge and informal ideas to sophisticated understandings of various concepts.  Also included throughout are common student misconceptions and a variety of models and representations that have been shown to foster particular understandings.   What I love about the trajectories is that they include the research that was used to build them.

So how do I use them?  Recently, I needed to prepare a lesson for fourth graders on angles so I went to the site and clicked on the Shapes and Angles Trajectory.  I was able to view suggested activities, such as the angle game where students stand and follow directions to rotate a half turn or quarter turn.  I also read about common student misconceptions, such as the fact that students sometimes think that an angle with longer rays is larger than an angle with shorter rays, and viewed questions that assess this misconception.   The references were also included in case I wanted to go back to some of the original papers and explore any of the ideas further.

Of course, it wasn’t the only resource I used when I was preparing the lesson.  I used my prior experience of teaching angles and other resources I have collected over the years.  But it was nice to have place to go that summarized the major research in a way that allowed me to think about how the students might develop a concept and the challenges they might face along the way.

I’d love to know of any resources you use that connect research and teaching.





Transforming the Teaching Profession: It starts with RESPECT


Articles titled “My profession no longer exists”  and “A warning to young people: don’t become a teacher” have received a great deal of attention recently.  Clearly, they are resonating with many teachers who seem to agree that the profession is being devalued and demeaned.

What’s sad to me is that while I continue to see these articles making the rounds on social media, I don’t see many posts or tweets about people who are offering a solution.  Quitting the profession or discouraging others from joining it cannot be the only ways to take action.

It seems to me that in order to really address the issues that those pieces so passionately point at, we need to have collaboration across different groups.  That’s why I was happy to learn that the President unveiled a blueprint for the RESPECT initiative today.  I think it is a great example of what is possible when those inside the classroom work together with those outside the classroom.

Although my main interests are research and teaching, I also spent a little bit of time in the policy world.   In 2008, I had the opportunity to be a Teaching Ambassador Fellow for the Department of Education.

It was a transforming experience for me.  I had a chance to be a voice for teachers within the department, learn about policy from those within the department and meet many amazing teachers from across the country.  However, I also became aware of the huge gap that exists between policy makers and teachers.  The fellowship was an attempt to bridge that gap.

The fellowship has continued to evolve since our initial group of fellows began in 2008.   The RESPECT initiative is one of the things that resulted from this continued collaboration between policymakers and teachers.

Fellows led hundreds of conversations across the country with teachers about how they envision transforming the teaching profession.   Input gathered from these interviews was used to help inform the blueprint.  I think it’s a great example of the potential that exists when different groups work together.  I hope it can help begin a conversation of how we fix the problem that has so many teachers feeling discouraged about the profession.

I encourage you to go to the educator homepage to check out the resources below and continue the conversation with those around you.


Two Different Worlds

I started my Ph.D. program thinking I knew a lot about teaching and learning.  After all, I spent 7 years in a classroom, went to many professional development sessions, attended courses, and read a lot of books and articles on education.   However, having the luxury to step away from the classroom and really immerse myself in thinking about teaching and learning exposed me to a whole new world.    I realized that there was all this research out there that would have helped me when I was teaching.

Here’s an example.  Recently, I was looking through the research on algebra.  One of the things I learned after reviewing the literature was that researchers have known since the 80’s that elementary students develop misconceptions about equality that cause major problems later on in algebra.  For example, when most elementary school students see the equal sign, they view it as a signal of where to write an answer or as a direction to work out an equation.  So “5 + 3=__” becomes a direction to add 5 and 3.  They don’t see the equal sign as relating two equivalent quantities or amounts.  Therefore, when they see a number sentence like:  5 + 4 = __ + 2, they often write 9 as their answer.   Various researchers have offered teaching interventions, such as using balance beams when first introducing the equal sign, to avoid this misconception.

What amazed me was that the research community documented this problem years ago and yet, I never heard about it when I was teaching.  Now maybe my fellow teachers were secretly reading lots of journal articles and not telling me about what they were learning, but I have a feeling that I was not the only one who was unaware of the research.

Why was this work that was being conducted in universities by people who had the time and money to study these things not being relayed to the teachers who needed it?  This seemed ridiculous to me.  Imagine if doctors in hospitals weren’t using the research that scientists did in the 80s.

I’ll mention this in other posts, but I want to make it clear that this lack of communication goes both ways.  Some researchers are completely disconnected from what goes on in classrooms today.   Just as I was surprised to learn about the research world, I would imagine some of these researchers would be surprised at what they would learn if they went into a classroom today and had to teach full-time.

So why does this happen?    Why is there this gap between these two worlds?  I’d love to hear your thoughts.



My 5 Core Beliefs about Learning

1. All children can learn

They may not learn at the same pace or in the same way.  They may not need to learn the same thing at the same time.   They may not have the same motivation or the same interests.   But they all can learn and they all can learn math.

When I tell people I was a math teacher, they often tell me they just don’t have a brain for math.  I don’t buy it.   We need to rethink the messages we are sending students both in school and at home because I rarely hear anyone say they don’t have a brain for reading.

2.  Things need to make sense to students.

Learning should be a way for students to make sense of the world around them.  In a math classroom, that means taking everyday situations and using math to help make sense of them.

When math is disconnected from the world students live in, it is viewed as a series of random calculations that don’t make sense.  As a result, students resort to tricks and memorization and then struggle when novel situations are presented.    Math needs to make sense to students.   It needs to be a way to organize the world around them.

3.  Build on what students know.      

All students bring experiences about the world to a learning situation.  These intuitive strategies students already have can be built upon to learn math in a classroom.  All too often there is this assumption that what students need to learn about math comes from a textbook or a teacher.  It is as if they have never had any experiences in their lives that could be used to think about mathematics.

We need to think about how we build upon what students know.  For example, I recently designed a lesson on inequalities.  Students have an intuitive sense of whether their brother has more cookies or fewer cookies than them.  They may not have the symbols or tools to represent these situations mathematically but they understand something about that relationship.  This can be built upon so that math becomes a way to record that situation and to make sense of other situations like that.

4.  There is not a single “best way” to foster learning

Some methods may be better than others but there is no one prescription for how to teach.  It would be much easier if there was and I see the temptation for schools or districts to mandate one way of doing things.  However, all students are unique and there can’t possibly be one right way for a variety of individuals to learn.   We need to question policies that demand one way of teaching.

That’s not to say that I don’t think that we need to radically change the way we approach or evaluate good teaching.  We do and we can talk more about what that looks like in later conversations.   But I am nervous when people prescribe one way to teach children.  I always remember someone telling me during my first year of teaching:  “There is more than one way to skin a cat.”    Now I’m not sure why you would want to skin a cat to begin with, but the sentiment stands.

5.  What students understand is not the same as what they are able to do.

Learning is about understanding.   Understanding is sometimes confused with what students can do.  If students can choose the correct answer on a standardized test, we sometimes say they understand a particular mathematical concept.  But all they might know is how to follow a procedure for a particular type of question.

Really thinking about what we want students to understand about a concept is an important step in designing tasks for students that will foster their learning.    It is also an important step in thinking about how we want to assess them.


So there you have it.  Those are the ideas that guide my work as a teacher and as a researcher.  For me, it’s important to be transparent about what my beliefs are.  It allows me to clear about what I value and it allows others to understand my perspective when I talk about teaching and learning.   Being explicit about these ideas also helps me decide who I want to collaborate with, where I want to work and what projects I want to take on.  In future posts, I’ll talk in more detail about some of the ideas and discuss the research that supports them.

What about you?  What are your core beliefs?

The subway ad that changed my life

It said, “You remember your first grade teacher’s name.  Who will remember yours?”  At the time I was working long hours as a market research analyst trying to figure out why people bought one toothbrush instead of another.    That day on the subway it became clear that I was wasting my time doing something I didn’t care about.

So I quit my job and joined the program advertised on the subway. I was placed in what was labeled a “hard to staff” school with no real teaching experience and little training.  Yet, despite the challenges and heartbreak, I fell in love with teaching.

I taught for seven years, starting at the elementary school level, then moving to middle school math, and eventually coaching math teachers.  I also dabbled a bit in the policy world as a Teaching Ambassador Fellow for the US Department of Education.   My quest to learn more about teaching and learning led me to a Ph.D. program where I am now researching student learning in mathematics.  I also continue to work with current teachers as well as those pursuing a degree in teaching.

I truly believe that teaching is one of the most important and difficult professions one can choose.   Although other jobs have paid more, offered better working conditions and were more respected by my peers, it is the profession I am most proud to have been a part of.    I may no longer be a full-time classroom teacher, but I will always be a teacher at heart.

Over the last few years as a Ph.D. student, I have had the time and opportunity to think about teaching and learning from a different perspective.  Unfortunately, I think that those outside the classroom often speak a different language than those inside the classroom.  As a result, messages get lost in translation.  I believe that only through true communication across these different groups can we bridge the gap so that change can occur.  I hope you come along on the journey as I explore what that means and how we can begin to do that.