Transparent and Organized Math Instruction

A puzzle for my math teaching this year was how to make our learning goals transparent and how to organize class work so that it could be easily coalated into a portfolio at the end of the year.  Then I learned about a resource called Betterlesson.com.  They hire master teachers to submit lessons in a variety of topics.  I liked one teacher's work in particular, especially the way her lessons featured student notes.

I appreciated the way she focused on her learning goals, both questions and mandated standards.  The "I will be able to" statements allowed students to understand what they were responsible for learning. The "Do Now" offered a way to kick off activities or to do mini-assessments.  The notes can contain information, activities, links, or resources.  It is easy to attach class or homework to them.  

I adapted this model for my own class and felt much more organized this year.  It was easy to integrate essential questions into our every day lessons.  It is also a useful system for differentiation as each student can get an individualized set of activities attached to the notes.   Students continued to need reminders to keep up with the notes and to use the notes to study for tests, but the system was in place for them to do so.  It was also useful during conversations with parents, as my expectations were clearly articulated in the student materials.  Here are some samples:

Solid Figure Notes

Integer Notes

Teaching Without Ego

What would it mean to

TEACH WITHOUT EGO?

Since returning from Lake Qinghai and the grasslands of Tibet. I have been contemplating this idea deeply, using it as a mantra to keep me grounded in a variety of scenarios.

Because, as a teacher, it's amazing how often I bump into my ego.  Not to mention the egos of my colleagues.  The worst is when you encounter two egos in the hallway duking it out, however politely. You know it's time to back away slowly.

What is "ego"?  It is a projection of the self, a self that is separate and different and special from the other selves in this world.  It is the source of identity and self esteem.  It is the driver of the question, "Who am I?"  Its importance is culturally constructed and particularly celebrated in modern America. 



It is also divisive and a barrier to compassion.  It is at the root of exclusion, indifference, and hate.  It breeds insecurity, fear, jealousy, and competition.   And as teachers,  it can be a detriment to our practice.

There are a good articles on this idea, from teachers of various disciplines:

"...the role of teacher is one in which you gain satisfaction through observing the learning of your pupils."

 "...one needs to teach with as little ego involvement as possible." 

But it's hard.  Because don't we teach who we are?  Don't we need to "guide [students] with a strong sense of self?"

The danger is this:

The ego can exist in two states – one, wherein it is aligned with the higher consciousness and the other, where it is not in alignment with this consciousness.

When the ego is aligned with the higher consciousness, there is just a quiet assertion of the individual self in a particular direction, which it knows intuitively is right for the fulfilment of its life purpose. This assertion is based on love and faith and leads to a harmonious state of goodwill and cooperation for the greater common good. When it is not aligned with the higher consciousness, the ego can really wreak havoc. It can misguide and misinterpret. Propelled by emotions based on fear and insecurity, it can cause action motivated by the desire for applause, attention, getting ahead of competition, reaching the finishing line first, proving one’s superiority over others, etc. Here, the ego is intensely conscious of the other and derives its identity in comparison with the other. The individual self asserts itself purely from the need to create a position of unassailability from which it cannot be dislodged.

~ Sangeeta Menon, "Are Teachers Allowed an Ego?"

"The inviolable principle must be that the interests of the student must at all times be foremost..."

When I joined my current school I entered into a team teaching situation that I greatly underestimated for its difficulty.  I have come to realize that 99% of any angst I feel at work can be traced back to my ego.  Desires to be recognized, respected, and right can be strong. Self-preservation seems sensible and competition seems healthy.  A twinge of resentment over a close relationship or individually achieved success seems normal. Suspicion of new colleagues and initiatives seems expected. Giving priority to the activities and topics I prefer seems logical. But when these emotional responses interfere with my ability to focus on what is best for my students and their learning, my ego is the one that needs to back down.  

In those moments I am learning not to fight, not to defend, not to protect, not to promote.  I am learning to let go.  When "I" am at the center of my practice, the world my students and I are creating together feels fragile and destabilized, tilted in the direction of a self-concerned perspective.  When my students are at the center of my practice, our world seems infinitely more solid.

In pursuing this idea, I meditate on these questions:

• In what ways does my ego support  the development of my teaching practice?
• In what ways does my ego impede the development of my teaching practice?
• In what ways does my ego improve my relationships with students and colleagues?
• In what ways does my ego make relationships with students and colleagues more difficult?
• In what ways can I work with greater compassion?

Reflection: Order of Operations and Geography Stories

So one of the deals with this reflective blog business is that I want to make my practice visible and vulnerable.  Best to start with today.  Beware, extremely detailed lesson talk ahead.

Over the weekend I replanned our geography unit and set up another round of Algebra week for our fourth graders.  These are both units that I have loved teaching because the ideas are accessible at many different levels allowing for much differentiation and the information is useful for a variety of tasks in and out of school. There are a lot of fun activities and what we do now will be helpful later on in their studies.

Today I taught a lesson on the order of operations.  Now this is truly an idea they will explore extensively in 5th and 6th grade so why bother in 4th?  We have a pretty wide range of mathematical abilities, but these beginning algebra lessons have a little something for everyone.  They are learning to reason algebraically, recognize and construct equations,  and apply basic principles such as balancing values and using inverse operations fluidly, while practicing all sorts of basic operations and reviewing their facts.  We talk about working systematically and efficiently.  We talk about challenge as a good thing.  We are previewing material many students find scary in a fun way with little risk.   I don't expect them to master this material.  We are just playing around with numbers.

Meanwhile we are asking some deep questions such as:

We will try and keep these lines of inquiry open and reflect on them as we review a variety of topics throughout the year.

The goals for the lesson were:

• To introduce the order of operations
• To practice writing equations that reflect our thought processes
• To apply memorized math facts in a novel situation

We started with the equation 18 + 6/3 x 2 and asked them to provide an answer they thought would work.  We received mostly 4 and 16 from the students.  Teachers then threw out 40, 19, and 22.  We discussed why this was happening, tossing around the idea that everyone was starting in their own place and following their own path.  I presented the idea of an order of operations that mathematicians had agreed upon.  Then we played hopscotch and learned a rhyme to go with how we might solve an equation: Parentheses, Exponents, Multiplication or Division Left to Right, Addition or Subtraction Left to Right.

Students then practiced on 3 more equations with partners, looking for the most accurate answer.  They encountered parentheses and exponents and I explained how those worked.  They had questions about how the order of operations might work within a pair of parentheses.  They wondered about whether to do multiplication or division first and I reinforced the idea of them being equal so we just go left to right.  We found instances where the answer wasn't dependent on the order of operations, but recognized it was important to not assume that would always be the case.  I am leaving the PEMDAS vs. GEMS debate to the middle school and did not introduce those terms at all.

Then I gave students the four 5's problem. I had made cards with the 4 operation symbols,  a pair of paratheses, four 5s, and an equal sign (which I had forgotten but they asked for).

I asked them to make as many equations as they could and find out how many different answers they could get.  I had found two versions of this activity and was trying to simplify one and make another more open ended.  I realized from their questions that I had taken away some of the structures that made the problem particularly challenging and interesting.  I had also not predicted and prepared contingencies for some of the directions they would go.    Were they allowed to make numbers like 55?  Did the answer have to be 5?  Did they have to use all of the cards?  What if they wanted to repeat an operation, but didn't have another card with that sign?  Could they make exponents?  Now I am pretty flexible and was mostly interested in watching them work it out:

It all went fine.  There were some thoughtful equations and some ripe for refinement.  Overall, I could see the benefit in going back to the original problems the way that they were presented, which should probably be the next step.  Instead of exploring the open range of equations (with no self checking mechanism) we can move towards puzzler models, uncovering equations that will allow us to get to 10 or 125 or 24.   

My writing lesson got smooshed between an art project and lunch.  As part of our geography unit we are going to write geography stories about places and geographical features that are meaningful to us.  I had prepared a pre writing activity, but after a vibrant work period involving watercolors, interspersed with PE and an emergency drill, I was left with only enough time to present the idea and then give it out for homework. Ideally we could have brainstormed in class, began the pre-writing activity, and shared our work.  This didn't happen. We will see the consequences of this on Thursday.

It may be that I get away with it, or that we will need to compensate with more coaching on the back end.  It's something I know I could have taught better and will teach better next time.  I am comforted by the fact that I am not the first teacher to run out of time or to sacrifice a darling for a greater cause. Nor am I the last.  It's an occupational hazard.  Luckily, we all come back tomorrow and the project will live to see another day.