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.

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