Category: Teaching Tuesdays

Our focus this year has been summarizing.  We have placed additional emphasis on using summaries to increase student learning in mathematics.

Today was our instructional rounds.  Administrators and coaches from throughout the district gathered at our school to observe in the classrooms and collect data related to 5 steps of summarizing.   These steps are:  comprehension, chunking, compacting, conceptualizing, and connecting.

I have struggled with the idea of summarizing in mathematics.   Not because I don’t think it is valuable, but because I am not sure what it looks like.   Today brought some clarity…I think.

Here are my thoughts…so far

1.  Parts of mathematics  …the algorithms, conditional statements, and formulas, etc…. are summaries.   They succinctly describe how to solve problems and define relationships.  So we teachers need to consider what and how we are asking students to write summaries.
2. Traditionally, we have made it easy on our students.  We have summarized the summaries, if you will.  We have explicitly taught the skill first then added the context.  If I were to change one thing about my past experience teaching mathematics, I would change this.   Instead of starting a lesson with a lecture, I would start with a task.   I would have the students then summarize their own thinking around the task.  Then collectively we would summarize a strategy related to a task.  And then, I would share the common algorithm or compare class thinking to common theorems or definitions.
3. If/when I asked my students to summarize around a standard algorithm, I would expect the summary to actually add back in the extra details.  Thus summary does not equal less words…especially in mathematics.
4. And the five steps of mathematical summary are more like an escalator in my mind.  It is difficult to distinguish where one step ends and the other begins.  You can slide right in to the next level without realizing it.
5. For the sake of clarification though, this is how a see a summary of mathematical problem solving within the 5 step framework.
   • comprehension:  The student reads and understands the problem.  This might call for an explanation of vocabulary.  It might also prompt further questioning on the part of the student.
   • chunking:  Students generate a list of smaller steps or tasks.  Based on the questions they generated in the previous step.
   • compacting:  Students begin to answer questions and simplify the process.  They eliminate unnecessary steps in the name of efficiency.  They use variables and generate a general process for solving similar problems.
   • conceptualizing:  Students use examples or models to verify and illustrate their process.
   • connect:  Students write a statement to formalize the problem solving process.  In connection to this, the student explains why the process works.

So in summary, mathematics is about problem solving.  When we create algorithms or make conjectures and defend these with a valid but succinct argument, we are in effect summarizing our problem solving practice.

Update:  In a moment of reflectioon I wrote the following at an ELL training as I considered language as action.

A summary is stripped of all the hard work (language and mental processing) that comes with producing it! To give a summary outside of context is meaningless. To not ask students to engage in summarizing (writing), is a disservice. It renders the content meaningless.