Noticing Subtleties…Reflection

So I wrote this post yesterday, and today we tried it out.

The purpose was to give students an opportunity to notice the subtleties in the language associated with three different mathematical scenarios they may need to represent.

We read each scenario out loud and then asked the students to compare and contrast each problem type with their table partner.

Some students began by discussing the similarities and differences of the contexts…noticing subtleties 1

Others began by making a list of what they noticed on the back…You can see that this student paid attention to more of the mathy parts~understanding what was meant by a one-variable versus a two variable equation.

noticing subtleties 2


And then you’ll see below where students were able to make sense of each scenario and the math required.  However, the first student used an equation in two variables for the first scenario and created a table to find the solution.

We didn’t get to the whole class conversation part of this lesson…I want to talk about each problem type and how to recognize the differences.  Notice that two of the students above wrote the equation for the two variable scenario, but the third student created a table.  I think we need to talk about why that is.  Also, I think I may want to do three more scenarios that would produce equations in standard form, to see if they would recognize the differences then.

I definitely think that this was a useful exercise and would do it again.

One little shout out…one student pulled out their phone because they wanted to check out the equations for the third scenario on their Desmos app!!!


July Challenge #10 Writing in the Math Classroom

Many years ago I read the book Writing to Develop Mathematical Understanding by David Pugalee.  I want to use this post to record some of the important points.

Pugalee wrote that

“The goal of writing in mathematics is to engage students in ways that require them to think deeply about the mathematics they are encountering.”

He suggested that, writing, or more generally, communication, happens along a continuum.



This continuum “is not about writing ability but about the level of cognitive engagement of the student.”

He offers many practical suggestions for incorporating writing into the daily routine of math classrooms.

Beginning of the lesson suggestions (pages 36-38):

  • Use a prompt such as “Write what you know about ________.”
  • Have students write a short description of how they solved a particular homework problem.

Middle of the lesson suggestions:

  • Have students write an example or draw a diagram or other illustration to demonstrate a key idea or concept.
  • Have students write a question about a concept or problem, then turn to another classmate and exchange questions.
  • If students are taking notes, pause and have them write a summary of an idea or concept in the margin.

End of the lesson suggestions:

  • Write the main idea from the day’s lesson.
  • Write definitions in your own words. This might also apply to a procedure or property.
  • Have students exchange notes, a practice problem, or another task.  Students can identify common elements and approaches as well as differences.

He shares a list of 50 Activities for Writing in Mathematics…a few of my favorites include:

  1. Construct test or quiz questions
  2. Write freely on any topic
  3. Create a dialogue between one student and another
  4. Defend a decision or action
  5. List characteristics or steps
  6. Write a mathematics word problem
  7. Prepare an outline of a lesson
  8. Identify personal goals for mathematics learning
  9. Write about what gave you difficulty on a particular task
  10. Write about how two problems are similar or different

Pugalee discusses the importance of creating a safe environment for communicating in the classroom.  He offers suggestions for promoting this culture.

Struggling writers might need “a framework or skeleton” as entry points into a communication task.  This might look like:

I need to…

I notice…

This means…

So, I need to… because…

Therefore, the…

A couple of quick ideas that I use for incorporating writing into lessons that were inspired by this book:

  • I might put these words on the board:  hypotenuse, Pythagorean Theorem, leg.  And then ask students to write 2-3 sentences using these words.
  • I like to create fill-in the blank responses.  So students have to make sense of what’s already on the paper…and how best to complete the sentences.

There is soooooo much out there on this topic.  But I don’t think enough of it has trickled down to implementation in the classroom.

I’m going to continue to add to this list below…but here are a few additional resources for learning about writing in the math classroom.