Essential Questions to Personalize Mathematics Learning

The biggest breakthrough I’ve ever had in teaching high school mathematics has involved doing drastically less lesson planning.  That probably bears repeating, in a slightly different way:

When I started spending less time planning lessons I was dramatically happier with what was happening in my classroom.

One key factor, and perhaps the backbone of the entire concept, was allowing time, a lot of time, for any example or question I put before the students.  By this I mean something along the lines of spending an entire class period on a single prompt.  It’s an approach which would have seemed preposterous to me 10 years ago, and yet it was absolutely the best idea I have ever tried in teaching HS mathematics.  I have now taught classes in which we spent a full 50 minutes on a single quadratic function as the prompt.  One function.  50 minutes.  And the students understood more from those 50 minutes, and retained more of that learning over the coming weeks and months, than any class I ever taught using multiple examples.

50 minutes on one prompt led to more and better learning

Perhaps now you can see where the reduced lesson planning time comes in.  If I only need one prompt to start class I don’t need a whole lot of time to come up with it.  There are days when I would spend more time devising the right prompt, in the hope of eliciting the types of questions and observations I was most hoping to move toward, but even then it is critical to remember that this whole approach relies heavily on having a high level of comfort with letting the class go in the direction it needs to go in, and even letting individual students go in the directions they need to go in.  I’ve never taught two classes which turned out the same way, even when they started with the same prompt, using this approach.

The students in the room are different, the learning will evolve to meet their needs, and each class will move forward as a group along a unique path.

Another key factor is an approach which consistently allowed me to start class by meeting students exactly where they were.  By this I absolutely mean that each student was able to start the learning process where they were, individually, when they walked in the door, and move forward from that point.

So, how exactly, does one structure 50 minutes of time spent on a single question, example or prompt? If someone had asked me this years ago I would have imagined that it depended on asking a rich enough question.  Any question that is sufficiently challenging will take the students a whole class period, or more, to complete, and will give them plenty to talk about.  But often it will often cause some students to drop out part way through.  Other students may stick with it by following along, but won’t feel that they have contributed their own voice in any meaningful way.

Each and every student must have a meaningful voice in the conversation, and must leave feeling that he or she contributed to the learning process.

So, a richer, more complex question isn’t always the best approach.  (I don’t mean to imply that it isn’t valuable, and even very important, as an approach some of the time!  More on that another time.)  Sometimes the best prompt is the simplest.  Just a single equation or function.  A description of a relatively simple scenario which can be represented with a linear equation. A sequence.  Nothing more.  And in particular no specific question.  Nothing that says, “Find the vertex of the graph of this quadratic function.”  Nothing that asks, “What is the slope of the line.” No specific question to answer about the scenario.  Perhaps just, “I went for a run yesterday, and it took me 45 minutes to run 5 miles.”  That’s it.

Then you let the students talk.  Once they get the hang of it they can start individually, or in small groups, but I found the first few times I sprung this on them it had to be done as a whole class.  I would start by asking,

“What do you notice?”

Ask the students to make some observations.  State what they see.  If the prompt was a quadratic function I would get responses like, “There’s a little 2 above the x,” or “It’s quadratic,” or “There are 3 terms.”  This is where the students get to start where they are, not where I think they ought to be.  And there are no wrong answers.  If it’s what they see then it’s correct.  The vocabulary they choose tells you a great deal about what’s going on for them when they see a quadratic function.  You can build on their observations, by first providing validation and commendation on what they have offered, and then restating it in a way which helps them move forward, or builds upon what they know.

When the observations start to die down, I ask, “What do you want to know?” or “What questions do you have?” And then, “What can you do with this?” or “What can you figure out or tell me about this?” This question, with a quadratic function as a prompt, will gradually lead to covering an entire board with information related to that one function.  And it can easily take 50 minutes.  Ideally the process will continue on to, “Are there other ways that you could think about it,” or “Are there other approaches that you could use find that information,” and, “How do you know that your work is accurate?”  Last, “What else would you like to know or do at this point?” or “Where could you take this from here?”

In summary, this highly personalized approach depends on a simple prompt and 6 essential questions:

  • What do you notice?
  • What questions do you have?
  • What can you do or figure out?
  • How have you verified that your work or answer is accurate?
  • Is there another way to approach this?
  • What else can you do or say, or what more would you like to know?

References:

Boaler, Jo. “Depth Not Speed.” http://www.youcubed.org/category/teaching-ideas/depth-not-speed/

Kress, Nancy. “Essential Questions for Mathematics: Developing Confident, Knowledgable, Creative Students.” Presentation, NHTM Spring Conference. 2014.

McTighe, Jay and Grant Wiggins. “Essential Questions: Opening Doors to Student Understanding.” Association for Supervision and Curriculum Development. 2013.

Richardson, Will. “Still Here… Sort of” blog post.  Comments include the following quote, “I think our whole concept of teaching is going to have to change in order for us to truly help kids flourish in this world. I think it’s all focused on how people learn, and in the teacher’s case, how he or she can create the conditions required for learning more deeply in the classroom. Anybody still doing lesson plans hasn’t gotten there yet.” This quote influenced my writing.  http://willrichardson.com March 18, 2015.

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