Transforming Procedural Math

This week, I headed out to Seattle with two of my colleagues to think about how we might be able to re-culture math for higher achievement.  The message and the research is loud and clear that the number one thing that increases math achievement is one’s belief that they can do it.  I went into this course hoping to develop a framework for designing challenging tasks in order to provide students with more opportunities to develop this belief.  So what about direct instruction? I went into the course thinking about challenging tasks as something separate from direct instruction of a more procedural nature.  But, what if they aren’t two separate things? We looked at two examples that provide a framework for transforming direct instruction into group learning tasks offering students autonomy and an opportunity to develop the belief that they can do it themselves.

Working backwards

Instead of teaching the procedure itself, allow students to work in groups to figure out the procedure by giving them only the start and end of the problem.  

Given:

(3x+5)(2x+7)—–>6x2 +31x+35

Work as a group to show the steps in between

Or the other way…

6x2 +31x+35—–>(3x+5)(2x+7)

Once students have had a chance to grapple with the problem, they can take ownership by presenting the process.  Just about any procedure could be taught this way instead of in a traditional lecture style.  Many people shared concerns about not every student getting it or how do you know? Well, does every student get it when it is taught directly through traditional lecture style? How do you know who gets it when it is delivered that way? Exit tickets, mini quiz, questioning? Any of these methods could be used to confirm who got it and who didn’t using either method of teaching.  

Bubble worksheets

A bubble worksheet is designed to ask questions about an example problem in order to help students learn the process.  They work together in groups to answer the questions and then generalize the concept based on their answers.  In a traditional textbook, the examples give you the steps.  Bubble worksheets require you to ask questions of each step instead.  Here is one taken from  Gordon-Holliday, B. W. (2008). Algebra 1. New York: Glencoe/McGraw-Hill and transformed into a bubble worksheet problem.  

Creating a bubble worksheet version of an example problem isn’t exactly easy but with a clear learning target, it is probably doable for just about any procedural math task.  Having a clear learning goal is the key to creating the right questions to get at the concept you are trying to teach.  In order to make sure students are getting the concepts, a teacher checkpoint can be used to prevent students from moving on before they are ready.