Tuesday, November 30, 2010

Less is Sometimes More

I had a meeting today...actually I had two really big meetings today; one with a newly appointed principal, and another with a group of teachers.  Both groups having the same focus...how do we get our students to do better on the state exams?  I had some thoughts, but what they all really wanted was a "quick fix" to a very deep and systemic problem.  It is kind of like using Band-Aids for a festering, and pus-filled wound; not effective and in the long term...deadly.

You know one of the biggest problems our students have...besides our antiquated educational structure...it is expressing, showing, and explaining their thinking.  As educators, we generally rebel against expressing our thinking.  Expressing our thinking requires work, sometimes mental, emotional, and/or physical.  Showing our thinking takes effort.  Explaining our thinking may leave us open to criticism.  Expressing our thinking is sometimes a slow and incomplete process that requires us to revisit ideas.  When do we do that (expressing, showing, and explaining their thinking) for ourselves? How can we do that (expressing, showing, and explaining their thinking) for our students?

Consistently using conceptually rich, academically rigorous problems can be a pathway to expressing thinking, but it begins with teachers being willing to embrace and express his/her own thinking publicly and in writing.

Step 1: Choose or create a problem. 
Step 2: Do the problem with your colleagues and use multiple representations to show your thinking. 
Step 3: Identify the "big ideas," math standards, and concepts/skills addressed. 
Step 4: Outline the various strategies that can be used to show and resolve the problem.
Step 5: Identify the possible student misconceptions. 
Step 6: Express what you would like to see in your student's solutions.
Step 7: What questions might you ask students to help them deepen their thinking and understanding?

By following these aforementioned steps, teachers will have a clear understanding of what students are thinking as expressed in their work.  Teachers will also have a better road map of student understanding and of how to move them to the next level.

Now what is meant by conceptually rich, academically rigorous problems?  Compare the two tasks below and think about which question allows you find a solution by taking multiple paths?  Which question pushes your thinking via the mathematical connections that you make?  What are the mathematical concepts involved? How might the concepts involved, change by grade?  Can you show multiple representations for either task?

Task #1: Write a number sentence using at least four different numbers for which 0.86 is the answer. (Source: OPEN-ENDED ASSESSMENT IN MATH by Heinemann)

Task #2: Solve the problem: 0.23 + 4.5 - 1.4 + .25

  • Select Task 1 or 2
  • Complete Steps 1 to 7
  • Group discussion: Ms. Jones, an educator, states, "While task #1 is richer, my students would not see that kind of question on the test. Besides, my kids would just get confused with all the different solution options. On the test there is only one correct answer."
  • How would you respond to Ms. Jones?