What is the Problem Solving Workshop?

The Problem Solving Workshop is a lesson design that is different than the conventional Math class.  The conventional approach to arithmetic has been to teach a procedure, provide time for individual student practice, and finally present problems that utilize the particular procedure.  An alternate approach is to start with problem solving situations.  The teacher poses problems that are accessible given the students’ current set of skills and concepts.  This gives the children an opportunity to figure out some mathematics on their own and use their own creativity and reasoning skills to decide how to approach the problem.  Students are allowed to work on the problem in any way they choose using whatever tools (manipulatives included) that are meaningful and helpful.  Students are then invited to share their strategies in a whole class meeting often called a “congress.”   There are three parts to the Workshop:  posing the problem, the work period, and the congress or sharing time.  Teachers choose which students they want to share and in what order.

The teacher’s role in the workshop is to help students refine their strategies through questioning and to help students learn to model their strategies using math notation.  In this way, the math notation is introduced in context when there is actually something to represent. 

The finished product for older students is a written write-up of the solution.  Younger students might verbally explain their thinking or show their thinking using manipulatives with the teacher doing the written recording as a model for the students.  I call these write-ups math reports.  A good math report not only answers the question of the problem but also gives evidence to convince the reader that the solution is correct.

The Problem Solving Workshop is a lesson design that puts students in an authentic Math Situation.  Thomas Romberg in his article (2001), Mathematical Literacy: What does it mean for School Mathematics? , writes that the general strategy used by mathematicians can be characterized as having five aspects:

1. Starting with a problem situated in reality;

2. Organizing it according to mathematical concepts;

3. Gradually trimming away the reality through processes such as making assumptions about what are the important features of the problem, generalizing and formalizing; 

4. Solving the mathematical problem; and

5.  Making sense of the mathematical solution in terms of the real situation.

Usually in school, students are only given the opportunity to do step 4, the actual calculations.  Even when teachers have used problems in the past, they were imbedded in a unit so the expectation was that all students would use one computation method or one problem solving strategy.  The teacher was imposing their method on the student.  Sometimes this is very appropriate, but sometimes we need to allow the students to make these decisions for themselves.

The problem with trying to teach in a strict linear method is people do not all start at the same place or even learn in the same order.  In the series of books about how children construct mathematical concepts by Catherine Twomey Fosnot and Maarten Dolk (2001), the current philosophy has been compared to a journey.  As you travel on your journey you can see the horizon in the distance.  As you move towards that horizon you pass certain landmarks.  Everyone who makes this journey will pass those same landmarks (learning the alphabet, learning to chunk syllables, etc.) but they may get to them in different orders or stay at one for a shorter or longer period of time than someone else.  As you make your way past these landmarks towards the horizon you realize that where the horizon used to be is where you are now, but in the distance is a new horizon to be reached.  “Thus, each of us is continually becoming more literate as we acquire new skills [pass landmarks], with new information and communication technologies like the Internet; we can no longer consider a fixed end point to achieving literacy.” (Leu)

In the Problem Solving Workshop, we are trying to give students the opportunity to become more mathematically literate. 

http://www.ncrel.org/sdrs/areas/issues/content/cntareas/math/ma300.htm

They are given a problem, then they are given time to work on the problem with the tools they choose (or don’t choose), and they share their thinking using representation that makes sense to them.  In the end, the class gathers for a meeting, sometimes called a ‘Congress’ or ‘Seminar.’  In this meeting, various strategies and representations are chosen by the teacher to be highlighted so that the students can learn from each other.  The goal is to structure the workshops over the year and over the years so students have the opportunity to construct their understanding of key math concepts not all at the same time, but as they are ready to.

Originally, I thought this lesson design was only used with story problems.  But recently, I read an article that changed my thinking. Here is an excerpt from that article:
“Avoid frantically searching for a clever problem for tomorrow’s lesson.  Instead, begin with the mathematics you want to teach. Many traditional textbook lessons can be changed to problem-based tasks with very little effort.  Think about how you and your textbook would typically explain the main idea in a lesson or unit to your class; then
create a parallel situation in which students are the sense makers instead of you.  For example, instead of showing students how to change mixed numbers to improper fractions, ask them to find a single fraction name for 3 2/3.”
(Taken from Designing and Selecting Problem-Based Tasks by John Van de Walle
 Chapter 5 from: Teaching Mathematics through Problem Solving (Pre-K through Grade 6).

One of the best outcomes of the Problem Solving Workshop is the development of number sense and computational fluency.  Students work with numbers in ways that make sense for them, so they use computational strategies that they understand.  There is an excellent article on this topic by Hyman Bass called, Computational Fluency, and Mathematical Proficiency: One Mathematician’s Perspective. In it he states, “Algorithms have qualities that are important to evaluating the algorithms’ usefulness.  These include the following:

·      Accuracy (or reliability).  The algorithm should always produce a correct answer.

·      Generality.  The algorithm applies to all instances of the problem, or class.

·      Efficiency (or complexity).  This refers to whether the cost (the time, effort, difficulty, or resources) of executing the algorithm is reasonably low compared to the input size of the problem.

·      Ease of accurate use (versus error proneness).  The algorithm can be used reasonably easily and does not lead to a high frequency of error in use.

·      Transparency (versus opacity).  What the steps of the algorithm mean mathematically, and why they advance us toward the problem solution.”

The goals of computational algorithms are that they are efficient, accurate, and transparent.  At the younger ages, more of a focus is on the concept of transparency (the student understands what they are doing and how it leads us to the answer) and as the students mature the focus gradually shifts to being more efficient.  This shift will occur at different times for different students.  In the Problem Solving Workshop students are working with algorithms they created so they are transparent.  This is quite different from the traditional method of teaching computation, which was to teach the rules or steps of the algorithm and then practice it in isolation of a context.  This is the major difference between the traditional linear United States approach and the Problem Based approach often called the Japanese lesson design.  As with all shifts, teachers will need time to adopt this philosophy as well as support through ongoing professional development.

The ultimate outcome should be students that can compute fluently but also with thoughtfulness.  The same student may use a variety of computational strategies based on the situation.  We want all of our students to be able to accurately compute, but we also want them realize that sometimes it is better to not use the standard algorithm because there is an easier way to the answer.  We also want them to realize that sometime an estimate is good enough and is much quicker and easier than calculating the exact answer.  In short we want students who are not only skilled but are also smart.  We want students who are numerically powerful.   (Please see the FAQ “What is Computational Fluency?” for a more in depth look at this topic).