Wednesday, November 9, 2011

Questions That Cultivate Mathematical Thinking


I just had the privilege of presenting at two of the 2011 NCTM Regional Conferences.  While attending both conferences, I noticed that a common theme in many presentations related to creating/developing mathematical thinkers.  This got me thinking about how the questions we ask, or more importantly, don't ask students on a daily basis can cultivate, hinder, or even prevent mathematical thinking. So, I've started a list of questions/question prompts that can/should be used regularly to help cultivate mathematical thinking and reasoning.
 
 Questions That Cultivate Mathematical Thinking  (this list in no particular order):
  • Why?...Why did you do that?...Why do you think...?...Explain your thinking. --- In my opinion, asking "why" is one of the most overlooked, undervalued, and important questions a math teacher can ask.  This question helps teachers by giving them a better understanding of where the student is coming from and how much they really do understand about a given topic, concept, or problem solving situation.  Asking "why?" requires students to think about and verbalize their thought processes.  When you ask students "why?", you gather information to help you ask other questions that address misconceptions and further mathematical thinking. 
  • What if...? ---- "What if" questions allow teachers to change constraints on problems/situations.  This furthers mathematical thinking by having students see patterns and relationships beyond the initial problem.  "What if" questions also help students to see that their initial thoughts about the answer to the question or their problem solving strategy may no longer apply to the new situation.  This understanding helps students learn to focus on the context of a problem rather than what they perceive as "set guidelines" for solving a particular type of problem.
Example of "What if" questioning:  The two tables below represent the sales from Fu Do Chinese restaurant in Anchorage, Alaska.  The premise of the problem is that Fu Do, a family owned and operated restaurant, had to close four days during the week of 10/9 - 10/15 due to a death in the family.
The questions: 
First Question:  Which measure of center (mean, median, or mode) best represents the sales of Fu Do on Oct. 2 - 8 (a typical week)? 
"What if"Question:  What if Fu Do had to close 4 days for a funeral?  Which measure of center best represents the sales from the week of Oct. 9 - 15?
In Table 2, the constraints of the problem were changed resulting in the possibility of a different answer to the question.  This type of problem and questioning provides a basis for rich, interesting discussions about which measure of center is the best representation and what factors impact this decision (range, the context of the situation, etc.).  Students begin to see that they need to consider many factors when answering a question like this.
This problem/discussion/question can be taken even further by asking another "What if" question that again changes the constraints of the problem.  
Another "What if" Question:  What if Fu Do only had to close 3 days during the week of Oct. 9 - 15?  Which measure of center would best represent the sales from Oct. 9 - 15?
Now the mathematical thinking and discussion can really get interesting!  By changing this one constraint on the problem (only 3 days closed), you've opened up new questions and factors that need to be considered when answering the question.  You can cultivate mathematical thinking even more by opening up the discussion to include comparisons between the 3 different scenarios. This may lead to asking other questions like "Under what conditions would mean be the best representation?...median?...mode?"
  • What patterns or relationships do you notice?...How can you use this pattern to solve the problem?...How do these patterns/relationships help us to think about the problem? --- Mathematics is all about recognizing and using patterns to answer questions and/or learn more about a situation.  As students get better at recognizing and understanding patterns they begin to develop number sense, see connections between mathematical concepts, and become better problem solvers.

    • Is this the most efficient way to solve this problem?...What's the most efficient way to solve this problem? --- It's important to have students explore various ways for solving a problem.  But then it becomes important to have students evaluate strategies for efficiency.  Some methods will always be inefficient and should be discarded as such. With other methods, efficiency may depend on the individual.  The method that's most efficient for you may not be the method I find most efficient for solving the same problem.  The key to this questioning is to get students to recognize that there are various methods for solving problems, that it's important to consider efficiency when choosing a strategy, that some methods are valid but inefficient, and that efficiency can depend on individual understandings and preferences.  
    In an effort to keep this post from getting too long, I'll stop elaborating on each question.  Below are a few more questions/question prompts that help cultivate mathematical thinkers.

    Questions that Cultivate Mathematical Thinking Continued:
    • What other ways can this problem be solved?
    • How could you represent this visually?...differently?  In what other ways can this problem be represented? (tables, graphs, equations, pictures, etc.)
    • Compare and contrast these two problems...How are these two problems different?...How do these differences affect how you would solve each problem?
    • How could you define or explain this without using numbers?
    • Based on ________, how would you approach this problem differently now? 
    • How has your thinking about this problem changed? 
     Tip:  If you're just beginning to incorporate these types of questions into your daily practice, write some of the question prompts on posters and place around the room.  They'll serve as a reminder if you draw a blank.  And, students will think you posted the questions to prompt their thinking.

    By no means is this an exhaustive list!  It's just a work in progress.  Let's continue to build this list together.  Leave a comment with your favorite questions that help cultivate mathematical thinking.  

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