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.  

Differentiation Tip: Questioning Strategies

Varying your questioning and targeting specific questions to specific students is a great way to differentiate instruction.  During the course of a class, you can ask higher level or more open questions to students are at a higher level of readiness.  You can direct lower level or more closed questions to students who have a lower level of readiness for the skill or concept being taught.

If you're doing a lesson that involves manipulatives or problem solving, you can provide more or less scaffolding depending on the needs of the student.  Many times opening (providing few parameters) or closing tasks (providing more parameters) can be an easy way to differentiate.

Example of Differentiating by Opening or Closing a Task:

Lesson:  Exploring Fractions with Multilink Cubes

Students are shown an example of a fraction tower made with multilink cubes.  They are then given tasks to create their own fraction towers.

This is a fraction tower that has one-half red cubes.

Task A:  Build a fraction tower that is made up of two-thirds red cubes. 

(This task is completely open.  The only parameter is that is must have two-thirds red cubes.  Students have to decide how many total cubes to use.  Some students might use 2 different colors and some students may use more than 2 colors to build their towers.)


Task B:  Build a fraction tower with 9 total cubes and is made up of two-thirds red cubes.

(This task is now somewhat closed because you have stated how many total cubes to use for the tower.  If students were having trouble with the Task A because it was too open, this would be one way to make it easier.  It does still have an open element to it because you haven't stated if the tower has to have exactly 2 colors or more than 2 colors.)


Task C:  Using 3 different colors, build a tower that represents two-thirds.

(This task is mostly open with 2 parameters:  3 colors and represents two-thirds.)


Task D:  Using 3 different colors and 6 total cubes, build a tower that represents two-thirds.

(This task is completely closed.  Since you are using 6 total cubes and representing two-thirds, you would not be able to create a tower with more than 3 colors.  If you had specified a value greater than 6 for the total cubes, there would still be an open element to the task.)

Task E:  Build a tower that is one-half red, one-third green, and one-sixth blue.

(This task has 3 parameters, but is open in respect to the total number of cubes used to build the tower.  If students were struggling with this task, it could be made easier by giving a total number of cubes or by scaling back on the given parameters.  For example, you could just say to build a tower that represents one-sixth.)

Adjusting questions and tasks can be a quick and easy way to differentiate instruction, but it does take some practice.  The great thing about this strategy for differentiation is that you have plenty opportunities to practice and fine tune it!

Creating Patient Problem Solvers

In the TED Talk posted May 2010, Dan Meyer discusses how to create patient problem solvers.

TED Talk --- Dan Meyer:  Math Class Needs a Makeover



In future posts, I will use Dan's ideas and reconstruct some typical Middle School Math Problems.



Creating Patient Problem Solvers by Love of Learning Educational Services, LLC is licensed under a Creative Commons Attribution-ShareAlike 3.0 Unported License.
Based on a work at www.ted.com.

What do Questioning Strategies have to do with Expectations?

I've been researching questioning stategies recently.  It turns out that I'm learning a lot more than I thought I would.  As part of my research, I've come across some fantastic examples of questioning by math teachers.  These examples got me thinking about our expectations of students and how our questioning techniques reflect these expectations.

What do we expect from our students?  Do we expect them to wait for us to work through the problem for them? Or, do we expect students to think through problems on their own?  Do we expect students to communicate mathematically?

The way we question students is directly related to our expectations of them.  The questions we ask, or don't ask, imply our expectations.  The following examples demonstrate questioning strategies of teachers who expect students to think and communicate about thier learning. 

Check out this masterful questioning of an Algebra student by David Cox.  He uses a series of questions to get his student past "I don't know how to do this."  How often have we heard that one?!  He helps the student focus on what they do know in order to answer the given problem.  His questioning techniques speak volumes of his expectation that the student do the thinking.  I would also guess that the student felt much more satisfaction and confidence at the end of conversation than they would have if David had worked the problem for them.

Tom Woodward's videos (see videos below) provide another example of excellent questioning in a math classroom.  This teacher is a master of facilitating mathematical discourse.  She continually requires the students to do the thinking and communicating about the problem they are solving.  What struck me was that whether the students were right or wrong in their thinking, the teacher never stopped asking questions to get them to communicate with each other.  At one point a student is speechless when asked a question by the teacher.  A few questions later, he is jumping right in with his thoughts. 

Here's the problem that is being solved by students in the following videos.  In his post, Tom said the teacher had the questions more clearly delineated.  I would also suggest reformatting the question before using it with students.

From Tom Woodward's Post

Math Questioning from Tom Woodward on Vimeo.


Refining Solutions from Tom Woodward on Vimeo.

The teacher in this video expected her students to discuss the problem among themselves.  Everything she did in her conversations with students indicated that expectation.  I noticed how excited the student's were about solving the problem.  They really seemed to be enjoying the process of problem solving.  Wouldn't we all love see students enjoying problem solving in math class?! 

My take away from these examples is that we must align our expectations and our questioning techniques.  If we expect our students to problem solve and communicate mathematically, we should use questions to help guide the way rather than giving in and doing the thinking for them.




What do Questioning Strategies have to do with Expectations? by Love of Learning Educaitonal Services, LLC is licensed under a Creative Commons Attribution-ShareAlike 3.0 Unported License.