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: Using Technology to Differentiate According to Student Interests

Current and emerging technology gives us many ways to differentiate according to student interests.  Most students these days are interested in technology, and they use technology to navigate their world. They use it to play games, watch TV and movies, communicate with friends, etc.  Just by using technology in school, we're probably tapping into an interest of most students.  At the very least, we are connecting students with their world.

 Ideas for Using Technology to Differentiate According to Student Interests:

• Keep student interests in mind when searching for videos to use with students.  If you know that many of your students like sports, try to find sports videos that go along with math concepts you're teaching.  If many of your students enjoy watching movies, try to find movie clips where math is being used.  

• Create a classroom blog, website, and/or Social Network (ex. Ning, Social Go).  These days Social Media is everywhere.  Today's students are used to being part of a social community.  Creating classroom blogs or websites is another way to help build and extend the community you create with students in the classroom.  Blogs and websites also allow you to involve parents in your classroom activities.  If you're worried about what students will post, most blogging and Social Network platforms give you the ability to monitor content.

• When having students create a product to demonstrate learning, allow them to choose a format for creating the product.  For example, students could create a GlogPrezi (zooming presentation editor), make a video about the concept, write a song about the topic, create a Voice Thread about the topic.  This list could go on for a while.  There are so many new Web Applications that allow students to create amazing products.  The idea here is to allow students to choose which type of the product they'd like to create based on their particular interest.  (interactive poster), create a

• Create a Live Binder based on the math topic you're teaching.  Fill the binder with websites, videos, images, and documents that students can use to learn about and practice the math concept.  You could require that some resources be used by all students, but then allow students to choose other resources which are helpful to them.  For example, some students may want to watch a video of someone demonstrating how to work out a problem, some students may want to play a game to practice their skill at solving problems, or some students may want to use virtual manipulatives to help them understand the math concept.  The added benefit to creating a Live Binder like this, is that students can access it from home! 

These are just a few ideas for Using Technology to Differentiate According to Student Interests.  We'll explore more ways to do this in upcoming Differentiation Tips.

Do you have any suggestions for using technology as a way to differentiate instruction?  If so, please leave a comment and tell us your ideas.

An Interesting Blog of Math Illustrations --- Picasso Math

Picasso Math is an interesting blog containing illustrations of math topics. These can be used in many ways. I think students would enjoy them. This would also be a great way to bring Novelty to the classroom.

Here are a few ideas:

• Use them as writing/journal prompts. Students could describe the math topic being illustrated or describe what the illustration means.

• Show one to students and have them create their own for a different math concept/topic. For example: Show this one about dividing fractions (see below) and have students create their own about multiplying fractions.

• Have students critique the image. Do they thing it best represents the math topic? If so, why. If not, why?

Happy Illustrating! Hope you enjoy this great resource!

via picassomath.blogspot.com

Excellent Resources from RealWorldMath.org

via realworldmath.org

Here's an example of a Line Graph that was created using Google Earth. This image is courtesy of Real World Math. This site is definitely worth checking out!

Collaborize Classroom - Online Education Technology for Teachers and Students

via collaborizeclassroom.com

Collaborize Classroom is an easy to use online discussion forum. I especially like the various ways you can post questions for class discussions.
This is an excellent tool for posting multiple choice questions and having students choose answers that do not make sense. You can also have them explain their choices in the Comments section which appears below the Posted Question.
I'll share more ways to use this tool with math students in future posts.

Connect a Million Minds - Videos

via connectamillionminds.com

This is a great video from Connect a Million Minds that demonstrates the type of things are possible because of Math and Science. If you have students who are not interested in math, start showing them videos like this. After seeing what's possible because of math, students who don't like math might just change their minds!!!