Hmm.. Mathematical Thinking?

Article Discussion: What is Mathematical Thinking?

by Barbara Ball

"A math teacher can decide to spend class time in two different ways: drilling students in routine operations, which would kill their interest and shut down their intellectual development or let them become curious about certain problems they are presented with at the right level, which would help them develop independent thinking skills." This was written by G.Polya in 1944, yet educators still face the same challenge of stepping away from teaching math in a closed way (computation-based) and encouraging students to engage in real mathematical thinking.

'Research on adults born in 1970 shows that their self-esteem and self-confidence at the age of 10 was as important as their academic ability in predicting later achievement." So, instead of killing students' interest, educators need to foster an environment where students learn to enjoy doing math. We want to raise kids who can think independently and critically so that they are able to solve problems in the real-world whether it be a simple task of choosing which detergent to buy to a complicated task of deciding which stock to invest in later in their lives. In order to do that, we need to define what mathematical thinking means and what its implications are in math education.

When engaging in a mathematical task, it's important to focus and pay attention to what we do before presenting a possible solution. The thinking that happens during the process is very valuable in developing critical problem-solving skills in students:

                - identify a task or situation to explore

                - break down a task

                - identify similar tasks which may help

                - identify appropriate knowledge and skills

                - look for a pattern or connection

                - identify assumptions

                - select a strategy that might be appropriate

                - consider alternative approaches

                - formulate a conjecture (a theory / guess)

                - test the conjecture and reformulate if necessary

                - prove the conjecture

                - make connections

Words That Describe Mathematical Thinking	Example Prompts
Exemplifying, Specializing	Give me one or more examples of... Describe, Demonstrate, Tell, Show, Draw, Find
Completing, Deleting, Correcting	What must / can be added, removed, altered? Tell me what is wrong with... What needs to be changed so that...
Comparing, Sorting, Organizing	What is the same and different about..? Sort or organize the following according to...
Changing, Varying, Reversing, Altering	What if...? Do... in two or more ways. What is the quickest, easiest...?
Generalizing, Conjecturing	What happens in general? Is it always, sometimes, never...? Describe all possible... What can change and what has to stay the same so that...is still true?
Explaining, Justifying Verifying, Convincing, Refuting	Explain why... Give a reason... How can be sure that...

The activity involving snap cubes that was done with a group of educators at a conference get at the heart of developing mathematical thinking skills. A teacher's role is extremely crucial: What kind of questions are you going to or not going to ask? How are you going to word those questions? How much guidance will you provide? It's not a matter of how difficult the required computations are to solve a problem, but how much demand there is on a high level of mathematical thinking that goes into formulating a final answer.

Dan Meyer points out how we can get through many math or physics textbooks just by using decoding skills. I also fell into this trap and used to think that I could "think mathematically" really well when all I ever did was pull out the numbers in a word problem and plug them into a formula to get an answer. When I was challenged with real questions, I felt frustrated and often resorted to leaving it up to others to solve the question because I just wanted the answer. This goes back to the idea of people being "impatient with irresolution". As a teacher, one thing to keep in mind is to ask the right probing questions to get students curious and thinking about how to approach problems so that they can formulate their own conclusions through patient problem-solving. I truly believe that if you're equipped with creative critical thinking skills, you can overcome a lot of difficulties that come your way.

Discussion Questions

1. Prior to discussion: What is your definition of "thinking"? What is thinking? What do you think is "mathematical thinking?"

2. What can we do to promote mathematical thinking? More importantly, how can we get students to become interested in math in the first place so that they are willing to think about it?

3. From the article: "The more children are tested and graded, the less motivated they become." We want to encourage kids to develop and practice critical thinking skills, but we still live in a society where people are graded and evaluated upon certain marks (e.g. SATs for students, LSAT and any other standardized exams for various professions). What is your opinion on standardized testing?

My group talked about many different things; what do we think mathematical thinking means, using more of open-ended questions and parallel tasks to promote critical thinking skills, how standardized testing can put some challenges for educators but also be used to check and reflect on teachers' performance as a whole. We came to a common understanding that mathematical thinking skills are not only applied in math, but it's also about developing transferrable skills which can be used in many different areas when problem-solving.