By Donna Wall
Assistant Professor, Mathematics at American Public University
In College Algebra, we include mixture problems, distance problems, interest rate problems, work problems, wind and air speed problems, etc. We provide math problems with a variety of options so that the students learn how to pick the best problem-solving option. One can reasonably argue that we are teaching thinking skills since a student would need to know when to use which equation and how to apply the variables to the equation.
Maybe we can learn something from the early 1900s about critical thinking. My grandfather understood this concept. He was born in 1896 and lived during a time when the teacher presented a problem to the class and the class would try to figure out how to solve it. As a result, he became an expert in critical thinking. He surprised us at age 85 by acing the GED so he could enroll in a college course after he was not able to track down his high school diploma.
Can we learn from the past? What if students were presented with a problem or problems in a forum, were given time to brainstorm, and later were given tools to solve the problem? Would it help with critical thinking skills and challenge them?
One scenario that comes to mind is the problem that was originally coined by the popular TV game show ”Let’s Make a Deal” and is a classic example of critical thinking. In the show, three doors are presented to contestants with a prize behind only one of the doors. If the contestant choses “door #1” but then “door #2” is opened showing no prize, is it better to stay with the original chosen door, or go with what is behind “door #3?”
If switching to “door #3” is the correct way to go, one must assume that the contestant must always have a second chance and a door is always revealed. What makes that statement true? Can you think of how you could apply this concept to a similar problem?
Do you think problems like this would make successful discussion questions? Why or why not?
About the Author
Donna Wall is an assistant professor with American Public University where she currently teaches College Algebra, Trigonometry, Contemporary Math, Statistics, Discrete Math, and Math Modeling. She received her BS degree in mathematics and MSIS degree with a concentration in mathematics from the University of Texas at Tyler. She has done some postgraduate work at Trident University. Originally she worked as a mathematician and programmer for Teledyne Geotech but now enjoys teaching and writing courses. Over the last 10 years she has written more than 15 courses in research, statistics, and mathematics.
Tags: challenging students, critical thinking, math, promoting critical thinking
I believe critical and creative thinking are both essential to doing math. Yet I believe both are relatively unexplored areas with our young student mathematicians.
Here is the lone reference to critical and creative thinking in the Ontario curriculum:
The star below the achievement chart is a footnote explaining that critical and creative thinking are present in some, but not all, math processes. It does not elaborate which! Obviously, this is not helpful – if the math processes are the actions of doing math, it makes sense then that these actions will, at times, encompass critical and creative thinking. Further compounding the problem, critical and creative thinking are, at best, ill-defined. The role of teachers in teaching critical thinking is debated- see Daniel Willingham’s piece here.
What is critical and creative thinking in the math classroom? What does it look like in the math classroom? I am starting from the presumption that all kids are capable of critical and creative thinking. My second presumption is that mathematical knowledge and skill gained as children grow older allows them to think creatively and critically. Third, I don’t buy the typical (and somewhat ill-defined) notion that creativity and critical thinking are only typical of “higher order thinkers.” It depressed me to no end when I did my literature review on these two topics and found that much of the work on these two types of thinking were done with gifted learners.
The other common line of thinking is that critical and creative thinking are somehow opposite, or at odds or competing with each other. I don’t buy this false binary. Typically this binary is set up as “making” versus “assessing” or “judging.” I believe that both are intrinsically tied together.
Here is an example I like to come back to. A student came up with his own method for predicting the career points scored of several hockey players. In his judgment, here they are:
Here’s a nice quotation on critical and creative thinking:
“These two ways of thinking are complementary and equally important. They need to work together in harmony to address perceived dilemmas, paradoxes, opportunities, challenges, or concerns (Treffinger, Isaksen, & Stead-Dorval, 2006).
Further, Poincare said something to the effect that mathematical creativity is simply discernment, or choice. Our young mathematicians will make judgements as they are solving problems, deciding which path to follow, and when. They will pick the best representations for their mathematical work, and their own idiosyncratic mathematical voice will come out. (Given a classroom culture of math talk, our students will find their voices. “Voice” is not just for the English classroom). Doesn’t that sound like critical and creative thinking, combined in one neat mathematical package?
I have a dislike for overly complicated frameworks and definitions that clutter and obscure important concepts. Einstein may have said something about how if you understand something, you can explain it to a child. If we can explain the quantum world without jargon, we can explain educational concepts without jargon, so here goes. Here are my personal working definitions of each:
Creative thinking: making something new.
Critical thinking: making sound judgements.
Yes, these are deliberately economical. Yes, you could add to these definitions if you wanted to. But if you are a student, and you are doing a mathematical problem or task, you are making something new every single time. There will be patterns and trends in the strategies and tools that individual students use that further differentiate more “unique” or “divergent” work which will perhaps “more” creative. I also maintain that, provided we don’t oversimplify our mathematical tasks to take students’ judgements away, they will be constantly hypothesizing, choosing, testing, and revising their work.
How does this happen in the math classroom? How can we harness these two powerful types of thinking?
In the first case, if we don’t see math as a generative process, a creative process, then we will not find creative thinking. Look closely at the picture I started this post with: both problem-solving and inquiry are mentioned. To the former: problem-solving classrooms will always have an element of creativity, unless we force our own methods, techniques and processes on our students. It will always be our job to consolidate purposefully, and to offer suggestions as to more efficient or effective solutions. The range and variety of the student work, with all its understandings and misunderstandings will lead us to that point. A balanced math program with strong foundations and a spirit of questioning will always lead to interesting lines of inquiry-questions, leading to more questions.
The beautiful diversity of student work:
Here is a video where we analyze the student work in our LearnTeachLead project,“Loving the Math, Living the Math.”
One of the best parts of really getting to know your students is starting to see inside their idiosyncratic mathematical thinking. For a long time, I felt like creativity was that certain “je ne sais quoi” of the math classroom, a “know it when I see it” type of thing. When I thought this, I probably didn’t have a broad enough definition of creative thinking. I was waiting to be bowled over by stunningly divergent solution paths. That does happen, but not always.
Since, I have been watching for more subtle evidence of creativity. Students using new thinking tools, or subtly tweaking a solution path or process they may have got from talking with their classmates. Creativity is there to be found in the math classroom.
Here is an example of a student finding a new use for Minecraft as a thinking tool to represent data:
Inquiry is also hidden in that little line in the picture from the curriculum above. Inquiry to me means: asking good questions. Are our students question askers? There are some astounding numbers floating around about the ratio of students asking questions, to teachers asking questions, in a typical math classroom. Question askers are typically critical thinkers. Once your classroom is an open space for wonder, your students don’t stop wondering! Questions lead to answers, leading to more questions (I once called this the “inquiry tumbleweed”).
The key thing is that students are becoming more confident in their judgements as young mathematicians. I want them to be able to use their mathematical thinking tools to decide “what’s best,” or “what’s fair.” I want them to justify their thinking. I want them always probing the mathematical world around them with their confident judgments.
This is one of my favourite things to tweet now and again:
This work came out of our LearnTeachLead project involving proportional reasoning: http://thelearningexchange.ca/projects/loving-the-math-living-the-math-part-1/. I found some very precision judgements happening, like students telling me a cup of pop was worth exactly $1.26. Not $1.25, not $1.27- $1.26. The power of their thinking led them to this conclusion.
There a nice quote in this book excerpt about how the “best way to think critically is to think critically.” We are risking circular logic there, but think about it: the best way to learn to think, is to think. That is why our classrooms should be open thinking spaces. If they are, our students will be constantly making judgments, testing them, revising them, and drawing meaningful conclusions about the important mathematical work of the classroom.
See: Treffinger, D. J., Isaksen, S. G., & Stead-Dorval, K. B. (2006). Creative problem solving: An introduction (4th ed.). Waco, TX: Prufrock Press.
“Loving the Math, Living the Math” on LearnTeachLead is here.
Originally posted on Matthew’s Olridge’s blog, here.