multiplication as resizingTo begin this blog, I want you, as the reader, to be a critical thinker.  You can imagine being in a math class and thinking!  You don’t have to be in math class though.   First, you have to be curious.  Critical thinkers cultivate an attitude of curiosity and surprise.    Second, you have to embrace skepticism.  You don’t just accept or reject.  You explore and consider evidence for yourself.    Third, you have to be able to handle uncertainty.  Don’t expect the answers to be straightforward.  You may have to struggle.

Now you can read on and remember to stay curious, explore and welcome the unknown.

Critical thinking is the ability to think rationally.  It includes the ability to engage in reflective and independent thinking.  Critical thinking is closely tied to logic, as a critical thinker has to be able to understand the logical connections between ideas.  Critical thinking requires following the rule of logic and rationality, but can be an essential part of creativity because we need to evaluate ad improve our creative ideas.  Critical thinking is quite compatible with thinking “out of the box”, challenging consensus and considering less popular approaches.  Critical thinking skills include identifying, constructing and evaluating arguments and solve problems systematically.  A critical thinker can see the relevance and the importance of ideas and reflect on the justification of one’s own beliefs and values.  In Noddings’ Philosophy of Education (Noddings, 2012), we are reminded that critical thinking is not about memorizing facts or rules or formulas and the critical thinker challenges their own assumptions and arguments.

I want my students to be critical thinkers and see that they are able to take information, deduce consequences from what they know, and make use of the information to solve problems.   Critical thinking can play an important role in cooperative tasks and constructive tasks.

What is logic and does it relate to critical thinking?   Briefly speaking, we might define logic as the study of the principles of correct reasoning.   As discussed in Noddings’ chapter on formal and informal logic, how logic is defined is actually quite a controversial matter (Noddings, 2012).  For example, there is the well-known Pythagorean Theorem proof without words.

pythag proof







What you experienced is the thinking that is associated with reasoning and proving from a mathematical viewpoint.  It is easy to see how logic is related to critical thinking since logic is really thinking about your own thoughts.  Memorizing facts and formulas didn’t help you with the proof.  It was the mathematical processes of representing, reasoning, and reflecting which brought you to the conclusion that the theorem is indeed correct.

The roots of critical thinking date back to the teaching practice and vision of Socrates 2500 years ago.  Noddings describes the method of questioning now knows as “Socratic method or Socratic questioning” in Chapter 1 and is seen as a critical thinking teaching strategy (Noddings, 2012).

A short history of critical thinking was given in an article by Ioana Marcut on critical thinking applied to teaching math (Marcut, 2005).  Plato and Aristotle emphasized that things are often very different from what they appear to be and it is the trained mind that can see through the surface and understand and make sense of what is beneath the surface.

During the Renaissance, scholars in Europe began to think critically about religion, art, society, human nature, law, and freedom.  Fifty years later, Descartes wrote the  “Rules for the Direction of the Mind” where he argues for the need for every part of thinking be questioned, doubted and tested.

Robert Boyle and Isaac Newton contributed to the spirit of critical thought in their work and philosophers of the French Enlightenment all began with the premise that the human mind, when disciplined by reason would be better able to figure out the nature of the social and political world.

In Nodding’s Philosophy of Education (Noddings, 2012) Robert Ennis’ conception of a rational thinker is described – exhibiting proficiencies in observing, inferring, generalizing, evaluating and reasoning.

Ennis also believed that students can learn something about thinking which can be applied across different subject areas, for example,  mathematics applied in science classes.  Noddings disagreed with this stating that for one to use math in science, one would need to know science well enough to decide when to use the math (Noddings, 2012).  This is consistent with her thoughts on knowing the procedures and skills before being able to solve problems.  I would disagree with this notion.  There are many examples where the application of math has helped explain a scientific concept or make predictions such as game theory or mathematical modeling.

Noddings talks of the informal logic movement being an effort in curriculum and instruction to emphasize skills over content.  This approach is assuming that there is a cognitive proficiency in these mathematical skills such as multiplying or dividing – which is separated from performing math (Noddings, 2012).    I would also disagree with this idea that skills are separate entities from problem solving.  In my work with math students at various grade levels, math skills are not prerequisite skills to problem solving or other mathematical processes.  An example would be teaching algebraic reasoning – where the understanding of algebra can be integrated in the context of linear or quadratic relationships and not necessarily taught on its own or prior to solving more complex problems.

What does critical thinking look like in a math classroom?   

I think that when students think critically in mathematics, they make reasoned decisions or judgments about what to do and think.  They do not use rules or formulas without assessing its relevance.

Example:  Which holds more?

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Instead of using a formula to calculate the volume of a cylinder – students should realize that a change in either the radius or the height affects both the volume and surface area, but in different ways. For example, doubling the height doubles the volume but affects the surface area differently.


Marcut, I. (2005).  Critical thinking applied to the methodology of teaching mathematics.  Educatia Matematica, 1(1), 57-66.

Noddings, N. (2012).  Philosophy of education (3rd ed.). Boulder, Colo.: Westview Press.