By Dr. Amina Eladdadi, Mathematics Department
“What was I thinking,” one of my Calculus students exclaimed when I pointed out the mistake he made while solving an applied math problem on free-fall motion that required both synthesis and analysis. “Well, I am glad you’re thinking at all, that’s a good place to start,” I replied with a sense of humor. Students do not develop problem-solving abilities, nor they become critical thinkers overnight. Critical thinking and problem solving are acquired skills that require instruction and practice, as well as time, involvement and devotion from both the students and instructors alike. Although the National Council of Teachers of Mathematics (NCTM) recommends that elementary and secondary mathematics instructions address problem solving, quantitative reasoning and critical thinking, many of us at the College level still struggle to engage students in critical thinking and problem solving activities. In this blog, I briefly reflect on how problem solving and critical thinking in mathematics – or any discipline for that matter – are intertwined.
Many students come to college ill-equipped to problem solving in mathematics as well as in other disciplines. Problem solving requires critical thinking and both are fundamental to learning mathematics. In fact, students must learn how to think critically to be able to acquire mathematical knowledge through problem solving. This is why NCTM advocates that mathematics instruction should include problem solving, quantitative reasoning, and critical thinking. The Principles for Mathematics Curriculum and Assessment (2009) states:
“… Students should have frequent opportunities to formulate, grapple with, and solve complex problems that require a significant amount of effort. They should then be encouraged to reflect on their thinking. Problem solving is an integral part of all mathematics learning.” (http://www.nctm.org/standards/content.aspx?id=23273)
While critical thinking has several definitions depending on the discipline, there is a strong consensus that critical thinking is the ability to use knowledge to conceptualize, apply, analyze, and synthesize information to successfully solve problems (http://www.criticalthinking.org/). Hence, for the students to be critical thinkers, they need to be able to both analyze and synthesize information. Mathematics can be either analysis or synthesis, and sometimes both depending on the math topic. Nonetheless, both require critical thinking in problem solving.
Many problem-solving models have been developed. Some of these models are specific to a given discipline while others are all-purpose models. Two models that are worth noting are the Polya’s and Wallas’ problem-solving models. In his best-selling classic How to Solve It (Princeton University Press, 1945), George Polya (1887 – 1985), a Hungarian mathematics educator, identifies the four main steps that form the basis of any problem solving. These steps are: understanding the problem (identifying what is being asked), devising a plan (formulating a set of strategies), carrying out the plan (executing the selected strategies), and looking back (checking and interpreting the results). Polya also argued that a mathematics problem should not end just because the answer has been found, instead, there should be a constant probing related to the problem. This practice not only helps the students to develop critical thinking skills, but also allows them to increase their confidence, inspire and engage them in the subject.
When I first started reading Diane Halpern’s (2014) text Thought and Knowledge: An Introduction to Critical Thinking, 5th ed. (Psychology Press), suggested to us by our colleague and provision fellow Prof. James Allen, I skipped straight to Chapter 9 on Development of Problem-Solving Skills. Halpern describes how psychologists think of the word problem as “a gap or a barrier between where you are and where you want to be.” She also gives a nice visual illustration of a “problem” in Fig.9.1 p. 453: one long rectangle/box divided by a vertical line into two blocks “X” and “Y” – you are at “X” (box left of vertical line) and want/need to get to “Y” (right box), how do you that? Well, you may be tempted to say, “Jump over that line!” – I can assure you that the “line” is so high for some students, that the “line” is the “problem” – Got the picture? …… Good!!
Halpern examines the stages in the model of problem solving proposed by the English psychologist Graham Wallas (1858 – 1932), which is commonly known as the model of the process of creativity. These four stages are: preparation (definition of issue, observation, and study), incubation (step back from the problem and let the mind contemplate and work it through), illumination (the moment when a new idea finally emerges), and verification (checking it out). Halpern argues that the incubation is the most difficult stage and the least understood and therefore devotes a whole section of this chapter to it.
Notwithstanding the many stages in the model, it all begins by looking for a clear statement of the problem, and defining it as accurately as possible. Getting the student to interpret the problem is the first important step in successful problem solving. Once the problem is well stated, students will be engaged to think critically about the solution – hopefully!
Both critical thinking and problem solving are intertwined and similar in a way that they both involve steps and processes to tackle thought-provoking challenges such as applying solid reasoning, understanding the interconnections among systems, framing, analyzing and synthesizing information. So, when students participate in problem solving in mathematics or for that matter any other discipline, they are engaged in critical thinking in their analysis of the problems and in the synthesis and application of previously learned concepts. Moreover, students’ critical thinking abilities are improved when the solutions require knowledge and problem solving skills from more than one discipline such as physics, business, psychology, sociology, etc., and when the problems are ill- defined, as is the case for most real-world problems.
Problem solving and critical thinking are not only vital skills in all academic disciplines, but also life skills that students will continue to use throughout their lives. It is important that our students are challenged in ways that engage them in critical thinking and be metacognitive, that is, that they think about their thinking.
In summary, I liked reading a couple of chapters from Diane Halpern’s text, which I highly recommend to anyone interested in integrating critical thinking into the classroom.
Likewise, I enjoyed re-visiting George Polya’s classic How To Solve It, which I have previously read (many times in French) during my undergraduate studies. Finally, I would like to close this blog with one of Albert Einstein’s remarkable quotes: “The value of a college education is not the learning of many facts but the training of the mind to think“
- Polya’s book: https://notendur.hi.is/hei2/teaching/Polya_HowToSolveIt.pdf
- Chapter 1 & 8 – from Diane Halpern (2014) Text:
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