“I thought of a reason why students must learn math.”
One of my dearest friends, Mark, said to me during our brief lunch period between classes.
“It may be because mathematics is the purest form of logic.”
Mark is a friend who would always bring up interesting topics to discuss. This time, the topic was the purpose of mathematics education.
Normally, when others (especially my students) ask me why to learn math, I would explain it is to promote students’ problem-solving abilities and math proficiency. With Mark, however, I knew he demanded to immerse deeper into the conversation.
Here is his opinion: in everyone’s life, logic is essential: making decisions, telling a story, persuading others, filing lawsuits, and numerous other situations. Through mathematics education, students learn to construct and critique various logical arguments in a mathematical – or, as Mark put it, the purest – form. For example, 1+1=2 could allow students to make a logical conclusion that 1+2=1+1+1=3. Such purest form of logic, Mark suggested, may be the reason why students should learn math.
Is Logic the Most Important Aspect of Mathematics?
My belief slightly differs from Mark’s. Without a doubt, logic is a major factor that constitutes math. In a usual primary and secondary classroom setting, logic is probably the best way to maintain good grades, thanks to fixed questions and reasonable answers. In fact, one of the practices that the Common Core State Math Standards aim to emphasize is for students to reason abstractly and quantitatively, and construct and critique arguments,1 so Mark’s opinion might actually be the suitable purpose of current math education. However, pure logic is often insufficient to explore the entirety of mathematics. A famous proof of the Four Color theorem illustrates a controversy between a creative method and traditionally valued logic.
The Four Color Theorem
The Four Color Theorem states that any map (or any plane consists of contiguous regions) requires no more than four colors to color in a way that no two adjacent regions share the same color. The conjecture was proposed in 1852, but it wasn’t until 1976 that the statement was proven to be true. The proof was introduced by Kenneth Appel and Wolfgang Haken using a computer-aided method. However, much controversy was engendered – wide criticism from mathematicians claimed that the proof was “inelegant,” meaning it relies on technology instead of intellectual/mathematical logic and arguments.
However, was Appel and Haken’s method invalid? Absolutely not. They first narrowed down the infinite possibilities into 1,936 maps by expanding the concept of reducibility, and they took an innovative approach to check those maps using computers. There was no error in the proof, and, notably, no one so far has proven the theorem in an elegant way.
Another conflict between conventional mathematical logic and creative approach takes place in the ongoing debate around Nautilus shells and the golden ratio.
Nautilus shells
On many occasions, a cross-section of Nautilus shells is believed to manifest a logarithmic spiral with the growth factor of the golden ratio. But such a belief is refuted widely by mathematicians, as the spirals of Nautilus shells have an average growth factor of around 1.33, while the golden ratio is Φ=1.618… In fact, comparing Nautilus shells with the Fibonacci spiral (the commonly-used term for the golden spiral) would show a significant difference between them.
Therefore, it seems to require only simple logic to conclude that the golden ratio is not found in Nautilus shells.
However, some still endeavor to find a meaningful relationship between Nautilus shells and the golden ratio. One of them is Meisner, who argues that there may exist more than one “golden” spiral. As an example, he presented a spiral that expands with 180-degree rotation (compared to the Fibonacci Spiral’s 90-degree rotation), while maintaining the golden ratio. As one can observe, Meisner’s spiral grows much slower than the conventional golden spiral. What is remarkable is that this variation of the golden ratio is very similar to Nautilus shells.2 Inspired by this work, another researcher, Cathy Williams, experimenting with a Nautilus shell and studying its ratio with different degrees of rotation, found that the model of 175-degree rotation particularly yields a close approximation of the growth factor of the golden spiral with less than 2% error.3 Then, can we really claim that Nautilus shells do not have the golden ratio? Or should we expand our views on how to define what is “golden”?
None of this should belittle the importance of logic
But logic is not all there is to math.
(Refer to Reference #2 for more information)
Examples above challenge the common belief towards mathematics – the belief that classifies mathematics as a rigid subject orbiting around some fixed set of logic. Surely, logic does not always entail fixed mindsets. Nonetheless, I am cautious about agreeing with Mark’s opinion. I worry that the claim may obscure the creative nature of mathematics. Could Appel and Haken have thought about the innovative method of utilizing computers if they had entirely relied on pure logic? Could all the meaningful discussions around the golden ratio have still elicited if everyone had simply agreed on the Nautilus Shell discussion because the formula says so?
Again, I certainly am not arguing that mathematical logic is insignificant (Because it is very, VERY important). However, I do hope the classrooms would place more value on the creative, explorative, and, as I wrote on my Purpose, poetic element of mathematics. Then, perhaps it would become more clear for students why they must learn mathematics.∎
1 National Governors Association Center for Best Practices & Council of Chief State School Officers. (2010). Common Core State Standards for Mathematics. Washington, DC: Authors.
2 Meisner, G. (2016, August 31). The Nautilus shell spiral as a golden spiral. Retrieved February 05, 2021, from LINK
3 Selbach-Allen, M., Williams, C., & Boaler, J. (2020). What would the nautilus say? Unleashing creativity in mathematics! Journal of Humanistic Mathematics, 10(2), 391-414. doi:10.5642/jhummath.202002.18