Over the past century, mathematics education had been firmly tied to two horses: conservative and progressive. Much like the medieval torture, the party that is hurt the most when the horses are whipped is the party that is tied to the ends: education.
The Progressivist Reform of the early 20th century challenged the established ideals of knowing math versus doing math - claiming that students must practice doing math via inquiry-based questions if they want to add to their knowledge pool. As a free-thinker who likes to ponder math in my own way, I associate with the progressive mantra of understanding; I try to emphasize that in my own teaching (vs plug-and-play).
New Math, headed by the School Mathematics Study Group (SMSG), a subsidiary of the American Mathematical Society (AMS) was successful in reverting math teaching back to a highly conservative model. The impetus of which was USSR's launching of Sputnik in 1957. Influenced by the Bourbaki group's disregard for geometry (or any diagrams) in favour of abstraction - New Math supported minimal progressive features as it assumed that every student was a potential rocket scientist (certainly not a viable/realistic viewpoint - learners should be treated as individuals).
The Math Wars over the NCTM (National Council of Teachers of Mathematics) standards followed for the next generation. I (unknowingly) was a part of this, as my elementary/high school was throughout the 1990s and early 2000s. Looking back, I was enrolled in classes that followed the back to basics approach, incorporated standardized testing, encouraged flexible problem solving skills, represented mathematical relationships in multiple forms, and/or embraced new technology (personal computers, internet resources). As we move further into the 21st century, I'm worried that with the advent of modern technology students will end up becoming so dependent on it that their (mathematical) imagination will be stunted. Many beautiful results in mathematics were discovered by thought alone (and later formalized with ink and paper). Imagine if Cardano had simply typed a general cubic polynomial equation ax^3 + bx^2 + cx + d = 0 into a supercomputer and saw the end result instead of depressing the cubic by removing the x^2 term, performing a clever substitution, etc. I'm not saying that we should compute the sample variance for 1000 data points by hand, but we must exercise caution when technology is introduced - it should be used as an aid to walking (e.g. cane), not as a replacement for your legs.
After a 1996 study revealed that the USA ranked 28th out of 48 countries in math, it was determined that deeper conceptual analysis was required to rise in the rankings. It's appalling how there are so many (unqualified) math teachers operating today who only emphasize rote memorization of material and only focus on questions and question styles that will be on the next test instead of broadening and strengthening the student's understanding. As aspiring mathematics teachers, we must be ready to embrace change as waves of pedagogical reform cascade through our classrooms.
It may be possible to adopt a custom hybrid approach with select features from both philosophies should be implemented in the classroom as the teacher deems appropriate, as there are multiple ways to learn new mathematical concepts. A potential problem with the hybrid approach is you risk bifurcating the class into those who desire mathematical rigour and do not receive it fully and those who would benefit from a more exploratory approach and who have difficulty handling abstract theories. Is it even possible to teach in this manner, given time constraints and diversification of students in a given class? A second, more radical approach would be offer two streams of math: one for those more comfortable with conservative ideals, and another for progressive-minded students. Some issues with this strategy are: once you're on a track, how easy it is to make a switch if you think they other stream is more suitable, how easy is it to switch back, when is it too late to switch (back), or will learners' minds be less complete if they're taught one side of the math coin only?
Will one side win out eventually, or will it be like our political parties - at the mercy of the swaying breeze?
Thanks for your thoughtful discussion here, Jordan!
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