US Politics Mega-thread - Page 3362

1. One example of a huge shift in how we approached elementary and secondary mathematics involved a push for applications and real-world contexts. There used to be a time when most math books were extremely proof-heavy and equation-heavy, with page after page of definitions and theorems and abstract exercises. With the exception of some timeless applications (perimeter, area, money, etc.), it was assumed that science courses were already going to take care of many relevant applications of mathematics, so most standard math courses could happily focus on the “pure math” side of things. Critics of incorporating real-world modeling into math curricula were concerned that the purity of math would be compromised, pushed out, and ultimately forgotten, replacing one of math’s unique characteristics with a redundant emphasis on scientific empiricism. But then, over many decades, applications of math became better integrated into math textbooks and curricula and standards and pedagogy, which not only *didn’t* kill off pure mathematics, but actually complemented and reinforced the learning of pure mathematics. Learning about mathematical applications alongside an equation or theorem or proof showed students that the endeavor for understanding pure mathematics wasn’t only an abstract, intellectual thought experiment, but that it could also be useful and practical. Some people appreciate mathematics for its purity and elegance, but there is also utility, and that makes the subject even more attractive for more people.

2. A second example of how mathematics education has changed over the years is that there is a greater emphasis on problem solving than ever before. Gaining some traction in the 1940s with George Polya, the ideas of learning *about* problem solving and learning *through* problem solving have led to a variety of new teaching strategies, especially student-centered ones. For a very long time, the standard was that teachers had all the knowledge and transferred it to passive students through lecturing. This was seen as a more teacher-centered approach to education, which had generally evolved into strategies like “direct instruction” (which are still far from useless). With more modern research into teaching philosophies that incorporated a more active-learning approach for students, such as “guided learning” and “flipped classrooms” and more social “collaborative learning environments”, many critics were concerned that these bottom-up approaches (starting with the students) would be less effective than the more traditional top-down approach of a teacher having all the control. This general schism is sometimes referred to as those who favor the older “sage on the stage” approach vs. those who favor the newer “guide on the side” approach, but it basically turned out to be a false dichotomy, as there are a variety of variables to consider (the teacher, the students, the lesson, the resources, etc.) that make it impossible to claim that one strategy will always have more success than any other strategy. It’s extremely conditional, and as a teacher, I find it much more beneficial to study all pedagogical strategies and pull from different ones at different times, than restrict myself to only one kind of method that may not resonate with all my classes. As with most things in education, the answer is “it depends”, and there are a million different ways to effectively teach any given lesson.

3. A third example of how math education has evolved has been a push for strategic competence. In other words, can students approach a problem in multiple ways, so that they have more flexibility when finally experiencing an adjacent problem that is slightly different. Solving systems of equations through graphing is fine if all you have is integer coefficients, but students are at a disadvantage if they can’t also solve through substitution or solve through the elimination (linear combination) method or represent systems as matrices, because real-world equations are messy and you can’t ignore decimals and fractions forever. The quadratic formula lets you solve any quadratic you want, but it’s slow and there are additional benefits for learning factoring strategies (more efficient, more generalizable to higher-degree polynomials, etc.). Same goes with basic multiplication techniques: the “traditional” approach might be ideal for 70% of the students, but if I can show one or two other methods (lattice method, partial products, etc.) that the other 30% better understand or prefer, then I’m making sure that 100% of my students are comfortable with the skill. On a more holistic level, the ability to model and represent math problems in a variety of ways is an incredibly useful skill to have. Can you take a situation and create a math equation for it? Can you use abstract, formal notation? Can you create a graph that represents it? What about a table or chart? Can you use manipulatives to model the scenario? What other representations can you use? The more strategies you have in your arsenal, the easier it’ll be to model new problems. This may all seem like common sense now, but there was a time when the mainstream approach was “there is a single best way to conceptually understand X, and it’s by doing Y”. Critics of this push for strategic competence point out that due to time restrictions, it’s often impractical to repeat the same problems three different ways, and that we should favor tried-and-true strategies and not waste time with redundancy. (After all, there are hundreds of proofs for the Pythagorean Theorem; how many do you really need to know?) Common rebuttals to those critiques are that students will understand the material more deeply by seeing multiple versions of solutions, and spending more time on more strategies early on can often save time at a later date, as students will be able to move through problems more efficiently with their preferred approaches.