Day 27: Thoughts on Math Education
Recently, my thoughts have been drifting to the overall state of math education in this country. While this seems very pretentious, I have been labelled as "the math guy" by my peers for quite some time, and that focus on math has lead me to some very interesting places and to my current strengths, so I am thankful for it. With this in mind, let's go onto the main topic of today's post: math education.
In many ways, the USA is a paradox of mathematics education. On the one hand, we do very well in the International Mathematical Olympiad, and do tend to place well with our contributors to mathematics. On the other hand, there are many, many, fields where American mathematics students tend to fail. It only takes one look at "The Art and Craft of Problem Solving" to see a chapter entitled "Geometry for Americans" to see this in action, as we tend to lag very behind other countries in teaching that part of mathematics. Additionally, due to many policies in place, and general school funding trends, mathematics education tends to get less and less rigorous, as high school students have to optimize their GPAs and their IB/AP/A/etc. credits to ensure even placement in the school that they want ( Rest in Piece IB Further Mathematics ). This leads to situations where students, once exposed to even somewhat challenging problems in mathematics, are unable to even begin to tackle it, unsure of how to proceed when the problems are in any way beyond the simple computations you might see in a freshman textbook. Additionally, proofs, the language for a lot of modern mathematics, tend to be placed on the back-burner till a few years into a potential math major's career ( assuming they had no credits coming in and could only take one class a semester of course, but that is the normal route and most colleges would prevent people from taking multiple math classes in a sequence in one semester ). The absence of proofs is crazy when I look back at mathematics, as so much of the field revolves around understanding why something is true rather than computation, and experience with proofs help students understand what pure mathematicians do in their spare time rather than thinking they just come up with random theorems and play mind games. Additionally, if we look at how we introduce proofs to students here, we tend to make them seem very mechanical and by-the-numbers, forcing students to apply theorems in very traditional manners to get out very traditional results. Now while I do see the value of some of these "two-column" proofs when one is starting, as it replicates the structure of formal logic proofs using a formal logic system and forces students to justify literally everything, I think it's more important for students to see some of the beauty in the proofs that math has as, at the least, they won't rail on mathematics as being cold and calculated when it can be anything but if someone sees the aesthetics in proofs like that of Cantor's Diagonal Argument.
Besides these notable exclusions, however, I think that the most notable exclusion from mathematics education in the early days is discrete mathematics. From graphs, to combinatorics, to probability and statistics, while we do cover some of it in elementary and middle school ( and highschool if you're in the IB program or taking AP statistics ), we do not cover any of the more difficult and unintuitive parts of the subject, focusing on the simple parts where, after enough training, even a dog could do the same. Thus, while we cover stuff like calculating the mean, median, mode, and sometimes the standard deviation of a set, along with some limited hypothesis testing ( though that tends to be more in collegiate-level classes like those in the AP and IB systems ), we do NOT cover stuff such as discrete random variables, concentration inequalities, the central limit theorem ( except in quick breath to just state what it means for experiments ), and the other theoretical parts of combinatorics, like generating functions and the Stirling numbers. These parts of mathematics, in my personal opinion, need to be included in more and more classes at the HS level for a few reasons. Firstly, if we are speaking solely about job growth, Computer Science exists. However, unlike most of the continuous math physics might need, CS requires a lot of discrete math for students to make the most of algorithms and data structures, from learning how to rephrase common problems into problems in graphs and problems in counting to learning how to take hard NP-complete problems and finding min-max algorithms to get decent results. As students, at college, then have to try to learn all of this content in a very quick manner, they tend to lose some of the understanding that proper exposure will give them, and thus lose out on the wonderful intuitions and wonderful parts that will help them in their carreers if they happen to solve computationally interesting problems. Secondly, consider the sciences and social sciences. In the wake of modern scenarios like p-hacking, the current technique-based approach to experimental analysis needs to be destroyed through careful understanding of mathematical laws, which is best done by teaching students the math behind the techniques rather than the techinques themselves. As a bonus, due to the more abstract nature of discrete mathematics, students will be pushed to have to make proofs to convince themselves of harder questions, thus also pushing us to get better as a group. Lastly, discrete mathematics is a good field for those people who wonder what the point of mathematics is. As higher levels combine analytical approaches from continuous mathematics, it becomes a matter of just ensuring people understand the problems with statistical analyses and the benefits of mathematical modelling to inspire people to see that math is not just some weird abstract field they'll never use, but a tool that will keep them sane and keep them making accurate assessments of reality till time itself terminates.
In many ways, the USA is a paradox of mathematics education. On the one hand, we do very well in the International Mathematical Olympiad, and do tend to place well with our contributors to mathematics. On the other hand, there are many, many, fields where American mathematics students tend to fail. It only takes one look at "The Art and Craft of Problem Solving" to see a chapter entitled "Geometry for Americans" to see this in action, as we tend to lag very behind other countries in teaching that part of mathematics. Additionally, due to many policies in place, and general school funding trends, mathematics education tends to get less and less rigorous, as high school students have to optimize their GPAs and their IB/AP/A/etc. credits to ensure even placement in the school that they want ( Rest in Piece IB Further Mathematics ). This leads to situations where students, once exposed to even somewhat challenging problems in mathematics, are unable to even begin to tackle it, unsure of how to proceed when the problems are in any way beyond the simple computations you might see in a freshman textbook. Additionally, proofs, the language for a lot of modern mathematics, tend to be placed on the back-burner till a few years into a potential math major's career ( assuming they had no credits coming in and could only take one class a semester of course, but that is the normal route and most colleges would prevent people from taking multiple math classes in a sequence in one semester ). The absence of proofs is crazy when I look back at mathematics, as so much of the field revolves around understanding why something is true rather than computation, and experience with proofs help students understand what pure mathematicians do in their spare time rather than thinking they just come up with random theorems and play mind games. Additionally, if we look at how we introduce proofs to students here, we tend to make them seem very mechanical and by-the-numbers, forcing students to apply theorems in very traditional manners to get out very traditional results. Now while I do see the value of some of these "two-column" proofs when one is starting, as it replicates the structure of formal logic proofs using a formal logic system and forces students to justify literally everything, I think it's more important for students to see some of the beauty in the proofs that math has as, at the least, they won't rail on mathematics as being cold and calculated when it can be anything but if someone sees the aesthetics in proofs like that of Cantor's Diagonal Argument.
Besides these notable exclusions, however, I think that the most notable exclusion from mathematics education in the early days is discrete mathematics. From graphs, to combinatorics, to probability and statistics, while we do cover some of it in elementary and middle school ( and highschool if you're in the IB program or taking AP statistics ), we do not cover any of the more difficult and unintuitive parts of the subject, focusing on the simple parts where, after enough training, even a dog could do the same. Thus, while we cover stuff like calculating the mean, median, mode, and sometimes the standard deviation of a set, along with some limited hypothesis testing ( though that tends to be more in collegiate-level classes like those in the AP and IB systems ), we do NOT cover stuff such as discrete random variables, concentration inequalities, the central limit theorem ( except in quick breath to just state what it means for experiments ), and the other theoretical parts of combinatorics, like generating functions and the Stirling numbers. These parts of mathematics, in my personal opinion, need to be included in more and more classes at the HS level for a few reasons. Firstly, if we are speaking solely about job growth, Computer Science exists. However, unlike most of the continuous math physics might need, CS requires a lot of discrete math for students to make the most of algorithms and data structures, from learning how to rephrase common problems into problems in graphs and problems in counting to learning how to take hard NP-complete problems and finding min-max algorithms to get decent results. As students, at college, then have to try to learn all of this content in a very quick manner, they tend to lose some of the understanding that proper exposure will give them, and thus lose out on the wonderful intuitions and wonderful parts that will help them in their carreers if they happen to solve computationally interesting problems. Secondly, consider the sciences and social sciences. In the wake of modern scenarios like p-hacking, the current technique-based approach to experimental analysis needs to be destroyed through careful understanding of mathematical laws, which is best done by teaching students the math behind the techniques rather than the techinques themselves. As a bonus, due to the more abstract nature of discrete mathematics, students will be pushed to have to make proofs to convince themselves of harder questions, thus also pushing us to get better as a group. Lastly, discrete mathematics is a good field for those people who wonder what the point of mathematics is. As higher levels combine analytical approaches from continuous mathematics, it becomes a matter of just ensuring people understand the problems with statistical analyses and the benefits of mathematical modelling to inspire people to see that math is not just some weird abstract field they'll never use, but a tool that will keep them sane and keep them making accurate assessments of reality till time itself terminates.
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