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I.e. mathematical history. I think one power of calculus (and everything else) is that you don't need the history. The same way I don't need the history of the

I.e. mathematical history. I think one power of calculus (and everything else) is that you don't need the history. The same way I don't need the history of the hammer to use a nail. What drives my interest is the examples. At the end, the book author talks about students not paying attention until the teacher starts talking about examples. To me, that suggests you should first contextualize and create a need for the tool, then describe the tool. Not to start introducing names of dead people, no matter how great they were. (At least where I grew up, this will always devolve into the curriculum requiring you to know those names and dates, draining focus from what actually matters.)

agomez314 4y ago
The comparison between a hammer and math doesn't make sense, math is much more than a tool. Personally, I'm able to learn best when I understand the context and the "why" questions that people had which led them to the answer. I know other people learn differently so to each its own. Also, I agree with you in that the traditionally-taught history is awful because it lacks relevance. It's now about "how is this useful for me" or even "why should I care" but rather "why did they decide to act that way?"
- deleted 4y ago
  [deleted]
- chongli 4y ago
  I think the dual role of means (tool) and end in itself is part of the reason mathematics is so difficult to motivate. There is this constant tension within the pedagogy between the desires to teach the beauty and the practicality of mathematics. From what I’ve seen, the beauty side tends to come from mathematics teachers and professors who have developed a passion for the subject and genuinely want to share that passion with their students. The practicality side, on the other hand, seems to be driven by industry, administrators, lawmakers, parents, and others involved in the curriculum-setting process. Ultimately, both sides struggle mightily for students’ attention and a typical result is a collective sigh from the class. If were to make any progress I think we need to recognize that student motivation comes from within. Some students are genuinely attracted to the beauty of mathematics while others are interested in different subjects (science and engineering) for which mathematics is only a prerequisite. Perhaps one approach for dealing with this disparity might be to offer students a choice of projects with differing mathematical styles (proof vs calculation).
  - kkylin 4y ago
    Another thing that makes calculus hard to teach today (and not unrelated to the points being made) is that while a couple generations ago, every calculus student also took calc-based physics, that is no longer true today. With the way we compartmentalize teaching today, it's hard not to teach entry-level math abstractly, and that's only going to work for a small subset of students.
egroo 4y ago
Look up Lockhart's lament, if you haven't heard of it
earlgray 4y ago
Learning names for the sake of learning names is never helpful. However the history of mathematics can help to contextualise the material and its motivation. For example, group theory has myriad modern applications. It can be (and usually is) presented in the standard mathematically-polished form, beginning with the axioms and perhaps using matrix groups or simple geometric symmetries as examples. For many learners this is a perfectly sensible way of approaching it and they wouldn't be interested to know more, yet I found it rather clinical and dry. As I'm sure you know, group theory arose as a novel yet natural way of investigating the conditions under which polynomials can have closed-form solutions, and when Galois began to sniff this out it allowed him to get to grips with the impossibility of a closed-form solution to the general quintic polynomial: a problem which had beguiled mathematicians for centuries, solved by a theory which crystallised a more structural way of approaching abstract mathematics. Even though Galois theory is a relatively niche topic within the broad context of academia, for me it draws out the true character of group theory in a way that matrix groups don't, and neither do contrived examples formed from the natural numbers under various quotients. And it clearly solves a problem that required a new way of thinking. There's certainly a balance to be struck here. If you fixate on historical origins then you aren't engaging with the reasons why the topic is still relevant today, so you risk missing the point. It also slows down the pace at which you can absorb the tools needed for applications. But if you only engage with the modern approach you risk building a disconnected, lifeless archipelago of knowledge, unable to see the beautiful links that unify so much of mathematics.
- flashfaffe2 4y ago
  I totally second this. For someone who has a tortuous relationship with mathematics, I always lovedthe beauty of som concepts but to a certain point, got lost mainly due to knowledge base not really understood. Getting to know some historical facts would have help to be more critical to myself ( rather than just copy and paste formula). Years later I surprised myself spending time on new maths problems thanks to YouTube maths contents(3brown1blue, Michael Penn, mindyourdecisions,etc...) Which make me reboot my knowledge. The challenge here is how is it inseminated during your scholarship. From the what I remember, complex number theory were introduced to me as a rule that need to be acknowledged ( with no incentive to look further). Welsh lab vid below made me reconsider my own shortcomings. https://youtube.com/playlist?list=PLiaHhY2iBX9g6KIvZ_703G3KJXapKkNaF https://youtube.com/playlist?list=PLiaHhY2iBX9g6KIvZ_703G3KJ...
alexb_ 4y ago
I disagree. The biggest thing that can help you learn something complex is by introducing it as it was discovered, and making someone feel like they could have come up with it on their own.
- msla 4y ago
  Things were usually discovered in a confusing way that kind of made sense at the time, but had false starts and conceptual limitations driven by people still being in an older mindset when they were trying to discover it. A good teacher sands off those edges, leading to an oversimplified history, but not starting from the firm conceptual footing we only got later on still makes things overly difficult to grasp. Good teaching builds on concepts, even if history took a messier route.
  - sifar 4y ago
    >> Things were usually discovered in a confusing way that kind of made sense at the time, but had false starts and conceptual limitations driven by people still being in an older mindset when they were trying to discover it To know this gives us some perspective on our own biases/limitations and not take things as gospel.
    - a9h74j 4y ago
      > some perspective on our own biases/limitations This is sometimes prescribed in such form as: "Fraction X of your reading should be of authors writing at least 100 years ago." Similar: In defense of the reading of old books by C.S. Lewis ... https://www.youtube.com/watch?v=jFHQDIE7CSw https://www.youtube.com/watch?v=jFHQDIE7CSw "All contemporary writers [of any period] share, to some extent, the contemporary outlook."
- tchaffee 4y ago
  That's not the biggest thing that can help you learn something complex. Practice is the biggest thing. And there are many variations of what motivates people to practice.
deleted 4y ago
[deleted]
ekianjo 4y ago
> he same way I don't need the history of the hammer to use a nail This has to be the world's worst analogy ever used.
replicant 4y ago
As others have argued, one can learn mathematics without its history, but it is very helpful to motivate many of its abstractions. A good example of this are some of the textbooks of David Bressoud. In a "Radical approach to Analysis", concepts are introduced in historical order. The book starts with the heat equation and its solution by Fourier series which was rejected at the time, in constrast to most books that start with the real numbers and limits.
LastTrain 4y ago
I think something quite different than you, but that isn't important. The important thing is that different people are learn differently and are motivated in different ways. Curriculums should take this into account and accommodate different learning styles. Unfortunately, they usually don't.

agomez314 4y ago

The comparison between a hammer and math doesn't make sense, math is much more than a tool. Personally, I'm able to learn best when I understand the context and the "why" questions that people had which led them to the answer. I know other people learn differently so to each its own. Also, I agree with you in that the traditionally-taught history is awful because it lacks relevance. It's now about "how is this useful for me" or even "why should I care" but rather "why did they decide to act that way?"

deleted 4y ago
[deleted]
chongli 4y ago
I think the dual role of means (tool) and end in itself is part of the reason mathematics is so difficult to motivate. There is this constant tension within the pedagogy between the desires to teach the beauty and the practicality of mathematics. From what I’ve seen, the beauty side tends to come from mathematics teachers and professors who have developed a passion for the subject and genuinely want to share that passion with their students. The practicality side, on the other hand, seems to be driven by industry, administrators, lawmakers, parents, and others involved in the curriculum-setting process. Ultimately, both sides struggle mightily for students’ attention and a typical result is a collective sigh from the class. If were to make any progress I think we need to recognize that student motivation comes from within. Some students are genuinely attracted to the beauty of mathematics while others are interested in different subjects (science and engineering) for which mathematics is only a prerequisite. Perhaps one approach for dealing with this disparity might be to offer students a choice of projects with differing mathematical styles (proof vs calculation).
- kkylin 4y ago
  Another thing that makes calculus hard to teach today (and not unrelated to the points being made) is that while a couple generations ago, every calculus student also took calc-based physics, that is no longer true today. With the way we compartmentalize teaching today, it's hard not to teach entry-level math abstractly, and that's only going to work for a small subset of students.

egroo 4y ago

Look up Lockhart's lament, if you haven't heard of it

earlgray 4y ago

Learning names for the sake of learning names is never helpful. However the history of mathematics can help to contextualise the material and its motivation. For example, group theory has myriad modern applications. It can be (and usually is) presented in the standard mathematically-polished form, beginning with the axioms and perhaps using matrix groups or simple geometric symmetries as examples. For many learners this is a perfectly sensible way of approaching it and they wouldn't be interested to know more, yet I found it rather clinical and dry. As I'm sure you know, group theory arose as a novel yet natural way of investigating the conditions under which polynomials can have closed-form solutions, and when Galois began to sniff this out it allowed him to get to grips with the impossibility of a closed-form solution to the general quintic polynomial: a problem which had beguiled mathematicians for centuries, solved by a theory which crystallised a more structural way of approaching abstract mathematics. Even though Galois theory is a relatively niche topic within the broad context of academia, for me it draws out the true character of group theory in a way that matrix groups don't, and neither do contrived examples formed from the natural numbers under various quotients. And it clearly solves a problem that required a new way of thinking. There's certainly a balance to be struck here. If you fixate on historical origins then you aren't engaging with the reasons why the topic is still relevant today, so you risk missing the point. It also slows down the pace at which you can absorb the tools needed for applications. But if you only engage with the modern approach you risk building a disconnected, lifeless archipelago of knowledge, unable to see the beautiful links that unify so much of mathematics.

flashfaffe2 4y ago
I totally second this. For someone who has a tortuous relationship with mathematics, I always lovedthe beauty of som concepts but to a certain point, got lost mainly due to knowledge base not really understood. Getting to know some historical facts would have help to be more critical to myself ( rather than just copy and paste formula). Years later I surprised myself spending time on new maths problems thanks to YouTube maths contents(3brown1blue, Michael Penn, mindyourdecisions,etc...) Which make me reboot my knowledge. The challenge here is how is it inseminated during your scholarship. From the what I remember, complex number theory were introduced to me as a rule that need to be acknowledged ( with no incentive to look further). Welsh lab vid below made me reconsider my own shortcomings. https://youtube.com/playlist?list=PLiaHhY2iBX9g6KIvZ_703G3KJXapKkNaF https://youtube.com/playlist?list=PLiaHhY2iBX9g6KIvZ_703G3KJ...

alexb_ 4y ago

I disagree. The biggest thing that can help you learn something complex is by introducing it as it was discovered, and making someone feel like they could have come up with it on their own.

msla 4y ago
Things were usually discovered in a confusing way that kind of made sense at the time, but had false starts and conceptual limitations driven by people still being in an older mindset when they were trying to discover it. A good teacher sands off those edges, leading to an oversimplified history, but not starting from the firm conceptual footing we only got later on still makes things overly difficult to grasp. Good teaching builds on concepts, even if history took a messier route.
- sifar 4y ago
  >> Things were usually discovered in a confusing way that kind of made sense at the time, but had false starts and conceptual limitations driven by people still being in an older mindset when they were trying to discover it To know this gives us some perspective on our own biases/limitations and not take things as gospel.
  - a9h74j 4y ago
    > some perspective on our own biases/limitations This is sometimes prescribed in such form as: "Fraction X of your reading should be of authors writing at least 100 years ago." Similar: In defense of the reading of old books by C.S. Lewis ... https://www.youtube.com/watch?v=jFHQDIE7CSw https://www.youtube.com/watch?v=jFHQDIE7CSw "All contemporary writers [of any period] share, to some extent, the contemporary outlook."
tchaffee 4y ago
That's not the biggest thing that can help you learn something complex. Practice is the biggest thing. And there are many variations of what motivates people to practice.

deleted 4y ago

[deleted]

ekianjo 4y ago

> he same way I don't need the history of the hammer to use a nail This has to be the world's worst analogy ever used.

replicant 4y ago

As others have argued, one can learn mathematics without its history, but it is very helpful to motivate many of its abstractions. A good example of this are some of the textbooks of David Bressoud. In a "Radical approach to Analysis", concepts are introduced in historical order. The book starts with the heat equation and its solution by Fourier series which was rejected at the time, in constrast to most books that start with the real numbers and limits.

LastTrain 4y ago

I think something quite different than you, but that isn't important. The important thing is that different people are learn differently and are motivated in different ways. Curriculums should take this into account and accommodate different learning styles. Unfortunately, they usually don't.