Grade-school elementary algebra presented in an abstract-algebra style?Are there elementary-school curricula...
Grade-school elementary algebra presented in an abstract-algebra style?
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I remember once hearing a (probably apocryphal) story about a university math professor that tried to teach a gradeschool class about algebra by telling them a few simple axioms and definitions and then making deductions. As the story goes, he thought he would be making things as easy as possible by minimizing the number of things the students would have to learn, but that turned out to be a bad idea because the way you get kids to do good on the SAT is by minimizing how much they have to think.
I am curious about what that lecture series would actually look like. I would like to find a presentation of elementary algebra that treats it from an abstract standpoint, but that requires no prior knowledge. Essentially, I am looking for algebra explained in the "professional" style that the above story depicts.
I have no idea where to find it. Textbooks that present the abstract algebra view of elementary algebra tend to assume you already know elementary algebra. University textbooks about elementary algebra written in the 1700s (when elementary algebra was a dominant research topic) come close, but abstract algebra was not around back then. Does such a book or series of notes exist? Can it exist?
soft-question
asked 9 hours ago
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show 1 more comment
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I remember once hearing a (probably apocryphal) story about a university math professor that tried to teach a gradeschool class about algebra by telling them a few simple axioms and definitions and then making deductions. As the story goes, he thought he would be making things as easy as possible by minimizing the number of things the students would have to learn, but that turned out to be a bad idea because the way you get kids to do good on the SAT is by minimizing how much they have to think.
I am curious about what that lecture series would actually look like. I would like to find a presentation of elementary algebra that treats it from an abstract standpoint, but that requires no prior knowledge. Essentially, I am looking for algebra explained in the "professional" style that the above story depicts.
I have no idea where to find it. Textbooks that present the abstract algebra view of elementary algebra tend to assume you already know elementary algebra. University textbooks about elementary algebra written in the 1700s (when elementary algebra was a dominant research topic) come close, but abstract algebra was not around back then. Does such a book or series of notes exist? Can it exist?
soft-question
asked 9 hours ago
Display NameDisplay Name
1192
New contributor
Display Name is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
$endgroup$
$begingroup$
"...the way you get kids to do good on the SAT is by minimizing how much they have to think." Is that a part of the story, or a claim you are making? If the latter, it seems unsubstantiated to me (at least the way you present it).
$endgroup$
– schematic_boi
9 hours ago
$begingroup$
That was the punchline of the story, as I remember it. I'm not saying it was a good story. ;)
$endgroup$
– Display Name
9 hours ago
2
$begingroup$
Maybe you'd like to look for textbooks from the New Math movement.
$endgroup$
– Nate Eldredge
9 hours ago
2
$begingroup$
Not exactly grade school, but in 1998–1999, Luc Illusie was going to teach a course on algebra and Galois theory in the first year at the École Normale Supérieure (I think that would be roughly “advanced undergraduate” or first year graduate in the US). A few of us who had taken his course in algebraic geometry the year before asked him how he planned to explain Galois theory, and he said “oh, very simply, merely as the equivalence of the topos of étale algebras over a field with that of sets under action of the absolute Galois group of that field”…
$endgroup$
– Gro-Tsen
9 hours ago
3
$begingroup$
This question makes me think of the paper pdfs.semanticscholar.org/b44b/… on carrying when making additions and group cohomology.
$endgroup$
– Daniel Robert-Nicoud
8 hours ago
|
show 1 more comment
$begingroup$
I remember once hearing a (probably apocryphal) story about a university math professor that tried to teach a gradeschool class about algebra by telling them a few simple axioms and definitions and then making deductions. As the story goes, he thought he would be making things as easy as possible by minimizing the number of things the students would have to learn, but that turned out to be a bad idea because the way you get kids to do good on the SAT is by minimizing how much they have to think.
I am curious about what that lecture series would actually look like. I would like to find a presentation of elementary algebra that treats it from an abstract standpoint, but that requires no prior knowledge. Essentially, I am looking for algebra explained in the "professional" style that the above story depicts.
I have no idea where to find it. Textbooks that present the abstract algebra view of elementary algebra tend to assume you already know elementary algebra. University textbooks about elementary algebra written in the 1700s (when elementary algebra was a dominant research topic) come close, but abstract algebra was not around back then. Does such a book or series of notes exist? Can it exist?
soft-question
asked 9 hours ago
Display NameDisplay Name
1192
New contributor
Display Name is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
$endgroup$
I remember once hearing a (probably apocryphal) story about a university math professor that tried to teach a gradeschool class about algebra by telling them a few simple axioms and definitions and then making deductions. As the story goes, he thought he would be making things as easy as possible by minimizing the number of things the students would have to learn, but that turned out to be a bad idea because the way you get kids to do good on the SAT is by minimizing how much they have to think.
I am curious about what that lecture series would actually look like. I would like to find a presentation of elementary algebra that treats it from an abstract standpoint, but that requires no prior knowledge. Essentially, I am looking for algebra explained in the "professional" style that the above story depicts.
I have no idea where to find it. Textbooks that present the abstract algebra view of elementary algebra tend to assume you already know elementary algebra. University textbooks about elementary algebra written in the 1700s (when elementary algebra was a dominant research topic) come close, but abstract algebra was not around back then. Does such a book or series of notes exist? Can it exist?
soft-question
soft-question
asked 9 hours ago
Display NameDisplay Name
1192
New contributor
Display Name is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
asked 9 hours ago
Display NameDisplay Name
1192
New contributor
Display Name is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
asked 9 hours ago
Display NameDisplay Name
1192
New contributor
Display Name is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
asked 9 hours ago
Display NameDisplay Name
1192
asked 9 hours ago
Display NameDisplay Name
1192
1192
New contributor
Display Name is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
New contributor
Display Name is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
$begingroup$
"...the way you get kids to do good on the SAT is by minimizing how much they have to think." Is that a part of the story, or a claim you are making? If the latter, it seems unsubstantiated to me (at least the way you present it).
$endgroup$
– schematic_boi
9 hours ago
$begingroup$
That was the punchline of the story, as I remember it. I'm not saying it was a good story. ;)
$endgroup$
– Display Name
9 hours ago
2
$begingroup$
Maybe you'd like to look for textbooks from the New Math movement.
$endgroup$
– Nate Eldredge
9 hours ago
2
$begingroup$
Not exactly grade school, but in 1998–1999, Luc Illusie was going to teach a course on algebra and Galois theory in the first year at the École Normale Supérieure (I think that would be roughly “advanced undergraduate” or first year graduate in the US). A few of us who had taken his course in algebraic geometry the year before asked him how he planned to explain Galois theory, and he said “oh, very simply, merely as the equivalence of the topos of étale algebras over a field with that of sets under action of the absolute Galois group of that field”…
$endgroup$
– Gro-Tsen
9 hours ago
3
$begingroup$
This question makes me think of the paper pdfs.semanticscholar.org/b44b/… on carrying when making additions and group cohomology.
$endgroup$
– Daniel Robert-Nicoud
8 hours ago
|
show 1 more comment
$begingroup$
"...the way you get kids to do good on the SAT is by minimizing how much they have to think." Is that a part of the story, or a claim you are making? If the latter, it seems unsubstantiated to me (at least the way you present it).
$endgroup$
– schematic_boi
9 hours ago
$begingroup$
That was the punchline of the story, as I remember it. I'm not saying it was a good story. ;)
$endgroup$
– Display Name
9 hours ago
2
$begingroup$
Maybe you'd like to look for textbooks from the New Math movement.
$endgroup$
– Nate Eldredge
9 hours ago
2
$begingroup$
Not exactly grade school, but in 1998–1999, Luc Illusie was going to teach a course on algebra and Galois theory in the first year at the École Normale Supérieure (I think that would be roughly “advanced undergraduate” or first year graduate in the US). A few of us who had taken his course in algebraic geometry the year before asked him how he planned to explain Galois theory, and he said “oh, very simply, merely as the equivalence of the topos of étale algebras over a field with that of sets under action of the absolute Galois group of that field”…
$endgroup$
– Gro-Tsen
9 hours ago
3
$begingroup$
This question makes me think of the paper pdfs.semanticscholar.org/b44b/… on carrying when making additions and group cohomology.
$endgroup$
– Daniel Robert-Nicoud
8 hours ago
$begingroup$
"...the way you get kids to do good on the SAT is by minimizing how much they have to think." Is that a part of the story, or a claim you are making? If the latter, it seems unsubstantiated to me (at least the way you present it).
$endgroup$
– schematic_boi
9 hours ago
$begingroup$
"...the way you get kids to do good on the SAT is by minimizing how much they have to think." Is that a part of the story, or a claim you are making? If the latter, it seems unsubstantiated to me (at least the way you present it).
$endgroup$
– schematic_boi
9 hours ago
$begingroup$
That was the punchline of the story, as I remember it. I'm not saying it was a good story. ;)
$endgroup$
– Display Name
9 hours ago
$begingroup$
That was the punchline of the story, as I remember it. I'm not saying it was a good story. ;)
$endgroup$
– Display Name
9 hours ago
2
2
$begingroup$
Maybe you'd like to look for textbooks from the New Math movement.
$endgroup$
– Nate Eldredge
9 hours ago
$begingroup$
Maybe you'd like to look for textbooks from the New Math movement.
$endgroup$
– Nate Eldredge
9 hours ago
2
2
$begingroup$
Not exactly grade school, but in 1998–1999, Luc Illusie was going to teach a course on algebra and Galois theory in the first year at the École Normale Supérieure (I think that would be roughly “advanced undergraduate” or first year graduate in the US). A few of us who had taken his course in algebraic geometry the year before asked him how he planned to explain Galois theory, and he said “oh, very simply, merely as the equivalence of the topos of étale algebras over a field with that of sets under action of the absolute Galois group of that field”…
$endgroup$
– Gro-Tsen
9 hours ago
$begingroup$
Not exactly grade school, but in 1998–1999, Luc Illusie was going to teach a course on algebra and Galois theory in the first year at the École Normale Supérieure (I think that would be roughly “advanced undergraduate” or first year graduate in the US). A few of us who had taken his course in algebraic geometry the year before asked him how he planned to explain Galois theory, and he said “oh, very simply, merely as the equivalence of the topos of étale algebras over a field with that of sets under action of the absolute Galois group of that field”…
$endgroup$
– Gro-Tsen
9 hours ago
3
3
$begingroup$
This question makes me think of the paper pdfs.semanticscholar.org/b44b/… on carrying when making additions and group cohomology.
$endgroup$
– Daniel Robert-Nicoud
8 hours ago
$begingroup$
This question makes me think of the paper pdfs.semanticscholar.org/b44b/… on carrying when making additions and group cohomology.
$endgroup$
– Daniel Robert-Nicoud
8 hours ago
|
show 1 more comment
3 Answers
3
active
oldest
votes
$begingroup$
It is a parody, but "Mathematics Made Difficult" by Carl Linderholm is this, using concepts from category theory in order to explain things like "counting" and "subtraction". The presentation is not strictly correct in terms of the mathematical concepts being wielded, so it's not a great source to actually learn about category theory or elementary arithmetic. People who do know about those things may appreciate the humor.
answered 9 hours ago
Math AppreciatorMath Appreciator
811
New contributor
Math Appreciator is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
$endgroup$
add a comment |
$begingroup$
Hardly "requires no prior knowledge," but:
Klein, Felix, M. Menghini, and G. Schubring. Elementary mathematics from a higher standpoint. Berlin/Heidelberg: Springer, 2016.
Springer link.
answered 6 hours ago
Joseph O'RourkeJoseph O'Rourke
87.1k16243718
$endgroup$
add a comment |
$begingroup$
You might ask Mark Sapir. I understand he had a series of lessons on words in semigroups with a 4th grade audience in mind. (I don't know which country.) He might know of similar efforts.
Gerhard "For Me , It Was Latin" Paseman, 2019.05.21.
answered 9 hours ago
Gerhard PasemanGerhard Paseman
9,18122247
$endgroup$
2
$begingroup$
Several years ago I attended a wonderful talk by Persi Diaconis in the Stanford Statistics Department where he used very sophisticated statistics to analyze properties of elementary-school arithmetic, such as the statistics of how frequently you must perform a carry when adding or multiplying multi-digit numbers. Not what you're seeking, but close. statweb.stanford.edu/~cgates/PERSI/papers/…
$endgroup$
– David G. Stork
7 hours ago
add a comment |
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3 Answers
3
active
oldest
votes
3 Answers
3
active
oldest
votes
active
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active
oldest
votes
$begingroup$
It is a parody, but "Mathematics Made Difficult" by Carl Linderholm is this, using concepts from category theory in order to explain things like "counting" and "subtraction". The presentation is not strictly correct in terms of the mathematical concepts being wielded, so it's not a great source to actually learn about category theory or elementary arithmetic. People who do know about those things may appreciate the humor.
answered 9 hours ago
Math AppreciatorMath Appreciator
811
New contributor
Math Appreciator is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
$endgroup$
add a comment |
$begingroup$
It is a parody, but "Mathematics Made Difficult" by Carl Linderholm is this, using concepts from category theory in order to explain things like "counting" and "subtraction". The presentation is not strictly correct in terms of the mathematical concepts being wielded, so it's not a great source to actually learn about category theory or elementary arithmetic. People who do know about those things may appreciate the humor.
answered 9 hours ago
Math AppreciatorMath Appreciator
811
New contributor
Math Appreciator is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
$endgroup$
add a comment |
$begingroup$
It is a parody, but "Mathematics Made Difficult" by Carl Linderholm is this, using concepts from category theory in order to explain things like "counting" and "subtraction". The presentation is not strictly correct in terms of the mathematical concepts being wielded, so it's not a great source to actually learn about category theory or elementary arithmetic. People who do know about those things may appreciate the humor.
answered 9 hours ago
Math AppreciatorMath Appreciator
811
New contributor
Math Appreciator is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
$endgroup$
It is a parody, but "Mathematics Made Difficult" by Carl Linderholm is this, using concepts from category theory in order to explain things like "counting" and "subtraction". The presentation is not strictly correct in terms of the mathematical concepts being wielded, so it's not a great source to actually learn about category theory or elementary arithmetic. People who do know about those things may appreciate the humor.
answered 9 hours ago
Math AppreciatorMath Appreciator
811
New contributor
Math Appreciator is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
answered 9 hours ago
Math AppreciatorMath Appreciator
811
New contributor
Math Appreciator is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
answered 9 hours ago
Math AppreciatorMath Appreciator
811
answered 9 hours ago
Math AppreciatorMath Appreciator
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811
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Math Appreciator is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
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New contributor
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add a comment |
add a comment |
$begingroup$
Hardly "requires no prior knowledge," but:
Klein, Felix, M. Menghini, and G. Schubring. Elementary mathematics from a higher standpoint. Berlin/Heidelberg: Springer, 2016.
Springer link.
answered 6 hours ago
Joseph O'RourkeJoseph O'Rourke
87.1k16243718
$endgroup$
add a comment |
$begingroup$
Hardly "requires no prior knowledge," but:
Klein, Felix, M. Menghini, and G. Schubring. Elementary mathematics from a higher standpoint. Berlin/Heidelberg: Springer, 2016.
Springer link.
answered 6 hours ago
Joseph O'RourkeJoseph O'Rourke
87.1k16243718
$endgroup$
add a comment |
$begingroup$
Hardly "requires no prior knowledge," but:
Klein, Felix, M. Menghini, and G. Schubring. Elementary mathematics from a higher standpoint. Berlin/Heidelberg: Springer, 2016.
Springer link.
answered 6 hours ago
Joseph O'RourkeJoseph O'Rourke
87.1k16243718
$endgroup$
Hardly "requires no prior knowledge," but:
Klein, Felix, M. Menghini, and G. Schubring. Elementary mathematics from a higher standpoint. Berlin/Heidelberg: Springer, 2016.
Springer link.
answered 6 hours ago
Joseph O'RourkeJoseph O'Rourke
87.1k16243718
answered 6 hours ago
Joseph O'RourkeJoseph O'Rourke
87.1k16243718
answered 6 hours ago
Joseph O'RourkeJoseph O'Rourke
87.1k16243718
answered 6 hours ago
Joseph O'RourkeJoseph O'Rourke
87.1k16243718
87.1k16243718
add a comment |
add a comment |
$begingroup$
You might ask Mark Sapir. I understand he had a series of lessons on words in semigroups with a 4th grade audience in mind. (I don't know which country.) He might know of similar efforts.
Gerhard "For Me , It Was Latin" Paseman, 2019.05.21.
answered 9 hours ago
Gerhard PasemanGerhard Paseman
9,18122247
$endgroup$
2
$begingroup$
Several years ago I attended a wonderful talk by Persi Diaconis in the Stanford Statistics Department where he used very sophisticated statistics to analyze properties of elementary-school arithmetic, such as the statistics of how frequently you must perform a carry when adding or multiplying multi-digit numbers. Not what you're seeking, but close. statweb.stanford.edu/~cgates/PERSI/papers/…
$endgroup$
– David G. Stork
7 hours ago
add a comment |
$begingroup$
You might ask Mark Sapir. I understand he had a series of lessons on words in semigroups with a 4th grade audience in mind. (I don't know which country.) He might know of similar efforts.
Gerhard "For Me , It Was Latin" Paseman, 2019.05.21.
answered 9 hours ago
Gerhard PasemanGerhard Paseman
9,18122247
$endgroup$
2
$begingroup$
Several years ago I attended a wonderful talk by Persi Diaconis in the Stanford Statistics Department where he used very sophisticated statistics to analyze properties of elementary-school arithmetic, such as the statistics of how frequently you must perform a carry when adding or multiplying multi-digit numbers. Not what you're seeking, but close. statweb.stanford.edu/~cgates/PERSI/papers/…
$endgroup$
– David G. Stork
7 hours ago
add a comment |
$begingroup$
You might ask Mark Sapir. I understand he had a series of lessons on words in semigroups with a 4th grade audience in mind. (I don't know which country.) He might know of similar efforts.
Gerhard "For Me , It Was Latin" Paseman, 2019.05.21.
answered 9 hours ago
Gerhard PasemanGerhard Paseman
9,18122247
$endgroup$
You might ask Mark Sapir. I understand he had a series of lessons on words in semigroups with a 4th grade audience in mind. (I don't know which country.) He might know of similar efforts.
Gerhard "For Me , It Was Latin" Paseman, 2019.05.21.
answered 9 hours ago
Gerhard PasemanGerhard Paseman
9,18122247
answered 9 hours ago
Gerhard PasemanGerhard Paseman
9,18122247
answered 9 hours ago
Gerhard PasemanGerhard Paseman
9,18122247
answered 9 hours ago
Gerhard PasemanGerhard Paseman
9,18122247
9,18122247
2
$begingroup$
Several years ago I attended a wonderful talk by Persi Diaconis in the Stanford Statistics Department where he used very sophisticated statistics to analyze properties of elementary-school arithmetic, such as the statistics of how frequently you must perform a carry when adding or multiplying multi-digit numbers. Not what you're seeking, but close. statweb.stanford.edu/~cgates/PERSI/papers/…
$endgroup$
– David G. Stork
7 hours ago
add a comment |
2
$begingroup$
Several years ago I attended a wonderful talk by Persi Diaconis in the Stanford Statistics Department where he used very sophisticated statistics to analyze properties of elementary-school arithmetic, such as the statistics of how frequently you must perform a carry when adding or multiplying multi-digit numbers. Not what you're seeking, but close. statweb.stanford.edu/~cgates/PERSI/papers/…
$endgroup$
– David G. Stork
7 hours ago
2
2
$begingroup$
Several years ago I attended a wonderful talk by Persi Diaconis in the Stanford Statistics Department where he used very sophisticated statistics to analyze properties of elementary-school arithmetic, such as the statistics of how frequently you must perform a carry when adding or multiplying multi-digit numbers. Not what you're seeking, but close. statweb.stanford.edu/~cgates/PERSI/papers/…
$endgroup$
– David G. Stork
7 hours ago
$begingroup$
Several years ago I attended a wonderful talk by Persi Diaconis in the Stanford Statistics Department where he used very sophisticated statistics to analyze properties of elementary-school arithmetic, such as the statistics of how frequently you must perform a carry when adding or multiplying multi-digit numbers. Not what you're seeking, but close. statweb.stanford.edu/~cgates/PERSI/papers/…
$endgroup$
– David G. Stork
7 hours ago
add a comment |
Display Name is a new contributor. Be nice, and check out our Code of Conduct.
Display Name is a new contributor. Be nice, and check out our Code of Conduct.
Display Name is a new contributor. Be nice, and check out our Code of Conduct.
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$begingroup$
"...the way you get kids to do good on the SAT is by minimizing how much they have to think." Is that a part of the story, or a claim you are making? If the latter, it seems unsubstantiated to me (at least the way you present it).
$endgroup$
– schematic_boi
9 hours ago
$begingroup$
That was the punchline of the story, as I remember it. I'm not saying it was a good story. ;)
$endgroup$
– Display Name
9 hours ago
2
$begingroup$
Maybe you'd like to look for textbooks from the New Math movement.
$endgroup$
– Nate Eldredge
9 hours ago
2
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Not exactly grade school, but in 1998–1999, Luc Illusie was going to teach a course on algebra and Galois theory in the first year at the École Normale Supérieure (I think that would be roughly “advanced undergraduate” or first year graduate in the US). A few of us who had taken his course in algebraic geometry the year before asked him how he planned to explain Galois theory, and he said “oh, very simply, merely as the equivalence of the topos of étale algebras over a field with that of sets under action of the absolute Galois group of that field”…
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– Gro-Tsen
9 hours ago
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This question makes me think of the paper pdfs.semanticscholar.org/b44b/… on carrying when making additions and group cohomology.
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– Daniel Robert-Nicoud
8 hours ago