I am an undergrad math student and work as a math tutor or elementary through university level students.
Project idea overview
In my work tutoring, I've found that there are a few major hurdles/milestones. Learning to count is hard, then learning to add and subtract numbers, then learning to add and subtract numbers without counting (divorcing the idea of a "number" from a count), long multiplication/division are hard because now you have all these intermediate steps that don't seem like they're directly related to the calculation. Then (elementary) algebra, you have to do math with numbers that you don't even know what they are yet, like x. Toward calculus you stop working with numbers, so to speak, and are now dealing with functions as your primary object of interest. A common thread through all of this though is that math is about doing things to "numbers." Those "numbers" eventually take the form of (differentiable) functions, but it's still about the numbers, so to speak.
Undergraduate introductory linear algebra is another one of these hurdles and it's qualitatively different. You're not just expanding on what you already know, like how algebra is just secret numbers and functions are just several numbers (a continuum even) put together. If you try to learn linear algebra with matrices as your "numbers" and row reduction/multiplication/etc as the things you do to them then you're going to have a rough time. Algebra, linear or otherwise, is better understood as an exploration of things to do with "numbers" and looking at what you can do those things to. Crudely, analysis takes a field (almost always $\mathbb{R}\text{ or }\mathbb{C}$) and looks for interesting things that happen in there. Ring theory takes the ring axioms, things you can do with "numbers," and looks for what other structures are compatible. When can or can't you define a unique greatest common factor or irreducible factorization? How can you get rings out of other rings? Algebra requires a qualitatively different approach than analysis and I want to make my project about how they're different, in particular through the lens of linear algebra because that's where most students first encounter this mode of thinking.
The project also (hopefully) touches on the idea that there are various meaningfully different fields of math. Analysis has a certain mindset to it, as does algebra, but also topology, combinatorics, geometry, etc. all have their own ways that are more or less productive to think about them to be more successful. There is no one "mathematical mode of mind," but in fact millions of mathematical modes of mind.
In particular, my salient points at present are
Up to this point, math has started with numbers and explored what can be done with them. Now it's about things you can do to numbers (vector space axioms) and looking at what you can do them to.
Matrices are not numbers. They kind of behave like numbers, look at how they form a noncommutative ring and nonabelian group (what those objects are would be explained from the ground up and not in those terms).
A bit of why group theory is useful; symmetry is another way of just preserving structure and algebra is just about preserving and identifying structure.
Analysis feels a bit (to me) like building a rubber-band powered racecar and sending it off, hoping that I calibrated my epsilons and deltas well enough that it hits the right point. You have a lot of direct control in the setup but it still has to work out overall. By contrast algebra feels like arranging iron filings on a table by moving magnets around below it. You don't get much direct control over any auxiliary structure of your group so you have to pull in Sylow theorems or something comparatively abstract and indirect to show why something must or can't be the case.
As you can probably tell I haven't written this in a form that's accessible to your typical sophomore in their first linear class but this is the general vibe. In particular point 4 there is more about how I'm thinking about this and less something that will actually go into If you have any ideas for additions, removals, or changes then this is also very go
Target medium
What I have now is the outline for something like an article or blog post (I've put most of what I have in the summary section tbh). I can readily put forth some ideas for illustrations or animations, but also it'd be great to have this as a video. I don't have much of a creative vision there, it'd follow the same narrative as the equivalent article.
All of this is to say I'm not super picky about a medium, I just don't have the skills to do much more than a wall of text. Any way to enhance that would be lovely.
Contact
If you're interested in helping out or have any thoughts generally I can be reached at danewmeade@gmail.com
About the author
I am an undergrad math student and work as a math tutor or elementary through university level students.
Project idea overview
In my work tutoring, I've found that there are a few major hurdles/milestones. Learning to count is hard, then learning to add and subtract numbers, then learning to add and subtract numbers without counting (divorcing the idea of a "number" from a count), long multiplication/division are hard because now you have all these intermediate steps that don't seem like they're directly related to the calculation. Then (elementary) algebra, you have to do math with numbers that you don't even know what they are yet, like x. Toward calculus you stop working with numbers, so to speak, and are now dealing with functions as your primary object of interest. A common thread through all of this though is that math is about doing things to "numbers." Those "numbers" eventually take the form of (differentiable) functions, but it's still about the numbers, so to speak.
Undergraduate introductory linear algebra is another one of these hurdles and it's qualitatively different. You're not just expanding on what you already know, like how algebra is just secret numbers and functions are just several numbers (a continuum even) put together. If you try to learn linear algebra with matrices as your "numbers" and row reduction/multiplication/etc as the things you do to them then you're going to have a rough time. Algebra, linear or otherwise, is better understood as an exploration of things to do with "numbers" and looking at what you can do those things to. Crudely, analysis takes a field (almost always $\mathbb{R}\text{ or }\mathbb{C}$) and looks for interesting things that happen in there. Ring theory takes the ring axioms, things you can do with "numbers," and looks for what other structures are compatible. When can or can't you define a unique greatest common factor or irreducible factorization? How can you get rings out of other rings? Algebra requires a qualitatively different approach than analysis and I want to make my project about how they're different, in particular through the lens of linear algebra because that's where most students first encounter this mode of thinking.
The project also (hopefully) touches on the idea that there are various meaningfully different fields of math. Analysis has a certain mindset to it, as does algebra, but also topology, combinatorics, geometry, etc. all have their own ways that are more or less productive to think about them to be more successful. There is no one "mathematical mode of mind," but in fact millions of mathematical modes of mind.
In particular, my salient points at present are
As you can probably tell I haven't written this in a form that's accessible to your typical sophomore in their first linear class but this is the general vibe. In particular point 4 there is more about how I'm thinking about this and less something that will actually go into If you have any ideas for additions, removals, or changes then this is also very go
Target medium
What I have now is the outline for something like an article or blog post (I've put most of what I have in the summary section tbh). I can readily put forth some ideas for illustrations or animations, but also it'd be great to have this as a video. I don't have much of a creative vision there, it'd follow the same narrative as the equivalent article. All of this is to say I'm not super picky about a medium, I just don't have the skills to do much more than a wall of text. Any way to enhance that would be lovely.
Contact
If you're interested in helping out or have any thoughts generally I can be reached at danewmeade@gmail.com