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I think this is an interesting question. In the US undergraduate mathematics curriculum, one often finds a sequence of courses "Abstract Algebra I" and "Abstract Algebra II." I think there is lots of variation. Typically groups are in the first course, but Sylow theorems might be in the second course, along with rings. And the second ...

5

How about this: Let $k$ be an infinite field, and let $f \in k[x]$. Assume $f(t) = 0$ for all $t \in k$. Assume to the contrary that $f$ is not the zero polynomial. Then $f$ is a polynomial of degree $n \geq 1$. Choose distinct elements $z_1, z_2, z_3, \dots, z_{n+1}$ of $k$. By repeated use of the linear factor theorem, we know that $f(x) = (x-z_1)(x-z_2)(... 4 The title says the student doesn't "retain," but your final summary at the bottom that he doesn't "understand." These seem like totally different things to me. From your description, it seems like the problem is that he doesn't understand, and therefore when he learns an algorithm, it's a bunch of arbitrary mumbo-jumbo, like being asked ... 4 You make the (common) error of emphasizing explanation/understanding as the KEY aspect of pedagogy. However, the key is PRACTICE, not preaching. Humans are not computers, to which an instruction set, given once, can be run indefinitely and perfectly, once scanned properly. Now, of course, nothing wrong with an explanation. And it can help. But if the ... 4 Group theory has lots of applications in the sciences, such as to crystallography and quantum mechanics. Rings and fields basically don't, except in trivial ways like doing linear algebra over the complex numbers rather than the reals. I assume that's the reason why the curriculum is normally arranged the way it is, so that STEM students have the option of ... 3 I think this certainly makes a lot more sense than teaching rings before groups. I think it might be a wash whether you teach all of the parts of groups and then all of the parts of rings and note the similarities along the way or if you collate them like this book does, but at least this seems pedagogically viable. I think my larger concern is how the ... 3 Opal: Your proposed scheme does indeed sound like the Second Chance Grading scheme I created and wrote about in PRIMUS (see the link and figure Brendan referenced). I'm happy to chat further with you -- and anyone else interested -- about the system, how it has fared since I wrote the article -- including during the pandemic (spoiler alert: it's still ... 3 Here are projects I have written and used. I assign these to student groups of 3-6 students, mostly Engineering majors, mostly sophomores. In class time is about 30 minutes per project to introduce the set up; then they work out of class for about 2 weeks per project (and come to office hours frequently during that time). Linear programming Camera matrices ... 2 Didactic stories should be simple, clear, and short. Think Aesop’s fables. If they become more complex, at some point they become case studies. (Case studies are good to use, but it’s a separate question.) Why repeat them? I don’t know Pearl’s book, so I am not familiar with his purposes in repeating stories. So I’ll offer reasons why I sometimes repeat ... 2 Given the equation $$\lvert2x\rvert=x-1,$$ rewriting the left-hand side as$\pm2x$results in the unique solution set$\{-1,\frac13\},$which nevertheless fails to satisfy the equation. (The equation has extraneous but no actual solutions.) 1 It surprises me that nobody has brought up the rule of Sarrus yet: For$\mathbf{u} \times \mathbf{v}$, make the table$\begin{array}{|ccc|cc} \mathbf{i}& \mathbf{j}& \mathbf{k}& \mathbf{i}& \mathbf{j} \\ u_1 & u_2 & u_3 & u_1 & u_2 \\ v_1 & v_2 & v_3 & v_1 & v_2 \end{array}$, then sum up products of diagonals ... 1 In my opinion$2 \times 2$matrices, particularly$2 \times 2$real matrices are fairly easy to understand as mathematical objects, even to folks who haven't been exposed to linear algebra.$2 \times 2$real matrices can be motivated by linear functions in$\mathbb{R}^2\$, (equivalently) as scaled rotations of the plane that fix the origin, as a convenient ...

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The question of whether formal relationships "make sense" is an old one in mathematics. People asked the question about whether negative numbers "made sense" (how can you start with three pebbles and take away six of them?) or complex numbers (what number when squared gives minus 1?). But we found that putting them in algebraic ...

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