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15

"Cheating Lessons" by James M. Lang argues (and has many references to back up) the claim that smaller, more frequent, lower stakes assessment both improves student learning outcomes and decreases the frequency of cheating.

14

Here's a silly example: Give students collections of the same type of thing, where each collection contains "good" objects and "bad" objects -- for example, a stack of Pokemon cards with both rare and common cards. We might assume common cards are worth $1$ and rare cards are worth $5$. Have ready another stack of cards that are all common -- call this the "...

10

I found out about the Prisoners' Dilemma as a kid from a book about the Harry Potter phenomenon, which had a chapter about the problem, but presented as a story about Harry and Draco being accused of breaking school rules. Each was offered the same deal as in the original problem, formulated with House Points being taken away instead of a prison sentence. ...

10

If you really want to understand why the curriculum is structured the way it is, and how it got that way, you might want to read a brief history of the discourse around geometry education for the past 150 years or so; I recommend Chapter 1 of The Learning and Teaching of Geometry in Secondary Schools: A Modeling Perspective, by Herbst, Halverscheid, Fujita, ...

8

I like Nick C's idea more than modifying the typical formulation. The notion of snitching on a friend, regardless of the severity of the "crime", has real-world ramifications beyond the punishment put out by the authorities. Depending on your student population, that is possibly going to spur a conversation that will overshadow the objectives of your ...

7

I cannot speak to curricula outside of the US, and I don't really buy your argument that geometry courses are "focused more on memorizing theorems rather than understanding where they come from." It is a poor geometry instructor indeed who admonishes students "Don't be an ASS" before first taking the time to discuss the failure of the angle-side-side ...

7

This has certainly been tried before. See for example, H. Jerome Keisler. Elementary Calculus: An Infinitesimal Approach. On-line Edition. This has also been published in print by Dover. Kathleen Sullivan. The Teaching of Elementary Calculus Using the Nonstandard Analysis Approach. The American Mathematical Monthly Vol. 83, No. 5 (May, 1976), pp. 370-...

5

I teach at community college. I often publish the homework problems at the beginning of the semester, listed by section. I have never had a student work ahead (that I know of). And I have had a few students who loved math, asked deep questions, and were interested in doing extra. I don't think that is likely to happen, except in very rare instances. One ...

5

My understanding of this question is that it proposes the idea of a "monolingual" freshman calc course in which students mostly learn the language of NSA, and limits are largely or completely neglected. This makes it different from this question by Mikhail Katz, which asks whether it's a good idea for students to be "bilingual." I have some experience ...

4

Speaking as a former student, though an engineering one . . . It was hard enough learning to integrate tricky expressions and solve differential equations, without having to learn a new number system as well. And I don't remember ever needing the rigorous definition of a limit, standard or nonstandard. The most I needed even in complex calculus was an ...

3

Perhaps: You and another classmate are together in an obstacle course. If you both make it to the end within a minute, you each get a free day off from school. If just one of you makes it to the end in a minute, they get a whole week off. You know that if you had the other student's help, you could easily finish in a minute, but without it, you ...

3

My mental picture of why angle-side-side doesn't work as a congruence condition doesn't involve trigonometry and seems to me much simpler than trigonometry: Imagine that you're given a line segment $AB$ to serve as one of the two specified sides of the triangle and you're given the angle at $A$, so you can draw a line $l$ through $A$ in the desired direction....

3

You may wish to view the MAA's report Transitions to Proof by Carol Schumacher, Susanna Epp, and Danny Solow: https://www.maa.org/sites/default/files/Transitions%20to%20Proof.pdf It has some suggestions for these kinds of courses, and gives references to some relevant books and articles. Ultimately, the course design is very open-ended, so it's hard to ...

3

It's not really that relevant since the bulk of a normal calculus course (e.g. AP BC, Thomas Finney, Stewart) just does a small amount of epsilon-delta (so student is exposed to it) and then moves to "x+h". The bulk of the course is about learning derivatives, antiderivatives, methods of integration, classic applied problems, a bit if polar coordinates, bit ...

2

2

What are the benefits of having experts create curriculum ? I gather the alternative you entertain is some sort of user-driven machine learning path which is custom fit to the student. Ok, so, if that automation is initiated and curated by experts in math then I don't see the distinction. In some sense it would be a return to the old apprentice system that ...

2

It sounds like you wish to protect your students from the violence and greed of the adult world, while still making the dilemma real enough to keep them engaged. To that effect, I offer two solutions. One, replace prison with detention. Make the crime something like using cell phones in class or throwing spitballs. Two, have them arbitrarily grade each ...

1

My suggestion would be to look at the existing literature. Blake, Rand, Tingley, and Warneken (2015) "introduce a novel implementation of the repeated Prisoner's Dilemma (PD) designed for children to examine whether repeated interactions can successfully promote cooperation in 10 and 11 year olds." Dealing with younger children (ages 6-11), Fan (2000) ...

1

Use of performance-enhancing drugs in sports is a good example. However, why the objection to money and capitalism? Real-life actors value money very much and it affects their behavior greatly. A very realistic and applicable example involving money is what happens when a group of friends goes dining, depending on whether each friend pays for themselves or ...

1

Anecdotal evidence from Germany: congruence is taught way before trigonometry (~7th grade vs ~9th grade for trig functions) Why is the course focused more on memorizing theorems rather than understanding where they come from. The fact that two triangles are congruent if their sides are pairwise of equal length has nothing to do with trigonometry in the ...

1

It's actually a very hard topic to teach. We don't want to teach them the Naive set theory definition of a cardinal number because Naive set theory is contradictory. In Zermelo-Fraenkel set theory, it's easy to define the property of there existing a bijection from one set to another set but the axiom of choice is not provable so it's very hard to define an ...

1

As someone who probably took undergraduate courses more recently than most, I can attest to the fact that applications are necessary. If not given, students very often get lost - not understanding the big picture. They may be able to apply specific operations to specific problems, but it is the applications that allows them to see the forest instead of just ...

1

Be it undergraduate or professor, you need hands-on examples to "see" why some concept or technique is worthwhile. Sure, you can take it that more advanced people are better able to come up with their own examples and applications, or have a richer experience to which new material relates. So examples, applications, cross-connections are certainly needed for ...

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