15

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 "...


12

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. ...


11

One well-known source is Project Euler. The concept behind it is that each problem is mathematical and designed to be solved by an efficient algorithm on a "normal" computer in less than a minute. The early problems are all extremely accessible. As the problems go on, they become (in my mathematical opinion) far more esoteric from either a mathematical or ...


11

Given 100% control, I would have one-to-one instruction. One instructor meeting individually with each student. That instructor can change the approach, the speed, the order of topics, the method of instruction, based on that individual student. But of course hiring (and training) enough instructors for that is probably way beyond any reasonable budget. (...


9

Without directly answering your question, you don't seem to have the background you need to be "improving" the undergraduate experience yet, and have some work to do. I think you're right in sensing that your question is too general Don't talk about the "difficulties in forming such a course", and spend your initial time finding out what aspects of the ...


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

Any puzzle game which requires students to plan the entire problem before executing it might help. Also, there are physical puzzles which can be solved algorithmically much more neatly than if they use 'trial and error'. In English, one resource which I have not seen mentioned yet is Code.org which has themed coding puzzles for all ages. Other puzzles, ...


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

Challenging question! Two ideas. (1) Calculate the Greatest Common Divisor of two natural numbers, not so easily accomplished by hand on moderately large numbers. The Euclidean algorithm could be used to illustrate recursion/induction. Here is Python3 code: trace = True # True turns on tracing prints. def GCD( a, b ): '''Returns the Greatest Common ...


4

It's about the learning experience. An issue with project based learning is the need for objective grading isn't going away. Since the first semester of mathematics is what serves to filter out under performing students there is a desire to have an unmovable bar. Without it you get grade inflation. This isn't to say there can't be projects. You just have ...


4

An "old school" answer (nearly 60 years old now!) which works for any age range is turtle graphics, which is (are?) implemented as a Python module. We can only guess how much of your curriculum is officially labelled "geometry", but it will certainly teach algorithmic thinking, and also be fun.


3

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 ...


3

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 ...


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 ...


2

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 ...


2

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 ...


2

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) ...


2

This isn't a resource, but a fun algorithmic problem that I remember solving back in 7th grade: Consider a long loop of train wagons, $1 ... N$ for some unknown $N$ where wagon $i$ is connected to wagons $i-1$ and $i+1$. You can walk from one wagon to its neighbors. In every wagon, there is a lightbulb connected to a switch. You start in a wagon and want to ...


2

These may be too hard, but the ACM's International Collegiate Programming Contest (ICPC) has a set of past programming problems that require algorithmic thinking. I took a class in college where we basically just worked on these for 3 hours a week. It was really good problem-solving experience. https://icpc.baylor.edu/worldfinals/problems


2

I'm an engineer who really likes and uses mathematics on work and daily life. Myself, I would try to make the course as much self-contained in classes, but still stick to some written material the students have access to. This allows the subject to be more complex and extensive while keeping down the difficulty settings. I'd prove theorems as much as ...


1

For a Calculus I class, I usually: Don't teach limits first. Limits, as they are currently taught, lead to confusion. You don't do anything with them except prove the derivative. After that they are largely forgotten, basically leaving students feeling they wasted their time, and they are quickly forgotten. Instead, teach limits at the end to justify ...


1

Select and use an open textbook. Students who wish to use a physical textbook will be able to get a copy run off and hole-punched at a local print shop, then put in a binder, for far less than the cost of a conventional textbook. You will save your students a pile of money and there are enough available options that you should be able to find something ...


1

Although some of the puzzles might be appropriate only for extra credit, have a look at Algorithmic Puzzles by Anany and Maria Levitin. The ad copy for this book gives a good description: While many think of algorithms as specific to computer science, at its core algorithmic thinking is defined by the use of analytical logic to solve problems. This ...


1

The de Casteljau algorithm for generating polynomially parameterized curves in the Beziér representation admits a simple geometric interpretation in terms of the control polygon that is easily implemented in Python. More precisely, a polynomially parameterized curve in the affine plane or affine three-space can be represented in the form $P(t) = \sum_{k = ...


1

Perhaps the problems of "make up" and "traffic" could also be used. Make-up: Assuming that the chemicals etc., damage your skin in the long run, we may state that constant make-up is "bad". And contrarily no make-up is better in the long run since your skin remains "naturally beautiful". Given any two individuals with the option of applying make-up or ...


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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