I often teach a course with the enigmatic title “Fundamentals of Mathematics I”, intended for liberal arts majors. This is usually the last encounter with math for students in non-scientific disciplines. The syllabus contains a decent amount of optional topics so it is quite possible to tailor the material to one’s taste and professional interests. As many of us are well aware, a course like this poses its unique challenges. Unlike calculus, where the topics are fairly standard, a “fundamentals” course must, in my view, depend much more on mathematical ideas and much less on drudgery computations. Of course, students need to compute, but the result of their computational efforts should be exciting and fun, rather than a more or less meaningless answer that matches the solutions page at the end of the chapter.
These are the topics I usually teach: set theory, logic, group theory, and combinatorial methods. All lend themselves to a great deal of enjoyment, where students are confronted with deep ideas (e.g. what is truth? what is counting? does infinity come in different “sizes”?). Some students feel bewildered when they discover how difficult it can be to “count”.
It is a time-tested favorite of mine to teach them about groups. But unlike a formal course in abstract algebra, I tell them briefly what a group is, how abstract notions can be fun and useful, and after showing them the cyclic groups, both infinite and finite, I proceed fairly quickly to the dihedral groups D3 and D4 (of orders 6 and 8 respectively). I construct them as the groups of rigid symmetries (rotations and reflections) of the triangle and square. D3 is such a revelation since it is the smallest group that happens to be non-commutative; and I usually spend several lectures drawing pictures and discussing composition of symmetries. It is really exciting to reveal to them how “multiplication” is not a universal idea and it can be defined as a non-commutative binary operation. After producing the multiplication tables of both D3 and D4 we set out to explore the orders of individual elements as well as the subgroup structures of each one of these groups. We emphasize the subgroups of rotations and reflections.
One of my favorite results in elementary group theory is Lagrange’s theorem: If G is a group of order n and H is a subgroup of order m, them m divides n. I take advantage of the sheer simplicity of this result and “test” it for D3 and D4. It is not magic, I tell them, it’s a theorem!… I also exploit Lagrange’s result by bringing the (finite) cyclic groups back and sharing the pleasant fact that the converse of Lagrange is true in that context: if Cn is the finite cyclic group of order n and m is any divisor of n, there exists a subgroup (necessarily cyclic) of Cn whose order is m. And yes, we “test” the truth of this and explore the consequences when n is prime.
Many topics presented in more advanced courses in combinatorics, abstract algebra, logic, etc. can certainly be made accessible to a liberal arts audience. The trick, of course, is to explain the ideas in layman’s terms, progress to some level of formalization, and work out many enlightening examples. Teaching this course has been very satisfactory indeed. My hope is and has always been to leave my students with some long-lasting interest in the ideas behind mathematics, as well as a taste of what mathematicians do.
I know this site presents quality depending articles or reviews
and extra material, is there any other site which gives such things in quality?
레플베이
온티티I savour, cause I found exactly what I was having a look for
I know this web site provides quality based content and additional information, is there any other web page which presents these kinds of things in quality?
온티티
레플베이Great post. I am facing a couple of these problems.
온티티Please let me know if you’re looking for a writer for your blog
Great article. I’m going through many of these issues as well..
온티티
It’s remarkable in favor of me to have a site,
which is useful in support of my know-how. thanks admin
레플베이
“Kurumsalx” haber yazılımı ile tüm bunların yükünden kurtuluyorsunuz.Sizi klişeleşmiş herkes de olan temalardan ve sabit tasarımlardan kurtarıp bütünüyle kendi arzu ettiğiniz gibi haberlerinizi sunmanıza olanak sağlamaktayız.
Bu haber yazılımını özellikle yerel habercilerin ulusal seviyede haber sitelerine sahip olabilmeleri için tasarladık.
Daha fazla bilgi ve detaylar için kurumsalx.com sitesini ziyaret edebilirsiniz.
visit it.
بلبرينگ در مقاومت آنها در برابر انواع بارهاي وارده است. انواع بلبرینگ ها و کاربرد آنها بطور کلي بار وارده بر انواع رولبرينگ ها و بلبرينگ ها به دو صورت شعاعي ومحوري مي باشد و بسته به موقعيت و شرايطي که بلبرينگ مورد استفاده قرار مي گيرد يکي از اين دو بار ويا هردو بار بر آن وارد مي شوند.در مواقعي که فشار از نوع شعاعي باشد، مي توان از رولبرينگ ها استفاده کرد، زيرا رولبرينگ ها در برابر اين نوع بارها بسيار مقاوم اند و درصورتي که هردو بار شعاعي و محوري وارد شود، از بلبرينگ ها مي توان استفاده کرد. تفاوت اصلي رولبرينگ و بلبرينگ در مقاومت آنها در برابر انواع بارهاي وارده است.
Huawei atlas 900 AI cluster
The Atlas 900 AI cluster set a new benchmark with its top cluster network, modular deployment, heat dissipation system, holistic design, performance, extensibility, and human-centric details.
Is this true. may i share this post with my friends
konveksi tas
I think it doesn’t matter like us pabrik tas
In a group must be able to solve the problems of each group.
konveksi tas
hello, my name is anes, as one who had studied in school, better to accept what lessons are given by teachers.
thanks for sharing an amazing and informative article
Best Nespresso Capsules for Lattes
Best!
best sushi in vancouver
a href=”https://trafoos.com/best-sushi-in-vancouver/”>best sushi in vancouver
learning methods are not rigid will make the students feel so much fun.
in learning is better there is a group
Really it very useful I just got to this amazing site not long ago. I was actually captured with the piece of resources you have got here. Big thumbs up for making such wonderful blog page!data science course in dubai
Really I Appreciate The Effort You Made To Share The Knowledge. This Is Really A Great Stuff For Sharing. Keep It Up . Thanks ForQuality posts is the crucial to invite the visitors to visit the web page, that’s what this web page is providing. data science course in singapore
Really I Appreciate The Effort You Made To Share The Knowledge. This Is Really A Great Stuff For Sharing. Keep It Up . Thanks ForQuality posts is the crucial to invite the visitors to visit the web page, that’s what this web page is providing. data science course in singapore
Best Tarot Card Reader In Bangalore With Affordable Price. Under Your Past To Heal Your Present And Plan For Future. No.1 Tarot Card Reading In Bangalore.
tarot card reader
Easy Home specializes in providing comprehensive architecture solutions from inception to completion. Our award winning team of architects, project managers and interior designers based in Bangalore work hand in hand all under one roof. We will design and construct your home with emotion, passion, and a genuine love for creating. As you view the finished product and home designs those same emotions will carve a space in your heart. Easy Home’s job is not complete until you and your family feel 100% comfortable in your new custom-built palace.
architects in bangalore
Superb article. Keep sharing such articles.
wow i love this blog you are so smart guys ^^
learning methods are not rigid will make the students feel so much fun, and easier to understand what will be shared by teachers
pertamina solusi bahan bakar berkualitas dan ramah lingkungan
hello my name is rohan, as one who had studied in school, better accept what lessons are given by teachers if in a state of happy, especially if there is created a kind of game it will be easier to absorb knowledge given
learning methods are not rigid will make the students feel so much fun, and easier to understand what will be shared by teachers
in learning is better there is a group that can exchange ideas so as to understand the subjects will be easier
Marcos! It seems you have a certain amount of freedom in choosing the content in your “fundamentals”-type courses, which I envy – ours are more-or-less proscribed by the department, via a detailed syllabus. But I still like the variety that these courses provide, and the there is great opportunity to do math you can “get your hands on” (like messing with triangles and squares!).
Marcos! It seems you have a certain amount of freedom in choosing the content in your “fundamentals”-type courses, which I envy – ours are more-or-less proscribed by the department, via a detailed syllabus. But I still like the variety that these courses provide, and the there is great opportunity to do math you can “get your hands on” (like messing with triangles and squares!). I think the focus on concrete examples is likely to have the most lasting effect at this level – these are the things that might just stick around in someones head beyond the completion of their degree. I want to take your class – it sounds fun!