Tour:General instructions (beginners)

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This is a general instructions page for the Groupprops guided tour for beginners. Get started with the tour.

How to use this guided tour as a first-time learner

General tips

  1. Keep a paper and pencil handy throughout the guided tour. At every stage of the tour, you'll be asked to do calculations or check statements that are being made. A paper and pencil is thus indispensable. You can also jot down your thoughts and doubts, making them easy to refer to later.
  2. Use the navigational links at the top (PREVIOUS, UP and NEXT) to go from one page to another. You are advised not to click on other links in the page, as these will take you away from the tour to the rest of the wiki. In case you do want to check out other things mentioned in the pages, open them in a new tab or window so that the flow of the tour is not disturbed.
  3. The guided tour is organized into parts. At the beginning of each part is a list of prerequisites, goals and general suggestions. Go through the prerequisite list and make sure you have a reasonable idea of these prerequisites. Also, make a note of the goals of the part.

Regular pages: definitions and facts

  1. Most pages of the guided tour have a box on top, below the navigational box, with a WHAT YOU NEED TO DO. This box gives information about how you're supposed to read the page. In some cases, the page simply asks you to read and understand the definition. On other pages, you may be asked to play a more proactive role, for instance, to read a statement in the page, and try proving it before reading the proof. It is best that you attempt doing so, using paper and pencil if necessary, before reading the proof.
  2. Sometimes, multiple equivalent definitions of a term may be given, and you may not immediately see why they are equivalent. If you are not asked to verify the equivalence in the WHAT YOU NEED TO DO, then it is likely that this equivalence of definitions will either be proved later or will become clearer after you have been exposed more to the material.
  3. On some pages, there is a WHAT'S MORE box after the main statement or definition or proof. You are not required to go through the material after this box. If this is your first exposure to the material, and you are finding the main material challenging, it is suggested that you do not go through it. The additional material may or may not make complete sense to you as a first-time learner, depending on your other background. You can ignore this section with no loss in flow.
  4. On some pages, there is a PONDER header, either with the WHAT YOU NEED TO DO or with the WHAT'S MORE. The PONDER header can again be skipped, but pondering over things suggested in this header can help the concepts settle.
  5. On some pages, there is an Expected time for this page header in the navigational box. This expected time is a loose upper bound.

Other pages

Apart from the regular pages giving definitions and facts, each part also has a number of special pages. These pages serve somewhat different functions, as discussed here:

  1. Confidence aggregator page: The confidence aggregator page for a part is an opportunity for learners to assess the level of understanding and confidence achieved. A typical confidence aggregator page, for instance, will look at each of the terms defined so far, and ask generic questions like: how do you use this definition? how do you judge whether an object satisfies this property? The questions are typically open-ended and refer to overall skill levels rather than specific bits of knowledge.
  2. Mind's eye test page (Further information: Mind's eye test instructions (beginners)): This page contains a number of standard and routine exercises and problems related to the material covered. However, there's an important difference between the mind's eye problems, and problems found in textbooks. The goal of the mind's eye test problems is to ensure a thorough understanding of definitions and facts, and hence most of the problems are very direct applications and variations of the definitions and facts. If you understand the definitions and facts clearly, you should be able to create complete solutions in your mind's eye without using paper and pencil, in a few minutes. Nonetheless, the first time you attempt these problems, you may need paper and pencil, and you are encouraged to use them if you're unable to solve the problems mentally. There are a few slightly harder mind's eye test problems, that require a little further thought. These are marked NEEDS SOME THOUGHT and NEEDS LOT OF THOUGHT.
  3. Hard nuts page: This page contains a number of harder and more challenging problems related to the tour material. These problems may be different in the following respects: they may require a stronger problem-solving ability, they may require ideas or ways of thinking that you have not yet been exposed to, and they may require you to come up with intermediate steps that are not immediately obvious. Problems in this page should always be tried after trying out the Mind's Eye problems, because many of them assume comfort with all the ideas used to solve the Mind's Eye problems.
  4. Interdisciplinary problems page: This page is optional and can be skipped without loss of continuity. The interdisciplinary problems page for a part relates material covered in that part with ideas from other parts of mathematics. Usually a very rudimentary knowledge and understanding of those other parts of mathematics is demanded.
  5. Examples peek page: This is intended for learners who like to learn through working out more examples. This reviews material covered in the part using questions based on specific examples. Again, this section can be skipped without loss of continuity, because a minimum level of examples is already covered in the main tour.
  6. Entertainment menu page: The page lists entertainment resources, survey articles, and other useful content.

Review and revisit

As a first-time learner, you may find, upon going through the mind's eye test and confidence aggregator, that your understanding of the definitions and facts is not sufficiently strong. You are encouraged to review the part again before going to a later part. Some suggestions for reviewing:

  1. If your understanding of the material is reasonably good but you want to improve it further, try doing the Examples Peek and Interdisciplinary Problems before revisiting.
  2. Each time you go through a part again, take a more proactive role. For instance, the first time, you may passively read and absorb the definition of group. The second time, try to recall the definition of group before reading, and make sure it matches.
  3. If you've been through the mind's eye test, you may also be able to make some more sense of the part of each page after the WHAT'S MORE section. If you've gone through the Examples Peek and Interdisciplinary Problems, you'll be able to make even more sense of it.
  4. See if all the doubts and questions you had during your first round are now resolved.
  5. Follow links within the wiki to solve further doubts and satisfy your curiosities.

Get started with the tour.