Tour:Mind's eye test instructions (beginners)

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This page gives information on the mind's eye tests for the Groupprops guided tour for beginners.

GO TO MIND'S EYE TEST PAGES: One | Two | Three | Four | Five

The idea behind mind's eye tests

Something is said to be in the mind's eye when it is completely effortless to recall in as much detail as necessary, and when it feels totally transparent. The mind's eye tests are designed as lists of problems whose solutions should be in the mind's eye of the learner after successful completion of learning. In other words, being able to recall solutions to these problems perfectly and effortlessly is a measure of how successfully learners have achieved the learning objectives.

Mind's eye tests are in many ways a departure from traditional end-of-chapter exercises or problems found in books, and the assignments given by instructors at the end of lectures. Some of these differences are discussed here.

Short and simple problems

The average mind's eye test problems are easy and can be solved quickly. More specifically:

  • An experienced person should be able to solve the problem completely in the mind in under a minute, without any use of paper and pencil, and should be able to, without much thought and without error, commit the solution on paper or explain it to others.
  • A first-time learner who has understood the material should be able to solve the problem with the aid of paper and pencil in under two minutes, and present a written solution after having solved the problem, in about five minutes. The learner should, after writing a complete solution, have the confidence to write this solution again without thinking through the problem.

For many mind's eye test problems, these bounds are upper bounds, and the actual time taken may be much less.

Some mind's eye test problems are marked NEEDS SOME THOUGHT. For these:

  • An experienced person should be able to solve the problem completely in the mind or using paper in under two minutes, and should be able to, without much thought and without error, commit the solution on paper or explain it to others.
  • A first-time learner who has understood the material should usually be able to solve the problem with the aid of paper and pencil in five to ten minutes, and present a written solution after having solved the problem, in about five minutes. The learner should, after writing a complete solution, have the confidence to write this solution again without thinking through the problem. However, the learner may get blocked while solving the problem the first time, and may need to think about it for some time in order to get the flash of insight.

Some mind's eye test problems are marked NEEDS LOT OF THOUGHT. For these:

  • An experienced person should be able to solve the problem completely in the mind or using paper in under five minutes, but might get blocked about the solution. Once solved, the person should be able to commit the solution on paper or explain it to others.
  • A first-time learner who has understood the material should usually be able to solve the problem with the aid of paper and pencil in about fifteen minutes, and present a written solution after having solved the problem, in about five minutes. The learner should, after writing a complete solution, have the confidence to write this solution again without thinking through the problem. However, the learner may get blocked while solving the problem the first time, and may not be able to solve the problem without further hints.

In all cases, once a first solution is obtained, the problem becomes easy.

Total clarity and straightforwardness

Mind's eye tests are not intended, by and large, to challenge learners or test their ingenuity and skill. The majority of mind's eye test problems are stated in a language that makes clear exactly what tools are needed to solve them. Further, efforts are made to place these problems in an order and context that makes the effort of solving them easier. Finally, adequate hints are put to prevent learners from getting stuck on problems.

The purpose of this is to allow learners to concentrate on the heart of the problem without being subjected to too many distractions. This also makes problems genuinely easy to store in the mind's eye. Some organizational features include:

  • The mind's eye test problems on a part of the tour are divided into sections and subsections. The title of the section or subsection itself makes clear what the problems within that section or subsection are about.
  • Within a section, problems are ordered in a way that later problems, if hard, can build on the ideas, techniques and results of earlier problems. There is no or little interdependence in problems between sections. In most cases, dependencies between problems within a section are explicitly pointed. out. For instance, Problem 3 in a section might read as Using the results of Problem 1 and 2, show that....
  • The action words in each problem are highlighted in bold. This makes it easy for learners to locate what they are asked to do. For instance, in problems asking for a proof, the Prove that words are in bold.

Ready link to solution and generalization

In cases where the statement of the problem is itself of some significance, there is likely to be a wiki page devoted to that problem. When such a page exists, a link to this page is provided with the problem. The wiki page is not designed as a solution page to the mind's eye test, but is designed independently according to the wiki's rules; nonetheless, a learner who gets stuck with a problem can use this page to learn the solution. This also allows learners to use a simple problem to begin a grander exploration.

In addition, generalizations to the problem are sometimes stated in parenthetical notes, and links to these generalizations may also be given.

Suggested approaches to mind's eye tests

Mind's eye tests are intended to help learners consolidate their learnings, to provide them an opportunity to evaluate their level of understanding, and to provide them an opportunity to correct and remediate a lack of understanding. This section discusses how learners can use mind's eye tests effectively.

Starting the mind's eye test

It is recommended that, before starting the mind's eye test on a part:

  • The learner go through all the pages of the part in sequence and make sure he/she understands them.
  • The learner go through the factsheet for that part, and make sure he/she is comfortable with all the facts, as well as the notation, and that all these facts seem familiar from the tour.

The mind's eye tests are best done alone, without discussion with others. The problems are streamlined for easy individual work and are not intended for group discussion. The learner should keep a paper and pencil handy.

Doing the mind's eye test

Learners are advised to go through all the problems within a subsection in sequence. It is also recommended that the learner go through the sections/subsections in the sequence in which they are presented, though this is not as necessary since the sections are relatively independent.

It is recommended that, for each subsection containing problems:

  • The learner look at the title of the subsection, and recall all the definitions/facts related to that subsection. This recall could be done using paper and pencil (a quick jotting down of the important definitions or facts) or purely at the level of the mind.
  • The learner try each and every problem, parsing the problem statement where necessary. If stuck over a problem, the learner should recall the proof techniques for similar facts seen within the tour, and go over those proofs line by line to see how they could be adapted to the given situation.
  • The learner review the relevant pages of the guided tour if he/she is getting blocked or facing difficulty with the problems.
  • For the problems marked NEEDS SOME THOUGHT and NEEDS LOT OF THOUGHT, the learner should try the problems for some time, but put a cap on the time spent trying the problems, and move on to other things if he/she is unable to solve them.
  • The learner should try writing complete solutions to at least some of the problems, specially the ones he/she had difficulty solving.

Once the mind's eye test problems are solved, learners are encouraged to follow the links to the page on the wiki giving detailed solutions and generalizations of the problem statements.

Reviewing the mind's eye test

For learners who have difficulty with the mind's eye test during the first round, a review round is recommended. This review round includes a quick whiz through that part of the guided tour, followed by another round of the mind's eye test. During the second round, the learner should be able to recall most solutions without much thought, or at least have a clear idea of where to begin each exploration.

Use of mind's eye test for guided learning/instruction

Since the mind's eye test consists largely of easy problems, the entire set of problems in the mind's eye test is not suitable for a written assignment. The following is suggested/recommended:

  • Instructors pick a combination of the problems marked NEEDS SOME THOUGHT with NEEDS LOT OF THOUGHT, as well as some problems they personally consider important, and assign these for submission in writing.
  • Instructors urge students to go through and achieve mastery of all the other problems, without having to write down the solutions. In a tutorial session, students should be willing to present the solution to any of these problems at a moment's notice, to the instructor and to other students.

The focus thus shifts from the formality of writing solutions to the need to master key ideas for a much larger class of (simple) problems.