Hints and Suggestions


There seems to be some of the following sentiment in class:  "What do we write on the problems?  Wouldn't we just be rewriting the comments in the text?"

These are good questions.  The basic answer is 'no'.  If you wish, use that as a starting point.  Starting is going to be the most difficult part of this writing  task, because after you have begun, I can work to lead you to expand.  The final goal will a deeper exposition than is presented in the text.  Both deeper mathematically and deeper personally.  It should eventually include your personal thoughts and reflections on the concepts.  Remember your intuition is a valuable tool, not to be discarded but developed.  

For more on these topics, I recommend reading the introductory materials in the textbook, where Henderson discusses these issues.  Particularly relevant to this are the introduction to the Preface:  pp. xv - xvii, How to Use this Book (including How I Use This Book in a Course [as we are doing almost precisely what Henderson has done]): pp. xxii - xxiv, and of course Message to the Reader:  pp. xxvii - xxxii.

As I have said, many of these questions do not have 'right' answers.  And it shouldn't be thought that the text has the right answer.  The best answer is the personal answer that incorporates your personal thoughts, this is what makes things 'wonderful' on the grading scale.  Again, there are not 'right' answers.  There are 'wrong' answers (which include statements that are not true) and incomplete answers (which do not answer the question or do not elaborate beyond what was said in class or written in the text).  We have discovered that these questions are deep and deserve extensive careful consideration.  Your writeups should reflect that careful consideration.

Make sure that you read the questions carefully and address at least all parts of the question.


Of note:  these drafts are not a one-step process.  The intent is to have a continual exchange of ideas, further refining our understanding.  In other words, do not submit a draft and then wait until the due deadline to address the questions again.  Keep working on the questions and resubmit so that we may discuss them.

Again, if there are questions, please ask, and check here for more information.

Frequent Comments

Make sure you answer all of the questions that are asked in each problem.  

Avoid the unnecessary use of degrees.  Try to restate in more basic terms.  Saying that a straight line has half-turn symmetry is more intuitive than that it has 180° symmetry.  And, furthermore, saying that "a straight line has 180°" seems difficult to understand and might require more explanation than a simple discussion of the point you are trying to convey.

In fact "a straight line has 180°" is a very problematic statement.  Aside from the fact that it doesn't make much sense, it begs the question "What does a curved line have?"  Think about this and please avoid saying "a straight line has 180°".

Be careful about the use of distance.  We haven't talked much about it.  At least think carefully before using it, and remember that it is a number, and not a geometric object.

If you are thinking about the concept of "shortest distance" be careful which logical direction you are using:  
if a path is the shortest distance between two points, then it is a straight line

if a straight line connects two points, then it is the shortest path connecting them
For angles, remember that it will likely be useful to distinguish where the interior of the angle is.

Remember that in general an intuitive explanation is much preferred over an explanation with technical terms.  In fact, it is more likely to be correct, as there is a good chance of misusing technical terms while trying to impress (thus impressing far less than the intuitive understanding).