Proofs and Logic

This assignment is due Thursday, September 12, at the start of class.

Assignment.  Choose ONE of the following two topics.  Write a reply to this post, responding to the topic.  Begin by telling us which topic you chose. (1-2 paragraphs).  Be sure to include your name in the post, and which section you are in (so I can give you credit).

Topics. There is a common experience that  many people describe having at various times when they are learning mathematics.  It is the experience of suddenly understanding a concept for the first time – something that might have seemed confusing, or hard, or just didn’t make sense,  all of a sudden becomes clear.  This can happen in class, because of something a teacher says, or because a friend explained it to you in a new way.  Sometimes it happens on your own, because you were thinking hard about a particular problem.  Afterwards,  the concept might even seem obvious, or simple – for some people it might even be hard to remember exactly why you found it so confusing.

1. Choose an experience you had in which you suddenly understood a math concept (it could be any kind of math, from elementary school up through college).  Describe what happened.  What was the mathematical concept?  Who was involved? How did it happen?  Finally, do you think you could explain it to others in a way that they could have the same flash of understanding?
2. If you have never had an experience like this, then talk about your own experiences in learning math.  Choose one mathematical concept that you feel you know well – how did you learn it?  What did your teachers/friends/parents/tutors/yourself/etc. do that helped you to learn it?  What was the most important factor that contributed to your understanding?

Extra Credit.  For extra credit, write a response to one of your classmates’ comments.  Do you feel the same, or different?  Did you learn anything?  Did you get any ideas about teaching, or about learning?

Why are we doing this, anyway?  Having progressed this far in your school career, you are familiar with many of the tools for learning math:  studying, practicing by doing problems, asking questions when you need help, and so on.  I’d like to talk about two activities that may NOT seem related to learning math — but research shows that engaging in these activities can dramatically increase the amount that you learn, and change the way you learn it.  The first is writing — something not typically associated with mathematics.  When you express your ideas in words, it forces you to think them through very carefully, detail by detail.  A great way to check and see if you really understand something is to try to explain it to someone else, either out loud or in writing.  Example: if you know how to add fractions, try teaching it someone who doesn’t know how.  The second is called metacognition, or “thinking about thinking.”  This happens when you think about what was going on in your head while you were working on a problem or trying to learn a new idea.  What train of thought did you follow?  Where did you get stuck, and what did you do next?  What were you feeling at the time? and so on.  Combining writing and metacognition can be a tremendously powerful tool in identifying the ways we learn best and the ways we make mistakes, and learning to improve.  However, like any skill, it takes practice.  That’s why we’re getting started by writing a little about our past experiences with learning mathematics.