How is mathematical knowledge formed?

 You need to find an article that contains one of the concepts below and be ready to explain it to the class.  Wait for a number.  Concepts:  Correlation  “studies show”  Poll  Standardized/standardization  Obviously biased article that includes some kind of statistical data – chart, graph, %, etc.

 After having read “A Father and Son ‘Discuss’ Math,” have you ever felt like this about math? Do you ‘like’ math?  To what extent do you think people’s beliefs about the value of mathematics are determined by their ability in the subject?  What counts as understanding in mathematics? Is it sufficient to get the right answer to a mathematical problem to say that one understands the relevant mathematics?

Tennis tournament A tennis club arranges a knockout tournament in which the winner takes it all. There are 843 participants. Round Matches No. 1 st In the first round of the tournament, the 843 players are matched; one odd player is left out in this round. This means that 842/2=421 matches are played nd In the second round, the 421 winners plus the odd player are matched. Thus, 422/2=211 matches are played rd In the third round, the 211 winners from the second round are matched; again one odd player must be left out. 105   and so on   Last In the last round, only two winners are left so only one match is played – the final! 1 Problem: How many matches were played before the final winner was found? (A relevant question if you have to pay the ballboys!) Sum ?

Lesson 2 5 Niels Østergår d, Birkerød Gymnasi um, May 2006 Better solution At the end of the tournament, all players except the final winner will have lost exactly one match. In each match there is exactly one looser. Therefore, the number of matches equals the number of players excluding the final winner. Suppose the tournament has n participants. Then, n-1 matches will be played before the winner is found!

 What QUALITIES make the second solution better than the first?  Let’s pause for a moment to watch a short video.video  What’s the issue with Ma & Pa Kettle’s mathematical problem solving?  Is there a clear-cut distinction between being good or bad at mathematics?

 What did you get from this reading?  Do you believe that basic math is inherent?  What are the greater implications of what this article is saying?  How have we come up with such a complex system that goes so far beyond our ‘natural’ abilities?

 Is there a correlation between mathematical ability and intelligence?  How would you account for the following features that seem to belong particularly to mathematics: some people learn it very easily and outperform their peers by years; some people find it almost impossible to learn, however hard they try; most outstanding mathematicians supposedly achieve their best work before they reach the age of 30?  Did you gain any insight to these questions from “What are Numbers, Really?” article?

 Take a moment and write down the equation I am about to read to you.  Did you all write down the exact same thing?  What does this tell you?  What did you learn from reading “What Happens When You Can’t Count Past Four?”  Can mathematics be characterized as a universal language?  To what extent is mathematics a product of human social interaction?

 Why is it that mathematics is considered to be of different value in different cultures?  How have technological innovations, such as developments in computing, affected the nature and practice of mathematics?  What use will you have for pure mathematics in your lifetime?  Why do we teach such “impractical” stuff in school? Is pure math really important?

 Answer the following question, making sure to use as many of the things we have learned this year as possible and including relevance to your life – try to weave your personal experiences around the argument.  “Mathematics may be defined as the subject in which we never know what we are talking about, nor whether what we are saying is true.” Bertrand Russell (1917)  To what extent do you agree with this quote?  This will be due on Monday, 9/28/09.

 Read the following:  “The Numbers of Life”  “Is mathematics the language of the universe?”  Again, REALLY read!