Planning our ToK essays

As we take a moment to look ahead at the quarter, we need to consider the best prompt for our ToK essay that will be submitted later in February.  Here are some things (and dates) to consider:

  • Students will select a title(prompt) to develop on January 26th and get started
  • Students will consider and craft the Knowledge Questions pertinent to their title
  • Students will begin to draft an outline of their essay on January 27th.
    • By the end of class on January 27th, student outlines will include:
    • The title inserted into the top center of the word document
    • Below the title, the essay’s Thesis (response to the title)
    • A list of the knowledge questions that will be addressed by the essay
  • Students will bring a paper copy of the initial draft of their essay to class on Jan. 28.  Students will be meeting in groups to discuss the possible issues with their titles and to consider approaches to the title they have chosen.
  • A thorough draft of their outline will be submitted in paper form during class on January 30th.

During the week of February 2nd, we will be discussing Thomas Kuhn’s The Structure of Scientific Revolutions, so, if you would prefer to take notes in your own book, you should purchase a copy and bring it to class on Monday, February 2nd.  If you don’t, you should check out the text from the Skyline library during the last week of January.  We will begin reading the text in class on Monday.

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Final Exam creation

Today we are working to review for the final… and to help craft a test that will assess whether or not students have mastered the most important material this semester.  During class, you should link to this Google Doc (Thanks, Nikki) and craft questions that:

  1. assess essential content from Semester 1 (i.e. Gestalt)
  2. link ideas to each main philosopher (i.e. Plato – Forms)
  3. compare ideas (a priori knowledge is different from Forms because…)
  4. determine whether or not students can identify central ideas in an excerpt

The questions below would be decent questions.  Questions that are plausible are most likely to become part of the final.  Please notice the details in the formatting so I can build the final with as few hiccups as possible.

Which author considered ideas to be the things most real in the universe?

A. Euclid

B. Mandlebrot

C. Da Vinci

D. Plato

E. Icarus

Which of the following philosophers crafted the idea “I think, therefore I am.”

A. Max Cohen

B. Immanuel Kant

C. René Descartes

D. David Hume

E. Plato

As we prepare to view the film Pi…

It is important to explore some of the ideas contained in the film.  Today, during class, try some of these exercises introducing Number Theory and explore some exercises designed to introduce Number Theory to you.  Show your discussion ideas and your solutions to these exercises in your journal.

Number Theory         

Number Theory is partly experimental and partly theoretical. The experimental part normally comes first; it leads to questions and suggests ways to answer them. The theoretical part follows; in this part one tries to devise an argument that gives a conclusive answer to the questions. In summary, here are the steps to follow:

  1. Accumulate data, usually numerical, but sometimes more abstract in nature.
  2. Examine the data and try to find patterns and relationships.
  3. Formulate conjectures (i.e., guesses) that explain the patterns and relation- ships. These are frequently given by formulas.
  4. Test your conjectures by collecting additional data and checking whether the new information fits your conjectures.
  5. Devise an argument (i.e., a proof) that your conjectures are correct.

All five steps are important in number theory and in mathematics. More generally, the scientific method always involves at least the first four steps. Be wary of any purported “scientist” who claims to have “proved” something using only the first three. Given any collection of data, it’s generally not too difficult to devise numerous explanations. The true test of a scientific theory is its ability to predict the outcome of experiments that have not yet taken place. In other words, a scientific theory only becomes plausible when it has been tested against new data. This is true of all real science. In mathematics one requires the further step of a proof, that is, a logical sequence of assertions, starting from known facts and ending at the desired statement.

Exercises

1.1. The first two numbers that are both squares and triangles are 1 and 36. Find the next one and, if possible, the one after that. Can you figure out an efficient way to find triangular–square numbers? Do you think that there are infinitely many?

1.2. Try adding up the first few odd numbers and see if the numbers you get satisfy some sort of pattern. Once you find the pattern, express it as a formula. Give a geometric verification that your formula is correct.

1.3. The consecutive odd numbers 3, 5, and 7 are all primes. Are there infinitely many such “prime triplets”? That is, are there infinitely many prime numbers p such that p + 2 and p + 4 are also primes?

1.4 It is generally believed that infinitely many primes have the form , although no one knows for sure.

  1. a) Do you think that there are infinitely many primes of the form Nsquared – 1?
  2. b) Do you think that there are infinitely many primes of the form Nsquared -2?
  3. c) How about of the form Nsquared -3? How about Nsquared – 4?
  4. d) Which values of a do you think give infinitely many primes of the form Nsquared – a?

1.5. The following two lines indicate another way to derive the formula for the sum of the first n integers by rearranging the terms in the sum. Fill in the details.

1 + 2 + 3 + · · · + n = (1 + n) + (2 + (n − 1)) + (3 + (n − 2)) + · · · = (1+n)+(1+n)+(1+n)+··· .

How many copies of n + 1 are in there in the second line? You may need to consider the cases of odd n and even n separately. If that’s not clear, first try writing it out explicitly for n = 6 and n = 7.

1.6. For each of the following statements, fill in the blank with an easy-to-check crite- rion:

(a) M is a triangular number if and only if is an odd square.

(b) N is an odd square if and only if is a triangular number.

(c) Prove that your criteria in (a) and (b) are correct.

Euclidean postscript

After our Socratic Seminar on Euclid’s Postulates, we read this essay by Doug Muder – a selection from his larger text “The Unreasonable Influence of Geometry.”  The essay asserts an argument about the impact of Euclid’s ideas – that the impact went far beyond the field of mathematics.  In class, students were asked to read, discuss, and debate his essay.  As a culminating step, students were asked to write a cogent paragraph evaluating his assertions.  The author asserts that Euclid, in part, made “self-evident truths” the goal for all thinkers.  In doing so, it minimized the role of experience in knowing what is true.  Do you think this is true?  Is this definition/assumption/theorem structure the best way to know?  Write a paragraph to explain your point.  Use examples to support your argument.

Exploring the Tower of Hanoi

Today, we are exploring the nature of Math – In IB/ToK terminology, we’re considering the mathematical methods as well as the scope and applications of mathematical patterns. Today, you are going to do an individualized search for knowledge. For context, we’re using the mathematical scope of the solutions for the Tower of Hanoi.  Before we address the challenge, we first have to recognize the relationship between the number of disks and the fewest number of moves to solve the puzzle.  How could this relationship be explained mathematically?  Please discuss with others and then write the expression.  Have “n” represent the number of disks.

Here are some possibilities for your exploration.  As a way to tackle the challenge, consider first the solutions for the Tower of Hanoi.  How can there be multiple solutions?  Why are there so many solutions to the same problem?  What is the conceptual mathematical difference between an ‘answer’ and a ‘solution?’ Are all of these solutions equal?  Are they different?  How?  What makes a solution “best?”   Is a solution always true?

Since there are multiple solutions to the Tower of Hanoi puzzle, what does this multiplicity suggest about the nature of mathematics?  How is truth determined in mathematics?

Consider these graphical representations of the solutions.  What does this particular pattern suggest about the nature of this task?  Why does this shape emerge?  What factors have influenced the evolution of this shape?  What would the ‘solution’ look like for 100 discs?  What is a Sierpinski gasket?  What are its mathematical relatives?  How/when (if ever) are they found in nature?  Here’s a possibility – Conway’s game of life.  Read about it here.  Then, give it a shot.  What do you think about it as a model of the natural world?

How does the Tower of Hanoi help us know?  Consider the use of the Tower of London as a way to determine whether or not someone has limited executive function?  How does executive function (or lack thereof) influence learners?