Day 02, MAT115

• Announcements:
  • Reminder: turn off your rectangles!
  • You have your first assignment, to turn in next Thursday.
  • If you wrote up a "solution" for the card trick, the please hand that in now. (We'll discuss your ideas later....)

• Today we're going to talk about primitive counting:
  How would primitive, numerically illiterate people have counted?

• If you want to talk primitive, how about muppets?! Where do kids learn to count, but Sesame Street!

  I asked you to do the reading "From Fish to Infinity" for today, in which we learn the answer to one of life's persistent questions:

  What is "six"?

  We learn something of that, anyway -- at least about six fish. But

  All numbers are interesting, but what about six makes it special?

  "Six is more ethereal than six fish.... It applies to six of anything..., the ineffable thing they all have in common." (Steven Strogatz, author of From Fish to Infinity)

  We learn that

  • "...numbers are shortcuts for counting by ones...."
  • Furthermore, "...Humphrey might realize he can keep counting forever." -- something that will be very important when it comes to talking about infinity.

  Did you notice Humphrey using any other cues to help him keep track of the number of fish?

  We encounter three important things in this example:

  1. To our minds, counting is the act of putting objects in one-to-one correspondence with the "counting numbers" (we'll call them the natural numbers -- 1, 2, 3, 4, 5, ....).

     The order is very important, of course!

  2. "Blocking" is important -- fish, fish, fish
  3. Prime decomposition

• Let's dive into today's new topic: How might "primitive" people have counted? (We'll use a mathematical concept called a "tree" (a type of graph) to help us. More on those later....)

  There are at least three good suggestions that I know of:

  • Tallies

  • One-to-one correspondence
    • using body parts -- "one hand" of sheep, say -- meaning five sheep.
    • cairns

  • And then an unusual method of "counting by partitions" that Patricia Baggett and Andrzej Ehrenfeucht described at the 2011 National Math Meetings (their work is derived from a 1827 book by John Leslie -- The Philosophy of Arithmetic -- in which Prof. Leslie describes how "savages" may have developed arithmetic.)
    • Let's begin by counting some students their way....

    • They proposed that primitive societies may have counted anything -- students, spark plugs -- well, maybe not -- this way. Let's suppose you need to let the King know how many sheep you have. Here's the process, in words:
      1. divide your sheep equally ("one for you, one for me") into two pens: either there is one left over, or not. You make a note of whether there is one left over or not.
      2. Send all the sheep in pen two (and any "left over") out to pasture, and then
      3. You divide the sheep in pen one into pens one and two: i.e., just do it again! And again, and again, and.... until you get down to a pen one with just sheep in it.
      4. Now let's see how we might record the results to send to the King.

         The easiest way to illustrate the counting method is via a tree, which is a mathematical object in its own right.

         Let's see how we might use a tree to represent the solution to the "22" counting problem: in the linked example, we would get 10110 by writing the remainders from left-to-right starting from the bottom of the tree. (The result should always start with a 1 if done correctly, since we always end with one sheep!)

         

         The answer will be written as 1, 0, 1, 1, 0

         That is, from the bottom up, left to right. This is important! We have to have a consistent scheme for writing.

         So how do we write

         • 9 sheep
         • 31 sheep
         • 54 sheep

         ---

         Ice Cream Card trick interlude....

         Before our interlude, however, I want to talk about our goals for this course, which are much more than just mathematics -- there are "Lessons for life" that we should focus on:

         1. Keep an open mind.
         2. Just do it. Jump in. Make mistakes and fail, but never give up.
         3. Often when we've done it once, we do it again. Follow up one good deed with another! Many times in this course you'll hear me say "Now do it again" -- you should have heard me say it already!
         4. Understand simple things deeply.
         5. Break a difficult problem into easier ones.
         6. Look for patterns and similarities.
         7. Explore the consequences of new ideas. Generalize.
         8. Examine issues from several points of view.
            Don't be a turkey -- be a dog! (this one may require a little explanation....)

         How did that trick work? What is the key? And how can we use some of those ideas above to figure this out?

         ---

      5. Now: can you go backwards? How many sheep is meant by
         1. 1,1,1,1,0,1
         2. 1,1,0,1,1,0,0
         3. 1,0,0,1,0,1,1,0,1

         Try making a tree with these remainders. Notice that each way of writing has a leading 1 on the left. These representations always do.

      6. and now let's count some coins:
         1. We'll be counting pennies - in pairs.
         2. Split your penny rolls into two parts (no need to do it evenly!), and each of you use this method.
         3. Then try to deduce the number of pennies your partner counted.
         4. Compare (and record) your answers.

fish fish fish
fish fish fish