Mathematical knowledge revisited: A Noah’s Ark Science Story
Noah wanted to make sure that Medina’s knowledge was in good hands, that is, minds. He asked himself, ‘Could he count on math knowledge being stored properly in the noggins of the Ark’s animals?’ Noah didn’t quite trust them when they said things like “Sure I know about numbers and how they work. That is easy. Algebra, that is a snap. Calculus, differential equations, set theory, imaginary numbers, ha, ha. Noah, no sweat. We all know math down one staircase and up another.”\
Noah asked Werner the Jumbo Blue Turtle to poke around and find out how much math the animal kefruzza knew, even simple math.
It was the German Land Lizard that told Noah that Werner was the perfect choice to test how much and how well math knowledge was known and understood by the Ark animals. He was also known to be able to teach even the densest dunderheads all sorts of mathematical knowledge. However, he did warn Noah to check on Werner regularly since he found schnaps far too delicious.
Werner knew how to ask math questions that were like mini muscle stress tests. So, on yet another rough stormy afternoon at sea, he got a bunch of the Ark’s animals to join him in the dining room. Werner loved to ask questions, especially the ones that were disguised as simple.
Werner started to talk to the animal gathered around him with this conclusion, “There is a big difference between knowing some math and knowing some math”.
The Purple Squeaker shouted, “Here we go again. Riddler time.” Werner like many of the animals on the Ark found the Squeaker to be a pain in the ass so they were pleased when Werner chose him to do his questions experiment.
Werner started with his signature gentle sneaky manner of asking questions.
“OK Squeaker. I bet you know all about simple arithmetic like how to add, subtract, multiple, and divide. You know all about numbers and how big or small they are. Right?
Squeaker was dumbfounded (curious word). “What do you think I am, stupid or something?
Werner assured him. “No, I don’t think you are an idiot but then again…best not to go there. Alright. Here is an easy question. Just, as an approximation, tell me how much is 50 time 53? Just take a wild guess and here are some choices. Is it about 5000, or 5 million, 25,000 or 500,000?”
Squeaker responded, “Give me a piece of paper and pencil and I’ll tell you the answer.”
Werner sighed. “Oh, come on. You don’t have to be exact, approximately …I gave you choices.” Werner repeated the choices as Squeaker sucked his thumb and then shouted, “got it. 500,000.” Werner looked puzzled and then said ‘nope.’
Let us try some division in your head using no pencil and paper. “How many one quarter inch licorice sticks could you cut from a long piece of licorice that is 9 feet long. Just approximately how many?”
Squeaker pleaded for some paper but Werner just shook his head. “Just take a wild guess.” Squeaker thought and then blurted out “More than a hundred.” and Werner responded, “Really?”
He also asked similar questions of some of the other animals in the dining room. Like Squeaker, they all had trouble approximating solutions to Werner’s calculation questions.
Werner understood that Squeaker, and most of the other animals, did not have a real sense of numbers, and a conceptual feel for using them in calculations. They knew the mechanics of calculation but had no real sense of the size of numbers.
Werner then took on a soothing, therapeutic tone. “Look Squeaker, you don’t have a real feel for manipulating numbers even though you can perform the mechanics of calculation. You really don’t understand the size of numbers and the comparative size of numbers. I bet you know what I mean especially if I give you the answers to the questions I just asked you.” He then told Squeaker and the others in the room, “I think I can help you. And if you don’t annoy me I will give you a sense of how you can also know all about algebra, I mean really know it and not just phony-baloney mechanical algebra for blockheads. Would you like to learn how to have a real conceptual sense of numbers and their manipulation in solving problems? And would you like to have a real intimate feel for how to think about algebra? ”
The animals sat there and seemed to nod yes. Werner wasn’t sure they meant it.
Werner went on. “And before we disappear today I will give you a preview of how you will learn algebra. I am going to ask you to put into words what you make of the graphs that I will draw here on this big piece of wrapping paper. The next time we meet I want you bring what you wrote, your descriptions of the graphs.
Werner preceded to draw a bunch of pictures of lines going in different directions in the space between what he called an x and y-axis. Pay close attention to the differences between these lines. I’ll give you a hint that may seem obvious. You can see that some lines move up to the left, others move up to the right. The lines cross the x and y-axis in different places. When we get back together we will take the words you used to describe these lines and show you a shorthand, in symbols and numbers, what you wrote about the lines. The symbols and numbers will be in the form of equations which will ‘say’ the something that you used words to describe these lines. Wait, you will be amazed at the power of that translation of words into symbols. So now let’s go to the bar and have some schnaps refreshments.