This week, we dived into Activity0102, which introduces two huge concepts.

The first is that there are different types of knowledge and therefore also different types of questions. Students are presented with a group of questions, let’s call them Group 1 Questions, and are asked to decide what they have in common. I’ve used this activity for so long that I can perfectly predict what they will say about this, and it is that these questions all have definite answers that are all fact-based. Unfortunately, this is not what I intend for students to land on. I won’t say here what I want them to land on, but it’s something else. Students are then presented with another group of questions, let’s call them Group 2 Questions, and are again asked to decide what these questions have in common. Once again, I can predict exactly down to the word what students will say about these questions. They always say that these questions have no definite answers and the answers they do have are all matter of personal opinion.

For theatrics, I set this up at a mind reading stunt. While I’m describing this part of the activity to the class, I tell them that I can predict what they will say about each group of questions with astonishing accuracy. I write my predictions on a sheet of paper, fold it, and seal it with tape and hand it to “the most honest looking student in the class” for safe keeping until I ask that student to open it and read it to the class. They then do their thing with the two groups of questions, and then I have them come to the room’s center for a checkpoint to discuss their decisions. After they go precisely where I predicted they would go (but not where I wanted them to go), I reveal my predictions.

I describe how my accurate predictions are troubling for me because it indicates to me that questions requiring more than regurgitation of facts are considered questions for which there is no objectively correct answer (don’t worry…I didn’t reveal anything there). This leads to one of the first really deep conversations of the semester.

After this semester, I will restructure this activity to make it more streamlined and to put the parts about what each list of questions has in common adjacent to each other, separate from the questions about tracing one’s knowledge back to…well…back to something (can’t reveal that here of course).

The second huge concept in this activity is that of **operational definition**. I was introduced to operational definitions by Arnold Arons and use them extensively in this course. Basically, an operational definition of a sequential list of step one must carry out in order to define a concept or a measurement. I use the obligatory silly example of the steps one must go through in order to define the concept of “peanut butter sandwich.” It’s astonishing how many people forget to open the peanut butter jar before putting peanut butter on the bread! I give each student a set of four, five, or six wooden sticks. I usually use barbeque skewers from a local grocery store, but toothpicks work just as well. The important thing is that each student’s sticks must be essentially identical. The activity then asks for students to formulate operational definitions for the concepts of “parallel” and “perpendicular.” I chose these concepts becuase students think they are already familiar with them, but quickly find there is more to these concepts than they realized (which is one of my goals of course). We will use geometry later in describing the behavior of shadows and in the geometry of the sky so students will definitely see these ideas again and again in the course. If I sense students are struggling with this, and they always do, I interrupt and offer a slightly more familar example. I give each table (there are six in my classroom) a cup and a handful of glass beads. I ask them to formulate an operational definition for “adding two small positive integers each of which is less then ten.” The use of small integers less than ten is entirely arbitrary on my part, but I don’t want to deal with fractions, large numbers, or negative numbers.

It’s very interesting to watch students realize that they’ve probably never really thought deeply about simple arithmetic (it’s not so simple) but eventually they land on what I need them to land on. If time permits and they need more practice, we do similar operational definitions of subtraction, multiplication, and division. It’s very eye opening for both students and me. As a bonus, they see how these operations are related to each other and how they really work. Of course, they ultimate goal is to construct good operational definitions.

After practicing with operational definitions of basic arithmetic operations, we then move to the actual activity in which students must formulate operational definitions of “parallel” and “perpendicular” using the sticks I gave them. The instructions list words students are not supposed to use (mainly because my experience is that these words represent partially understood concepts in most cases). For theatrics, I move around the room attemptning to break the operational definitions students formulate. They make it a competition to see whose operational definition can survive “The Joe.” It’s fun!

Incidentally, one interesting aspect of operational definitions must be culturally independent. For example, if a students write a step in his/her operational definition calling for the reader to “make a letter H” this won’t work becuase in, for example, Japanese there is no H. Therefore, the operational definition must not rely on something not shared by all cultures.

Given that most students have simply memorized trivial definitions of “parallel” and “perpendicular” and that when they try to use these previously memorized definitions operationally, they find that I can always break them. Therefore one of my goals is to get them to abandon those old, mostly meaningless definitions and find new ones. As a hint, I tell them that Nature sometimes uses geometry to talk to us. Most then go on to where I need to be and are happy about it.

I welcome questions, comments, feedback, and constructive criticism.