- Being comfortable with the material
- Pacing material appropriately
- Knowing material, having a plan
- Knowing what questions to ask
- anticipating questions.
At the last big group meeting, some of these skills were compiled and there was a collective shift in teaching priorities.
The black asterisks denote markers from the first meeting;
the green tallies were generated at the mid-quarter meeting.
Specifically, the following skills were given the most votes for "most important."
- Getting them on track, keeping people focused
- Pacing material appropriately / Knowing material, having a plan
- Knowing what questions to ask / anticipating questions.
An undergrad noted that comfort is not equivalent to competence; another undergraduate responded that "if you never struggle with the material, you may not be able to think about how to explain it." This is related to an observation that Deborah Ball develops in her work on mathematical understandings(#): even when preservice teachers feel comfortable with the material, they may not actually be competent at it. Furthermore, while teachers who feel uncomfortable with mathematics risk having students who also fear mathematics, teachers who are falsely comfortable risk having their confidence prompt students to imprint on false ideas about math. The primary risk that Ball describes is a satisfaction with the explanation akin to "that's just the way it is"--uttered with sufficient confidence.
During the second half of the meeting, we sat in a circle and took on roles to narrate two dialogues from Cai-Lester's work on solution representations (@). The first dialogue was a translated transcript taken from a class taught by a Mr. Liu.
Mr. Liu: (Mr. Liu drew two figures on the blackboard. The top one was a rectangle and the bottom one a square. He pointed to the blackboard and asked his students). What is the figure on the top called?
S1: A rectangle.
Mr. Liu: What about the bottom one?
S2: A square.
Mr. Liu: What is the perimeter of a rectangle? What is the perimeter of a square? Who is willing to come to the front
and explain to us? (He saw many hands and pointed to S3). O.K., S3?
S3: (Pointed at the rectangle) The perimeter of a rectangle is the surrounding edges.
Mr. Liu: Please be more precise.
S3: The perimeter of a rectangle is the length of the four sides.
Mr. Liu: That’s right. What about a square?
S4: The perimeter of a square is also the length of the four sides.
Mr. Liu: So, how is the perimeter of a rectangle related to the length and width of the rectangle?
S5: The perimeter of a rectangle is two times the sum of the length and width of the rectangle.
Mr. Liu: That means that we can represent the perimeter of a rectangle using two times the sum of the length and
width. What about a square?
S6: The perimeter of a square is four times the length of a side.
Mr. Liu: So how can we represent the relationship between the perimeter of a square and its sides?
S7: Just multiple a side by 4.
Mr. Liu: How can we use letters to represent the perimeter of a rectangle?
S8: The sum of the length and the width multiplied by 2.
Mr. Liu: Think about it. How can we use letters to represent this?
S1: Equals to the quantity a plus b times 2.
Mr. Liu: What about the formula for a square?
S2: 4a.
Mr. Liu: If the length of a rectangle is 4 cm and the width is 3 cm. What is the perimeter?
S4: The perimeter of the rectangle is 14 cm.
Mr. Liu: What is it? Provide the process.
S4: Equals to the quantity 3 plus 4 times 2.
Mr. Liu: Can we say the quantity 3 plus 4 times 2? What is the result?
Group: 14.
Mr. Liu: What about the perimeter of a square with a side equal to 3 cm?
S8: 4 × 3 = 12 cm.
The undergraduates remarked that Mr. Liu seemed to have a constant stream of questions, and that he broke down the process of finding the perimeter to a great degree.
We then narrated the second dialogue, transcripted from the (English-speaking) classroom of one Mr. M.
Mr. M: What is the circumference of a circle?
S1: Isn’t that, like, kinda beginning going around?
Mr. M: Ok, the distance around. So if you circumvent your mother’s rules about being out late, what does that mean?
S2: Like, to bend a rule.
Mr. M: I could bend them or break them if I don’t follow them, I go around them. So the circumference of the circle is the distance around.
The undergraduates noted that the students here may have sounded less put-together than the ones in the first classroom because their words were not translated. One of the undergraduates touched on the issues of learning styles: she felt that Mr. M's explanation would appeal more to visual learners rather than linear learners. The undergraduates then discussed how Mr. M's explanation, while arguably not executed well, had the good intention of explaining the "big picture" and "intuition"; in contrast, Mr. Liu's excerpted dialogue did not touch on holistic aspects of the material.
As the final portion of the meeting, we discussed strategies for working with a hypothetical high school student we called "S1". This student throws papers at his fellow students and acts distracted during class.
The strategies that the undergraduates brainstormed.
The undergraduates brainstormed strategies that addressed S1 individually as well as ones that changed the structure of the class as a whole. An undergraduate suggested "using Mr. Liu's strategy of asking lots of questions" to S1 and his table, reasoning that engaged students are less likely distract each other. It was noted that we could adopt a "divide and conquer" strategy among the different tables of high school students because of the exceptionally high staff-student ratio, which at Math Circle varies from 1-3 to 1-7.
(#)Deborah Loewenberg Ball. The Mathematical Understandings That Prospective Teachers Bring to Teacher Education. The Elementary School Journal, Vol. 90, No. 4. (Mar., 1990), pp. 449-466.
(@)Jinfa Cai, Frank K. Lester Jr. Solution representations and pedagogical representations in Chinese and U.S. classrooms. Journal of Mathematical Behavior 24 (2005) 221-237.
