Mathematical Content Literacy Philosophy
Sam Otten
ED 321
Fall 2005
Teachers of mathematics currently maintain their title as posterchildren of transmissive learning. Math instructors are famous for the daily routine – students sitting quietly in rows, lectures given everyday, homework exercises from the book every night, and procedurally based tests at the end of every chapter.I think Jeffrey Wilhelm had them in mind when he wrote about teachers who “assign and evaluate, but do not actually teach.” Is it any surprise that mathematics is seen asboring and impersonalby so many high school graduates? I believe that teachers of the past (and some of the present) have done the field of mathematics a great disservice, and it will be my job as a future teacher to cultivate in students an appreciation and an engagement in the subject. Articulating my philosophical framework is a foundational part of that task.
Mathematics, more than anything else, is the careful and precise use of our rational capabilities for the purposes of problem solving and deductive reasoning. It is not the memorization of skills, procedures, or computations, but instead, the understanding of why certain procedures and problem solving strategies work so well in our world. Math is not the memorization of the multiplication table, it is the appreciation of how frequently multiplication occurs in everyday life. It is not the regurgitation of geometric definitions, but the realization that a “line” can represent a road, the edge of a desk, a piece of string, the path of a beam of light, or countless other real life phenomena. The former notions can be (and often are) taught transmissively, but the latter concepts can only be achieved through transactive, learning-centered instruction.
Additionally, mathematics is a discipline that should empower people by tapping into deep reserves of their mental functioning. But when the teacher stands in front of a passive audience and lectures constantly, it is only the teacher’s mental functioning that is being truly activated. Instead, a transactive approach should be implemented where students take up much of the cognitive load, student inquiry guides the specific content, and Rogoff’s community-centered learning model is present. Students can be “transformed through participation” because the teacher is no longer lecturing, but acting as the expert guide for the student investigators.
At this point it needs to be said that reading and writing are a vital component of this transactive mathematical classroom, a component that is conspicuously absent in the traditional transmissive model.As Siegel, Borasi, and Fonzi point out, language becomes vitally important in inquiry-based, constructive instruction because it provides “the symbolic resources for members of a community to negotiate meanings and representations of their world” (p. 379). Reading should no longer be simply a means for the student to receive knowledge from the outside, and writing should no longer be simply a function for displaying procedural knowledge. Language in its various forms, reading, writing, and talking, will take a more vibrant place in my future classroom.
Consider the following scene adapted from The Teaching Gap, by Stigler and Hiebert.
“Okay, class. With the last few minutes here today I’m going to give you all a copy of a hexagon. I want you to measure the interior angles and then add them up to find a total.” Mr. Jones handed out his worksheet and then paced the rows of his classroom, keeping his students on task. As he noticed a few students finishing he addressed the class once again, “Okay, what did you find? Did everyone get something around 720 degrees?” Hearing a murmur in the affirmative, Mr. Jones continued. “Good. A hexagon will always contain 720 degrees. Even if the hexagon were like this,” he draws a new shape on the board, “would it still have 720 degrees?” Another affirmative murmur. “Good. Now, there’s not time today, but tomorrow you will learn a formula. Here’s a hint - if you take the number of sides and subtract two, then multiply that number by one hundred eighty, you will get the total degrees for the figure. Like today, we had six sides. Six minus two is four, and four times one hundred eighty is 720. We’ll go over it again tomorrow and I’ll have some exercises for you to do.”
This scenario may seem painfully familiar if you attended an average mathematics class in the United States. Even though there was something resembling an activity in the hexagon hand-out, the students were told exactly what to do and they were even told exactly what they had “learned.” Rather than allowing the students to investigate a new concept through inquisition and curiosity, Mr. Jones told them the formula and virtually guaranteed that it would be rote memorization and mindless procedural practice in the lessons to come.
Here is how the same material may occur in a transactive classroom.
“I’m giving a different hexagon to each small group,” Mr. Smith announced as he passed out the manipulatives. “But before you get to work, I want each of you to write in your math journal, answering these questions: What do I know about this hexagon? What would I like to know about this hexagon?” The students were then given a few moments to individually jot down ideas. As Mr. Smith noticed a few students finishing he addressed the class once again, “Now you can go ahead and talk amongst your group about what you wrote.” As the groups were talking, Mr. Smith was able to circulate and eavesdrop on the conversations. A few groups mentioned the hexagon’s angles, which is precisely what Mr. Smith wanted to hear. He quietly asked one student to raise their point about angles in the full-class setting.
After a class discussion, the students decided that the first thing they wanted to investigate with regard to the hexagons was the measure of the interior angles. “Okay, let’s take out our protractors and get to work.” When several groups had completed their measurements, Mr. Smith informed them that they could compare their results with other groups. The rumor started to spread that different groups with different hexagons were arriving at the same result. At this point, Mr. Smith asked the entire class to give him their attention. “The number 720 seems to be popping up in all the groups. Is anyone prepared to make a conjecture at this point?” One student offered the conjecture that all hexagons have an interior angle total of 720 degrees. “Okay, mathematicians, we have an open conjecture in the room. Let’s all copy it down into our journals and get to work. Remember to look back in your journal for any information that we may be able to use. I think last week we determined something about the interior angle total of triangles.” And the class got to work.
Wouldn’t you have been more engaged in this type of math environment? You probably would have learned more, as well. First of all, student inquiry is the driving force behind the movement of the lesson, even if it is discretely and deftly manipulated by the instructor. There is a cooperative feeling as the entire classroom, including the teacher, works toward a solution. The students are playing an active role in constructing their knowledge. They generated the questions that were asked, they physically investigated the situation, and they will eventually arrive at a proof of the result. Furthermore, a culture of mathematical literacy permeates throughout the entire scene. The students are reading mathematics, they are writing mathematics, and they are participating in mathematical dialogue as the teacher guides them in discussions of not only content, but the deductive process that mathematicians espouse.
I would like my future classroom to be a part of the progressive movement that breaks the mathematical stereotype. I would like to steal a page from the reading instructor’s book, where it is written that students should construct knowledge by participating actively, thinking creatively, and engaging thoroughly. I believe that this is the only way I will be able to cultivate students who are mathematically literate.
REFERENCES
Siegel, M., Borasi, R., & Fonzi, J. (1998). Supporting students’ mathematical inquiries through reading. Journal for Research in Mathematical Education, 29, 378-413.
Stigler, J. & Hiebert, J. (1999). The Teaching Gap: best ideas from the world’s teachers. Simon and Schuster Inc., New York, NY.
Wilhelm, J., Baker, T., & Dube, J. (2001). Strategic Reading: guiding students to lifelong literacy, 6-12. Heinermann.
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