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 as boring and impersonal by 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
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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.”
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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
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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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