Building Representations of Children's Meanings

Maher, C. (1990). Building representations of children's meanings. Journal for research in mathematics education, 6, 79-90.

Summary:

This chapter walks you through a lesson in a fifth grade math classroom where the teacher is including problem solving group work on a regular basis in her classroom. Many of the tasks she gives her students encourage them to make models of their solutions.

Much of the chapter focuses on two students discussing possible solutions to the following problem: "At Pizza Hut each large pizza is cut into 12 slices. Mrs. Elson ordered two large pizzas. Seven students from Mrs. Elson's class are to eat one piece from each of the pizzas. What fraction of the two pizzas was eaten?" The two boys in this scenario are at odds because one chooses to solve the problem using models and one uses paper and pencil computations. They can not seem to understand each other's interpretations of the problem. To further complicate the situation, the teacher misinterprets one of the boy, Brian's thinking on the problem even though he was correct. In the teacher's mind, one pizza was "the whole". In the Brian's mind, both pizzas formed "the whole" together. When Brian answered that the class would eat 14/24 of the pizza, the teacher viewed that as him incorrectly adding the denominators instead of viewing the problem a different way. The teacher led the boys through a discussion to get them to come around to her way of thinking instead of attempting to understand where they were coming from. The boys eventually changed their answer, but were still unclear as to why their original answer was "wrong".

A year later, the boys were given the same problem. Both of them remarked that they were familiar with the problem, but didn't remember how they solved it. When Brian solved the problem this time, he still got the answer 14/24 indicating that the teacher's reasoning had not stuck with him. This is because her answer had no meaning to him and it did not make sense. The chapter asserts that, "a more promising approach is for the teacher to try to understand what the students were doing and why, and then to provide them with an opportunity to see their own faulty reasoning."

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Quotes:
"...one of the tasks that a teacher faces is to construct in her or his own mind a mental representation that matches the student's mental representation." -page 82

"A teacher's failure to recognize the way a student is thinking about a problem can at the very least end up wasting time in mutual misunderstanding." -page 90

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White Privilege: Unpacking the Invisible Knapsack

McIntosh, P.  (1990)  White privilege:  Unpacking the invisible knapsack.  Independent School, 49(2), 31-35.

Summary:

This article takes a perspective on racial disadvantage as a "white privilege".  It discusses how people in the majority group are willing to accept that certain races, genders, sexual orientations, etc, are socially disadvantaged in a way.  These same individuals are hesitant though to except the reflective statement, that majority groups receive unearned privileges due to their race, gender, etc.  In the article, McIntosh lists several “privileges “ that she has received by merely being white. Some jump out as something other than “privileges”, and more as standards that we should be striving towards as a society. An example is, “I can go shopping alone most of the time, pretty well assured that I will not be followed or harassed.“ Through white-privilege we have acquired these standards more easily and do not need to give anything up to spread the wealth. McIntosh describes these as “positive advantages”, but then there are “negative advantages” that instill a structure of power and is holding down a disadvantaged group.  She also differentiates between "earned strength" and "unearned power".  Unearned power through privilege can look like strength when it is in fact a form of oppression.  The ultimate message from this piece is that not all types of oppression are clearly visible.  In order to make true change in this oppression, we must accept these unseen factors and tackle them head on.

Quotes:

(1) “I have often noticed men's unwillingness to grant that they are overprivileged, even though they may grant that women are disadvantaged. They may say they will work to improve women's status, in the society, the university, or the curriculum, but they can't or won't support the idea of lessening men's. Denials that amount to taboos surround the subject of advantages that men gain from women's disadvantages. These denials protect male privilege from being fully acknowledged, lessened, or ended.” (McIntosh, 1990)

(2) “One factor seems clear about all of the interlocking oppressions. They take both active forms, which we can see, and embedded forms, which as a member of the dominant group one is taught not to see. In my class and place, I did not see myself as a racist because I was taught to recognize racism only in individual acts of meanness by members of my group, never in invisible systems conferring unsought racial dominance on my group from birth.” (McIntosh, 1990)