Normative Practices: Expectations for how students and teachers act or respond to particular situations. They form the basis for the way tasks and tools are used for learning and they govern the interactions around the task. The norms can be manifest through overt expectations or through subtle messages that permeate the classroom environment. Norms exist in a classroom even when the teacher has not taken explicit steps to establish them.


For me, the normative practices section of the lesson proved to be the most difficult because there are already norms set up in Ms. H’s class as well as throughout Lea. Before beginning the lesson, I reminded students that we would be working as a group and asked them for their input for the best way to do that. In this conversation, we decided it would be important to be respectful and to listen and respond to everyone’s ideas. The goal of making the lesson discussion based was to remove me from being the main authority in teaching the lesson. Instead, students were able to explain their own thinking and make sense of the material on their own. Because this is also stressed in Ms. H’s class, for the most part, students were able to respectfully discuss their ideas.

After attempting the array method, K used the standard algorithm to solve to problem.  However, after seeing that her two answers were different, she did not revise either method.
After attempting the array method, K used the standard algorithm to solve to problem. However, after seeing that her two answers were different, she did not revise either method.

However, while I thought to discuss norms for cooperative, discussion-based learning, I did not think to discuss norms for solving math problems. While I wanted to mainly focus on the process of the array method in solving multiplication problems, many of the students just wanted to get the answer, and some were even reluctant to explore an additional method for solving multiplication problems. Because the culture of mathematics at Lea, as well as at many other schools, is to get the right answer, this was the students’ first priority. For example, at the beginning of the lesson when I asked students what type of problem my story represented, many of them instead gave me the answer to the question it asked.

Although C set up the array to represent the problem, he used the standard algorithm to find the solution.
Although C set up the array to represent the problem, he used the standard algorithm to find the solution.

Furthermore, many of the students have become comfortable using the standard or lattice method to solve multiplication, even if they do not have a “relational understanding” of it. As Fosnot and Dolk (2001) explain, there is “strong evidence that the algorithm actually works against the development of children’s understanding of place value and of number sense.” As illustrated by my group, when given a multiplication problem, many students immediately solved it using the standard method but did not see how that method was related to the array method or why it made sense in general. Although it is hard to change a norm with a single lesson, I should have emphasized that I was more interested in process than product at the beginning of the lesson.

 

References:
Class notes.
Fosnot, C. T. & Dolk, M. (2001). Algorithms Versus Number Sense. In Young Mathematicians at Work: Constructing Number Sense, Addition, and Subtraction. Portsmouth, ME: Heinemann.
Skemp, R.R. (1976). Relational Understanding and Instrumental Understanding. Mathematics Teaching, 77, 20–26.

 

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