Time to Develop Understanding

                                     Time to Develop Understanding

Exerpts from “Beginning the Conversation: Time to Develop Understanding”, Harmon, Shannon, and Angela Barlow, Dimensions in Mathematics, Spring 2012, 32(1), 4-9

“As states move forward in adopting the Common Core State Standards in Mathematics (CCSSM), we must support students in gaining both conceptual understanding and procedural skills.” ”Teachers have recognized that in order to properly implement CCSSM, their students need time to develop understanding.” ”The purpose of this article is to begin the conversation regarding how time is a key component to supporting students’ understanding.”

This is the story of Demitri, a third grader, who is expected to understand two fractions as equivalent (equal) if they are the same size, (CCSSI, 2010, p. 24)  As you read the story, consider the element of time.

At the conclusion of the third day of representing fractions as parts of wholes, manipulatives were displayed to the class to view.

Demitri: Are those equal?

Teacher: What do you mean, Demetri? 

Demetri: Well, Ansley said that this one’s 2/6 and Derrick said the other ones 1/3 but they look the same to me.

Teacher: Who thinks they understand what Demetri means when he says they look the same?

(pause) Meredith?

Meredith: I think that he means that the two green blocks and the blue block are the same size.

Teacher: Interesting. Let’s all take a moment to talk with our partner about Demitri’s idea.

Are  2/6 and  1/3 the same?

“This exchange allowed Demetri and his classmates to begin thinking about the concept of equivalent fractions. Over the next several days, students continued to explore this concept. Through the explorations, Demetri deepened his understanding of equivalent fractions and ultimately began moving towards defining the concept.

Demetri: I don’t think that just ‘cause the blocks are the same size that it means the fractions are the same.

Karl: What?

Demetri: Look, referring to the pattern blocks. These are the ones we had the other day but these fractions ain’t equal.

Karl: Yes, they are. They’re the same size.

Demetri: No, the blocks are the same size size but they ain’t the same amount.  See, the green blocks are over half of the red block, but the blue block is less than half of the yellow block. Those fractions ain’t equal.

Following this discussion, Demitri shared his reasoning with the class.  The next day, the teacher began the lesson by having the students record in their math journals what they thought it meant for two fractions to be equal.  Demetri wrote, “I think that equal fractions means that two fractions are the same amount and they have to come from the same size hole.

While it is clear that Demetri is developing a deep understanding of equivalent fractions, the article wants to focus on the time it took to support this level of understanding.

First, several times, Demetri was given the opportunity to think about the mathematics and share out his thoughts.  Secondly, by allowing the exploration of the mathematics to take place over several days, Demetri had the time needed for his mathematical ideas to develop.  Third, during the lesson, Demetri was given time to explain why he believed his mathematical ideas were true.

CCSSM outlines the mathematics that all students should learn and it explicitly states that  students are to learn mathematics with understanding, indicating that “mathematical understanding and procedural skill are equally important.

FAQs:

Question 1: Is spending more than one day on a concept worth the time?

If you want children to internalize the math and not just memorize the math, you would have to give them more time to internalize it as they make links between their background knowledge and the new material.  Learning Math has to be more than memorization.

Question 2: Given the pressure to cover the curriculum within a predetermined time frame, how are you able to slow down and spend more than one day on a concept?

 When students have the opportunity to understand the mathematics, the need to reteach and review for the end-of-year is greatly reduced.  This reduction of re-teaching time will give you additional time.

So, with the conversion to CCSSM, how do we support our students in gaining not only procedural skills but also conceptual knowledge?