Profound Understanding of Fundamental Mathematics (PUFM)

1. Profound Understanding of Fundamental Mathematics (PUFM) María E.Torres Summer 2005

2. PUFM • Ma, L. (1999). Knowing and teaching elementary mathematics: Teachers’ understanding of fundamental mathematics in China and the United States. Mahwah, NJ: Lawrence Erlbaum Associates. • Comparative study of American and Chinese teachers of mathematics • Finding: Chinese teachers had a “Profound Understanding of Fundamental Mathematics [PUFM],” and are “not only aware of the conceptual structure and basic attitudes of mathematics inherent in elementary mathematics but able to teach them as well.”

3. “Packing” Knowledge • Chinese teachers talked about a group of pieces of knowledge rather than a single piece of knowledge. To “pack knowledge”—that is, to see mathematics topics group by group rather than piece by piece—is a way of thinking. Not everyone “packs” knowledge the same way.

4. Procedural Understanding Conceptual Understanding Structure of the Subject attitudes attitudes Liping Ma: Chinese Teachers’ Conceptual Understanding of the Subject

5. Procedural Knowledge Procedural topic Procedural topic Pseudo-conceptual understanding Liping Ma: American Teachers’ Understanding of the Subject

6. 52 91 -25-79 Subtraction with regrouping of large numbers Subtraction without regrouping Subtraction with regrouping of numbers between 20 and 100 The composition Of numbers within 100 Addition without carrying Addition and Subtraction within 20 The rate of composing a higher value unit Addition and Subtraction Within 10 The composition of 10 Composing and decomposing a higher level unit Addition and subtraction as inverse operations Liping Ma: A knowledge package for subtraction with regrouping.

7. Identifies “ingredients” Identifies the grade levels where the “ingredients” are introduced and/or maintained Assists in developing lessons with procedural understanding and conceptual understanding of the mathematics As a tool, a knowledge package:

8. A knowledge packages does not: • Contain strategies • Contain language like “how to subtract,” or “how to multiply” Instead, it contains the ideas toward the procedural understanding.

10. Making Connections Exercise 1: Construct a knowledge package. Think of the necessary mathematics concepts or skills that are needed to master the mathematical task identified, then do the following: • Decide on the one concept/skill that is the most crucial for student mastery of the new concept or task and place in the larger shaded ellipse that lies just below the “new learning” ellipse. • If there are one or two other concepts/skills that are deemed especially critical, place those in the other shaded ellipses. • Place other essential concepts in the unshaded ellipses. • It is not necessary to fill in each and every ellipse! Exercise 2: Underline those topics in your knowledge packet that you feel are addressed strongly in the state content standards such as grade level expectations (GLEs) and state assessments.

11. Exercise 3: Identify the standard for every connecting concept, idea, etc. you have in your knowledge package for a specific mathematics skill (idea, concept) you have identified as being in need of development. • Exercise 4: Map it out across grade levels. Enter the standard at the grade level found. Suggested table shown below.