Profound Understanding of
Fundamental Mathematics
(PUFM)
QuickTime™ and a
TIFF (LZW) decompressor
are needed to see this picture.
María E.Torres
Summer 2005
1
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
QuickTime™ and
a
TIFF (LZW) decompressor
are needed to see
this picture.
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.”
2
“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.
QuickTime™ and a
TIFF (LZW) decompressor
are needed to see this picture.
3
Procedural Understanding
Conceptual
Understanding
QuickTime™ and a
TIFF (LZW) decompressor
are needed to see this picture.
attitudes
Structure of the Subject
attitudes
Liping Ma: Chinese Teachers’ Conceptual Understanding of the Subject
4
Procedural Knowledge
Procedural
topic
Pseudo-conceptual
understanding
Procedural
topic
QuickTime™ and a
TIFF (LZW) decompressor
are needed to see this picture.
Liping Ma: American Teachers’ Understanding of the Subject
5
52
-25
91
-79
Subtraction with
regrouping of
large numbers
The composition
Of numbers within 100
Subtraction with
regrouping of numbers
between 20 and 100
The rate of composing
a higher value unit
Addition and Subtraction
within 20
Addition and Subtraction
Within 10
Composing and
decomposing
a higher level unit
Subtraction without
regrouping
Addition without carrying
The composition of 10
QuickTime™ and a
TIFF (LZW) decompressor
are needed to see this picture.
Addition and subtraction
as inverse operations
Liping Ma: A knowledge package for subtraction with regrouping.
6
As a tool, a knowledge package:
• Identifies “ingredients”
• Identifies the grade levels where the
“ingredients” are introduced and/or
maintained
QuickTime™ and a
TIFF (LZW) decompressor
are needed to see this picture.
• Assists in developing lessons with
procedural understanding and conceptual
understanding of the mathematics
7
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.
QuickTime™ and a
TIFF (LZW) decompressor
are needed to see this picture.
8
QuickTime™ and a
TIFF (LZW) decompressor
are needed to see this picture.
Knowledge package adapted from Liping Ma’s Knowing a nd Teaching Elem entary
9
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.
QuickTime™ and a
TIFF (LZW) decompressor
• It is not necessary to fill in each and
everytoellipse!
are needed
see this picture.
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.
10
• 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.
Math Focus
Generating data
Measurement
6th
7th
8th
9th
GLE 37
QuickTime™ and a
TIFF (LZW) decompressor
are needed to see this picture.
Circumference
Rates of change
GLE 18
11