A DEEPER LOOK INTO
PROGRESSIONS
Team Task
Choose 1 Progression to READ:
1. 3-5 Number and Operations--Fractions
2. 6-7 Ratios and Proportional Relationships
Develop “Content” Knowledge Package
Knowledge Package
BIG IDEAS that are foundational for understanding
Facts or procedures that help in solving problems fluently and efficiently
Visual models that make concepts and relationships most clearly visible
Approaches for accomplishing a task that help make sense of the mathematics
Vocabulary, terminology, or phrases that are necessary for participation in
discussion about the topic
Unpacking YOUR Knowledge
169 ÷ 14
Your homework was to solve the division problem using
two strategies other than the conventional algorithm.
You explained and represented your thinking using
symbols, words, and diagrams, as appropriate for each
strategy.
Unpacking Division Reflections
4 Square Reflection share out
“Knowing mathematics sufficiently for
teaching requires being able to
unpack ideas and make them accessible
as they are first encountered by the
learner.”
-Ball, 2003
READ and REFLECT
Liping Ma’s Knowledge Packages
 Liping Ma used the term “knowledge package” when speaking about the subject matter knowledge of
teachers. She explains that when a teacher begins to teach a new topic, that teacher has an idea in her
mind about where this idea is situated in the field of mathematics. Thus, “given a topic, a teacher tends to
see other topics related to its learning,” and such topics comprise the knowledge package for the topic to
be taught. In knowledge packages, there are “key pieces,” which are certain related topics that are viewed
as being more important to the comprehension of the topic at hand. Knowledge packages for any topic can
contain both procedural and conceptual elements, and Ma asserts that the two are interrelated. Ma found
that teachers with a conceptual understanding of a topic viewed related procedural topics as being
essential to student understanding. “In fact,” Ma emphasizes, these teachers felt that “a conceptual
understanding is never separate from the corresponding procedures where the understanding ‘lives’. Ma
believes that knowledge packages are important because it is from this information that a teacher attempts
to construct a cohesive and comprehensive picture of a mathematical topic. With underdeveloped
knowledge packages, it can be very difficult for a teacher to plan and facilitate a course of study for their
students.
 Learning Paths – a linear progression?
Knowledge packages also contain implicit sequences of student learning, where a student is expected to
know and understand key related pieces before they can grasp the topic at hand. Ma explains, “the
teachers believe that these sequences are the main paths through which knowledge and skill about the…
topic develop” . While this may seem to be a linear progression of learning (you need to know x, y, and z,
before you can learn topic A), Ma clarifies that topics in a knowledge package are interdependent, and that
“linear sequences, however, do not develop alone, but are supported by other topics”. Thus, the learning
paths generated through teachers’ knowledge packages are similar to the learning trajectories.
Knowledge Package Structure

Organizing our thinking about a mathematics topic around
the Knowledge Package, provides us with a structure for
asking these same five questions for all mathematics topics.

These categories allow us to understand what is important
for students to know, to teach with clarity and intentionality,
and then to assess student understanding with minimal
ambiguity.
Knowledge packages provide an important
pattern for learning and teaching mathematics.
Multiplication Knowledge Package – 3rd Grade
Knowledge Packages include Five
Key Components:
1. Conceptual Understanding: Big overarching ideas that are
foundational for understanding
2. Skills: Facts or procedures that help in solving problems
fluently and efficiently
3. Representations: Visual models that make concepts and
relationships most clearly visible
4. Strategies: Approaches for accomplishing a task that help to
make sense of the mathematics
5. Mathematical Language: Mathematics vocabulary,
terminology, or phrases that are necessary for participation
in discussions about the topic
Division Package of Knowledge
Teams:
Unpack knowledge needed to solve this problem
Whole Group:
Record team responses on Chart Paper
TEAM TASK:
Record “knowledge responses” on post-its
Sort “Knowledge” into 5 Key Components
Create “Knowledge Package” (justify and defend)
Purposes of Knowledge Packages
Liping Ma’s “knowledge packages” provide us with a useful way to
clarify the ideas that students must understand in mathematics
content. Mathematics standards are typically listed in linear form.
They tend to reduce mathematics to disconnected lists of
knowledge and skills.
In contrast, knowledge packages highlight connections between and
among ideas. Knowledge packages provide the context and the
glue for mathematics standards, bringing ideas together into a
more comprehensible whole.
Knowledge packages may address a large topic, such as
multiplication, or describe a task within a topic, such as comparing
fractions. The specific contents of a knowledge package vary
according to the grade or developmental level being addressed.
Using Knowledge Packages
Knowledge packages are not static words
on a page, but instead can be the focus
of discussions as teachers discuss
relationships between standards,
prioritize ideas, and seek out new and
better ways to teach their students.
It is the connections among these ideas,
concepts, and procedures that enable
teachers to portray
mathematics as a unified body of knowledge
rather than as isolated topics.
(Ma 1999)
Knowledge Packages: Simple Tools of Change
The simplicity of Knowledge Packages allows schools to turn on a dime
and realign their thinking about mathematics. When teachers come to
understand the relationships within knowledge packages, they find
their voices. They learn what is important for students to understand,
and they ask the right questions about how concepts, skills, strategies,
representations, and mathematical language relate.
When educators throughout the school ask these same questions, student
experiences from class to class, from year to year, connect. Everyone is a
link in the chain, and it takes time for links to forge. That forging
happens when educators gain outside support and when they engage in
school-based research about knowledge packages.
Knowledge Packages are the chains that bind our thinking about
mathematics and the test of their strength is student success.
CREATING YOUR OWN
KNOWLEDGE PACKAGE
IF THE HAMMER IS THE ONLY TOOL YOU GIVE
STUDENTS . . . . EVERYTHING BEGINS TO LOOK LIKE
A NAIL!