Plan for the Presentation
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Review our Algebra Problem
Examine why it is such a hard problem
Offer a content analysis of school algebra
Argue why solving the algebra problem must begin
early, with a reformulation of elementary mathematics
including the several root aspects of algebra
• Offer a three-dimensional, teacher-centered
“Algebrafication” (transitional) strategy for deep and
generative change of elementary mathematics
• Describe some specific techniques for the
algebrafication of elementary school maths activities
Our Algebra Problem
• Most students do not learn algebra very well
• Most teachers do not teach algebra very well
• Researchers do not understand algebra as
content, or its teaching and learning, very well
• Its breadth and depth are not systematically
reflected in school mathematics experience so
that School Algebra does not serve School
Mathematics in the positive ways that Algebra
has historically served Mathematics.
• Its relevance & importance are in doubt and in
change (technology, applics & demographics)
It’s a Hard Problem for Four Sets
of Reasons
• Institutional/historical Reasons, with deep
socio-cultural and historical roots
• Content Reasons: The complexity and
pervasiveness of algebra as mathematical
content
• Learning & Learning Reasons: Its
conceptual/semiotic complexity & subtlety
• The changing goals for algebra, as content
and demographically
Institutional/Historical Reasons
• Algebra is a Socio-Political-Economic Institution
as well as a web of skill and knowledge that
complexly interact with other knowledge and skill
• Algebra the Institution co-evolved with our
systems of education, so are deeply intertwined
with them and cannot change independently of
them
• Algebra and arithmetic had different historical
origins, served different social purposes, served
different populations and were historically
instantiated as separate curricula for
demographically separate populations
Content Reasons Why It’s Hard
• The pervasiveness of the roots of algebraic
reasoning - It is not just a topic or topic
strand: Embodying generality and the
systematic expression of generality, it is an
intrinsic way of being mathematical.
• The several aspects of algebra as content
intersect with most other mathematical
topics
• Content choices depend on extrinsic factors
and hence can vary by location & time
But What Content?
What Do We Mean By
“School Algebra”
A Content Analysis of School Algebra
• Root Aspect #1: Algebra as Generalizing and Systematically
Expressing Patterns & Constraints, especially, but not
exclusively, Algebra as Generalized Arithmetic and Quantitative
Reasoning
• Root Aspect #2: Algebra as Reasoning from the Forms of
Syntactically-Defined Statements and the Syntactically Guided
Manipulation of those Formalisms to Gain Further Insight
• Algebra as the Study of Structures and Systems Abstracted from
Computations and Relations (beginning with properties of numbers &
operations and continuing through abstract algebra)
• Algebra as the Study of Functions, Relations, and Joint Variation
(beginning with simple patterns and continuing through calculus)
• Algebra as a Cluster of Modeling and Phenomena-Creating &
Controlling Languages (in static/inert & dynamic/interactive media)
Narrow conceptions of algebra cannot solve
the Algebra Problem
Nor can late approaches to algebra
But each aspect of algebra makes sense at
the elementary school level
Hence, we suggest solving the problem
beginning at the elementary level and
addressing the full richness of algebra
throughout elementary school mathematics
How to Start? - Teacher-Centered
Algebrafication of Elementary School
Mathematics
Algebrafying Instructional Materials: Teach how to engineer algebraic reasoning
opportunities using both existing text materials & new materials
Understanding Student Development of Algebraic Reasoning: Build
teachers’ “algebra eyes and ears” so that they can identify opportunities for generalization
and systematic expression of that generality and then exploit these as they occur across math
topics
Classroom Practice & Culture: Create classroom practice and culture to encourage
and support active student generalization and formalization within the context of purposeful
conjecture and argument, so that algebraic reasoning opportunities occur frequently and are
viable when they do occur
Work with education administrators & policy-makers to support
algebrafication: The institutional side of the Algebra Problem needs to be addressed
directly - it involves more than students and teachers
How to Algebrafy Elementary School
Mathematics Activity?
Algebrafy one-answer arithmetic tasks by systematically varying the givens of the
tasks to create sequences of such - a very powerful, learnable, and flexible strategy
that can serve a wide range of teacher and curricular agendas, including the
development of number sense, introducing variables, building modeling capability
(by replacing a single answer “word problem” by a sequence of such - see Mason),
studying the different kinds of variation, ...
Organize sequences of tasks and number sentences:
Promote habits of mind
and of representation that seek generality and its expression whenever possible
Treat numbers algebraically: Frequently use numbers out of computational
range, to draw explicit attention to their place in numerical statements rather than
the numerical value of the expressions involved
Treat operations algebraically: Leave numerical statements, especially
sequences of these, in unexecuted form so that students learn to reason from the
forms of arithmetic statements - that is, to reason algebraically