Plan for the Presentation 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

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 Start? - Teacher-Centered Algebrafication of Elementary School Mathematics

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 How to Algebrafy Elementary School Mathematics Activity?