## The algebra learning model

The algebra learning model was created by the research team to describe the different ways students demonstrate important algebraic ideas before they have a complete understanding of algebra. The algebra learning model represents the process of how many high school students develop understandings of algebra. The algebra learning model was developed from findings of previous research conducted by members of our team and from international research literature. We found that as students learn ideas about algebra, they demonstrate a range of problem solving strategies. Because students demonstrate these strategies in predictable ways, the strategies students use could be classified into stages of learning.

The algebra learning model was used to design diagnostic tests, an interpretation guide with scoring rubrics and practical instructions for teachers to use in their mathematics classrooms. The algebra learning model contains ideas that we are continually reconsidering as we use them in practical ways with our students. We welcome your feedback as you use the components of the algebra framework.

Algebra is too often regarded, both by students and teachers, as a new topic concerned with letters rather than numbers that has no connections to previously learned mathematics. However, the NZ Curriculum now has combined Number and Algebra as one strand and there is a wealth of international research that has documented intimate connections between algebra and other branches of mathematics, particularly arithmetic. Some of our previous work has shown that knowledge of arithmetic structure, knowledge of inverse operations and understanding of equivalence are important prerequisites for learning algebraic strategies. Knowledge of basic facts is absolutely crucial! In the past we have regarded this knowledge as arithmetic, but we are now beginning to viewing it as early algebra. The Numeracy Development Projects (NDPs) have been a strong influence on our efforts because they give us a clear perspective of the strategies that children use in arithmetic and the knowledge required for these strategies. We have been examining the strategies that students use to solve equations, find relationships and express generality, and what prerequisite knowledge they use when doing so, and have found clear stages of learning.

**Garry discusses that the diagnostic tests show him student thinking**