# Algebraic thinking with and without algebraic representation: a three-year longitudinal study

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## Abstract

An Algebraic Thinking Test was given to 116 students aged 12–14, at the end of each of three years. This age span crosses two levels of school in New Zealand. This test assessed their ability to represent compensation in the four arithmetic operations both numerically and with letters for variables. The analyses of these results, together with the results from separate interviews designed to report individual progress of students in the New Zealand Numeracy Project, showed that students who had developed advanced mental strategies for dealing with additive, multiplicative and proportional operations, were the students who were capable of making full use of the alphanumeric symbols of algebra. These results, taken together with earlier studies by the authors, led to a proposal for a “pathway for algebraic thinking” accessible to all students.

## Keywords

Operational Strategy Professional Development Program Lead Teacher Algebraic Thinking Decimal Fraction## Notes

### Acknowledgments

We are grateful to the Ministry of Education for funding this research project. We also express our thanks to R. J. Irwin for some of the statistical analysis.

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