Translating equations and inequalities: from visually supported problems to algebraic symbolism

Resumen

Within the context of a design research teaching experiment focused on students’
development of algebraic thinking, we attend to the structures that 3rd and 4th grade
students show when translating equation and inequation problems stated with visual
support into algebraic symbolism. Referring to and generalising structure are our
focuses as components of algebraic thinking (Kieran, 2022) which have proven to have
an impact in promoting students’ success in mathematics as well as into their
preparation for the later study of formal algebra (Blanton et al., 2018).
We qualitatively analysed the written responses and oral discussions carried out by 51
students when solving an equation problem and an inequality problem. Subsequently,
we conducted semi-structured interviews to 12 of these students when solving other
similar problems. From the answers obtained, we analysed the structures perceived by
the group. The results inform of the elements that students included in their
expressions: relationship of equality and inequality, indeterminate quantities, the use
of operations, as well as the composition, decomposition, and re-composition of
numbers. They perceived and expressed multiple structures and equivalences between
them, both for equations and inequalities, and reasoned about the equivalence of
structures. Differences between the case of equations and inequations revel difficulties
to be addressed in teaching. The role of visual aids is highlighted as a way to contribute
to facilitate structural reasoning even without specific instruction. It was a key element
in recognising the equivalence of different symbolic representations of a problem.