Teaching mathematics from the concrete to the abstract

Researchers in the project aim to test a model to see whether elementary school students 
can cope with more advanced middle school mathematics if they have followed a program in early grades designed to build long-term understanding of concepts.

It is a widely held belief that mathematics teaching in the early school years should progress from the concrete to the abstract. “The abstract” refers to the ability to reason mathematically using the specific system of mathematical symbols. However, there is a lack of firm theories explaining how “the concrete” – i.e. the ways chosen to introduce mathematical concepts in early school years – should be selected.

Yet research has clearly established that poorly chosen methods for introducing basic concepts in early grades can create misconceptions about multiplicative and other structures in later school years. These misconceptions are difficult to correct because the first impression of a new phenomenon tends to become deeply rooted in the mind.

Over time, the constraints of these concrete models have caused middle school mathematics to become increasingly trivial. Instead of teaching the mathematics that students need to learn, the mathematics taught is constrained by the early entrenched models. The concrete models introduced in early grades hinder the transition to abstract models in middle school.

The project team’s hypothesis is that if middle school mathematics is to address abstract models, early-grade mathematics must introduce concrete models whose validity endures all the way to middle school without creating cognitive conflicts and misconceptions.

The team has developed a theoretical model to determine the characteristics concrete thought models in early grade mathematics need to display to lay the foundation for long-term mathematical progression. They aim to ascertain whether the model works in practice and whether students can cope with more advanced middle school education if they have followed a program in the early grades aimed at building long-term conceptual understanding.

Due to the time span between early and middle grades, it is not usually feasible to investigate an issue of this kind within the timeframe of a research project. However, in 2019 a Swedish municipality implemented a theory-based mathematics program of the kind described above in early grades, thereby making this project possible.

All students in the municipality who started first grade in 2017 (Group 1) and 2018 (Group 2) received traditional instruction. All students who started in 2019 (Group 3) received instruction in line with the project’s theoretical model.

The project team will study the students in the eighth grade, focusing on multiplicative structures (multiplication, division, fractions, ratios, proportions, and linear relationships). They will use a research-based test to assess students’ ability to manage multiplicative relationships.

Over three years, they will test Groups 1, 2, and 3 at the beginning of the eighth grade. They then plan to use a specially designed 15-lesson instructional sequence taught by municipal mathematics teachers to Groups 2 and 3 but not Group 1. A follow-up test will be conducted at the end of the eighth grade.

The researchers expect their instructional sequence to have some effect on Group 2, which is expected to perform better than Group 1. The central hypothesis is that Group 3 will clearly perform better than Groups 1 and 2. This would be an important indication that the main hypothesis is correct, i.e. that it is worthwhile to conduct early-grade instruction according to their model, whereby concrete models in early grades lay the foundation for abstract mathematical reasoning in middle school.