Supporting 5 Year Olds to Represent Their Mathematical Reasoning

Oct 15, 2026

A seven-lesson classroom study examines how 5-year-old students develop mathematical representations from concrete manipulatives toward more sophisticated external and symbolic forms. Teacher support emphasizes high expectations, structured collaborative discussion, and close attention to students’ reasoning and representational choices. Findings highlight the instructional actions needed for young learners to connect informal “how many/longer-shorter” experiences to formal curriculum practices, including explanation, justification, generalisation, and representation.

LanguageEnglish

CategoryResearch & Report

FormatPDF

Pages8

Reading Time20 min

AccessFree to download

Lesson sequence for representation development

From concrete manipulatives to formal ideas

Teacher instructional actions

Expectations and collaborative discussion

Monitoring reasoning and extending representations

Connections to curriculum and prior experiences

Mathematical practices: explanation and justification

Supporting 5 Year Olds to Represent Their Mathematical Reasoning Emily Pearce Massey University [e.pearce1@massey.ac.nz](e.pearce1@massey.ac.nz) Generosa Leach Massey University [g.leach@massey.ac.nz](g.leach@massey.ac.nz) Jodie Hunter Massey University [j.hunter1@massey.ac.nz](j.hunter1@massey.ac.nz) This paper illustrates how young students aged 5 years old were supported to develop their mathematical representations. In seven lessons, students progressed from using concrete manipulatives to representing their mathematical ideas in more sophisticated ways. The teacher utilised specific instructional actions including holding high expectations for all students to engage with collaborative discussion, paying close attention to the students’ reasoning and representations and drawing on these to extend them further. The findings offer potential for how teachers of young students can support them to develop reasonable mathematical representations. The New Zealand Mathematics Curriculum (Ministry of Education, 2024) outlines expectations that students are supported to engage in mathematical practices such as explanation, justification, generalisation and representation. New Zealand children begin school when they are 5 years old and bring with them a wealth of prior life experiences and skills. These include making informal representations to show “how many” or “which are longer or shorter”, however, the process of connecting these to formal mathematical concepts requires support (Bobis & Way, 2018) . Mathematical representations can be made in a range of different ways, for example, using concrete materials, manipulatives, or drawing pictures or diagrams, or by using number sentences or formal equations. The purpose of representing mathematics in these ways is to show or explain mathematical reasoning (Bicknell et al., 2016; Bobis & Way, 2018; Pape & Tchoshanov, 2001). Accurately representing mathematical thinking is particularly challenging for young students as representing mathematical reasoning is a complex process requiring students to transfer internal thought to an external medium (Pape & Tchoshanov, 2001) . For young learners, there is often an emphasis placed on using some type of materials to support mathematical learning. However, depending on how and why certain materials are used, some students may find it challenging to reasonably represent their mathematical ideas. Teachers often use material representations when modelling or showing students how to do certain things such as counting using counters or blocks or using tens frames to count and group to ten (Bicknell et al., 2016; Obersteiner et al., 2014). In their study, Donovan and Alibali (2021) investigated how students aged 7-8 years-old used non manipulatives (toys) as “tools for doing maths”(p. 296) . While the study showed success in the students manipulating these materials, this occurred through the researcher (in the teacher role) specifically instructing the students to use the toys as tools for mathematics. Donovan and Alibali concluded that it is necessary to make explicit to students how materials are to be used to represent mathematical reasoning. ANew Zealand study by Bicknell et al. (2016) examined how the teacher made deliberate decisions about which manipulatives 5-year-old students would use in a mathematics lesson involving multiplication and division. For example, when working out the number of egg cartons required for a selected number of eggs, egg cartons and counters were provided to support the students in their representation of division. In a follow-up interview, the teacher in Bicknell et al.’s study shared that at times the students would revert to using representations that they were comfortable with rather than engaging with the formal representations modelled by the teacher. To support the students the teacher would either co-construct number equations with the students or remind the students to represent us

FAQs

What does the paper investigate about 5-year-old students?

It investigates how young students just beginning school can be supported to develop mathematical representations for communicating their reasoning.

How does the instruction progress across the lessons?

Across seven lessons, students progress from using concrete manipulatives toward representing mathematical ideas in more sophisticated ways, guided by teacher support.

What teacher actions are emphasized in supporting mathematical representations?

The teacher holds high expectations for collaborative discussion, closely attends to students’ reasoning and representations, and uses insights from students’ work to extend their representational thinking.

Oct 15, 2026

Uploaded by Taylor Morgan