C – Concrete

During the concrete stage, students learn to solve mathematical problems using concrete objects or manipulatives to support their conceptual understanding. Educators should present the use of concrete tools through explicit instruction, and should monitor student progress to ensure they achieve mastery before moving to the next stage.


Manipulatives are necessary tools for supporting the effective learning of mathematics by all students. These concrete learning tools invite students to explore and represent abstract mathematical ideas in varied, concrete, tactile, and visually rich ways. Moreover, using a variety of manipulatives helps deepen and extend students’ understanding of mathematical concepts [i].

The use of manipulatives increases the number of sensory inputs a student uses while learning the new concept, which improves the chances for a student to remember the procedures to solve a problem [ii].

Click here to access the list of EQAO-approved mathematics manipulatives for primary junior divisions.

The following video demonstrates how teachers at Sagonaska Demonstration School use math manipulatives to help students at all grade levels improve their mathematical reasoning and problem solving. The accompanying viewers’ guide provides activities and resources to enhance your professional learning.

Click here to view the transcript of this video.

Cover Page of the Viewers'guide

Click here to access the viewers' guide.

Virtual Manipulatives

There is emerging evidence for the use of virtual math manipulatives within the CRA framework [iii]. Virtual manipulatives may be online or app-based versions of concrete manipulative objects.

Some benefits of virtual manipulatives include [iv]:

  • they can be used discreetly on a tablet so as not to draw attention to students with LDs, who may need to use manipulatives for a longer period of time in the learning process compared to their peers without LDs
  • ease of managing and transporting materials, which benefits busy educators as well as students with LDs who may have challenges with organization and other executive functions
  • they can be accessed by as many students as required on their personal devices, at school or at home
  • many virtual manipulatives come with prompts and scaffolds to support greater student independence

Student and teacher doing math on an interactive whiteboard

Click here to access the learning tools and virtual manipulatives available on Mathies.

Click here to access the National Library of Virtual Manipulatives.

Click here to access the Math Learning Center’s list of virtual manipulative apps.

To view some examples of virtual math manipulatives in action, click here to access the webinar recording Math and LDs – Using Assistive Technology and Project-Based Learning to Support All Students.

Concrete Stage in Action: Using Straws to Help Students Understand Place Value

The following activity demonstrates one possibility for exploring the concrete stage in the elementary classroom.

Click here to access the original article Using Straws to Help Students Understand Place Value on the LD@school website.

Preview of Lesson plan document

Click here to access the printable lesson plan and student handout.


[i] Ontario Ministry of Education, 2005, p. 25

[ii] Garforth & Siegel, 2014

[iii] Bouck & Sprick, 2019

[iv] Bouck & Sprick, 2019