This month’s curricular competency focus is communicate mathematical thinking in many ways. This curricular competency is the same across grades K-9 and is included in the Grades 10-12 courses as “explain and justify mathematical ideas and decisions in many ways“.
This competency falls under the organizer of ”Communicating and Representing” which includes the following related competencies:
Elaborations are suggestions for educators to consider as they plan for developing this curricular competency:
- communicate using concrete, pictorial and symbolic forms
- use spoken or written language to express, describe, explain, justify and apply mathematical ideas
- use technology for communication purposes such as screencasting and digital photography and videography
There are clear connections between the Core Competency of Communication with this grouping of curricular competencies. A one-page table showing the language of both types of competencies can be downloaded here:
An important part of communicating mathematical thinking in many ways is to be able to use different forms such as concrete (materials or math manipulatives), pictorial (drawings, diagrams, tallies) or symbolic forms (numerals and symbols).
An example from primary classrooms of how students may move from concrete to symbolic notations is with the use of materials such as base ten blocks. Students may communicate their understanding of numbers by creating that number with materials and then recording the symbolic notation. The following are some examples from a grades 2&3 classroom at Cook Elementary that show how children used concrete, pictorial and symbolic forms to help them solve and communicate their solutions for mathematical problems.
As students begin to understand a concept, such as multiplication, they usually construct a representation with materials to build understanding. These representation may then be recorded pictorially and then labels are added using symbolic notation. This fluency between forms is important and the connections between representations is essential to conceptual understanding. A student may be presented with a symbolic form (such as an equation) and asked to show a concrete form or pictorial form that “matches”. The following are examples from a grades 2&3 classroom at Tomsett Elementary.
For our intermediate and secondary students, it is still important to be using concrete materials, especially when students are developing their understanding of a new concept such as fractions, decimals, or integers. The following are examples from a grades 4&5 classroom at Homma Elementary
and also more fraction investigations with a grades 4&5 class at Steves.
In our curriculum, the terms “concrete, pictorial and symbolic” are used in ways for students to think about concepts but also to communicate and represent their thinking. In some other jurisdictions around the world, the term CRA is used to reference an instructional approach to concept development, standing for Concrete, Representational and Abstract. More information can be found HERE. There is some overlap between the the CRA framework and how our curriculum focuses on concrete, pictorial and symbolic communication of mathematical thinking and understanding.
Another area of focus in our district is using iPad technology for students to communicate their thinking and learning. One of the most common uses of the devices in math is to use screen casting apps such as doceri, ShowMe, Explain Everything or 30Hands. When students screencast, they can take a photograph or video of what they are doing and then annotate with arrows, words etc and then orally describe their problem-solving process or thinking. For example, in a grade 8 class at Hugh Boyd Secondary, students took images of number balances they used to develop their understanding of equivalence in algebraic equations and then communicated their thinking by orally explaining their understanding.
Some questions to consider as you plan for learning opportunities to develop the competency of communicating mathematical thinking in many ways:
How is the core competency of communication noticed, named and nurtured during the teaching and learning of mathematics?
What different materials are students learning to use, think through and represent with? What materials are mathematically structured and what other types of materials might we offer to students?
What opportunities are we providing for students to share their thinking in different ways? Are students provided with choices and is there a balance in the different ways students can communicate their mathematical thinking?
How might technology provide access for students or transform the way they are able to communicate their mathematical thinking?