Report: The Literacy of Mathematics

PETAA – Primary English Teaching Association Australia

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The basis of representational theory in mathematics education is that mathematics consists of abstract ideas, so we must create representations in order to access and work with these ideas (Goldin & Kaput, 1996).

‘Representational fluency’ in mathematics generally refers to the flexible and adaptive use of multiple representations of the same idea, such as a pattern in a sequence of numbers being described
in words, generalised symbolically in a ‘rule’, and plotted on a graph (Heinze, Star & Verschaffel, 2009).

When students use representation as a learning tool they engage in the personalised, dynamic interplay between internal thoughts and images, and external signs and symbols that embody emerging concepts (Thom & McGarvey, 2015). When students use representation for communication, they reflect on the idiosyncratic features of their creations and consider the meaning-making needs of others.

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Underlying each of the teaching strategies for developing representational competence are four pedagogical aims. To:
● Promote student engagement with learning
● Focus on conceptual understanding
● Encourage reasoning
● Develop communication skills

The emergence of mathematical drawing

emergent mathematical features such as shapes and partitioning

With adults regularly asking children, ‘What did you draw?’, children begin to realise that people expect drawings to represent external objects, rather than only be self-expressions or personal explorations of form.

Recent research has emphasised that young children’s drawings are a strong indicator of awareness of pattern and structure, which is critical to developing conceptual understanding across most areas of mathematics (Mulligan & Mitchelmore, 2009).

Storying with drawings

If the representation is being used as a thinking tool, then storying might occur as ‘thinking out loud’ while the drawing is in progress. If the representation is for communication purposes, then the drawing and the narrative might be more summative or reflective

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For example, students might be asked to solve the story-problem of ‘Five birds sat on a fence. Two flew away. How many were left on the fence?’, and record their solutions by drawing and writing. The images suggested by the story-problem typically prompt young students to illustrate the story by drawing the fence and quite detailed birds (an artistic challenge). The mathematical challenge is to represent the dynamic operation of ‘take-away’. Children rarely draw separate ‘before and after’ scenes, instead attempting to depict the change in the number of birds in a single drawing.

The drawing by five-year-old Lily (Figure 1) is mathematically ambiguous because it shows seven birds, and is suggestive of addition rather than subtraction. However, in telling the story of her drawing, Lily clearly articulated that, “Two birds flew up in the sky. Those two. [covers 2 birds on the left with her hand]”. Lily’s verbal and gestural representation clarified the drawn representation and revealed her conceptual understanding of take-away subtraction.

Daniel (5 years) used a different representational approach (Figure 2). He decided that the fence was not important, and only drew the five birds. He used the diagrammatic device of arrows to indicate the movement of two birds away from the group, and his story confirmed that he knew three birds remained.

the teacher can use spoken words to identify the birds on the fence, without drawing a picture of the scene. Instead, the teacher might just draw five circles (the birds), and cross two out.

Communicating mathematical ideas and reasoning: Purposeful use of student talk

Communication is an essential component of learning mathematics (Chapin & O’Connor, 2007) and underpins the four competencies (understanding, fluency, problem-solving and reasoning)

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How can talk support students’ learning of mathematics?

Talk can support students’ learning of mathematics by:
● revealing understandings or misconceptions and gaps in their reasoning;
● repeating important ideas to allow deeper understanding;
● developing more sophisticated mathematical language;
● responding to the ideas of others, resulting in elaboration and/or clarification of their own thinking.

Academically productive talk is that which engages students both behaviourally and intellectually, leading to deeper understanding of mathematical concepts and higher order thinking.

What can we talk about in mathematics?

• Number talks
• Mathematical representations (diagrams, concrete materials, symbolic notations etc)
• Mathematical concepts
• Compare and contrast problem solving or computational strategies
• Mathematical reasoning
• Mathematical terms and their definitions

The basic procedure of number talks involves:

• A number problem is posed to the class
• Thinking time is allowed for a mental solution
• Readiness to share is indicated by individual students raising a thumb unobtrusively against their chests, (and raising one or more fingers if they think of other solutions)

Guiding principles for classroom discussions

• Discussions must have a clear mathematical goal, and the discussion plan should vary according to this goal;
• Students need to know how and what mathematics to discuss; and
• Teachers must establish classroom norms whereby all students are encouraged to participate equitably and respectfully.

Clear goals

The goal of an open strategy discussion might be for students to simply share as many strategies for solving the same problem as possible. A targeted discussion focuses on finger-grained details, such as the efficiency of one strategy over another or to clarify the meaning of a mathematical term.

requires teachers to anticipate student responses. Anticipating likely responses allows teachers to better prepare how they will direct students’ attention to the important mathematics in question, thus ensuring that student talk does not simply become ‘show-and-tell’ or meander through a range of unrelated
and already known content.

Knowing what and how to discuss

Students can only learn how and what to discuss in mathematics with explicit support from teachers.

prompts or sentence-starters (e.g. “I agree with X, because …” and “I would like to revise my thinking because …”)

It is important that teachers provide explicit opportunities for students to rehearse what these talk moves might sound like during whole class, small group or partner discussions.

Classroom norms for discussion

Norms in the classroom that emphasise respect for each other and the right for everyone to express their thinking are important to model and discuss to ensure there is a shared understanding of their purpose.
Such expectations include:
● Every student will listen to what others say;
● Every student has the right to speak;
● You may respectfully agree or disagree with a person’s comments—not the person.

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Comprehending and creating diagrams

Knowing about the use of diagrams and being able to apply that knowledge appropriately has been referred to as diagram literacy, and identified as a component of visual literacy (Deizmann & English,
2001).

in developing visual knowledge, students must “… understand how visual elements create meaning”.

When students effectively create their own diagrams during problem solving, they engage in a process of translating their reasoning and internal images into concise external representations.

Diagram literacy is an essential part of students’ development of both mathematical proficiency and visual literacy and, therefore, requires deliberate attention by teachers.

Diagram talks

Diagram-talks are based on the increasingly popular teaching strategy of number talks

diagram talks place greater emphasis on the teacher recording a student’s thinking using a diagram, while making explicit the connections between components of the student’s described strategy and elements of the drawn representation. In other words, the teacher models how to construct a diagram that effectively communicates the student’s reasoning and solution. The key pedagogical characteristic of this approach is that the teacher is modelling the student’s own thinking, not demonstrating the teacher’s way of thinking.

Using diagram talks to develop the use of empty number lines

Example 1: So, you started at 50 and added on 20 to get 70. Then you added 8 and 2 (which you knew was 10), and that made 80.
Example 2: 58 plus 2 makes 60, plus 20 makes 80

Finally, the teacher drew another number-line diagram that represented a different strategy (Figure 4) and asked the students to interpret it, and explain how the strategy worked.

The teacher conducted several similar diagram talks involving basic addition and subtraction problems over two weeks, then encouraged the students to begin drawing their own diagrams. Once the class had gained familiarity with using the number-line diagram as a communication tool, the teacher’s goal shifted to encouraging the students to use it as a thinking tool.

The students were asked to record their problem-solving process on paper before solutions were shared and discussed. During discussion, the construction of the diagram and the reasoning behind it, were highlighted.

Conclusion

● Build on a student’s existing and emerging representations of mathematics;
● Student progress in representational fluency requires deliberate attention and explicit instruction by teachers;
● Provide regular opportunities to work with a variety of representational forms for the same mathematics concept; and
● Use representations as both thinking tools and as forms of mathematical communication.