Mathematics AchievementThe role of reading and writing

Stephen Lerman

London South Bank University

UK

Mathematics achievement is unevenly distributed across populations of students.

This is of great concern to students, parents, teachers, and researchers alike.

As researchers, what can we say about this?

What perspectives can we take on the possible causes and therefore actions that might be taken to improve the achievement of students?

In particular, given the theme of this conference, how might we theorise the role of reading and writing mathematics in this?

Vygotsky (1986, pp. 180-183) on writing:

• Differs from oral speech in both structure and mode of functioning

• In conversation every sentence is prompted by a motive

• Motives for writing more abstract, more intellectualised

• Addressed to an absent or imaginary person• Must be consciously directed• Follows development of inner speech, whereas

oral speech precedes it

The psychological functions on which written speech is based have not even begun to develop in the proper sense when instruction in writing begins.

Performing/doing preceding understanding.

Writing: compression of meaning

Reading: decompression of meaning

Issues:

• Reading and writing to regulate students’ knowledge/identities

• Using reading and writing to develop mathematical knowledge – ‘writing-to-learn’, and other approaches

What achievement means and how it is realised in pedagogy, curriculum and especially in evaluation (assessment) changes what is expected.

Issues:

• Reading and writing to regulate students’ knowledge/identities

• Using reading and writing to develop mathematical knowledge – ‘writing-to-learn’, and other approaches

Reading

Note the difference between these:

1. Solve simultaneously:C + P = 90 and 2C + P = 145

2. A drink and a box of popcorn together cost 90p. 2 drinks and a box of popcorn together cost £1.45 What does a box of popcorn cost?

In the second form of question one child answered:

I said to myself that in a sweetshop a can of coke is normally 40p so I thought of a number and the number was 50p so I add

40p and 50p and it equalled 90p

It is quite likely (research suggests) that this child would have been able to solve the task in its abstract (esoteric) form of the simultaneous equations.

The first question we must ask is: what explanation is there for this response?

The second question is: what can we do about it?

Learning school mathematics is a process of acquiring the recognition and realisation rules.

‘Recognition’ means knowing which game you are involved in.

‘Realisation’ means knowing how to produce a legitimate text.(Bernstein, 2000)

Being taught to ‘read’ mathematics, therefore, is not how to utter the words but:

• that they signify the mathematical register (recognition)

• what, in a mathematical sense, they signify (strong grammar)

• and what the student should do, what is appropriate (realisation).

What should the student do when faced with the following question?

This is the sign in a lift at an office block:

In the morning rush, 269 people want to go up in this lift.How many times must it go up?

This lift can carry up to

14 people

We can conceptualise learning mathematics at school as ‘getting used to it’, or ‘becoming mathematical’.

How to ‘read’ mathematics and how to write it are key to mathematical identity and, hence, achievement.

As Jo Boaler (1997) has shown, different forms of pedagogy, driven by different notions of what desired achievement in mathematics might mean, lead to different mathematical identities.

Forms of pedagogy

• Performance (traditional)

• Competence (liberal/progressive)– Individual cognitive development

(constructivism)– Social/cultural (ethnomathematics)– Emancipatory (criticalmathematics)

• New performance mode - Vygotsky

Forms of pedagogy

• Performance (traditional)

• Competence (liberal/progressive)– Individual cognitive development

(constructivism)– Social/cultural (ethnomathematics)– Emancipatory (criticalmathematics)

• New performance mode - Vygotsky

Achievement is framed, more or less explicitly, by the teacher (rarely

freely).

‘Less explicit’ classrooms, what Bernstein (2000) calls invisible

pedagogy, such as one sees in reform classrooms, can disadvantage some

students.

Looking at research:

Amongst others:

• Pimm (1987, 1995)

• Waywood (1992)

• Borasi and Siegel (1994)

• Morgan (1998)

• Ntenza (2004)

• Pugalee (2004)

Andrew Waywood: Journal writing

• Across 4 years, grades 7 to 11• Part of the ‘normal’ learning programme, to be

assessed• Journals to be used to:

– Summarize– Collect examples– Ask questions– Discuss

• Over years, journal writing enhances students’ learning, but only with conscious teaching effort

Raffaella Borasi & Marjorie Siegel: Reading Counts

• Reading to Learn Mathematics for critical thinking (RLM)

• Countering the ‘mining’ metaphor (Pimm) which emphasises the author not the reader

• Why:– as a means of learning from text

– as a means of supporting and enhancing students’ mathematical inquiries

– as a means of negotiating a learning community

• What:– Rich texts (essays, stories etc.)– Cartoons, tables of data, videos, – Multi-media– But how used is crucial

• How:– Say something– Cloning an author– Sketch-to-sketch

Candia Morgan: Writing Mathematically

• The myth of transparency• Emphasising certain features:

– Words over algebra– Introduction– Diagrams– Reading ‘ability’

• Learning-to-write• Critical language awareness

S. Philemon Ntenza: Teachers’ perceptions of the benefits of children’s writing

Hierarchy (Davison and Pearce, 1988):

• Direct use of language – e.g. copying

• Linguistic translation – symbols into words

• Summarizing/interpreting

• Applied use of language – e.g. story problems

• Creative use of language

David Pugalee: Writing in problem solving supporting metacognition

• Students who wrote about their problem solving processes produced correct solutions at a significantly higher rate than when using think-aloud processes.

• Students who wrote descriptions of their processes produced significantly more orientation and execution statements than students who verbalized their responses.

Revisiting Bernstein’s view:

Reading and writing mathematics, as presented in the research above, are

potentially indicative of weak framing, and an invisible pedagogy.

Teachers play a key role in students’ mathematics

achievement

Borasi, R. & Siegel, M. (2000). Reading Counts: Expanding the Role of Reading in Mathematics Classrooms. New York: Teachers College Press.

Morgan, C. (1998). Writing Mathematically: The Discourse of Investigation. London: Falmer.

Ntenza, S. P. (2004). Teachers’ Perceptions of the Benefits of Children Writing in Mathematics Classrooms. For the Learning of Mathematics, 24(1), 13-19.

Pugalee, D. K. (2004). A Comparison of Verbal and Written Descriptions of Students’ Problem Solving Processes. Educational Studies in Mathematics. 55(1-3), 27-47.

Waywood, A. (1992). Journal Writing and Learning Mathematics. Foe the Learning of Mathematics. 12(2), 34-43.