I attended a seminar at Monash University last night that is part of the Centre for Science, Maths and Technology research colloquium series. Professor Helen Doerr from Syracuse University in the US presented her research under the title: “It’s Not Only Talk! Learning to Support Students as Mathematical Readers and Writers”
Her research project involved 5 teachers of lower secondary mathematics over 3 years as they grappled with the requirements of the ‘new’ NCTM standards – particularly relating to the requirement that students be able to read and write about mathematics. The questions that formed the focus of the investigation were:
(1) What does it mean for students to be good mathematical readers and writers?
(2) How can students become better mathematical readers and writers?
(3) What instructional strategies can support this development?
The NCTM standards call for all students to be able to communicate their mathematical thinking. Curricular materials that are provided to teachers presented a challenge to teachers and students as they were texts that needed to be interpreted and understood through talking, reading and writing.
This is something I have believed for a long time – that mathematical thinking is about looking for patterns, testing these for the circumstances under which they prevail (the nature of mathematical proof) and abstracting them in order to generalise and predict. In order to inculcate such thinking in our students, I believe we need to build on ‘the Cs’ – Connections, Coherence, Creation, Collaboration and Conversations. These will only come if students (and their teachers!) can articulate what mathematics and mathematical thinking is about….a focus on the communicative process.
Helen said that the NCTM standards had put emphasis on worthwhile mathematical tasks that related to mathematical ideas, allowed mutliple methods of solution, allowed mutliple representations and afforded students the opportunities to justify, explain and further their understanding through verbalising their thinking. Core characteristics of the tasks were depth, making connections between ideas and sense-making through meaningful activities. Students were meant to be engaged in solving problems, representing ideas and making their thinking visible through the written word.
In the initial stages of the project, Helen found that teachers’ attitudes towards the curricular materials that included these tasks formed a barrier to create opportunities for the students to learn mathematics. As the school in which they worked had a large number of ESL and poor language students, the group realised that the first thing they needed to do was lift the literacy opportunities in their mathematics classes before they could engage their students in worthwhile mathematical learning through the vehicles of these tasks that were language dense and vocabulary packed.
Teachers developed some strategies to increase literacy standards in their classrooms and worked on ways in which they could provide more opportunities for the reading and writing of mathematics. These included:
(1) “Quick Writes” where students wrote a sentence or two about their prior knowledge of something, their understanding of some aspect of the work or what they had learnt in a lesson
(2) RAVE where R = Restate the question, A = Answer the question asked, V = Vocabulary used, E = Explain or give Examples
(3) Scaffolded writing tasks where the teacher prompted thinking in a particular way with the use of deliberate, targeted questions to focus students’ writing
(4) Writing Over Time where students might have been asked to write answers to the same question over the course of a unit to show how their understanding has developed eg. What does it mean for two objects to be mathematically similar?
In this way, students learnt how to read and write about mathematics – a different skill to how to read and write in other disciplines. Throughout these exercises, the correct formal mathematical language was used to give students practice and confidence in its use. (I have often thought that it actually disadvantages ESL students to use words such as ‘flipped’ instead of ‘reflected’)
However, although the use of these strategies certainly provided more opportunities for students to engage with mathematical reading and writing, the teachers found that the students who were good at it already, got better, but the students who were struggling initially didn’t improve..in fact, their writing sometimes got worse. So the provision of more opportunites wasn’t enough to ‘add value’.
The research group realised that the teachers had, in fact, been doing too much modelling for the students. Students could only do the problems when led through the task by the teacher and had developed a very dependent approach, not ‘game’ to try anything without having someone interpret the task for them. The teachers then thought about how they could gradually release responsibility to the students.
The methodology they developed for doing this has strong parallels to the way in which I want to set up curriculum documentation for mathematics syllabi. The teachers started thinking about their writing prompts in terms of what the essential mathematical idea was they wanted the students to understand. So each writing task was planned and thought about first. The planner was split into three columns:
(1) TASK – which was a description of the actual task itself. For example, it could be the question “What do the numerator and denominator of a fraction tell you?” This task had to relate to a key mathematical idea.
(2) RATIONALE – which was basically the answer to “Why did you choose this prompt?”
(3) INSTRUCTIONAL APPROACH – which described the strategies /activities that would lead to the desired understanding
This approach was hugely successful. Not only for the students and their reading and writing of mathematics and mathematical thinking, but also for their understanding of mathematical ideas. Additionally, the teachers, having been compelled to think about what the big ideas were for each unit, also developed their own understanding of what was important in their discipline and in their teaching. It was a methodology that allowed them to focus on the development of mathematics through writing about the mathematics and thinking about what were ‘essential’ understandings.
Perhaps too much emphasis currently is placed on students being able to perform the skills of mathematics without thinking enough about the literacy needs of students in a mathematics classroom. “The words” matter just as much in a mathematics class as in any other and we have to find ways of increasing mathematical literacy.