Linking Language in Math Practice

Alfredo Ferreira, Lecturer in UBC Vantage College

VANT140 Math is one of several coordinated ‘Content and Language Linked Tutorials’ in the UBC Vantage College ‘Vantage One’ program, which is a full-credit first-year program with streams in Arts and Sciences in its inaugural year 2014-15. UBC Vantage College students are international UBC students who meet the university’s academic requirements but do not quite meet its language requirements. They take full credit loads in their respective programs as well as credit-bearing courses in the VC Academic English Program: a foundational academic writing course and content-integrated language courses designed to accelerate their academic language development. In Vantage One, generally speaking, there is one VANT140 tutorial course linked to each content-area course that the students take, whether in the Arts or Science streams.

VANT140 Math is linked to two accelerated first-year mathematics courses in the Science stream, Math 100 and 101. The general aims of the VANT140 Math course was to help students describe and explain, using English language, the mathematics concepts and reasoning involved in the math coursework. While this may at first seem counter-intuitive, the language and the math instructors find value in students’ facilities for translating between mathematics and language, and recognize that language, while clearly not central, is integral to much mathematics discourse. The math instructors also identified weaknesses in UBC Vantage students’ performance in explaining mathematics processes in ways that are typically valued in university math classes and tutorials.

Initially the VANT140 syllabus involved analysis of the language portions of the mathematics textbooks. However, these classes tended to lack direct engagement with the mathematics problems and reasoning that the students were encountering in their coursework.

How did you use CLAS in your course and what made you decide to do this?

I (the Vantage College instructor assigned to the VANT140 Math course) was introduced to CLAS by the Vantage College Curriculum Manager, Brian Wilson, with whom I attended a familiarization workshop. I saw the potential of CLAS in VANT140 Math: Math 100/101 instructors would be videoed solving mathematics problems, and the students would annotate the videos. The annotation tasks could be regulated in various ways to support students’ practice and understanding of the links between language use and mathematics practice.

With the first video, the focus was on getting students comfortable with the mediating tools; they were asked for example to put pre-written descriptions and explanations of stages in a math solution in a suitable order after watching the video. Later, they would be given just the time profile of the (say) six stages towards the math solution; at the beginning of each stage students would describe and explain what was going on (this accounts for the long columns of annotations occurring at a single time-point in many of the CLAS videos). Later students were asked to watch the video and break down the solution into stages themselves; we noted that this resulted in a much more diffused pattern of annotations, showing variation in both the level of delicacy of their descriptions as well as their facilities for representing mathematics processes and positioning their explanations rhetorically. An extension of these tasks that supported notions of disciplinary literacy in mathematics espoused by the lead math instructor, Dr. Fok-Shuen Leung, was to have students describe and explain not only the mathematics processes undertaken towards the solution, but also to describe the mathematical choices not made, and to consider why these paths may not have been taken. This model overlaps in important ways with the language literacy models in play, in which language is viewed as a meaning potential, meaningful practice in social context entails choice from the potential, and learning entails expansion of the learner’s functional range.

In the final lessons, in an effort to extend students’ autonomy and critical engagement, students were asked to assess each other’s annotations, recording their assessments on a Likert scale. This led to student-led microstudies of each other’s practices in describing and explaining mathematical processes. This is arguably a sophisticated meta-cognitive task in content-linked language learning. This task also parallels several other micro-research projects students undertook in the first two terms of their program, which aims to develop students not only as users of knowledge but also as creators of valued knowledge.

The types of annotations made ranged from descriptions of a mathematical procedure at one time position to the breakdown of a multiple-stage mathematical solution using annotations at several time positions. Students were also asked to assess each other’s annotations to practice reflecting critically on linguistic representations of mathematical processes

What has been the result?

CLAS has been a tremendous support in a challenging instructional context mainly because it allows instructors, through collaboration between subject-area and language specialists, to scaffold students’ disciplinary learning and practice, and link this learning to language, in principled ways. Although no formal studies have been undertaken, many students appeared to appreciate the tool; for example, they were often eager to see the math instructor solving problems similar to problems they would soon be expected to solve themselves. Furthermore, students’ work in CLAS appears to show developmentally-relevant engagement in and reflection on mathematics practice using language. The language and subject-area instructors considered the experiment with CLAS in the first two terms of the Vantage One program generally successful (noting with thanks that this would be impossible without the wonderful support available to instructors using CLAS!). Additionally, the work with CLAS has potential for much extension, including for example through use of transcriptions of the videos (currently being undertaken) and the use of video recordings of students solving mathematical problems, to be followed by peer review and annotation using CLAS.