# Secondary and university mathematics: Do they speak the same language?

The decline of tertiary enrolments and the low retention rates in mathematical sciences is a national and international concern.

The project will examine symbol use, a key aspect of students’ mathematical experience at school and university, and measure its impact on progression rates. Research, at secondary level, shows that mathematics symbols’ conciseness and abstraction can be a barrier to learning. Discontinuities in how symbols are used can make mathematics seem like a foreign language so students lose confidence. This project will explore the equivalent issue at the university level, when mathematics becomes more symbolic and its writing more subtle, requiring increased ‘flexibility’ from students. It will ascertain the impact on progression rates in mathematical sciences.

The project’s outcomes will include illustrated descriptions of similarities and contrasts between conventions and use of symbols in secondary and university mathematics and across mathematical sciences. It will provide evidence of links between students’ response to increased symbolic load and their confidence to continue studying subjects with high mathematical content at university, thus grounding potential teaching action plans to smooth secondary-university transition. The outcome will be innovative teaching practices to retain students in mathematical sciences, and hence increase the STEM workforce.

The project has been approved by the University of Melbourne Human Research Ethics Committee and is funded by the Australian Research Council under the Discovery Project scheme for 2015 (DP150103315).

## Rationale

Mathematics derives much of its power from the use of symbols (Arcavi 2005), but research at secondary level has shown that their conciseness and abstraction can be a barrier to learning (Pierce, Stacey & Bardini 2010; MacGregor & Stacey 1997). We anticipate that many students have difficulty with the new and more intense ways in which symbols are used at university (these will be described below), with the consequence that they do not understand the mathematical content as well as they did before, which leads to a decrease in positive affect, which in turn discourages enrolment in further mathematical subjects.

## Team

Dr Caroline Bardini

Leading Chief Investigator

Ph: +61 3 90353679

Email: caroline.bardini@unimelb.edu.au

A/Professor Robyn Pierce

Co-Chief Investigator

Ph: +61 3 83448519

Email: r.pierce@unimelb.edu.au

Emeritus Professor Kaye Stacey

Co-Chief Investigator

Ph: +61 417399 745

Email: k.stacey@unimelb.edu.au

Mrs Meredith Begg

Project Manager

Email: meredith.begg@unimelb.edu.au

### Advisory panel

- Dr Deborah King (University of Melbourne)
- A/Prof Michael Page (Monash University)
- A/Prof Helen Chick (University of Tasmania)
- Dr Alasdair McAndrew (Victoria University)
- Dr Jason Giri (Federation University)
- Janine McIntosh (Australian Mathematical Sciences Institute)

With the collaboration of Victorian Secondary Schools

## Publications

### Current publications

Pierce, R., Begg, M., Chin, K. E. (2017, November 14). *Symbolic Synonyms*. AMSI Calculate Teacher Resources.

Pierce, R., Begg, M., Chin, K. E. (2017, November 01). *Met Before*. AMSI Calculate Teacher Resources.

Pierce, R., Begg, M., Chin, K. E. (2017, October 17). *Communicating Like a Mathematician*. AMSI Calculate Teacher Resources.

Vincent, J., Pierce, R. & Bardini, C. (2017). Structure sense: a precursor to competency in undergraduate mathematics,* Australian Senior Mathematics Journal*, 31(1), 38-47.

Begg, M. & Pierce, R., McAndrew, A. (2017). Incomprehensible workings followed by correct answers. In V. Barker, T. Spencer & K. Manuel (Eds.) *Proceedings of the 26 ^{th} Biennial Conference of the Australian Association of Mathematics Teachers. *pp 64-70. Adelaide: AAMT.

Pierce R., & Begg, M. (2017). First year university student’ difficulties with mathematical symbols: the lecturer/tutor perspective. In A. Downton, S. Livy and J. Hall (Eds.), 40 years on: We are still learning! *Proceedings of the 40 ^{th} Annual Conference of the Mathematics Educational Research Group of Australasia. *(pp. 413-419) Melbourne, MERGA.

Pierce, R. & Bardini, C. (2017). Mathematical writing and writing mathematics: The transition from secondary to university mathematics. In A. Downton, S. Livy and J. Hall (Eds.), 40 years on: We are still learning! *Proceedings of the 40 ^{th} Annual Conference of the Mathematics Educational Research Group of Australasia. *(pp. 608-611) Melbourne, MERGA.

Bosch, M., Bardini, C., Rasmussen, C. & Trigueros, M. (2016). Current interactions between mathematicians and researchers in mathematics education. *First conference of International Network for Didactic Research in University Mathematics*. Montpellier, France. <hal-01337872>.

Bardini, C. & Pierce, R. (2015). Assumed mathematics knowledge: the challenge of symbols. *International Journal of Innovation in Science and Mathematics Education, 23*(1), 1- 9.

Vincent, J., Bardini, C., Pierce, R. & Pearn, C. (2015). Misuse of the equals sign: an entrenched practice from early primary years to tertiary mathematics. *Australian Senior Mathematics Journal, 29*(2), 28-36.

Bardini, C., Pierce, R. & Vincent J. (2015). Contemplating symbolic literacy in first year mathematics students. In M. Marshman, V. Geiger & A. Bennison (Eds.), Mathematics Education in the Margins*. Proceedings of the 38 ^{th} annual conference of the Mathematics Education Research Group of Australasia*, pp. 77-84. Adelaide, Australia: MERGA.

Bardini, C. (2015). The reader and the writer perspectives or the subtleties of symbolic literacy. In K. Beswick, T. Muir & J. Wells, Mathematics education: climbing mountains, building bridges. *Proceedings of the 39 ^{th} Conference of the International Group for the Psychology of Mathematics Education*. Hobart, Australia: PME.

### Prior related publications

Pierce, R. & Bardini, C. (2015). Computer algebra systems: permitted but are they used? *Australian Senior Mathematics Journal, 29*(1), 32-42.

Bardini, C., Pierce, R. Vincent, J. & King, D. (2014) Undergraduate mathematics students' understanding of the concept of function. *IndoMS-Journal of Mathematics Education, 5*(2) 85-107.

Bardini, C. Vincent, J. Pierce, R. & King, D. (2014) Undergraduate mathematics students' pronumeral misconceptions. In J. Anderson, M. Cavanagh & A. Prescott (Eds.) Curriculum in Focus: Research Guided Practice, Sydney: MERG.

Bardini, C., Oldenburg, R., Stacey, K., & Pierce, R. (2013). Technology prompts new understandings: the case of equality. In V. Steinle, L. Ball & C. Bardini (Eds.), *Mathematics education: Yesterday, today and tomorrow*. *Proceedings of the 36 ^{th} annual conference of the Mathematics Education Research Group of Australasia*), pp. 82–89. Melbourne, VIC: MERGA.

Bardini, C., Pierce, R. & Vincent, J. (2013). First year university students' understanding of functions. Over a decade after the introduction of CAS in Australian high schools, what is new? In L. Rylands, B. Loch and D. King, *Proceedings of the 9 ^{th} Delta conference on teaching and learning of undergraduate mathematics and statistics*. pp. 2–11.

## Conferences and seminars

### 2017

**First Year in Maths Workshop 6: Supporting undergraduate mathematics education**

Dr Alasdair McAndrew attended the FYIMaths workshop, which was held at the University of Melbourne (July 13 - July 14, 2017).

Alasdair’s presentation shared key findings from the project and suggested implications for teaching first year university mathematics. It drew on data collected from five Universities, including interviews with University lecturers and first year University students, and reported on findings related to:

- Student difficulties with symbols use.
- Student difficulties reading and writing mathematics
- Differences in the use of mathematical symbols across the mathematical sciences
- Symbol use in other STEM disciplines such as physics, chemistry and engineering, and the opportunity for confusion when these symbols (or their usage) differ from mathematical convention.

**Australian Association of Mathematics Teachers (AAMT) Conference**

Associate Professor Robyn Pierce and Meredith Begg attended the AAMT Capital Maths conference in Canberra (11 July to 13 July 2017).

Their presentation titled “Incomprehensible workings followed by ‘correct answers’” focused on problems identified in student workings, and drew on data collected through interviews with University lecturers and students, and survey data completed by first year University students. A paper related to this was included in the conference proceedings.

**Abstract:** Lecturers and teachers lament over the difficulties their students have writing coherent mathematical statements. Students’ habitual misuse of mathematical symbols, combined with an inability to explain what mathematical steps are being taken and why, is a concerning observation. At best, this represents a lack of mathematical rigour and at worst, may undermine learning. This paper reports on findings to-date of a project which investigates the use of symbols in mathematics and related sciences, as students progress from secondary school to university, with a focus on implications for secondary school teaching and assessment.

**Mathematics Education Research Group of Australasia (MERGA) Conference: Symposium**

Associate Professor Robyn Pierce and Meredith Begg attended the MERGA 40 conference at Monash University in Melbourne (2 July - 6 July 2017).

__Symposium__

Robyn was a member of the symposium panel, discussing Transitions in Mathematics Education. A paper related to this was included in the conference proceedings.

**Abstract:** This paper reports findings from research on students’ use and understanding of mathematical symbols, which has been recognised as playing a major role in students’ success in mathematics. One goal is to identify potential trouble spots for the usage of symbols as students travel the path between school and university mathematics. We present a framework that supports a fine-grained analysis allowing us to better apprehend the subtle differences in students’ experiences with symbolic expressions. This is illustrated by considering some first year mathematics students’ written responses to questions and insights from interviews with senior secondary teachers and university lecturers and tutors.

__Research Presentation__

Separately, Robyn and Meredith presented on First year university students’ difficulties with mathematical symbols: the lecturer/tutor perspective, drawing on experiences and observations reported by lecturers and tutors across three different Universities during interview. A paper related to this was included in the conference proceedings.

**Abstract: **School and university mathematics: do they speak the same language? Our university mathematics students come from amongst the successful school mathematics students but what difficulties with symbols do their lecturers and tutors observe? This paper reports on data from interviews with twenty-one first year lecturers and tutors from four universities. Key emerging themes focused on mathematical communication: the importance of comprehending mathematical symbols and of composing a mathematical narrative consisting of both explanatory words as well as symbols.

**University of Tasmania Seminar: Associate Professor Robyn Pierce and Meredith Begg**

Presenting the findings of the project to date to mathematics and education staff at the University of Tasmania (1 February, 2017)

### 2016

#### The Mathematical Association of Victoria (MAV) Conference

Associate Professor Robyn Pierce attended the MAV annual conference held at La Trobe University in Bundoora, Victoria (December 1 – December 2, 2016).

Robyn's presentation was titled "Symbol sense: from school to university". In this well-received talk, a synopsis of the project and findings to date were conveyed, with a focus on appropriate examples of student difficulties with symbols, followed by a discussion of the implications of this to teaching and learning and some practical ideas for teachers to take away to the classroom.

A copy of the presentation is available upon request to the Project Manager.

#### International Congress on Mathematics Education (ICME) Conference

Dr Caroline Bardini attended the ICME conference in Hamburg, Germany (July 24 – July 31, 2016).

Caroline's presentation was titled "Déjà vu in mathematics: What does it look like?".

**Abstract:** Analysis of mathematical notations must consider both syntactical aspects of symbols and the underpinning mathematical concept(s) conveyed. We present here the construct of 'syntax template' as a theoretical framework for analysing students' written responses to mathematics problems. We give an illustrative example of symbol-related errors and lack of attention to symbolic structure from a pilot study of a 3-year project on students' understanding and use of mathematical notation.

**First Year in Maths Workshop 5: Evaluating Learning and Teaching**

Dr Caroline Bardini and Associate Professor Robyn Pierce attended the FYIMaths workshop held at the University of Melbourne (July 11 - July 12, 2016).

Caroline's presentation was of the project findings to date and was based on combined data from three Universities.

#### Monash University Seminar: Dr Carolyn Bardini and Dr Jill Vincent

Presenting the findings of the Semester 1 2016 student survey data (June 24, 2016).

#### Indrum 2016 First Conference of the International Network for Didactic Research in University Mathematics

Dr Caroline Bardini attended as an invited speaker at the plenary panel session, held at the University of Montpellier, France (March 31 - April 2, 2016).

Indrum 2016 is the first in a series of biennial conferences on mathematics education in higher education.

### 2015

#### MGSE Seminar: Dr Caroline Bardini

Do university students mean what they write and write what they mean? (October 6, 2015).

#### Australian Mathematical Society (AustMS) Conference

Dr Caroline Bardini attended the Adelaide conference of AustMS (September 28 - October 1, 2015) and presented on the topic Symbols: do university students mean what they write and write what they mean?

**Abstract:** Analysis of mathematical notations must consider both syntactical aspects of symbols and the underpinning mathematical concept(s) conveyed. We argue that the construct of 'syntax template' provides a theoretical framework for analysing undergraduate mathematics students' written solutions, where we have identified several types of symbol-related errors and lack of attention to symbolic structure. A focus on syntax templates may address this issue of an under-developed symbol sense of many tertiary mathematics students.

#### Psychology of Mathematics Education (PME) Conference

Dr Caroline Bardini, Dr Robyn Pierce and Dr Helen Chick attended the PME Conference in Hobart (July 13 - July 18, 2015).

Caroline's presentation was titled The reader and the writer perspectives or the subtleties of symbolic literacy.

**Abstract:** This paper draws on a larger study (Bardini, 2003) where epistemology is envisaged as a complementary tool for didactic analyses of students' use and understanding of algebra. We will focus on one particular epistemological idea explored in our work namely 'the author and the reader' perspectives (Serfati, 2005), here referred to as 'the reader and the writer' perspectives. We will describe the potentials offered by such an epistemological lens when it comes to better understanding what underlies the construct of algebraic expressions and highlighting the multiple facets that constitute what we may call 'symbolic literacy'. As a practical implication, we will show how the epistemological framework provides a fine-grained tool for selecting and devising appropriate tasks aimed at assessing students' symbolic literacy.

#### Mathematics Education Research Group of Australasia (MERGA) Conference

Dr Robyn Pierce and Dr Helen Chick attended the MERGA conference on the Sunshine Coast (June 28 - July 2, 2015).

Robyn's presentation, Contemplating symbolic literacy in first year mathematics students, provided an overview of the research construct of 'syntax template'.

**Abstract:** Analysis of mathematical notations must consider both syntactical aspects of symbols and the underpinning mathematical concept(s) conveyed. We argue that the construct of syntax template provides a theoretical framework to analyse undergraduate mathematics students' written solutions, where we have identified several types of symbol-related errors. A focus on syntax templates may address the under-developed symbol sense of many tertiary mathematics students, resulting in greater mathematics success, and with the potential to improve retention rates in mathematics.

## Teacher resources

- Communicating like a mathematician (PDF 779KB)
- Symbolic Synonyms (PDF 667KB)
- Met Before (PDF 671KB)

## Student resources

- Suggestions for Students (PDF 626KB)