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Mind your [Maths] Language

It is no secret that South Africa has a notorious track record when it comes to mathematics education. The Trends in International Mathematics and Science Study (TIMSS) places South Africa virtually at the bottom of the pile of participating countries. South Africa’s own Annual National Assessment (ANA) reports show that of the national cohort of Grade 9 learners between 2012 and 2014, at most three percent achieved 50% or more on the assessment with the highest national average being 14% for this period. The National Senior Certificate (NSC) school exit level examinations annually report very poor results. For the nine-year period 2008 – 2016, an average of 33,5% percent of the learners who sat the examination passed with 40 percent or more, so 66,5% of candidates scored less than 40%.

In its reports on both the ANAs and NSC results the Department of Basic Education acknowledges that learners are ‘unfamiliar with mathematical terminology and properties and use them incorrectly’ and that they have poor algebraic skills. The reports further recognise that learners do not engage successfully with mathematical problems that require conceptual understanding. The perennial and lingering unsatisfactory mathematics performance by a vast majority of learners across all grades in South Africa’s classrooms is indicative of a systemic crisis. Add to this that teachers of mathematics are under- or unqualified, that their mathematics competence has been shown to be wanting, and by implication, their ability to teach the subject content is therefore also questionable, we have a dismal picture of mathematics education.

Mathematics education in South Africa is in crisis, has been for decades and the endemic must surely beg the question ‘For how long?’ An elusive answer must lie somewhere. Research identifies what is wrong, but seems not to have the power to ameliorate or remediate a serious problem. Annual reports of the Department of Basic Education (DBE) similarly diagnose in ritual fashion what leads learners to fail but they provide no prognosis or treatment of the symptoms. No amount of hard or soft technology thrown at mathematics classrooms, or malignant diagnoses, has produced a change to national mathematics performances.

The perpetuation of the status quo has several implications. Mathematics literacy compared to mathematics over the past four years alone has on average had 16% more candidates of the total enrolment for the NSC examinations. The reason for this difference is perhaps told by the comparison of the subjects’ pass rates. For the four-year period 2013 – 2016, mathematical literacy has an average of 53,2% of learners who wrote the exam achieving 40% or more, whilst for mathematics, the equivalent statistic is 35,2%. Schools’ pass rates are also increased by diverting learners toward mathematical literacy. There is the other push factor toward mathematical literacy – a paucity of teachers with the requisite subject knowledge to teach mathematics. Consequently, more learners opt for mathematical literacy because it offers a better chance of obtaining a matric certificate but the downside is that mathematical literacy has less buying power in terms of accessing science-related fields of study, or access to university.

Mathematics is important whether it be deemed elitist or not. It is a prerequisite for key vocational and occupational drivers of the economy, mathematics teachers among these. The sad thing is that mathematics is feared, and its elitist maligning comes as a result of those who speak about it, for it, or on its behalf – teachers – who do not know it or understand its language. They, therefore, misrepresent it.  At a level of personal justice, the biggest injustice is toward those who have an innate mathematical acuity but are denied a chance of being a champion of the subject. From perspectives of social justice mathematics in its current crisis cannot impact the economy and begin to relieve poverty.  Constitutionally education is called to free the potential of each person, and mathematics as part of that education fails horribly. There is no justice in mathematics education.

I allude to language in the previous paragraph and it is language that I wish to make the central issue of this article since I believe that language has the potential of realising personal and social justice in mathematics education. Being South African since 1994 means that we are familiar with eleven official languages. Of these English is the language of teaching and learning in most of South Africa’s classrooms, and according to students under my tutelage, the preferred language of teaching in initial teacher education courses. This means of course that most of South Africa’s learners are not taught in their mother tongue.  And in mathematics classrooms, mathematics, the universal language of the sciences is a language within the language of instruction. The compounded effects of the complexities of English as a second language, and the sophistication of mathematics as a scientific language causes me to posit that South Africa is not measuring up in mathematics because it is doubly, and possibly triply, misunderstood in the case of English as the language of teaching, an African language as mother tongue, and mathematics the universal language of science being in the discourse of any single classroom.

Language has been the focus of much research both nationally and internationally over the past decades. In the context of multilingual South African classrooms, the languages of learning and teaching are seen as resources where shared languages enable learners and teacher to revoice their understanding of mathematical content for the purpose of agreed meaning. Code-switching, the alternating of languages while in dialogue, for example, may use the term ‘idenominator’ and refer to its position in the fraction as ‘phansi’ meaning ‘down’ or ‘under’. In this case, the alternating languages would be English and isiXhosa. As useful as code-switching may be theoretically, language as a resource in its broadest application has the potential of creating confusion amongst learners in instances where the language of mathematics is tarnished by incorrect vocabulary.

For dialogue to be meaningful, there is the subliminal presumption that the object of the conversation has the same meaning for its participants. What one communicant accepts and understands as the object under discussion must be exactly the same as the other’s acceptance and understanding. I illustrate this point by an example of what transpires frequently in initial teacher education courses that aim at helping prospective mathematics teachers develop sound pedagogical practices. In mathematics, an expression and an equation are fundamentally different constructs, the former associated with a process of simplification and the other with a process of solution. One, therefore ‘simplifies expressions’ and ‘solves equations’. The terms, however, are used synonymously by a majority of my student teachers, with a leaning toward all mathematical objects needing to be solved, or a hybrid concoction of ‘solve the expression’. This shows that there is no agreed meaning of the words ‘equation’ and ‘expression’ when I am in dialogue with students. Transfer this scenario to an episode of teaching where a teacher does not distinguish between an equation and an expression and an entire class is then placed at risk of cognitive dissonance.

The language ‘of’ and ‘in’ mathematics is in sharp contrast to the resourcefulness of multiple languages used in the classroom. If the former is wanting then the authenticity of the latter is questionable. If one considers the language of the entire mathematics curriculum becoming progressively more sophisticated with increasingly advanced concepts, then the consequences and implications for understanding mathematics are far reaching. A fine-grained vocabular[1] knowledge is core and integral to understanding mathematics.

The Department of Basic Education recognises that learners in the NSC examinations (Matric) are not able to engage successfully with mathematical problems that require conceptual understanding. I suggest strongly, that for teachers and learners, an insufficient and inaccurate vocabulary store with which to assimilate and accommodate concepts is the root cause. We think in words, and construct concepts with them. And it is with these vocabularies that we are able to reason about mathematical objects and their relationships. That we think and reason with words and construct concepts with them are well respected Vygotskian notions on thought and language and we seem as mathematics educators not to have imbibed them.

The Department of Basic Education’s national annual diagnostic reports also acknowledge that the correct use of mathematical language in classrooms is problematic. Acknowledgment is insufficient though – it needs a strategy for implementation.

Paulo Freire’s work champions dialogic education. He emancipated the Brazilian proletariat from their muteness through transforming an education system from one characterised by the deposition of knowledge to one of authentic dialogical engagement between teachers and learners. What he achieved in the nineteen seventies is what needs to be achieved in South Africa in the sense that we need to liberate learners from their muteness, whether that muteness has its roots in oppressive practices, or deficient language with which to engage with one’s critical consciousness and contribute in communicative episodes. To realise this in mathematics classrooms congruent vocabularies of teacher and learner is an essential prerequisite. The practice of using exact mathematical language for teaching will empower learners to ask critical questions that are clearly understood.

The congruency of teacher and learner vocabularies is not only important for meaningful conversation. It carries responsibilities of ethical practices and moral obligation. As teachers, we cannot speak carelessly or sloppily about the mathematics we teach. Learners are dependent on us as being the more informed. They trust us to teach with intellectual integrity. We need to know the difference between and an equation and an expression and be able to explain it. It is incumbent on us to grapple with the essence of the words and language of mathematics we use as teachers to ensure meaningful discourse.

So how does South Africa measure up in mathematics? By current statistics on learner performance, and the capacity and competence of our teachers we do not measure up nationally or internationally in all grades across the school system. The hypothesis proposed here is that a broad focus on language without attention to precise use of terms, and understanding, will not make a substantial difference. By contrast, specific focus on concepts, and the associated terms, and the expression of these concepts verbally and in writing will provide the groundwork for more advanced mathematics.

Indeed, our studies into how language impacts the classroom, without the detailed attention to the precise use of mathematical terms, over the past two decades have not made a difference to the national landscape. The perennially poor outcomes in terms of the quantity and quality of our matric results are evidence of that. It may be time to lift a magnifying glass to scrutinise the exact language that teachers use in their classrooms. We also need to review textbooks with this precise use in mind. We may find that we are inadvertently involved with mathematical deception instead of nurturing mathematical conception.

 

Dr. Pete van Jaarsveld
Mathematics Education Lecturer, University of the Witwatersrand

Image credit: Lawrence University (lawrence.edu)

[1]  ‘Vocabular’ is coined as an adjective to qualify knowledge.

 

 

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