Midstream #1: Four anecdotes
I find I am entering my anecdotage3 (a term I came across either referring to Hans Freudenthal talking about instances of his grandson’s actions or comments, or perhaps him using that term about himself). But I hope that these brief-but-vivid (Mason, 2002, p. 46) short stories can also be seen pointed narratives.
- (a) When I was in graduate school at the University ofWisconsin-Madison in the late 1970s, I had an older friend there whose brother was an algebraic topologist. She could not possibly read the vast majority of each article he had published due to its intense mathematical notation, so she had him read them to her. She found the language both rhythmic and engaging, so decided to write an opera with this English-language transformation of the mathematics as its libretto.
- (b) While working at the Open University in the 1980s, students came to a university every summer for a single week to be in the face-to-face and audible presence of teachers and other students which for some was their only plural contact. Repeatedly, I would hear them observe, “I didn’t know how to say (or read) this until now”.This was surprising for me then, because I had been surrounded by mathematical speech and writing combined for many years, but had not consciously realised its significance. But, even now, when looking at certain mathematical texts where I do not recognise the symbols I am placed in a similar silenced position with regard to being unable to read the text aloud. I do not know what to say.
- (c) My uncle, who lived in Australia, regularly read books aloud for the local radio, especially for those who were unable to read themselves for various reasons. Rather than commit to a specific reading (with verbal emphasis, tonal variation and expressed emotion, among other things), he read them as completely flat and uniform as possible (just as the print on the page customarily is). This too was during the time when I was working at the Open University, which started making regular audio-recordings of the written mathematical units of each course for visually impaired students. On the first occasion I had opted to do this, expecting to be able to read them fluently and straightforwardly (though neutrally), I was confronted by a real-world graph on the second page of Unit 1 which silenced me (a different form of silence than in the previous anecdote). After an awkward pause, I produced a mix of description and regeneration, which I felt necessary to transform a two-dimensional image into one-dimensional speech (being unable to say two things simultaneously, the ordinality of speech being a paramount feature).
- (d) In 2000, Lynn Paine (a former non-mathematics education colleague at Michigan State University) and I were in Shanghai for two weeks, among other things attending both regular and public mathematics lessons, while working on a project about new teacher induction (for more on this, see Britton, Paine, Pimm, & Raizen, 2003). Unlike me, who had and needed a local mathematics graduate student by my side whisper-translating in real time, Lynn is highly fluent in Mandarin and so just listened to the classes directly. In a grade eleven mathematics class, however, something came up, an expression that she did not recognise, and she murmured to my translator (in Mandarin),“What does xxx mean?” (she was perfectly capable of echoing the statement without understanding it), who replied (in English), “A dihedral angle which is formed by the intersection of two planes”. And then Lynn turned to me and asked (in English),“And what does tlial mean?”Two very different meanings for mean!
Midstream #2: Somewhat picky transcriptions
This starting point begins quite small and specific, and, although not intended to be, might feel a little picky. It concerns, at its base level, an aspect of transcribing mathematics-related speech - but for me it also reflects some significant differences between mathematical speech and writing, differences that relate to the mathematics register. Without needing to cite anyone in particular as erroneous individuals to point a finger at (myself included, over time), it is commonplace to find mathematical symbols (including, but certainly not limited to, numerals and arithmetic operations) presented in classroom transcriptions, as if they had been spoken thus.
There is an obvious convenience in so doing, not least because one aspect of mathematical notation is compression (as well as easing manipulation) and, hence, it reflects a certain economy. But there is an unnecessary and unhelpful ignoring of what was actually said and so the transcript does not accurately reflect it. For instance, if r2 was presented in the transcription, the speaker could actually have said, “r squared”, “r to the power two”,“r to the little two”, or perhaps “the radius times itself” or even “the area of a square whose side length is r”. And on and on.
Here are another couple of very small examples: people never say ‘100’. In English, we say “hundred” or “a hundred” or “one hundred” (or perhaps even “one zero zero” or “one oh oh”, what I termed ‘spelling pronunciation’ - see Pimm, 1987, p. 181). People never say *+’. In English, we say “plus” or “add” or “added to” or ... Obviously, different things (sounds)4 would be said in French or Russian or Mandarin or Hindi in response to the same written symbol. If you see in a transcript, for example,“6 sq. in.”5 or “6 sq in” or “6 in2” (where the double quotation marks do indicate a quotation6), this is most unlikely what a student actually said. In the last instance, a student might have said (in English) “six inch squared” or “six square inches” or (if less clear, perhaps, about the notation) “six inches to the second power” or “six inches to the power two”. And another set of instances could include fractions: e.g. “13/24”, rather than “thirteen twenty-fourths” or “thirteen over twenty-four”.7
However, such occurrences are not limited to numbers, arithmetical operations and unit abbreviations. One only has to look at transcriptions of secondary- or tertiary-level classes. Yet transcriptions are supposed to be both accurate and precise, reflecting the spoken language (in some instances including dramatic ‘audio’ detail - see, for instance,Jefferson, 2004 - which some refer to as prosodic aspects of speech) - each one claimed to be a subsequently written version of the actually said speech. So why is it that mathematical notation, not specific to a particular language (though certainly created by people who had specific native languages and that frequently reflect both syntactic and semantic aspects of those languages), is included in a transcript as if it were?
The above is about transcriptions being textual presentations created subsequent to prior recorded speech and not, for example, lesson plays (Zazkis, Sinclair, & Liljedahl, 2013), where the purported classroom ‘speech’ is actually written first, nor Lakatos-like scripts (Lakatos, 1976). Having said this, I think that teachers should follow this proposed transcription criterion too, not least as it could draw their attention to aspects of mathematical symbolism in relation to language in their classrooms. And when teachers, either pre-service or in-service (see Stenner, 2019), actually ‘play out’ their constructed lessons, transcribing the actual lesson might also bring certain aspects into play. There is also the inverse work of Sue Staats (2008, 2017, 2018), where she creates/recreates written transcripts in a poetic form to try to communicate aspect of the oral form better. Any transcription is, necessarily, among other things, a re-presentation.
For one of the things about speech is that speech is always spoken within a natural language (or, in the case of sociolingustic code-switching, or of sociocognitive translanguaging, across more than one natural language: see, for instance,Setati, 1998, for the former, and Garcia and Kleyn, 2016 - as well as the Planas and Chronaki (2021) chapter in this book ‘Multilingual mathematics learning from a dialogic- translanguaging perspective’ - for the latter). But even the sophisticated plurality of translanguaging does not affect the non-inclusion of mathematical notation (which does not belong to any natural language, spoken or written).
Therefore, every single word (or character or...) in any transcription should be exactly that, namely written elements in the language(s) of the speaker (or a language of translation to match the language of the article or book). Which raises for me a reversing question: since written mathematical notation does not belong to any natural language, how can written mathematical notation be ‘read’ (aloud), and thereby incorporated into one (or more) language, considering that it does not belong to any natural language, although it is primarily alphabetic (which many natural languages are not8). In order to be said aloud, with both grammatical and semantic elements, it needs to be invited (projected?) into a (specific) natural language.
My response, needless to say, features the Hallidayan mathematics register (about which more soon), while stressing the fact that any particular mathematics register (that in English, Hawaiian, Russian, Arabic, German, Basque, Chinese, Maori,...) is, of course, a (proper) subset of a specific natural language (both spoken and written). It may be the case, however, that some individuals’ personal mathematics registers may well cross languages. And while a transcription is written, it needs to be completely within the language. Mathematical notation is not part of it, nor are diagrams of whichever sorts (geometric and otherwise). An under-studied aspect of mathematics education - perhaps with the exception of Maori (see, for instance, Barton, Fairhall, & Trinick, 1988, or Meaney, Trinick, & Fairhall, 2012; also, in relation to a first nation Canadian language, Mi’kmaq, see Lunney Borden, 2011) involves the development of any mathematics register over time, both its successes and failures (e.g. Robert Recorded failed attempts to generate anglophone terminology, rather than Latin or Greek derivations, with respect to the English mathematics register - see Fauvel, 1989). Most of the European language mathematics registers are only some 500 years old.
While words are also symbols, mathematical symbols are not words. They are not part of any particular language (though they are not completely independent, structurally, of features of the languages spoken by the creators of such symbols), nor are images (such as geometric diagrams or graphical depictions) for that matter. Which is why the mathematics register in any specific language is required to be developed to a point where it enables articulating mathematics orally into whichever natural language is required. Mathematics registers, among other things, allow us to ‘lin- guify’ mathematical symbols. Though, of course, in multilingual classes each student (and teacher) needs to engage with and work through the nature of the so-called ‘language’ of instruction (whether monolingual or polylingual in fact, including how mathematical terminology and grammar is deployed).
Here is a small example of mathematical authors deliberately drawing attention (both of pronunciation and grammar) to an individual aspect of the English mathematics register that relates to a specific mathematical notation (from a textbook recommended for my first-term undergraduate lecture course on real analysis in 1971, given by Rolph Schwarzenberger, at the University of Warwick):
If an entity is in a given set, it is said to be an element or member of the set. There is a notation for this; if the element is called t and the set is called S, we write teS, which is read as‘t is in S’ or ‘t is a member of S’. The symbol G is derived from a Greek e, but is usually written slightly differently to avoid confusion. For reasons of grammatical convenience, teS is sometimes written S3t, which is read as ‘S contains f. (Notice that in the sentence ‘S3t’ the verb
‘contains’ is in the active, whereas if the sentence ‘teS’ is read as‘t is contained in S’, the verb is in the passive.This is in accordance with the usual mathematical habit of writing the passive of a verb denoted by reversing that symbol.)
(Moss & Roberts, 1968, pp. 1-2)
These mathematical symbols are included in a piece of text written primarily in English, a text written in English about how to read mathematical notation in English. But this is actually a brief meta-text, explaining how to ‘read’ (in the sense of‘read aloud’, but not just pronounce or describe) a piece of (written) notation that is not English-specific. And there are two alternatives offered, among other things indicating that this is not a singular symbol with a one-to-one correspondence with an English word or expression, as well as mention of its grammatical features that the particular language requires.
