The mathematics of writing

A mathematician, like a painter or a poet, is a maker of patterns… The mathematician’s patterns, like the painter’s or the poet’s must be beautiful; the ideas like the colours or the words, must fit together in a harmonious way. Beauty is the first test.

GH Hardy

How are most children taught writing?


Eight weeks ago I took over an AS English Language class in which none of the students had a clear understanding of the difference between a noun and a verb. How is that they have got so far through formal education with absolutely no explicit understanding of how sentences work? The answer, my friend, is that teachers’ own language skills are just not up to snuff.

I had an argument with Phil Beadle recently in which he maintained that he’d never met an English teacher who a) knew what a sentence was and b) knew how to use a comma. I was shocked. Could this really be true? Obviously I proceeded to demonstrate my own understanding in true show off style but this merely disguises the problem he was trying to describe. It really doesn’t undermine his argument to say, I’ve only met one English teacher who knows what a sentence is. (See below for definitions.)

Like most English teachers, I’m a graduate of English Literature and, like most people my age, I escaped any hint of grammar teaching in my own education. My great good fortune was to teach English as a Foreign Language (EFL) before becoming a ‘real’ teacher. I had to get to grips with my trusty copy of Michael Swann’s Practical English Usage in fairly short order to be able to field the steady stream of questions about present participles and phrasal verbs.

As products of this system, the modern English teacher is very comfortable discussing metaphor, alliteration and other literary techniques but is often rather out of their depths with semi colons and conjunctions. Needless to say, if we don’t know these things, there’s little chance they will!

My personal bête noir is the lie that you put a comma where you take a breath. I’ve lost count of the number of children that I’ve had to disabuse of this misapprehension: it is simply not true. That said, knowing that punctuation marks were originally notation for actors on how to read scripts does give some credence to this theory and while it’s still fairly useful advice that you might take a breath where you see a comma, it’s certainly bad advice for our putative writer. So what to do?

Well, the teaching of punctuation deserves a post of its own; here it is my intention to demonstrate how approaching sentence construction from the logical and precise stand point of the mathematician might be helpful. Basically, one has to start by knowing that a sentence contains the following elements:

  1. A subject. This is the noun (or noun phrase) about which the sentence is about
  2. A verb. This is the process by which the subject interacts with the object. It is not a ‘doing word’.
  3. An object. This is (usually) the noun (or noun phrase) with which the subject is interacting. Sometimes it isn’t, so if you’re not happy with object, refer to it as ‘other’. It’s all good.

For instance: I (the subject) am (the verb) a teacher (the object).

The observant among you may have noticed that I failed to label ‘a’ (an indefinite article) and that’s deliberate. For one, I don’t want to over burden anyone and also they aren’t required in a sentence. A better, purer example perhaps might be:

David (subject) loves (verb) English (object).

This understanding of the SVO structure can then be applied to existing sentences. Here’s one entirely at random from earlier in the post:

Like most English teachers, I’m a graduate of English Literature and, like most people my age, I escaped any hint of grammar teaching in my own education.

Now, this is a fairly complex sentence made up of 4 different clauses which I’ll try to deconstruct into its component parts:


And other stuff:

If you were then to transcribe this sentence as part of an equation it would something like this:

P O, S V O C, P O, S V O.

The commas are doing a similar job to that of + signs and help us see the different clauses within the sentence.

This is useful when marking students’ writing and you encounter something like this:

As I ran down the street.

Because it starts with ‘As’ (a coordinating conjunction) it’s a subordinate clause and cannot therefore be a sentence in and of itself. Our response to this fragment (unfinished sentence) is to scream, “What? What happened when you ran down the street?” Students need to know that where this happens a sentence needs to look like this:

C S V O, S V O.

And that anything that is left merely as C S V O is wrong. It’s also worth knowing that if a subordinate clause begins a sentence it is always followed by a comma. So in effect what you would be doing is leaving the comma dangling and then just starting a new sentence with a capital letter. Obviously, you wouldn’t do that because the vast majority of people know that you end a sentence with a full stop. But what if commas were understood in the same way? What if we knew, deep in our souls, that you only ever use a comma to divide single items (nouns or adjectives, or phrases if you want to get technical) in a list or when a subordinate clause begins or is embedded in a sentence?

If we knew that we could teach it. If we knew that a sentence describes the relationship between a subject and its object then maybe we’d have more luck communicating this knowledge to our students.

We could then instruct them to write a sentence which did this:

S V O; S V O.

Or this:

V, S O.

And, by God, they’d know how to do it!

But language is messy. Maths on the other hand is neat and ordered. If algebra makes sense to you, it is a realm of certainties. So, could writing harness some of this logic and precision (while remaining mindful that “fiction that does nothing but follow rules is cold arithmetic”?) Can we, as English teachers (and don’t forget that every teacher in English is a teacher of English) give students the mental tools to be able to construct technically accurate sentences? And does it even matter?

Some may argue that all this emphasis on grammar stifles creativity. To them I say, pah! We wouldn’t value a mathematician so focused on a creative solution to a problem that they couldn’t add up, or an architect whose ‘creative’ buildings were unbuildable. We value precision in so many other fields, why is it OK for writing to be sloppy?

I’m pleased to report that after 8 weeks of an intensive crash course in grammar, my AS class are now able to write. They are so much more thoughtful about how they’re writing rather than just dumping their thoughts on the page. I would argue, and so would they, that this has allowed them to be much more confident and creative in their writing. Most of all, it’s allowed them to decide when, where and why they might want to break the rules.

And crucially, none of this need be dull. Just as there are bucket loads of creative, exciting maths teachers out there, so too can there be regiments of outstanding grammarians. Take a leaf out of the wonderful Dancing about Architecture for some excellent ideas on how to combine the physical with the abstract.

And finally, a plea

Please note that am in no way an expert in grammar. I do my ‘umble best with the little I know. I accept that having the temerity to write about these things opens me up to all sorts of criticism from the sort of people I’d fear to meet down a desk alley. Be gentle with me. I know that ‘object’ is often wrong and that we could dance for hours discussing predicates and compliments. But to what end? This level of knowledge is sufficient to vastly improve students’ knowledge of and ability to improve writing. And that’s what matters. I thank you.

Related posts

Slow writing: how slowing down can improve your writing
Is grammar glamorous?
Does creativity need rules?
More useful stuff on commas here
Why Fiction Writers Should Learn Math

Lisa Jane Ashes on the fusion of Maths and English: Manglish!


31 Responses to The mathematics of writing

  1. Stuart Lock says:

    As a Maths teacher, I have some time for this post. However, it also exposes something I’ve long felt and not expressed about my experience of the teaching of English when I was a student in comparison to what I see and read as an adult.
    When I was taught English, I can hand on heart say that I did not know what an adjective was. I barely knew of a verb as a “doing word” and a noun as a “person, place or thing”. I certainly didn’t know what a sub-clause, present participle or any of that stuff is (and don’t now, if I’m honest).
    I was also the most able in my English class. Or rather, I used to write some of the best essays. I went on to take an essay-based subject in my degree and MA (Philosophy) and I still now regularly proofread the work of other adults I work with.
    Now I grew up in a household where despite a high level of poverty, I used to loan books from the mobile library. I would stay up until 3 or 4am reading a book I was enjoying (before getting up at 5am for my paper round). It was my number two favourite pastime behind playing football. I’m sure I absorbed a lot of the grammar rules in reading.
    So I don’t understand why, increasingly in my experience, we’re obsessed with teaching the formal rules to students. Isn’t English supposed to be creative? Aren’t we better off telling and reading stories via English? Isn’t it better that we say “wouldn’t it sound better like this?” rather than “you must include a sub-prime-present-participle-adjunct-clause”?
    I can tell the sentence “As I ran down the street” isn’t finished, but not because I know that it starts with a co-ordinating conjunction (that sounds made up to me) but because I would think, and hence say “you did what as you ran down the street?” in exactly the way you describe. I think that adding that it is a co-ordinating conjunction doesn’t really add to the feedback one might give the learner here.
    As English teachers, aren’t we trying to get students to use language to make what they write, or say, sound better?
    I just don’t get the formal rules of writing English, and I don’t think I’ve ever needed to.
    I realise there’s a significant risk that a sentence or two in this post fail to meet the conventions of formal English!

    • learningspy says:

      Stuart, I too knew nothing about grammar when taught English at school. I too instinctively wrote well. Unlike you though I’m troubled by my lack of knowledge in these areas. It’s not enough for me to say, “Wouldn’t it sound better like this?” because I always want to know why. You wouldn’t accept this as a maths teacher would you? Don’t you have to show your working?

      I think that just because you and I have an implicit understanding of grammar doesn’t mean that the student we teach do. One of the most important things we can do is to make our implicit understanding explicit. I wouldn’t say I was ‘obsessed’ with teaching rules, but I know that any knowledge of grammar makes an immediate and lasting difference to the quality of writing of all my students but most especially the less able.

      Thanks, David

  2. A Smart says:

    Having studied Lit and Lang for my main degree and following that with an MA in Applied Linguistics, I see a noticeable lack of Language graduates in English teaching. I totally agree that because teachers are wary of the grammar, they focus less, if at all, on these topics in the classroom. Grammar is complicated but a simple awareness of it really can help students understand the crafting of their language. Teachers need to be happy to field potentially difficult questions by admitting ‘I’m not sure, let’s look it up.’ Something which can happen an awful lot when studying grammar!

    P.S. In the sentence ‘I am a teacher’, ‘a teacher’ is a complement rather than object. Equation: I = teacher, if subject relates to same thing as what is in ‘object’s place’ through use of linking verbs then it’s complement 🙂

  3. Ant Heald says:

    Your diagnosis about English teachers not knowing language structure is, I think, in large measure correct. Part of the problem in putting it right is a lack of agreement over the terminology and methods of analysis involved.

    My jaw dropped a little at your contention that a sentence must contain an object. However you analyse it, that isn’t true. Where’s the object in ‘The man fell over’ or ‘I am happy’? And if there’s no object in the latter then is there one in ‘I am an English graduate’? You can say that ‘an English graduate’ is an object because it’s a ‘thing’ (a noun phrase), but then you’re back to defining grammatical units by their meaning rather than function, leading to the kind of confusion you get when you chuck a sentence like ‘Cycling is my favourite sport’ at students, ask them which is the ‘doing word’, and then ask them which is the verb. If both ‘happy’ and ‘an English graduate’ are telling you what ‘I am’, then grammatically they must be doing the same job, and the grammar I’m familiar with refers to that as the ‘complement’, (not object) of the clause.

    At this point we start getting into the kind of ‘naming of parts’ confusion that Stuart Lock is objecting to. But I don’t think the idea that you can just ‘pick this stuff up’ holds water. Do you think the same applies to Maths, I wonder Stuart? I’m sure some people can just ‘intuitively’ pick it up, but most of us need rigorous structured teaching and practice. I think the same applies to English structure.

    Discussions about grammar crop up a lot on the English Language List email group I’m on which is blessed by regular contributions from Dick Hudson, emeritus professor of linguistics at UCL, and a long time contributor to work on teaching grammar in schools. He recently referred to a symposium discussing the development of a more coherent approach to grammatical terminology. I hope something can be done on that front. The fact you refer to ‘as’ as a coordinating conjunction rather than a subordinating conjunction shows the importance of accurate terminology here. The concept, as you rightly say is key to what’s going on in structures like this, and I think having a shared understood terminology would be immensely valuable. Without it, as Stuart suggests, the use of terminology can seem like an unnecessary layer of confusion. However, with the concepts that lie behind the terminology all sorts of things become clear that otherwise are muddled or left to chance. (How do you know that ‘must of done’ is wrong if you don’t know how verb phrases work. It sounds right, dunnit?) Stuart: “wouldn’t it sound better like this?” Only works when the student thinks it would sound better like this. 2+2=4, and 2×2=4. Easy this numbers game, so 3+3=6, therefore 3×3=6, yes? I don’t need to be told that multiplication is a different kind of operation than addition. In fact, why use fancy words like ‘addition’ and ‘multiplication’? Doesn’t that just confuse the issue, and we can’t be filling their dear little heads with *difficult* stuff, can we?

  4. Rob Spence says:

    I sympathise, and have undergraduates studying English who can’t confidently identify nouns and verbs. But I disagree on the history of the problem. The explicit teaching of parts of speech has been enshhrined in the National Curriculum for twenty odd years now, so everyone under the age of 40 must have had some teaching on this. I taught English in secondary schools from 1977-1992, and I never came across any English teacher who didn’t teach these basics. Phil Beadle likes to make these remarks, and imagines that all English teachers are like him. They’re not. I really don’t accept that there’s been no teaching of grammar for years. There was a short period in the late sixties when this might have been true (and only then in certain schools – I was doing tree diagrams in my sixties school)so there must be another reason why there is this perceived lack of knowledge. All power to you for getting your students back on course, but I think that the fact it takes only eight weeks to get them doing quite sophisticated analysis suggests that there was already some latent knowledge.

  5. Madeleine Morris says:

    Having spent the last 10 years teaching in Vietnam (not teaching English), I can attest to just how well ESL training equips teachers to teach English to native speakers. The CELTA course may be grueling, but it’s excellent.

  6. Ant Heald says:

    Just because something’s enshrined in the National Curriculum doesn’t mean it’s been taught! There hasn’t been adequate training for teachers, and the time for systematically building up these skills isn’t there because of an overcrowded curriculum and, for most of that time, a SATs testing regime that didn’t actually value these skills explicitly enough. Your experience from 1977-92 would largely have been working with teachers who had been educated knowing this stuff (I was fortunate to have been taught by such people in my early secondary school years in a former secondar modern turned comp), but the rigorous teaching of English grammar (I nearly automatically wrote ‘structure’ rather than grammar as it can still be seen as a bit of a dirty word) largely fell by the wayside at least in part *because* of the National Curriculum.

  7. Ant Heald says:

    Apologies for typos- they never look good when you’re defending more rigorous teaching of English. I blame my ipad. And my eyes.

  8. Rob Spence says:

    Well, I taught in secondary schools of all types for 15 years, and then led a PGCE course for eight years. We taught grammar, and we taught parts of speech. I’m not saying it isn’t an issue – it clearly is, as this article suggests –
    I can remember discussing frequently with colleagues how to make grammar lessons more accessible and interesting – so we were teaching it!

  9. Ant Heald says:

    Thanks for the link, Rob. It looks interesting from the abstract: I’d like to read it. It’s a shame that most teachers can’t readily access such academic material.

    I hope it didn’t come across as if I was suggesting you we’re fibbing by the way. I know there’s been plenty of ‘knowledge about language’ teaching, much of which has covered grammar topics, and ‘parts of speech’ (or should we call them ‘word classes’), but as I think we’re agreed, it’s been fairly unsystematic.

  10. Robert Mohr says:

    Hi David, It’s very good to find and read your thoughts and others’ about teaching what traditionally we called an ‘applied grammar’. I’m living in Dublin for the past 21 years but grew up in and around San Francisco. As a boy,I learned pure grammar, but it didn’t teach me to write. Not until I took a course at SF State Univ, Projects in Teaching Writing in 1975 (part of the Writing Project with UC Berkeley) did I learn how to apply grammar to composing. “Why didn’t someone tell me this before?” I cried out. Well, when I did learn the phrase and clause structures and joining techniques, my writing speed increased as did my control. I believe the meta-language helps hugely.

    So I revived the old writing course in Dublin and taught it all over the university and town. I even published a course book, How to Write: Tools for the Craft (UCD Press, 1998). And recently I’ve been the instructional designer/literacy specialist in the making of an app unlocking the vocabulary of academic English in the context of science topics:
    This is a language and literacy support for both native and ESOL students between 11-16 years. It’s a terrific project.

    Aside from agreeing that ‘as’ is a subordinating conjunction, I wanted to say that the craft of writing resembles the craft of music: 8 notes in the scale is like the 8 parts of speech; A-B-A form mirrors Intro-Devel-Recap/Conclusion; and both have technical language that give strength and depth to a composer’s expression. In a music school, students and teachers don’t recoil from technical language with an intake of breath: “Ooo, that’s technical. I just want to express myself.” No, musicians learn the terms and practice, practice. So young writers should as it is as venerable a craft as musical composition. There, I’ve said it. Thanks for indulging me.

    Wishing everyone the best in reviving the venerable old craft of writing.

    Kind regards,
    Robert Mohr

  11. Xris32 says:

    Great post. I think you already know my thoughts about ‘the grammar issue’. The more we talk about it openly the less it becomes an issue. Our language is as rich as it is complex. Teachers favour the ‘rich’ bit and neglect the ‘complex’ bit – grammar.

    There is a lot of fun to be had in the complex stuff. Embrace the darkside and enjoy grammar.

  12. Robert Mohr says:

    Hi Xris32.
    Yes! A young writer without a knowledge and terms of structure is like a sailboat without a rudder, rudderless.

    Yes. People used to recoil from grammatical terms in class (1995-2000ish), but now they ask. Imagine my delight.

    And yes again: we should keep talking about it, make a buzz, practice guerrilla grammaticalism.


  13. learningspy says:

    Hi Ant
    I take your point about ‘objects’ although they are a useful lie. I did add the caveat in the post that O could (should) stand for Other.

    However, as you say, avoiding difficult stuff is not the answer but there has to be a point at which enough meta language is enough. I think Object is more useful than predicate, but, hey, what do I know?

  14. learningspy says:

    Rob, you are clearly a very special individual. Most English teachers don’t know this stuff and I’ve watched Phil prove it! Staggering but true.

    Please accept that I was never burdened by any knowledge of grammar at school – everything i know I’ve pieced together subsequently. I am not at all unusual in this regard. So how can teachers be expected to pass on what they don’t know? I only found out about tree diagrams last year and got very over excited. How is it that our experiences are so different?

    And as regards my AS class: they are bright, motivated students. They had an implicit understanding of how language works and that allowed them to do well at GCSE but the ability to parse their own writing has improved it exponentially.

  15. Rob Spence says:

    Fascinating discussion. I’ve not taught school age children regularly for years, though I sometimes teach taster classes to university applicants. If the ubiquitous Mr Beadle is right, then it’s very depressing. On the PGCE course I ran, we did a subject audit at the beginning, where trainees identified their strengths and weaknesses regarding topics they would have to cover. They then had to demonstrate progress in their weak areas. We also had a whole strand of the course based on teaching language, including grammar. Of course, back then, we had more control of what we offered, and more time. Now trainees are in schools much of the time, in the care of busy teachers who might not have the time, inclination or skill to guide them.
    To explain the difference between us – that’s quite simple, I think. I am old.

  16. DebbieH says:

    Nice article.
    FYI, your lead in quote is by G.H. Hardy, the mathematician, not Thomas Hardy the novelist and poet.

  17. Robert Mohr says:

    Hi learningspy. Rob here. I was taught grammar right before the cultural revolution of the mid-late 60s, when rote learning was tossed. (I was born at the back end of 19951.) No one is to blame for that revolution, but we can claim back some form and even memorization a bit. It’s good brain’s ‘skeleton’. As I said before, pure grammar did not teach me to write; it merely gave me terminology to see, discuss and use structure in my written expression: expository and otherwise. While Bill Robinson was training grad students for the slave trade of composition teachers at SFSU, he found most didn’t know the grammar, so Eng 657 was a semester long training in the phrase and clause structures, sentence patterns, conjunctions, etc. that make up the nuts and bolts of the sentence. Add to that three strategies for paragraph construction, and voyla, you have an applied grammar for the purpose of composition. I’m far from the scene in the States here in Dublin, but my book is my own reworking of that training: How to Write: Tools for the Craft (UCD Press, 1998). It’s cheap and handy: under ‘Study Guides’. Sorry for the plug. I don’t really profit. But it’s my best answer to the question, “What can a teacher do to get this training?”

    Respect to everyone struggling with the literacy problem, especially when it comes to teaching writing. Whew, uphill work and yet a joy when they grab ahold.
    Rob from Dublin

  18. Robert Mohr says:

    Argh! Sorry for the typos in my last post. It’s hard to see everything in this tiny composition box. Well, as they used to say in the furniture moving world where I got my summer work during college: “Results expected, excuses unaccepted.” Still, me hat’s in me hand with a shy stoop.

  19. kristianstill says:

    As a non-specialist English teacher, I am learning and teaching in all phases of my teaching (planning, delivery and assessment). It can be exhuasting and every week I tried and set aside at least an hour of CPD.

    The more I teach the subject, the more I consider English a practical subject. In this regard I see grammar as one of the ‘basic skills.’ More recently I have started to teach in the lower school in much the same way I coached football at our local Academy.

    We introduced strict routines, tranistion tasks, periods of technical repetition to grove ‘technical ability’ before setting up small practices to apply the technique (very rarely more that 4v4). We were relentless in our pursuit of technical mastery and building confidence in our players. Every session (20 mins of 80 mins), personal practice (homework) and part of every pregame warm up. We most certainly fostered mastery before understanding, though we always knew that understanding we be developed when their thinking skills allowed it. It was a question of readiness. Even once a skill had been mastered, it sometimes took 6-8 weeks before a player tried a new technique in a game situtation and we would have to be very patient.

    What I am trying to say (in a rather long winded fashion) is that within our English dept at least, we tend to teach the game of English and not commit to teaching and mastering the techniques. Passing, control, 1 v 1 to beat / defend a player, creating / closing space; reading, writing, spelling, punctuation, grammar. The more I teach English, the more I feel that repeated teaching SPAG will secure not only English skill but student confidence. Just anticipate that even a concentrated and committed effort to technical mastery takes more time to bloom in free form writing then you would think. I also forecast that it would mean that my Y11 students would get more from the self and peer assessment then they currently do.

  20. […] the debate surrounding David’s ‘The mathematics of writing,‘ post. There was a lot to digest and I bookmark the post to come back to read it again. I […]

  21. […] creativity need rules? The mathematics of writing The Grand Unified Theory of […]

  22. Chipperfield says:

    In a nutshell, that you can teach to 5 year olds, a sentence names something, then tells more. Thank you Siegfried Engelmann.

  23. Geraldine Carter says:

    Yes, an 80s grammar-free education left one of mine in shocked ignorance when EFL teacher in Milan. A fast learning curve..

  24. JT says:

    This is a really interesting article, and I have a lot of time for the author and his theories – one that could really mark up the standard of English grammar in our young people as they develop from even the youngest of ages. I do however find it quite ironic that amidst a plethora of detail about grammatical correctness, there are two spelling errors (line 22 – ‘where’ is used instead of ‘were’, and line 53 – ‘and’ is used instead of ‘a’).

  25. […] The mathematics of writing  – an attempt to make teaching sentence structure more logical Thinking like a writer – advice on improving writing skills How to get students to value writing […]

  26. Suki says:

    I totally agree with this post: there are rules, and logical reasons for writing.

Constructive feedback is always appreciated

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