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<div id="main-column" class="span9"> <article id="about-probability" style="margin-top: 16px;">
<h1 class="section-title">About probability</h1>
<ul class="breadcrumb"><li><a title="Home" href="../../../index.html"><i class="fa fa-home"></i></a><span class="divider">/</span></li><li><span class="gray">Language System</span><span class="divider">/</span></li><li><span class="gray">Language of Instruction</span><span class="divider">/</span></li><li><span class="gray">Language of caution </span><span class="divider">/</span></li><li><span class="active">About probability</span></li></ul>
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<h2>Calculating the chances...</h2>
<p>Probability - and its close relative, possibility - are fundamental areas of thinking. As such, they crop up in all areas of study, from Maths to the Arts. That statement immediately suggests that the related concepts are treated in very different ways - very largely because they are handled in radically different types of 'language'. This page sets out to address this issue, principally in relation to the way that English may be used in Maths, and the way that mathematical concepts may be used to underpin the effective use of English.</p>
<p>In thinking this through, I came to realise that the English language (like all languages, I presume) has evolved to express degrees of probability/possibility across a very wide range of ways of thinking, and a very wide range of situations and contexts.</p>
<h3>An overview</h3>
<p>The diagram below is a considered view of what the field, the general conceptual area, looks like ...</p>
<p style="text-align: center;"><img alt="" class="noborder" src="../../../ib/englishb/images/Probability%201.jpg" style="width: 700px; height: 534px;"></p>
<p>... but of course it is not as simple as that! A number of the elements of the diagram assuredly need a little amplification.</p>
<p style="margin-left: 40px;"><span style="font-size:14px;"><em><strong>'Conditional type' </strong></em></span> ... the left-hand column makes use of the standard categories of the Conditional tenses and relationships, and the essential language features are listed at right centre in italics. The labels in bold are my attempt to name the intellectual functions that these grammar structures represent and express.</p>
<p style="margin-left: 40px;"><span style="font-size:14px;"><em><strong>'Maths versus Fiction' </strong></em></span>... the vertical axis between mathematics and the fictional imagination may be seen as a continuum between what <em><strong>is</strong></em> and what <em><strong>could be</strong></em>. At one extreme, the Maths of Probability sets out to state truths about reliable logically-verifiable relationships, while at the other extreme, the Fiction of Imagination sets out to explore everything else that can be thought. I would emphasise that this is a continuum, and that all subjects, all fields of human knowledge, can be placed at some point along this range. Economics, for instance, blends hard logical probability with all sorts of imagined possibilities and presuppositions - it should properly be placed between Prediction and Speculation in the diagram, and it is vital to determine the exact blend of these two ingredients in any economic theory.</p>
<p style="margin-left: 40px;"><span style="font-size:14px;"><em><strong>'Hard versus Soft logics'</strong></em></span> ... the right-hand element suggests another continuum in parallel : from Hard logic at one pole to Soft logic at the other, with the blend of the two varying as you move along the range. Hard logic uses fixed rules with determined quantities; Soft logic makes up the rules as it goes along and usually functions best with vague or representative quantities. To put it another way, 'Hard' thinking makes connections on a strict <em>logical</em> basis, while 'Soft' thinking makes connections on a <em>rational</em> basis (where 'rational' may vary according context, tradition, current values, etc).</p>
<p style="margin-left: 80px;"><span style="color:#006400;"><em><span style="font-family: comic sans ms,cursive;">There is of course plenty ... but plenty ! ... of thinking which is neither logical nor rational, but such 'Squishy' logic could not be incorporated helpfully into the diagram. You could imagine it lurking fetidly beneath the surface ... </span></em></span></p>
<p>I imagine that we would all accept as self-evident that the educated mind should ideally be able to handle all of these modes of thought competently. Accordingly, the English B teacher's job is to provide students with the tools and the skills to handle the modes linguistically ... which necessarily involves intellectual development so that the students can apply the language intelligently.</p>
<h4>Illustration : the diagram applied</h4>
<p>Here are some examples to illustrate how the diagram's sense can be expressed in concrete linguistic and intellectual terms :-</p>
<p style="margin-left: 40px;"><strong>#0 – Probability</strong></p>
<p style="margin-left: 120px;"><em>“If you let go of something, it falls.”</em></p>
<p style="margin-left: 80px;">(i.e. stating a general rule – 'always')</p>
<p style="margin-left: 40px;"><strong>#1 – Prediction</strong></p>
<p style="margin-left: 120px;"><em>“If you let go of that glass, it will fall.”</em></p>
<p style="margin-left: 80px;">(i.e. giving a specific case of a general rule – 'now')</p>
<p style="margin-left: 40px;"><strong>#2 – Speculation</strong></p>
<p style="margin-left: 120px;"><em>“If you were to let go of that glass, it would fall … but I don't think you're so stupid.”</em></p>
<p style="margin-left: 80px;">(i.e. considering something that is possible, but may well be unlikely – 'maybe')</p>
<p style="margin-left: 40px;"><strong>#3 – Probability in the past</strong></p>
<p style="margin-left: 120px;"><em>“If you had let go of that glass, it would have fallen and smashed ...but of course you didn't.”</em></p>
<p style="margin-left: 80px;">(i.e. considering alternatives to what really, actually happened)</p>
<h3>A shopping list</h3>
<p>So what specific language items should we concentrate on to support discussion of probability/possibility ?</p>
<p>In general terms, covering the full field as described in the diagram above, it must be vital to teach a thorough coverage of all Conditional tenses, their relationships, and varied special usages.</p>
<p>In the concrete case of helping the Maths Department to teach Probability, however, it is likely to be better to concentrate on certain restricted and specific language items. I offer three packets of language aspects, ranging from simple/basic through to quite complex:-</p>
<p><span style="font-size:14px;"><strong>Simple modifiers</strong></span> ... the teaching of Probability, it seems, begins with getting the students to recognise that the probability of something happening ranges from 'certain' to 'impossible'. It would therefore seem useful to provide language that helps to express<em> degrees</em> of certainty or impossibility. Thus :-</p>
<p style="text-align: center;"><img alt="" class="noborder" src="../../../ib/englishb/images/Probability%202.jpg" style="width: 600px; height: 177px;"></p>
<p><span style="font-size:14px;"><strong>A wider range</strong></span> ... following the same principle, it may help to provide more vocabulary along the same lines, in diagrammatic form so as to help students to grasp how exactly the words might be applied. Thus :-</p>
<p style="text-align: center;"><img alt="" class="noborder" src="../../../ib/englishb/images/Probability%203.jpg" style="width: 650px; height: 403px;"></p>
<p>Notice that words such as 'certain' or 'unlikely' are helpful for students' general understanding, but that the Maths teacher will necessarily move on to replacing the approximate expression 'fairly unlikely' with precisely-quantified expression such as 'a one in six chance'. This does not matter - the approximate expressions of language can, and should, coexist in practical reality with the precise expressions of numbers. Which takes us back to the educated mind that can swim between the various modes of thought ...</p>
<p><span style="font-size:14px;"><strong>Appropriate verb structures</strong></span> ... a more challenging area of language to teach is that of making sure that students have the full range of verb structures to be able to express accurately the varying degrees of probability. This means that they not only have to internalise relationships between verb forms which are sometimes quite complex, but also have to have an accurate sense of when they should properly be used. Thus :-</p>
<p style="text-align: center;"><img alt="" class="noborder" src="../../../ib/englishb/images/Probability%204.jpg" style="width: 600px; height: 484px;"></p>
<p>This is actually a skimmed, reduced and adapted version of the many and various ways of expressing Probability and Possibility suggested in the grammar book <em><strong>English Grammar for Communication</strong></em> (De Devitis, Mariani, O'Malley, 1991). This excellent reference work takes a notional & functional approach which I find very useful in considering what the English B teacher ought to teach.</p>
<p>.</p>
<div class="pinkBg">
<h3>qBank resources</h3>
<p><span style="color:#B22222;"><strong>Framing & certainty</strong></span> ... use of words and phrases to frame and modify the strength and certainty of statements</p>
<p><span style="color:#B22222;"><strong>must & can't , may, might & could</strong></span> ... exercises to practise modals of logical deduction - with degrees of certainty</p>
<p><strong><span style="color:#B22222;">must, have (got) to, should & ought to</span></strong> ... exercises to practise modals of requirements versus suggestions</p>
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<p>.</p>
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