However, there is a remarkable difference between polygons with more than three sides (such as quadrilaterals), and polygons with exactly three sides (i.e. triangles). Any two *equilateral* triangles of different sizes are similar—and for much the same reason as any two squares are similar: (i) all angles are equal to 60˝ , so are certainly "equal in pairs", and (ii) all three sides of one triangle have equal length (say *a*), and all three sides of the other triangle have equal length (*b* say), so the ratios of corresponding sides are all equal to *a* : *b*. But, unlike the case of squares (where both conditions are needed), one of these conditions for equilateral triangles comes for free. As the name implies, for *△ABC* to be equilateral, all we need is

"that the three sides are all equal: *AB* " *BC* " *CA*".

The fact that the three angles are all equal to 60˝ then comes for free—thanks to the SSS congruence criterion (since *△ABC* " *△BCA*, so =*ABC* " =*BCA* and =*BCA* " =*CAB*). Hence, to check the claim that any two equilateral triangles are similar, it is enough to observe that the second bullet point is satisfied (and the first then comes for free).

The same is true whenever we apply the idea of "similarity" to triangles in general. Officially two (ordered) triangles *△ABC* and *△DEF* are *similar* (which we write as *△ABC* " *△DEF*) if

*•* corresponding *angles* are equal: =*A* " =*D*, =*B* " =*E*, =*C* " =*F*,

#### **and**

*•* corresponding *sides* are proportional: *AB* : *DE* " *BC* : *EF* " *CA* : *FD*.

Yet the challenge, to think of

leads to a surprise.

This fact is unlikely to make sense if simply stated in the way we have stated it here. So pupils need prior experience of drawing and measuring that makes this important statement meaningful and plausible: the *similarity criterion* states that, **for triangles**, each of the above bullet points implies the other. (See, for example, "Problem 0" in Section T13 of *Extension mathematics Book Gamma*.)

Initially pupils need the idea of similar triangles for simple trigonometry: i.e. only for *right-angled* triangles.  If we fix the angle at *O* =*BOC* " *θ*, and choose *C* 1 to lie on the line *OC*, then the angles of the two triangles *△OBC* and *△OB*1*C* 1 are equal in pairs (namely to *θ*, 90˝ , and 90˝ ´ *θ*); and we can establish as a fact of experience (by drawing and measuring; or partly of deduction—first for *k* " 2 or *k* " 1 2 , then for any integer or unit fraction, and finally for any fraction) that

if *OB*1 " *k* ¨ *OB* (so to get from *O* to *B* 1 we go "*k* times as far **along**" as we did to get to *B*) then *B* 1*C* 1 " *k* ¨ *BC* (so to get to *B* 1 from *C* 1 we go "*k* times as far **up**" as we did to get from *B* to *C*)

It follows (by *Pythagoras' Theorem*) that *OC*1 : *OC* " *k*. So the first bullet point implies the second. Hence if =*BOC* " =*B* 1*OC*1 " *θ* (and =*OBC* " =*OB*1*C* 1 " 90˝ ), then corresponding sides are in proportion:

$$\mathcal{OB}' \colon \mathcal{OB} = \mathcal{OC}' \colon \mathcal{OC} = \mathcal{B}'\mathcal{C}' \colon \mathcal{BC}.$$

Cross-multiplying shows that

*B* 1*C* 1 : *OB*1 " *BC* : *OB*,

so the quotient " opposite adjacent " depends only on the angle =*BOC* " *θ*, and not on the choice of triangle. Hence we can safely write it as "tan *θ*"—that is as a *function* that only depends on the angle *θ*. (See Section C20 in *Extension mathematics Book Gamma*.)

Similarly *B* 1*C* 1 : *OC*1 " *BC* : *OC*, so the quotient " opposite hypotenuse" depends only on the angle =*BOC* " *θ* and not on the choice of triangle, so we can safely write it as "sin *θ*"—that is, as a function of *θ*. And the ratio *OB*1 : *OC*1 " *OB* : *OC*, so the quotient " adjacent hypotenuse" depends only on the angle =*BOC* " *θ* and not on the choice of triangle, so we can safely write it as "cos *θ*". (See Section C33 in *Extension mathematics Book Gamma*.)

The congruence criterion and the parallel criterion allow one to transfer *exact* relations (such as *equality* of line segments or of angles) from one place to another. The *similarity criterion* goes beyond this world of *exact equality* to allow one to deal with ratios, scaling, and enlargement. Hence this criterion is probably best delayed until the basic consequences of congruence and parallelism have been sufficiently explored, and until pupils are sufficiently confident in working with ratio. (The *similarity criterio*n may be thought of as a substitute for the evidently false "AAA congruence criterion". The criterion can also be re-formulated as SAS-similarity: (see Section C13 of *Extension mathematics Book Gamma*).

As hinted above, special cases of the similarity criterion can actually be proved using the congruence criterion and the parallel criterion—namely where the ratio between corresponding sides in the second bullet point is a fraction. The most important example occurs when this ratio is equal to 2 (or to 1 2 ) and is called the *Midpoint Theorem*, which says that:

if in *△ABC*, *M* is the midpoint of *AB* and *N* is the midpoint of *AC*,

then *MN* is parallel to *BC* and *BC* : *MN* " 2 : 1.

That is *△ABC* " *△AMN*, with the corresponding scale factor

$$AB:AM = AC:AN = BC:MN = \mathbf{2}:1$$

(see section T13, Problem 6 in *Extension mathematics Book Gamma*).

The third requirement listed at the start of Section 3.4.7 concerns applications of these ideas. Once we know *Pythagoras' Theorem* we can use it to find lengths exactly (in surd form). An equilateral triangle of side 2 has height equal to ? 3. A square *ABCD* of side 1 has diagonal *AC* of length ? 2. A regular pentagon *ABCDE* of side length 1 has diagonal *AC* of length 1` ? 5 2 . A regular hexagon *ABCDEF* with sides of length 1 has two different length diagonals—a diameter *AD* of length exactly 2, and a shorter diagonal *AC* of length exactly ? 3. The square of side 1 allows one to write down the exact values for tan 45˝ " 1, and for

$$
\sin 45^\circ = \frac{1}{\sqrt{2}} = \frac{\sqrt{2}}{2} = \cos 45^\circ.
$$

In the equilateral triangle of side 2, the perpendicular from the apex to the base bisects the apex angle into two angles of 30˝ , and meets the base at its midpoint. Hence we can write down the exact value for

$$
\sin 30^\circ = \frac{1}{2} = \cos 60^\circ,
$$

for

$$
\sin 60^\circ = \frac{\sqrt{3}}{2} = \cos 30^\circ \text{ }
$$

for tan 30˝ " ? 1 3 , and for tan 60˝ " ? 3. One can also use *Pythagoras' Theorem* to find the distance between any two points whose coordinates are given (in 2D or in 3D).

Wherever right angled triangles appear, one can use sin, cos and tan (or similar triangles) to find missing angles or lengths. Classical applications include

	- (i) the pupil's height and
	- (ii) the distances from our eye to the pupil's feet, and to the base of the tree.

One would also like to see other applications of angles which do not involve right angled triangles directly (e.g. angle problems involving bearings).

## 3.5. The remaining requirements

These two final requirements look very much like a collection of "remnants". Both seem to relate to rather late in Key Stage 3 or even to Key Stage 4.

*Pythagoras' Theorem* and similarity (or trig) feature in solving problems relating to regular polygons or familiar figures in 3D—whether calculating the lengths of ladders leaning against walls, or the height of some point above the ground or table, or surface areas and volumes. However, the examples listed seem better suited to Key Stage 4 than to Key Stage 3. Nevertheless one would love to see problems at some stage that involve finding and using the slant height of a cone, or the height of a pyramid, or the distance between two opposite corners of a cube, or the angles between lines in 3D figures, or the angle between a slanting face and the base of a pyramid.

The final requirement is admirable as a general idea. But it is also rather too vague for us to try to interpret it reliably here.

## 4. Probability and Statistics

The requirements under these headings leave many questions unanswered. It is not always clear how to interpret them as they stand, so we have tried to suggest "alternative readings". We have also taken the opportunity to discuss some of the background which needs to be borne in mind when devising a scheme of work.

## 4.1. Probability

#### **4.1.1 Introduction**

– **record, describe and analyse the frequency of outcomes of simple probability experiments involving randomness, fairness, equally and unequally likely outcomes, using appropriate language and the 0–1 probability scale**

Our understanding of how to teach probability is less well developed than our understanding of how to teach geometry. So it is difficult to know exactly where the problems lie. But there would seem to be considerable potential for confusion here between

the language of messy "experiments" in the real world,

and

language that belongs to a pristine mathematical universe (namely *probability*).

This confusion is especially awkward given the explicit mention of "using appropriate language".

Of course the mathematical universe often has its roots in the real world, so terms and expressions may at times inhabit both worlds. Nevertheless it may be easier to interpret the above official requirement if one imagines added quotation marks (and the extra word "eventually") roughly as follows:

– record, describe and analyse the frequency of outcomes of simple "probability experiments" involving "randomness", "fairness", "equally and unequally likely outcomes", using appropriate language and [eventually] the 0–1 probability scale.

"Record, describe and analyse the frequency of outcomes of simple [. . . ] experiments" is an excellent requirement: pupils need such experience in order to develop their ideas of variability, and to understand how these are ultimately captured by the universal model of a sample space p*S*, *p*q, where *p* assigns values between 0 and 1 to subsets of *S* according to certain rules (e.g. for a single toss of a fair coin, *S* " t*H*, *T*u, with *p*p*H*q " *p*p*T*q " 1 2 ). However, this step lies some way off—though it is alluded to vaguely in the next batch of requirements, where we read (see Section 4.1.2):

"generate theoretical sample spaces for simple and combined events with equally likely, mutually exclusive outcomes".

The idea also features in the GCSE *Subject Criteria* using curious, non-standard language

"construct theoretical *possibility* spaces for single and combined experiments with equally likely outcomes" [emphasis added].

In contrast, there is no hint of "sample spaces" in the Key Stage 4 programme of study.

However, given the explicit mention of "theoretical sample spaces" in the next official requirement (see Section 4.1.2), we assume that the "experiments" referred to in the first Key Stage 3 requirement are intended to open up *informal* consideration of questions involving "fairness", "randomness", and the crucial idea of "equally likely". And if these informal considerations are to lead (eventually!) to the idea of a sample space p*S*, *p*q, we may need a shift of focus from arbitrary real-world experiments to more carefully chosen settings (such as coin tossing, or dice rolling, or equally divided spinners), where a theoretical analysis is possible.

Hence, if pupils' understanding of probability is to progress, we may need to distinguish three separate settings:

experiments in the *real world* of messy data;

experiments and analysis in the *in-between world* of controlled data (fair coins, dice, etc.);

and

the *mathematical* world of theoretical probability.

We may choose to start in the real world of messy data: for example, with pupils examining the apparent likelihood of being born on each day of the week. The obvious "sample", or experiment, (namely, collecting all the results for pupils in the class) leads first to the need for them to use their known birthday and age to discover the day of the week when they were born; the class can then record numbers for each day of the week; and finally one can introduce the idea of using "relative frequencies" as a better measure than the raw numbers. The resulting distribution will inevitably raise the question of "fair sampling" and "randomness" (for it is almost bound to contradict pupils' gut feeling by deviating from the expectation that each day should be "equally likely"). More representative data—if it can be procured—is just as likely to challenge this understandable assumption.

The use of "relative frequencies" introduces the idea of a 0–1 scale (though not at this stage a "*probability* scale"). And one can emphasise the fact that the relative frequency of those born on a weekday (say) is obtained by adding the five separate relative frequencies for Monday-Friday.

But relative frequencies only tell us what **was** observed—**once**; and this would seem to tell us nothing about what *will* be observed *in the future*. This is the whole point of non-deterministic data. We may know that the recorded relative frequencies add up to 1; and that the relative frequency of a combined event is equal to the sum of the ingredient relative frequencies. But this only tells us *what happened last time*. We cannot calculate with observed relative frequencies to learn anything more general—as one can to some extent with probabilities. So it should soon become clear that this is not a mathematical world, where one can answer more interesting questions using exact calculation.

Classical science is deterministic, and reported results in classical science must be replicable: if you or I repeat a deterministic experiment as it was reported, we expect to replicate the stated results. And if we fail, then we have to question either the reported result or our own attempted replication. But with stochastic processes, the situation is completely different. When we repeat a "probability experiment", the observed outcomes vary considerably. Yet within the observed variations one can discern certain clear *trends*. This new science is no longer to be judged by, or analysed through, the outcomes of a single experiment, but through patterns in the variation of the outcomes of repetitions of the experiment. Single snapshots are of little relevance; instead we try to summarise the background reality that lies behind what we observe by integrating *all possible snapshots* into a single model of "probabilistic reality".

#### **4.1.2 Theoretical probability**

So there are strong reasons to move beyond messy real-world "experiments", and to focus on a more restricted (or more artificial) *mathematical* universe—such as coin tossing or dice rolling—where everything is much more clearly defined. Here one can perform *repeated* experiments relatively easily. And one can also analyse the background situation precisely—by counting.

Even in this restricted world, there are elephant traps to be identified and avoided. For example, when tossing two coins, time is needed to clarify the expected ratios of the three possible outcomes—"two Heads", "two Tails", and "one Head and one Tail". But unlike the messy real world, it is now natural to imagine an idealised version where one thinks of a "fair coin", with Head and Tail truly "equally likely" (or a "fair die", where each of the six outcomes is "equally likely"). Experiments show that the observed "relative frequencies" of Heads and Tails (or of the six possible outcomes for rolling a die), vary significantly. But they always add to 1, and can always be combined to find the relative frequencies of compound events (such as "rolling an odd number").

More importantly, experimental results can now be compared with what one would "expect" on the basis of the idealised model. This background "expectation", based on counting within the idealised model, is quite different from "the recorded results of experiments". And whereas the results from successive experiments will vary, the "expected" results stay the same. It is as though the calculated expected results are some kind of "ideal summary", and each experiment is only an approximation to, or a flickering shadow of this ideal summary. For example, within the model we can *count exactly*: there are 24 possible sequences of 4 tosses of the coin, and exactly 4 of these sequences have just one "Tail"—which seems to say that, if we record 100 such sequences of 4 tosses, then we should "expect" 4 16 of them to have just one "Tail". The reality will of course usually be different; but pupils may gradually come to realise that the existence of the idealised model provides a *fixed reference point* with which we can compare the results of different experiments, and provides the key to making sense of their variability (as "deviations from the expected ideal").

There are many advantages in working within these carefully chosen, controlled settings. In particular, they clarify the difference between observed "relative frequencies" from a single experiment and the expected frequencies, or "theoretical probabilities". The theoretical model also allows us to see more clearly how the cumulative results of many experiments tend to "average out", and how this long-term average tends to approximate the theoretical probability ever more closely.

An experiment, and the associated set of observed frequencies, is like a single snapshot *of a ghost*, or *a shadow* of some hidden object. This is especially true if the experiment involves messy real world data. The snapshot gives one a record of vague outlines—hints of something substantial. Yet one cannot be sure of the precise outline or shape which gave rise to this impression; that is, one may at first have no knowledge of the "background reality" that caused the impression or shadow. The observed results (of say "days of birth") may suggest a surprising pattern, but it is only a hint: the actual reality that lies behind the observations remains elusive. Subsequent snapshots of apparently the same object may vary greatly from each other—and yet between them reveal patterns that suggest that there really is some "background reality" that lurks out of sight.

This is a classic instance of Plato's parable of *The Cave*. We can only discern shadows of some presumed "Platonic reality" ("theoretical probability" in this instance), and must somehow *infer* what we can about the hidden reality that is casting the shadow, or leaving a ghostly impression. And the test for any inferred "reality" is whether it explains the shadows that we do see, and *why we do* **not** *see the shadows that we do* **not** *see*. If the observed shadows were always the same shape (as would be the case if the object were a solid statue, and the light source remained constant), then the "Platonic reality" might be a classical numerical measurement from elementary mathematics (like "the height of Nelson's column").

Probability and statistics are different, in that the observed "facts" differ each time we look. Yet there is still something substantial behind the observation. A single experiment, or sample, and the associated set of "observed relative frequencies", is but a single shadow of an elusive, moving object. And our inferred "Platonic reality" must somehow combine all *conceivable* observations into a single idea, which somehow incorporates the observed variability, and explains how each snapshot arises as a single view, or aspect of it. That is the role played here by the idea of a **sample space**: a set of atomic outcomes, with a *probability* assigned to each, so that their sum is 1.

Elementary mathematics can be largely summarised as *the art of exact calculation* with numbers, symbols, geometrical entities, etc. If we wish to find "the height of Nelson's column", though we do not know the answer, it is natural to *assume it has a definite value*, and then use the methods of elementary mathematics to calculate this presumed "definite value" using other known facts (e.g. properties of similar triangles). That is, the objects to which this "art of exact calculation" applies—whether represented by numerals or letters—are usually assumed to have *definite* values (possibly unknown). The associated mathematical universe may be abstract; but its objects have specific values, which remain constant throughout any subsequent calculation. Such entities are relatively tame, and static; they can be imagined relatively easily.

However, the numerical data related to probability and statistics is more elusive than this—though the fact that it is clearly still numerical (in some sense), may tempt us to overlook its more elusive character. Consider, for example,

```
the number of "Heads" obtained in a sequence of 4 coin tosses.
```
Each particular "instance" (toss a coin 4 times and keep track of the number of "Heads") gives rise to a *single value*—namely, the number of "Heads" obtained. So this "number of Heads in 4 tosses" is superficially *like* "the height of Nelson's column". However, the object of thought is not the individual value that we obtained on this one occasion, but

the *totality* of all possible "numbers of Heads" that could be obtained if we repeated the experiment,

together with

the way these "numbers of Heads" are *distributed* between 0 and 4.

This object of thought is multi-layered: there is a **sample space** *S* (the set of integers between 0 and 4), with each member having an attached *number* (the relative frequency with which this number of "Heads" occurs). If we shift from repeated experiments and observed "relative frequencies", we can use the idealised model of a "fair coin" with

$$p(H) = p(T) = \frac{1}{2}$$

*to calculate exactly the expected frequency* for each number of "Heads". This "expected frequency" varies with the number (though 0 "Heads" turns out to be exactly as likely, or as unlikely, as 4 "Heads", and 1 "Head" turns out to be exactly as likely as 3 "Heads"). We therefore get a new probability *P* for this sample space *S*, where

$$P(0) = P(4) = \frac{1}{16}, \quad P(1) = P(3) = \frac{4}{16}, \quad P(2) = \frac{6}{16}.$$

The nature of variability, and the difference between

can be explored in the messy real world. But at this level, the analysis of stochastic data is largely restricted to discussion and qualitative statistics. So one needs to move the field of play to the in-between world of controlled data (fair coins, dice, etc.). Here one can still do experiments; but one can also analyse things in a way that offers a bridge to *theoretical probability*. One can construct a natural God-given model, and can compare its predictions with the results of experiments to see just how variable things can be. In particular, it makes didactical sense to choose in-between examples with *finitely many* atomic outcomes, and to focus on examples where symmetry guarantees that the atomic outcomes are *equally likely*. **Everything then reduces to counting.** And one can compare the relative frequencies that arise in experiments with the "God-given" relative frequencies derived from counting (which form the model for our idea of a "sample space and probability").

With luck it may now be a bit clearer why we questioned the informal mix of words "probability experiment", "randomness", "fairness", "equally and unequally likely outcomes", "0–1 probability scale" in Subsection 4.1.1 and suggested there was a danger they might blur the distinction between

The requirement in Subsection 4.1.1 used language that ultimately belongs to our inferred Platonic model, and imposed it upon the world of shadows, or observed data. Pupils should definitely "record, describe, and analyse the frequency of outcomes from simple [. . . ] experiments", whose results are "non-deterministic" in that the data vary from one experiment to the next; but these are not really *probability* experiments. Repeated coin tossing, or dice rolling, or drawing pin tossing force us to address the underlying issue of "variable outcomes" and to nurture ideas of probability. But they can only be described as "probability experiments" in retrospect—once we have the background notion of an underlying sample space *S* and a probability function *p*. The language used in this official requirement is the language that should emerge as a result of a carefully chosen sequence of such experiments and analysis: it should not really be used "up front". Hence its use is best interpreted as a summary of what is needed—used informally for ease of communication between parties who are already "in the know".

We end this rather heavy digression on a lighter note. During a test in which all the questions required only a true/false response, a pupil was observed to be repeatedly tossing a coin until he had answered every question. When asked what he was doing, he replied: "I have no idea of what is going on in this course. So I flip the coin; if it turns up Heads, I choose true and if it turns up Tails I choose false." The invigilator tried to keep a straight face, and moved on. Later, the invigilator announced: "You have five minutes remaining" and was surprised to see the pupil madly tossing a coin once more.

#### 4.2. Statistics

No-one should doubt the increasing importance of statistics in the modern world. But it is less clear how this fact should influence the school curriculum—and in particular, the school *mathematics* curriculum. The world is awash with data. But the information available to decision makers in government, in business, in management, in operating public utilities, etc. only tells part of the story. They may collect "random samples" to try to eliminate bias, but the result is still an incomplete "snapshot". What can one infer about the true situation from such a snapshot? And how much confidence can one place in the resulting inference? If one takes a second sample from a different source, or at a different time, it is bound to differ from the first. But when are the differences such as to suggest that "something has really changed"? These are the kinds of questions addressed by statistics. It is one thing to suggest (rightly) that the mathematics curriculum must think carefully about how to prepare pupils so that they have a chance of making sense of the way statistics is used at Key Stage 5 and beyond; it is quite another thing to suggest that significant chunks of elementary mathematics should be sidelined, or de-emphasised, in order to make room at Key Stage 1–3 for possibly premature, low grade statistical content.

The situation we face should sound familiar. No one disputed the realisation in the 1950s and 1960s of the increasing importance of a kind of "modern mathematics" that was very different from school mathematics as then taught. However, the inference that school mathematics should be re-formed into something closer to the said "modern mathematics" proved to be thoroughly misguided—and it took us two decades before we finally admitted this fact. In much the same way, no-one doubted the claim that the 1970s and 1980s witnessed the beginning of a revolution in computational technology, that led to a marked shift in the way mathematics was being used in the outside world; yet the assertion that primary school mathematics "therefore" needed to be radically re-formed to incorporate calculators proved once again to be misguided. Claims were made at the highest level that pupils no longer needed to "learn their tables"; and it again took twenty years for us to discover that "learning one's tables" is important **not** so that we can compete with a calculator, but because it is part of the way young minds internalise an understanding of *the way numbers work*—and so is essential if we are to help pupils prepare to make use of the new technology **later**.

Hence, while welcoming the commitment and enthusiasm of those with a special interest in statistics, it is important not to repeat the same mistake of unquestioningly accepting the claims of those who may have allowed their enthusiasm to run away with them. We would all like the next generation to be well-placed to use statistics intelligently in adult life. But the experience of the last 25 years should convince us that this is not likely to be achieved by neglecting parts of elementary mathematics that are needed for the subsequent effective analysis of statistical methods in favour of low-level qualitative methods of little lasting value. In particular, the basic framework for statistics depends on having a firm grasp of theoretical probability.

Given the ubiquity of statistical data, some understanding of the associated problems deserves attention. But it remains unclear how this experience should be embedded within the wider curriculum, and how much of it, and which aspects, are best treated in the time allocated to *mathematics* (and at what stage). The dilemmas were clearly indicated in the analysis and recommendations of the Smith report *Making mathematics count* (2004)—see para 0.28, paras 4.16–4.18 and **Recommendation 4.4**. 21

Curricula since 1988 have allocated significant amounts of classroom time to *Handling data* many years before pupils master the mathematics that is needed for statistical calculation. As a result, the content listed under *Handling data* has been largely restricted to *descriptive statistics*. Whilst there is some value in using common sense to extract simple information from statistical data in all subjects, and to use this to draw pupils' attention to misconceptions, we need to consider carefully how much of the necessary time should be taken from that allocated for *mathematics*. There is a balance to be struck between on the one hand alerting pupils to the challenge presented by statistical data, and on the other developing the mathematical tools that will subsequently allow pupils to engage in some more significant analysis of problems—including statistical problems. If the necessary tools are not mastered, pupils are likely to be reduced to applying cookbook procedures which they cannot possibly understand. Moreover, this contradicts the declared *Aims* of the curriculum, and the idea that one should insist on *meaning* and *understanding*. So we should perhaps look for ways of treating this material at a later stage when pupils can make sense of it using mathematics that they understand.

#### **4.2.1**

**– describe, interpret and compare observed distributions of a single variable through: appropriate graphical representation involving discrete, continuous and grouped data; and appropriate measures of central tendency (mean, mode, median) and spread (range, consideration of outliers)**

21 http://www.mathsinquiry.org.uk/report/MathsInquiryFinalReport.pdf

**– construct and interpret appropriate tables, charts, and diagrams, including frequency tables, bar charts, pie charts, and pictograms for categorical data, and vertical line (or bar) charts for ungrouped and grouped numerical data**

The listed requirements have acquired a fairly standard interpretation in current textbooks and assessments. Yet it is worth asking how well this standard interpretation *prepares* pupils to understand the more serious statistics that is used in many subjects *beyond Key Stage 4*.

Most of the elementary mathematics we have covered so far can be summarised *as the art of exact calculation* with numbers, symbols, geometrical entities, etc. Suppose we wish to find "the height of the school building", or "the height of Nelson's column". Though we do not know the answer, we assume it has a definite value. We then use the methods of elementary mathematics to calculate this value. That is, the objects to which this "art of exact calculation" applies (whether represented by numerals or by letters) can be assumed to have *definite* values, which remain constant throughout any calculation. Such entities are static, and can be imagined relatively easily.

However, stochastic, or statistical data—though still numerical—is not quite like this. Consider, for example, "the height of a UK adult male in 2014". Each particular instance of such data ("choose one adult UK male, then measure and record his height") gives rise to a *single value*—the height of that particular individual. So one might think that "the height of a UK adult male in 2014" is like "the height of Nelson's column". But the object of thought here is not the single value obtained by choosing and measuring the height of one adult male: we are interested in the *totality of individual heights*, and the way these individual heights are distributed throughout the whole population of "UK adult males in 2014". This object of thought has several 'layers':

*•* this attached number varies as one varies the choice of individual, and does so in such a way as to give rise to a *distribution* of possible values, where each "height" occurs with its own *frequency*, or **probability**.

Later, these multi-layered objects will be formalised as *random variables*, and captured via *distributions*. No matter how they may eventually be formalised, all we need to notice here is that they are clearly more elusive than the numbers studied elsewhere in elementary mathematics.

And this is just the easy part of the story. The harder part is that we rarely know the underlying *distribution* precisely. So we try to draw inferences about the underlying distribution on the basis of some more-or-less representative *random* **sample**! (The word "random" deserves a whole mini-essay of its own; but it indicates that the sampling is done in a way that avoids giving a systematically false impression of the population being sampled.) Or we may want to decide whether the apparent differences between two different random samples can be explained by "natural variation", or whether the differences suggest that something significant has changed.

The specific (possibly unknown, but fixed) numbers of more familiar elementary mathematics have here been replaced by *distributions*, where a range of possible values can occur—each with its own frequency. The background *distribution* may be unknown—and instead all we know is information from one or more *samples*. And the goal is to decide what one can infer about the (unknown) background *distribution*, or whether the differences between two different samples are significant. This is an important art. But it is very different from (and conceptually much more demanding than) the mathematics of numbers, measures, symbols, or functions that is studied elsewhere in Key Stages 1–3.

#### **4.2.2**

**– describe simple mathematical relationships between two variables (bivariate data) in observational and experimental contexts and illustrate using scatter graphs**

Any tabulation, or graphical representation, involves two variables! In Section 4.2.1 there was an initial imagined "single variable", whose *frequency* of occurrence was being recorded. So one was dealing with two linked variables: the original variable, and the counting numbers. But in some sense the counting numbers did not have an independent interest. In Section 4.2.2 we are concerned with two independently existing variables which may be related (such as height and weight among adult males), and where we wish to understand the possible linkage better.

An obvious trick is to plot linked pairs p*x*, *y*q, with one variable along the *x*-axis, and the other variable along the *y*-axis. The resulting collection of points in 2D is called a *scatter graph*. This is not the graph of a function, since

The idea that there might be a "connection" between the two variables then translates into the idea that the scatter graph may reveal some structure.

The simplest imaginable structure would be for the plotted points to lie along some straight line, or to reflect some other functional dependency of one variable upon the other. A non-statistical example might plot the temperature in "degrees Fahrenheit" against the temperature in "degrees Centigrade": here because the relationship is deterministic and exact, the data sits along a perfect straight line *y* " 9 5 *x* ` 32. But statistical data is never quite so well-behaved.

When trying to spot a hidden relationship with messy data it can help to impose an additional **constraint**. For example, we may consider whether there is some special point that should be forced to lie on any possible curve which links the two variables *x* and *y*. The data points themselves are all as reliable, or as unreliable, as each other. But examples can be used to support the idea that the point pAvp*x*q, Avp*y*qq, where Avp*x*q is the average of all the *x*-values, Avp*y*q is the average of all the *y*-values, serves as a kind of "representative centre" for the set of data points, and so should lie on any resulting curve. In particular, if we decide that the relationship is approximately linear, then requiring the line to pass through the point pAvp*x*q, Avp*y*qq makes it much easier to choose the gradient "by eye" so that we get a line that seems to follow the data approximately, and which leaves deviant data points p*x*, *y*q in some sense "equally distributed" above and below the line. The whole thrust of this analysis is to try to see patterns in the data that might not be apparent from a mere list of numbers. However, the analysis remains at best weakly "mathematical": we are not yet sufficiently well-placed to engage in genuine calculations.

In the relatively tame world of elementary mathematics we have already highlighted the difference between *direct* calculation, where the answer can be ground out deterministically, and *inverse problems*, whose solution forces us to "work backwards" from some "output" in search of some direct calculation that might give rise to the given data (see Part II, Section 1.2.3, and Part III, Sections 1.2.2, 1.2.4). The art of analysing statistical data mathematically would seem to be an important instance—and a rather subtle instance—of such *inverse problems*. This art is therefore doubly challenging. Not only are the objects of the relevant *direct* statistical calculations more subtle than those we meet in the rest of elementary mathematics; but *handling data* is useful precisely because statistical problems are *inverse* problems—we typically know only selected information (from some presumed *random sample*), and we need to assess what we may infer from this *sampled* data about the unknown background *distribution* of the whole *population*—and what degree of confidence we may attach to such inferences.

Despite the difficulties, this material plays such an important role in modern society that it is natural for educators to try to find ways of introducing pupils to the underlying ideas. It is not easy to summarise the experience of the last 25 years; but it is probably fair to say that the rhetoric has been consistently ahead of the reality. Thus there are many outstanding issues which a programme of study, or a scheme of work, needs to weigh up and resolve. Three important questions concern

*•* the time that is needed to make the engagement with statistical questions worthwhile at a given stage, the likely progress that might be made at that stage, and (crucially) *what other topics would have to be sidelined* in order to make that time available.

## IV. A sample curriculum for all—written from a humane mathematical viewpoint

What follows stems from an attempt to consider what a "humane mathematician and educator" might expect to see included at each Key Stage of a National Curriculum—under the assumption that:

In particular, the sample curriculum tries to set realistic (rather than ambitious) goals for primary mathematics. It also tries to restrict the extent of abstraction at Key Stage 3, on the assumption that for a significant group of pupils this material may form the core of work at secondary level (age 11–16), along with consolidation of material from upper primary school. More abstract material has been delayed wherever possible until Key Stage 4, where it serves as a transition from elementary mathematics to higher mathematics for all pupils who intend to proceed to further academic studies beyond age 16 (in any subject). Hence teachers may choose to blur the boundary between Key Stages 3 and 4 for many pupils, taking much longer to cover the listed Key Stage 3 material for some, while others may treat Key Stage 3 material more abstractly than is suggested here in preparation for Key Stage 4.

Having produced a *Brief Version* (which was already fairly detailed—see Section 2 below), we then drafted two further versions:

We have chosen to reproduce here (and to improve) the *Brief Version* and the *Very Brief Version*, to provide an easily accessible reference for the reader of Parts I, II, and III. Some words that appear in **bold type** have a technical meaning that will have been explained in detail in Part II or Part III. If some aspect remains unclear, we recommend that readers refer to the *Fuller Version*, which is freely available at The De Morgan Forum.22

## 1. Very Brief version

#### **1.1 Key Stage 1**

By the end of Year 2 pupils:

#### **Counting, reading and recording number**

#### **Recalling facts**

*•* have instant recall of addition and subtraction facts for numbers to 10; have instant recall of ˆ2, ˆ5, ˆ10 multiplication tables, and derive corresponding division facts

22 http://education.lms.ac.uk/wp-content/uploads/2012/02/KS 1-4 DMJ.pdf

#### **Calculating**

#### **Describing shapes and measuring**

#### **Solving problems, reasoning, and using language and symbols**

#### **1.2 Key Stage 2**

By the end of Year 6 pupils:

#### **Place value**

*•* handle place value to represent and order numbers to 10 000 and beyond; extend this to negative integers and decimals; work with decimals and measures involving tenths, hundredths and thousandths; multiply and divide integers and decimals by 10, 100, 1000

*•* round integers, decimals and measures to the nearest "ten", integer, or tenth

#### **Recalling facts; correct use of language and symbols**

#### **Structural arithmetic**

#### **Calculating**

#### **Geometry and measures**

*•* use and calculate with money; tell the time to the nearest minute; read scales—interpolating between marks; convert between related units

#### **Solving problems**

*•* tackle and solve **word problems** and simple **multi-step** problems involving numbers, measures, and shapes; make sensible estimates; make **connections** between topics; explain their reasoning

#### **1.3 Key Stage 3**

Key Stage 3 revisits important Key Stage 2 material—partly for revision, but mainly to reinterpret old material from a new "viewpoint" (extending the written algorithms to decimals, shifting the focus from bare hands mental methods to structural arithmetic and the simplification of expressions, etc.).

By the end of Year 9 pupils:

#### **Place value**

#### **Calculating**

#### **Structural arithmetic**

*•* use **place value**, factorisation, and the algebraic structure of arithmetic to **simplify** and to evaluate numerical expressions and calculations—including with fractions and negatives

#### **Simplification of algebraic expressions; solving linear equations**

#### **Geometry and measures**

#### **Solving problems**

*•* tackle and solve **word problems** and simple **multi-step** and **inverse** problems involving numbers, measures, symbols and shapes

#### **1.4 Key Stage 4**

Key stage 4 revisits important Key Stage 3 material—partly as revision, but also to interpret it afresh. For some pupils, this re-working and strengthening of Key Stage 3 material (together with consolidation of Key Stage 2 material) will be their main goal at Key Stage 4; other pupils may supplement revision of Key Stage 3 material with a programme that covers selected parts of what is summarised here. Those who expect to continue to further academic studies beyond Key Stage 4 should aim to cover everything summarised here.

By the end of Year 11 pupils who complete the Key Stage 1–4 programme:

#### **Number and measures**

#### **Calculating and simplifying**

#### **Algebra (expressions, formulae, equations, identities) and graphs**

#### **Geometry**

#### **Solving problems**

*•* solve **word problems** and simple **multi-step** and **inverse** problems

## 2. Brief version

#### **2.1 Key Stage 1**

#### **Breadth of study**

1. During the Key Stage pupils should be taught the required *Knowledge, Skills, and Understanding* through:

#### **Knowledge, Skills, and Understanding**

Teaching should ensure that appropriate connections are made between the section *Number and measures* and the section *Shape, space, and measures*.

#### **Ma1 Number and measures**

## **1.** *Using and applying "Number and measures"*

Pupils should:

#### **Solving problems**

(a) explore, interpret, develop flexible approaches to, and persist with problems involving number and measures in a variety of forms

#### **Communicating**

#### **Reasoning**

#### **2.** *Numbers and the number system*

#### Pupils should: **Counting**

#### **The base 10 number system**

(a) understand the groupings into units and 10s (and later into 100s) that underpin **place value**; know what each digit represents (including 0 as a number, and as a placeholder), and how the "value" represented by each digit is determined by its position

#### **Number patterns and sequences**

(a) create, describe, and explore basic number patterns and sequences—including odd and even numbers, multiples of 2, multiples of 5, and multiples of 10

#### **3.** *Calculation*

Pupils should:

#### **Number operations and the relationship between them**

#### **Mental, informal, and standard written methods**

#### **4.** *Solving numerical problems*

Pupils should:

(a) choose sensible calculation methods to solve simple **word problems** involving whole numbers—including problems involving money or measures, drawing on their understanding of arithmetical operations

#### **5.** *Processing, representing, and interpreting data*

Pupils should:

(a) solve suitable problems using simple lists, tables, and charts to sort, classify, and organise information; discuss the methods they use and explain what they find

#### **Ma2: Shape, space, and measures**

## **1.** *Using and applying "Shape, space, and measures"* Pupils should:

#### **Solving problems**

(a) follow instructions to construct simple 2D and 3D objects; represent 3D objects via 2D drawings

#### **Communicating**

#### **Reasoning**

(a) recognise simple spatial patterns and relationships; sort and classify shapes according to given criteria

#### **2.** *Understanding properties of shapes, position, and movement*

Pupils should:

#### **3.** *Understanding measures*

Pupils should:

(a) use direct comparison to order objects by size, using appropriate language; put familiar events in chronological order

#### **2.2 Key stage 2**

#### **Breadth of Study**

**1.** During the Key Stage pupils should be taught the required *Knowledge, Skills, and Understanding* through:

#### **Knowledge, Skills, and Understanding**

Teaching should ensure that appropriate connections are made between the section *Number and measures* and the section *Geometry and measures*.

#### **Ma1 Number and measures**

**1.** *Using and applying "Number and measures"*

Pupils should:

#### **Solving problems**

#### **Communicating**

#### **Reasoning**

#### **2.** *Numbers and the number system*

Pupils should:

#### **Counting**

#### **The base 10 number system**

(a) use **place value** in representing numbers first up to 1000, then up to 10 000 and beyond; extend to decimals with up to three decimal places

#### **Number patterns and sequences**

(a) recognise two- and three-digit multiples of 2, 5 and 10; find the factors of a given integer, and the common factors of two given integers; find the HCF and the LCM of two given integers; recognise prime numbers to 50, and square numbers to 10ˆ10; find factor pairs and all the factors of any two-digit integer; double or halve any two-digit integer

#### **Integers**

#### **Integers and decimals**

(a) use decimal notation for tenths, hundredths, and thousandths; order a set of numbers or measurements

#### **Fractions, percentages and ratio**

#### **3.** *Calculation*

Pupils should:

#### **Number operations and the relationship between them**

(c) know and use the conventions for the order of operations; understand and use **structural arithmetic** to **simplify** calculations; write numerical expressions involving brackets; group related terms in a sum and related factors in a product to simplify, and hence evaluate, numerical expressions

#### **Mental methods**

#### **Written methods**

#### **Measures**

#### **4.** *Solving numerical problems*

Pupils should:

#### **5.** *Processing, representing, and interpreting data*

Pupils should:

#### **Ma2: Geometry and measures**

#### **1.** *Using and applying "Geometry and measures"*

Pupils should:

#### **Solving problems**

#### **Communicating**

(a) use geometrical notation, terminology, and symbols correctly; interpret solutions to problems involving geometrical figures and measures; organise work and record findings clearly

#### **Reasoning**

(a) analyse standard 2D and 3D figures; calculate efficiently and make simple deductions with angles, lengths, areas, volumes, time, and other measures

#### **2.** *Understanding properties of shape*

Pupils should:

(d) make and draw shapes with increasing accuracy, and analyse their geometrical properties

#### **3.** *Understanding properties of position and movement*

Pupils should:

#### **4.** *Understanding measures*

Pupils should:

#### **2.3 Key stage 3**

#### **Breadth of Study**

**1.** During the Key Stage pupils should be taught the required *Knowledge, Skills, and Understanding* through:

#### **Knowledge, Skills, and Understanding**

Teaching should ensure that appropriate connections are made between the section on *Number and algebra* and the section on *Geometry and measures*.

#### **Ma1 Number and algebra**

**1.** *Using and applying "Number and algebra"* Pupils should:

#### **Solving problems**

#### **Communicating**

#### **Reasoning**

(a) understand that some statements can be clearly **proved**, and that other statements can be shown to be false

#### **2.** *Numbers, the number system, structural arithmetic, simplification, and algebra*

Pupils should:

#### **Counting and numbers**

#### **Sequences; powers and roots**

(c) find specified terms of a sequence given a **term-to-term** or a **position-to-term** rule; guess the simplest position-to-term rule for the *n*th term given the first few terms of a sequence

#### **Integers and decimals**

#### **Fractions, percentages and ratio**

#### **3.** *Calculation*

Pupils should:

#### **Number operations and mental methods**

#### **Structural arithmetic**

#### **Algebraic simplification**

(a) substitute numerical values into **formulae** and **expressions** and evaluate; multiply out brackets, collect like terms, identify and take out common factors to simplify expressions; recognise that different-looking expressions may be identical; prove simple algebraic **identities**, and explain why two given expressions are not identical

#### **Written methods**

(a) relate decimal arithmetic to integer arithmetic; use **standard written methods** in column format for addition and subtraction, short and long multiplication, short (and long) division of integers and decimals

#### **Inequalities**

(a) solve simple linear **inequalities** in one variable and represent solutions on a number line

#### **Measures**

#### **4.** *Algebra: equations, formulae, identities, and functions*

Pupils should:

(a) set up and solve linear equations in complete generality [e.g.

$$2 - \frac{3}{4}x = \frac{2 - 4x}{5} \text{[:]}$$

reduce a linear equation in two variables to standard form (*ax* ` *by* " *c*, or *y* " *mx* ` *c*); eliminate a variable from two simultaneous linear equations in two unknowns; solve linear inequalities in one unknown

#### **5.** *Solving numerical problems*

Pupils should:

(d) use algebraic formulae; set up and solve equations

#### **6.** *Processing, representing, and interpreting data*

Pupils should:

#### **Ma2: Geometry**

#### **1.** *Using and applying "Geometry"*

Pupils should:

#### **Solving problems**

#### **Communicating**

#### **Reasoning**

#### **2.** *Constructing and analysing geometrical configurations*

Pupils should:

#### **Know and analyse**

#### **Constructions and congruence**

(a) use ruler and protractor to draw triangles with given data; extract and apply the basic congruence criteria (SAS, SSS, ASA; RHS) to prove standard results (including that the base angles in any isosceles triangle are equal, and the converse)

(b) draw specified figures using "ruler" (i.e. *straightedge*) and compasses only; use the basic **ruler and compass constructions** to complete other constructions

#### **Area and Pythagoras**

#### **Circles**

#### **Volume and 3D**

#### **Scaling and enlargement**

(a) draw figures to scale; interpret distances, angles, and areas on maps and other scale drawings

#### **Loci**

(a) interpret a circle as a locus; interpret the perpendicular bisector of a given line segment as a locus

#### **3.** *Coordinates and graphs*

Pupils should:

#### **2.4 Key stage 4**

#### **Breadth of Study**

**1.** During the Key Stage pupils should be taught the required *Knowledge, Skills, and Understanding* through:

#### **Knowledge, Skills, and Understanding**

Teaching should ensure that appropriate connections are made between the section on *Number and algebra* and the section on *Geometry*.

#### **Ma1 Number and algebra**

#### **1.** *Using and applying "Number and algebra"*

Pupils should:

#### **Solving problems**

#### **Communicating**

#### **Reasoning**

#### **2.** *From numbers to algebra (including calculation)*

Pupils should:

#### **Numbers and arithmetic**

#### **Measures**

#### **Bounds and estimation**

#### **Integer factorisation, fractions, and surds**

$$
\sqrt{12} = \sqrt{2^2 \times 3} = 2 \times \sqrt{3} \text{[}
$$

#### **Fractions**

#### **Fractions and decimals**

#### **Surds**

(a) recognise ? *k* (for *k* ą 0) as the **exact** positive real number whose square is equal to *k*; given *k* ą 0 find the exact or approximate value of ? *k*; use the algebra of surds—including rationalising denominators [e.g.

$$\frac{1}{\sqrt{2}+1} = \sqrt{2} - 1[\ddots$$

use surds (and *π*) to calculate *exactly* in geometric contexts; give lengths arising from applications of *Pythagoras' Theorem* and solutions to quadratic equations in exact (mixed surd) form

(b) use the standard notation for, and calculate with, cube roots;

#### **Powers, roots, and the index laws**

#### **Fractions, decimals, and percentages**

work freely with percentages; use the multiplicative character of percentage increase and decrease; solve problems involving percentage change (including inverse problems and compound interest)

#### **Sequences**

#### **Ratio and proportion**

#### **Algebraic expressions**

$$\frac{1}{\mathfrak{x}+1} = \frac{\mathfrak{x}-1}{\mathfrak{x}}\mathfrak{y}$$

the solutions of the general quadratic

$$y = a\mathbf{x}^2 + b\mathbf{x} + c;$$

use this formula to solve quadratic equations; deduce the symmetry of the graphs of quadratic functions

#### **3.** *Coordinates, graphs, and functions*

Pupils should:

#### **4.** *Processing, representing, and interpreting data*

Pupils should:

$$prob(A \cup B) = prob(A) + prob(B)$$

for disjoint events *A*, *B*; use simple counting to calculate probabilities in discrete sample spaces; understand and use the inclusion/exclusion formula

$$prob(A \cup B) = prob(A) + prob(B) - prob(A \cap B)$$

for events which are not necessarily disjoint

#### **Ma2: Geometry**

#### **1.** *Using and applying "Geometry"*

Pupils should:

#### **Solving problems**

#### **Communicating**

#### **Reasoning**

#### **2.** *From naıve construction to Euclidean geometry ¨*

Pupils should:

#### **Ruler and compass constructions revisited and organised**

#### **Parallel lines and angles in a triangle**

#### **Similarity**

#### **3.** *Geometric calculation*

Pupils should:

#### **Trigonometry**

$$\text{area}(ABC) = \frac{1}{2}ab\sin{\text{C}};$$

deduce the *Sine Rule* and use it to "solve triangles"; prove that

$$\frac{a}{\sin A} = \text{2R}\_{\prime}$$

where *R* is the circumradius of triangle *ABC*

(e) show that on the unit circle with centre at the origin *O*, the point *P* for which the radius *OP* makes an angle *θ* with the positive *x*-axis has coordinates pcos *θ*, sin *θ*q; apply *Pythagoras' Theorem* to derive the identity

$$
\sin^2 \theta + \cos^2 \theta = 1;
$$

use this identity to find values of cos *θ* given the value of sin *θ* (and *vice versa*), and the value of tan *θ* given the value of cos *θ*

#### **2D and 3D figures**

#### **Circles**

(a) understand and use the terms centre, radius, chord, diameter, circumference, tangent, arc, sector, segment

## This book need not end here...

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This book is intended to help candidates prepare for STEP examinations. STEP (Sixth Term Examination Paper) is an examination used by Cambridge colleges as the basis for conditional offers. They are used by Cambridge, Warwick, and many other mathematics departments recommend that their applicants practice on the past papers even if they do not take the examination. Advanced Problems in Mathematics is also recommended as preparation for any undergraduate mathematics course, even for students who do not plan to take the Sixth Term Examination Paper. The questions are all based on recent STEP questions selected to address the syllabus for Papers I and II, which is the A-level core (i.e. C1 to C4) with a few additions. Each question is followed by a comment and a full solution. The comments directs the reader's attention to key points and puts the question in its true mathematical context. The solutions point students to the methodology required to address advanced mathematical problems critically and independently. This book is a must read for any student wishing to apply to scientific subjects at university level and for anybody interested in advanced mathematics.

## Teaching Mathematics at Secondary Level

## Tony Gardiner

*Teaching Mathematics* is nothing less than a mathematical manifesto. Arising in response to a limited National Curriculum, and engaged with secondary schooling for those aged 11 ̶ 14 (Key Stage 3) in particular, this handbook for teachers will help them broaden and enrich their students' mathematical education. It avoids specifying how to teach, and focuses instead on the central principles and concepts that need to be borne in mind by all teachers and textbook authors—but which are little appreciated in the UK at present.

This study is aimed at anyone who would like to think more deeply about the discipline of 'elementary mathematics', in England and Wales and anywhere else. By analysing and supplementing the current curriculum, *Teaching Mathematics* provides food for thought for all those involved in school mathematics, whether as aspiring teachers or as experienced professionals. It challenges us all to reflect upon what it is that makes secondary school mathematics educationally, culturally, and socially important.

Tony Gardiner, former Reader in Mathematics and Mathematics Education at the University of Birmingham, was responsible for the foundation of the United Kingdom Mathematics Trust in 1996, one of the UK's largest mathematics enrichment programs. In 1997 Gardiner served as President of the Mathematical Association, and in 2011 was elected Education Secretary of the London Mathematical Society.

*Cover image: Tisseur De Tolle, 'Un phare en coquille' (2007). CC BY-NC-SA 2.0*

As with all Open Book publications, this entire book is available to read for free on the publisher's website. Printed and digital editions, together with supplementary digital material, can also be found here at www.openbookpublishers.com