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ma4
What should be done about "data-handling" in the mathematics national
curriculum?
A position paper from The Mathematical
Association
 word
version
Summary
The Cockcroft report Mathematics counts (1982) drew the
attention of the mathematical community to the importance of ensuring that
pupils master simple statistical ideas. This led to "data-handling" being
identified as one of the 15 strands in the original 1988/9 version of the
mathematics national curriculum. There were culls to the stranding in 1991,
1993, 1995, and 1999, and yet "data-handling survived (out of all proportion, in
our view) to the extent that in the current 4-strand version it ranks as a
separate "attainment target" alongside Ma1: Using and applying
mathematics, Ma2: Number and algebra, and Ma3: Shape space and
measures.
Adrian Smith’s report Making mathematics count (2004)
contained clear evidence of the resulting distortions which the present
curriculum and assessment arrangements have caused. Among his recommendations a
few were flagged as being "more urgent", in that their wording used the striking
word "immediate". These more urgent recommendations included
Recommendation 4.3 "… that there
should be an immediate review … of coursework in GCSE mathematics and, in
particular, the data handling component etc."
and
Recommendation 4.4 "… that there
should be an immediate review … of the future role and positioning of Statistics
and Data Handling within the overall 14-19 curriculum. This should be informed
by … a recognition of the need to restore more time to the mathematics
curriculum for the reinforcement of core skills such as fluency in algebra and
reasoning about geometrical properties…"
These recommendations were the result of numerous inputs to the
inquiry from many different constituencies. They were not new; but they had
never before been confronted as squarely or as publicly. The heart of
recommendation 4.4 and hence of any "review" must be the need to find ways of
restoring more time to the Mathematics curriculum for Algebra and Geometry. The
directness and clarity of the recommendations were welcomed in school staff
rooms and in universities up and down the land.
To help stimulate this "immediate review" this paper seeks to
identify those aspects of Ma4 (and directly related work) which most
mathematics teachers would agree should be taught within the national curriculum
for mathematics come what may. In short, we conclude that:
(i) The current version of the national
curriculum fails to reflect the fact that important elementary statistical ideas
fall naturally into two quite different categories.
The first consists of relatively simple examples, which can be
used both to reinforce elementary mathematics and to establish links between
"school mathematics" and "making sense of the real world", using basic
applications of elementary work on number and measures (including some graphical
representation and a little algebra). This work may deserve more attention than it currently receives - not under a separate heading, but
within mainstream mathematics.
The second category moves into new territory, seeking to
develop the "science of statistics". As is illustrated by the wealth of
paradoxes in probability and statistics, most of this latter material involves
unavoidable subtleties, which can only be mastered intelligently given a
suitable grounding in number (including fractions, ratio and proportion),
algebra and geometry, and appropriate prior "bare hands" experience with data
(which cannot easily come from within mathematics). A mathematical treatment of
this material may be premature at KS3/4.
(ii) The distinction between these two categories, and other
pressures on the mathematics curriculum (including Adrian Smith’s comments in
Recommendation 4.4), suggests the need for more careful consideration before we
accept that there should be a separate attainment target for data-handling
content at KS3/4.
(iii) While some good work has been done to extract worthwhile
"mathematical" activity from the current data-handling requirements, the
professional consensus among good mathematics teachers has been clear for many
years: the inclusion of "data-handling" encourages a welcome (if limited kind
of) "use of mathematics" for less able pupils, but the way the curriculum is
assessed, imposes unwelcome pressures, undermines teachers' ability to teach
mathematics well, and guarantees that fewer students master what should be
core material in any mathematics curriculum.
(iv) When one reads the data-handling sections of the English
National Curriculum, one is struck by the amount of "padding", which arises when
a potentially valuable - but relatively modest - activity is treated as a
fully-fledged, self-sufficient, free-standing Attainment Target.
(v) Some topics in the mathematics curriculum (such as
arithmetic, order, coordinates, etc.) are relevant to the analysis of data, but
also demand to be included for all sorts of other reasons. We do not attempt
to list such topics. Instead we list important topics, or particular aspects
of broader topics, which support, and which are strongly (and naturally)
reinforced by, serious aspects of data-handling which can be handled
effectively at KS3 and KS4 (e.g. percentages). Thus we list essentially
"mathematical" topics (leaving the reader to imagine the significant
applications to simple probability, statistics and data analysis), and topics
which are essentially rooted in "probability" or "data/statistics" yet are worth
addressing systematically within the mathematics classroom. (We anticipate that
these topics will be treated through appropriate activities within Ma2, Ma3 and
Ma1.)
QCA have recently initiated their response to Adrian Smith’s Recommendation
4.4 (but not yet his Recommendation 4.3). They are doing so by contracting the
Royal Statistical Society’s Centre for Statistical Education, the same agency
which is responsible for designing the current arrangements and for advising on
the existing curriculum. We would like to suggest that a small working group,
including carefully chosen practising teachers alongside seasoned observers,
might be better qualified to assess the nature of the problem and to make
objective recommendations.
Particulars
THE ELEMENTARY MATHEMATICS NEEDED FOR SIMPLE DATA
ANALYSIS
Note: Some themes in the current Ma4 (e.g. inter
quartile ranges) are not mentioned explicitly in the suggested curriculum
list, but are subsumed under some general heading (e.g. measures of
spread).
Some themes (which may pervade the current version of Ma4, such
as the "data-handling cycle") are omitted on the grounds that they deserve no
canonical place within the Mathematics curriculum. Some of these omissions may
perhaps deserve attention elsewhere - outside the mathematics curriculum; others
may be best omitted altogether in their current form, or may be best replaced by
practical "formative" experiences, which embody the ideas in a more appropriate
way. (For example, the "data-handling cycle"; while this may be a useful tool it
is not something which is justified in having the same status within the
mathematics curriculum as say Pythagoras’ Theorem.)
Listed topics are to be addressed naturally, and explicitly,
within Ma2, Ma3 and Ma1.
As one moves down the list, the depth of treatment and the
expected level of mastery becomes increasingly dependent on the target
audience.
1. "Reading" (i.e. interpreting and using) tables (timetables,
tables of data, etc.)
Frequency (counting, tabulating, graphing, summing, etc.)
Distinguishing between situations where individual outcomes
(e.g. tossing a "fair" coin) are "equally likely" (where counting offers an
alternative strategy) and situations where individual outcomes are "not equally
likely" (biased coin, or real life!).
3. Recognition that many familiar numerical "measures" applied
to many natural "populations" (such as height for a typical group of
people/children) give rise to a "roughly symmetric, unimodal distribution".
4. Using ratios and simple percentages in number problems.
Using averages (means) in number problems.
[Later: (i) mode and median, and (ii) measures of spread for
statistical data.]
5. Systematic listing; sample spaces (e.g. for two dice);
simple and (for some) compound "events"
Simple counting problems; sum rule for disjoint sets.
[Later (for some): (i) inclusion-exclusion |AÈB| = |A| + |B| - |AÇB|; and
(ii) product rule for counting ordered pairs.]
7. Whole and parts of a whole; fractions; ratios; percentages;
relative frequencies for "events"
Full blooded arithmetic of fractions.
Percentages as operators/multipliers; hence non-additive
(operands may differ, or sets may overlap)
9. Fractions less than 1 (relative frequency) as measure of
"likelihood" or "probability". Summing relative frequencies for disjoint and
overlapping events.
10. Distinction between individual events (random,
unpredictable) and larger samples or longer sequences of events (naive Law of
Large Numbers: "on average")
The idea that we can usefully calculate with known
"probabilities" even when particular values are not known.
Sample spaces/models where individual events
(i) are "equally likely" and
(ii) definitely "NOT equally likely".
Simple calculations using 8. and 9. for discrete sample spaces
in which the "atomic events" are equally likely (and, for some, not equally
likely).
12. Producing and interpreting simple mathematical/graphical
representations of data (of which pie charts, histograms and bar charts are
merely special cases, rather than distinct curriculum items). Switching between
frequency tables/graphs and cumulative
frequency tables/graphs.
13. Naive approach to "dependence" and "independence" in simple
examples.
(For some) Conditional probability and "independent" events -
based on relative frequency (unchanged relative frequency relative to a subset
as "interpretation" of - and hence as formal definition of - the psychological
notion of "independent").
"E independent of E*" same as "E* independent of E".
Probability calculations involving conditional probability and
independent events; sequences of repeated simple events (coin tosses, card
choosing, etc).
14. (For some) Similarities and differences between discrete
and continuous data, between finite sample spaces and infinite sample spaces.
Background
When originally injecting "statistical ideas" into the
mathematics curriculum melting pot, the Cockcroft report was exceedingly modest
in what it suggested might be a sensible objective (para 458, final section). It
was also completely open in admitting (if much later in para 774!) that such a
recommendation was more in the way of a tentative suggestion than a reflection
of a consensus professional view: "surprisingly few of the submissions which
we have received have made direct reference to the teaching of
statistics".
Thus, the inclusion of "statistical ideas" in Cockcroft's
"Foundation List" was speculative, but also modest:
"One aim should be to encourage a critical attitude to
statistics presented by the media ...
Emphasis should be placed on the relevance of probability to occurrences in
everyday life as well as to simple games of chance ... pupils should not
necessarily be expected to use the words mean, median and mode."
One reason for the ambivalent response to Recommendations 4.3
and 4.4 in the Smith Report stems from confusion about what the recommendations
meant. It seems to us unlikely that a statistician would propose that difficult
statistical ideas should be "explicitly taught in other subjects". Rather, one
suspects that Smith intended to emphasise three things:
First, that one serious effect of having a separate Attainment
Target for data-handling has been to reduce the time available to teachers and
students in order to achieve mastery of important basic techniques.
Second, that while the most basic material (see our tentative
list above) fully deserves to be included within the mathematics national
curriculum (under Ma2 and Ma1), much of the additional material is either
"padding" (stem and leaf, box and whisker, etc.), or is too subtle for a
suitable mathematical treatment to be given at KS3 and KS4.
Thirdly, that before attempting an educationally useful
mathematical analysis of this harder material, students need much more
experience of the subtleties of data in other subject contexts.
For example, one very natural (and historically pertinent)
place where all students should "experience" the normal distribution - long
before specialists choose to attempt a mathematical analysis - is in making
repeated experimental measurements of definite physical quantities. All
students, irrespective of their future mathematical "pathways", can
appreciate the fact that the resulting bell-shaped distribution of results which
arises in all such cases suggests some universal "law of errors". We need to
find simple, reliable examples in the same spirit whereby other subjects might
provide experiences and opportunities on which to base a "common sense" feeling
for certain important features of "the way data can behave" - experiences
which can then be used as examples in any subsequent mathematical
treatment.
Another reason for the delay and confusion in responding to
Recommendations 4.3 and 4.4 is that the mathematical community lacks agreed
notions (i) of what is meant by "elementary mathematics" and by "data-handling",
and (ii) of what is needed in order to teach statistical ideas well. If we had
such shared notions, we could examine dispassionately the relative importance of
material from "elementary mathematics" and from "data-handling" and come to some
agreement concerning which statistical ideas can be effectively taught,
and which need to be taught, at KS3/4 (and to whom), and also which
pre-mathematical experiences should be used to establish the necessary prior
"intuitions".
In the absence in practice of such agreed professional norms,
we have proceeded – in as broad-minded a spirit as possible – on the basis of
what we believe those norms should be. We have then tried to propose a core of
ideas, which we think most teachers and end-users would wish to see retained
(and even strengthened).
Concern about the role of "data-handling" within the
Mathematics National Curriculum has been around since its inception in 1988/9,
but has been made more obvious by the recent imposition of compulsory GCSE
Data-handling coursework. This brought to the surface many of the problems
implicit in Ma4. But coursework is in fact a completely separate problem from
that which we seek to address here - as Adrian Smith recognised in separating
Recommendation 4.4 from Recommendation 4.3. The coursework issue is serious ("in
particular the data handling component" to quote Recommendation 4.3) and it is
clearly very important that the recommendation of an "immediate" review of the
role of coursework be acted on. Nevertheless it was not our business here to
comment on the matter of data-handling coursework.
Nor is it within our brief to suggest how statistical material
should be integrated with other subjects. (However there are obvious things we
might wish to contribute to such a debate, such as the observation that much
more thought needs to be given to the simple analysis of data
arising in other contexts - using percentages, simple proportion, straight line
graphs, etc. - before students are likely to be in a position to benefit from
more mathematically sophisticated approaches. Thus it might make sense for us to
work with other subjects to replace their use of correlation, confidence
intervals, regression, or whatever, by elementary methods which provide the much
the same sort of insight.)
Our instructions were to step back and consider
which aspects of the existing curriculum relevant to
probability, data-handling and statistics are so much part of the stuff and
responsibility of elementary mathematics that they should be retained in some
form within the mathematics curriculum if possible.
We hope our attempt to respond to this challenge will stimulate
a rational debate of how, and how much, "data handling" material should be
included in the mathematics curriculum. The fact that there does not seem to
have been a serious professional debate along these lines stems in part from the
tendency to present the case for "data-handling" as if it were "self-evident"
that life in the modern world "demands that such material be included". The
reality is more complex. Data – like love and calculators – is all around us.
Yet deciding how much of the mathematics curriculum should be given up to ideas
related to "data-handling", and at what stage, requires a mature assessment of
priorities, and of what is realistically achievable.
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