Collins,+J+D+and+Quigley,+M+-the+mathematics+fallacy


 * Future curriculum planning—the fallacy of mathematics for all**

Collins, J D and Quigley, M (2010), Future curriculum needs -the mathematics fallacy, paper for the 14th World Congress for Comparative Education Societies, Bogazici University, Istanbul, Turkey, 14th - 18th June.

This paper examines and challenges the belief, held by many nations, that a nation’s

economic growth depends significantly on the mathematical skills of all its people.

Drawing upon local research and evidence from sources such as the Programme of

International Student Assessment (PISA) and the current Gross Domestic Product

(GDP) of countries, the authors argue that there is little evidence to support such a

contention. In particular they argue that, for a large proportion of children, there is

little justification for requiring that mathematics be taught beyond that expected by

the age of 14 years in a typical high school programme.

//The benefits of education//

For at least a century economists have tried to find a link between education and

economic growth. The search has embraced both the benefits accruing to an

individual and the benefits accruing to the state. In the former case, if we allow for the

actual cost of schooling plus the amount of income foregone, education is a good deal

for the individual. For instance, if we consider two people aged 50 in the UK, one

with a master’s degree (or better) will be earning on average about twice as much as

someone who left school without any qualifications.

Worldwide the Mincerian wage model (Mincer, 1974) has been applied to most

countries with fairly consistent results. It appears that the rate of return of investment

in education to the individual is between 5 and 15 per cent, with the UK roughly in

the middle of the pack. There are other benefits of qualifications too. People in the

UK with no qualifications are three times more likely to be unemployed than degreeholders,

and the figure for the US is about five times.

However, we should not conclude from this that more education is necessarily a

“good thing”. There are good reasons to believe that people who complete more

formal education may have other qualities, such as high IQ or greater motivation,

which would have led them to higher incomes anyway.

At the country level, the picture is rather different. Benhabib and Spiegel (1994)

reported that “human capital accumulation fails to enter significantly in the

determination of economic growth” using the standard Cobb-Douglas production

function. Other researchers, such as Krueger and Lindahl (2001), argue that increases

in education do not necessarily contribute to economic growth. As Wolf (2002)

suggests, whilst it is clear that wealthy countries tend to have well-educated

workforces, it is less clear that the education was a driver of the prosperity. She points

to studies by the World Bank which suggest that across the developing world,

countries which have done the most to improve the standard of education have

actually grown less fast economically than those which spent much less.

Evidence from the National Child Development Study (a longitudinal study of all the

children born in one week in March 1958, and tracked ever since) suggests that it is

years of education which matter, rather than any particular academic subject—with

one exception: mathematics. Using this data, Doherty (2003) compared the apparent

impact of levels of numeracy and literacy on earnings. The results suggested that

numeracy, rather than literacy, has a highly significant effect on earnings. Mostly this

is through its effect on college attainment, but also directly, and its effect may be

increasing year by year at a rate of 6% per year. Clearly education in mathematics is

important but this applies in the main to those pursuing education beyond age 16

years; in particular to those who gain from a college education, at most 40 per cent of

school leavers. Adelman (2006) amplifies this position further referring to the college

bound in the US:

"The highest level of mathematics reached in high school continues to be a

key marker in pre-collegiate momentum, with the tipping point of

momentum toward a bachelor's degree now firmly above Algebra 2. But

in order for that momentum to pay off, earning credits in truly collegelevel

mathematics on the postsecondary side is de rigueur… The math

gap is something we definitely have to fix." (Adelman, 2006, p. 20)

However, looking more towards earning capacity, the report of the New Commission

on the Skills of the American Workforce (2007) provides data showing that high

earnings are not just associated with people who have high technical skills such as in

mathematics and science. In fact, even if literacy is not, mastery of the arts and

humanities is just as closely correlated with high earnings, and, according to their

analysis, that will continue to be true. In other words history, music, art and

economics will give American students an edge just as surely as mathematics and

science will.

Clearly, there is ample evidence to support the contention that investment in

education, especially mathematics education, is a good deal for the college bound

individual. We agree totally with higher mathematics education for such students. But

the question we wish to answer is whether mathematics should be taught compulsorily

to all students after age fourteen years and if so to what extent it contributes to the

growth of the national economy?

Some 40 years ago Thomas Kuhn (1962) introduced us to the idea of a paradigm. At

the time he was referring specifically to the ideas, perhaps beliefs, which permeate an

entire scientific community. The present belief that nations should teach mathematics

to all school students until they reach the age of 16 years, and beyond, is one such

paradigm. It is, for example, one of the recommendations of the RAND mathematics

study panel (Ball, 2003) that

"The United States needs to improve the mathematical proficiency of all

students in the nation’s schools."

This belief is based on several assumptions. Firstly, that high-level achievement in

mathematics (perhaps as a minimum the equivalent of a National Vocational

Qualification (NVQ) level 2, grade C GCSE) is a pre-requisite for nations to grow

their economy. Secondly, that everyone is capable of learning and applying

mathematics at quite sophisticated levels and thirdly, the belief that mathematics

provides certain transferable skills.

As with many paradigms, to create a paradigm shift is difficult and likely to happen

only as a result of a crisis in which the ideas underpinning the paradigm are robustly

challenged. We believe that we are at that stage in the teaching of mathematics, in

other words, the teaching and learning of mathematics is approaching a crisis; a view

also expressed in the influential report Making Mathematics Count (Smith, 2004).

Is a minimum level of mathematics achievement for all such as NVQ level 2

necessary for nations to grow their economy?

One has to ask where is the evidence that a nation’s economy depends in any way on

the mathematical skills of all its people beyond those skills acquired by the age of 14

years in a typical high school programme? We venture to suggest that there is no

supportive evidence. However, there will be a significant mathematical requirement

for some professions in order to facilitate technological progress in a wide range of

fields. It certainly will not be the whole population nor even half the population that

will require this. What is actually required by more and more people are skills such as

problem solving, logical thinking, conceptual ability, communication, data handling

and interpretation, using computers and engaging in research. These are skills that can

be developed through the study of very many subjects and not only through

mathematics. This has become clear from work done in the United States as shown in

Figures 1 and 2 below.

Fig. 1: Changes in Computer use and Occupational Structure (Note from Rebecca - this diagram won't copy. It shows some interesting trends in the use of computers by different professional groups in society from 1982-1999 (i.e. pre-internet))

Fig. 2: Changes in demand for different categories of employment - 1960-2000 (US). (Showing an increase in demand for jobs which are non-routine).

The above charts make abundantly clear that skills such as the use of computers and

non-routine interactive and thinking tasks are increasingly required in the work place.

However, in neither case do the authors specify mathematical thinking or mathematical

skills and in the text refer to critical thinking skills, problem solving, and

interpretation skills that may or may not require mathematics at a level beyond that of

the 9th grade.

Yet there have been many calls for increasing the importance of mathematics in

school, either on the grounds that mathematics is a qualifying subject for postsecondary

education, or else for the purposes of employment. For instance, Evan,

Gray, and Olchefske (2006) bemoan the achievement of American students in international

comparisons. They argue that the key to improving the proportion of students

graduating with “high-level, globally competitive skills” is to “dramatically increase

the number of students who achieve proficiency in Algebra in their middle or early

high school year” (p2). Their argument is that “successfully passing Algebra early in a

student’s career—no later than 9th grade—greatly improves the chances of the student

graduating from high school, going to college, and graduating from college” (p9).

What this means is that there is a correlation between completion of ‘Algebra’ at high

school and subsequent graduation from college. But there is no demonstrable

correlation between the content of ‘Algebra’ and college graduation. It may well be

that completing ‘Algebra’ is no more than a proxy for problem solving ability, and

has little to do with algebra itself. It may also be the case that the correlation exists

purely because ‘Algebra’ is usually a prerequisite for high school courses designed for

the college-bound. On this basis it isn’t unreasonable to predict that students who do

not gain access to these courses are not very likely to graduate from college.

Evan, Gray and Olchefske also describe the apparent pressing need for highly

educated entrants to the workforce, pointing out that the US Department of Labor

estimates that “by 2008 there will be 6 million job openings for scientists, engineers

and technicians” (p.5). This figure seems large at first sight, but the working

population of the US is roughly 140 million.

An on-going study carried out by one of the authors surveyed, amongst other things,

the jobs done by the parents of 267 children in Birmingham. Each identified job was

given a rank based partly on the International Standard Classification of Occupations

and partly on the National Qualifications Framework for England, Wales, and

Northern Ireland. The rough amount of education required for each and the frequency

of the ranks, are shown in table below. Table 1: Educational Level of Parents' Jobs (Rebecca's note - I've tried formatting this table with spaces but it doesn't stay. However you can work out what the figures mean with some effort)

Level of education Frequency Valid Percent Cumulative Percent

No formal qualification 55 19.2 19.2

GCSE or skill licenceship 133 35.1 54.3

Scottish CSYS; A level 29 4.6 58.9

Advanced/higher certificate 46 18.5 77.5

Ordinary degree 17 2.0 79.5

Honours degree; higher diploma 35 14.6 94.0

Master’s degree, postgraduate diploma 8 3.3 97.4

Doctorate or higher doctorate 9 2.6 100.0

Total 332 100.0

(Quigley, O’Sullivan & Al Balooshi, 2010)

We can see that more than 50% of jobs entail no more than GCSE level. On

examining the mathematical requirements of these specific jobs, few entail any more

mathematics than that described in the Cockcroft Report paragraph 85, nearly 30

years ago. It seems that not much has changed since then.

"… We believe that it is possible to summarise a very large part of the

mathematical needs of employment as ‘a feeling for measurement’.

This implies very much more than an ability to calculate, to estimate, and

to use measuring instruments, although all of these are part of it. It implies

an understanding of the nature and purposes of measurement, of the many

different methods of measurement which are used and of the situations in

which each is found; it also implies an ability to interpret measurements

expressed in a variety of ways."

We know that there is a positive correlation between a country’s Gross Domestic

Product (GDP) and its achievement in science as assessed by the Programme for

International Student Assessment (PISA, 2006). However, that correlation is of the

order of 0.3 and certainly does not provide evidence of causality. For example,

Finland and New Zealand score higher on the PISA science than would be predicted

on the basis of their GDP per capita if the relationship were linear and deterministic

whereas US scores far lower than might be predicted from its high GDP. Of particular

note is that according to the OECD data, Finland, with an average per capita GDP, is

clearly the top performer, while the US, with the highest per capita GDP, performs

below the OECD average.

In mathematics a similar situation exists but with a remarkable difference. If we take

twelve of the top countries rated by GDP per capita for which we have PISA data and

match these with the corresponding PISA scores for mathematics we find a negative

correlation (around -0.6).

Table 2: Comparison of PISA scores and GDP for selected countries

Data from PISA Maths 2006 and CIA World Factbook 2006 (Note from Rebecca - I'm afraid this graphic won't copy)

Whilst this probably has more to do with the expenditure on education for that

particular cohort of students, also negatively correlated, it does raise the question of

whether or not we are right to assume that a country’s success depends on the

mathematical competence of its working population. Chen and Luo (2009)

investigated the link between test scores (mathematics and science) and cross-country

income differences using data from PISA and the Trends in International Mathematics

and Science Study (TIMSS). They examined whether or not test scores are good

indicators of labor-force quality. The analysis suggested that after properly controlling

other variables that are typical in cross-country economic growth study, the strong

link between test scores and cross-country income differences disappears. Perhaps of

greater importance is that they demonstrated that variables such as Research and

Development researchers (per capita) or Scientific and Technical journal articles (per

capita) can better account for cross-country income differences.

//Is everyone capable of studying and achieving mathematics at level 2?//

The drive by countries to increase the number of skilled well-educated workers is

clearly having a positive effect if statistics in the US are any indication of what is

happening worldwide. As the chart below indicates, there is a steady and almost

continuous year on year increase in the percentage of the population with 3 and 4-year

college degrees and a corresponding decline in the percentage of high school dropouts

to around ten per cent. But the proportion of workers who are college graduates is still

around thirty per cent and the figure in many other countries is similar. The US

Bureau of Labour Statistics report in 2001 forecast US labour needs and required skill

levels to 2010 (Hecker, 2001). This indicates that in the US only 21.8 per cent of jobs

are forecast to require a university degree or higher up to 2010.

Figure 3: Change in educational success 1978–2003 (note from Rebecca - graphic wont copy - it shows the rising trend of % workers with higher level qualifications)

Yet this is still not sufficient for politicians who continue to push the boundaries

beyond reasonable levels. For example, in 1999 in the UK the New Labour target for

the proportion of the population that study in Higher Education was set at 50 per cent.

“In today's world there is no such thing as too clever. The more you know, the further

you will go… So today I set a target of 50 per cent of young adults going into higher

education in the next century” (Blair, 1999). Ten years later it has increased by 0.6

percentage points from 39.2 to 39.8 per cent, a trend which, if continued, would take

very much in excess of 100 years to meet the target if at all.

The Leitch report (Leitch, 2006) highlights the necessity for 95 per cent of adults to

be functionally literate and numerate by 2020 and 90 per cent should achieve level 2.

The latter broadly equates to 5 GCSEs A*–C or equivalent though surprisingly,

neither mathematics nor English are specified requirements. However, we know from

the DFES 2003 survey that for adults in the age range 16–65 this is about 18 per cent.

Currently around 46 per cent of 16 year olds obtain level 2 including mathematics and

English. Even so, the Leitch target still seems challenging when we learn from the

numeracy report DFES, 2003 nearly half (47 per cent) of all adults aged 16-64 were

classified at Entry level 3 (roughly what a 9 year old is expected to be able to do) or

below in at least one of literacy or numeracy. Only one in five (18 per cent) achieved

level 2 or above for both literacy and numeracy. This Government research suggests

that 15 million workers struggle to grasp basic calculations and many also have

functional literacy problems.

This latter statement is difficult to reconcile with the same survey findings that less

than two per cent felt their weak numeracy skills, had hindered their job prospects, or

led to mistakes at work. Furthermore, 67 per cent with entry level 1 or lower (roughly

what a 7 year old can do) felt that they were fairly good at number work, presumably

because on a day-to-day basis they coped adequately with any numeracy problems

they faced. Interestingly, the percentage of those at higher management level

requiring or using level 2 numeracy is less than 60 per cent. It is time for politicians

to recognise that everybody cannot be at higher management level and that the great

majority of those working at below management level will require

mathematics/numeracy at level 1 or below. We believe this is more than adequately

met by the mathematics taught in high school in most countries by age fourteen and

certainly by the National Curriculum for England at Key Stage 3.

Figure 4: Adult (age 16–65) numeracy levels and occupation category, England,

2002/03 (this shows that people in higher level - less routine jobs are more highly qualified in maths).

What is difficult to understand is why politicians, and some educators, cannot accept

that there are people for whom mathematics, beyond that taught for Key Stage 3 (11–

14 years), is virtually a closed book; furthermore, as the chart above shows, this does

not mean that they are disadvantaged in the labour market. All the more reason to be

astounded that following this same observation made publicly by the then President of

the National Association of Schoolmasters and Union of Women Teachers in 2003,

the matter was debated in the House of Commons in a remarkably narrow fashion,

focusing largely on the benefits of quadratic equations for all! Mathematics teachers,

the present writers included, have long sought ways to motivate children across the

ability and age range to learn and achieve well in mathematics. Varying degrees of

success have been achieved for many but never the whole cohort. What mathematics

teachers know from experience is supported with clear evidence from twin studies

viz: some children are not genetically predisposed to learn mathematics. We should

accept that there are, as Alarcon, Knopik and Defries (1999) put it, issues of

heritability and genetic influence that for some children at the outset prejudice growth

in their mathematics learning beyond a certain level. The question then is what

proportion is ‘some’; the authors venture to suggest that this would be no more than

60 per cent but more likely 40 per cent.

Also worthy of note is that there is a distinct difference between learning and using

language and learning and using mathematics. For example, learning a language is

usually acquired through immersion over time in a sea of vocabulary and once gained

can be used creatively by almost everybody. Furthermore, native speakers can

produce and understand a vast number of new words and sentences that are

appropriate to their circumstances. This is not observed to anything like the same

extent in mathematics learning. That language acquisition ‘apparently happens’

through immersion leads Chomsky (2003) to point to the problem of knowing how

children learn the ‘diverse and complex system needed to support this amazing

activity so smoothly in such a diverse circumstances’. Mathematics ‘acquisition’, if it

happens at all, is clearly different and generally people do not make progress in the

subject simply by being exposed to it. Indeed, the levels of anxiety that are so often

produced by such unstructured exposure are generally regarded to be

counterproductive to learning (Buxton, 1981). Yet it should be pointed out that

children can acquire simple numerical skills very early on in infancy even after a few

months of life (Barth, Beckmann, & Spelke, 2008).

The belief that mathematics provides certain transferable skills

During the latter part of the 19th century there was a notion that study of subjects such

as Latin and Greek, some include mathematics, somehow prepared the human brain

adequately for both current and future learning. This was presumably because these

subjects would ‘exercise and develop’ necessary skills and learning processes to such

an extent that the benefactor would be able to apply their minds to just about

anything; the analogy being that of the brain as a muscle which you exercised. In the

case of mathematics, such beliefs imply an ability to transfer both skills and

knowledge to other subjects or non-mathematical problems but there is no research

evidence to support such belief. For instance, many people have suggested that

learning mathematics develops logical thinking. However, when “reasoning abilities

of groups of mathematics students in secondary schools were compared with those of

other equally talented students who had not had the same mathematical training, no

differences in general logical effectiveness were observed between the groups”.

Indeed, as far back as 1924 Thorndike showed this was the case and others

(Detterman 1993, Evans 2009) have repeatedly confirmed his findings and regardless

of how appealing transfer of learning is as a theory, modern experimental tests have

also refuted it. Yet it seems that the influence of this 19th century Zeitgeist remains.

Part of the confusion lies in a redefinition of “transferable skills”. We no longer think

of transferable skills in the sense of transfer of training, rather they are skills of a nonacademic

nature used in the workplace. Curry and Sherry (2004) explored the views

of various organizations as to transferable skills. They investigated which skills were

considered important and which skills were most highly developed amongst young

people entering the job market. Their results are summarized in the table below. (The

listing of the skills in each panel is a combination of the rankings of that skill in both

importance and development.)

Table 3: Important versus developed skills (this table is reorganised to clearly show it's four sections as a list)

__Above average development and Above average importance__

Oral communication Teamwork Information Presentation Skills Coping with multiple tasks Managing One's own learning Written Communication Planning Problem Solving Critical thinking

__Above average development and Below average importance__

Analytical ability Information management Research skills

__Below average development and Above average importance__

Time management Decision making

__Below average development and Below average importance__

Negotiation Customer service Project management Career management Leadership Networking Numeracy Fluency in a second language

Curry and Sherry p. 14

We see that numeracy is rated to be of below average importance (actually -1.6 on a

scale of -2 to 2) and of below average development (-1.4 on a scale of -2 to 2). It is

hard to see any direct connection of the other ‘transferable’ skills to mathematics,

with the possible exception of problem solving. But even here, the problems that

children typically solve in mathematics have little relevance to the workplace. This

observation is reinforced by evidence to the Smith report that ‘being in possession of

an Application of Number qualification rarely results in candidates having

transferable mathematics skills of any worth.’ (Smith 2004, p91)

In any case, what is the point of trying to get students to learn, say, presentation skills

by teaching them mathematics? Would it not make a great deal more sense to simply

teach them presentations skills directly?

//Conclusions//

We believe there is more than sufficient evidence in the arguments presented to

justify reconsidering the place of mathematics in the curriculum for all students post

age 14 years. As recommended in the Smith report (Smith, 2004), what seems to be

needed is a well-researched and definitive statement of the mathematical requirements

of all students by age 14 years followed by a review of the expectations set by post-14

mathematics syllabuses preparing students for higher education in areas requiring

mathematics. These should be evidence-based statements that properly inform

political decisions concerning standards of mathematical proficiency to be met by

students. They should be directly related to realistic workplace requirements as well

as to the requirements of further and higher education. The joy of mathematics for its

own sake has a place for all who can appreciate it but realistically, we should bear in

mind that at least one in five school students does not enjoy learning mathematics

(International average, TIMSS 2003 p.160) and in the UK, evidence from focus

groups run by the QCA for the Smith report revealed that for many students GCSE

Mathematics seems irrelevant and boring. We could speculate that, unlike the

teaching of English and most other subjects of the curriculum, this is because

mathematics in schools is rarely taught in the context of real applications.

The current examinations at ages 16 and 18 years present only moderate challenge to

the most able students, many of who might enjoy the intellectual challenge of

increased mathematical rigour and more advanced content. Anecdotally, one of the

authors recently spent time observing mathematics lessons for 16-17 year olds in

North American, Indian, British and Arab schools in the Middle East. Whilst there is

considerable overlap in what is taught, the expectations in terms of rigour and content

of both Indian and Arab students was significantly higher than that for either British

or American students. For example, a class of twelve Indian 17 year olds was

expected to be able to use applications of differentials in real examples, to use partial

differentiation to demonstrate Euler’s theorem, to use Lagrange Undetermined

multipliers to find maxima and minima in three dimensions. Their construction and

applications of second order differential equations included discussion of such real

applications as how time of death could be established and how the growth and decay

of bacteria could be analysed. These are topics not generally found before the first

year of mathematics degree courses yet were handled with considerable mathematical

maturity and competence by the students; the issue, we believe, is one of expectations

rather than actual content. Are such topics, differential equations perhaps excepted,

beyond the capabilities of mathematically able British and American mathematics

students? Of course not, so why are the expectations lower? Perhaps because by

attempting, some might say unsuccessfully, to raise the standard of the majority we

are neglecting the capabilities of the minority whose potential contribution to our

nation’s growth is huge. In our view, the recommendations in the Smith report do not

go far enough. Mathematics should be optional after the age of fourteen and only

those who have shown the capability and motivation to benefit from further study in

mathematics should continue to do so; in England this might be judged by those who

obtain a secure level 5 in the standardized assessment tests at age 14 years. If after

nearly ten years of education students are not sufficiently numerate to meet the

challenges of everyday mathematics in the workplace we contend they are very

unlikely to do so with a further two years of study at an even higher level.

There is a compelling case for re-examining the place of mathematics in the school

post-14 curriculum and for an honest appraisal of the contribution mathematics

education actually makes to the growth of nations. Are our politicians capable of such

appraisal and are they brave enough to take the appropriate action? If not, we are

committing generations of children to spend excessive time learning material that will

be irrelevant to a large proportion of them instead of focusing on more of the things

that really matter in their lives.

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Psychology, 38 (1), pp. 63-77

Autor, D, Levy, F, and Murnane, R (2003) The Skill Content of Recent Technological

Change: an Empirical Exploration The Quarterly Journal of Economics, November

Ball, D. (2003). Mathematical Proficiency for All Students: Toward a Strategic

Research and Development Program in Mathematics Education, Rand Education.

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John Collins and Martyn Quigley are British mathematics educators with extensive

experience of teaching at school, college and university level in England and

internationally. Martyn currently teaches on the EdD and MEd programmes at the

British University in Dubai (BUiD) and leads the BUiD Postgraduate Diploma in

Education. John is former Professor of International Education at University of

Northern Kentucky and at Assumption International University, Thailand and is

currently working from England as a freelance education consultant and inspector of

schools.