My
apologies for the gap in posting to this blog, a combination of life being very busy and problems with my website provider which are now solved. I look forward to reflecting on working in school again shortly, meanwhile,
I realised that I'd written the piece below a while ago, based on my conference
talk in April, and thought you might be interested to see it. As always,
very pleased to hear what you think!IntroductionIn the
1980s comedy series, “Yes, Minister” there was an episode “The Moral Dimension”
in which it became apparent that a British contract in a fictional Middle
Eastern country had come about through bribery. The Minister, Jim Hacker,
initially took a highly censorious line, saying, “Sin is not a branch of
geography” – so if it is wrong to obtain a contract through bribery in the UK
it is wrong to do so anywhere else in the world. The Permanent Secretary, Sir
Humphrey Appleby, argued the opposite, sin is a branch of geography,
what may be considered to be wrong in some parts of the world is considered an
essential oiling of the wheels in another. What appears to be an absolute
standard, in fact, is not.Meanwhile,
through my career I have been privileged to work in a variety of very different
contexts across the world. This includes starting as a school teacher in
UK comprehensive and other schools, then engaging in British teacher training,
spending some time working with mathematics and science teachers in Jamaica,
and most recently have spent 4.5 years working in Dar es Salaam, Tanzania, with
the Aga Khan University, engaging with education professionals from across
Tanzania, Kenya and Uganda on a Master of Education programme and also doing
outreach courses for primary teachers on interactive mathematics teaching in
Kenya and Uganda. There is
no getting away from the fact that, in practice, mathematics teaching and
learning looks very different in different parts of the world. But, when
surface features are stripped away, is good mathematics teaching a branch of
geography? Are there different notions of ‘good practice’ determined by
local and national culture? Or are there absolutes within what we would
consider to be good practice in mathematics teaching irrespective as to where
we are in the world? In
examining this question, principally drawing on my recent experiences in East
Africa, I wish firstly to look at policy statements from across the world and
the extent to which they are similar. I then look at Tanzania as an
example of a country very different to the UK, considering some aspects of the
nature of education in Tanzania and the culture, particularly attitudes towards
authority, which give rise to the suggestion that the nature of good
mathematics teaching differs across the world, therefore is a branch of
geography. In conclusion some examples are given of classes I taught in
neighbouring Uganda in which I argue that, in the long term, there are elements
of ‘good practice’ which transcend cultures. Some
policy statements from across the worldBelow are
four policy statements from Government publications or professional
organisations, edited only to obscure spelling differences in different types
of English. Before proceeding, I ask you to guess which country each
comes from. *It is expected that if concepts rather than algorithms are properly taught, the need for re-teaching, remediation and revision will be seen to be much less than was previously the case*
*Learning mathematics is maximised when teachers focus on mathematical thinking and reasoning. Progressively more formal reasoning and mathematical proof should be integrated into the mathematics programme as a student continues in school.*
*The main objectives of teaching mathematics in secondary schools are:*
- To promote the development and application of mathematical skills in
interpreting the world and solving practical problems in daily life;
- To provide pupils with mathematical tools and logical thinking which they
can
apply in understanding better other subjects;- To develop a foundation of mathematical knowledge, techniques and skills
for
studying mathematics and related subjects at higher levels of education.*Mathematics is a creative and highly interconnected discipline that has been developed over centuries, providing the solution to some of history’s most intriguing problems. It is essential to everyday life, critical to science, technology and engineering, and necessary for financial literacy and most forms of employment. A high-quality mathematics education therefore provides a foundation for understanding the world, the ability to reason mathematically, an appreciation of the beauty and power of mathematics, and a sense of enjoyment and curiosity about the subject.*
Does it
help if I tell you that there is one each from Tanzania, UK, USA and
Jamaica? Answers are given at the end of this paper. Unless
you happen to recognise these statements, I suggest that it is not possible to
work out where in the world they come from. I would go further – each of
these could reasonably be a policy statement of the ATM itself! These are
not carefully picked statements, I encourage you to check for yourself that
statements like this can very easily be found from across the world using
Google. The
commonality to be found in these policy statement as to what we are trying to
achieve in the mathematics classroom would imply that the answer to the
question in the title of this paper, ‘Is good mathematics teaching a branch of
geography?’ is no, there is considerable agreement across the world at policy
level as to what mathematics is, why it is important and how we go about
teaching it. There is, apparently, consensus in the mathematics education
world which is not found, I suggest, in other areas, including literacy and
early childhood education. Let us
now look at this question from a different perspective, for which I take
Tanzania as the main comparator for the UK in further probing the question as
to whether good practice is the same across the world. Case
study: TanzaniaTanzania
is a developing country on the east coast of Africa with a population of
approximately 54 million people. Broadly consistent with other developing
countries, the population pyramid for 2016 looks like this: This
narrowing of the pyramid right from earliest age corresponds to considerable
neo-natal deaths with every age group losing appreciable numbers of its members
through illness and accident. This means that the overall age profile of
the population is young, with 44% of the population aged 14 or less. Meanwhile,
the population pyramid for the UK for 2016 looks like this: As can be
seen, it is not until the age of about 70 that there are consistently fewer
people in older age brackets, with fluctuations in birth rates making a bigger
difference to the size of the groups than attrition rates up to that age.
In contrast to the 44% of the Tanzanian population which is aged 14 or less, in
the UK the corresponding figure is 17%. So,
Tanzania has hugely more youngsters needing education, with correspondingly
fewer adults in the working population to become teachers. Putting these
two things together, I calculate that, to achieve the same teacher pupil ratios
as there are in the UK, Tanzania needs 4 times as many teachers relative to the
working population. On top of low levels of resourcing, both human and
physical, this hugely greater need for teachers from the working population
represents yet one more challenge in providing a much needed quality education
in the developing world. Of
course, teacher – pupil ratios are not the same or anywhere close.
Classes are much larger, fewer children are in school, particularly girls in
secondary school, and the qualifications and pay rates of teachers are low. It is
worth noting also that, when Universal Primary Education was introduced in
1974, there was a massive and sudden increase in the number of teachers needed,
with the primary school population growing from 1.3 million in 1973 to 3.5
million in 1981 (Oketch & Rolleston, 2007: 16). The response to this
need included the introduction of what was called the ‘Certificate 3C’ route
into teaching, that necessitated only that new primary school teachers had a
primary school education themselves with initial teacher training consisting of
a non-residential course of only a few weeks (Komba & Nkumbi, 2008).
It was only in 2015 that all certificate routes were discontinued, the minimum
requirement now being a diploma. Meanwhile, according to Tanzanian
Government figures, 70% of primary teachers in 2013 had only a certificate
qualification (PMO-RALG, 2014: 38). That poor
qualifications come alongside poor subject and pedagogical knowledge was
exemplified for me in a class in neighbouring Uganda for primary
teachers. I was somewhat perplexed to discover that, way beyond the
Ugandan National Curriculum, primary teachers were teaching formal algebra to 8
year olds. When I made some enquiries about this, it turned out that they
were dealing with bright children but themselves lacked the subject and
pedagogical knowledge to devise interesting, challenging activities around
numerical structures working towards algebra, instead, in effect, turning the
page in the textbook and going onto the next thing rather than enriching
youngsters’ experience. It can
readily be seen that there are considerable difficulties here, given the need
for a large number of teachers being met by allowing low qualifications with
low rates of pay, alongside also poor resourcing of schools in other respects.
But there is another issue to explore, which I call ‘ Shikamoo
culture’.‘Shikamoo
culture’: the respect for age and authorityWhen
people I know go to Tanzania I tell them that if they only get to learn two
words of Swahili, those two words need to be, “ Shikamoo” and “Marahaba”.
Shikamoo is a highly respectful greeting word, with its use largely
determined by age, one says Shikamoo to people who are approximately ten
or more years older than oneself. “Marahaba” is then the required
response.These two
words fall within a culture which shows great respect for age, experience and
authority. When one first arrives in East Africa this is a very
attractive part of what one sees. In many ways it is excellent that
teachers come within this, with respect for teachers being the norm. And yet
there is a very big price to pay here. The position of authority teachers
have means that what they say goes unquestioned. So, on one occasion the
mathematics students on the Aga Khan University Masters programme came to me
with their issue of the moment: is 0 an even number? To which I
responded: you tell me your definition of even number and I’ll tell you if 0
comes within that definition. It became apparent that this was not an
acceptable response as far as they were concerned, as the teacher I was the
authority, it was therefore for me to decree an answer which would then become
gospel truth. Answering questions with questions does not fit into this
very easily! I can
give many further examples illustrating the tension between authority on the
one hand and meaningful learning on the other. An examination question
was an algebra problem in words which at first sight is fine – but when one
tries to solve it it becomes apparent that it has teenagers aged 25 years old
and old age starting at 35. The associated examiners’ report noted that
the problem had, in general, been done badly, and gave advice to teachers on
the need to improve algebraic skills and converting word problems to
equations. That there was a problem with the question itself was not
mentioned, corresponding to the authority of the examiners not being up for
discussion. Another similar incident arose from a class I was doing with
primary teachers in neighbouring Uganda, where I was emphasising understanding
how methods worked, using Gelosia for long multiplication as one example.
They really liked Gelosia but concluded that they couldn’t teach it to their
pupils for fear that examiners, faced with a correct answer but a method they
didn’t understand, would mark the work wrong. Again, there is an
unquestioning acceptance of authority, that this is the way it is – alongside
also an unquestioning wielding of authority of people who have achieved senior
positions. These
examples look to illustrate the tension between the respect for teachers on the
one hand – which is good – and the treatment of teachers as the absolute
authority who cannot be questioned on the other – which, I suggest, is bad and
makes the use of methods of teaching which look to put ownership of the
learning with the learner difficult if not impossible to implement. As
Hardman et al. (2009) in the context of the whole of Sub-Saharan Africa, found,
learning tends to be highly ritualised with considerable use of choral
responses, copying from the blackboard, and routine practice. Changing
what happens in the classroom needs to be in the context of what happens in
society at large. So, is
good mathematics teaching a branch of geography?Putting
together all the points made above, I would wish to argue that, at least in the
short-term, arguably also the medium-term, good mathematics teaching is a
branch of geography. What one can reasonably expect to happen is determined
by local context, history, availability of resources both human and physical,
beliefs about learning and the nature of authority, and applying also the
principle of Vygotsky’s zone of proximal development, it would not seem to make
sense to be trying to achieve too much too quickly. I would
also wish to put forward the idea that, in looking to encourage teachers to
improve their practice, this needs to be done in the context of a thorough
understanding of what is involved. So, if wanting to promote a problem
solving approach, it would not seem to make sense to have a rigid marking
scheme tying down what youngsters need to do, as I saw in Jamaica. And
if, in an early years context, learning through play is interpreted as meaning
that the teacher goes to the staffroom whilst children are left to get on by
themselves, as I heard second hand in Tanzania, again, what is in principle a
really important part of children’s learning is not being handled effectively
through teachers not fully understanding the underlying principles, seeing only
the surface features of what is happening. Care
needs to be taken, then, in looking to make progress with teachers, too much
too quickly means either practice goes back to what it was or, as in the
examples above, the input becomes counter-productive. Put another way,
better, I suggest, a ‘good’ didactic teacher in the short term, rather than a
teacher trying to use innovative methods without really understanding them. Work done
with teachers in UgandaIn my
role working for the Aga Khan University, I was privileged to conduct week long
courses with primary teachers in Kenya and Uganda – not in Tanzania as Swahili
is the medium of instruction in primary schools and associated teacher
training, then switching to English for secondary school and above. The
underlying language policy and its implications – particularly that children
are expected to learn, including at primary school, in a language not their
mother tongue or even sometimes their second or third language – is a whole
issue in itself, alas beyond the scope of this paper but another reasons why
providing education in Tanzania – and also the developing world more generally
– is hugely challenging. In trying
to implement the principles of incremental change, I started with the Ugandan
National Curriculum – which, in accordance with the points above about
considerable commonality across the world as to what constitutes ‘good’
mathematics learning and teaching, actually represents a very good place to
start, enabling me to say, OK, this is what you need to be teaching, how might
we go about this? One topic
still very much on the curriculum for young children is sets. So I marked
out part of our classroom and used ‘people maths’ approaches, with the inner
sets being things like ‘numbers in the 2 times table’ and ‘numbers in the 3
times table’ , or ‘girls and women’ and ‘people who wear glasses’. It
looked something like this: The
approach went down well, and was very much in the spirit of looking to communicate
a straightforward step which could be implemented relatively easily from where
teachers are. Another
approach I took was ‘geometric art’, getting the teachers drawing pictures
using geometric shapes as the foundation, and then to talk about the shapes
that they were using. This activity is useful also as an opening activity
taking as much or as little time as needed – if the stated start time is 9am
then one can reasonably expect some to be there at 8am and others still
arriving at 11am. Here are
two examples of the type: And we
made clocks: In each
case deliberately keeping the ideas low key, enjoyable and active, looking to
work from where teachers are and ensuring a level of enjoyment and interest. But in
the longer term…In keeping
ideas low-key, recognising low levels of qualifications and training, the
nature of authority of the teacher and the syllabus to which they are working,
there is a sense in which the work I was doing was within the spirit of
regarding good mathematics teaching as a branch of geography. One is, in
effect, working to a different model of what constitutes good mathematics
teaching than one might do in other parts of the world. However,
as noted above, Government policy statements across the world communicate a
clear and consistent message about what represents ‘good practice’ in
mathematics learning and teaching: problem-solving and practical approaches,
learner ownership of their learning, relational understanding, connections
between topics, meaningful links with other school subjects and the outside
world. Do these things represent a standard of good practice,
irrespective of culture, resourcing and practice? With appropriate
caution, I would incline to the answer, yes, they do. To put the question
more in an East African context, is it possible to have the best of all worlds,
respect for the teacher who encourages youngsters to try things out for
themselves and find things out for themselves? With concerted effort by
all involved - community, teachers, headteachers, examination boards,
policy makers, syllabus writers – in principle I think it is possible to retain
respect with the repositioning of the teacher to facilitate a model of learning
more in line with the policy statements across the world. Ultimately, I
do not think that good mathematics teaching is a branch of geography, there are
elements of good practice which transcend culture. But this is a hard
path to follow. Where the
policy statements come from: 1
Jamaica Ministry
of Education and Culture (1998). Curriculum guide grades 7-9 for
career education, mathematics, language, arts, science and social
studies. Kingston, Jamaica: MOEC. Page 86.2
USA http://www.nctm.org/About/At-a-Glance/Statement-of-Beliefs/ Accessed 14 March 2017 3
Tanzania http://tanzania.elimu.net/Secondary/Tanzania/TCSE_Student/Mathematics/TZ_O-Level-Mathematics%20Syllabus-Objectives.htm accessed 14 March 2017 4
UK DfE
(2013). Mathematics programmes of study: key stage 3. London:
HMSO. Page 2.References: Hardman,
F., Abd-Kadir, J., Agg, C. Migwi, J., Ndambuku, J. and Smith, F. (2009).
Changing pedagogical practice in Kenyan primary schools: the impact of
school-based training. Comparative Education, 45 (1), 65 - 86.Komba, W.
L., & Nkumbi, E. (2008). Teacher professional development in Tanzania:
perceptions and ctices. Journal of International Cooperation in Education,
11(3), 67–83.Oketch,
M. O., & Rolleston, C. M. (2007). Policies on free primary and secondary
education in East Africa: a review of the literature. London: Institute of
Education, University of London. PMO-RALG.
(2014). Pre- primary, primary and secondary education statistics 2013.
Dar es Salaam: Prime Minister’s Office, Regional and Local Government. |

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