Geoff Tennant - Promoting access to mathematics for all
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29/10/17: is good maths teaching a branch of geography?

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!
In 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 world
Below 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: Tanzania
Tanzania 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 authority
When 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 Uganda
In 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
3              Tanzania
4              UK
DfE (2013). Mathematics programmes of study: key stage 3.  London: HMSO.  Page 2.
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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