The problem with Problems
Problem Solving has lain at the heart of the Maths Curriculum in Primary schools for many
years and indeed the new curriculum 2014 places a greater emphasis on this element.
However as it is all too easy to take the bland label of some supposed innovative issue and
apply it in such a shallow fashion that it rips the heart out of what was truly intended as an
excellent piece of curriculum thinking. I do feel that quality teaching of problem solving has
all too often fallen into a deep, dark educational crevasse becoming morphed into
something that is so diluted as to become unrecognisable in terms of good classroom
practice.
Word Problems or just words?
Many have come to equate problem solving with those written questions found at the end
of a given chapter in the Maths textbook. “Tom has 4 cats and 3 fish. How many pets does he
have altogether?” The expected answer is 7, but this assumes that none of the cats have
eaten any of the fish in which case it could be any number between 4 or 7. And here is the
issue; problems should be set in the real world solving (surprisingly enough!) real problems.
The clue is in the title, and yet so many word problems are simply a linguistic presentation of
a standard algorithm.
As Jo Boaler point out this leads to children developing a context for their Maths learning in
an arena she calls “Mathsland”. She cites the following example: A pizza is divided into fifths
for friends at a party. Three of the friends eat their slices but then four more friends arrive.
What fraction should the following slices be divided into? No doubt there will be a
technically correct numerically based answer in
the teacher’s handbook that the teacher might
use to mark the work in the evening. However,
surely in the real world the answers are more
likely to be those Jo Boaler offers namely that “if
extra people turn up at a party more pieces are
ordered or people go without slices” She
concludes that “Over time children realise that
when you enter “Mathsland” you leave your
common sense at the door” The picture to the
right illustrates this perfectly!
The truth is that not all problems based on algorithms translated into words have meaning
when solved mathematically. As Mike Askew points out the question; “If Henry VIII had 6
wives then how many did Henry IV have?” implies that he had 3 but this is another number
problem with a one-way ticket to Mathsland.
So bearing in mind that most of these problems lack all sense of reasons and reality what do
children need to learn to successfully solve word problems in a classroom context. Well, the
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first key skill is to realise what Ofsted noted that “Standard problems tend to be associated
with exercises at the end of chapters, where pupils know the operation to use is, say,
multiplication because the chapter was all about multiplication”. (Recent Research in
Mathematics Education 5-16, Ofsted) The second survival strategy in the modern Maths
classroom is to work out the likely operation from the context of the numbers. Mike Askew
tells how he has a Chinese textbook which is written solely in Mandarin apart from the
numbers, and whilst he does not read one word of mandarin he writes “I am confident that
the problem involving 25 and 14 is most likely to be multiplication while the one with 3007
and 1896 is almost certainly subtraction.”
Teaching in such contexts leads to a bland form of numerical understanding that sells
children woefully short of the richness that number and Mathematics can offer. Ramya
Vivekanandan analysed the PISA results for Maths in 2012. He showed that those children
who relied purely or significantly on memorisation of formulaic strategies to solve
calculations were consistently amongst the low achievers, whereas those students who
focused on a breadth of Mathematics and developed the “Big Ideas” concepts were the high
achievers. Jo Boaler in a lecture at Oxford University (18th December 2014) pointed out that
the UK lay 64th out of the 65 countries for having the largest percentage of “memorisers”.
Only Ireland scored lower. There was a consensus amongst the delegates that it is the UK’s
obsession with testing that allows children to thrive on such bland, shallow word based
material because it is that style of questioning that forms the basis of the tests at the end of
Key Stage 2.
So if it is true that many word problems are simply, as Mike Askew points out “calculations
wrapped up in words” what is problem solving all about and does it, or should it, have such a
prominent place in the Maths curriculum?
What is Maths all about anyway?
As with most issues related to education it increasingly seems to me that the key is to stand
as far back from the issue as you can to allow the fullest perspective. Then when the macro
elements are resolved in the mind, one can drill down to the micro elements of what makes
up a scheme of work, a weekly plan, a daily lesson and down to the progress of individual
children. Because our profession hinges around the classroom and its practice there is
always the danger that teachers start with the question: What should I teach tomorrow? But
this has no context if one has not engaged with the deeper and broader questions in the first
instance.
So what is “real mathematics?” Volumes have been written on the theme too numerous to
replicate here but here is a selection of thoughts. Galileo stated that “Mathematics is the
language with which God wrote the universe” – not bad for starters! Oswald Veblen, a
mathematician living in the last century said “Mathematics is one of the essential
emanations of the human spirit, a thing to be valued in and for itself, like art or poetry.” To
these men Maths was so much more than a few calculations undertaken in the lesson before
break on a daily basis, to them Maths was a language to live by and a means of making sense
of the world.
It is often hard to recognise our “calculation-dominated” curriculum with either of the
comments above but to be fair the Programmes of Study for the previous National
Curriculum held philosophical statements for the teaching of each subject and calculation
was not a major focus (indeed not even mentioned) in its rationale, It said: “Mathematics
equips pupils with a uniquely powerful set of tools to understand and change the world.
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These tools include logical reasoning, problem-solving skills, and the ability to think in
abstract ways.” It went on to include the following quote from Professor Ruth Lawrence,
University of Michigan “Maths is the study of patterns abstracted from the world around us –
so anything we learn in maths has literally thousands of applications, in arts, sciences,
finance, health and leisure!
Recently I watched the film “Enigma”, a story about the codebreakers in the war based at
Bletchley. Those seeking to decipher the codes were all mathematicians. At one point in the
film one of the girls turns to Tom Jericho and asks him; “Why are you a mathematician? Do
you like sums?” Tom replies “No, I like numbers, because with numbers, truth and beauty are
the same thing. You know you're getting somewhere when the equations start looking
beautiful”
To those well versed in the subject, Mathematics is not about calculation. Calculators
calculate and we carry these in pencil cases or access them on our mobile phones.
Mathematicians are those who see the big picture, the patterns and relationships in
numbers and how these relate to situations and problems in their world.
Conrad Wolfham is a mathematician and the head of Wolfram Alpha that seeks to develop IT
applications. His TED talk has had over 1 million hits and is entitled “Teaching Kids real
Maths” It seeks to establish a clear rationale for the teaching of the subject. His premise is
that any Maths problem is founded on four stages:
1. Posing the right questions (The key to finding any solution is to ask the right question)
2. Real World to Maths Computation (Taking a problem and turning it into a Maths based
problem for analysis)
3. Computation (Crunching the numbers)
4. Maths formulation real world verification (Putting the solution back into the real world
for verification)
He maintains that schools are focusing their energies on the wrong elements of these four.
“Here is the crazy thing” he says “in Maths education we are spending about 80% of the time
teaching people to do step 3 by hand but this is the one step that computers can do better
than any human… instead we ought to be getting students to focus on the conceptualising of
problems and applying them and getting the teacher to run through with them how to do
that.” I would contend that there is more than an element of truth in what he advocates. We
live in the throes of a technological revolution that is transforming the world and
transmuting every aspect of society. This new dawn is one based on logical thinking and
reasoning more than any other skill and Maths is the native language of that digital world.
Problem Solving set in this context
In light of the above we need to be developing a fresh form of Maths that engages children
on a totally different level. It is tragic to think that one girl said of her Maths lessons “In
maths you just have to remember in other subjects you think about it” (quoted in Jo Boaler’s
book “The elephant in the classroom”). But the truth is that much of our maths curriculum is
“calculation focused” preventing children embracing the true Maths that those more
conversant with the subject readily recognise.
Quality problem solving resolves this tension. It has at its heart the observation of patterns.
It develops persistence and risk taking alongside logical thinking processes that encourage
children to tackle tasks systematically. They also develop the ability to recognise numerical
relationships in their initial findings.
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Jane Jones is the Chief HMI for Maths and in the “Ofsted Better Maths” series of conferences
she stated that “The degree of emphasis on problem solving (and conceptual understanding)
is a key discriminator between good and weaker provision”. This may well simply be due to
the fact that it is these schools that have engaged most with the Maths teaching rationale
and therefore are generally more cogent in their planning but whatever the reason good
problem solving lies at the heart of good Maths teaching. It is interesting to note that none
of this is a new message; the Cockcroft report published in 1982 described “problem solving
as lying at the 'heart' of mathematics”
The National Primary Strategy produced a document in 2004 in which they outlined the five
types of problem solving they felt children should be engaged in:
 Finding all possibilities;
 Logic problems;
 Finding rules and describing patterns;
 Diagram problems and visual puzzles;
 Word problems.
For reasons I have articulated above I would raise an objection to the inclusion of “word
problems” and would place it back into the calculation section of the curriculum where I
believe it truly belongs, but that aside the above offers a starting point for a framework.
From here teachers need to source problems, either within the context of real world
scenarios or from sources which deliver the type of Mathematician that we need to secure
for the 21st Century. Without wishing to labour the obvious the quality of learning will be
dictated by the quality of the problem the teacher sets for the children. But if we keep
before us the attributes we wish to develop through these activities which are the key
elements of; resilience, risk taking, creativity, posing questions, exploring through trial and
error and latterly through a more systematic process it should be self-evident which will
deliver these. As most teachers are aware the DfE Primary Strategy materials provide a good
starting point and NRich is a quality resource, as is the newly created SPEAR Maths which
seeks to present not just problems in themselves but also a framework to work within.
Teaching Problems
One of the points the Primary Strategy made to good effect in its document is that teaching
needs to reflect the outcomes from the task. It states that, “Children need to be taught the
strategies and to be shown how they can apply these systematically to problem solving. For
many children the hit-and-miss approach they use when gathering information and their
poor management of information limits their
ability to work systematically.” (Problem
Solving Primary Strategy 2004) Therefore
teaching should focus on either creating and
extending the open ended nature of the task
or developing cogent strategies for children
to solve such problems.
Jane Jones showed the following problem to
delegates at a recent conference. The
problem is not complex in and of itself but
she was encouraging teachers to focus on the
strategies they used to solve the puzzle. The
likelihood is that you spotted that the 3 cows
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in the middle row must be multiples of 3 that total 15. From this point the puzzle starts to
unlock. We should not assume that children will necessarily see this without a secure
amount of scaffolding to support them. The lesson therefore needs to focus on the
development of the systematic skills required rather than leaving children with the
impression that the key feature is “finding out the value of the pig”
All too often it is easy for teachers to lose sight of what the problem solving lesson is truly
for and it reverts subliminally into a calculation style of lesson where the primary focus can
become “finding the answer” But just as good calculation lessons should focus on the
process and explore the most “effective and efficient” method (wording of the Numeracy
Strategy), so the teaching of problems should not focus solely on the answer but on the
mathematical skills and processes the children are gaining along the way. It is essential that
children build on these throughout the Maths curriculum.
Alan Schoenfeld makes an interesting observation related to this when he says that whilst
the idea of problems has been always been a key element in the maths curriculum, problem
solving has not. What he is seeking to clarify is that children need to develop a “toolkit” of
“problem solving” strategies rather than accessing a series of answers and solutions to an
eclectic group of problems unrelated in any form to each other. The Primary Strategy
document sought to emphasise this with its advice regarding medium term planning. It
suggested “Over a week, the children learn how to apply the strategies they have been
taught in a particular lesson to similar problems in the following lessons.” There is more than
an element of wisdom in this approach.
There are of course a range of strategies that children should acquire under the umbrella of
Maths Problem Solving. This might include:
 Trial and improvement – This is a valid starting point but we should ensure that children
don’t stay there.
 Working systematically – finding all possibilities within a given problem
 Working logically – maintaining some inputs whilst changing selected variables
 Pattern Recognition – looking for rules within numbers, shape patterns etc.
 Conjecturing, generalising and improving – These are skills that are often generic to any
problem solving solution but they undergird the thinking process for children
Conclusion
Having drawn from the old National Curriculum document in the early part of the document
it appeals to my sense of symmetry to quote the new one at the end. The document
outlines what it calls the “Purpose of Study” and states the following; Mathematics is a
creative and highly inter-connected 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.
Without wishing to labour the point the word “calculation” is not used but again the fullest
and richest view of Maths is painted on the broadest of canvas. This is not to say that
calculation is not important, as we all know it is a key building block in a child’s mathematical
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development but it must not remain the sole raison d’etre for the primary curriculum. This
would be to sell children, and indeed the subject of Maths itself, woefully short.
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