Malta published the new Learning Outcomes Framework for school mathematics
http://www.schoolslearningoutcomes.edu.mt/en/subjects/mathematics
In my opinion, it is representative of current trends in mathematics education around the world and deserves a wider open discussion.
A random bit from Level 5:
COGNITIVE LEARNING
31. I can use equivalent fractions to discuss issues of equality e.g. gender.
I believe in power of mathematics and I am convinced that comparing numbers (for example, salary) reveals a lot about gender inequality (and other, frequently hidden, inequalities in the world — just recall the Oxaca Decomposition and its role in fight against discrimination of any kind). But equivalent fractions? 1/2 = 2/4 = 3/6? How are they related to gender issues?
I am a teacher of mathematics; when I hear a strange statement from my student, my first duty is to try to analyse my student’s way of thinking.
I found that the “Learning Outcomes Framework” triggers in me the same Pavlovian reflex of trying to figure what the authors of the “Framework” have meant. In this particular case, I cannot come up with anything better than a conjecture that perhaps the authors of “Learning Outcomes Framework” associate the words “equivalent” and “equality” a bit too closely. Every teacher of mathematics knows that mixing similary sounding terms is one of more common stumbling blocks for weaker students. The standard pedagogical remedy is to help the student to separate the concepts by asking him/her a splitting (or separating) question, for example
Equivalent fractions are also known under the name “similar fractions”. Why does the learning outcome
31. I can use similar fractions to discuss issues of equality e.g. gender.
appear to be less coherent and less convincing?
My main concern about “Learning Outcomes Framework” is that an official governmental document of a souverign nation of proud historic past has to be analysed using didactical tools (such as “separating questions”) reserved for work with struggling students.
Malta is a small country, and contributions to the debate from mathematics education experts from around the world might happen to be useful to our Maltesean colleagues. Please post your comments here.
Alexandre Borovik
De Morgan Gazette 
I have had a quick look – good heavens!
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A response is published as a post:
Response to “Malta: new Learning Outcomes Framework”.
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It is is really like a nightmare. The humanity is under attack and they
are everywhere!
Who are they? Brain-washed believers of the dogma that living beings are
just a complex sort of finite-state automata? Or in most of the cases
worse: just conformists who are ready to go willingly to wherever the
mainstream takes them.
Frankly to speak, I don’t know how to respond. A few weeks ago I
received a similar list of “assessable outcomes” from the university
administration, supposedly meant for the self-evaluation of the
university. They were asking for comments and suggestions. I did not
answer, because objecting to this or that specific item seemed to me
like collaborating with them. The only honest answer I could give (and
obviously didn’t) was that I consider this kind of reductionism as a
crime, which needs to be rejected on principle.
How can we support the Maltesean colleagues? And all others to come?
Can we open a front by writing something like manifesto against
outcome-based education and starting a campaign on the internet?
Frankly, I do not believe much in such methods, but still, one could try.
Outcome-based methods have been adopted in education systems around
the world, at multiple levels. Australia and South Africa adopted OBE
policies in the early 1990s but have since been phased out. The United
States has had an OBE program in place since 1994 that has been adapted
over the years. In 2005 Hong Kong adopted an outcome based approach for
its universities.[6] Malaysia implemented OBE in all of their public
schools systems in 2008. The European Union has proposed an education
shift to focus on outcomes, across the EU. In an international effort to
accept OBE The Washington Accord was created in 1989, it is an agreement
to accept undergraduate engineering degrees that were obtained using OBE
methods. As of 2014 the signatories Australia, Canada, Taiwan, Hong
Kong, India, Ireland, Japan, Korea, Malaysia, New Zealand, Russia,
Singapore, South Africa, Sri Lanka, Turkey, the United Kingdom and the
United States.
As one can immediately recognise from the paragraphs above (Wikipedia:
outcome-based education), it is a highly political issue and it is long
decided for by the powers that be. There are already new economic
sectors feeding from these policies. Just while I was writing this mail
I received a phone call from a friend who came out from a meeting with
an internet platform provider for such purposes. The topic is apparently
very hot and is evolving too fast.
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Your randomly picked example
COGNITIVE LEARNING
31. I can use equivalent fractions to discuss issues of equality e.g. gender.
made me really jump. It shows that everything I have been criticising about popular science is becoming a legitimate part of the formal education.
In my opinion, examples like the above sentence should be given to pupils to demonstrate how concepts can be easily confused and why mathematical accuracy needs to be observed in order to avoid them.
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A response is published as a post: Outcome Based Education
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It feels like a lifetime’s work to put all the arguments against these Maltese outcomes, and OBE in general, and I only have a few minutes, so will offer a very truncated response. I think the original purpose of an outcomes-based approach was fine – that it gave common expected goals for learning a subject in the school curriculum, rather than allowing some teachers to carry on ploughing through mathematics in their own favourite way, irrespective of whether anyone was learning anything from them. Its introduction in South Africa was to level the playing field so that all students could expect to have access to the same mathematics (unlike during apartheid) because they were all expected to realise similar outcomes. Educating everyone towards some agreed purposeful worthwhile goals justifies education as being a state funded concern, rather than the concern of individuals. In conjunction with a market-place view of education this becomes poisonous because for the market to operate outcomes need to be measurable, and in order to be measured the outcomes have to be easily tested on a large scale and the possible disciplinary and human centred desirable outcomes of an education in mathematics have to either be ignored, or watered-down, or mangled. This is what happened in UK in the late 90s when ‘mathematical investigation’ became ‘find a formula relating variables in a situation by applying inductive reasoning to a table of integer values’. That is when it all goes pear-shaped and turns into little gobbets of procedural performance that, accumulated over time, do not magically turn into mathematics, nor a mathematically-competent populace. (Understanding multiplication to be repeated addition, for example, as well as being mathematically slovenly will not get you very far in mathematical competence). Meanwhile the ‘assessability’ juggernaut tightens a grip on the world via the global companies such as Pearson who, presumably, promulgate the OBE approach because it can be manipulated to be testable. One of the things that amazes me about the Maltese version is the colour coding. Did anyone notice that the colour coding is predominantly red in year 5, i.e. what they call ‘cognitive learning’ but as you progress you apparently do less and less cognitive learning and a higher proportion of other things. So just as the concepts become harder and harder and applicability and transformability becomes more necessary the OBEs turn into ‘managing learning’ and ‘reading and writing’ rather than ‘cognitive’. I don’t know about other people but in my mathematical studies I became more and more cognitively challenged and more and more dependent on changing and developing my mathematical thinking as I met more and more abstract ideas. How have Malta managed to decide that in their world maths becomes less and less cognitive? If you substitute ‘easily testable learning’ for ‘cognitive learning’ I think it is more obvious what is going on here.
I am stumped about what to do about it, since, like in our own government, personal political whim trumps long term development by knowledgeable people, properly trialled and with a supported teaching force.
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You are quite right, we do not need to start from a blank page in 2015. State-funded elementary schooling has been established in Malta since 1798, and education was first made compulsory in 1946. So we have been developing curricula for quite some time, and, I would say, based rather on the British model, so certainly not in some isolated vacuum. But it seems that when EU funds are available what is important is to spend them not what you might get in return. So now, it seems, we are actually starting from scratch with a curriculum drawn up by foreign consultants who might not even dream of proposing this in their home country.
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Surely, there are existing national curricula that small nations, like
Malta, might (with permission) use as a solid starting point? I can’t
believe these colleagues need to start with a blank page in 2015?
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I am afraid that it is the European Commission who wants to start from a blank page. I understand the project is funded by the EU and is run as a compliance exercise with the EU policies.
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My experience with OBE is limited – largely because it contradicts everything I know about learning mathematics, so I avoid it. However, I once addressed the South African national conference of mathematics teachers when OBE was being imposed there, and used the opportunity to explain, and to illustrate, how misguided it was. The occasion was unique in that afterwards several of those in the audience showered me with kisses!
So although I cannot speak with authority, I would distinguish two kinds of flaw in what I know of OBE. The first flaw is the whole idea that one can specify and test short-term “objectives” in mathematics learning. The second flaw seems to be the attempt to challenge the distinctive, technical character of elementary mathematics, by integrating it into a more holistic educational scheme (gender, AIDS, equality, or whatever).
Anne Watson is far too kind in judging the original goals of OBE to be OK in principle (by which I assume she means the first of my two ‘flaws’ – before the social engineering flaw is added to the mix).
There are important objectives in mathematics teaching; but those that really matter are all long-term objectives. Lower primary lays the foundations for upper primary; upper primary lays the foundations for lower secondary; lower secondary lays key foundations for upper secondary; and so on. In particular, though teachers need to keep one eye on progress, and though change over time can be significant, the important stages cannot be summarised as short-term objectives (whether declared at the outset of a single lesson or chapter, or of a school term, or even of a school year). Instead teachers need the insight and the authority/responsibility to work towards key long-term objectives. This long-term, ‘global’ progress may be achieved through a succession of smaller steps – individual lessons, and homework/tests that assess progress ‘locally’; but the crucial global objectives are distinctively different, and cannot be characterised simply as ‘the sum of smaller steps’.
However, the world (including the world of Higher Education) is increasingly dominated by ‘teaching specialists’, whose teaching experience is either limited, or is restricted to a single phase (lower primary, or upper primary, or lower secondary). The kindest thing to say is that “they draw the wrong conclusions” (conclusions which just happen to please the new breed of politicians and middle managers). Sadly, neither the ‘teaching specialists’, nor their superiors, ever seem to notice the wretched long-term consequences of teaching for short-term success. Some are mere Snake Oil salesmen. But some truly believe in their message: they produce detailed schemes-of-work, that purport to break down the messy process of ‘learning mathematics’ into steps reminiscent of B.F. Skinner. Their pupils may even make measurable short-term progress (of a kind). And though one can be sure that most of them crash at the next stage, no-one seems to join the dots (by following up progress from one phase to the next).
Short-term, objectives-driven schemes ignore the integrated character of elementary mathematics. They simply list atomic fragments – assumed to be the steps, or rungs, in some long ‘ladder of progress’. Pupils may then progress at their own pace – reaching certain specified ‘levels’ as they go along. But because the model pays no attention to the crucial long-term goals, the approach guarantees ultimate failure: each step may be completed successfully; but the grasp remains superficial, and the succession of steps rarely results in effective mastery of elementary mathematics. Each idea tends to be tackled in a spirit that is ‘backward-looking’ – adapting familiar ideas and methods to get by, rather than learning new methods whose pay-off lies in the future. (The most blatant example in the UK is the way fractions, ratio and proportion have been reduced to an Ersatz, that avoids the need to address the subtleties on which mathematics beyond age 14-15 depends.) This phenomenon is completely general in any system which tries to define “progress” in terms of ‘objectives’ and ‘levels’.
Thus the truly scary thing about OBE is not the social engineering (which is in some sense laughable), but the pernicious lie that education is about short-term, measurable ‘outcomes’, and trivial short-term ‘progress’ (which now seems to be almost universally accepted).
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Yes Tony, one key of the problem is what you’ve just mentioned: “measurable outcomes”. We have experienced it also at University level, and I met colleagues at prestigious UK universities who have even had to cross swords with the concept. Basically, following a document written by some “Bologna Expert”, we were given a set of verbs which we could not use in our course descriptions. The only one I remember is “understand” because since then I have been trying to use it as much as possible. So, if you write something like “This course will help the student understand how group theory can be used to solve certain enumeration problems,” somebody will mark the word “understand” in red telling you not to use it because it is not a measurable outcome (they had a stock phrase which they kept repeating, but I cannot remember it exactly). Now, this might not cause such a great harm at university level (although in the long run it will, I think) because we draw up our syllabi and we can teach what we think should be taught around that syllabus irrespective of the verbs which the curriculum accountants force us to use. Our ultimate judges are our external examiners, and I have not yet met one who is a Bologna freak.
But schools are different. Teachers are watched over, their teaching schemes are regularly checked, and a teacher who steps out of line sticks out. Some years ago the rule apparently was that grammar should not be brought into a language lesson for primary schools. A very dedicated primary school teacher who I know thought that his class could benefit with knowing what things like verb, subject and object meant, so he introduced them slowly in his lessons. A university lecturer can take such impromptu decisions most of the time with impunity, but this teacher was stopped by a superior who came to check his lesson notes, although his class by then were doing fine in reading and writing.
I think that those of us who are not at the front line in schools do not realise how nefarious some educational practices can be, especially if their strongest appeal is something like short-term measurability. I myself plead guilty. I never thought about any of this before my own children went through primary and secondary schools. Most of my colleagues in mathematics and the sciences have that same attitude. Thinking and writing about the pedagogical aspect of their field seems to be an inferior type of activity compared with research in the subject. You, Tony, have shown that this is not true. But there are not many who think that way or are able to produce what you did.
I am not saying that university academics have all the answers to all educational problems. But they surely do have some answers. In my opinion, their absence has created an imbalance and a void in some of the prevailing educational theories.
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