I seemed to stir up a hornet's nest with regard to maths education here based on a YouTube short video I posted yesterday.
Most commentators seemed to think I was wrong to suggest that the teaching of largely uncontextualised, formulaic, trigonometry in schools is wrong when there is so much that is vastly more useful, in my opinion, that can be taught about maths that will be much more relevant to most young people.
My opinion is based on a lot of discussion. It has to be said, the comments offered on this blog are way out of line with those I am offered almost everywhere else - including by many mathematically competent, well-trained people as well as a lot of young people still suffering the shock of their maths education.
I offer two observations. The first is to ask what is the use of this question posed by EdExcel in a GCSE paper:
I wonder how many of those saying I had missed the point could answer this question posed of sixteen year olds. Admittedly, this was in the higher paper for those wanting the better grades of GCSE, but is knowing this really a requirement at 16?
Now let me pose a question I often put on the board when starting a discussion of maths in the context of the political economy of data with second year undergraduates. It was this:
2 + 2 = ?
I have never yet had a student give the right answer.
They all say 4.
It isn't.
It's 5. Both those figures written as 2 were 2.49 rounded to the nearest whole number. The sum of the two is 4.98, which rounded to the nearest whole number is 5. The answers the students gave me were almost 25% out in all cases.
And these wrong answers came despite children being taught about rounding to whole numbers in primary schools, just about anywhere in the world.
It would also have been extraordinary for a student to have arrived in my class without having used a spreadsheet, where such rounding is commonplace.
And, to be candid, you would have thought that if I asked such a question they would have assumed the answer was not going to be 4, or why would I have asked it?
But no one has ever wondered if I was asking for an answer in base 3.
No one has ever thought about rounding.
No one has ever suggested anything but 4.
And that answer is wrong.
And then I explained the relevance of this. In accounting - financial, management, national, whatever - figures are usually rounded to thousands, millions or even billions. So the figure presented is always representational. They often do not not add up as a result. Taking them at literal face value is wrong in that case. They always represent a range.
And that's before we ever get near understanding the uncertain assumptions on the basis of which they are almost always estimated.
My point was to emphasis that the blind mathematical assumption they had all been taught and absorbed that there are right answers to questions expressed in numerical form is wrong, and often deeply dangerous because grossly misleading conclusions can be drawn if the quality of the representation that a figure conveys to the literal mind is not questioned.
My whole point was to make clear that we do not live in closed, discrete systems. We exist within open, malleable, uncertain environments where certainty is almost always an unknown luxury and the assumptions we make - in this case the entirely false one that both of the figures were discrete numbers - can profoundly impact our conclusions, often making them completely incorrect.
Based on this thinking I used to set essays on the appraisal of data quality as a pre-condition to actually undertaking any calculation using it. The world would, I suggest, be a vastly better place if this issue was better understood.
So why do I think trigonometry so unhelpful to the vast majority of the population who will, as a matter of fact, whatever some have claimed here, never use it in the rest of their lives? It teaches that there are certain answers to any question that involves numerical data. Bizarrely, people carry this belief with them for life. And so they are conned by neoclassical economists and so many other snake oil merchants.
So, most certainly I want people to learn maths. But I want them to learn the maths that matters in the world where we live, where there are very few right answers.
One final comment. I remember discussing the maths involved in bridge construction with a bridge engineer once upon a time. I asked what happened when all the safety tolerances had been calculated. “Then we double them”, they said. But why is doubling enough, was my question. They had no answer, and yet that was where the risk really was.
When we think maths gives answers we can rely on we are in trouble. And yet most of it is taught as if that is the case. And that, in my opinion, is wrong. Maths is staggeringly useful, but only when it is understood that it only provides answers within a range, at best. My example yesterday highlighted a case where the exact opposite is implied. And that's of very limited real world benefit, as even the bridge engineer knew, even if they had also failed to question their assumptions.
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You are absolutely right that errors and rounding in maths is vital. We spent as lot of time, studying physics, having this reiterated many times. Physicists generally quote numerical results along with an error, usually one standard deviation.
Units also important.
I wish these would be emphasised more.
But (and you knew it was coming), I must take issue with you about your 2+2 example (you knew someone would). The information you gave did not state the numbers had been rounded. For many applications numbers are not rounded (cryptography, error control coding from my field of work). But, even allowing for rounding, the best, most probable, answer is still 4. Yes, those numbers might have been rounded. Both may have been rounded up, one up and one down, or both rounded down. Without further information we do not know. By your reckoning the answer could be 3, or 4, or 5. That’s why physicis and other scientists quote the most probably answer plus an error bar.
Bear in mind not all numerical results have this type of error. Counting is precise.
And I must also common your GCSE question. I thought it was rather good. Of course no one needs the numerical answer. But the question was to see if the students could think and solve a problem. No one expects them to know such things by rote. The question even gave a hint to help them. I would have thought this was precisely what you wanted, pupils being able to think and solve problems. That’s certainly what I want.
I think that EdExcel questiion utterky crass and totally irreevant tio life
We will have to doffer on this
And you totally miss the point of my 2 + 2 question. I was asking them to question – not telling them to expect to be spoonfed the answer – which as you note, even EdExcel did. That is not what the real world does.
Of course 4 was the most likely answer to my question. But that none could see that anything from almost 3 to 5 could be was my whole point. Did you not get that?
We must conclude we see the world in very different ways, Tim. And I do not think yours is one that involves problem solving at the level that is needed. or even the application of basic knopwledge – which is all I was asking for.
Problem solving,and lateral thinking is exactly what I want. That’s my business, having been in engineering research and development for decades. You probably even use one of my inventions at home.
Of course the GCSE question was not real life; that’s extremely difficult in a short exam. But it did have a problem which tested understanding not just rote learning.
We’ll have to agree to differ.
All the best 🙂
@ Tim
Absolutely
It is the processes and thinking that are crucial in problem solving that are the central issues, and not only in Maths teaching.
Nor do examples all need to be real world, and hence familiar, to achieve ‘relevance’.
In Maths assessments the marks given for showing each logical step in solving the problem are higher than getting the ‘right’ answer.
This is why Maths examinations have “Show Your Working” in the rubric.
Reducing schooling to some utilitarian notion of immediate practical value is essentially nihilistic.
It entirely ignores the fundamental curriculum imperative of ‘working from the known to unknown’ – and with increasing depth – that Bruner’s Spiral Curriculum requires.
You entirely miss the point.
Most of the time we live with many unknowns – most especially when it comes to supposed data on which a the national operations are the performed
What is more useful (given we have to make choices)? Dealing with world scenarios when it comes to something as important as maths – or an alienating and unexplained world of mathematical fantasy that maybe a few percent o the class might relate to?
As one who has actually taught Maths / Numeracy at both primary and secondary levels, so through the differing levels of curricular complexity, I can assure you that I have not missed the point.
The book blurb for Bruner’s “Process of Education” covers it :-
“For Bruner, the purpose of education is not to impart knowledge, but instead to facilitate a child’s thinking and problem-solving skills which can then be transferred to a range of situations. Specifically, education should also develop symbolic thinking in children.”
Your issue seems to be with the ‘symbolic thinking’ end of Numeracy/Maths education, whereas you are obviously happy with the more basic and mechanistic arithmetical areas of day to day numeracy.
Your approach seems to be firmly wedded in the early ‘concrete’ stage, whereas Bruner saw a continuous process that developed through to higher level problem solving skills, yet not as a simple linear progression, but across the curriculum, hence the spiral nature of the model.
For example, I would use maths skills in getting a Geography class to undertake modelling a Flood Risk and Management case study from precipitation data.
Thankfully, given climate change, this level of study is required in the Scottish Higher syllabus. We are going to need more applied hydrologists.
This spiral curricular process is not subject specific, though Maths syllabii have been your chosen target.
I really would recommend Bruner and the logic and practice of the spiral curriculum.
His thinking is still profoundly influential and probably even more relevant now to anyone involved in fornal education – certainly than the ‘facts and knowledge’ reductive practices of successive Tory Education ministers over the last 45 years.
Teaching trigonometry is ability learning to comply with rules
I am interested in teaching thinking
I think Bruner and I would get on
Where you fit into your own argument I do not know
Tim
In discussing the 2+2 question, you point out that the information did not state that the figures had been rounded. I use spreadsheets quite a lot. As you know the decision on whether to round and to what level is set in the cell format. It is never visible on looking at the spreadsheet, it is not knowable if looking at a spreadsheet that has been printed. So using figures other people have produced, one needs to be aware of the level of precision, not assume it.
Precisely
Always check your priors, as would be said in the US
It was a wholly appropriate example as a result
A year or so ago my oldest son bought me
https://www.goodreads.com/book/show/62837453-the-maths-that-made-us
Now while it does have some maths in it it explains why maths
Clearly if you start by understanding why then what math (and physics) is becomes much more interesting and relevant.
Clearly later in life I now understand why the engine in Waverley is triple expansion, why a 4REP* gave electrical engineers the heeby jeebies, how to calculate tax & NI manually etc.
Had I been taught ‘back to front’ in the first place then I might have understood maths and much else much better.
Finally one Sunday a friend of mine who taught ‘use of number’ ina FE College said ‘I have a class of car mechanics to teach tomorrow, what on earth can I do with them.
My reply was ‘discount for cash’ which went down as well with him as it doesnt with you
* 4REP 4 coach train with 8x365hp motors running on a 750V traction supply, draws about 3000A at full cry
Agree entirely, John. I wish that when I was at school someone had given me a book focused on maths in the way you describe, or indeed if maths teachers had taken the time to teach ‘why maths’ before trying to ram ‘numbers’ into our brains. The result, for me and many others, was simply to be turned off maths. Fortunately, many, many years later, I worked with people and on subjects (in academia) where I learnt why and how so much in maths is crucial to making sense of so much of the world in which we live.
I think maths phenomenally useful
That’s why I want it – and which issues are taught within its broad umbrella – to be taught so much better – in ways that empower people.
I quite agree with what you have said about the teaching of maths. I recall similar lessons on obscure trigonometry which I found rather boring and which I have never used. I used a lot of economic data at work and it was obvious that you had to recognise the assumptions behind the figures and that often a figure was an estimate (for good reasons), and that you had to assess that a change in the underlying assumptions and interpretation could make a significant difference to what the figures represented. Yes we need a revamp of the system.
Thanks
Any specific maths question may have little relevance to a person’s day-to-day activity.
But I guarantee that life is full of problems, and being able to solve them is useful.
This is an example of transferable skills.
Of course there may be better examples and more useful skills.
We did matrices for O level maths. I asked what the point of them was and the teacher’s answer was along the lines of, “because it’s maths”. Their sole point was to create another Matrix with no practical use at all. We did do percentages, but they were taught with no reference to life.
As for your 2+2=?, that has given me food for thought.
Your own 2+2 question makes no reference to rounding or different bases. Your students are absolutely right in using the most common and most well understood interpretation of what you might mean. If they failed to realise which of many non-standard interpretations you were referring to, that’s a failure in your communication.
One very important lesson taught in maths classes is that the precision of answers is dependent on the precision of the initial data. You were imprecise.
Trig may not be used by many people in their day to day life. But the same applies to the accounting you were teaching. You had to make explicit that in your specialist field context is not the same as applied generally. You can’t expect non-specialist or new starters to automatically think like you.
As a secondary point, you imply that education should only relate to future utility. I think learning shouldn’t always be practical or applicable. Trig is useful in science and engineering, but it shouldn’t only be taught to those students,just as you argue basic accounting should be available more widely.
My question used a real world situation – where you are not given all the data.
Why do you think eduction should be about knowledge in an artificial environment?
When I started Uni (some time ago, before mobiles were common) a lecturer asked us a hypothetical question to judge how we might react without the full set of information in the real world : “You agreed to meet a friend in town tomorrow but lost the details and need to wing it. Where and when do you go to meet them?”
The best answer is to try to use the most likely place and time that the other person might think of, eg train station or significant landmark at midday. Not perfect, but the point is that common assumptions are required for good communication.
Your lesson went the opposite way. Your students found the most common/likely answer, but you chose something obscure. You would’ve missed that meeting.
Artificial environments are how we model things. We extrapolate to real world scenarios. That’s generally good practice, because similar principles can be applied widely. Much easier than learning on a case by case basis.
I wed a real world situation and a very commonplace problem they had all encountered using skills they had all been taught. I did so to get them to think about their assumptions. Politely, you’re asking nonsense.
There is a story about a Soviet corporation requiring an accountant. This is one version remembered from my 70s degree reading.
They asked the first candidate to add two thousand units of a product to two thousand from another factory and tell them the total. He answered 5 and was told they appreciated his loyalty to the revolution but could not use his services.
The next answered 3 and was told he was a potential traitor to the revolution , undervaluing the efforts of the Proletariat and party. He was sent away.
The third said ‘whatever you want it to be.’
He got the job.
I am not sure if it’s relevant but it is probably not restricted to the Soviet Union. I expect our Colonel Smithers has seen similar.
Richard,
I reacted negatively to your post yesterday but agree with your more tightly defined statement about presenting maths topics in an “uncontextualised, formulaic” manner.
That is what needs addressing.
With regard to trigonometry, apart from enabling me to answer the question as to whether the tree opposite is a threat to my house, discovering the graphical form of trig functions as illustrated in, for example,
The Incredible Sine Wave and its Uses
is engaging and eye opening for many youngsters and nowadays accessible to them on their smart phones through programs such as
https://www.desmos.com/calculator
There’s no reason why a maths curriculum nowadays should to be focussed purely on the computational aspect as IT can handle that. As you point out the underlying feel for number and pattern enabling a judgment about what a solution should look like is much more relevant. And why are arithmetic, algebra and geometry treated separately as if there are no links between them. Then there is the history of maths, the geographical and linguistic background to different number systems. There are so many different aspects of maths that can open doors for different individuals.
As written, the answer to your question is clearly 4. You ‘cheated’ by then adding extra information… you could have said “but what if it’s not base-10?” or but what if its adding two cells in a spreadsheet?”, etc… A fairer challenge to your students would have been to write “2 + 2 = 4”, and ask them if that is always true.
As a teaching moment your approach has value, but it doesn’t demonstrate any failings on the part of your students.
On the more general point of what is taught in maths I think you are partially correct. A similar objection could be raised about any subject — have you ever had any practical reason to know about drumlins and eskers? How many students make practical use of their foreign languages? Part of early education should be to give students a ‘taster’ of what they will encounter if they take a subject further. And some proportion of students will make use of everything that is taught (yes, I have used trigonometry — and calculus — in my work). Or do you advocate a return to Secondary Modern schools where pupils are taught only those subjects which will be ‘useful’ to them in their limited lives? I don’t know what the right answer is, but I do know that it’s not easy to find.
I was not trying to fail my students
It was not a test
No one was marked
I introduced an idea that they could not handle, despite all their prior training that should have let them do so.
I asked them to question their assumptions in other words – which is the foundation of judgement
I remain staggered that those challenging me think I am asking for dumbed down education
I am asking for education that inspires people to engage with the world
Appraising data quality – the subject I was introducing – is not easy. There was nothing dumbed down about it.
And the question worked because it introduced an idea they had not thought about.
I should add I always got very high teaching quality marks from my students (4.9 out of 5 from one class of 80 one year – who I would gave asked this question to) They liked my approach.
Ah, the additional context that you were teaching ‘appraising data quality’ does rather change the nature of the question you asked! Students should definitely have been on the lookout for the catch.
You presented us with a diagram. My first reaction?
Count the sides of what were claimed to be octagons. Twice – once clockwise and once counter-clockwise. Each one.
Oh well, too much contemplation of lying politicians.
I was educated at what was then one of the top 10 public schools (currently 7th) and of course we did trigonometry. Whether any of my classmates went on to use that knowledge in real life is was up to them, but the point is it was taught should the need arise. It was part of the education process. For me, I went on to do additional level pure and applied mathematics in which I got a matriculation, then advanced level in further education. Similarly, did we really need to know that salt is composed of sodium and chlorine in real life or whether the chemical symbols are Na and Cl? For my part I started life in research chemistry, and did indeed need to know those things in what had become my real life. But most people just need to know is comes out of the ground or the sea. I later changed to design engineering and, again, needed trigonometry in my real life. Not, of course, when I reinvented myself as an accountant in which, I hasten to admit, I have no formal qualifications other than Sage Accreditation.
I think you miss the point that education is about equipping students up for all sorts of real lives, some of which they may use and a lot of which they wont. Up to them. They won’t use trigonometry if they choose to become an accountant or lawyer, but they most certainly will if they become a design engineer.
BTW I don’t claim to be any academic genius. I was consistently in D stream (of 4), and when I was moved up to C I asked to be put down again because I didn’t like being bottom of the class instead of top.
My precise point is that schools are not preparing children for life – they are teaching them to follow rules, are killing creativity and are making subjects that are of marginal relevance to a few core curriculum issues when there are so many more important things that should be there.
Which I accept. Nothing more clear than my French lessons which did not prepare me for being in France. I learned more French by being forced to speak it than at school. And doing Latin might seem useless for real life, but I’m glad I did it. Nevertheless the academic grounding in the language was sufficient to help me on the way.
So let’s try to summarise where we’ve got to:
– It seems that Richard isn’t criticising trigonometry per-se …. but any non-functional skill being taught without a ‘real world’ context.
– However, it is clear that teaching of techniques / methods in some subjects (including trigonometry) is required for those who want to study some subjects at a higher level (this includes most scientists, engineers, mathematicians, etc).
– If pursuing some subjects further, people will require exposure to some abstract concepts at an early age (because human brains are bad at dealing with some things if left till post 18). This includes at least languages, maths, music, art.
– We don’t know what things will appeal to some students (or the range of things people will do in the later life).
It seems that exposure to some aspects that some people will find ‘not useful’ or will ‘never apply’ is a natural consequence of trying to expose people to some of the skills they others may need. We’d don’t have a crystal ball to know who’ll do what.
BUT the real beef that some folk seem to have is the lack of context or engagement in the way that the material has been taught. This is a very different issue than objecting to the teaching of trigonometry, by itself. Ironically, both this thread and the 2+2 example have been presented in a context-poor manner.
Now: On the exam question posed: It’s quite a good example to test higher level GCSE pupils ability to: recognise shapes within patterns, and to segment the problem, factorisation, expansion and surds. Very little trigonometry (1-1-root(2) triangle, 45 degrees — simple triangle shapes that people could use to lay out shed bases – I think the shape used in ancient Egypt). As for the answer being ‘spoonfed’ — it highlights that they want the working to be done using surds – and the method of presenting the correct answer. It would be difficult to establish the correct value of ‘p’ without answering the question, as intended. I’m happy to provide the working!
We shouldn’t object to this type of question if we accept that for the subset of students who’ll need to develop maths skills, they need to have knowledge of this type of problem solving and the methods of manipulation required to answer the question. Time-limited owing to brain plasticity. This is less the case for some other subjects.
So the issue is: (In England) Is there a need to introduce a ‘functional maths’ for all, and maintain a ‘proper maths’ for those who may need to develop maths skills? What is the correct age for this split? And how can the teaching be improved with teachers skilled in maths and able to provide insight / interest. I note that many state English Schools do not have qualified maths teachers and that PE teachers are being encouraged to teach maths – and they admit the PE teachers are essentially 1 lesson ahead of the students. This is unlikely to allow for insightful or useful examples — but the rote-learning approach we all agree is unhelpful.
And, perhaps Richard knew this …. but the most likely answer to both questions posed —- is 4!
(In absence of any other information or context, the most likely answer to 2+2 is indeed 4). But clearly a series of clarification questions could allow an more suitable answer suitable for a given application / use ……………… I’d suggest the following is a better question: What additional information do I need to help me derive an answer to what is 2+2 ?
What I do know is that your pedantry does not help your cause, but does clearly indicate you cannot comprehend the points I was making to my students, or here.
I’m sorry – but if you really think that existing education works then the problem is all yours, and you are doing nothing to solve it.
That’s a cheap and crass criticism.
I have pointed out the need for functional skills — but also some more abstract material (for those that may need it (?), at an earlier age than you had identified (18+)). I also pointed out a problem with under-qualified teachers – exactly those who will struggle to make the subject relevant.
To deliberately misconstrue my comments, then to claim ‘So I’m right’, and ‘Let’s not talk any further on this’ does you no credit.
This subject is far to important to assume that it about you ‘being right’ and ‘its time to stop now’. Many people have made very valid points in opposition to your original assertion. I tried to summarise where we’d got to …. you could have developed that. I’m normally interested in your comments – but I think you’ve failed to take valid issues on board here.
I made my point
I have related my point
I think you are very wrong on this issue
I have nothing to add
I am absolutely sure we waste children’s time with a very bad maths education that prepares few of them for anything of use. And that’s precisely why people won’t teach it. There is nothing to add.
I had to check what the term surds means – because it is not a term I can ever recall hearing
Autocorrect does not recognise it
And it is being taught as if it is core material.
And why do schools not have maths teachers – precisely for all the reasons I say – the curriculum is so bad using material so irrelevant that no one wants to teach it
I think I have had enough of this now. I am sure you did not intend too say everything I have said right – but you have provided the evidence that I am.
Shall we end this debate here?
Thank you for your work and communication to make education more connected with real life.
Below are two education related quotations which may be relevant.
Banking Model of Education: the perception and associated attitudes which views learners as deficient, inferior, passive recipients of a teacher’s delivery of a set knowledge structure.
(From the Rollins School of Public Health)
Critical Pedagogy: habits of thought, speaking, conversation, pictorial representation, reading etc. that go beneath the surface meaning, first impressions, dominant myths, official pronouncements, traditional cliches, received wisdom and fashionable opinions etc to achieve some grasp of deeper meanings, root causes, contexts etc.
(From Ira Shor)
I think you can guess which camp I am in.
It does remind you of that old joke, where you ask a mathematician, an engineer, and an accountant what 2+2 equals. The mathematician says 2, the Engineer says 2, +/- your uncertainties, and the Accountant says…well, what do you want it to be…
A little example with a big application. Politicians are very fond of quoting averages, especially ‘average earnings’. Students do not usually get taught the critical point that the ‘average’ (arithmetical mean) is distorted by outliers, unlike the median. “The average wage has increased 5% under the Tories” because CEO wages have spiralled upwards etc. The median wage, usually much lower and less affected is hardly ever mentioned. RM may also say the whole is meaningless unless you contextualise it with inflation and other factors. There are other such examples abounding, such as close examination of axes and timelines in graphs, due to distortions and selection.
As someone dealing with students (of all ages at times) who are not maths specialists, even simple calculations need contextualisation and critical examination – which is largely not provided to them at GCSE. Very, very few go any further than that level, and many who attain that level quickly lose any facility they may have had.
I suspect that RMs point has a great deal of political relevance.
I agree