Education administrators and governments seem to be totally detached from any understanding of the real world when it comes to designing curriculum. Nor do they seem to care how making students study something they are ill suited and unmotivated for will destroy self-confidence and create negative internal dialogue. It also wastes time that could be more productively spent. An example of this (though not the only) is the way maths is taught. The traditional math pathway taught in almost all high schools is designed to prepare students for higher education studies in STEM (science, technology, engineering and math)[1]. Some will follow this path, but many will not.
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What is Taught
The following is a broad summary across many countries. Details obviously vary from country to country, but what follows is a reasonable average.
Primary schools (usually the first six or seven years of schooling) generally cover arithmetic, and introductions to geometry and statistics. Most will also cover real numbers and percentages. While there are many students who are severely challenged by this, such as those with dyscalculia, it is probably not unreasonable to expose most students to this level of maths. Arithmetic and some level of geometry and statistics is useful to understandings in many areas, provides a nice early introduction to logical thinking and are sufficiently useful across most careers and vocations to be worth covering.
High school is usually the last six years of school. In most countries the first four are mandatory. In some nations the last two are also compulsory, but the type of studies can be varied significantly. The topics covered in high school math look something like this:
· Algebra 1: Real numbers; solving, writing, and graphing linear equations; quadratic equations and functions; polynomials
· Geometry: Plane and solid geometry including constructions, formulas for measurement, and formal proofs
· Statistics and probability
· Money and business math
· Algebra 2: Continuation of the concepts taught in algebra 1, including a more in-depth study of graphing and solving equations, inequalities, and functions
· Trigonometry: Applies algebra and geometry skills to circular and periodic functions. NOTE: Trigonometry is usually not its own class, but is often taught during algebra 2, geometry, or pre-calculus
· Pre-Calculus: Series and sequences, probability, statistics, limits, and derivatives
· Calculus: Continuation of the concepts taught in pre-calculus, with an emphasis on integration and differentiation
The last two may well not be covered in the ‘core’ math subjects that all students do. In Australia, for example, they sit in the last two years of high school where students have control over what type of maths they study or if they study maths at all.
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History
In the Western world the teaching of basic mathematics to everyone was triggered by the needs of the industrial revolution. Basic skills in handling currency, counting stock and such was required. The need for seamen who could navigate led to wider teaching of trigonometry. During the Victorian era topics further expanded. In the UK it was the second half of the 19th Century that integrated the alignment with STEM requirements that we see to the present day. By the beginning of the 20th Century mathematics was compulsory in all developed countries and most developing countries were at least heading the same way.
Justifications
Many justifications are put forward for making all students do mathematics. Broadly, they can be summarised as[2]:
· Mathematics is personally satisfying and empowering
· It forms part of our cultural heritage
· Workplace mathematics, which is needed for work and the economy
· Advanced mathematics for the technical and scientific professions.
One of the guiding principles of education is that later life choices shouldn’t be cut out too early by excluding important topics.
Broadly speaking politicians have seen a link between a better educated society and economic growth.
Another oft put argument is that mathematics teaches logical thinking and that we need more logical thinking in society.
Now for a Different Viewpoint
As I said earlier in this article, I think we can agree that some knowledge of arithmetic, as well as some geometry and statistics, as taught in primary school, makes sense.
But even here, we need to take a higher perspective. All teaching tends to follow historical patterns, and nobody normally questions them. I’m about to. Some children have a real issue with arithmetic. Is the need in later life really worth making them think they are stupid?
A huge amount of time in primary school is taken up with learning the times tables, to pick on example. My daughter, who is dyslexic, never got the hang of them and never even learned to tell the time on a normal clock. Yet she is a film producer and spends a lot of time with Excel spreadsheets, project management and budgets. Clearly the lack of those skills from primary school has not impacted her ability to operate in the modern world.
We live in a word today where everyone has a calculator with them all the time. Spreadsheets are available online and Wikipedia is only a click away if we forget how to do something.
When it comes to high school level mathematics, the situation is even more extreme — most people will use less than 10% of the maths they study in high school and some will use none of it.
The justifications fall apart when you analyse them from a first principles perspective. First principles thinking is a way of breaking down complex problems, ignoring what authority has said ‘must be’, removing assumptions and thinking creatively.
Most high school maths is not required in most workplaces. You can teach the cultural place of mathematics without teaching all the math. Logical thinking can be learned without mathematics, and in fact would probably better impact some of societies’ ills if we did so.
Most universities offer or support what are called bridging courses. Such courses offer an intensive way to cover material to make up for a student who did not do particular subjects at the end of high school or who did not get high enough marks.
In examining what is available I found existing bridging courses that in threeto six weeks, or even as little as 10 days of full time study covered the equivalent of the last two high school years’ worth of specialised maths.
Much of workplace training is moving towards a just-in-time learning approach, where you cover just what is needed in bitesize pieces when someone has the need. Surely this is more broadly applicable. Combining just-in-time with the bridging course concept might let students who decide on a STEM career to get all the math they need just before they need it at university. For those taking a STEM career path might this not be all that is needed?
When you look widely at high school students, some reasonable proportion will know at the start of high school the broad direction they want to take. But a much greater proportion will, within the first year of high school, know whether STEM is in their future or not, even if they are not yet clear on the final destination.
Also in high school we currently spend too much time on how to do and not enough on why we do. Students see no context and reason for learning what they do.
What is more important today is knowing how to find information and having enough basic understanding so you can apply that knowledge to whatever drove you to look for it in the first place. Though I did maths through to university level, and continued to use it through my research career and in other aspects of my life, I never worried about forgetting how to do some advanced integration, for example. What I have always had next to my desk are a few well-chosen reference books on maths so I could look up what I needed.
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An Alternative
There is slow development of a multiple pathway model for maths in high school that is developing from more extensive innovation in pathways at the university level. A concern I have about the efforts so far is that they still keep too much math in every students path.
Others are considering these questions. UNESCO’s Futures of Educationinitiative is doing exactly this in an international framework.
This is the situation I would like to see. It will sound quite radical, but I ask that you really consider it before getting angry:
When children start school there is screening of everyone to identify any issues like dyslexia and dyscalculia before negative experiences create self-confidence issues.
Primary school teaching of math applies modern ideas of active learning and appropriate context.
Integrate technology immediately and eliminate the rote learning of things like times tables.
Primary school math is very suitable for gamification. Gamification is the idea of building games that teach while they are played. Work needs to be done on really cool, interesting games that kids actually want to play. Most of the gamification attempts so far have been pretty lame. You’ve made it when the kids want to play outside of class.
The first year of high school should teach math in context, what it actually does in society, its importance to our historical developments and the different ways maths is actually used in the real world. How to learn more maths should also be taught, with students introduced to resources like Khan Academy (a great site with excellent online courses in topics like math).
After that, all math should be optional. Different types of math should be offered from that point on, offering a business stream as well as the more STEM ones.
Logic should be taught as a compulsory subject at certain points throughout high school. Logic is the study of reasoning. Essentially it is about how to think rationally and reason thinks out in a methodical way. This should not be mathematically based.
Even the STEM oriented advanced maths subjects should not take as much time as they do now. Rather a math intensive bridging course should be standard for all students entering STEM studies at university. This means a student should be able to skip all optional maths at school and still do a STEM university degree if they want to and are thus sufficiently motivated. This embraces just-in-time learning.
The time freed up by the above should allow time to cover the things that high school in many countries do not have time to teach, like emotional intelligence development, self-directed learning, motivation and mental health, to pick just a few examples.
Also in the time freed up we should be teaching students how to really relate to technology. This is not done at present, probably because most teachers don’t, but needs to be. Students need to understand that technology is a constant byproduct of our culture, they need to know how it actually impacts almost every aspect of their lives and that you don’t need to be able to build it to understand it. There are so many other life skills that should be taught in high school, but that we don’t because of lack of time.
The above is a pretty radical rethink of math education in schools. The key benefits are freeing up time to teach what is not currently covered but desperately needs to be, removing negative self-image issues for many children, and making better use of modern ideas of education. The old production line model of school education is an industrial revolution invention that does not reflect good educational practice as we now know it, and needs to die a painful death.
Making this change will be tough because all our politicians and ‘experts’ have grown up knowing nothing else. Plus first principles thinking is not something that comes naturally to most people. So there will be resistance, there will be denial, there will be refusing to change. It will be seen as the end of civilisation as we know it.
It should be seen as change and change means life.
Let’s stop torturing our children. Let’s stop setting so many up with a self-image that includes the word stupid. Let us recognise that every child is different, has different needs and different aspirations. Let’s give our children options instead of mandated conformity. Let’s give our children control.
I’ll be revisiting aspects of this and beyond in future articles.
[1] https://hechingerreport.org/opinion-we-can-make-math-less-traumatic-by-ensuring-every-student-is-on-the-right-pathway/
[2] https://my.nctm.org/blogs/matthew-larson/2018/02/21/why-teach-mathematics