I love a good argument about mathematics education. Whether it be in a staff meeting or at a dinner table conversation, when somebody makes a sarcastic remark about the usefulness of what they learnt in high school maths, I can’t resist biting. “Well actually”, I begin, “I think a basic understanding of calculus and statistical distributions is something that every 21st century citizen should have”. Usually, a challenge then follows: “Oh yeah? Give me one example of something I learnt in high school maths that I need to use in my life”.
I’ll get to the specifics of those later in the article. First, though, I want to examine the pervasive “I don’t need to learn maths” attitude that seems to pervade affluent societies.
The disconnect between maths class and real mathematics
I attribute the planting of the seeds for this article to my friend and colleague Billy Applebaum forwarding me a copy of Lockhart’s Lament in 2015. For those not familiar with said tome (and let’s face it, that’s probably most people reading this), I would summarise the main ideas in it as the following:
- Because maths education is compulsory, there hasn’t been much pressure to make it fun, interesting or useful – we’ve mostly relied on the “well, you’ve got to learn this to get into college” mantra to motivate students.
- In subjects like music, art and design, students actually get to DO those things – they make music, art and products, respectively. Even in science, students get to experience at least something similar to what scientists do. In maths; however, students rarely get to experience what mathematicians actually do.
The combination of these two factors has led to many students around the world despising maths class (especially those who aren’t “good at it” – more on that later), when it really doesn’t need to be that way. When visiting the doctor the other week, I experienced an all too common reaction in response to my answer to the “so what do you do here?” question. Either it’s “oh, you’re a maths guy! You must be smart! I really hated that subject/sucked at it when I was at school”, or it’s something along the lines of “oh, my son/daughter is studying at <insert name of international school>, but they’re not a maths person. They won’t be taking your class when they get older!”
“That kid’s just not a maths person”
In this case, it was the latter response. When I asked the good doctor why he felt that way, his response was (and again, I’m paraphrasing), “well, she’s in 6th grade now, but she doesn’t seem to really understand what she’s doing. I just don’t think there are many good maths teachers out there”. Thanks pal. Putting the implications for my professional competence to one side, though, the story illustrates an important reason for the reluctance to engage with difficult maths concepts that many children in the developed (or perhaps English-speaking) world seem to have – the “I’m not a maths person” belief is perpetuated by many parents, and in some cases even primary educators.
This isn’t just anecdotal either. People are doing large-scale studies on it, and governments are setting up departments dedicated to addressing the problem. In the readings for my M.Ed in STEM, I recently stumbled upon a concept called “(mathematical) horizon content knowledge“. It put a lot of my conversations with upper primary and middle school colleagues about the usefulness (or otherwise) of certain high school mathematics topics in perspective. These people are highly professional educators. They are no doubt proficient at teaching the mathematical concepts that their respective grade level curricula require (and probably do a better job than I could of teaching foundational concepts like multiplication, division and fractions), and they certainly don’t have anxiety about it. But for them, the mathematical horizon that they are preparing their students for is…more advanced maths in high school.
Why are we teaching this anyway?
We argue that teachers need a treatment of advanced mathematics that is conducted from an “elementary perspective,” one that provides an understanding of the role of important topics in the discipline, an intuitive handle of concepts, and the resources needed to recognize and use such knowledge in teaching.
Jakobsen, Thames, Ribeiro and Delaney
It’s hard for these teachers to convince their students that trigonometry is worth learning when they themselves haven’t seen (or have forgotten) what’s on the other side and how it can actually be useful (not necessarily directly, but because of the understanding of periodic functions it facilitates).
So, the first hurdle is making more educators aware of how useful advanced mathematics can be. But even when you have committed and competent secondary maths educators who have studied (or are aware of the usefulness of) maths at advanced levels themselves, there’s still the issue of out of date curricula. For example, we probably don’t need kids to learn angle laws and geometry proofs any more, unless they’re planning to study a tertiary mathematics course.
Long division is another candidate that can be left alone until kids who choose advanced maths study polynomial division. Working with quadratic models can probably be made a little more intuitive with the help of technology. Perhaps we can go a little easier on the factorising, again, until kids have decided whether they want to pursue studies in advanced maths or not. Removing these topics would then allow room for things like processing and interpreting large data sets, iteration and algorithms.
A vicious cycle
Now suppose we’ve got the non-maths teachers on board and updated the curriculum. Then we still have to deal with that other elephant in the room – maths anxiety. This is interdependent with the “I’m not a maths person” belief.
“Highly maths-anxious individuals avoid maths…They can study less at home, take part less in class, and talk themselves out of even so much as attempting a problem. And with practice in maths being so vital to success, we end up in a vicious cycle with avoidance leading to poorer performance, which inevitably adds to the existing anxiety.”
Craig Barton, summarising the work of Mark Ashcraft
We now know that neuroplasticity is a thing. Therefore we can construct the rational argument that if you work harder at mathematics (on the right things), then you will inevitably get better at it. But the rational self falls to pieces when you’re suffering from high-stakes test anxiety:
“Timed tests seem to cause anxiety…This may seem obvious, but it in fact poses us with two major problems. Firstly, every major exam that students encounter has a timed element to it. Secondly, without time pressure students are unlikely to develop the kind of automatic knowledge of key number facts that they need to free up capacity in working memory to solve more complex problems.”
Craig Barton, again summarising the work of Mark Ashcraft
A catch-22?
Ashcraft’s work suggests that there are two opposing forces at work that are very difficult to reconcile:
- Timed maths tests, the main way that we assess mathematical competence, are a major contributing factor to maths anxiety and the so-called “vicious cycle”.
- At the same time, some level of time pressure when doing maths problems is necessary in order to develop fluency in the foundational skills and concepts that allow one to access more advanced skills and concepts.
Perhaps some thoughtful pruning of the curriculum as suggested above will make more room for anxiety-reducing strategies and give students more “runway” when working towards timed assessments. But then again, re-examining my claim from the opening paragraph, why does Jo(e) Citizen even need to learn said advanced skills and concepts? Won’t he or she get by in life just fine without ever having taken an introductory calculus course?
This seems to be the consensus among most educators I talk to, unless they teach a STEM subject themselves (refer to my earlier point about Horizon Content Knowledge). Author and middle school English teacher Jody Stallings, in his article about updating an allegedly outdated US mathematics curriculum, writes:
“Calculus (whatever it is) is apparently really hard. It involves the kind of high-level thinking that the average person simply can’t or is unwilling to do. Those who can tend to go into fields where they will – how shall I put this gently? – make money.”
Jody Stallings
Stallings’ viewpoint is fairly representative of the comments I’ve encountered during conversations with non-STEM educators about advanced mathematics education over the last 9 years (disclaimer: I haven’t (yet) come across any published research that suggests this is the case on a larger scale).
Unknown unknowns
Although the article makes several other valid points and suggestions, I disagree with Stallings on this one. I think you need to study something at a level above whatever level of understanding you’re hoping to retain and assimilate into your thinking for the long-term. For example, I studied maths at 2nd year university level. What I’ve retained long-term; however, and the level of understanding that I’ve truly mastered and am able to use in the way I think about the world, is more like the understanding that a 1st-semester university student has, or perhaps a capable high school graduate who took advanced maths.
Interpolating, if you only study maths until Grade 12, hopefully, you’ll remember concepts from at least Grade 10 and 11. Concepts like non-linear proportionality, compound interest and exponential growth, sinusoidal variation, mean and standard deviation, and the difference between correlation and causation. All critically important to understanding the science and technology that surrounds us in our modern world. Taking it even further, I’d argue that every 21st century citizen should take an advanced maths course up to Grade 12 level that includes the following:
- Non-constant/instantaneous rates of change
- Continuous distributions and confidence intervals
- Conditional probability and Bayesian thinking
- Understanding “computational effort”, like how the number of computations required by an algorithm varies with scale
The first two points cannot be understood without at least an intuitive understanding of calculus. The latter two, I would argue, are key survival skills in a technologically advanced society. The corollary, of course, is that not understanding these concepts makes you more easily fooled by popular media and pseudoscientific claims when they use poorly understood words such as “exponential” and “quantum” in order to give an appearance of credibility – stay tuned for future posts on these very topics!
Making maths great again
So to summarise, we need a citizenry that understands maths, but the majority of them have anxiety about learning it and using it. They don’t perceive it as useful, and in fact often perceive it negatively. Try doing a Google image search for “math class”. You won’t have to scroll too far to find a picture of a confusing jumble of symbols on a blackboard. This perpetuates the myth that maths is something inaccessible and only understood by a select group of genii.
So what is the solution then? Other than the curriculum updates and teacher training already mentioned, there’s another piece of this puzzle (or should I say “side of the equation” – pun intended) that, in all the debate about how to improve the teaching and learning of mathematics, is often overlooked. We need to do a better job of convincing people of how meaningful, relevant and useful maths can be in the first place. Enter Karim Ani and his recent opus, “Dear Citizen Math”.
Towards authentic mathematical thinking
“…it might help to teach kids what engineering IS before we teach them how to do the complex math it involves.”
Jody Stallings
Ani’s proposal seems obvious but is one that is easier said than done in a system of grades, standardised tests and a packed curriculum to “get through”: we need to engage students as early as possible in using maths to create and debate solutions to relevant and contemporary issues. Examples from the book typically involve applying quantitative reasoning to the fairness of various systems in society – from health insurance premiums, to overselling flights, to the non-standard placement of boundary fences in Major League Baseball (the “issue” here being that players who spend their careers at smaller home grounds generally have better batting averages).
These are great starting points for making maths class more engaging and will work well as stand-alone lessons in just about any Grade 6-10 maths curriculum, but I would argue that we can do better still – by explicitly involving students in using mathematics to make decisions about things that directly affect the environment around them.
The power of the “back-of-the-envelope calculation”
There are many situations in life in which a little bit of first principles thinking and some “back-of-the-envelope calculations” can help individuals or groups to make good choices – how much to charge for something; what size to make something; how much time or money to spend on something. Granted, these probably aren’t things that most (typical) teenagers contemplate on a daily basis, but think of the possibilities for connections with design and service learning projects: how should the dimensions of this 3D printed part be determined? How can I program this drone so that the acceleration stays within limits? How should we set our pricing or spend on advertising to optimise the reach of our website?
These are all questions that mathematical thinking can (help) answer. I don’t purport to have a fully fleshed out solution that can replace the current status quo in mathematics education; these are just examples that pop into my head as I think about how to scale the examples and methods in Karim Ani’s book across an entire secondary mathematics curriculum (perhaps also the subject of a future post).
What I am sure of, though, is that the curricula and assessments that typify our current mathematics education system could be doing a lot more in the authenticity department.
A step in the right direction
It won’t be easy. It will require discarding all (or most) previous assumptions about what “should” be in a mathematics curriculum. We’ll need to start with the kind of mathematical thinking that we want a 21st century citizen to be able to do as a basis, and then see what follows from there. Then again, designing authentic assessments that effectively bridge from the “sandbox” of high school to the real world of adult work roles never was easy.
In the grand(er) scheme of things, if we want more of our population to move from conventional to post-conventional moral development, then I would argue that honing our ability to think rationally and analytically is an essential prerequisite. This is essentially the big picture argument that Karim Ani is making in his book – that having a citizenry that is able to think in this way is essential for preserving modern democracy. Let’s at least start with the low-hanging fruit by getting rid of the least relevant and most outdated parts of the curriculum and giving maths teachers some more breathing space to get more creative with their assessments.
Which parts of most modern-day maths curricula would you get rid of and why? On the flip side of the coin, which topics would you add to make maths class more relevant and future-oriented?
Leave your thoughts in the comments below!
