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Last week, I shared my thoughts on a book about Statistics. While it is an excellent introduction to basic statistical concepts, there is so much more to Math than just learning how an average works or how probabilities may be used in deceiving ways. After a long-lived hatred for numbers, I came to realise Math is only half about numbers. The other half is about mental models.
Unfortunately, many educational systems only focus on teaching the rules of Math without explaining the reasons why they came to be or why they are useful. Understanding the reasoning behind the rules is crucial to truly grasp the concepts and applying them in real-life situations.
Mental models are an essential part of our problem-solving process. We each have our own unique way of interpreting complex information, breaking it down into simpler terms, and using systems we created in our minds to interpret them. Most of the time, we do it automatically, without even realising. As we go through life, we learn a bunch of basic reasoning concepts along the way and use them to create mental models that help us solve problems.
Mathematics helps us understand and form these mental models. By providing a framework for understanding complex concepts, Math serves as a tool for problem-solving and critical thinking. So, it's not just about following rules but about developing a deeper understanding of the world around us and the mental models we use to make sense of it.
Here’s a fun example: how do you divide 1920 by 16 in your mind without the help of a calculator or pen and paper?
Think about it, and note the steps you take.
I asked my partner. The way he did it is like this:
“1600 by 16 is 100.
I’m left with 320
16 times 2 equals 32; 32 times 10 is 320.
2 times 10 is 20.
100 + 20 = 120.
120 is the result.”
I was amazed to hear that because the way I calculated it in my head was like this:
“1920 and 16 are both divisible by 4.
So it would be as dividing 480 by 4
4 times 12 is 48.
so, 12 times 10 = 120.
120 is the result.”
So, I decided to ask a friend too (cause why wouldn’t I).
May, a mathematician, does the following:
“First, I try to multiply 16 until I find a number that fits 1920 nicely: 16 x 2 = 32 x2 = 64 x 2 = 128.
Then, I sum 64 to 128 = 192
1920 x 10 = 1920
It leaves me with (16 x 8) and (16 x 4)
8 + 4 = 12, times 10 = 120
120 is the result.”
Kerry, a fellow data viz person, does hers like this:
“I know 1920 is bigger than 1600, so I do 1920 - 1600 = 320
320 divided by 16 = 120”
And Fernando, who is a designer, replied:
“I don’t. My God. No”
(next day) Ok, wait… I got a visit from the number gods in my dreams. They told me:
100 times 16 is 1600.
and 10 times 16 is 160.
If I go adding 10 I’ll get there.
160 is too little. Let’s do 20 times 16 = 320
huh, yeah 1600 + 320 is 1920.
It’s 120. 120 times 16 is 1920.”
All of us get to the correct result. All of us do it rather quickly, without needing a calculator or a piece of paper. However, each of us follows an entirely different path to get the same result. Each of us has a different mental model for dividing a number by another, and all of them have equally served us well.
Math as the extension of common sense by other means
How Not To Be Wrong by Jordan Ellenberg is another book that helps us learn about everyday applications of Math in the real world. It is my second recommendation about basic numeracy concepts in this blog series. But it is a substantially different book from my previous recommendation, Naked Statistics, by Charles Wheelan. While Wheelan talks about applied statistics and gives us examples of how we can use each of those concepts in practical terms, How Not To Be Wrong is more about mathematical thinking and how developing and fine-tuning our sense of logical comprehension can help us make sense of the world, with Math as a tool.
A lot of what we do when working with and interpreting data is precisely that: we grab something complex, try to break it down into more familiar steps and look for ways to make sense of it, very likely based on previous bits of knowledge we collected through our lives and organised in the form of a mental model. That is why biases are a thing: this whole process is not a conscious effort for us. Rather, it happens almost instantly and automatically. Our previous experiences create a sort of muscle memory in our brains, which we employ when trying to make sense of the world around us. If we only ever get exposure to a handful of experiences of the same type among the same groups of people, our mental models will be solely based on that, leaving us completely blind to diverse ways of reaching similar conclusions. When we share these mental models based solely on experience with fellow humans in a social arrangement, we create what philosophers and thinkers alike named Common Sense.
Jordan Ellenberg goes a bit deeper than explaining how particular mathematical theories work - he seeks to help us understand how mathematical thinking works and how it can be beneficial to us in all sorts of situations. He argues that Mathematics is just common sense, explained through a methodical and formalised language so that it can be reproduced and easily shared among those initiated in how the whole system behind it works. So, when you see a formula like a + b = b + a, that is just Mathematical jargon to say that if you sum two things, no matter in what order, the result will always be the same. It is common sense codified as a formula. Once the method was there, each next mathematician built on top of the previous ones, and the complexity of the field grew - that’s how any field evolves.
But as Jordan Ellenberg argues next in his book, not everyone needs to know about all of these more complex developments to be well-versed in how Maths can be helpful to them. He splits Mathematics in a neat quadrant chart, with 4 areas: simple to complicated in one axis and shallow to profound in the other. Most of us leave school somewhat stuck in the simple and shallow quadrant. This becomes more evident when we’re faced with the obscene amounts of data, metrics, KPIs, charts, forecasts and projections we see everywhere. It makes us feel like these concepts are somewhat out of reach for us non-Math people. How Not to Be Wrong is about bringing you on the journey from the simple and shallow to the simple and profound quadrant. Things that are not complex to grasp but which will help you get a deeper understanding of how to see and use Maths in the real world without wasting time arguing about the complicated stuff.
Jordan Ellenberg walks us through multiple examples of how mathematical thinking can help us expand our understanding of different situations. And he does that in plain, understandable, accessible English without relying on complicated equations, theorems or formulas. Even though professional mathematicians use those tools in their arsenal to explain things, this book is about learning how to ask good questions, look for good answers and how to interpret them. The core of the book is mathematical thinking, not mathematical rules.
All the ways we encode and decode data
I create data visualisations for a living. One of the biggest challenges in what I do is to choose how I will encode the data I have using shapes, colours and positions on a plot. Some of my clients incorrectly assume that tools make those decisions for us. They don’t. They provide me with a range of options, and they facilitate my work when I want to implement those options - but the choices of what’s an appropriate representation of data are made by the humans operating the tools. Tools sometimes make suggestions of encodings - like Excel or Tableau in the “show me” tab. Those are precisely that: suggestions - and they’re often not very good ones.
But why does successful data visualisation still need human input despite all the BI tool promises we see? The answer lies in the way we, as humans, encode and decode data. When we collect and record data, we are trying to capture a phenomenon or situation so we can analyse it later. This collection process is usually flawed or biased, full of human input and decisions made along the process. Those decisions and biases are often invisible and go unnoticed until further analysis is carried out.
When I am then asked to visualise this data, I first need to understand what this data actually represents. I have to go back to reality, out of the numbers and tech and make sense of the process or situation that originated this data and why these records are the way they are. Ideally, this process involves understanding the mental models of the people involved in collecting, keeping and wrangling this data. Sometimes, I won’t have all that information, so I’ll have to fill in the gaps. That means making assumptions. These assumptions mean I am now starting to decode the meaning of the data and trying to make sense of it so that I can encode it again, to help someone else (my audience) to see what I see.
After I have an insight, message or story to tell, I’ll ask a bunch of questions: who is this for? Why do they care about it? What will they use this for? Are they making decisions with this? Which types of decisions? What are the expected outcomes? What does good or bad look like in this context? Does it matter to them? All of these questions will vary wildly in answer depending on who’s on the other side, and they are crucial to define which choices will best fit the purpose of the visualisation I am building.
There is another set of questions I often ask to better understand my audience. They’re a bit deeper into my craft but no less relevant: how will my audience use this? How much are they used to using Data Viz to perform these tasks? Are they expecting to interact with this information in any way? How? If so, how long will they likely spend interacting with it? What’s their affinity with technology like? Have they used a similar BI product before? What did it look like? Did they have a hard time using it? What were their struggles like? Do they wish something was different from that experience? How can I make this more valuable and accessible to them?
Once I have uncovered what the data can tell me, I must discover how my audience will receive such a message. I need to have a better understanding of my audience’s context and how their mental models work because those mental models will be the ones used to decode the data visualisations I so carefully design for them - and they are just as crucial to communicating what the data have to say, as the data’s message itself. If you write a message as a code (encode) and you don’t know if your recipient has the key, they won’t be able to read it (decode).
This is often seen as me being too precious about my work, and it is: I am precious about my work. There’s enough of us hastily churning out dashboards out there. There’s a whole graveyard of business analytical tools developed that no one uses. Making people interested in using something they find difficult is hard work. And most people find analytics difficult. I want to show them it doesn’t have to be this way. It can be pleasant, it can be fun, and it can be delightful. But that means asking and answering a bunch of questions first.
Mathematics, as Jordan Ellenberg explains in his book, does something similar. It tries to translate the world’s complexities into a simplified form - a number or metric - that can be easily communicated to others who understand the symbols used. So, when another mathematician sees a theorem or a formula, they can decode the thought process of the mathematician who wrote this piece of information. Math helps to organise the thought process behind a logical statement by establishing logical models. Just like Information Design does to Data.
Sometimes the right answer is “there is no clear answer”
One of the points that the author drives home in different chapters is the one of becoming comfortable with uncertainty. We have a strange obsession with accuracy, no matter what it takes. As an analyst working in a business setting, I often faced the dilemma of a boss asking me for a number - any number - to explain something that has happened or to predict something that might occur in the future. Sometimes there is no clear answer. Sometimes the noise outweighs the signal. Especially when trying to analyse unstable forecast models - like the complex long-term effects of COVID-19 on a supply chain, for example - often there is no clear answer. There are just too many moving parts and too many unknowns, and the correct answer may as well be “there is no clear answer”.
This is often the cause for outrage. How come all this data we so carelessly hoarded over several years, keeping it in a range of fragmented legacy systems with little to no support, cannot tell me right now, when I need it, exactly the answer of what’s going to happen in the next 12 months in a volatile world-changing situation? How dare I, as an analyst, say that I do not know something because it is too uncertain? That cannot be; that cannot do. Bosses will demand answers. Executives must have projections to show their shareholders. The clock is ticking, and even if your uncertainty rate is 50% to either side of the number, we must have a number to go with.
In my review of The Tyranny of Metrics by Jerry Z. Muller, I discussed the phenomenon of relying too heavily on metrics in decision-making. Jordan Ellenberg also addresses this issue towards the end of his book in a chapter aptly titled “How To Be Right”. Ellenberg emphasises that while it may be tempting to assign a level of confidence to one's answer, sometimes the right answer is simply "there is no clear answer" due to contextual factors and limitations in available data. While data can provide valuable insights, it can't completely eliminate all uncertainty. We must learn to become comfortable with this level of uncertainty and recognise that mathematical thinking can help us better navigate the complexities of reality. By expanding our comprehension of numbers beyond just black-and-white, straightforward, precise answers, we can better gauge reality's complexities and make better assessments and decisions.
Should you read it
How not to be Wrong by Jordan Ellenberg is an incredibly insightful and thought-provoking book that explores the power of mathematical thinking in our daily lives. While it is certainly a more involved read, it is well worth the effort for anyone who is interested in expanding their basic mathematical knowledge and gaining a deeper understanding of the world around them.
One of the things that sets this book apart from others on the topic is its focus on how mathematical thinking can help us update our mental models and better interpret the world. The author delves deep into numerous examples from a variety of fields, including Philosophy, Politics, and Administration, to show readers how Math can be used to put complex concepts into context instead of just explaining a bunch of theoretical rules. By doing so, the author makes a compelling case for why Math is such an essential tool for critical thinking in today's world.
If you are looking for a more basic introduction to Math and Statistics concepts that you can use in more practical situations, I recommend checking out my review of Naked Statistics by Charles Wheelan first. If, however, you are already comfortable with basic mathematical concepts and are curious about deepening your knowledge, this is the perfect follow-up book.
One thing that I particularly appreciate about How not to be Wrong is the author's use of a wide range of examples from numerous historical periods. Rather than relying solely on contemporary examples like sports and current politics, the author brings up examples from the early 20th century, the 1929 crisis, and breakthrough theories from the Greeks to put the evolution of mathematical theories into perspective. By doing so, readers gain a deeper appreciation for the fact that all knowledge builds upon previous discoveries, and they are able to develop a more nuanced understanding of how Math has evolved over time.
Overall, How not to be Wrong is an excellent book that will challenge you to think more deeply about the intricacies of how our world works. By exploring the power of mathematical thinking and its applications in various fields, the author provides readers with a rich and engaging introduction to the subject that will leave a lasting impression.
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