Welcome to Mathematical Reasoning and Investigation.
Or, as I like to call it, mathematics for people who think they’re bad at it.
Why mathematics is important
Mathematics can be a powerful way to investigate, analyse and make sense of the world around us. Most of us are aware that mathematical reasoning underpins a great deal of research in science, medicine and technology, but are less familiar about the ways in which it can be used to solve the kinds of problems that confront us every day in our workplaces and society more generally.
So many of these issues can be informed by mathematical analysis. Learning about mathematics can help us to make better-informed decisions and develop critical insights into effects of the decisions made on our behalf. Indeed, some of our biggest unresolved social questions can, and should, be informed by mathematical reasoning and investigation. Questions like how we can live in a more equal world, how we can best use and distribute the resources around us, and how we should understand and respond to changes in the climate all rely on numbers to point us in the right direction. We’re not suggesting for a moment that numbers are the only thing that should inform our decisions. However, when you’re trying to tackle a difficult problem, being able to reason mathematically is an important tool to have at your disposal.
Mathematical reasoning
This is what makes the capability of mathematical reasoning so essential. People who have developed the ability to reason mathematically are able to make reasoned conjectures, test them and then come to reasoned conclusions. The process of doing so will prompt us to challenge our assumptions, keep a mind that is open and strategic at the same time, justify our own thought processes and think about what it means to solve problems in the first place. If there’s one thing that practicing mathematical reasoning can teach us, it’s the skill of patiently working through problems creatively, systematically and methodically.
Who is this book for?
This book has been written for
- Adults who want to improve their own skills in mathematical reasoning and investigation
- Teachers who want ideas about how to teach mathematical reasoning using a problems-based methodology.
If you are in the first category, try not to stress (see the section below). We all need to start from somewhere, and this book makes very few assumptions about the base knowledge that people will come in with. Anyone with a basic level of mathematical knowledge should be able to start at the beginning of the book and work their way through. If you can use a basic calculator, add, subtract, multiply and divide then you’ll be good to go. As with any area of mathematics, and most things in life, the more you know the easier it will be for you. If you’re comfortable with algebra, that’s a great start. But if not, we’ll walk you through it and you’ll be using it to solve problems in no time.
If you are a teacher, you may be able to adapt sections of this book to your classroom and, perhaps more importantly, use it to inspire learning activities that explicitly develop a student’s ability to reason mathematically, by taking a problems-based approach. As you’ll see, often when taking a problems-based approach, getting to the correct answer (if there is one) is a secondary objective to developing the ability to plan, test conjectures and generalise findings. Working through the book may also help give you the grounding in mathematical reasoning you need to be confident setting activities for your students.
Starting out with mathematics can make you nervous
If we can agree that maths is a tool that can help us to make better informed decisions, then Western societies in general have a problem. The thought of doing mathematics makes us feel a little nervous.
I (Chris Rawson) believed I was bad at mathematics well into my adult years. I believed that I wouldn’t be able to do it, that I didn’t like it, that it was fundamentally procedural and even that it dampened creativity. It was this more than anything else that held me back from attempting to learn more. But by working through the content that forms the bedrock of this book, I was able to move beyond my own anxiety and false assumptions.
Once I gave mathematics a go, I discovered that even if I’m not a maths wizard like Simon James, I am able to competently tackle some quite tricky problems. This book is based on Simon’s methods – the overwhelming majority of it is Simon’s work. One of the core reasons I encouraged its publication as an open resource was because working through this content had taught me so much and made me feel more confident and able to tackle problems that previously I would simply not have attempted. It’s been liberating, and that is something I want to share with others.
According to research published in 2018, 93% of adults in the USA report that sometimes the thought of doing mathematics makes them feel anxious. The problem isn’t confined to the United States, and it’s not just adults either. According to the OECD’s 2012 PISA report that looked at 34 different nations, 59% of students aged 15 and 16 often worry that maths class will be difficult for them.
This isn’t helped by some of the stereotypes and media portrayals of what it looks like to be good at maths. Many people in the movies who are good at maths are born with a gift with numbers and can imagine complex patterns emerging (sometimes quite literally) where most of us see nothing. Frankly, they are geniuses and the rest of us will never get near them. The assumption: mathematical ability is a gift that you’re born with. People with this gift, in the movies, also happen to be almost exclusively white males.
It’s important we don’t make these same assumptions for ourselves, our students or those around us. Mathematical reasoning is a skill that can be developed like anything else, it just takes practice. It’s a little bit like starting to learn a musical instrument: it’s normal to think ‘I’ll never be able to do this’ at the beginning, but with persistence you will get there. It’s interesting to recognise that this advice doesn’t only hold true for people who started out thinking that, like Chris, they were allergic to mathematics. It’s just as true of people with a deep mathematical knowledge and affinity for numbers, like Simon James. One of the most powerful moments in this book is when Simon describes starting out on geometry, struggling to solve problems, feeling frustrated when he has trouble and then cheating by looking up the answer on YouTube (then promptly forgetting the answer). We don’t nominate this moment for special attention because we want to cut Simon down to size – rather it’s to point out that even someone who has a PhD in mathematics can struggle when they encounter something for the first time. The trick is knowing when to persist, and when to try another strategy.
Mathematics is more than just methodology
Mathematical reasoning is a process of making reasoned conjectures, testing them and seeing if we can justify our solutions to sometimes complex or fairly undefined problems. This can be an incredibly creative and satisfying process. Many people don’t associate mathematics with creativity – indeed some people consider it the opposite of creativity, seeing it as a fundamentally rigidly structured process of memorising abstract rules. Worse, some people get the impression that mathematics is purely abstract, a secret code only understood by scientists and with otherwise little real-world application. This is a very sorry state of affairs.
Perhaps some of this comes down to the way we were taught mathematics. Memorising complex methods by rote, solving abstract equations and checking to see if we got the right answer is bound to seem abstract, difficult and ultimately irrelevant to most students. While these kinds of activities remain a fundamental and necessary part of developing numeracy in children, teachers should not resile from the task of demonstrating how and why mathematics can be used to solve issues both in our day-to-day life and, as children get older and more socially aware, our society in general. However, if you’re not confident doing this on your own, it’s difficult to teach it to others.
The kind of problem-based learning modelled throughout this book is a great place to start. We seek to propose an alternative to the back of the book. Instead, we attempt to pose problems that ask you to come up with your own methods, to make conjectures, test how reasonable they are and to justify your answers. Finally, we ask you to work towards the ability to generalise – that is, to be able to answer any similar problems that come up.
Just to emphasise the point, the authors are not suggesting that there is no place for teaching mathematical methods explicitly. In fact, we do so ourselves at times throughout the course of the book where it promotes problem solving. Instead, think of learning mathematics as being analogous to learning a new language. In this analogy, methods are like grammar – a series of rules that effectively need to be learned by experience and somehow internalised or at least memorised. But as important as grammar is to language (the words will make no sense without it), you can’t learn to communicate and interact with others or achieve your goals just because you know all the grammatical rules. Similarly, the ability to reason mathematically goes above and beyond knowing mathematical methods in an abstract sense – we need to show that we can apply it to help us navigate our lives in the real world and achieve our objectives. As many of us ask ourselves as teenagers: what’s the point of algebra if we can’t do anything with it?
About the authors
Simon James is Associate Professor of Mathematics at Deakin University. Chris Rawson assisted in the development of this book in the capacity of Senior Educational Designer, also at Deakin.
Conclusion
We hope that you and your students enjoy working through the problems in this book. More importantly, we hope it helps you to see new possibilities for mathematical reasoning and investigation.
