Reasons to Reason in Primary Maths and Science

Alison Borthwick, Alan Cross, Author

• Publisher: Learning Matters
• Year: 2018
• Print ISBN: 9781526435040, 9781526435033
• eBook ISBN: 9781526451101
• Edition: 1
• Topic: Education
• Subtopic: Teaching Mathematics

How can teachers help children to develop reasoning skills?
What is reasoning and how do we teach it?

Much is being said in schools and education about the importance of reasoning skills. This book explores what reasoning is and what it is not. It includes examples of how reasoning in primary mathematics and science classes can develop. It shows how a connection between the ?skills? of mathematics and science can help children to gain a better understanding of reasoning.

• What is a conjecture
• What makes you think?
• What makes you think about your thinking?
• What does reasoning look like?

With links to classroom practice and examples of effective teaching throughout, this book not only provides an exploration of what reasoning is and why it?s important—it also show you how to develop children?s reasoning skills in your classroom.

1 A rationale for reasoning

In this chapter

By the end of this chapter you will:

• have considered definitions of thinking and reasoning;
• understand how thinking and reasoning are similar yet different;
• be able to begin to identify skills of reasoning;
• have reflected upon the place of reasoning in at least two primary STEM subjects.

Introduction

This chapter will introduce you to a number of important ideas around how primary-aged learners can, do and could reason within their mathematics and science education. As well as explanations and illustrations the chapter will give examples and opportunities to reflect and reason yourself.

The ability to question, pose, investigate and solve problems is at the heart of mathematics and science. However, until a problem has first been understood, learners may struggle to engage with it. This presents learners with many difficulties, because there is a great deal involved in solving a problem; a fact that is often unappreciated. There are many skills involved in unravelling what a problem is about, which concepts and skills are needed and how to make use of them in finding a possible solution. Some of the required knowledge and skills will be subject-specific, some will be common to mathematics and science – for example, measurement.

Even when learners are equipped with several different investigative and problem-solving strategies they still need toreason which one to use. Reasoning is essential to children learning more and solving problems. It is the one element that binds together all the different skills needed to solve a problem (skills such as pattern spotting, offering conjectures, generalising). Without reasoning, learners may be simply following procedures, applying rules, ignoring patterns and missing opportunities. While this book is about reasoning, we acknowledge that this is one of several strands that learners need to draw on to be mathematically or scientifically proficient. Kilpatrick et al. (2001) offer five different strands: conceptual understanding, procedural fluency, strategic competence, productive disposition and adaptive reasoning (see Chapter 7). They view these strands as equally important and interdependent. Being able to draw on each of these stands when solving problems and carrying out investigations is key to being successful in these two STEM subjects – and without the reasoning strand, learners may struggle to make progress, connect ideas and knowledge and reach a level of understanding that is not simply about producing a formula or memorising a fact. So, what is reasoning?

What is reasoning?

The word ‘thinking’ is used in everyday language and we use it in different ways. For example, Anne is sure to think of a solution (an action); What do you think about the decision to cut down these trees to enable a housing developing? (seeking an opinion); Why didn’t you think of that before you went ahead with that idea? (a kind of foresight).

Types of thinking have been described by different people. We do not have space here to examine these in detail but Frank Williams’ (1969) taxonomy lists eight types of creative thinking:

• fluency: the capacity to generate ideas, possible responses to a problem or situation
• flexibility: coming up with alternatives, different ideas
• originality: generation of new unique solutions
• elaboration: the development or expansion of ideas to make them more comprehensible/interesting
• risk-taking: experimenting, trialling, challenging ideas
• complexity: applying logic, establishing order, identify missing parts
• curiosity: wondering, puzzling
• imagination: the ability to see a mental picture, new ideas, new possibilities

It is interesting to note that reasoning is not explicitly mentioned in Williams’ list.

In talking with teachers and reflecting about thinking and reasoning, we have settled on a definition that, for us, is a point of reference which we can draw on when exploring reasoning in primary classrooms:

Reasoning is thinking,

but it is thinking in a logical, purposeful, goal-directed way.

We have selected the words carefully within our definition. To apply some logic assumes that alternative viewpoints have been considered; to have purpose implies that we are focused and determined in pursuing our line of enquiry; and to have a goal suggests that there is an objective in mind, but this does have to be a final answer or a proof – this could be the process we have been engaged with or the knowledge we have gained along the way. We purposefully did not include the word ‘correct’ in our definition. This is because, for us, reasoning is how we navigate through a problem, to try and make sense of the situation, to select the tools to help us and to reach some sort of resolution, even if this is to continue to reason! We also believe that all learners can reason, despite researchers such as Inhelder and Piaget (1958) suggesting that learners’ ability to reason is quite limited until around the age of 12 years old.

So, for us, reasoning is another type of thinking, some may even say a subset of thinking. In some cases it is about finding reasons for things, making decisions, considering cause and effect, wondering, What is the reason for this happening? While reasoning is not mentioned in Williams’ (1969) list of thinking, it would contribute to all his types of thinking if we applied our definition to it.

How do reasoning and thinking fit together?

Thinking is that exercise of the mind which forms ideas, conceptions, conditions, recognitions, revisions, considerations, connections, opinions, judgements, imaginings and intentions. Since our brains are so very active it is perhaps no surprise that we have so many terms associated with thinking. Reasoning is a form of thinking; words such as logical and systematic often preface. But thinking can be disordered, illogical and messy. Reasoning is, according to Fowler and Fowler (1984), the action of thinking in a logical, sensible way even if the outcome if incorrect. Reasoning brings with it a certain order (often subjective and pertinent to that person); it suggests clarity and logic.

Reflection

When do you reason in your day-to-day life? When making a purchase? Choosing what to eat? Which route to take to work?

The terms ‘reasoning’ and ‘thinking’ are often used interchangeably. On one level this is understandable and acceptable, but on another, it de-values the importance of reasoning and the skills associated with it.

To assist your understanding, take a moment to look at the options below. For each, consider does this require thought? Does it require the logical, purposeful, goal-orientated thought we call reasoning?

Do these situations require reasoning?

1. Choosing a birthday present for a friend.
2. Deciding how to vote in a referendum.
3. Selecting a new pair of shoes.
4. Deciding how to pay for the purchase of a house.

Each scenario requires thought. Options a) and c) might, in some circumstances, require little thought or in others more than that. Options b) and d), it might be argued, require more thought and the weighing of options and consideration of rationale we associate with reasoning. Equally, different people will employ different skills of reasoning for each scenario. For some, buying a pair of shoes is a fairly straightforward experience which may require some reasoning, but for others it may require many more reasoning skills.

By now you are probably thinking I need one definition for thinking and one for reasoning. You are reasoning about reasoning!

What do others say about thinking and reasoning?

There are many varied and interesting definitions of thinking and reasoning.

Thinking is using thought or rational judgment while reasoning is to persuade, or move by argument, to express in logical form.

(Fowler and Fowler, 1984)

Mathematical thinking is more than being able to do arithmetic or solve algebra problems. In fact, it is possible to think like a mathematician and do fairly poorly when it comes to balancing your checkbook. Mathematical thinking is a whole way of looking at things, of stripping them down to their numerical, structural, or logical essentials, and of analyzing the underlying patterns. Moreover, it involves adopting the identity of a mathematical thinker.

(Devlin, 1991)

Probably the single most important lesson is that being stuck is an honorable state and an essential part of improving thinking.

(Mason et al., 2010)

Reasoning is fundamental to knowing and doing mathematics. Reasoning enables children to make use of all their other mathematical skills and so reasoning could be thought of as the ‘glue’ which helps mathematics make sense.

(Pennant et al., 2014)

Adaptive reasoning refers to the capacity to think logically about the relationships among concepts and situations. Such reasoning is correct and valid, stems from careful consideration of alternatives, and includes knowledge of how to justify the conclusions.

(Kilpatrick et al., 2001)

In ...

Table of contents

• Cover
• Half Title
• Publisher Note
• Title Page
• Copyright Page
• Contents
• Acknowledgments
• About the Authors
• About this book
• 1 A rationale for reasoning
• 2 Frameworks to promote reasoning
• 3 Reasoning in mathematics and science in the Early Years
• 4 Using questions to promote reasoning
• 5 Using prediction, conjecture and hypothesis to promote reasoning
• 6 Noticing pattern to promote reasoning
• 7 Using problem-solving and investigation to promote reasoning
• 8 Noticing and evidencing reasoning
• 9 A toolkit of ideas to promote a culture of reasoning
• 10 Final remarks
• Appendix
• References
• Index