Reasoning: a dog's tale.

Holton, Derek ; Stacey, Kaye ; FitzSimons, Gail 等

We illustrate three basic types of reasoning used in mathematics by showing how they operate in practical and mathematical situations. The importance and function of the different types of reasoning in each situation is outlined. As a consequence we note that while introducing new techniques by example is good from a pedagogical viewpoint, several examples by themselves do not provide a proof that the techniques are correct.

Coco the dog

Grandparents were 'babysitting' Alex and Alyssa and their dog, Coco, while their parents were away. While the grandparents were in sole charge, Coco learned a new way to get 'treats'. Coco liked getting hold of anything that was lying around the house. She would grab the TV remote control, a pen, a child's toy, anything that was loose and within her reach. Being a big dog she could reach pretty well anything that was lying around the room at whatever height. Initially she probably grabbed things because she was bored and because it got her a great deal of attention. Because the grandparents wanted to have most of the household goods intact when the parents got home, they retrieved the grabbed object by offering Coco a food treat from the kitchen. She soon understood the game and before the end of the second day, she would grab something and run round the house with it until the grandparents went to get the treat. Then Coco would drop the object and go to the kitchen to wait for her reward.

Three types of reasoning

We use this story to illustrate three basic kinds of reasoning that are used in all human thinking, including mathematics. We suggest that such reasoning is inbuilt in humans for use in everyday situations, and some may even be natural to dogs (1) and other animals. The task at school is to teach children to develop these underlying human abilities in the special ways that mathematics requires. The three kinds of reasoning that we want to discuss are induction, deduction, and abduction. All of these are based on three things: a case, a specific example where something holds; a rule, a general if ... then ... statement or its equivalent; and a result, a further specific example that depends on a case linked to it by a rule (Reid & Knipping, 2010).

Induction is the kind of reasoning that produces generalisations based on examples. Having seen that some examples work a particular way, you conclude that all examples will work in this way. Here, a case and a result lead to a rule. More often in mathematics and science, several cases and results are needed before the rule is inducted. This was probably the situation with Coco too. For her, a case would be "Coco stole a remote control" and a result would be "Coco got a treat". From this observation and similar ones, she inducted the rule "If Coco steals an object, then Coco will get a treat". After stealing a few objects and getting treats, Coco made the inductive generalisation that she could get a treat by stealing any object that the grandparents valued.

We show induction diagrammatically in Figure 1. The boxes with solid edges indicate what is known and the box with dotted edges indicates what is induced. The act of induction is to frame the rule from the case and the result.

In a mathematical problem, we might see that a property holds for some numbers, even a lot of numbers, and conclude by reasoning inductively that it works for any number. For instance, after noting several cases like "2 + 3 + 4 is divisible by 3" and results like "234 is divisible by 3", we might induct the rule that "if the sum of the digits of a number is divisible by 3 then so is the number".

However, we cannot be absolutely sure that a rule obtained this way will always work, but that is not part of induction. In science and many other parts of life, induction offers new insights, based upon the best evidence available. But, in mathematics, the purpose of looking at examples is to give us some evidence about what might be true. It is an important tool for making conjectures. We make a conjecture inductively, but then we need to find a way to justify it absolutely for all cases. For example, we might see many cases and results like this: case "2 + 4 + 0 is divisible 6" result "240 is divisible by 6"

and induct the false rule that "if the sum of the digits of a number is divisible by 6 then so is the number". Coco's rule has the defect that it does not work if the grandparents are not in the room to see her steal the object. So, both the mathematical and canine rule must be revised before they have a chance of being proved. Of course, when teachers introduce a technique by using several specific examples, the teacher knows that they are not presenting a proof of that technique. But do the students know this?

Deduction involves using laws of logic to justify or find new conclusions from previously known results. Once Coco had convinced herself of the general theory that stealing objects produced treats, then she was able to use deduction. She now had a case: "Coco steals a cushion", and a rule: "If Coco steals an object, then Coco will get a treat". This produces the result: "Coco will get a treat".

Deduction is the type of argument that uses case plus rule to get a result. So Coco said to herself: "Stealing objects produces treats. This cushion is an object. So, if I steal this cushion, I'll get a treat".

Coco knows a general result which she assumes to be true about a class of objects. Then she chooses a member of that class and so she can deduce that the result is true for that member. This is one of the laws of logic that children know from an early age. We show this in Figure 2. Again, the dotted box is the one that is produced by the deductive process from the known solid edged boxes.

Reasoning deductively is very common in all mathematical practice. For example, the following known rule, is given above: "If the sum of the digits of a number is a multiple of 3, then that number is a multiple of 3." So, taking the case "the sum of the digits 5, 6, 7 is 18", we can deduce the result "the number 567 is a multiple of 3". Using another law of logic (this one much less intuitive), we can deduce from our rule that the sums of the digits of powers of 2 (e.g., 32, 64, 128) will never be multiples of 3 because powers of 2 are not multiples of 3. To see this, we note that "if the sum of the digits of a number is a multiple of 3, then that number is a multiple of 3" implies "if a number is not a multiple of 3 then the sum of the digits of a number does not add to a multiple of 3". Since powers of 2 are not multiples of 3, then the sum of the digits of a power of 2 is also not a multiple of 3. So it is possible to use the laws of logic on our rule and then apply deduction.

With a mathematical result, we can always be sure of the conclusion obtained by deduction from a true premise with a true rule. So, deduction is used, in chains of reasoning, to prove results/theorems (that may have been obtained initially by induction), and it is used to support our thinking in many aspects of our day to day experience. A special characteristic of mathematics is that all results follow from stated assumptions by deductive reasoning: of course, the chains of reasoning are sometimes immensely long.

Abduction is a third type of reasoning, in everyday life and mathematics. This is commonly described as a way of reasoning by detectives. In this reasoning, a result plus a rule lead to a case. Grand-daughter Alyssa notices that Coco has a treat. She knows the rule "If Coco steals an object, then Coco will get a treat" and reasons abductively "Coco has just stolen an object".

We show the general abductive structure in Figure 3. As before, the boxes with solid edges are known and the dotted box is the one that is abducted from the other two.

In mathematics you might have a rhombus that you want to show is a square. So the result you want is "this rhombus is a square". But you know the rule: "If a rhombus has a right angle then it is a square." So the case that you abduct is, "the rhombus has a right angle". This now gives a direction to the investigation: you can now try to show that the rhombus you have does have a right angle, so then it is a square.

In general, then, abduction works like this. You want to cause or explain a particular phenomenon (result) in the presence of a rule. What general idea will do this? You therefore guess the something (a case) that might make it happen. So, the abducted case gives you a direction to approach your proof of the result you want--it produces an idea that can be tested. This is an important creative function of abduction, both in and out of mathematics.

As with induction, there is no guarantee that the abducted guess is true. For instance, Alyssa was not around when Coco performed a different trick for Alex, and was rewarded with a treat on this occasion also.

Reasoning in mathematics

These three types of reasoning are fundamental to mathematics and are used continually in problem solving situations by students in schools and by research mathematicians. However, they are not all to be found in great abundance or emphasis in text books or in much of classroom work. The most common form of reasoning used in texts and in explanations by teachers is induction. For example, if a method for solving simultaneous equations is being taught, then several examples lead to the statement of a solution technique, but generally that technique is left unproved. When students are working on text book exercises, they are applying the method that they have just been taught. While they may initially use deduction to justify each step, the method soon becomes routine and the individual steps may no longer require any conscious thought or reasoning.

It is only when some openness comes into students' work or they get stuck when solving a problem, that students need induction or abduction. Induction is useful in finding patterns and so is likely to be used in problems like matchstick problems where they want to know how many matchsticks are needed to make a given number of squares, say. Students may use it in any situation where they are looking for patterns. On the other hand, abduction is useful at sticking points because students know the result that they want and they know the rules they have but they may not know in what direction to go. However, students are generally given few opportunities to use induction or abduction creatively in mathematics classes.

Conclusions

The examples above suggest that some animals, such as dogs, appear to use the three kinds of reasoning: induction, deduction and abduction. What's more, small children also go through similar routines for attention or treats, in the way that Coco did. So we know that the basic elements of reasoning in everyday situations are well in place at the beginning of school (Hollister Sandberg & McCullough, 1999). Indeed they assist learning. However, there are many aspects of using reasoning in mathematics that need careful attention.

In this paper we have considered examples of three types of reasoning. We have noted that deduction is the reasoning that establishes the truth of important mathematical theorems such as Pythagoras' Theorem; abduction generates ideas that might enable proofs to be found or problems to be solved; and induction is a useful conjecture-producing tool. Neither abduction nor induction can be guaranteed to produce true results. All three processes--abduction, induction and deduction--are creative, and help us to find new mathematical results.

It is valuable for students to understand that these three types of reasoning exist even if they don't know their names. This is because it enables them to think about ways to justify results, ways to produce conjectures, and ways to produce ideas that may explain things. It is also important in mathematics that students understand the logical gap between induction and deduction; induction--observing a pattern in some examples and inducting that it might always hold, and deduction--proving that the pattern really works in all cases. Many teachers find that students are more likely to remember a result if they have explored specific cases in advance and guessed at the rule, rather than simply being shown it. However, as students progress through school, they need to become increasingly aware that this induction process is partly how mathematics is created, but not how mathematical truth is established.

In recent years, the process of doing mathematics has become of greater importance in national curricula. While content is still absolutely important it is now recognised that mathematics has another aspect. So we find problem solving and reasoning to be key aspects in the curricula of Australia (ACARA, 2012), England and Wales (DEE & QCA, 1999), the EU (EACEA, 2011) , New Zealand (Ministry of Education, 1992) and the USA (NCTM, 2012). In this article, we have tried to underline the importance of reasoning in mathematics, and to offer insights into its various aspects, with examples of how it works in everyday and mathematical practices.

References

Australian Curriculum, Assessment & Reporting Authority [ACARA] (2012). Retrieved from http://www.australiancurriculum.edu.au/Mathematics/Content-structure

Education, Audiovisual & Culture Executive Agency [EACEA]. (2011). Mathematics in education in Europe: Common challenges and national policies. Brussels: Author. Retrieved from http://eacea.ec.europa.eu/education/eurydice

Hollister Sandberg, E. & McCullough, M. B. (2009). The development of reasoning skills. In E. Hollister Sandberg & B. L. Spritz (Eds), A clinician's guide to normal cognitive development in childhood (pp. 179-198). London: Routledge.

Department for Education and Employment, Qualifications and Curriculum Authority. (1999). Mathematics: The National Curriculum for England (Key Stages 1^1). London:. Author. Retrieved from https://www.education.gov.uk/publications/eOrderingDownload/QCA-99 460.pdf

Ministry of Education. (1992). Mathematics in the New Zealand curriculum. Wellington: Author. Retrieved from http://www.minedu.govt.nz/~/media/MinEdu/Files/EducationSectors/ Schools/MathematicsInTheNZCurriculum.pdf

National Council of Teachers of Mathematics [NCTM] (2012). Process standards. Retrieved from http://www.nctm.org/standards/content.aspx?id=322

Reid, D. A. & Knipping, C. (2010). Proof in mathematics education: Research, learning and teaching. Rotterdam: Sense.

Derek Holton, Kaye Stacey & Gail FitzSimons

University of Melbourne

<[email protected]>

(1) Of course, we do not know if Coco or any other dog really has the thoughts attributed to Coco here. However, Coco certainly acted as if she was reasoning this way.