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INVESTIGATING THE DIFFERENCE BETWEEN GENERIC PROOFS AND

PURELY EMPIRICAL VERIFICATIONS

Leander Kempen (University of Paderborn, Germany)

The teaching of generic proofs has produced interest across the globe. Besides the advantages generic

proofs offer for learners, the use of concrete examples in the context of general verifications might

foster students’ misconceptions about the epistemological value of purely empirical verifications. The

study presented in this paper is about the results of a proof questionnaire exploring pre-service

teachers’ ability to distinguish different usages of examples in the context of mathematical proof. The

study shows that a specifically designed course making use of generic proofs may help students to

issue correct judgements on the epistemological value of concrete examples in the context of proving.

INTRODUCTION

The concept of generic proof has produced interest in the context of teaching and learning

mathematical proof. Its use seems fruitful both in the context of school mathematics (e.g., Karunakaran

et al., 2014) and at university level (e.g., Selden, 2012). Kempen and Biehler (2019a) highlight the

advantages of teaching generic proofs to foster first-year pre-service teachers’ proof competencies.

However, while establishing an inquiry-based transition-to-proof course at university, these authors

faced various problems concerning the incorporation of generic proofs (the complete research process

is reported in (Kempen, 2019)). In their own proof attempts, some students seemed to mix up purely

empirical verifications with generic proofs. Accordingly, the authors decided to investigate the open

question, if students were relying on popular misconceptions concerning the epistemic value of

concrete examples in the context of proving. This phenomenon would hinder students to grasp the

concept of generic proofs. To answer this question, a proof questionnaire was constructed, where

students were asked to rate the quality of different kinds of given “generic proofs”, including different

usages of examples, ranging from a single test of a proposition to general valid generic proofs

containing generic examples. In this paper, we will report on the results of part of this proof

questionnaire. Finally, we will discuss first-year pre-service teachers’ abilities to evaluate the usage of

examples in the context of proving.

THEORETICAL BACKGROUND

Since Mason and Pimm’s (1998) discussion about generic examples, the concept of generic proof has

gained international attention. As Kempen and Biehler (2019a) stress, a generic proof consists of a

presentation of generic examples and a corresponding argument that verifies the given claim in general.

These generic examples display a general (“generic”) scheme that can be used to verify the given

claim. It is this general overall scheme that makes the generic examples go beyond purely empirical

evidence. (An example of a complete generic proof is given below.) This difference is very important

as generic proofs provide a sufficient argument for a claim, but empirical evidence by one or several

examples does not.

Kempen, L. (2021, July). Investigating the Difference between Generic Proofs and

Purely Empirical Verifications [Paper presentation]. 14th International Congress on

Mathematics Education (ICME-14), Shanghai, China.

Kempen

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Martin and Harel (1989), however, showed that more than half of the pre-service elementary teachers

in their study accepted an inductive argument as a valid mathematical proof in the context of a familiar

and an unfamiliar statement. We took the study of Martin and Harel as a basis for our research interest

and transferred their study in the context of our learning scenario.

Research Question

As pointed out above, we are interested in students’ ability to distinguish the usages of examples in

the context of empirical arguments and generic proofs. Moreover, we aimed to investigate if the

students overestimate the epistemological value of purely empirical verifications. Accordingly, our

research question was: “How do pre-service teachers rate different kinds of reasoning involving

different usages of examples in a familiar and an unfamiliar context?”.

METHODOLOGY

In winter term 2012/13, we constructed a proof questionnaire based on Martin and Harel’s (1989)

work. We adopted their claim used in the unfamiliar context (“For all 𝑎, 𝑏, 𝑐 ∈ ℕ: If 𝑎 divides 𝑏, and

𝑏 divides 𝑐, then 𝑎 divides 𝑐.”) and used the following claim for the familiar statement: “The sum of

an odd number and its double is always odd”. We formulated different so-called “generic proofs” for

these claims relying on Martin and Harel’s (1989) different types of purely inductive reasoning.

“Examples” implies the single check of a given claim by testing two particular instances. “Pattern”

indicates a chart containing a sequence of examples supporting a claim. “Big number” makes use of

large numbers and is meant to represent an “arbitrary chosen” number. In “Example and non-example”

a single example is given supporting the claim, but afterwards, a non-example is presented in the sense

that if the conditions do not hold, neither does the conclusion. Finally, we added different kinds of

“generic proofs” that differed concerning the quality of explication of the generic argument: (i)

“complete generic proof” in the sense that the argument given links the given premise with the final

conclusion, (ii) “incomplete generic proof” and (iii) “paraphrase” (an example for each type will be

given below). In the proof questionnaire, all kinds of reasoning were called “generic proof” to avoid

giving hints. In the following, we will specify some items of the questionnaire, concerning the familiar

statement.

“Generic proof (a)” [a correct and complete generic proof]

9 + 2 ∙ 9 = 9 + 18 =27, 13 +2∙13 =13 +26 =39, 17 +2∙17 =17 +34 =51

The following principle can be detected: Since two times an odd number is always even, one always

obtains the sum of an odd number (the initial number) and an even number (“its double”). Since the

sum of an odd and an even number is always odd, the result must be an odd number. Accordingly, the

claim is true for any odd number.

“Generic proof (b)” [a generic proof, where the argument is not completed]

9 + 2 ∙ 9 = 3 ∙ 9, 13 +2∙13 = 3 ∙ 13, 17 + 2 ∙ 17 = 3 ∙ 17

The following principle can be detected: If one adds an odd number and its double, the result must

always be three times the initial number. Accordingly, the claim is true for any odd number.

“Generic proof (c)” [“Paraphrase”: a wrong generic proof based on a redundant explanation]

Last names of the authors in the order as on the paper

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9 + 2 ∙ 9 = 9 + 18 =27, 13 +2∙13 =13 +26 =39, 17 +2∙17 =17 +34 =51

The following principle can be detected: If one adds an odd number and its double, the sum is always

odd. This is true for any natural number. Accordingly, the claim is true for any odd number.

“Generic proof (d)” [example and non-example]

9 + 2 ∙ 9 = 9 + 18 =27 is odd, 4 + 2 ∙ 4 = 4 + 8 = 12 is even

The following principle can be detected: If one adds an odd number and its double, the sum is always

odd. But if one adds an even number and its double, the sum is always even. Accordingly, the claim

is true for any odd number.

For both statements seven so-called “generic proofs” were given in the questionnaire in an arbitrary

order. The students were asked to rate the quality of each of the given 14 “generic proofs” on a five

point Likert scale ([1] insufficient … [5] very good) and to write a short comment to explain their

choice. The study was conducted in the second to last session of the course in winter term 2012/13.

The students (n=94; 74% female; 77% first-years) had one hour to work on the questionnaire.

RESULTS

We summarize students’ ratings as follows: The ratings [1] and [2] on the Likert scale are taken

together as “negative”, rating [3] corresponds to “neutral” and [4] and [5] are combined as “positive”.

The corresponding results are given in table 1.

Familiar statement

Unfamiliar statement

negative

neutral

positive

negative

neutral

positive

inductive reasoning

Examples

90.3

6.5

3.2

78.3

18.5

3.3

Big Number

89.4

8.5

2.1

90.2

7.6

2.2

Pattern

74.2

15.1

10.8

75.3

18.3

6.5

Non-example

66.3

22.5

11.2

83.5

14.3

2.2

generic ‚proofs‘

Paraphrase

62.8

20.2

17.0

35.2

37.5

27.3

Generic proof (incompl.)

20.2

39.3

40.4

20.7

31.5

47.8

Generic proof (compl.)

18.3

20.4

61.3

6.5

25.0

68.5

Table 1: Percentage of students’ ratings of inductive reasonings and generic ‘proofs’ (n=94).

The numbers for the inductive reasonings (“Example”, “Big number”, “Pattern” and “Non-example”),

show that the vast majority of students give negative rating to these kinds of reasoning both in the

familiar and in the unfamiliar context. Concerning the given ‘generic proofs’, the wrong generic proofs

containing only a paraphrase of the given claim is considered as “negative” by 62.8% in the familiar

context but only by 35.2% in the unfamiliar context. Accordingly, this redundant argument seems to

raise the quality of the ‘proof’ in the eye of the students, especially in the unknown context. (This

result is in line with the given norms in the lecture that for a generic proof, a narrative argument has

to be added to generic examples. In this sense, the reasonings categorized as “paraphrase” are indeed

better ‘generic proofs’ than the former inductive ones.) The incomplete generic proof was rated

considerably higher than the “paraphrase”, both in the familiar context (“positive”: 40.4% vs. 17.0%)

and in the unfamiliar context (“positive”: 47.8% vs. 27.3%). The complete generic proof obtains the

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highest percentage of positive ratings (known context: 61.3%; unknown context: 68.5%). However,

the results for the generic proofs were lower than expected, considering that these proofs indeed are

valid and complete generic proofs.

DISCUSSION AND FINAL REMARKS

First, we would like to be clear that we do not know how the students understood the given reasonings

in detail. (Due to the size of this paper, we do not report on the comments students gave. However,

these comments hardly explain their interpretation of the given reasonings.) For example, a reader of

the reasoning “pattern” might detect an overall general scheme in the listed examples and accept it as

an incomplete part of a valid generic proof. However, the results do not support such interpretation.

To sum up, the majority of the students did not consider the inductive reasonings to be valid

mathematical proofs. In this sense, they do not mix up the idea of generic proofs with purely empirical

verifications. The given ‘generic proofs’ achieve better ratings according to their quality (“paraphrase”

– “incomplete” – “complete”). Accordingly, students seem to realize the aspect of valid arguments and

complete argumentations. This result supports our claim that our specifically designed course may

help students to issue correct judgements on the epistemological value of concrete examples in the

context of proving and works against corresponding popular misconceptions. However, the complete

generic proofs did not achieve an agreement higher than 70%. This phenomenon points to the concept

of “proof acceptance” we elaborate on in Kempen and Biehler (2019b, p. 31). There, proof acceptance

is conceptualized as “the extent to which an individual perceives verification, conviction and

explanation when reading a mathematical proof combined with the extent, the reader does consider

the reasoning to be correct mathematical proof”.

References

Karunakaran, S., Freeburn, B., Konuk, N., & Arbaugh, F. (2014). Improving preservice secondary mathematics

teachers’ capability with generic example proofs. Mathematics Teacher Educator, 2(2), 158-170. Retrieved

from http://www.nctm.org/publications/article.aspx?id=40725. (last accessed 11.09.2019)

Kempen, L. (2019). Begründen und Beweisen im Übergang von der Schule zur Hochschule. Theoretische

Begründung, Weiterentwicklung und Evaluation einer universitären Erstsemesterveranstaltung unter der

Perspektive der doppelten Diskontinuität. Wiesbaden: Springer Spektrum.

Kempen, L., & Biehler, R. (2019a). Fostering first-year pre-service teachers’ proof competencies. Zentralblatt

für Didaktik der Mathematik. Retrieved from https://link.springer.com/article/10.1007/s11858-019-01035-

x. (last accessed 11.09.2019)

Kempen, L., & Biehler, R. (2019b). Pre-service teachers’ benefits from an inquiry-based transition-to-proof

course with a focus on generic proofs. International Journal of Research in Undergraduate Mathematics

Education, 5(1), 27-55. Retrieved from https://doi.org/10.1007/s40753-018-0082-9. doi:10.1007/s40753-

018-0082-9. (last accessed 11.09.2019)

Mason, J., & Pimm, D. (1984). Generic examples: Seeing the general in the particular. Educational Studies in

Mathematics, 15(3), 277-289.

Martin, W. G., & Harel, G. (1989). Proof frames of preservice elementary teachers. Journal for Research in

Mathematics Education, 20, 41-51. Retrieved from http://www.jstor.org/stable/10.2307/749097. (last

accessed 11.09.2019)

Selden, A. (2012). Transitions and proof and proving at tertiary level. In G. Hanna & M. de Villiers (Eds.),

Proof and Proving in Mathematics Education: The 19th ICMI Study (pp. 391-422). Heidelberg: Springer

Science + Business Media.