The Co-Development of the Idea of Proof and Students’ Sense of

Authority

Michael N. FriedProgram for Science and Technology Education,

Ben Gurion University of the Negevmfried@bgu.ac.il

Basic thesis:

• The emergence of students’ sense of what proof means runs parallel to their developing sense of authority in a mathematical community.

Structure of the Talk

• Part I: The idea of authority in general and authority in two 8th grade classrooms

• Part II: A micro-analysis showing the interplay between a sense of authority and the idea of proof.

Starting Points: 1

• Amit & Fried (2002, 2005) on authority:– Demonstrated the prevalence of authority

relations in students’ mathematical worlds– Suggested that “...by shifting the emphasis

from domination and obedience to negotiation and consent...” these relations are fluid and are, in fact, a sine qua non in the process of students’ defining their place in a mathematical community.

• Amit &Fried (2005) Authority and Authority Relations in Mathematics Education: A View from an 8th Grade Classroom. Educational Studies in Mathematics, 58, 145-168

Starting Points: 2

• Studies of proof and justification within and outside mathematics education:– Empirical tendency in mathematical

justification (e.g. Chazan, 1993; Fischbein, 1982)

– Need for empirical evidence not appreciated in scientific justification (e.g. Kuhn, 2001)

• The inconsistency suggests: – Our own sense of students’ understanding of proof

and, more importantly, that students’ own understandings of proof may be highly dependent on context.

– In students’ minds such understandings are much less fixed, defined, and univocal than characterizations as ‘empirical’ or ‘explanatory’ might imply.

– And, therefore, that these understandings are likely to be continually shaped by circumstances and interactions.

Starting Points: 3

• Sociomathematical norms (Yackel & Cobb, 1996)

• Harel & Sowder’s “authoritative proof scheme” (1998)

• Boaler’s “Dance of agency” (2003)

– Further reinforced the possibility that the formation of a notion of proof may coincide with a developing sense of agency and of the dispensation of authority

Starting Points:4

• Learners’ Perspective Study (LPS)– Focus on happenings, notions, practices in

mathematics classrooms from the students’ point of view.

The LPS program

• International effort involving twelve countries (Clarke, 2001).

• Expands on the TIMSS video study: instead of examining exclusively teachers and only one lesson per teacher (see Stigler & Hiebert, 1999), the LPS focuses on student actions within the context of whole-class mathematics practice

• It adopts a methodology whereby student reconstructions and reflections are considered in a substantial number of videotaped mathematics lessons.

LPS Classroom 1

• LPS classroom 1 was taught by a dedicated and experienced teacher, whom we shall call Danit.

• Danit’s 8th grade class is heterogeneous and comprises 38 students, mostly native born Israelis, but also new immigrants from the former Soviet Union and one new immigrant from Ethiopia.

• All lessons observed concerned the representation and solution of systems of linear equation.

LPS Classroom 2:

• LPS classrom 2 was taught by a new immigrant teacher, Sasha, from the former Soviet Union.

• Sasha has a strong mathematics background, has several years’ experience teaching in Israeli schools and much experience teaching mathematics in Russian schools.

• His 8th grade class is a high-level class and comprises 30 students.

• All lessons observed concerned geometry and, therefore, were particularly appropriate for examining students’ ideas of proof.

Part I

What is authority in theory and what are its manifestations in the LPS

classrooms?(Amit & Fried, 2005)

Authority

WeberianAuthority

Power

Classical treatmentbased on

LegalTraditionalCharismatic

Authority in Educational Contexts

Includes

Where flexible, responsive, and dependent on the will of the learner

Includes

Expert Authority

Shared Authority

Where obedience is based on a recognition of legitimacy

Cooperative learning & constructivist-pedagogies

Necessary for

Information, guidance

Provides

Instruction

Attenuates the role of

Teachers Replaced by

Discipline of mathematics: community of practitioners of mathematics

Non-localized authority &Benne’s Anthropogogic Authority

Leads ultimately to

Exemplified bySource of

Source of

Provides

Classical Scheme:Max Weber

Legal Authority

Traditional Authority

Charismatic Authority

Weber (1921/22): Wirtschaft und Gesellschaft (Trans.: Henderson & Parsons (1947), The Theory of Social and Economic Organization).

Five Theoretical Points on Authority

1.Authority involves obedience, but, unlike mere power, obedience to authority is based on a recognition of the legitimacy of the person or group holding authority. Authority relations thus say as much about the subjects of authority as about the agents of authority.

Power Authority

Ceasar Augustus:

After that time, I exceeded all in authority but had no more power than others who were also with me as colleagues in the magistracy.

Post id tempus auctoritate omnibus praestiti; potestatis autem nihilo amplius habui quam ceteri qui mihi quoque in magistratu conlegae fuerunt. (Res Gestae §34)

2.In educational contexts, a key species of authority is expert authority. Knowledge is the ground of the expert authority’s legitimacy. Expert authorities are sources of information and guidance; one turns to them for instruction.

3.Authority can be responsive and flexible and to some extent dependent on the will of those over whom it has authority. This opens the possibility of shared authority.

4.Shared authority, or at least, the attenuation of teachers’ authority, is necessary in cooperative and constructivist-pedagogies. Such pedagogies assume, however, that the reduced authority of teachers is replaced by increased authority of the discipline of mathematics.

5. The authority of the discipline of mathematics is not the authority of mathematics itself but of the community of practitioners of mathematics (or those who see themselves as part of this community) within itself. This kind of non-localized authority is the final expression of shared authority.

In the classroom

Teachers, friends, parents and siblings form a web of authority; when one is unavailable students turn to another. For example, if Yara in Sasha’s class cannot get help from Sasha, she turns to one of her friends:

Interviewer: And if your friend doesn’t know?...Yara: [If my friend doesn’t know], I ask someone else…I can go to my parents, my father...

: אם החברה לא יודעת את התשובה, מה אז? מה מאתם עושים?

: מסבירים לה. יערה: לא, לא, נגיד שאת מתקשרת לפנינה, כן, ואת שואלת מ

ואת אומרת לא הבנתי איך לעשות את שאלה הזאת, והיא אומרת, אני לא יודעת.

: לא יודעת איך להסביר?יערה: כן, היא לא יודעת את התשובה. מ

: אז אפשר להתקשר למישהו אחר. יערה: כן. מ

: ואם לא אפשר לגשת להורים, אבא שלי ולשאול יערהאותו איך עושים.

The authority of the teacher is supreme—and it is not only expert but charismatic authority

• Consider the following dialogue with two of Danit’s students, Michael and Saul, as to whether the teacher can ever err:

Michael: If I get a answer for one and a different answer for the other, then you’ve got to check. If I get the same answer, then I’ll believe it’s correct. But if there’s, maybe, still some doubt in my mind, I ask Danit.

Interviewer: What does Danit have that other people don’t?Michael : She’s a teacher, she can help; if you make a mistake,

she corrects it!Interviewer: And if she errs?Michael : She doesn’t err.Saul: She studies everything at home before she comes to

class.Michael : Otherwise she couldn’t correct—she’s a teacher!Interviewer: But she did make a mistake at the board.Saul: She got mixed up because she substituted wrong.Michael : Those are nonsense things she gets mixed up about,

but real things [gestures to show the weightiness of the things he has in mind]—if two exercises are supposed to get the same answer or not, it doesn’t seem to me she’d get mixed up about that.

מה זה שיש בדנית שאין במישהו אחר?:מ היא מורה. :שאול

// היא מורה, היא יכולה לעזור, אם יש טעות אז היא מתקנת. :מיכאלואם היא טועה? :מ

היא לא טועה :שאול[צוחק] :מיכאל

היא לא טועה? :מאני לא יודע. :מיכאלהיא לומדת בבית את החומר קודם לפני שהיא מלמדת :שאול

אותנו. דקות(31)כדי לתקן בגלל זה יש מורה. :מיכאל

היום תיקנו אותה על הלוח. :מלא, היא התבלבלה בגלל שהציבה לא נכון. :שאול

זה דברים שטותיים שהיא מתבלבלת, אבל בדברים :מיכאלממש...לא נראה לי שהיא תתבלבל.

The teacher’s authority extends to other authorities

Asked whether a salesman who explained to customers how much they should pay given such and such a discount had to show his/her work, Ben, from Danit’s class, replied:

“I can rely (somech) on him--for sure lots of people come to him and saw 18 percent—there must be ones who know that stuff, know how much that is, and they rely on him, so I can rely on him too. Why shouldn’t I rely on him?”

18: אני סומך עליו, בטח באו מלא אנשים וראו בןאחוז, יש כאלה שיודעים את הדברים, יודעים כבר כמה זה והם סומכים עליו, אני גם סומך

דקות(31)עליו. למה לא לסמוך עליו?

Most remarkably, and significantly, students related to other students as authorities ad hoc

Some conclusions:

• Authorities are everywhere for students—and these authorities are typically other people than themselves.

• This can get in the way of their ability to reflect and think for themselves.

• This can also get in the way of their ability to find their own place in a community of thinkers—of mathematical thinkers.

This does not mean we should try and get rid of authority

• Authority has many forms • Authority is not only a matter of obedience but

also of negotiation and finding one’s place in a community.

• And although authority relations in the classroom may appear fixed (even between the teacher and student), they are most likely in the process of formation. We need to recognize that authority relations are fluid and dynamic.

Kenneth Benne’s idea of anthropogic authority (Benne, 1970)

“The ultimate bearer of educational authority is a community life in which its subjects are seeking fuller and more valid membership. Actual bearers and subjects of this authority must together build a proximate set of mutual relationships in which the aim is the development of skills, knowledges, values, and commitments which will enable the subjects to function more fully and adequately as participants in a wider community life which lies beyond the proximate educational associations” (p.401).

Benne, Kenneth D. (1970). Authority in Education. Harvard Educational Review, 40, 385-410

Part II

Co-Development of the sense of authority and the idea of proof

That the sense of authority is dynamic and continually being formed is given plausibility and a significant context simultaneously, namely, in the emergence of students’ idea of proof.

Conversation with Yana and Ronit about proof

• Yana and Ronit are from Sasha’s class.• They are very bright, spirited, and talkative girls• They are very good friends: they are generally

attentive to one another but also allow one another the independence to disagree and qualify one another’s remarks.

• These characteristics are evident in their discourse style and typify their own interactions.

The conversation The opening

Interviewer: [38 min] Tell me now, are there also proofs in the book [the workbook], things you have to prove?

Ronit: To prove?Interviewer: Yes.Ronit: Umm.Interviewer: Did you meet up with something you

had to prove yourself? Yana: There are exercises here, what do you

mean? I don’t understand, like, prove what...like what was on the board?

: תגידו רגע, יש פה גם הוכחות בספר כי יש מדברים שאתם צריכים להוכיח? יש דברים

שאתם צריכים להוכיח בספר?

: להוכיח?רונית

: כן. מ

: אממ... רונית

: נתקלתם במשהו שאתן צריכות לבד להוכיח? מ

: כן, יש פה תרגילים, מה זאת אומרת? לא יאנההבנתי. כאילו, להוכיח מה ש-, כמו שהיה הרגע

על הלוח?

Definition d0:“Proof is saying whether something

is correct or incorrect & explaining what you say.”

Ronit: [Referring to interviewer’s question above] Like correct and not correct.

Interviewer: Yes?

Ronit: But you have to write if it is correct and not correct and to prove why this is correct and why this is not correct.

Yana: Explain why you say [what you say].

: כמו הנכון ולא נכון. רונית

: כן?מ

: אבל צריך לכתוב אם זה נכון או לא נכון רוניתולהוכיח למה זה נכון או למה זה לא נכון.

: להסביר למה אתה אומר... יאנה

T / W: They and We

• Ronit: “What you have to write…” i.e. What the teacher or book tells you to write…THEY

• Yana: “Explain why you say” WE

Definition d1:“To argue is saying why you think

something; to prove is showing how something is supported by what you have already learned.”

Interviewer: Is ‘to argue’ and ‘to prove’ the same thing?Yana: Uh...depends on the caseRonit: No, ‘to argue’ is to say why you think this way. Yana://No, it depends... [~39.5 min.] Interviewer: Hold on, Yana. Roni [indicating to her to go on].Ronit: ‘To argue’...Yana: All right [laughs]Interviewer: No, it’s ok, yes.Ronit: ‘To argue’ is to say why you think that way [emphasis

added], and ‘to prove’ is, umm, to find something to support what you say.

Yana: Something that [you] already...Ronit: Something existing, something you already learned

[emphasis added].

: האם לנמק ולהוכיח זה אותו דבר? מ: אה... תלוי באיזה מקרה. יאנה: לא, לנמק זה להגיד למה אתה חושב ככה, רונית: // לא, זה תלוי... יאנה

: רגע, יאנה, רונית. מ: לנמק, רונית: טוב. [צוחקת] יאנה

: לא, זה בסדר, כן. מ: לנמק זה להגיד למה אתה חושב ככה ולהוכיח זה רונית

אממ, למצוא משהו שיתמוך בדברים שלך. : משהו שכבר...יאנה: משהו קיים, משהו שכבר למדת. רונית

T / W

What is proved rests on what has been learned that is, what came from an external authority. [THEY]

What is argued rest on what you yourself think. [WE]

The ‘learning paradox’ and subsequent shift from T / W to W

Yana: [40 min.] But if you try to prove something new? Then that’s not something that’s written...

Ronit: Yes.Interviewer: I don’t understand.Yana: No, if you want to prove something new,

like, that no one’s ever proven before, then that can’t be written, so...I don’t know

Interviewer: That is, what you are doing then is...?Ronit: You [4 secs. pause] prove. [Ronit and Yana

laugh]

: אבל אם את מנסה להוכיח משהו חדש? אז יאנהזה לא משהו שכתוב.

: כן.רונית

: לא הבנתי. מ

: לא, אם היא מנסה להוכיח משהו חדש יאנהכאילו שאף אחד עוד לא הוכיח קודם, אז זה לא

מהכתוב. אז... לא יודעת.

: יכולים להוכיח מידע קודם שלך. רונית

: כלומר, מה אתה עושה אז? מ

שניות) מוכיח. 4: אתה (רונית

Definition d3:“Arguing is the same as proving.”

Yana: ...For me, I don’t know what the difference is between an argument and a proof.

Interviewer: Any conjecture, then?

Yana: If you write for me ‘argue’ or ‘prove’, I will write the same thing.

Interviewer: [~51.5 min.] The same thing?

Yana: Yes

: אה, כן. נראה לי, לא יודעת. אני מבחינתי יאנהלא יודעת מה ההבדל בין נימוק והוכחה.

: יש לכן איזו השערה? מ

: אם יכתבו לי נמק או הוכח אני אכתוב אותו יאנהדבר.

: אותו דבר? מ

: כן. יאנה

With definition d1 in mind, the distinction between THEY and WE is swept away by definition d3.

Are we witnessing progress?Have Yana and Ronit discovered their own agency and authority

and, therefore, no longer in need of THEY?

No! d1 and T / W return in force!

Yana: The argument is your opinion, what you think, and the proof is...Roni://That is what I think [what I do]Yana: And the proof is what they write? Like, what others write?Ronit: No, in fact when you are asked why you think that way, so,

umm...Yana: You are not asked why you think that way, they tell you, argue [!]Ronit: Come on, that’s the same thing. So in fact when you are asked

you answer, umm, you think this way because of what you have learned, I think. So, it comes out the same thing since in proof you write what you’ve learned before. [54 min.]

Yana: No, for an argument you write, like, what you say [i.e., what you mean]—that for an argument, that you think this way because of what you have learned and in a proof you write what you have learned...that’s what I understood. [Both laugh]

: הנימוק זה דעה שלך, מה שאת חושבת והוכחה זה...יאנה: // זה למה אני חושבת. רונית: והוכחה זה מה שהם כתבו? כאילו מה שאחרים כתבו? יאנה: לא, בעצם כששואלים אותך למה את חושבת ככה אז רונית

אממ... : לא שואלים אותך למה את חושבת ככה, הם שואלים אותך יאנה

נמקי. : נו זה אותו דבר. אז בעצם כששואלים אותך את תעני, רונית

אממ, את חושבת ככה בגלל מה שלמדת, אני חושבת. אז זה יוצא אותו דבר כי גם בהוכחה את רושמת את מה שלמדת

דקות(54 )לפני זה.: לא, בנימוק את כותבת, כאילו מה שאת אמרת זה שבנימוק יאנה

שאת חושבת ככה בגלל מה שלמדת ובהוכחה את כותבת את מה שלמדת. ... זה מה שהבנתי.

: איך את הבנת את זה ? מ: [צוחקות] יאנה + רונית

Yana and Ronit’s discussion of proof was inseparable from concern with THEY and WE.

The discussion went back and forth, like a dance…

…perhaps, like Boaler’s (following A. Pickering) ‘Dance of agency’

Summing up

On the empirical level, • Authority relations are ubiquitous in the 8th grade

mathematics classroom• These relations are very likely fluid and

developing• Students’ thinking about proof is connected to

authority• The developing relations of authority may go

hand in hand with developing ideas of proof

Regarding instruction,

• Teachers must still attend to logical aspects of proof: quantifiers, implication, conversion, etc.; however, they must also realized that these logical aspects are not the only aspects of proof.

• Recognizing that there can the sort of co-development of proof and authority described in this talk, teachers must be awake to moments (or find ways of encouraging them) in which that co-development comes to the fore and take the tide at the flood.