Students' Epistemology on Proof in Mathematics Education

Students’ epistemology on proof:

What do we want students to believe

and how should we measure it?

Keith Weber

Rutgers University

Outline of lectures

•

I am a mathematics educator who specializes in how

undergraduates and mathematicians learn and use proof in their

practice and in the classroom.

•

I’ve recently engaged in philosophical work in how proofs are

written, read, and understood in logic, set theory, and recursion

theory.

•

My goal is to give a survey of mathematics education research on

proof, focusing at the university level, and draw themes that you

might want to consider as you bring automated theorem proving to

the classroom.

Caveats

•

I have absolutely no expertise in automated theorem proving.

•

There is a substantial degree of heterogeneity in mathematics

education research, especially on theoretical issues related to

proving.

•

There are biases and orientations that depend a lot on the region

where the researchers reside.

–

Everything I say can be interpreted with the qualifier, “at least in the

United States”.

Outline of lectures

•

Lecture 1: Mathematicians’ and students’ epistemology with proof.

•

Lecture 2: How are proofs read and understood?

•

Lecture 3: The use of diagrams and instructions in proofs

Today’s lecture

•

Mathematics educators’ focus on epistemology

•

How do mathematicians obtain conviction?

–

And what is the key role that proof plays?

•

(Time permitting) Why don’t undergraduates justify claims

deductively?

Mathematics education and their initial

reluctance to investigate proof

•

In the 1980s, there was a big interest in how students and

mathematicians solved (non-routine) problems.

–

Why can’t students marshal the facts and skills that they have to

solve problems? How can we teach them so they do so better?

•

Proof played an ancillary role in all of this:

–

A proper solution to some problems involved a proof that the answer

was correct.

–

A proving task itself can be viewed as a special type of problem.

Mathematics education and their initial

reluctance to investigate proof

•

In the 1990s, there was a shift away from proof.

•

In the United States, the National Council for the Teaching of

Mathematics put out their 1989 

Principles of School Mathematics

,

a comprehensive volume on what and how mathematics should

be taught in the classroom.

•

Proof was virtually absent. The word “proof” only appeared four

times in the document. To the extent proof was mentioned, it was

to say proof should receive 

less

 emphasis in secondary geometry.

Mathematics education and their initial

reluctance to investigate proof

•

Mathematics educators’ views of proof were influenced on how

proof had been treated in the school curriculum: two-column

proofs in geometry, verification of trigonometric identities, and

(caricatures of) the “New Math” curricula in the 1960s, which

emphasized set theory and logic.

•

Proof was viewed by some as:

–

A formal symbolic object based on rules of logic

–

Banned various non-logical types of sense making, including the use

of examples and pictures

–

An object whose correctness and permissibility was based on form,

not conceptual content

–

An object that purportedly objectively “guaranteed” the truth of a

proposition

A rebirth of proof

•

The downplaying of proof was met with much criticism from

mathematicians (e.g., Wu, 1996) and mathematics educators (e.g.,

Hanna, 1995, Schoenfeld, 1994; also Balacheff and Dreyfus).

•

A key criticism was that the description of proof– as a sequence of

symbols that can be mechanically checked– did not accord with what

mathematicians did. Proof was a tool for gaining conviction and

understanding.

–

Proof didn’t squash creativity and autonomy. If taught correctly, proof can

enable it (Hanna, 1995).

•

By 2000, the NCTM reconsidered and considered proof as a central

practice that should be taught to all students. But what was counted on a

a proof for young students was expanded.

Knuth, E. (2000). The rebirth of proof in school mathematics in the United States. 

Lettre de la

Preuva. 

Mai/Jun, 2000.

A broader conception of proof for

students

•

Justifications using informal natural language, or involving

appeals to visual diagrams or kinesthetic actions, could qualify

as proofs (at least in some contexts or for some age groups).

•

There was a greater emphasis on the roles that proofs could

play, especially conviction and explanation. 

(e.g., deVilliers, 1990)

•

Proving was a social process involving debates between

students. 

(e.g., Alibert & Thomas, 1991)

•

Proof was often developed with technological tools, like dynamic

geometry software. 

(e.g., Mariotti, 2007)

•

There was a push by some to have proof emerge from students’

sense-making activity. 

(e.g., Boero et al., 1998).

Students’ difficulties with proof are

sometimes framed as an epistemological

problem

•

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“Despite this subjective definition, the goal of instruction must be

unambiguous– to gradually refine students’ current proof schemes

toward the proof scheme shared and practiced by mathematicians

today”.

•

One interpretation is that students are asked to seek and obtain

certainty in mathematical statements using the same evidence by

which mathematicians seek and obtain certainty.

My table has shifted somewhat

•

”Relativist” epistemologies still prevail, but truth is less personal and

about viability, and more social or institutional and established

through negotiation.

•

There is a greater focus on activity and participation, than on

personal cognition.

•

Personal understandings and reasoning are now augmented or

replaced by a focus on language.

•

Especially

 in the United States, there is an equity-based critical

orientation, looking at how mathematics instruction, and

mathematics as a discipline, excludes underrepresented groups.

Taking stock

and questions

•

What questions do you have about what mathematics educators

value with respect to proof?

•

How could Automated Theorem Proving allow for building conviction

and understanding, collaboration, social processes and

collaboration?

•

Can ITP allow for idiosyncratic representations in the classroom or

accommodate relativist epistemologies? Should we care if ITP

cannot?

Mathematicians’ views on

proof and certainty

Weber, K., Mejía-Ramos, J.P., & Volpe, T. (2022).The relationship

between proof and certainty in mathematical practice. 

Journal for

Research in Mathematics Education

, 53, 65-84.

A student, Emily, is shown a correct proof that 6 divides 

n

3

 – 

n

 for

every natural number 

n

. She is asked if the proof is fully correct.

She answers yes. She is then asked if additional numerical check

would be necessary to increase her confidence in the statement.

She again answered yes.

A student, Emily, is shown a correct proof that 6 divides 

n

3

 – 

n

 for

every natural number 

n

. She is asked if the proof is fully correct.

She answers yes. She is then asked if additional numerical check

would be necessary to increase her confidence in the statement.

She again answered yes.

What do you think is going on? What was Emily thinking? What

skills, understandings, or beliefs is Emily lacking?

A student, Emily, is shown a correct proof that 6 divides 

n

3

 – 

n

 for

every natural number 

n

. She is asked if the proof is fully correct.

She answers yes. She is then asked if additional numerical check

would be necessary to increase her confidence in the statement.

She again answered yes.

These questions were asked to a large number of Israeli high

school students. 81.5% of the students said the proof was correct.

Of those students, 70% desired an additional empirical check. Only

30% did not.

Fischbein, E. (1982). Intuition and proof. 

For the Learning of Mathematics

, 

3

(2), 9-

24.

Solid findings in mathematics

education

Students will not be certain that a theorem is true after reading a

proof of the theorem.

They will continue to seek empirical evidence, or appeal to an

authority, after reading the proof.

“[When mathematicians are determining if a statement is true], at a

certain moment this search process stops and a new situation

appears: 

the mathematician has found a complete proof for his

solution or theorem. Such a proof is the absolute guarantee of the

universal validity of that theorem. 

He believes in that validity”

Fischbein, E. (1982). Intuition and proof. 

For the Learning of Mathematics

, 

3

(2), 9-

24.

Caveats

•

Some mathematics educators have argued against tying proof

and certainty together.

–

deVilliers (1990): 

“proof is not necessarily a prerequisite for

conviction—to the contrary, conviction is probably far more

frequently a prerequisite for the finding of a proof” (p. 18).

–

Hanna (1983, 1991) said things like the reputation of the author and

the consistency of the theorem with known results were 

more

important to the acceptance of a theorem than the proof itself.

Methods:

Rationale

•

Studies like Fischbein’s can be viewed as a type of

“epistemological test”, where some students’ responses to a

task are taken as evidence of a deficient epistemology.

•

What would happen if we gave the same epistemological test to

mathematicians?

•

Do mathematicians gain certainty in theorems at all? If so, how

do they do it?

Methods:

Obtaining confidence from evidence

•

16 3

rd

 to 5

th

 year math Ph.D students who passed their

qualifying exams from a top 25 nationally ranked US

mathematics department.

•

In the first part of the interview, participants were asked how

confident on a scale of 0 through 100 they were in the truth of

the following identity:

–

Initially with no information

–

Then given unlimited time to study a challenging proof of the

theorem (that only used ideas from undergraduate calculus)

–

They were given access to an Excel file illustrating how the series

converged to 



 quickly.

–

Then the 

Amer Math Monthly

 article in which the theorem appeared

Methods:

Methods:

Understanding mathematicians’

Understanding mathematicians’

confidence

confidence

1.

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Results:

Task-based interview

Results:

Task-based interview

Results:

Task-based interview

•

I couldn’t be certain that the proof was correct (N=12):

“

Because I worked it out, and things checked out, 

and I gave it not

a 100 just in case there was some infinite sum trickery that I didn’t

catch

. I was pretty sure it was all okay because z stays between 0

and 1, and isn’t going to, and those sums were going to converge

pretty quickly, but just in case I gave it a 95, not 100”

Results:

Task-based interview

•

I couldn’t be certain that the proof was correct (N=12):

Even if I myself have given a proof, then I need someone else to

actually check it. Because I make mistakes here and there; all

kinds of people make mistakes here and there

. There are basically

different levels in my approval. First level is actually reading some

other’s proof. So if I simply read this piece of paper, without my

own computation, I will probably raise it to 60 or something. But

then I have made some key computations, and that will make me

raise it to 80. But then even if I have made some computations,

there might still be issues that I’m overlooking. So that’s why I’m

not going to give it 100%. Unless actually I’ve checked all these

kinds of things and then I’ve reproduced the whole thing totally by

myself. And if I manage to do that, I will probably give it a 95. Once

I reproduce everything I have to ask someone else to read the

proof and see if I have missed anything.

Results:

Task-based interview

•

I couldn’t be certain that the proof was correct (N=12):

Even if I myself have given a proof, then I need someone else to

actually check it. Because I make mistakes here and there; all

kinds of people make mistakes here and there. There are basically

different levels in my approval. First level is actually reading some

other’s proof. So if I simply read this piece of paper, without my

own computation, I will probably raise it to 60 or something. But

then I have made some key computations, and that will make me

raise it to 80. But then even if I have made some computations,

there might still be issues that I’m overlooking. So that’s why I’m

not going to give it 100%. 

Unless actually I’ve checked all these

kinds of things and then I’ve reproduced the whole thing totally by

myself. And if I manage to do that, I will probably give it a 95. Once

I reproduce everything I have to ask someone else to read the

proof and see if I have missed anything.

Results:

Task-based interview

Results:

Task-based interview

•

The Excel file complemented the proof and reduced doubts I

had about its validity (N=7):

Because you have already shown me very clearly that, I know this

is actually convergent and this actually shows me very clearly that

pi is very likely that thing.

If I only read the proof, it may have some error, and may not equal

to pi. 

It's equal to some number, but may not equal to pi.

 But if I see

the calculation I will know that it will not miss pi too much,

 so I'm

more convinced that that number is exactly pi

Results:

Task-based interview

Results:

Task-based interview

•

The publication status file reduced doubts I had about its validity

(N=9):

I trust the review process (N=7): I’m convinced because this is a

serious journal and 

somebody has refereed the paper

.

I trust the journal (N=6): So I think seeing 

a paper written in a

famous MAA journal

, that would be enough for me to believe it.

I trust the author (N=3): Because Stan Wagon’s name on it. I’ve

never met him, but I know his work. 

So I think he’s a reputable

mathematician.

Open-ended interview:

Results

•

Participants were asked to imagine that they were working on

an open conjecture with a colleague. The colleague presented a

proof of the conjecture. They read the proof and could find no

problems with it. Would they still retain some doubt that the

theorem is true.

•

15 out of the 16 participants answered yes.

“If I haven’t seen any flaws, that doesn’t mean that none exist”

“In a sufficiently difficult proof, I might not just trust my own ability to

validate it completely”.

“I might just not know enough to enough to verify everything that

my colleague does”.

Open-ended interview:

Results

So in real life I would look at the conjecture itself first, and try to run

it and try to do a computer check with the conjecture. 

In almost all

cases, I work in representation theory in abstract algebra, it

requires some effort, but it’s possible. Now once I have a certain

level of the trust of the statement itself, only then I try to produce

my own proof, and see how one can prove this. Once you gave a

sketch of the proof, then you start looking at the proof given by the

collaborator at the very last step. Now when you know that the

statement is true, when you know that there is a proof for the

statement, and approximately know how the flow of the proof goes,

you look at the particular step just in case he messed up, you

know, like something that can be fixed. That’s the way you do it.

You

 cannot do it in the opposite way. That’s a recipe for disaster.

Open-ended interview:

Results

So in real life I would look at the conjecture itself first, and try to run

it and try to do a computer check with the conjecture. In almost all

cases, I work in representation theory in abstract algebra, it

requires some effort, but it’s possible. Now once I have a certain

level of the trust of the statement itself, only then I try to produce

my own proof, and see how one can prove this

. Once you gave a

sketch of the proof, then you start looking at the proof given by the

collaborator at the very last step. 

Now when you know that the

statement is true, when you know that there is a proof for the

statement, and approximately know how the flow of the proof goes,

you look at the particular step just in case he messed up, you

know, like something that can be fixed. 

That’s the way you do it.

You

 cannot do it in the opposite way. That’s a recipe for disaster.

Open-ended interview:

Results

•

Participants were asked to imagine that they read a proof of an

open conjecture and they checked that it was correct. Would

they still want to increase their confidence by checking if the

open conjecture held for examples.

•

15 out of the 16 participants answered yes.

–

Oh sure. Yes. Definitely.

–

Definitely. I would also ask other people to check the proof.

Checking with examples is common. I try to do that always.

–

Sure, that’s usually what mathematicians do.

–

That happens all the time.

–

Of course. That’s precisely what I will do.

–

Sure. I mean if at that point we haven’t generated some examples

already, absolutely. Absolutely you go through and you check

specific examples.

Open-ended interview:

Results

I had something that ended up being really pretty at the end, and

so if something’s really pretty, it is a little surprising. 

And even

after

I had proved it and even after I knew it was pretty and everything

looked like it fit together, I still had doubts

. 

And even when I was

doing an example that I thought wasn’t in order to show that it

worked, because I thought I had already done

that, I still had these

doubts. 

And I still breathed a sigh of relief when it all worked, even

with this other example. Because there is

always some doubt in

there. 

I feel like. I mean, I know it’s true, I don’t know. I would say

that process of making 

examples to verify results that happen that

are surprising would be common.

Summary:

Main findings

•

8 participants were certain that a claim was true by the end of

the interview.

•

Only 2 participants were certain after reading the proof.

•

Of the 8 who gained certainty, 6 did so through a mix of

deductive, empirical, and testimonial evidence.

•

The majority of participants increased their confidence with

empirical evidence and testimonial evidence (publication status)

after reading the proof.

Summary:

Implications for mathematics educators

•

The claim is not that students do not have problematic views

about the relationship between proof, generality, confidence,

and certainty.

•

However, mathematicians provided coherent and compelling

rationales for making the same judgments for which students

are negatively evaluated.

•

This suggests that as a field, we need to think harder about the

relationship between proof and certainty.

Taking stock

•

What should the role of proofs written in ITPs play with regard to

conviction?

•

If they provide conviction, what is the role of (ordinary) proofs?

Reconceiving students’ use of example

based arguments

Weber, K., Lew, K., & Mejia-Ramos, J.P. (2020). Using expectancy

value theory to account for students’ mathematical justifications.

Cognition and Instruction

, 38, 27-56.

Jaime was asked to decide whether the following statement was true:

The difference between the squares of consecutive numbers is odd,

and equal to the sum of those consecutive numbers

Jaime writes:

36 – 25 = 11.

9 – 4 = 5.

So the statement is true.

Jaime was asked to decide whether the following statement was true:

The difference between the squares of consecutive numbers is odd,

and equal to the sum of those consecutive numbers

Jaime writes:

36 – 25 = 11.

9 – 4 = 5.

So the statement is true.

What do you think of Jamie’s response? What understandings,

conceptions, or beliefs (if any) does Jamie lack?

Jaime was asked to decide whether the following statement was true:

The difference between the squares of consecutive numbers is odd,

and equal to the sum of the consecutive numbers

Jaime writes:

36 – 25 = 11.

9 – 4 = 5.

So the statement is true.

40% of the 429 university students gave a response of this type.

Recio, A. M., & Godino, J. D. (2001). Institutional and personal meanings of

mathematical proof. 

Educational Studies in Mathematics

, 

48

(1), 83-99.

On ”solid findings” in mathematics

education research

“Many students provide examples when asked to prove a universal

statement” (p. 50)

“The majority of students enter high school have empirical proof

schemes” (p. 51), which students continue to evoke through college.

–

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)

The authors’ hypothesis is: “Students’ specific problems with regard to

proving are part of a more general challenge: to make a distinction

between reasoning in mathematics and reasoning in everyday life” (p.

52).

Do theorems admit exceptions? Solid findings in mathematics education on empirical

proof schemes. 

European Mathematical Society Newsletter. 

2011, 50-52.

Do empirical justifications imply

deficient epistemologies?

•

Knuth, Choppin, and Bieda (2009) asked 400 middle school

students to justify six statements.

–

The hard statement received empirical justifications 81% of the time.

–

The easy statement received empirical justifications 36% of the time.

–

“Students likely had no recourse but to examples as their means of

justification (i.e., a verbal or symbolic argument may have been too

difficult)” (p. 361).

•

Stylianides and Stylianides (2009) taught a class of preservice

teachers to recognize the limitations of empirical justifications and

to see the power and generality of deductive justifications.

–

For a justification tasks, 9 teachers still gave empirical justification

tasks.

–

But 4 said they knew their solution was wrong.

Do empirical justifications imply

deficient epistemologies?

Coe and Ruthven (1994) explored students’ justifications in a high

school classroom.

“Sometimes, however, 

certainty

appeared to be gained 

by checking

a relatively small number of cases” (p. 50)

Do empirical justifications imply

deficient epistemologies?

Coe and Ruthven (1994) explored students’ justications in a high

school classroom.

“Sometimes, however, 

certainty appeared to be gained

 by checking

a relatively small number of cases” (p. 50)

The authors cite only one example to illustrate this.

Bill makes a conjecture about the Billiards Task based on six

instances. (What will happen to the bounces of a billiard ball going

diagonal on an n by m table).

Examples from the literature:

Coe and Ruthven (1994)

Bill: It’s safe to make a conjecture.

Interviewer: Well it’s certainly safe to make a conjecture, it depends

how… what sort of certainty would you put behind that, say if I

forced you on that?

Bill: Well, pretty sure, I think.

Interviewer: High nineties?

Bill: Yes, high nineties.

Interviewer: OK, so you’d be very surprised if anyone found a

counterexample to that, would you?

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Examples from the literature:

Coe and Ruthven (1994)

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conjecture

.

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how… what sort of certainty would you put behind that, say if I

forced you on that?

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 high nineties

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very surprised 

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counterexample to that, would you?

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Examples from the literature:

Coe and Ruthven (1994)

Bill: It’s safe to make a

 conjecture.

Interviewer: Well it’s certainly safe to make a conjecture, it depends

how… 

what sort of certainty would you put behind that

, say if I

forced you on that?

Bill: Well, 

pretty sure, I think.

Interviewer: 

High nineties

?

Bill: Yes, high nineties.

Interviewer: OK, so you’d be 

very surprised

 if anyone found a

counterexample to that, would you?

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An alternative theory:

Expectancy value theory

Bandura (1997) raised an important distinction that (to my

knowledge) has been ignored in proof-based research:

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I stop with an empirical justification because…

•

I really believe this is the best form of evidence.

•

I don’t care enough to seek certainty. Likely is good enough.

•

I don’t have the motivation to do the work of seeking a proof.

•

I probably won’t find a proof if I try so I might as well stop here.

Expectancy value and mathematical

practice

Mathematicians and choosing not to work on Fermat’s Last Theorem:

•

David Hilbert: Before beginning I should have to put in 

three years

of intensive study

, and I haven’t that much time to spend on

probable failure.

•

Andrew Wiles (on why he didn’t pursue FLT in graduate school):

I realized the only techniques we had to tackle it [FLT] had been

around for 130 years, and it didn’t seem they were really getting to

the root of the problem. 

The problem with working on Fermat 

is you

could spend years 

getting nothing.

Methods

(high-level)

•

11 Pre-Service Teachers and Practicing Teachers were in a

teacher education course on problem-solving.

•

They worked in four groups:

–

Four problems with one hour to work on each

–

Some had multiple parts, so there were seven questions to answer

•

After obtaining a solution to their satisfaction, they were asked:

–

On a scale of 1 through 100, how confident are you in your answer.

–

[If 100] Why are you so certain?

–

[If under 100] Why do you still have doubts about our answer?

–

What further evidence could provide you with more conviction?

–

Why aren’t you seeking this evidence?

A study on bounded rationality:

Preliminary results

A study on bounded rationality:

Preliminary results

Episode #1

[Student claim: There are no 3-regular graphs of diameter 3 with an

odd number of vertices]

I: Okay, on a scale of 0 through 100, how confident are you that

your solution is correct.

Dan: 

90 percent

. I’m 90 percent positive.

Darlene: I feel like 

85

.

I: So then why are you not fully certain in your answer?

Dan: For me,

 personally in my experience in math, there’s always

one strange exception to a rule that could possibly work.

 […]

Darlene:

 I would feel 100 percent confident only if I have like a

proof telling me as to why it doesn’t work

, so if I worked out all of

the steps to a proof to say, okay, this is why, then I’d be more

confident. I think we have the basis for why it doesn’t work.

Why did students doubt the empirical

arguments?

Why do you still have doubt?

N = 16

The student wanted a reason or a proof for why

9

   the pattern held

The student was not sure the result would

8

   generalize to all cases

The students desired a closed form formula

4

   (rather than a recursive formula)

There might be a mistake in how the examples

3

   were calculated

Episode #1

I: So you guys said that a proof would give you further confidence.

Can I ask why you aren’t seeking that evidence?

Darlene: We were still working on diagrams at that point so like,

just the concrete diagrams I guess rather than the abstract of kind

of trying to find a proof. 

So I think that would be our next step

. Once

we finally exhausted all the diagrams we wanted.

I: So then are you saying that if you had more time, you would

continue to work on this until you had more confidence in this?

Dan: 

Absolutely not!

Darlene: 

Yeah. No, no, no.

I: Okay, so you say that this would be your next step, but you

wouldn’t take it?

Darlene: 

Yes. That’s exactly what I’m saying. My motivation has

gone down.

Episode #2

Int: So why aren’t you looking for a proof?

Abby: I was trying. I told you I was trying. 

I didn’t know what would

constitute a proof. 

I was trying 20 different ways. All of them failed.

Do you know what I’m saying?

Episode #3

Int: Is there a reason you are not seeking that evidence [a proof]?

Becky: We tried.

Brenda: I tried. That’s why the answer is no. But 

we hit a roadblock

.

Becky: We’re pretty much maxed out.

Why didn’t students continue looking

for a proof?

Why didn’t you look for this proof?

N = 14

The student lacked the motivation to search

6

    for a proof

The student reached an impasse and did not know

5

   how to proceed

The student did not know what a proof of their 

3

   answer would look like

The student was looking for a proof but ran out

2

   of time

Summary:

Main results

•

Students predominantly offered empirical justifications, but

students were usually aware that empirical justifications could

not provide certainty.

•

In 14 of the 18 empirical justifications, the groups stated that a

proof would convince them that the statement was true.

•

Yet in 12 cases they stopped seeking the proof.

–

This was due to low motivation (cost outweighed value)

–

And because they could not see how to find a proof (low likelihood

of success)

Summary:

Implications for mathematics education

•

These results do not necessarily generalize to other students.

Students in other studies may indeed gain certainty from

empirical evidence or not see the value of proof.

•

Rather, these results show that the inferential leap from

empirical justifications to empirical proof schemes is not valid

and that expectancy value theorem may be a useful lens to

evaluate students’ behavior.

•

Asking students if empirical justifications provide them with

certainty or if their justifications are limited in any way is a

methodological tool to increase validity.

Time for questions