Minding your speech
while speaking about mind
On some pitfalls of the ways we talk
as mathematics teachers and
researchers Anna Sfard
University of Haifa
“Sapienza Visiting Professor” (Sett-Ott, 2016)
XXXIII Convegno UMI-CIIM
Criticità per l’insegnamento della matematica nella scuola di oggi
Pavia, 9 ottobre 2016
10/10/16
1
As teachers or
researchers
we ta
lk
about how
people think
October 10, 2016
Today, I’d
like us
to think
e
w
w
o
h
t
u
abo
talk
r
o
s
r
e
h
c
a
e
t
as
researchers
2
Mathematical facts
can be stated in different ways
one way
another way
The minimum
of function x2
is (0,0)
When I multiply a
number by itself, the
smallest value I can get
is zero and I get it when I
multiply zero by itself.
October 10, 2016
3
Mathematical facts
can be stated in different ways
one way
If I extract a square root
from x and raise the
result to the third power,
I get the same result as
when I raise x to the 3rd
power and extract
October 10, 2016
square root from it
another way
The 3rd power of
square root equals
square root of the
3rd power
4
Stories about math students
may be told in different ways
one way
another way
X is an
able
student
(has a
gift)
In the majority
of school tests
and activities, X
attained above
average scores
5
Stories about math students
may be told in different ways
one way
Z has
learning
disability
(LD)
Z has been failing
one math test after
another in spite of
her being taught
mathematics just
like everybody else.
6
QUESTION 1
e
h
t
o
d
t
e
a
h
t
h
n
W
o
s
n
o
i
t
?
p
n
i
o
r
c
m
m
des
o
c
n
i
e
v
a
?
t
h
f
t
e
l
e
h
righ
t
n
o
e
s
o
h
t
d
An
What’s the difference?
Mathematical facts
can be stated in different ways
one way
another way
When I multiply a number by itself, the The minimum
smallest value I can get is zero and I
of function x2 is
get it when I multiply zero by itself.
(0,0)
s i
n
m
u
l
! x and
r
co root
If I extracthias square
from
e
n
T
n
i
h the third power, I
raise the result
to
t
h
c
u
get the m
same result as when I raise x
to the 3rd power and extract square
Octoberfrom
10, 2016it
root
The 3rd power
of square root
is equal to
square root of
the 3rd power 8
Statements about studnts
may be told in different ways
one way
another way
In the majority of school tests and
activities, X attained above
average scores n is m
u
l
!
o
r
c
e
s
none math test
hi failing
Z hasTbeen
n
i
h
t
h
c
after another
in spite of her trying
u
m
X is an able
student
(has a gift)
to learn mathematics just like
10/10/16
everybody else.
Z has
learning
disability
(LD)
9
QUESTION 1
s
i
e
r
e
But th e!
mor
What’s the difference?
Statem
right a ents on the
re sh
or te
r
t
u
Spea
o
b
Mathematical
facts
a
g
n
i
k
king Spea
abou
s
e
s
can
be
stated
in
different
ways
s
o
e
t bjec
proc
ts
one way
another way
When I multiply a number by itself, the The minimum
smallest value I can get is zero and I
of function x2 is
get it when I multiply zero by itself.
(0,0)
If I extract a square root from x and
raise the result to the third power, I
get the same result as when I raise x
to the 3rd power and extract square
Octoberfrom
10, 2016it
root
The 3rd power
of square root
is equal to
square root of
the 3rd power 11
Statements
about
studnts
S
w
peak
o
h
t
u
o
b be told in differentinways
a
g
g
n
i
abou
k
may
a
e
p
S
ob
t ses
proces
one way
In the majority of school tests and
activities, X attained above
average scores
Z has been failing one math test
after another in spite of her trying
to learn mathematics just like
10/10/16
everybody else.
ject
s
another way
X is an able
student
(has a gift)
Z has
learning
disability
(LD)
12
t
u
o
b
a
e
r
a
7
e
l
e h
t
n
o
o
d
e
l
Those
p
o
e
p
t
t
u
a
o
b
h
t a
w
h
g
i
r
s
e
h
e
t
i
t
n
r
o
e
e
p
s
o
o
r
h
t
p
d
r
i
an
e
h
t
d
n
a
s
t
c
obje
QUESTION 1
What’s the difference?
Statem
right a ents on the
re sh
or te
r
QUESTION 2
n
o
:
i
s
n
a
s
r
t
e
(
”
Do
n
o
i
t
a
c
i
f
i
r
t
e
c
t
t
e
j
a
m
)
“ob
s
n
u
o
n
o
t
s
b
r
e
v
t
m
o
u
r
f
o
b
a
lk
in talk ab
out the
lear ner
?
?
in ta
g
n
i
n
r
lea
Plan of this talk
1.  Why does the way we talk matter?
- the case of objectifying
2.  To objectify or not to objectify
when talking about math learner?
3.  To objectify or not to objectify
when doing mathematics?
4.  To objectify or not to objectify
when doing research on learning?
5.  A lesson on objectifying
for math teacher & researcher
10/10/16
EARLI 2011
15
Does it make a
difference?
y
n
a
y
b
e
A r os
e
m
a
n
r
othe
l
l
e
m
s
d
l
wou
t
e
e
w
s
as
William Shakespeare, “Romeo and Juliet” h
t
i
w
e
r
o
m
g
n
,
i
t
y
n
a
a
t
r
S
o
p
m
i
s
y
a
w
l
a
s
i
e
g
a
less e
h
t
n
i
y
l
l
a
i
c
e
p
s
.
w
o
and e
fl
r
e
v
o
n
o
:
a
m
r
o
of inf
In math
ematic
s, this is the nam
e of the g
ame More about it in a moment October 10, 2016
17
What difference objec:fica:on makes in talk about mathematics
learner
October 10, 2016
18
Example 1
Difficulty in learning as
a manifestation of
learning disability
10/10/16
19
n
i
s
e
:
l
u
c
ffi
i
d
t
u
o
b
a
k
a
e
e
h
t
p
s
g
u
n
i
h
t
e
m
o
s
When yo
m
o
r
f
g
n
:
l
u
s
e
r
y
l
s
p
a
g
m
i
n
i
u
n
o
y
,
lear
s
a
h
r
o
s
i
r
e
n
r
lea
•  transcendence: the ‘object’
exists in the world, beyond us
•  depersonalization:
it is given, not man-made
•  permanence: as a property of an actor,
not of the action, it is rather constant
October 10, 2016
20
n
i
s
e
:
l
u
c
ffi
i
d
t
u
o
b
a
k
a
e
e
h
t
p
s
g
u
n
i
h
t
e
m
o
s
When yo
m
o
r
f
g
n
:
l
u
s
e
r
y
l
s
p
a
g
m
i
n
i
u
n
o
y
,
lear
s
a
h
r
o
s
i
r
e
n
r
lea
•  transcendence: the
‘object’
t
o n
s
i
t
i
t
i
:
n
e
o
g
i
•  depersonalization
:
n
s
a
u
l
h
c
c
n
it is given,
not
man-made
Co us to
o
t
•  permanence
: as a property
of
an
actor,
p
u
exists in the world, beyond us
not of the action, it is rather constant
October 10, 2016
21
n
i
s
e
:
l
u
c
ffi
i
d
t
u
o
b
a
k
a
e
e
h
t
p
s
g
u
n
i
h
t
e
m
o
s
When yo
m
o
r
f
g
n
:
l
u
s
e
r
y
l
s
p
a
g
m
i
n
i
u
n
o
y
,
lear
s
a
h
r
o
s
i
r
e
n
r
lea
:
t
l
•  transcendence
: thet‘object’
h su
e
i
R
w
exists in the world,ebeyond
us
a
s
o
o
h
t
t
”
t
•  depersonalization
:
y
c
t
i
e
l
r
i
y
i
r
b
dman-made
o
it is given,enot
a
t
s
i
c
W ing d traje
•  permanence
:
as
a
property
of
an
actor,
n
e
r
f
i
a
l
e
l
not of“the action,ra
it tiserather constant
a
p
se
October 10, 2016
22
Example 2
grad
es
t
s
o
m
g
n
o
m
a
e
ar
”
s
r
e
i
f
i
t
c
e
j
b
o
“
l
powerfu
10/10/16
Means
for turn
ing talk
about a
ctions
into tal
about p
k
roperti
es
of the a
ctor
23
Example 2
grad
es
t
s
o
m
g
n
o
m
a
e
ar
”
s
r
e
i
f
i
t
c
e
j
b
o
“
l
powerfu
10/10/16
rather than being a mere assessment of something a person did, they become part of this person’s identity. 24
Example 2
grad
es
t
s
o
m
g
n
o
m
a
e
ar
”
s
r
e
i
f
i
t
c
e
j
b
o
“
l
powerfu
As such, they may be more
harmful than helpful
10/10/16
25
Dubious “truths” about grades
Grades
are
given to
the
learner
in her or
his best
interest
10/10/16
On the website
“Get through to
ugh :mes” “[if you get low grades] you might be feeling …. disappointment, anger,
numbness, guilt,
confusion, sadness,
physical sickness or
hopelessness.”
26
Dubious “truths” about grades
Grades
promote
learning
10/10/16
On the website
“Get through to
ugh :mes” Advice for those who get Somlow grades: ‘lower your
e
t
i
eseverybody
ownm
and
,
g
r
a
d
obselse’s
e s
tr ucexpectations.’
t lear n
ing/
No men:on of addi:onal different/more effec:ve learning 27
Dubious “truths” about grades
On the website
Grades
“Get through to
reflect
ugh :mes” reality in
s
i
h
“Even though your grades m
ight t
r
o
f
a
d
e
t for, ewere hoping not be what yoou o
n
n
n
reliable,
d
i
e
d
b
they d
on’t m
ean t
hat y
our
d
s
l
o
u
e
t
d
objective re wo
t
a
r
e
k
g
e future
is
hopeless, t
hat f
c
i
h
i
T
r
t
way laime
e
h
t
e
you’re
a
failure, o
r you
c
r
s
s
i
u
a
d
t
u
e
f
v
r
s
be
e’ able to achieve
sewon’ton
great things.” 10/10/16
28
!
l
l
i
w
Dubious “truths”
about
grades
s
e
d
a
r
g
d
a
b
On the website
y
n
Grades
a
m
t
“Get through to
u
B reflect
ugh :m
Grades not
just
es” reality in
reflect reality –
a
“It's l
ikely o
ne b
ad they
actually
reliable,
grade w
on't c
hange shape it!
objective
the c
ourse o
f t
he way An
d not a future.” lways i
n the
most b
enefici
al way
10/10/16
29
Objectification is
both
and dan
10/10/16
gero
us 30
.
10/10/16
or not to
objectify?
31
Plan of this talk
1.  Why does the way we talk matter?
- the case of objectifying
2.  To objectify or not to objectify
when talking about math learner?
3.  To objectify or not to objectify
when doing mathematics?
4.  To objectify or not to objectify
when doing research on learning?
5.  A lesson on objectifying
for math teacher & researcher
10/10/16
EARLI 2011
32
While speaking about the .
10/10/16
learner try to speak in verbs (about doing) rather than or not to
objects (about objectify?
what one is or has) 33
Stories about math students
may be told in different ways
one way
Z has
learning
disability
(LD)
Z has been failing
one math test after
another in spite of
her being taught
mathematics just
like everybody else.
34
10/10/16
35
Yes! 10/10/16
In Chinese,
people say
only in verb
s much of
what in Eng
lish is said
with nouns
(Perry Link) e
w
…
s
e
g
a
u
g
n
a
l
“in Western o
t
s
n
u
o
n
e
s
u
s
e
some:m
e
w
n
e
h
w
s
g
in
h
t
e
conceiv
) k
in
L
y
r
r
e
P
(
”
o
t
d
e
e
don't really n
36
Yes! In Chinese,
people say
In Dong-Joong
onlyKim
in verstudy: bs much of
what in Eng
lish is said
with nso
Korean-­‐speaking tudents ave uns (Perryh
Link) difficulty using the noun infinity, although they d
o u
se e
w
…
s
e
g
a
u
g
n
a
l
in Western “
the adjec:ve infinite a
nd t
he o
t
s
n
u
o
n
e
s
u
s
e
some:m
e
w
n
e
h
w
adverb infinitely. s
g
in
h
t
e
conceiv
10/10/16
) k
in
L
y
r
r
e
P
(
”
o
t
d
e
e
don't really n
37
Will re
10/10/16
moving
o
bjects from my ta
l
k
make t
hem di
sappea
from r the w
orld?
38
disca
lculia
cure m
y stude
n
t from this
10/10/16
disab
ility
39
dysca
lculi
a
10/10/16
include
d,
u
o
y
s
s
e
l
n
u
t
do not exis
m
e
h
t
t
u
o
b
a
k
a
e
sp
40
More precisely,
are discursive
constructs Many of the objects we talk about They are just our
way of talking
10/10/16
about phenomena 41
We find objects
everywhere in research
In natural science & mathema:cs •  Force, energy
•  Number, function, set
In human sciences • 
• 
• 
• 
• 
• 
October 10, 2016
Knowledge, concept, meaning
Belief, attitude, value
Personality, character, identity
disability, gift, discalculia
Ego, superego, id
meme
42
We find objects
It is dinifficult
everywhere
researchto
imagine w
e c
ould t
alk In natural •  Force, energy
about the relevant science & • 
Number,
function,
set
mathema:cs phenomena (processes) without a
r
eference t
o In human •  Knowledge, concept, meaning
these o
bjects. sciences •  Belief, attitude, value
• 
• 
but it
• 
possible! • 
October 10, 2016
is
Personality, character, identity
disability, gift, discalculia
Ego, superego, id
meme
43
More
precisely,
DISCLAIMER
The request to a
void Many of t he the word “dyscalculia” objects we about does not mean talk denying are discursive
the e
xistence
of
the
constructs phenomena that They are our gave way
of talking
raise to this word. 10/10/16
about phenomena 44
Stories about math students
may be told in different ways
one way
Z has
learning
disability
(LD)
Z has been failing
one math test after
another in spite of
her being taught
mathematics just
like everybody else.
45
More precisely,
conclusion
Many of the It is up to us objects whether we to objectify are discursive
constructs talk about (to speak i
n n
ouns, They are our way of
not verbs) or talking
not 10/10/16
about phenomena 46
Plan of this talk
1.  Why does the way we talk matter?
- the case of objectifying
2.  To objectify or not to objectify
when talking about math learner?
3.  To objectify or not to objectify
when doing mathematics?
4.  To objectify or not to objectify
when doing research on learning?
5.  A lesson on objectifying
for math teacher & researcher
10/10/16
EARLI 2011
47
In .
mathematics you have no choice: you have to objec:fy! Mathematics
10/10/16
simply dor oes nnot
ot exist to without objec:fica:on objectify?
48
What is mathematics?
•  Just like biology is the
activity of telling useful
stories about living
things (plants, animals)
•  and as physics is telling
useful stories about
natural things (moving
bodies, light, etc.)
so is
mathematics
an activity of telling
useful stories about
mathematical
objects
(numbers, sets,
functions,
geometric figures)
49
What is mathematics?
f
o
e
s
a
c
e
h
t
n
i
e
k
i
l
n
But u
n
i
,
s
c
i
s
y
h
r p
•  bJust
is the
y obiology
iologlike
e
s
e
h
t
l
l
a
s c
activity
of
telling
useful
:
a
m
e
h
t
a
mstories
ed
t
a
e
r
c
about
living
e
ar
s
t
c
e
j
b
o things (plants,
f
o
animals)
t
c
a
in the
!
f
l
e
s
t
i
g
n
i
l
l
e
t
y
r
sto
•  and as physics is
tellinguseful stories about
natural things
(moving bodies, light, etc.)
so is
mathematics
an activity of telling
useful stories about
mathematical
objects
(numbers, sets,
functions,
geometric figures)
50
So… o
d
y
h
w
d
n
a
w
o
H
s
t
c
e
j
b
o
l
a
c
i
t
a
mathem
?
g
n
i
e
b
o
t
n
i
e
m
co
Let’s take number as an example What makes us say:
“These pictures present the same person”?
What makes us say:
“These pictures present the same person”?
The
sam
is sh e obje
ct (p
own
erso
in al
n)
l the
pict
se
ures
What is “the same”
about these six things?
Is th
e sa
in al me obj
l the
ect
show
se p
ictu
n
res?
What is “the same”
about these six things?
The
sam
impl
e PR
icat
O
ed in CESS
i
a
l
l the s
pict
ures
s
e
Conclusion: Number is but a metaphor
It is an objectification of the process of coun:ng
10/10/16
56
Objectification
/
d
r
o
w
e
h
t
usinge ta
symbol
(word)
y
a
s
n
e
h
w
s
t
n
e
s
e
r
as
if
it
signified
p
e
r
symbol
t
c
e
j
b
o
an object
in
the
world
e
th
existing independently of us
10/10/16
57
Objectification
(a special type of metaphor) discursive process
that makes us use
mathematical
words and symbols
as if they signified
discourseindependent
objects;
it includes:
10/10/16
reification
- replacing
verbs with
nouns
+
alienation
- removal of human
subject
58
Why do we objectify number?
Imagine y
ou Suppose you
were go on cannotto use
number words
developing
ONE, TWO,
mathematics
THREE,… except
based on the in counting.
descrip:on on Howright!
would you
the then say what the
following equality
says?
3+4 = 7
10/10/16
•  If I have a set so that whenever I
count its elements I stop at the
word three,
•  and I have yet another set such
that whenever I count its
elements I stop at the word
“four”,
•  and if I put these two sets
together,
then
•  if I count the elements of the new
set, I will always stop at “seven”.
59
As an aside:
if numbers (sets, functions, etc.) are
processes turned objects….
?
=
10/10/16
60
As an aside:
if numbers (sets, functions, etc.) are
processes turned objects….
the idea of objectification
?
may help us understand
=
why so many people don’t
understand mathematics
10/10/16
61
Mathema:cs of Mathema:cs teacher as teacher a juggler as seen through
hisostudents’
eyes mathema:cal bjects 10/10/16
62
Plan of this talk
1.  Why does the way we talk matter?
- the case of objectifying
2.  To objectify or not to objectify
when talking about math learner?
3.  To objectify or not to objectify
when doing mathematics?
4.  To objectify or not to objectify
when doing research on learning?
5.  A lesson on objectifying
for math teacher & researcher
10/10/16
EARLI 2011
63
What difference speaking “in objects” makes to learning scientists
(those who do research on mathema:cs learning) October 10, 2016
64
Two LSs are reporting on their studies
One researcher says: The other says: Children under the age Children under of five o7en do not the age of five realize that it is o7en do not the last word A
in rte
he theyrealize that t
u
o
b
a
g
n
process osf pcoun:ng
i
k
ea
repeated coun:ng ?
g
n
i
h
t
elements o
f a
s
et
e
m
of the same set the sa
must end with that is the same the number of number word elements in the set 10/10/16
65
Talking about learning k
a
e
p
s
y
e
One t do th Another researcher: haresearcher:
W
age Children nder ut?the Children under abuo
of five o7en do not realize that it is the last word in the process of coun:ng elements of a set that is the number of elements in the set 10/10/16
the age of five o7en do not realize that repeated coun:ng of the same set must end with the same number word 66
Talking about learning ? is it
t
a
h
W
.
e
c
n
e
r
fe
if
talk a
bout a
d
a
e is
therresearcher:
Still,One Another researcher: Children under the age of five o7en do not realize that it is a
t
u
o
b
a
k
n
l
the l
ast w
ord i
n t
he
a
a
t
f
o
y
t
r
e
process of coun:ng p
o
r
p
t
c
e
elements of a set obj
that is the number of elements in the set 10/10/16
property of a Children under discursive the age of five process
o7en do not realize that repeated coun:ng of the same set must end with the same number word 67
?
r
e
t
t
a
m
e
c
n
e
Does this differ
One researcher: Children under the age of five o7en do not realize that it is the last word in the process of coun:ng
the whale elements of a set that is the number of the largest mammal
elements in the set 10/10/16
Another researcher: Children under the age of five o7en do not realize that repeated coun:ng “whale” and of the same et “the largest msammal”
must with can be eund sed the same interchangeably number word 68
e
m
a
s
e
h
t
t
o
n
cts is
e
j
b
o
t
u
o
b
a
es m
a
n
Speaking
r
i
e
h
t
t
abou
g
n
i
k
talk a
bout a
a
e
p
s
s
a
a
t
u
o
b
a
k
n
l
a
a
t
f
o
Children u
nder the age y
t
r
e
p
o
r
pof five oj7en t
c
e do not b
o
realize that it is the last word in the process of coun:ng
the whale elements of a set that is the number of the largest mammal
elements in the set 10/10/16
property of a Children under discursive the age of five process
o7en do not realize that repeated coun:ng “whale” and of the same et “the largest msammal”
must with can be eund sed the same interchangeably number word 69
Speaking about objects is not the same
as speaking about their names If you say:
Children under the age of five o7en do not realize that it is the last word in the process of coun:ng
the whale elements of a set that is the number of the largest mammal
elements in the set 10/10/16
you imply:
The child can have some direct experience of whales
before she knows of some of their proper:es
70
Speaking about objects is not the same
as speaking about their names If you say:
Children under the age of five o7en do not realize that it is the last word in the process of coun:ng
the whale elements of a set that is the number of the largest mammal
elements in the set 10/10/16
you imply:
The child can have some direct experience of numbers
whales
before she knows of some of their proper:es
71
The first text implies that in the life of a child, numbers exist prior to the talk on numbers
The second text is in concert with the claim that numbers arise from discourse
10/10/16
72
The first tFrom ext implies the 1stt that ext:i n the life the child learns prior of child, n
umbers exist from the world itself. to the talk on numbers
The second text in tcext:
oncert with From the is 2nd
the claim The child learns first tahat nd foremost from people
around her
numbers arise from discourse
10/10/16
This difference may have far-­‐
reaching implica:ons for our understanding of learning & the practice of teaching 73
Plan of this talk
1.  Why does the way we talk matter?
- the case of objectifying
2.  To objectify or not to objectify
when talking about math learner?
3.  To objectify or not to objectify
when doing mathematics?
4.  To objectify or not to objectify
when doing research on learning?
5.  A lesson on objectifying
for math teacher & researcher
10/10/16
EARLI 2011
74
Lessons about the way we talk
•  We all need to take care of how we talk
This is true of all mathema:cians, math students, math teachers, researchers those who deal with mathema:cs 75
Lessons about the way we talk
This is true of the •  We all need to take care mathema:cian of how we talk
who invents new •  In mathematics, mathema:cs
we must learn how to objectify
and of the student who tries to become a par:cipant of the exi:ng mathema:cal discourse
76
Lessons about the way we talk
•  We all need to take care of how we talk
•  In mathematics, we must learn how to objectify •  As teachers, we need to –  help learners objec:fy –  avoid objec:fica:on while talking about learners And what about learning scien:sts who study mathema:cs learning? 77
Learning
scientist
who asks
“What do children know about /do with numbers?” Is like a historian who asks
“What did medieval
people know about/do with cars?” The
historian’s can mo
ve as fa
report
st as 2 00 km/ho
ur
People did not know that cars ….
They did not use cars when it would be most helpful
The learning
scientist’s
historian’s report
can gro
w infinite
ly
Children People did not know that cnumbers...
ars ….
numbers
They did not use cars when it would be most helpful
The learning
scientist’s
historian’s report
Children People did not know that cnumbers...
ars ….
numbers
They did not use cars when it would be most helpful
The
learning
This creates discourse
of
deficit in w
hich the learning scientist’s
historian’s scien:st reports on what ignores children don’t do and what they actually do
report
Children People But just like people did travel
dideven notwhen know
that c
ars …
.
numbers...
there were no cars, do not
children do cars w
ithout numbers
They so did
use
numbers t
hings
they
will
when it would be most helpful
later do with numbers
For instance, children……
The learning
Make choices, some of scientist’s
historian’s them quantitative….
report
It is in the activity of choosing
that the s tudy of the development of Learners
People numerical t
hinking s
hould b
egin
did not know that cnumbers...
ars …. and it should con:nue as a numbers
They did
not
use
cars careful follow-­‐up of when would be most of
helpful
theit development
their routines of choosing
Lessons about the way we talk
,
t
r
o
h
s
•  As lk
learning
•  We all need to take !
In care a
t
scientists, r
u
o
y
we need to d
•  In mathematics, n
i
oscillate M
we must learn between the of how we talk
how to objectify perspec:ves of Wou we need to •  As teachers, insider and ld yo
outsider to our –  help learners oubjec:fy own abou
–  avoid objec:fica:on t
while talking about it now mathema:cal ?
learners discourse mind
talkin
g
84