developing preschool teachers’ knowledge of students’ number conceptions

Developing preschool teachers’ knowledge of students’number conceptions

Pessia Tsamir • Dina Tirosh • Esther Levenson • Michal Tabach •

Ruthi Barkai

Published online: 14 December 2013� Springer Science+Business Media Dordrecht 2013

Abstract This article describes a study that investigates preschool teachers’ knowledge

of their young students’ number conceptions and the teachers’ related self-efficacy beliefs.

It also presents and illustrates elements of a professional development program designed

explicitly to promote this knowledge among preschool teachers. Results indicated that

teachers’ knowledge of students’ number conceptions improved and that their related self-

efficacy increased. Furthermore, prior to participating in the programs, a significant neg-

ative correlation between knowledge and self-efficacy was reported. However, no signif-

icant correlation was found after the program.

Keywords Preschool teachers � Professional development �Number concepts � Knowledge of students

Introduction

The premise for the current study is that we now know a great deal about children’s

mathematical learning and what mathematics preschool children can learn. Thus, we need

to know more about what support is needed for teachers, so they can implement a richer

vision of early mathematics. Researchers agree that it is essential for teachers, including

preschool teachers, to be knowledgeable of their students’ conceptions, misconceptions,

and ways of thinking (e.g., Ball et al. 2008). It is also beneficial to have a high sense of

self-efficacy that goes along with knowledge. While professional development for pre-

school teachers has addressed the issue of knowing students’ conceptions, few studies have

focused specifically on investigating this knowledge as well as their associated self-effi-

cacy beliefs. This article presents a study which first describes two elements of a profes-

sional development program which specifically intended to promote practicing preschool

P. Tsamir � D. Tirosh � E. Levenson (&) � M. Tabach � R. BarkaiTel Aviv University, P.O. Box 39040, 69978 Tel Aviv, Israele-mail: [email protected]; [email protected]

123

J Math Teacher Educ (2014) 17:61–83DOI 10.1007/s10857-013-9260-5

teachers’ knowledge of their young students’ number conceptions and counting skills. The

first element focuses on the instructors, and the way research related to children’s early

number reasoning can be introduced to teachers. The second element focuses on the

teachers’ reflections of mathematical activities implemented with children. The study then

investigates the teachers’ knowledge of young students’ number conceptions, as well as

their related self-efficacy beliefs (e.g., teachers’ beliefs in their ability to identify common

mistakes made by children as they learn number concepts), before and after participating in

professional development.

Related literature review

There is wide agreement among researchers and educators that knowledge of students—

their conceptions, misconceptions, and ways of thinking—is an essential element of the

knowledge teachers need for teaching mathematics. In his seminal article on teachers’

knowledge, Shulman (1986) suggested that pedagogical-content knowledge includes

knowing ‘‘the conceptions and preconceptions that students of different ages and back-

grounds bring with them to the learning of those most frequently taught topics and les-

sons… learners are unlikely to appear before them as blank slates.’’ (pp. 9–10). Fennema

and Franke (1992) also stressed the importance of teachers’ knowledge of students. They

found that knowledge of students’ processes and thinking may impact on teachers’ deci-

sion-making, allow teachers to attend to individual students, and influence students’

learning outcomes. Even and Tirosh (1995) differentiated between two types of knowledge

of students: knowing that and knowing why. Knowing that refers to research-based

knowledge concerning students’ common conceptions and misconceptions. In the pre-

school, for example, this type of knowledge would include knowing that a five-year old

may orally count ‘‘… twenty-seven, twenty-eight, twenty-nine, twenty-ten.’’ Knowing why

refers to knowing the sources of these conceptions and misconceptions. For example,

saying ‘‘twenty-ten’’ makes sense to a child who has learned that ten follows nine but has

not yet learned that the twenties end with twenty-nine (Baroody and Wilkins 1999). An

et al. (2004) pointed out that knowledge of students’ conceptions includes knowing how to

use this knowledge in the classroom. Thus, knowing students’ thinking includes being able

to build on students’ ideas, address students’ misconceptions, engage students in mathe-

matics learning, and promote students’ thinking about mathematics. Schoenfeld and Kil-

patrick (2008) claimed that a proficient teacher knows students as thinkers and as learners.

In addition to the above studies, Ball et al. (2008) identified knowledge of content and

students (KCS) as a sub-domain of pedagogical-content knowledge (Shulman 1986). KCS

is ‘‘knowledge that combines knowing about students and knowing about mathematics’’

(Ball et al. 2008, p. 401).This includes anticipating and predicting what examples students

might find confusing or difficult and what tasks students might find interesting or moti-

vating. Within the context of counting objects, for example, it might mean knowing that

counting a small number of items arranged in a row is easier than counting a larger

collection, with no specific order (Baroody and Wilkins 1999). KCS also means being able

to interpret students’ ideas, even when they are expressed incompletely or in non-math-

ematical language.

Preschool mathematics education has gained increased attention in recent years. While

it was once thought that young children are hardly capable of learning mathematics (e.g.,

Piaget et al. 1960), studies have now shown that young children spontaneously engage in

play that involves mathematical concepts (Seo and Ginsburg 2004) and that several

62 P. Tsamir et al.

123

mathematical competencies develop during the early years (Baroody et al. 2006). The

context of this study is number conceptions and counting skills developed during pre-

school. In line with Clements and Sarama (2007), we differentiate between verbal and

object-counting skills. While on a day-to-day basis, the term counting is used for both

verbal and object counting, in Israel there is one term used for verbal counting and a

different term used for object counting. These two terms appear in the Israel mathematics

preschool curriculum (INMPC 2008) and help preschool teachers to distinguish between

the different skills.

Knowing to count verbally is more than a rote skill. It includes being able to say the

number words in the proper order and knowing the principles and patterns in the number

system as coded in one’s natural language (Baroody 1987). The relationship to language

may be seen in the difficulties of English-speaking (and Hebrew-speaking) children when

learning the number words from 11 till 20, and going from 29 to 30 (Han and Ginsburg

2001). Fuson and colleagues (Fuson 1988; Fuson and Hall 1982; Fuson et al. 1982) further

described two distinct but overlapping phases in learning to count verbally. The first phase

is the acquisition phase. Children during this phase not only learn the conventional number

words, but also learn to produce them in order and consistently. Thus, a common error of

children in this phase is to recite the conventional number names, in a consistent fashion,

but not in order. For example, reciting 1, 2, 3, 5, 4 and when asked to count again,

repeating the numbers 1, 2, 3, 5, 4 as before. The second phase is the elaboration phase

when children become aware that the chain of numbers can be broken up and that parts of

the chain may be produced starting from a number other than one. Counting backwards is

learned more slowly and is based on first mastering the forward sequence.

Object counting refers to counting objects for the purpose of saying how many. Gelman

and Gallistel (1978) outlined five principles of counting objects: the one-to-one corre-

spondence principle, the stable-order principle, the cardinal principle, the abstraction

principle, and the order-irrelevance principle. Competence in object counting may be

related to the amount of objects to be counted, as well as how the objects are set up (Briars

and Siegler 1984; Gelman and Gallistel 1978). In addition, children may show knowledge

of one principle while violating another principle, for example, erring with regard to the

one-to-one correspondence principle, but showing understanding of cardinality (Geary

et al. 1992). Related to counting objects is the understanding that wholes consists of parts,

and in general, number composition and decomposition. Several researchers have sug-

gested that encouraging children to compose and decompose numbers may enhance their

number sense as well as their ability to solve addition and subtraction problems (e.g.,

Baroody et al. 2006).

Along with increased research in the field of preschool mathematics come new demands

for teachers. A joint position of the National Association for the Education of Young

Children (NAEYC) and the National Council for Teachers of Mathematics (NCTM) rec-

ommended that teachers of young children ‘‘build on children’s experience and knowl-

edge’’ and ‘‘base mathematics curriculum and teaching practices on knowledge of young

children’s cognitive, linguistic, physical, and social-emotional development’’ (2002, p. 4).

In other words, knowledge of one’s students’ mathematical experiences and conceptions is

just as important for preschool teachers, as it is for other teachers.

Agreeing that knowledge of students is an essential element of teachers’ knowledge,

different studies investigated various ways of promoting this knowledge among teachers.

In what has come to be known as the CGI (Cognitively Guided Instruction) program,

Carpenter et al. (1989) offered a workshop to first grade teachers which familiarized those

teachers with research-based analysis of the development of addition and subtraction

Developing preschool teachers’ knowledge 63

123

concepts and problem-solving skills among young children. Teachers were provided with

specific readings that synthesized related research and discussed these readings during the

workshop. Teachers were also invited to view videotapes of children solving problems.

Results indicated that teachers who participated in the workshop spent significantly more

time teaching problem solving than teachers in a control group. In addition, students of

teachers who participated in the CGI program performed equally as well on tests of

computational skills as did students in control classes and scored higher than students in

control classes on number facts recall. Schifter (1998) described a professional develop-

ment program for elementary school teachers which also focused on students’ thinking.

Teachers analyzed written materials illustrating student work and studied videotapes of

interviews with individual children and classroom discourse. In addition, teachers wrote

about episodes of students’ thinking which occurred in their own classrooms and discussed

these episodes in small groups. Results indicated that teachers not only increased their

knowledge of mathematics and of children’s mathematical thinking, but also developed a

disposition to inquiry.

Regarding professional development for preschool teachers, Starkey et al. (2004)

described workshops for preschool teachers that provided teachers with the background,

rationale, instructions, and even scripts for implementing a mathematical intervention with

children. During workshops, an overview of early mathematical development for each

topic was provided, informing teachers of young students’ mathematical thinking. While

the study did not specifically assess teachers’ knowledge for teaching mathematics,

interviews with the teachers indicated that they gained an appreciation of the mathematics

that may be taught to young children and that the amount of mathematical activities

implemented in class had increased as a result of the intervention. Building Blocks (Cle-

ments and Sarama 2007) is a different mathematics curriculum which addresses preschool

mathematics skills. Along with the curriculum, learning trajectories were designed to

develop teachers’ content knowledge as well as knowledge of students’ developmental

progressions (Clements et al. 2011). During professional development, preschool teachers

were introduced to the goals of the curriculum, learn about developmental progressions of

levels of thinking, and view multiple video segments of task enactments in class. Bobis

et al. (2005) described three professional development programs for teachers of five- to

eight-year-old children. Although each program was slightly different, common to all was

the use of a framework based on research of young children’s mathematical learning, with

the intent of familiarizing teachers with young students’ ways of thinking. In one of the

programs described, an explicit aim was for teachers to use their knowledge of students’

mathematical development in order to analyze the interactions they had with their students

when implementing mathematical activities. In another of the programs described, teachers

were guided in conducting individual interviews with their students in order to assess their

students’ mathematical thinking. Teachers who participated in the programs reported

increased confidence and greater enthusiasm for teaching mathematics, and their teaching

was focused on the children’s thinking. A different study, focusing on increasing pattern

and structure awareness in preschool, found that professional development, especially one

which provides consistent support and practical resources, can effectively addresses a

range of student difficulties of which teachers were previously unaware (Mulligan et al.

(2006). With regard to prospective preschool teachers, Ginsburg and his colleagues

(Ginsburg et al. 2004) structured a course which entailed introducing prospective teachers

to mathematical content and curricular materials, as well as enabling them to observe and

analyze children’s mathematical thinking and learning by making use of technology and

web-based technology.

64 P. Tsamir et al.

123

In addition to promoting teachers’ knowledge, it is important to promote teachers’ self-

efficacy for teaching mathematics. Bandura defined self-efficacy as ‘‘people’s judgments of

their capabilities to organize and execute a course of action required to attain designated

types of performances’’ (1986, p. 391). Teacher self-efficacy may be conceptualized as ‘‘a

teacher’s individual beliefs in their capabilities to perform specific teaching tasks at a

specified level of quality in a specified situation’’ (Dellinger et al. 2008, p. 752). In terms of

the focus of this article, we were interested in teachers’ beliefs in their ability to assess

students’ mathematical conceptions. Teacher self-efficacy has been related to a variety of

teacher classroom behaviors including the effort put into teaching and persistence and

resilience in the face of difficulties with students (Ashton and Webb 1986). Studies report

that teachers with a high sense of self-efficacy are more enthusiastic about teaching

(Allinder 1994) and are more committed to teaching (Coladarci 1992). There also seems to

be a relationship between knowledge and self-efficacy and professional development.

Wheatley (2002) claimed that doubts in one’s teaching abilities may cause a feeling of

disequilibrium which in turn may foster teacher learning. Similarly, another study found

that the higher the teacher’s sense of self-efficacy, the higher the motivation to actively

participate in professional development and that an increase in knowledge was likely to

increase self-efficacy (Brady et al. 2009).

Framework

For the past several years, we have been providing professional development for preschool

teachers guided by the Cognitive Affective Mathematics Teacher Education (CAMTE)

framework (e.g., Tirosh et al. 2011; Tsamir et al. 2014). Acknowledging that knowledge

and beliefs are interrelated and that both affect teachers’ proficiency (Schoenfeld and

Kilpatrick 2008; Torner 2002), the framework and program takes into consideration

teachers’ knowledge as well as their self-efficacy for teaching mathematics in preschool.

The framework is presented in Table 1. In Cells 1 and 2, we address teachers’ mathematics

knowledge. Within the context of number, these cells address teachers’ knowledge of, for

example, different ways for comparing the number of objects in a set and evaluating which

of those ways is most appropriate for different situations. In Cells 5 and 6, we address

associated mathematics self-efficacy. Hackett and Betz (1989) defined mathematics self-

efficacy as, ‘‘a situational or problem-specific assessment of an individual’s confidence in

her or his ability to successfully perform or accomplish a particular [mathematics] task or

Table 1 The cognitive affective mathematics teacher education framework

Subject-matter Pedagogical-content

Solving Evaluating Students Tasks

Knowledge Cell 1: Producingsolutions

Cell 2: Evaluatingsolutions

Cell 3: Knowledge ofstudents’ conceptions

Cell 4: Designing andevaluating tasks

Self-efficacy

Cell 5:Mathematicsself-efficacyrelated toproducingsolutions

Cell 6:Mathematicsself-efficacyrelated toevaluatingsolutions

Cell 7: Pedagogical-mathematics self-efficacy related tochildren’sconceptions

Cell 8: Pedagogical-mathematics self-efficacy related todesigning andevaluating tasks


123

problem’’ (p. 262). In Cells 3 and 4, we address teachers’ pedagogical-content knowledge.

In our program, this included knowledge of students’ number conceptions and common

counting mistakes as well as designing tasks to promote children’s number conceptions. In

Cells 7 and 8, we address their associated pedagogical-mathematics self-efficacy. We

define pedagogical-mathematics self-efficacy as a teacher’s belief in his or her ability to

carry out teaching tasks related to the teaching of mathematics (e.g., self-efficacy for

designing tasks that will assess students’ counting skills).

In general, the CAMTE framework described in Table 1 served as an organizing tool

and as a set of checks and balances for the program. We used it to ask ourselves—What do

preschool teachers need to know in order to teach mathematics in preschool? Are we

paying attention to different types of knowledge? Are we devoting time to each of the

different elements signified by the different cells? Although each cell specifically focuses

on a different piece of the knowledge and self-efficacy puzzle, promoting the different

elements often occurs together. A more detailed discussion of the framework may be found

in Tsamir et al. (in press).

Focusing on teachers’ knowledge of children’s conceptions during professionaldevelopment

In this article, we focus on Cells 3 and 7 of the CAMTE framework. In Cell 3, we focus on

developing teachers’ knowledge of preschool children’s number concepts including verbal

and object-counting skills. In Cell 7, we focus on teachers’ self-efficacy for identifying

elements of preschool children’s number conceptions such as self-efficacy for identifying

common mistakes children make when learning to recite the numbers as well as self-

efficacy for pointing out challenges children face when learning object-counting skills.

Although several elements of the program combined to enhance teachers’ knowledge

and self-efficacy related to students’ number conceptions, due to space constraints, we

describe and illustrate two elements of the program designed to explicitly promote

teachers’ knowledge of young students. We chose to present these two elements because

they are related. The first element describes teachers observing the instructors as the

instructors simulate children’s counting practices. The second element describes teachers

observing their own young students as the students engage in counting and number

activities. In addition, the contexts of the two elements of the program, verbal counting,

object counting, and number decomposition are directly related to items on the ques-

tionnaires which were handed out to teachers before and after the program.

Simulating children’s actions

The participants in our program were all practicing preschool teachers, and almost all had

experience engaging children with number activities. Our challenge was to direct teachers’

attention to aspects of children’s knowledge that could be observed as children engage in

number activities. Toward this goal, we developed the Pair-Dialogue (PD) approach to

instruction, a specific form of team teaching in which two instructors teach cooperatively.

The dialogues are semi-structured, allowing for both prepared-in-advance and in-action

adaptations to different populations of teachers. In our interactions with teachers, we use a

blend of pair performances (e.g., thought-provoking dialogue episodes) and discussions

that involve the teachers and employ various modus operandi of the PD approach:

Sometimes, both characters offer correct (or erroneous or a mix of correct and erroneous)

66 P. Tsamir et al.

123

ideas. In other cases, one teacher educator acts as a ‘‘model learner,’’ presenting students’

dilemmas, and the other acts as the knowledgeable guide. The roles are altered occa-

sionally, to refrain from creating a ‘‘clever, always-right’’ character and a ‘‘puzzled-erring’’

character, thereby avoiding irrelevant hints that may take away the mathematical essence

of the situation. Below, in Table 2, we illustrate how this approach was used to simulate

different possible ways children may engage in a counting activity. I1 is the instructor

playing the role of the teacher, and I2 is the instructor playing the role of the child.

In the above dialogue, we see five different examples of how children may respond when

asked to count out loud. These examples were based on previous research related to chil-

dren’s verbal counting, as discussed in the literature review (e.g., Baroody 1987; Fuson

1988), as well as from our own research with preschool children (Tirosh and Tsamir 2008).

For instance, while previous research has shown that children may consistently skip one

number when counting verbally, we have found that when counting in Hebrew, the number

which children tend to skip is six, perhaps because it is the only number with one syllable.

While the dialogue was being enacted, a third instructor wrote the different responses on the

board. After the last exchange, the second instructor turned to the teachers and said, ‘‘What

do you see? … What do the children know and what knowledge is missing? … Many times,

what jumps up at us is only what the child does not know but it is just as important to be

aware of what the child does know.’’ One teacher pointed out that the first child (rows 1–3 in

Table 2) consistently skips the number six. While skipping the number six is, of course, a

problem, being consistent is a positive development. Not all of the children were consistent.

Another teacher points out that Jerry (rows 4–6) at least recited the numbers in increasing

order, as opposed to Rotem (rows 7–8) who was not only not consistent, but also recited the

numbers without regard for increasing order. When discussing Norma (rows 9–12), it also

becomes evident that the other children all knew what types of words should be included

when counting (number words) and what should not be included (letter words).

To summarize, simulating children’s responses allows for a controlled way of pre-

senting children’s common errors that may occur as they learn to verbally count, as well as

highlighting aspects of counting that are correct and necessary. The laughing on the part of

the teachers indicates that not only are they engaged in the performance but also that they

are relaxed. The simulations are done in a non-threatening manner, which we feel con-

tributes to teachers’ positive attitude toward participating in the program which in turn may

promote positive self-efficacy. The dialogues are not a replacement for watching actual

videos of children counting. However, it allows the teachers to focus on the specifics of

what the ‘‘child’’ is saying, without being distracted by irrelevant incidents. It also allows

us, as instructors, to build sequences that gradually reveal different layers of knowledge.

In the next section, we describe how teachers observed their own students engaging in

counting and number activities. In the specific program described here, teachers chose

several assessment tasks to implement with individual children in their kindergarten

classes, video-taped the task implementations, and then brought the video clip back to the

course to be viewed and discussed with the other teachers and with the instructor. This is

discussed in the next section.

Viewing teachers’ video clips of children engaging in number activities

In this section, we describe a session where the teachers and instructor viewed together

video clips of the participating teachers assessing one of their students’ knowledge of

counting and number decomposition. During the program, teachers encountered object


123

Ta

ble

2D

ialo

gu

esw

hic

hsi

mu

late

exam

ple

so

fch

ild

ren

’sco

un

tin

g

I1—

the

firs

tin

stru

cto

rp

lays

the

role

of

the

pre

sch

ool

teac

her

I2—

the

seco

nd

inst

ruct

or

pla

ys

the

role

of

the

chil

dre

spo

ndin

gto

the

teac

her

1I1

:S

o,

Pen

ny

(th

atis

the

nam

eI1

giv

esto

the

fict

itio

us

chil

d),

can

yo

up

leas

eco

un

t?D

oy

ou

kn

ow

ho

wto

cou

nt?

I2:

To

cou

nt?

2I1

:Y

es.

Ple

ase

cou

nt

I2:

1…

2…

3…

4…

5…

7…

8…

9…

10.

(I2

smil

esto

I1,

sati

sfied

wit

hher

acco

mpli

shm

ent.

)

3I1

:C

any

ou

ple

ase

cou

nt

agai

n?

I2:

Ilo

ve

toco

un

t.M

yfa

ther

alw

ays

cou

nts

wit

hm

e.1…

2…

3…

4…

5…

7…

8…

9…

10

4I1

:O

k.Je

rry

?(I

1n

ow

call

sI2

by

the

nam

eo

fJe

rry

tosi

gn

alth

atsh

eis

no

wta

lkin

gto

ad

iffe

ren

tch

ild

.)C

any

ou

ple

ase

cou

nt?

I2:

1…

2…

4…

7…

10

.(T

he

teac

her

sw

ho

are

list

enin

gto

this

exch

ange,

laugh.)

5I1

:C

any

ou

ple

ase

cou

nt

agai

n?

I2:

Fro

mth

eb

egin

nin

g?

6I1

:Y

es,

fro

mth

eb

egin

nin

gI2

:A

h.

1…

2…

3…

5…

8…

10

7I1

:N

ow

,R

ote

m,

ho

war

ey

ou

?Y

ou

hav

ea

ver

yp

rett

yd

ress

tod

ay.

Can

yo

uco

un

t?Y

es?

I2:

(I2

no

wp

lay

sa

dif

fere

nt

chil

d,

by

the

fict

itio

us

nam

eo

fR

ote

m.)

3…

4…

2…

7…

8…

10

8I1

:C

any

ou

cou

nt

agai

n,

ple

ase?

I2:

7…

4…

5…

1…

3…

2…

10

9I1

:H

ello

No

rma

(ev

ery

on

ela

ugh

s).

Can

yo

up

leas

eco

un

t?L

et’s

hea

ry

ou

cou

nt

I2:

(I2

no

wp

lays

the

role

of

yet

ano

ther

chil

d.)

1…

2…

Lik

eth

is?

10

I1:

Yes

.L

ike

that

I2:

3

11

I1:

Ok

I2:

5…

a…4…

10

12

I1:

Can

yo

up

leas

eco

un

tag

ain

,N

orm

a?I2

:5…

2…

3…

6…

4…

a…b…

c…1

0

13

I1:

An

dla

st,

bu

tn

ot

leas

t,w

eco

me

toY

air

I2:

(I2

no

wp

lays

the

role

of

Yai

r.)

Id

on

’tli

ke

toco

un

t.1…

2…

3…

4…

5…

6…

7…

8…

9…

10

14

I1:

Can

yo

uco

un

tag

ain

?I2

:1…

2…

3…

4.

Th

eyal

way

sas

km

eth

esa

me

thin

g(e

ver

yo

ne

lau

gh

s).

5…

6…

7…

8…

9…

10

68 P. Tsamir et al.

123

counting and decomposition activities and were introduced to literature related to such

activities, such as those mentioned in the literature review of this article (e.g., Baroody

et al. 2006; Gelman and Gallistel 1978). In order to focus on the observation and structure

feedback, we prepared in advance ‘‘a guided observation sheet’’ (see Fig. 1 below). Section

a of that sheet was aimed to help the teachers focus on the children’s knowledge. For

example, if the child was requested to count to 30 but succeeded only as far as 13 and then

started making mistakes and ended with the number 18, it was thought that the teachers

would write that the child knew how to count to 13, but did not know how to count to 30,

and it could not be known if the child knows how to transition from 29 to 30. Sections

b and c were open for general comments that might arise as the teachers viewed the video.

For example, under the heading ‘‘problems that arose,’’ we expected teachers to possibly

comment on technical difficulties associated with videoing children in class. The sheets

were filled in by the teachers as they viewed the video clips and were used to stimulate the

discussion which followed the viewings. After the discussion, the instructor collected the

sheets and handed them to the teacher, whose video was viewed, enabling that teacher to

continue reflecting on her implementation of the assessment task and her colleagues’

comments.

The instructor handed out the guided observations sheets and gave the following

instructions:

Pay attention to the difference between a child who demonstrates knowledge and a

child who cooperates. They are not the same thing. On the handout you may com-

ment on what you observe… what the child knows, what the child does not know,

and what is not clear. That is, it may not be clear if the child does or does not know

something. What difficulties arose during the activity? What interesting ideas arose

from watching the video?

The guided observation had several aims. First, filling in the observation sheet served to

focus teachers on the students’ responses to the task, separating what could be learned

about the specific child’s knowledge and what remained unknown and needs further

investigation. Second, by viewing several clips of purportedly the same task being

implemented by different teachers with different children, the follow-up discussions

could highlight the relationship between what the child is asked and what can be

learned about that child’s knowledge. Subtle differences between how the teachers

implemented similar tasks enabled teachers to hone in on different aspects of children’s

number knowledge. Finally, in Israel, preschools are often separated physically from

each other, and thus, preschool teachers do not usually have the chance to consult with

and receive feedback from their peers. We hoped these sessions would not only

promote teachers’ knowledge of their students, but also would provide a new way for

teachers to share their experiences and build a common language for talking about

students’ knowledge, which might be continued even after the course was finished via

computer forums.

Below, we present transcripts of two segments of a clip presented by one teacher, Lydia

(a pseudonym) who was assessing Mat’s (also a pseudonym) knowledge of counting, such

as counting forward and backward, knowledge of the number symbols, and number

decomposition. Mat is a child in Lydia’s kindergarten class; it is the year before entering

first grade. The segments were viewed by 12 teachers, and the comments they wrote on

their observation sheets are reported following each segment.


123

1. Lydia Up to what number can you count?

2. Mat Up to a hundred

3. Lydia Would you like to try?

4. Mat One, two, three, …twenty-eight, twenty-nine, thirty…(Mat is about to continue but Lydia stops him.)

5. Lydia You can stop. It was amazing. I enjoyed listening to you.Now, I want us to count, count forward

6. Mat One

7. Lydia Wait, counting on, from zero to ten. Go ahead

8. Mat Thirty-one. [Mat looks a bit puzzled.]

9. Lydia Wait; please listen to my question, counting on, from zero to ten

10. Both Zero

11. Mat One, two, …, ten

12. Lydia Thank you. Now I would be very happy if you would count from,from seven to ten. Go ahead

13. Mat Seven, eight, nine, ten

14. Lydia Thank you. Now listen carefully. Count backwards. From ten to zero

15. Mat Ten, nine, …, zero

16. Lydia Please count backwards from seven to zero

17. Mat Seven, [hesitates], six, five, …, zero

While viewing the above segment, teachers were asked to write comments on their

observation sheet in alignment with the outlined categories (see Fig. 1). For each category

listed in the observation sheet (Fig. 1), we gathered what the teachers wrote and listed them

under the category heading (see Table 3). The number of teachers who made each com-

ment is given in the parenthesis.

As we can see from Table 3, when assessing what Mat knew, teachers’ comments ranged

from the very general to the very detailed and in between. One teacher merely wrote that Mat

knew how to count. However, seven teachers noted that Mat could count to 30, and four teachers

specified that he counted from 0 to 10. While it might seem superfluous to write that Mat could

count to 10 when he demonstrated so well that he could count till 30, those teachers were

perhaps stating that he could count from 0 forward and not just from 1 forward. In other words, it

was another detail of Mat’s knowledge worthy of notice. Teachers did not specifically mention

that Mat successfully made the transition from 29 to 30, a notable accomplishment at this age,

especially in light of studies which show the difficulty in this transition (Baroody and Wilkins

1999). Mat did not make any mistakes in his counting and, as expected, teachers did not make

any comments under the heading ‘‘The child did not know.’’ Regarding what was ‘‘not clear,’’

we had intended this section to point to aspects of children’s knowledge which remain unclear

and perhaps untested. For example, in the case of Mat, although he claimed he could count till

100 we are not clear if he can truly do so. The teachers, however, wrote comments in this section

related to Lydia’s instructions, which indeed were not always clear. The teachers pointed out

that the term ‘‘counting on,’’ which is typically used to refer to counting up from some number

other than 1, was not used correctly. For example, Lydia asked Mat to ‘‘count on from 0 to 10.’’

Teachers also pointed to this issue on the section of the observation sheet reserved for problems.

Because of the confusion in the instructions, teachers did not claim that Mat was at fault when he

said ‘‘31’’ after being requested to count on from 0 to 10. Instead of attributing the questionable

response to Mat, teachers correctly noted that if the teacher’s request is confusing, then we

cannot make conclusions about the child’s knowledge.

70 P. Tsamir et al.

123

In the following segment, Lydia is shown assessing Mat’s knowledge of decomposing

the number five. She uses five identical bottle caps and a game of ‘‘hide-and-seek’’ to

discuss with Mat number combinations which make up the number five.

Observation Sheet

Video-clip of: __________ Comments by: _______ Date: _____

a. Assessing the child's knowledge

The child knew … The child did not know … It is not clear…

b. Problems that arose during the interview

c. Interesting ideas

Fig. 1 Guided-observation sheet

1. Lydia [Lydia arranges five identical bottle caps on the table in a straight line]. I have here caps. Let’scount

2. Mat One, two, …, five

3. Lydia How many caps do we have?

4. Mat Five

5. Lydia Five caps. The caps would like to play ‘‘hide and seek’’ with you… I am taking the five caps[takes the caps], yes? And I put my hands behind my back [Lydia puts her hands behind herback.] I am dividing the caps between my hands, and I have in one hand, [Lydia brings onehand back to the table, opens her palm, and shows two caps], how many caps?

6. Mat Two

7. Lydia And how many do I have left in my other hand?

8. Mat Umm… [Mat hesitates] four.

9. Lydia Let us check [brings her other hand back to the table and opens it]. How many?

10. Mat Three

11. And how many altogether?

12. Five

13. Lydia I have five … and they would like to stay five. But they would like to be in differentquantities, because they would like to hide. Let us check, again. [Lydia moves both handsbehind her back again, and then brings one hand forward to the table and opens it showingfour caps.] How many caps are here?

14. Mat [Mat counts to four silently using his fingers.] Four

15. Lydia And how many do I have behind my back?


123

Lydia continues to play this game with Mat showing him 2 caps in one hand. Mat

responds incorrectly that there will be 4 caps in the other hand. She then reverts back to the

first example, showing him 2 caps in one hand and this time he responds correctly that

there will be 3 caps in the other hand.

Regarding the above segment, teachers made several comments on their observation

sheet. As was done for the previous case, comments made for each category (see Fig. 1)

were gathered and are written below in Table 4, along with the number of teachers who

wrote each comment (in parenthesis).

First, it is important to note that the main aim of this task was to assess the child’s

knowledge of decomposing five. That is, does the child know which numbers can be

combined to make five? Perhaps because of this aim, Lydia asked Mat to say how many

caps were in her open palm before requesting him to tell how many caps were hiding. This

allowed Lydia to separate his knowledge of counting caps from his knowledge of the

decomposition of five. Perhaps, this is what led to 11 comments related to Mat’s ability to

count objects (i.e., ability to say the counting words in order, use of the one-to-one

correspondence, cardinality principle, subitizing). It also points to teachers’ awareness of

the complexity of counting objects and knowledge of the counting principles (Gelman and

Gallistel 1978). In other words, although the teachers were aware of the aim of this task,

Table 3 Teachers’ comments of Lydia’s first segment*

Category Comments

The child knew… Counting (1); Counting on (2); Counting up to 10 (3); Counting from 0 to 10(4); Counting up to 30 (7); Continued counting from 30 (3); Stable andacceptable counting beyond 30 (1); Knew from 7 to 10 (7); Countingbackwards (5); Counting from backwards (1); Knows to count from—didnot understand at first (1); From 10 to 0, well done! (7); From 7 to 0, welldone! (4)

The child did not know… –

It is not clear Unclear instruction counting on (there is a confusion in the instruction) (2);Counting on was not presented right (1);

Problems that arose duringthe interview

Understanding instructions (2); No need for help (1); Counting on from 0 to10?? (2); Counting on—a not clear task (1);

Interesting ideas Positive feedback (1)

* The number of teachers who made each comment is listed in parenthesis

continued

16. Mat Ah, one

17. Lydia Let’s check. [Lydia brings her other hand to the table and opens it.Mat smiles.] Let’s play again, ok? How many do I have altogether?

18. Mat Five

19. Lydia Five caps. [Puts her hands behind her back and brings one hand to the table,opens it with five caps]. How many caps do I have in my hand?

20. Mat Five

21. Lydia And how many do I have in my other hand?

22. Mat Zero

23. Lydia Let’s check [brings her other hand to the table and opens it, Mat smiles]

72 P. Tsamir et al.

123

they were still attentive to additional aspects of Mat’s knowledge which could be observed

while implementing this task and did not only focus on the decomposition of five. When it

came to analyzing Mat’s knowledge of decomposing five, we see some very general

comments as well as some more specific comments. Looking back at the video transcript,

we can see that Mat knew what to add to four, to make five, and what to add to five to make

five. This does not mean that Mat would know what to add to 1 to make five or what to add

to zero to make five. In fact, Lydia did not try out these combinations with Mat, and thus, it

might be expected that teachers would note this in the section of ‘‘it is not clear.’’ Yet, there

were no comments related to lack of clarity. Furthermore, stating that Mat knew that five

may be decomposed into 1 and 4 is not accurate. Nor is it accurate to say that Mat knew the

combination of 0 and 5. Most of the teachers did not make this distinction.

Taking together the two comment sheets, we see that teachers are in the process of

learning how to assess students’ knowledge. They are aware of some details, but not others.

This could be related to the specific content as well as to the implementation of the task by

Lydia. During the discussion which followed the viewing, teachers were interested in

comparing what they saw in this clip with other clips viewed during the same session. In

particular, they were interested in comparing the different ways teachers gave instructions.

The instructor used this opportunity to highlight what teachers may learn about students

from different questions. For example, a different teacher asked her young student to count

backwards from 10 and gave the following instructions: ‘‘Count backwards from ten. Ten.

Nine…’’ In other words, she started the process of counting backwards with the child,

whereas Lydia did not. The instructor clarifies the difference:

I want to emphasize the distinction between the instructions. Lolly (a different

teacher) said that the child in her class did not respond when she was requested to

count backwards and so Lolly thought that maybe she should have started off the

child by saying 10, 9. Lydia’s child responded right away to the instructions. But

what happens when a child does not respond? Does it mean that the child does not

Table 4 Teachers comments of Lydia’s second segment*

Category Comments

The child knew… Counting objects (1); Counting up to 5 objects (3); One-to-onecorrespondence (2); The principle of cardinality (3); Subtilizing 1 and 3(2); Decomposing the number (1); Decomposing the number 5, but doesnot know 2 and 3 (1); Decomposing the number 5, Mat knew 1–4 and 5–0(1); Decomposing number 5 as 2 ? 3, 4 ? 1, 3 ? 2 (1); Can complete thequantity (2)

The child did not know… Decomposing the number (1); Decomposing 5, Mat did not know 2–3, buthe did know 1–4 (2); Decomposing the number 5—when there were 2 inone hand he said 4 are missing (1); Decomposing number 5 if in the onehand there is 3, he said 4 are missing (1); Decomposing the number 5while 3 caps were shown, he did not know that 2 are needed to complete[the 5] (1); Did not understand how to compose 5 the first time (hadtrouble with 2 and 3) (1)

It is not clear –

Problems that arose duringthe interview

Decomposing the number (1)

Interesting ideas The teacher explained the task in a nice and clear way (1); Clear instructions(1)

* The number of teachers who made each comment is listed in parenthesis


123

know how to count backwards or maybe the child just does not understand the

instructions, what it means to count backward? One option in that case is to say

10…9… and then ask the child to continue.

In general, the observation sheets served to focus the teachers on the questions which were

asked as well as children’s actions, and the discussions which followed afforded the

instructor additional opportunities for discussing the rich and varied mathematical

knowledge displayed by the children.

In the next section, we describe the method and results used to investigate teachers’

knowledge of young children’s number conceptions, before and after participating in our

program.

Investigating teachers’ knowledge and self-efficacy

Method

Participants in the study were a group of 25 teachers who took part in our programs. At the

time of the study, all of the teachers were teaching 4–6-year-old children in municipal

preschools. All had a first degree in education from a credited teaching college. At the

beginning of the program and again at the end, teachers were requested to fill out a two-

part questionnaire. The first part began with six questions related to participants’ self-

efficacy for identifying specific aspects of children’s conceptions of numbers. That is, did

teachers believe in their ability to point out common mistakes children make when learning

to count? Did teachers believe in their ability to differentiate between counting tasks that

may be easier or more difficult for preschool children to perform? A four-point Likert scale

was used to rate participants’ agreements with positive self-efficacy statements: 1—I

strongly agree that I am capable; 2—I agree that I am capable; 3—I somewhat agree that I

am capable; 4—I do not agree that I am capable. The actual questions are presented in

Table 5. After participants completed this part of the questionnaire, they handed it in and

received the second part of the questionnaire.

The second part of the questionnaire consisted of knowledge questions. These questions

followed the self-efficacy questions in order to allow the participants to evaluate their self-

efficacy before actually engaging in the task. Participants were asked to assess how many

children at the end of kindergarten, before entering first grade, would be able to complete

various number-related tasks—almost all children, many, about half, few, or almost none?

For both of our questionnaires, two independent referees, experts in the field of mathe-

matics education, evaluated the use of Likert scales and agreed that the values were equally

distant. All questionnaires were completed in the presence of the researcher. The actual

questions are presented in Table 6. Many of the items on the questionnaire were based on

our previous experiences with young children and our investigations of their number

conceptions (Tirosh and Tsamir 2008). All items on the questionnaire were consistent with

the requirements of the mandatory mathematics preschool curriculum.

It is important to note that while the teachers’ estimates of children’s knowledge were

compared with the knowledge of children not in these teachers’ kindergartens, our previous

studies took place in the same geographic location as the kindergartens of the teachers in

the current study. In addition, the children who participated in our previous studies came

from the same socio-economic background as the children learning in the kindergarten

classes of the teachers in our current program.

74 P. Tsamir et al.

123

Data analysis included calculating the means and standard deviations (SD) of the self-

efficacy and knowledge scores on both the pre and posttests and carrying out paired-

samples t tests to compare the scores (details are given in the next section). The use of

these techniques is supported by many studies that used Likert scales to investigate par-

ticipants’ self-efficacy and then analyzed the results by reporting on means and SD. For

example, Tschannen-Moran and Hoy (2007) reported on the means (and SD) of teachers’

self-efficacy beliefs based on a Likert scale questionnaire and went on to use t tests to

compare the difference between the self-efficacy beliefs of novice teachers versus career

teachers. Zimmerman et al. (1992) used Likert scale questionnaires to investigate students’

self-efficacy for self-regulated learning and academic self-efficacy. They too, computed

means and SD based on the collected data. Additional studies that used Likert scales to

investigate self-efficacy, where analysis of the results included calculating means, SD,

Table 5 Teachers’ self-efficacy related to knowledge of students, pre and post

Item Question: I am capable of identifying … Pretest Posttest

M SD M SD

3 …which combinations of numbers that add up to 7 children find difficultto learn

2.38 .97 3.08 .76

6 …counting skills that most children are competent performing (whenconsidering counting up till 30)

2.52 .87 3.12 .67

5 …which number children find difficult to say the number which comesimmediately beforehand

2.67 .76 3.08 .70

1 …which number symbols from 1 to 9 children find difficult to recognize 2.76 1.13 3.12 1.09

2 …different arrangements of 8 items which children find difficult to count 2.92 .97 3.36 .57

4 …which numbers children find difficult to say the number whichimmediately follows

3.16 .62 3.32 .69

Table 6 Teachers’ estimates of students’ abilities to perform various tasks, pre and post

Item Question: How many students will be able to… Pre Post

M SD M SD

6 say that changing the position of objects to be counted does not changethe amount there are?

2.58 .83 3.40 .91

10 say how many apples to add to 3 apples in order to make 7 apples? 2.76 .93 3.13 .81

8 say which number comes right before 6? 3.56 1.12 4.16 .85

3 count from 6 to 15? 3.79 1.18 4.13 .90

2 count backwards from 7? 3.79 1.22 4.20 .82

7 say which number comes right after 6? 3.96 1.08 4.36 .64

9 identify the number symbol for 9? 4.08 1.06 4.24 .83

1 count from 1 to 30? 4.16 1.10 4.56 .51

5 say that it does not matter if you count objects from the leftor from the right?

4.25 .99 4.16 .80

4 to count 8 bottle caps places in a straight row? 4.32 1.11 4.80 .41


123

Pearson correlations, and/or t tests include Bates et al. (2011) and Skaalvik and Skaalvik

(2007).

Results

We begin this section with the results of the first part of the questionnaire related to

teachers’ self-efficacy. We then present the results of investigating teachers’ knowledge of

students. Finally, we compare teachers’ self-efficacy with their knowledge.

Self-efficacy

We begin by reporting on the results of the self-efficacy questions. Table 5 reports the

means and standard deviations on both the pretests and posttests for each self-efficacy

question. Statements are arranged according to the level of self-efficacy reported by the

practicing teachers on the pretest, from low to high. Item numbers represent the order in

which they were presented on the questionnaire.

Cronbach alpha was used to measure internal consistency. A coefficient of a = .730 on

the pretest and a = .805 on the posttest indicated that the items most likely formed a

coherent group. We thus configured for each participant a mean self-efficacy score. A

paired-samples t test indicated that teachers’ self-efficacy after the participating in the

program (M = 3.18, SD = .54) was significantly higher than their self-efficacy prior to the

program (M = 2.74, SD = .57), t(24) = 3.08, p = .005. In other words, teachers

increased their belief in their ability to identify students’ number conceptions and skills.

Teachers’ knowledge of students’ number conceptions

Regarding teachers’ knowledge of children, as described above, teachers were presented

with various number tasks and asked to estimate on a scale from 1 to 5 how many students

(1—almost none, 2—a few, 3—about half, 4—many, 5—almost all) would be able to

complete the task correctly. The means of participants’ estimations of students’ abilities for

each task and standard deviation are shown in Table 6. The table is arranged according to

the teachers’ estimations on the pretest, from the tasks estimated to be most difficult to

those estimated to be least difficult. The item number reflects the order of the questions in

accordance to how they appeared on the questionnaire.

In general, even before participating in the program, teachers believed that for most

tasks, more than half of kindergarten children at the end of their kindergarten year would

be able to correctly solve the tasks. The exceptions were items 6 and 10. For those tasks,

teachers believed that less than half of the children would be able to succeed. After

participating in the program, teachers increased their estimates of children’s abilities for all

items except item 5. This point will be discussed further in the discussion at the end of the

article.

In order to assess teachers’ knowledge of kindergarten children’s abilities to perform

number tasks, we compared the participants’ estimates with previous research we had

conducted with kindergarten children (Tirosh and Tsamir 2008). This research was much

in line with studies by other researchers (e.g., Baroody and Wilkins 1999). For example, in

our previous research we asked children (N = 82) to count from 1 to 30, to count back-

wards from 7, and so on. In those studies, we configured the percentage of children who

succeeded on the task. We then reconfigured the 1–5 scale the teachers used, to reflect the

76 P. Tsamir et al.

123

0–100 % scale used when configuring results of the children’s performance. Reconfigu-

ration was carried out in the following way. The lowest score on both scales was 1 and

0 %, respectively, and the highest score was 5 and 100 %, respectively. We transformed

the 1–5 scale by using the linear equation: y = 25(x - 1) where x represents the scale used

for teachers and y the scale used for children. We could then compare teachers’ estimates

of how many children would succeed at a task, with results of children’s actual perfor-

mances. This is presented in Table 7 where the order of the tasks is once again presented in

increasing order of teachers’ estimates.

In order to assess in general, if a teacher improved her knowledge regarding students’

ability to perform the above tasks, we calculated for each item the absolute difference

between the teachers’ assessment and the students’ actual performance. In other words, we

were interested in knowing how close teachers’ estimations were to actual students’ per-

formance. Each participant was then given a mean knowledge score on the pretest and a

mean knowledge score on the posttest, representing the mean absolute difference between

that teacher’s assessments and children’s actual performances. A paired-samples t test

indicated that the knowledge score on the posttest (M = 20.27, SD = 10.23) was signif-

icantly lower than the knowledge score on the pretest (M = 24.58, SD = 5.87),

t(24) = 2.38, p = .026. Teachers came closer to assessing children’s actual abilities after

participating in the program.

Despite the general improvement of teachers’ knowledge of their students, it is still

worthwhile to note where a rather large difference between teachers’ estimates and chil-

dren’s performance remained. The most notable occurrence was for item 1, counting from

1 to 30. There are several reasons why teachers might have overestimated students’

abilities for this task. First, if a child makes one or two errors when counting to 30, the

teacher may still feel that overall, the child can count to 30. One can almost imagine a

preschool teacher thinking, what is one error when counting all the way to 30? On the other

Table 7 Teachers’ estimates compared with children’s performance

Item Question: How many students will beable to…

Mean estimates(Pre) translatedto percents

Mean estimates(Post) translatedto percents

Percent ofchildren (N = 82)who succeeded

6 say that changing the position ofobjects to be counted does notchange the amount there are?

39.5 60 65

10 say how many apples to add to 3apples in order to make 7 apples?

44 53.25 52.5

8 say which number comes rightbefore 6?

64 79 59

3 count from 6 to 15? 69.75 78.25 68

2 count backwards from 7? 69.75 80 60

7 say which number comes right after 6? 74 84 94

9 identify the number symbol for 9? 77 81 88

1 count from 1 to 30? 79 89 49

5 say that it does not matter if you countobjects from the left or from theright?

81.25 79 77

4 to count 8 bottle caps places in astraight row?

83 95 93


123

hand, in our research, if a child made even one mistake, we coded his counting as incorrect.

Another possible reason for teachers’ over estimation may stem from the circumstances of

teachers’ assessment. From our observations of preschool classrooms, we found that

teachers often practice counting in unison and with the teacher participating. Thus, their

estimates of children’s counting skills may not stem from listening to each individual child.

In addition, during the program, teachers became aware of the problems children may have

when counting and were motivated to encourage children to practice this skill. Thus, after

participating in the program, their estimations of their own students may have indeed been

accurate while our estimates were based on preschool children who learned in preschools

where not all of the teachers participated in professional development focused on math-

ematics. It could be that the children we investigated had less practice and made more

mistakes than children learning in the participants’ classes. This last reason for the

teachers’ overestimation may also account for their overestimation of children’s other

skills, such as counting backwards from 7.

Comparing knowledge with self-efficacy

When investigating both self-efficacy and knowledge, it is also important to analyze the

relationship between the two. That is, does a very knowledgeable person have a corre-

spondingly high self-efficacy or might the self-efficacy be low, despite having a solid

knowledge base? Likewise, if a person is not very knowledgeable, is that person’s self-

efficacy low? In order to investigate this relationship, we compared participants’ mean

knowledge score with their mean self-efficacy both for the pretest and the posttest. Results

indicated a significant negative correlation between the two variables on the pretest

r =-.449, n = 25, p = .024. However, no significant correlation was found on the

posttest. These results are curious and somewhat unexpected. Regarding the pretests, it

seems that teachers who were knowledgeable of their students, never-the-less had a low

self-efficacy regarding their ability to assess students’ knowledge while teachers who were

less knowledgeable had a high self-efficacy. In short, on the pretest, teachers did not

accurately assess their own knowledge. After the program, we hypothesized that teachers

would not only increase their knowledge of students’ number conceptions (which they

did), as well as increase their self-efficacy (which they also did), but also would do so in a

corresponding manner. However, no correlation between knowledge and self-efficacy was

detected. That is, participants’ self-efficacy did not necessarily increase in proportion to

their knowledge. While this might not seem as a positive outcome, in fact it is an

improvement over the negative correlation shown on the pretest. Furthermore, if we look at

the self-efficacy scores, on the pretest only 10 participants had a self-efficacy between 3

and 4, whereas on the posttest, 19 teachers had a self-efficacy score between 3 and 4. Thus,

it could be that the lack of a positive correlation was due to a ceiling effect which may have

constrained the results of the self-efficacy score. To summarize, although we cannot say

that teachers’ knowledge and self-efficacy were correlated, we may say that at the end of

the program, teachers have begun a process which has shaken up the negatively correlated

self-efficacy with which they began.

Putting it all together

The significance of this article lies in the details which are provided to the reader. Many

articles related to preschool professional development, such as those discussed in the

78 P. Tsamir et al.

123

literature review, describe in general how teachers come to be familiarized with students’

conceptions. Our study offers concrete examples and excerpts which can be used by other

professional development providers as they plan programs for preschool teachers. In

addition, few of the previous studies analyzed specific aspects of teachers’ knowledge of

young students along with related self-efficacy beliefs. This article investigates both of

these issues. This too, may be helpful to professional development providers. Knowing that

teachers may over- or under-estimate children’s number abilities and knowing specifically

which abilities are over-estimated and which are under-estimated, may allow program

developers to address this issue. Knowing that professional development may impact on

self-efficacy, may also influence the way a program is delivered. In this section we discuss

how the elements of the program were related and how the results of the questionnaires

may shed light on the program.

Regarding the first element of the program described above, Shulman (1986) pointed

out that knowing students’ conceptions, including their preconceptions and misconcep-

tions, is an essential element of teachers’ pedagogical-content knowledge. Teachers should

be able to anticipate and predict what examples and tasks students might find confusing or

difficult (Ball et al. 2008). Thus, while most preschool teachers know, in general, that

counting skills may be, and perhaps even ought to be, developed during the preschool

years, not all teachers are aware of what these skills entail and that although a child may be

able to recite the counting sequence, that child may not be able to count a group of items or

know that the order of the elements to be counted is not important. Thus, the first element

of the program described in this article, was essential. According to An et al. (2004),

knowledge of students’ conceptions includes knowing how to use this knowledge in the

classroom. This includes being able to implement tasks which can promote as well as

assess children’s knowledge. The tasks which the teachers implemented with their young

students were designed together during program sessions. In order to build appropriate

tasks, whether they are assessment tasks or tasks intended to promote knowledge, the

designer must be aware of common conceptions. Once teachers were aware that some

children may acquire the one-to-one correspondence principle but not quite realize that the

last number recited represents the number of elements in the set, teachers appreciated the

necessity of asking the child again, after the counting process was over, to tell how many

objects were in the set. Once teachers realized that children find it difficult to count on from

some number other than one, they began to think of counting tasks which included this

skill. In a related study (Tsamir et al. 2014), we found that promoting preschool teachers’

knowledge of appropriate mathematical tasks is interrelated with promoting their knowl-

edge of students. In this article, we also claim that when designing tasks with teachers,

especially assessment tasks, we may in the process promote teachers’ knowledge of their

students and enhance their ability to observe different aspects of children’s knowledge.

This leads us to the second element of the program illustrated above. Being able to analyze

children’s mathematical thinking and learning is an essential aspect of knowing students

(Ginsburg et al. 2004; Schoenfeld and Kilpatrick 2008). However, teachers do not always

know what to look for. This brings us back to the first element discussed, knowing stu-

dents’ common conceptions. Yet, even when teachers are aware of students’ common

conceptions, and are implementing appropriate tasks, they need guidance in order to dis-

cern the subtle differences between their students’ knowledge. Thus, we believe, that the

elements of the program described and illustrated in this article were necessary as well as

related to each other. But were they sufficient?

In attempting to analyze the program and what aspects of knowing students may have

been less developed or even missed, we return to the results of our study. Recall that


123

teachers’ estimates of their students’ abilities increased as a result of participating in the

program. In general, we see this as a positive result. Teachers who believe in their students’

ability to successfully perform tasks and learn new conceptions are more likely to spend

time developing those skills and conceptions. This is especially important in light of

studies which found that preschool teachers spend little class time on mathematics-related

activities (Farran et al. 2007; Ginsburg et al. 2008). Furthermore, it has been shown that

teachers’ beliefs in their students’ abilities can positively impact on students’ motivation to

succeed, in turn affecting student achievement, even when those students come from

disadvantaged backgrounds (Halvorsen et al. 2009; Mayer 2008). Thus, raising the pre-

school teachers’ expectations of their students may have had additional positive effects.

We also saw that teachers’ improved the accuracy of their estimations related to students’

abilities to perform number-related tasks. Teachers’ knowledge of analyzing students’

thinking was less studied. While we engaged teachers in observing the implementation of

mathematical tasks, this element of the program was used to promote teachers’ knowledge

of their students and was not used as a tool to assess teachers’ knowledge. In the future, we

may wish to investigate more explicitly, teachers’ analysis of students’ ways of thinking.

Despite the overall success of the program, we also noticed gaps in teachers’ knowledge

even after the program was finished. For some of the tasks, teachers overestimated stu-

dents’ abilities. While we offered some reasons for this possibility in the results section,

and while we believe that in general, it is more beneficial for teacher to overestimate

students’ abilities than to underestimate them, we still need to consider how the program

might be adjusted in order for teachers to develop a more realistic view of children’s

abilities. For example, while the teachers in this program viewed video clips of themselves

engaging children in number tasks, it might be beneficial for the teachers to view video

clips of other children engaging in number tasks. Maher (2008), for example, reviewed

several studies where video cases and collections, including collections originally made for

research, were used in teacher education. She noted that collections have the benefit of

illustrating diversity in setting and in practice. It might be that this diversity was lacking in

our program.

Finally, we consider the self-efficacy issue. In general, as pointed out in the literature

review, studies have shown that teachers’ self-efficacy is related to enthusiasm for and

commitment to teaching (Allinder 1994; Coladarci 1992) and yet few studies have

investigated preschool teachers’ self-efficacy, specifically their self-efficacy for teaching

mathematics. While it was not surprising to find a medium level of self-efficacy prior to the

program, we were surprised by the negative correlation between self-efficacy and

knowledge. During the program, we attempted to promote teachers’ self-efficacy related to

students’ knowledge by using teaching methods which were non-threatening and which

were collaborative. While we succeeded in promoting teachers’ self-efficacy and while we

succeeded in breaking the negative correlation, we cannot say that at the end of the

program knowledge and self-efficacy were positively correlated. It could be that our

implicit methods for promoting self-efficacy were too indirect. Self-efficacy beliefs have

several sources (Bandura 1986). One of these is appropriate feedback. In our efforts to

encourage teachers to implement more mathematical tasks in their classroom, we may not

have given enough explicit feedback regarding their knowledge of their students. In the

future, we may consider using the observations sheets described above as a tool for

evaluating teachers’ knowledge of their students as well as a teaching tool. Instead of

merely collecting the sheets, we may provide comments on the sheets, pointing out to each

teacher what she did notice about her student as well as did not notice. We also note that

only a small aspect of teachers’ self-efficacy was investigated and perhaps a more

80 P. Tsamir et al.

123

comprehensive study is required in order to capture the variances in teachers’ self-efficacy

related to their students’ number knowledge.

Recently, a special issue in this journal was dedicated to developing preschool teachers’

knowledge for teaching mathematics (Tsamir et al. 2011). In that issue, it was pointed out

that while progress is being made, much is still to be done. None of the articles in that issue

studied teachers’ self-efficacy beliefs. None of the articles in that issue specifically focused

on teachers’ knowledge of their students. In this article, we highlighted the very important

aspect of teachers’ knowledge related to knowing their students as well as incorporated

into our investigation teachers’ self-efficacy related to students’ knowledge. We

acknowledge that number concepts is but one domain of mathematical knowledge

developed in preschool and that additional studies should investigate teachers’ knowledge

of their students’ geometry and pattern skills, including related self-efficacy beliefs, as well

as perhaps other domains developed during the early years.

Acknowledgments This study was supported by the Israel Science Foundation (Grant No. 654/10).

References

Allinder, R. M. (1994). The relationship between efficacy and the instructional practices of special educationteachers and consultants. Teacher Education and Special Education, 17(2), 86–95.

An, S., Kulm, G., & Wu, Z. (2004). The pedagogical content knowledge of middle school mathematicsteachers in China and the US. Journal of Mathematics Teacher Education, 7, 145–172.

Ashton, P. T., & Webb, R. B. (1986). Making a difference: Teachers’ sense of efficacy and studentachievement. New York: Longman.

Ball, D., Thames, M., & Phelps, G. (2008). Content knowledge for teaching. Journal of Teacher Education,59(5), 389–407.

Bandura, A. (1986). Social foundations of thought and action: A social cognitive theory. Englewood Cliffs,NJ: Prentice Hall.

Baroody, A. J. (1987). Children’s mathematical thinking: A developmental framework for preschool, pri-mary, and special education teachers. New York: Teacher’s College Press.

Baroody, A. J., Lai, M. L., & Mix, K. S. (2006). The development of young children’s early number andoperation sense and its implications for early childhood education. In B. Spodek & O. Saracho (Eds.),Handbook of research on the education of young children (2nd ed., pp. 187–221). Mahwah, NJ:Erlbaum.

Baroody, A. J., & Wilkins, J. L. M. (1999). The development of informal counting, number, and arithmeticskills and concepts. In J. V. Copley (Ed.), Mathematics in the early years (pp. 48–65). Reston, VA:National Council of Teachers of Mathematics.

Bates, A. B., Latham, N., & Kim, J. A. (2011). Linking preservice teachers’ mathematics self-efficacy andmathematics teaching efficacy to their mathematical performance. School Science and Mathematics,111(7), 325–333.

Bobis, J., Clarke, B., Clarke, D., Thomas, G., Wright, B., Young-Loveridge, J., et al. (2005). Supportingteachers in the development of young children’s mathematical thinking: Three large scale cases.Mathematics Education Research Journal, 16(3), 27–57.

Brady, S., Gillis, M., Smith, T., Lavalette, M., Liss-Bronstein, L., Lowe, E., et al. (2009). First gradeteachers’ knowledge of phonological awareness and code concepts: Examining gains from an intensiveform of professional development and corresponding teacher attitudes. Reading and Writing, 22(4),425–455.

Briars, D., & Siegler, R. S. (1984). A featural analysis of preschoolers counting knowledge. DevelopmentalPsychology, 20(4), 607–618.

Carpenter, T. P., Fennema, E., Peterson, P. L., Chiang, C. P., & Loef, M. (1989). Using knowledge ofchildren’s mathematics thinking in classroom teaching: An experimental study. American EducationalResearch Journal, 26(4), 499–531.

Clements, D. H., & Sarama, J. (2007). Effects of a preschool mathematics curriculum: Summative researchon the Building Blocks project. Journal for Research in Mathematics Education, 38, 136–163.


123

Clements, D., Sarama, J., Spitler, M., Lange, A., & Wolfe, C. B. (2011). Mathematics learned by youngchildren in an intervention based on learning trajectories: A large-scale cluster randomized trial.Journal for Research in Mathematics Education, 42, 127–166.

Coladarci, T. (1992). Teachers’ sense of efficacy and commitment to teaching. Journal of ExperimentalEducation, 60, 323–337.

Dellinger, A. B., Bobbett, J. J., Olivier, D. F., & Ellett, C. D. (2008). Measuring teachers’ self-efficacybeliefs: Development and use of the TEBS-Self. Teaching and Teacher Education, 24(3), 751–766.

Even, R., & Tirosh, D. (1995). Subject matter knowledge and knowledge about students as sources ofteacher presentations of the subject matter. Educational Studies in Mathematics, 29(1), 1–20.

Farran, D. C., Lipsey, M. W., Watson, B., & Hurley, S. (2007). Balance of content emphasis and childcontent engagement in an early reading first program. In Paper presented at the American EducationalResearch Association, Chicago, IL.

Fennema, E., & Franke, M. L. (1992). Teachers’ knowledge and its impact. In D. A. Grouws (Ed.),Handbook of research on mathematics teaching and learning (pp. 147–165). New York: MacmillanPublishing Company.

Fuson, K. C. (1988). Children’s counting and concepts of number. New York: Springer.Fuson, K. C., & Hall, J. (1982). ‘The acquisition of early number word meanings: A conceptual analysis and

review. In H. P. Ginsburg (Ed.), Children’s mathematical thinking (pp. 49–107). New York: AcademicPress.

Fuson, K. C., Richards, J., & Briars, D. J. (1982). The acquisition and elaboration of the number wordsequence. In C. J. Brainerd (Ed.), Children’s logical and mathematical cognition (pp. 33–92). NewYork: Springer.

Geary, D. C., Bow-Thomas, C. C., & Yao, Y. (1992). Counting knowledge and skill in cognitive addition: Acomparison of normal and mathematically disabled children. Journal of Experimental Child Psy-chology, 54(3), 372–391.

Gelman, R., & Gallistel, C. (1978). The child’s understanding of number. Cambridge: Harvard UniversityPress.

Ginsburg, H. P., Jang, S., Preston, M., VanEsselstyn, D., & Appel, A. (2004). Learning to think about earlychildhood mathematics education: A course. In C. Greenes & J. Tsankova (Eds.), Challenging youngchildren mathematically (pp. 40–56). Boston, MA: National Council of Supervisors of Mathematics.

Ginsburg, H. P., Lee, J. S., & Boyd, J. S. (2008). Mathematics education for young children: What it is andhow to promote it (pp. 1–22). XXII(I): Social Policy Report.

Hackett, G., & Betz, N. (1989). An exploration of the mathematics self-efficacy/mathematics performancecorrespondence. Journal for Research in Mathematics Education, 20(3), 261–273.

Halvorsen, A., Lee, V., & Andreade, F. (2009). A mixed-method study of teachers’ attitudes about teachingin urban and low-income schools. Urban Education, 44(2), 181–224.

Han, Y., & Ginsburg, H. P. (2001). Chinese and English mathematics language: The relation betweenlinguistic clarity and mathematics performance. Mathematical Thinking and Learning, 3(2–3),201–220.

Israel national mathematics preschool curriculum (INMPC) (2008). Retrieved April 7, 2009, from http://meyda.education.gov.il/files/Tochniyot_Limudim/KdamYesodi/Math1.pdf.

Maher, C. (2008). Video recording as pedagogical tools in mathematics teacher education. In D. Tirosh & T.Wood (Eds.), International handbook of mathematics teacher education: Tools and processes inmathematics teacher education (Vol. 2, pp. 65–83). Rotterdam: Sense Publishers.

Mayer, A. (2008). Expanding opportunities for high academic achievement: An international Baccalaureatediploma program in an urban high school. Journal of Advanced Academics, 19(2), 202–235.

Mulligan, J. T., Prescott, A., Papic, M., & Mitchelmore, M. (2006). Improving early numeracy through apattern and structure mathematics awareness program (PASMAP). In Building connections: Theory,research and practice (Proceedings of the 28th annual conference of the Mathematics EducationResearch Group of Australia) (pp. 376–383).

National Association for the Education of Young Children & National Council of Teachers of Mathematics(NAEYC & NCTM) (2002). Position statement. Early childhood mathematics: Promoting goodbeginnings. Available: www.naeyc.org/resources/position_statements/psmath.htm.

Piaget, J., Inhelder, B., & Szeminska, A. (1960). The child’s conception of geometry. London: Routledgeand Kegan Paul.

Schifter, D. (1998). Learning mathematics for teaching: From the teacher’s seminar to the classroom.Journal for Mathematics Teacher Education, 1(1), 55–87.

Schoenfeld, A. H., & Kilpatrick, J. (2008). Toward a theory of proficiency in teaching mathematics. In D.Tirosh & T. Wood (Eds.), The international handbook of mathematics teacher education: Tools andprocesses in mathematics teacher education (Vol. 2, pp. 321–354). Rotterdam: Sense Publishers.

82 P. Tsamir et al.

123

http://meyda.education.gov.il/files/Tochniyot_Limudim/KdamYesodi/Math1.pdf

http://meyda.education.gov.il/files/Tochniyot_Limudim/KdamYesodi/Math1.pdf

http://www.naeyc.org/resources/position_statements/psmath.htm

Seo, K. H., & Ginsburg, H. P. (2004). What is developmentally appropriate in early childhood mathematicseducation? Lessons from new research. In D. H. Clements, J. Sarama, & A. M. DiBiase (Eds.),Engaging young children in mathematics: Standards for early childhood mathematics education (pp.91–104). Hillsdale, NJ: Erlbaum.

Shulman, L. S. (1986). Those who understand: Knowledge growth in teaching. Educational Researcher,15(2), 4–14.

Skaalvik, E. M., & Skaalvik, S. (2007). Dimensions of teacher self-efficacy and relations with strain factors,perceived collective teacher efficacy, and teacher burnout. Journal of Educational Psychology, 99(3),611.

Starkey, P., Klein, A., & Wakeley, A. (2004). Enhancing young children’s mathematical knowledge througha pre-kindergarten mathematics intervention. Early Childhood Research Quarterly, 19, 99–120.

Tirosh, D., & Tsamir, P. (2008). Starting right: Mathematics in preschool. Unpublished research report. InHebrew.

Tirosh, D., Tsamir, P., Levenson, E., Tabach, M., & Barkai, R. (2011). Prospective and practicing preschoolteachers’ mathematics knowledge and self-efficacy: Identifying two and three dimensional figures. In17th MAVI (Mathematical Views) conference. Bochum, Germany. Retrieved May 7, 2012 from http://www.ruhr-uni-bochum.de/ffm/Lehrstuehle/Roesken/maviarticles.html.

Torner, G. (2002). Mathematical beliefs. In G. C. Leder, E. Pehkonen, & G. Torner (Eds.), Beliefs: A hiddenvariable in mathematics education? (pp. 73–94). The Netherlands: Kluwer.

Tsamir, P., Tirosh, D., & Levenson, E. (2011). Special issue: Windows to early childhood mathematicsteacher education. Journal for Mathematics Teacher Education, 14(2), 89–92.

Tsamir, P., Tirosh, D., Levenson, E., Barkai, R., & Tabach, M. (in press). Facilitating proficient mathematicsteaching in Preschool. In Y. Li & J. N. Moschkovich (Eds.), Proficiency and beliefs in learning andteaching mathematics—learning from Alan Schoenfeld and Gunter Toerner.

Tsamir, P., Tirosh, D., Levenson, E., Tabach, M., & Barkai, R. (2014). Employing the CAMTE framework:Focusing on preschool teachers’ knowledge and self-efficacy related to students’ conceptions. In C.Benz, B. Brandt, U. Kortenkamp, G. Krummheuer, S. Ladel & R. Vogel (Eds.), Early mathematicslearning—selected papers from the POEM 2012 conference (pp. 291–306). New York: Springer.

Tschannen-Moran, M., & Hoy, A. (2007). The differenetial antecedents of self-efficacy beliefs of novice andexperienced teachers. Teaching and Teacher Education, 23, 944–956.

Wheatley, K. F. (2002). The potential benefits of teacher efficacy doubts for educational reform. Teachingand Teacher Education, 18(1), 5–22.

Zimmerman, B., Bandura, A., & Martinez-Pons. (1992). Self-motivation for academic attainment: The roleof self-efficacy beliefs and personal goal setting. American Educational Research Journal, 29,663–676.


123

http://www.ruhr-uni-bochum.de/ffm/Lehrstuehle/Roesken/maviarticles.html

http://www.ruhr-uni-bochum.de/ffm/Lehrstuehle/Roesken/maviarticles.html

developing preschool teachers’ knowledge of students’ number conceptions

Documents