Mathematical knowledge and overall practice in initial teacher education of early childhood teachers

Godoy, Tatiana Cecilia Goldrine; Reyes-Santander, Pamela; Ayarza, Raimundo Olfos

doi:10.1590/S1413-24782021260061

Tatiana Cecilia Goldrine Godoy

Pontificia Universidad Católica de Valparaíso, Valparaíso, Chile.

http://orcid.org/0000-0002-7377-4221

Pamela Reyes-Santander

Universidad de Bielefeld, Bielefeld, Germany.

http://orcid.org/0000-0002-3422-2627

Raimundo Olfos Ayarza

Pontificia Universidad Católica de Valparaíso, Valparaíso, Chile.

http://orcid.org/0000-0002-9886-4282

Tatiana Cecilia Goldrine Godoy has a doctorate in Educational Psychology from the Universidad Nacional Autónoma de México (México). She is a professor at Pontificia Universidad Católica de Valparaíso (Chile). E-mail: tatiana.goldrine@pucv.cl

Pamela Reyes-Santander has a doctorate in Sciences from the University of Augsburg (Germany). She is a professor at Universidad de Bielefeld (Germany). E-mail: reyes.santander.pamela@gmail.com

Raimundo Olfos Ayarza has a doctorate in Education from the Universtiy of Wales (United Kingdom). He is a professor at Pontificia Universidad Católica de Valparaíso (Chile). E-mail: raimundo.olfos@pucv.cl

Conflicts of interest: The authors declare they don’t have any commercial or associative interest that represents conflicts of interest about the manuscript.

Authors’ contributions: Conceptualization, Formal Analysis, Data Curation, Writing - First Draft, Writing - Review and Edition, Investigation, Methodology, Validation and Visualization: Goldrine Godoy T. Conceptualization, Writing - First Draft, Writing - Review and Edition: Reyes Santander, P. Project Management, Conceptualization, Investigation, Funding Obtainment: Olfos Ayarza R.

Component		Dimensions
Component		Teacher Knowledge	Teacher Practice
CK		Appropriate mathematical concepts and semiotic representations for Early Childhood Education.	Teaching mathematical notions appropriate to Early Childhood Education. Mathematical objects and representations.
PCK	PCK-T	Focuses on mathematics teaching. Sequence of mathematical tasks. Manipulative and representative materials.	Organisation of teaching. Creation and implementation of learning opportunities. Analysis of manipulative and representative materials.
PCK	PCK-MTEC	Teacher knowledge of mathematical thinking in children. Stages of learning mathematical notions. Knowledge of children’s actions, strategies, and difficulties.	Teacher mediation of children’s actions, strategies, and difficulties.

Trainee teacher	Knowledge Test (%)	Concept Map (%)	Classroom Observation Form (%)
Helen	93	77	92
Lauren	43	30	36

Activity: How many chickpeas do we need for lunch?

Event: Presentation of the problem

H: Children, do you remember the activity we did yesterday? What did we have to do?
Student 1 (S1): Count dots and get the same number of chickpeas.
H: Let me tell you, my mom wants to cook a delicious lunch […] made with chickpeas, but she needs an exact quantity [of chickpeas]. […]. There are bags on each chair, so you will have to look to see how many chickpeas are missing until my mom has the exact number she wants - some bags already have some chickpeas in them. How many chickpeas do we need to get the desired quantity? [Children worked in pairs. The chickpeas were on another table, and the cards presented the numbers 13, 14, and 15.]

Event: Chickpea counting and calculation of chickpeas needed to reach the required quantity

H: How many chickpeas are there? [To a couple of children].
S2: Seven.
H: How many do we need in the bag in total?
N2: Thirteen.
H: If there are seven, how many more do you need?
S2: Thirteen.
H: Are you sure you need thirteen? […] Let us count again. How many do we have?
S3: One, two, three, four, five, six, seven.
H: If we have seven, how many more do we need to get up to thirteen chickpeas? Let us keep on counting from seven.
S3: Eight, nine, ten, eleven, twelve, thirteen.
H: So, how many more do we need to get there? [She raised a finger for each number said by S3].
S3: [No reply.]
H: How many do I have in this hand? Count them for me.
S3: One, two, three, four, five, six.
H: Good. Go and get six chickpeas, and then we will see if this gets us there. We will wait for the others to finish, and then you can share your strategy with the rest. How did you do it? [i.e., solve the problem]?

Activity: Build a collection of objects according to a given number series

Event: Activity setup

L: We are going to engage in a math activity on numbers […]. In this bag, there are many numbers. For instance, we can make the number ten. Put your hand into the bag and grab a number […]. We are going to put [the number] on the table […]. This bag contains stuffed animals, and this other bag contains Legos and other figures […]. How many items [i.e., stuffed animals, Legos, etc.] do you have to select according to this number? [Children are sitting at their desk and one child stands up to perform the activity.]

Event: Activity implementation: take a number and build a collection

L: Alexis, come forward […] and take a number.
S1: (Selects the number 1 and shows it to Lauren)
L: […] but this number needs company, so take another number.
S1: (Selects the number 6.)
L: (Inverts the number.) This needs to be turned around so that it shows the number 6, because 9 has already been selected. How many items do we need to put on the table, Alexis?

Event: Ending the activity

L: To end the activity, we are going to review all of the numbers we saw […]. I will show you the numbers, and you will have to tell me which number it is that I am showing.
L: […] And after 9, which number comes next?
Class: Ten.
L: Ten […]. Which numbers make up ten?
Class: One and zero.
L: One and zero, very good. […]. Let us count, but I want it to be really loud (points to a number sequence hanging from the ceiling.)
Class: One, two, three, […] ten.

Question	Answer
1) Why did you select this activity for the children?	I had already done this activity [with this group of students] when we were asked to conduct a problem-solving activity, but [I had previously] set the numeric sequence too low […]. So, I said […], what would happen if I did it again but with a higher numeric sequence? […] [The students] did not pay much attention to the numbers [they did not understand] and paid much more attention to the configuration of the dots. […]. [To make the students] establish the relationship, I tried to use a strategy to get to the numbers […], it was challenging.
2) Do you think this was a problem-solving situation?	It was challenging […] because their background knowledge did not seem to allow them to add and count the way they have always been doing it.
3) Did you include elements not incorporated in your initial lesson plan?	Problem-solving […] is meaningful if [the students] ask appropriate questions and the teacher does not give away the answer. [With] learning [through] logical mathematics, [teachers] can start working [with students] from a very young age, maybe not with numbers or quantities, but by classifying, ordering, […]. Later you can introduce the concept of numbers little by little.
4) What is the relationship between the implemented activity and number learning?	[…] if the children just repeat a number, it means nothing […], I said seventeen and of course the number seventeen was the one that came after sixteen, which itself followed fifteen, so they repeated the numbers as they had done previously in the oral series. However, with the proposed dots configuration, I think they [the children] were starting to build an association between the quantities and the number-words.
5) What advantage does the work proposed in your activity have for children?	I think that it helps reciprocally, to use a different strategy, […] or the children think of a similar strategy, they go on trying until they achieve it […]. It develops children’s logical thinking and language, since they explain what they are doing.
6) How do you think children learn math?	[…] There are many stages children have to go through. These stages must be developed little by little so that they can reach mathematics as we know it […]. Since we already know the principles of counting, it is obvious to us, but for children, it is not so obvious. These stages are very marked [mathematically], so you have to go on proposing challenging activities so that they can develop their learning little by little.

Question	Answer
1) Why did you select this activity for the children?	In the school system, mathematics are not applied dynamically; lessons are way too structured […]. I felt like I had to do something more dynamic, something that would get the children’s attention easily […]. I feel they liked [the activity] and that they had a good time […].
2) Do you think this was a problem-solving situation?	A problematic situation was presented, and they faced it without difficulty.
3) Did you include elements not incorporated in your initial lesson plan?	The children were always attentive, and I liked that because I felt that it was authentic learning for them and that it was not something they were accustomed to, as they did not use their workbooks.
4) What is the relationship between the implemented activity and the number learning?	I believe that everything is less structured and that they can use mathematics in a more didactic way, and this is how we should work in math.
5) What advantage does the work proposed in your activity have for children?	To understand that a number is not just a number, but that it has a purpose. I can draw the number symbol, but at the same time, I can identify the quantity to which that number corresponds.
6) How do you think children learn math?	I believe that in mathematics, we need an illustrative system for children. For us, we did not like mathematics in school. Math was something so structured and always the same, there was a formula to solve the exercise. On the other hand, with a more illustrative activity that provides real interest, children may like math in the future.

Component	Items
CK	Two children put their dolls to bed. They place one doll with hair, another one with no hair; one with hair, and another one with no hair. Teresa, a pre-service early-childhood teacher, thinks the children are putting a mathematical notion into practice. What do you think? Mark one of the responses.
PCK-T	Mercedes, an early-childhood teacher, attended an advanced training course and learned that it is important for children to become familiar with different representations of quantity. She wants to buy some materials. Can you help her decide which materials are the best options? Mark one of the responses.
PCK-MTEC	Mercedes, an early childhood teacher, observes a child that does not state the total when counting objects. She asks herself, “What counting principle should I work on with this child?” What do you think? Mark one of the responses.

Component	Evaluated criteria	Achieved (2 points)	Somewhat achieved (1 point)	Not achieved (0 points)
CK	Principal concepts of logic and number	Presents number and logic as principal concepts	Presents number or logic as principal concepts	Does not present number or logic
	Subordinate concepts of logic	Presents classification, seriation, patterning, one-to-one correspondence. (3-4 notions)	Presents classification, seriation, patterning, one-to-one correspondence. (2-4 notions)	Presents classification, seriation, patterning, one-to-one correspondence. (0-1 notions)
	Subordinate concepts of number	Presents oral series, quantifiers, numerals, ordinal numbers, cardinality, counting, conservation, sum/difference. (5 to 8 notions)	Presents oral series, quantifiers, numerals, ordinal numbers, cardinality, counting, conservation, sum/difference. (3 to 4 notions)	Presents oral series, quantifiers, numerals, ordinal numbers, cardinality, counting, conservation, sum/difference. (0 to 2 notions)
	Concepts of number representations	Presents pictographic, iconic, and symbolic representations	Presents pictographic-symbolic or iconic-symbolic representations	No, or only pictographic, representations
CK	Grouping of concepts	Groups and adequately relates concepts	Groups, but does not adequately relate, concepts	Concepts not grouped
CK	Connections between concepts	Uses arrows to adequately relate concepts	Uses arrows, but does not adequately relate concepts	Concepts not related
PCK-T	Subordinate concepts of teaching	Presents tasks or scenarios linked to problem-solving	Presents tasks or scenarios generally linked to problem-solving	Tasks or scenarios not presented
	Grouping of concepts or ideas	Groups and adequately relates concepts	Groups, but does not adequately relate, concepts	Concepts not grouped
	Connections between concepts or ideas	Uses arrows to adequately relate concepts	Uses arrows, but does not adequately relate concepts	Concepts not related
PCK-MTEC	Subordinate concepts of cognitive development	Presents tasks or scenarios linked to cognitive development	Presents tasks or scenarios generally linked to cognitive development	Not linked to cognitive development
	Grouping of concepts or ideas	Groups and adequately relates concepts	Groups, but does not adequately relate, concepts	Concepts not grouped
	Connections between concepts or ideas	Uses arrows to adequately relate concepts	Uses arrows, but does not adequately relate concepts	Concepts not related
	Concept map structure	Fully organised and easy to understand	Simple organisation and hard to understand	Linear organisation or illegible structure

Component	During an initial student teaching opportunity, the trainee teacher:
CK	Uses mathematical language correctly
CK	Uses more than one representation of number (pictographic, iconic, and symbolic representations)
PCK-T	Presents a problem-solving situation to children
	Asks a central question that poses a task to be performed by children
	Confirms comprehension of the task
	Organises children individually, or in groups of two to four, and as an entire class to complete the given task
	Synthesises findings, achievements, and/or mathematical notions present during the activity
	Presents an activity with expected learning goals, in line with the educational programs of respective student levels
	Promotes exposure of resolution strategies between children
	Uses manipulative materials and representations
	Presents an activity that will allow children to judge the results of their actions for themselves
PCK-MTEC	Asks about prior knowledge
PCK-MTEC	Provides feedback, responses, or actions to children’s questions without giving away the strategy and/or answer

[1] Tatiana Cecilia Goldrine Godoy has a doctorate in Educational Psychology from the Universidad Nacional Autónoma de México (México). She is a professor at Pontificia Universidad Católica de Valparaíso (Chile). E-mail: tatiana.goldrine@pucv.cl

Brasil

Brasil

Mathematical knowledge and overall practice in initial teacher education of early childhood teachers

CONHECIMENTO MATEMÁTICO E PRÁTICA EM GERAL NA FORMAÇÃO INICIAL DE PROFESSORAS DA EDUCAÇÃO INFANTIL

CONOCIMIENTO MATEMATICO Y PRACTICA COMO UN TODO EN LA FORMACION INICIAL DE LAS MAESTRAS DE EDUCACION INFANTIL

ABSTRACT