Presentation of the project "Functional thinking in students at elementary education as an approximation to algebraic thinking"- Spain, held during the 8th Science Projects' Networking Event, Brussels, 16 October 2015

1. Functional thinking in
students at elementary
education as an
approximation to algebraic
thinking
Aurora del Río Cabeza
adelrio@ugr.es
Universidad de Granada

2. Who we are?
 International group of professors and reseachers:
 Tufts University (USA)
 University of California, Davis (USA)
 National Institute of Education (Singapore)
 Universidad de Granada (Spain)
 Department of Mathematical Education
 Preliminary, Primary and Secondary School Teacher’s Training on Mathematics
 Research about teaching and learning of mathematics, in order to contribute to a better understanding
of it.
 Focus in algebra at primary school
 PROYECTO DE INVESTIGACIÓN I+D (Excelencia): EDU2013-41632P
Functional thinking in students at elementary education as an approximation to algebraic
thinking

3. Our main objectives
 To contribute to the characterization of functional thinking in the early
grades, to show evidences of students’ functional thinking, and to design
tasks of use to promote functional thinking in classrooms.
 We highlight the theoretical deepening of the cognitive contruct functional
thinking for early grades students.
 This will be usefull for curricular innovation, enriching the teaching-learning
of mathematics in consonance with the early algebra innovation curricular
proposal.
 Concerning teacher training, a better understanding of students’ cognitive
procedures will be usefull to perform a cognitive guided teaching process.

4. The context
 The project is placed within the early algebra proposal, relevant nowadays
from an international viewpoint, that advocate the integration of algebraic
thinking from the early school years.
 Previous international studies call into question the limitations traditionally
supposed to elementary students’ capacities to work with algebraic elements.
 They reveal the capacities of those students to reason about functions, to
identify relationships, and even to generalize; and provide evidences of the
contribution of early experiences with tasks involving functional relationships
to a later formal learning of algebraic notions

5. What is functional thinking?
 Is a kind of algebraic thinking that focuses in the relation between two or
more quantities that covaries
 Is the act of thinking in terms of and about functions
 It includes the process to construct, to describe and to reason in terms of and
about functions
 Working with functions depends on and contributes to an understanding of
variables, formula’s manipulations and the relation between several
representation systems (tables, verbal language, algebriac symbolism, graphs,
etc)
 It requires that students identifies patterns and to express this patterns as a
general relation

6. Method
 Teaching experiment in which we designed and implemented generalization
tasks that involved linear functional relations.
 We elaborated and applied an instructional design with 3 groups of primary
education students of a Spanish educative centre, with the aim of deepening
into descriptive elements of functional thinking through the analysis of
students’ productions.
 We used participant observation, semi-structured interviews, and written
questionnaires.
 We implemented 4 to 5 sessions of one to one and a half hours, following a
similar methodology for each group combining written work and whole group
discussions.
 One of the researchers taught the function lessons while the regular
classroom teacher was also present in the classroom.

7. The tasks
 All tasks involved functions with natural numbers presented though a familiar
context
 The first questions asked students to explore the functional relation by
considering (nonconsecutive) particular cases.
 Next, we included questions that directed students’ attention beyond
particular cases towards the relation that connected both variables
mentioned in the context.
 Thus, we guides the student to follow the process of inductive reasoning
 The order of questions is presented increasing the difficulty, according to the
involved mathematical content and the results of previous research.
 In some questions we asked the students to make use of different
representations such as tables, verbal language and letters.

8. The tasks (1º grade)
Session Task Functional
relation
Kind of questions
1 Dogs and collars y=x True or false
2 Dogs and collars y=x+5 Finish the sentence
3 Dogs and collars y=x+5 Fill the blanks in the table
4 Ages of Alvaro
and Carmen
y=x+5 Fill the blanks in the table
and answer questions
5 Ages of Alvaro
and Carmen
y=x+5 Fill the blanks in the table
and answer questions

9. The tasks (3th grade)
Session Task Functional
relation
Kind of questions
1 Ages of Raúl and
María
y=x +5 Answer and explain
2 School trip y=3x True or false, and why
3 School trip y=3x Fill the blanks in the table
and explain
4 The problem of
floor tiles
y= 2x+6 Answer and explain

10. The tasks (5th grade)
Sessio
n
Task Functional
relation
Kind of questions
1 School trip y=3x Answer and explain
2 School trip y=3x; y=2x+15 Answer and explain.
Comparasion
3 The deal of the
grandmother
y=2x; y=3x-7 Answer and explain.
Comparasion. Open
problem
4 The problem of
floor tile
y= 2x+6 Answer and explain

11. Preliminary Results
 Use of letters in first grade
 Capacity of students to establish correlation and correspondence relations,
using differents strategies in first grade
 Appearence of functional thinking in 5th grade througth various
representations systems, even algebraic simbolism, and correlation and
correspondence relations