**By Angela Murphy
Gardiner and Katie
Sawrey, posted on February 15, 2016 –
**

What does it mean to think functionally, and what does functional thinking look like in an elementary school classroom? Until very recently, the study of functions has been treated as something to be learned in middle or high school mathematics. We believe that children can learn to think functionally in much earlier grades and that this thinking should be nurtured in elementary school classrooms. Broadly speaking, functional thinking is generalizing about relationships between quantities; representing and justifying these relationships using natural language, tables, graphs, and variable notation; and reasoning to understand and predict functional behavior.

Back in 2012, our research team worked with six classrooms of K–grade 2 elementary school students on an early algebra project focused on functional thinking. Before our study, we were unsure to what extent young children might engage in these activities and how their learning would progress. Research literature gave evidence that children—primarily in grades 3–5—notice, represent, justify, and reason with functional relationships; use variable notation; and create function tables to organize function data. Our study in functional thinking pulled those algebraic strands together and introduced them to children at the beginning of formal education, in K–grade 2.

Admittedly, when we first began working with kindergarten and first-grade students, we were unsure what they would do when it came to functional thinking tasks. Research suggested that young children are capable of the algebraic thinking practices, and thus, potentially, functional thinking (Blanton and Kaput 2004), but a systematic exploration of functional thinking had not been done yet with this age group.

In general, each lesson was designed around a particular task. One task used in the sequence was People and Ears, where the relationship was between the number of people and number of ears (assuming each person has two ears). Students were easily able to talk about how many ears one person had, two people had, and so on. With the guidance of the classroom teacher, students created a function table (t-chart) to organize that information (Common Core Standards of Mathematical Practice [SMP] 5). Once students had an organized way to look at their data, they noticed relationships and patterns among numbers in the t-chart (SMP 8). Some of those patterns and relationships, in turn, could be developed into descriptions of functional relationships (SMP 7). In general, we were amazed at students’ capacity for functional thinking and their ability to notice, describe, and write functional relationships.

Functional thinking doesn’t happen overnight. It starts with the introduction of a good task and builds with the organization of data so students can see clearly where patterns and relationships lie. As you begin to introduce functional thinking problems in your classroom, we encourage you to try some of the techniques below that fostered functional thinking among our students.

**Choosing Tasks That Foster
Functional Thinking**

While we were able to map a variety of ways that young students can generalize, we also saw that generalizing was dependent on problem contexts in subtle ways. Part of our advocacy for an early introduction of functional thinking is that we believe that student be given many opportunities to systematically interact with these ideas through a variety of representations and experiences so that this way of thinking becomes part of how students mathematize.

**Recording Data (in an Organized
Way)**

Mathematically exploring a problem context is an important first step to functional thinking, and it can be messy. A goal is for students to develop a systematic way to collect their data about two related quantities. We found that students as young as five were able to successfully record data in a function table and use their table to interpret data and predict future data. In our project, we chose to organically introduce this tool. After initial exploration, the classroom teacher might say, “I need a way to organize my data so I don’t lose track of it. I think I am going to use a table.”

By modeling the construction of the function table with a specific purpose in mind and adding data to the table with students, you can support students in learning how to construct tables and hopefully, as was the case in our project, they will adopt this tool for organizing data. Figure 1 shows a student’s work on the Height with a Hat task (Blanton, Brizuela, et al. 2015).

**Now It’s Your Turn!**

We would love to hear about your students’ experiences with functional thinking. Try developing an activity from the table of functional thinking tasks above that would work with your students. Start by allowing students to explore the data and get comfortable with the use of a function table to organize their data; see what patterns and relationships they begin to notice.

**References**

Blanton, Maria, Bárbara M. Brizuela, Angela Murphy Gardiner, Katie Sawrey, and Ashley Newman-Owens.
2015. “A Learning Trajectory in Six-Year-Olds’ Thinking about Generalizing
Functional Relationships.” *Journal for Research in Mathematics
Education* 46 (November): 511–58.

Blanton, Maria, and James J. Kaput. 2004. “Elementary Grades Students’
Capacity for Functional Thinking.” In *Proceedings of the 28th Conference of
the International Group for the Psychology of Mathematics Education*, Vol. 3,
edited by Marit Johnsen Hoines and Anne Berit Fuglestad, pp. 35–42.
Bergen, Norway.

Blanton, Maria, Ana Stephens, Eric
Knuth, Angela Murphy Gardiner, Isil Isler, and Jee-Seon Kim. 2015. “The Development of Children's
Algebraic Thinking: The Impact of a Comprehensive Early Algebra Intervention in
Third Grade.” *Journal for Research
in Mathematics Education* 46 (January): 39–87.

Brizuela, Bárbara M., Maria Blanton, Angela
Murphy Gardiner, Ashley Newman-Owens, and Katie Sawrey. 2015. “A First-Grade
Student’s Exploration of Variable and Variable Notation /Una Alumna de Primer
Grado Explora las Variables y su Notación.” *Estudios
de Psicología: Studies in Psychology* 36 (1): 138–65.

Brizuela, Bárbara M., Maria Blanton, Katie
Sawrey, Ashley Newman-Owens, and Angela Murphy Gardiner. 2015. “Children’s Use of
Variables and Variable Notation to Represent Their Algebraic Ideas.” *Mathematical
Thinking and Learning* 17:1–30.

Carraher, David W., Analúcia
D. Schliemann, and Judah L. Schwartz. 2008. “Early Algebra Is Not the Same
as Algebra Early.” In *Algebra in the
Early Grades*, edited by James J. Kaput, David W. Carraher, and Maria
Blanton, pp. 235–72. Mahwah, NJ: Lawrence Erlbaum and Associates.

*The research reported here was supported
in part by the National Science Foundation *(*NSF*) *under DRK-12 Award No. 1415509.
Any opinions, findings, and conclusions or recommendations expressed in this
blog are those of the authors and do not necessarily reflect the views of NSF.*

Angela Murphy Gardiner is a senior research associate at TERC in Cambridge, Massachusetts. Before joining TERC, she was an elementary school educator. Her primary research interests include teaching algebra in the elementary grades and exploring students’ thinking and understanding of functions. She is also interested in designing and implementing professional development programs in early algebra for elementary school educators. Katie Sawrey is a doctoral student in the Tufts STEM education program. Her research interests include mathematical discourse and function representations in early algebra, with a particular focus on students’ sense making.

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