By Angela Murphy Gardiner and Katie Sawrey, posted on February 29, 2016 –  

In our last post, we described functional thinking as using algebraic thinking practices (generalizing, reasoning, justifying, and representing) to explore relationships between covarying quantities. Young elementary school students (K–Grade 2) are ready for functional thinking explorations. Teachers can choose simple covarying relationships and still offer their students rich engagement in functional thinking.

For example, the multiplicative relationship between any number of people and the number of hands can be described by the relationship x + x = y. Young students are eager to provide cases that fit this relationship. If a teacher suggests a group of three people, students will figure out that six hands would be in that group. This functional relationship is one from their daily lives and can be modeled in the classroom. Capturing specific cases in a function table is an organized, accessible way for young students to move beyond thinking about individual calculations and focus on the covarying relationship. This shift in focus starts by encouraging students to think about numerical relationships they might see in the table.

Noticing Functional Relationships, in Addition to Recursive Patterns

The function table for people and hands might look something like the left side of this table. Students will often notice recursive patterns in the values going down the columns. The values in the Number of People column increase by one each time, and the values in the Number of Hands increase by two each time. Using this table, a student noticing the recursive pattern might say, “I notice that the number of hands is going up by twos.”

We want students to move away from looking down the columns of the function table and begin to notice the relationship between number of people and number of hands. To accomplish this, we begin to ask them questions about specific cases from the table. For example, “How would we get from one to get to two?”

Once a student realizes that we have to add one to itself to get two (in the case of first graders) or double the one to get to two, we write that explicitly to the right of our table, as shown above. Repeat this process with other cases from the table, and several students will notice the common structure to these calculations. Highlighting the math facts embedded in the relationships between quantities allows students to notice, develop, and use structure in the tables to further understand the relationships between the quantities.

Generalizing the mathematical relationships

Oftentimes when students start to generalize about a mathematical relationship they have noticed, they will use a particular example (“If there are three people, I will do three plus three to get six hands”) instead of a more general relationship (“Because everyone has two hands, I need to count the number of people twice to know how many hands there are”). Teachers can move the discussion to a general level by asking about the change to get from one quantity to another: “What do you have to do to find out how many hands any number of people have?” Students’ natural language descriptions are a form of generalizing.

In a student’s own words

One task we shared with students was the Growing Train task:

A train goes along a track, and at each station, it picks up two train cars. So, at station one, it picks up two train cars; at station two, it picks up two more for a total of four, and so on.

If the train engine is not included in the total count of train cars, the cases of stops one through four will result in a function table very similar to people and hands above. In the excerpt below, one of our students, Rebecca, has created a function table (see her written work below [Blanton, Brizuela, et al. 2015]) and is describing relationships she sees in her data. This conversation with Rebecca happened at the end of our eight-week teaching project. The excerpt is powerful because it shows the ease with which a first grader discusses functional relationships after only a brief exposure to functional thinking.

Additional Benefits to Introducing Functional Thinking Tasks in Your Classroom

As you can see from the excerpt of Rebecca’s conversation, this one functional thinking activity afforded the use of seven of the eight Common Core’s Standards for Mathematical Practice (SMP). Functional thinking is a domain in which almost all the SMP can be addressed in a single lesson. So, functional thinking tasks not only allow for fun explorations for young learners but also align themselves nicely to meet many of the SMP.

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.