The K-5 Algebraic Thinking Initiative explores algebra from three different perspectives: relational thinking, patterning, and functions. We begin by considering what algebra is in the elementary classroom and what it looks like when students think algebraically. As we explore each of the three perspectives, teacher participants also consider how algebra lessons can be designed and implemented to concurrently support the development of arithmetic. An important outcome is for teachers to see that algebra and algebraic thinking is not an add-on to their curriculum. Rather, they can develop student understanding of algebra while teaching arithmetic and that most algebra-based activities support arithmetic as well.
A secondary focus of the K-5 Algebraic Thinking Initiative is to provide teachers opportunities to practice and refine teaching strategies that promote a learner-centered philosophy coupled with inquiry-based instructional practices in mathematics. The Toledo Public School district’s union provides math teachers in Title Schools with additional support using The Thinking Mathematics Program. This program focuses on pedagogy rather than content. Since most of the project teachers did not teach in Title I schools and did not receive Thinking Mathematics support, instruction on the Ten Principles of Thinking Mathematics was incorporated into the workshop activities. The Ten Thinking Mathematics Principles are:
- Build from intuitive knowledge.
- Establish a strong number sense through counting, estimation, use of benchmarks, mental computation skills, and understanding effects of operations.
- Base instruction on situational story problems.
- Use manipulative and other representations to represent the problem situation; then link concrete to symbolic representations.
- Require students to describe and justify their mathematical thinking.
- Accept multiple correct solutions and when appropriate, more than one correct answer.
- Use a variety of teaching strategies.
- Balance conceptual and procedural learning.
- Use ongoing and new types of assessment to guide instruction.
- Adjust the curriculum timeline.
The videos and written narratives provided here are examples taken from project teachers’ classrooms where they were implementing lessons with the potential to bring out algebraic forms of reasoning. We have developed these videos and cases for use with preservice and inservice teachers in future projects or coursework where K-5 algebra is the focus. Please note that some of the video clips are designed to provoke discussion around specific algebraic ideas. For example, the Missing Addend video, poses a question to the viewer. This question is intended to support a discussion about the algebraic nature of this second grader’s reasoning. Each of the written narratives includes the Thinking Mathematics Principles addressed in the lesson. A question to consider when viewing videos is which principles are being addressed in the lesson.
Recent research shows that students conception of the equal sign may be deterrent in students understanding of algebra. Many students do not see the equal sign as expressing a relation between what is to the left and to the right. In other words, the equal sign symbolizes the equivalence of what is on either side. When students see the equal sign as meaning “the same as” they can then begin to use relational thinking. As students develop greater sophistication with relational thinking, they can use relationships rather than rote computations to find missing or unknown values. An example of this can be found in the Missing Addends video when Savanah solves an open number sentence without having actually calculated the sums on the left and right side of the equal sign. Prekindergarten and kindergarten students can by explore variables as unknowns. By finding multiple ways to make a number students can learn begin to see that six for example, is equal to many sums. As students understanding of number develops through the school year, they can begin to solve missing addend problems.
Patterns and Functional Thinking
The Handshake Problem is a classic algebraic thinking task. It is often highlighted because it can be used in different ways depending upon the goals of instruction and the mathematical level of the learners. Project leaders have used this problem with secondary math teachers and with kindergarten teachers. The provided cases show how different teachers implemented the task with their students. As is to be expected, in these early elementary classrooms the focus was on helping students learn to collect data, record data and analyze the data. The analysis focused on looking for patterns in the data and when appropriate for the learners, to find patterns in the data that could be used to make predictions (a generalized statement) about what might happen for any number of people engaging in the handshake task and then use this generalize statement to predict what would actually happen. When students are able to use data to make predictions about what will happen in future cases, or for any nth case, they are starting to use functional thinking.
These narratives are from lessons implemented early in the school year. Note that
the students benefited from the teachers highly
structured lessons that incorporated strategies for collecting data in a systematic
way. By acting out the problem, collecting data about what
happened quantitatively, recording this data using some form of notation (tallies,
numbers, pictures), analyzing the data for patterns, and
discussing how the patterns can be used to make predictions about additional data
that had not yet been collected supported students as they
learned to look for relationships in the data. The focus in how quantities are related
to each other is an important form of thinking used in
algebra. The more experience students have generating patterns from situations, the
more adept they will become at recognizing patterns and
describing how they change. Analyzing how quantities change is a fundamental aspect
of formal algebra that can be supported in the early grades.
The video of first-grader students working on The Polar Express problem provides a
viewable example of a lesson that, like the Handshake Problem,
supports functional thinking.