# What is a Mathematical Circuit?

What is a Mathematical Circuit?

Circuit training is a popular form of exercise at the gym. A circuit is designed by an experienced trainer to meet a physical goal (e.g. strength, stamina, flexibility) through a set of assigned exercises, completed in a prescribed order. The intent is to keep the participant focused and engaged during the workout, without getting bored with the routine. In a gym, a circuit may include sit-ups, lunges, jumping jacks, and push ups. Individually, each exercise may not be very difficult, but put together, one gets a great workout. My students and I have had success and fun doing mathematical circuit training
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Principles to Actions (National Council of Teachers of Mathematics, 2014) calls for teachers to “establish mathematics goals to focus learning” (p. 10). A mathematical circuit is designed by an experienced teacher to meet a mathematical goal through a set of assigned exercises, completed in a designated order. An AP Calculus colleague, Nancy Stephenson from Houston TX, first exposed me to the concept of a mathematical circuit when she forwarded a calculus review circuit she wrote. I used the circuit for several years and noticed the format kept my students engaged. In the Spring of 2013, I wrote three algebra circuits in preparation for the Mississippi state-mandated subject area algebra test
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Since then, I have written over 60 circuits in many different areas of mathematics (algebra, trigonometry, calculus, etc.). With the encouragement of University of Mississippi professor Dr. Joel Amidon, I began creating focused circuits that did not just serve the purpose of review, but probed the material at a deeper level as students advanced in the circuit. For example, my reaction to a PARCC assessment item in the Fall of 2013 prompted me to write an Algebra I circuit “Finding Structure in an Equation.” I thought about what my students needed to master before having success with that particular item and those cognitive hurdles became the problems in the circuit. My use of the circuit format has allowed me to see my students advance toward that goal at their own pace, without even realizing that they are doing it! It has also allowed me to transition from so much direct instruction into the role of facilitator since students are self-checking their work
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I have shared these circuits and the design framework with my network of math colleagues both locally and throughout the country through word of mouth, blogging, my Teachers Pay Teachers site, and presentations. Many have reported back with their own classroom success stories, using either circuits I wrote or circuits they have crafted themselves. Their students stay engaged! Their classrooms hum with conversation! Their students master concepts and procedures!

Most circuits I design typically have between 12 and 28 problems, however, the students do not feel that they are staring at a blank paper, moving at a snail’s pace through their work. Students move around on the page, similar to moving around the gym, which helps keep them engaged. Some students may need to do 20 exercises for a workout, whereas some students may need 10, regardless they are getting a mathematical workout. The structure of the circuit allows students to advance at their own pace without too much direct comparison to where other students are in the circuit. The circuit format can be used for review, for weaving two concepts together, or for practicing a specific skill (e.g. factoring quadratic trinomials or order of operations). Circuits can even be used to introduce new vocabulary allowing students to apply that vocabulary during the problem solving process.

How Can a Teacher Use Mathematical Circuits?

Independent Practice

Students enter the mathematical circuit on an entry-level problem, solve it, and then search for their answer to locate the next problem in the circuit. For example, a factoring circuit might ask students to factor the binomial 3ab – 6b, and then to advance in the circuit the student must find either 3b or a – 2. Only one factor will be available so that it makes working backwards from answers (a valuable strategy, but one I do not want my students using in light of these new assessments) tricky. As students move forward in the circuit, the problems become progressively more difficult and challenge their previous skills and knowledge. I ensure that many of the answers are similar, so that students need to be accurate and precise. This makes the circuit format terrific for differentiated instruction.

About every fourth problem the circuit “levels up”, which means it becomes harder and/or incorporates students’ common errors and misconceptions. If students cannot find their answers, or if they pick the wrong answer and break the circuit (which would cause them to finish with problems unworked), they know they made a mistake. This element of self-check enables opportunity for questioning their own reasoning and attending to precision of their work. In addition, learning progressions can be incorporated into a circuit, see McCallum (2014), so that the assigned problems address significant cognitive hurdles in the understanding of important mathematical content.

If a circuit is designed according to a learning progression, then it can be used as a pre-assessment to see where students are lacking understanding. Similarly, it can also be used as a post-assessment to gauge procedural fluency of the unit’s concepts. Finally, if the circuit can be used as guided notes that a student completes at his or her own pace successfully, then understanding has been achieved.

Cooperative Groups

Having students complete a circuit in a group is a great way to add support, in addition to those already designed into the circuit. The structure of the circuit allows for students to self-check their results, but they may be confused on where they went wrong or how to remedy the situation. Having peers in multiple locations in the circuit allows for assistance to be on hand/on demand, or, having students work through the circuit together can allow for students to see multiple strategies and work to understand multiple students’ point of view. My students lean over effortlessly to discuss mathematics and explain things to each other when they are engaged in circuit training.

Complex Instruction

Another way to use the Circuit Training framework with groups is in what Lotan (2003) would describe as a “group-worthy” task. This is following the design standards of Cohen (1994) in complex instruction. Students are handed the circuit in pieces with the included task card. Students are to work through the circuit together (distributing ownership of the problems) with the focus being to develop multiple strategies for finding the group-generated solution. Establishing these connections will help students make sense of the solutions (e.g. what does a tabular, graphical, or algebraic solution to a quadratic look like?) , make connections across different representations, and identify the optimum method for solving particular types of quadratic equations (see Tools and Technology Guiding Principle (NCTM, 2014) and CCSS-M Standards for Mathematical Practice #5 (Common Core State Standards Initiative, 2010)).

Professional Teaching/Learning Communities

Another great way to use the circuit format is for a teacher to write his or her own circuits and share them with others. One of the highest predictors of student achievement (after controlling for socioeconomic status) is teacher content knowledge. If teachers are challenging their content knowledge by writing circuits and then sharing these circuits with their department members, all of the students benefit. Several of my colleagues have written their own circuits, and I have used some of these circuits with a lot of success with my own students. In fact, one of my colleagues in a different state writes what he coined “Mobius Circuits” which closes with 2, 3, 4, or 5 problems not completed and these remaining problems create a mini circuit. Students love a game, and unlike many games we play in high school mathematics classes, the circuit provides a written record of what they have worked on in class or at home. I have also written circuits at the middle and elementary levels and have shared them with the teachers in our district and in other districts around the country. In addition, a challenge for students and preservice teachers could be asking them to write a circuit toward a particular mathematical goal.

Final Remarks

After teaching 25 years, many teachers are ready to retire. I feel just the opposite. I love working with my students every day. The circuit format has breathed new life into my teaching. It is not the only resource I use to help bring my students to better conceptual understanding and procedural fluency, but it is the one I am most excited about at this time.

Virginia Cornelius
Lafayette High School
Oxford, Mississippi
April 2015

References
Cohen, Elizabeth G. 1994. Designing groupwork: Strategies for the heterogeneous classroom. Teachers College Press.
Common Core State Standards Initiative. 2010. Common Core State Standards for Mathematics. Washington, DC: National Governors Association Center for Best Practices and the Council of Chief State School Officers.