How to Use Tasks to Promote Mathematical Reasoning in Middle School
By Dr. George Roy, Professor of Mathematics Education and Reveal Math Author
Early in my career as a middle school math teacher, I rarely gave my students tasks that promoted mathematical reasoning. When I did, they were regulated to “fun” Fridays or right before a school break. As I reflect upon this instructional choice in my role years later as a teacher educator and researcher, I wonder what my hesitation was in not using them. What mathematical benefits did my students miss earlier in my career? What was I indirectly telling my students about what I thought they could do in the mathematics classroom?
We know that the mathematical tasks middle school teachers use with students can have a profound impact on how their students view mathematics. We also know that tasks that require students to perform a procedure in a routine manner lead to one type of opportunity for students, whereas tasks that demand engagement with concepts and stimulate students to make mathematical connections lead to different opportunities for students (Stein & Smith, 2011).
We’ve found that when effectively implemented during instruction, tasks that promote mathematical reasoning can challenge our students to investigate, explore, apply, expand upon, extend, and discover mathematics in meaningful and memorable ways. Moreover, the tasks that promote reasoning can provide the opportunity for all learners to develop deep mathematical understanding, understand and critique the world through mathematics, and experience the wonder, joy, and beauty of mathematics, which together contribute to a positive mathematical identity (National Council of Teachers of Mathematics, 2020).
To help you avoid the same hesitation that I had as a young educator, I’ve collected some lessons I’ve learned from using and writing about a wide range of tasks that promote mathematical reasoning in middle school students:
The mathematics students learn in middle school contains some of the most practical and foundational mathematics for future studies as well as life itself. It also contains some of the mathematics not understood well by adults (e.g., rational number concepts) (National Center on Education and the Economy, 2013; Organization for Economic Co-Operation and Development, 2016). My initial take away is that context matters! The context of the task must be relatable to students. But just finding a context that is interesting to students is not enough. Importantly, an effective task must provide the incentive and rationale for them to engage in the mathematical reasoning and sense making that we desire as teachers.
It is vital that we engage our students by making tasks problematic, while being within their mathematical grasp. As Battista (2017) suggests, “Students must be successful in solving challenging but doable problems.” (p. 3) Picking tasks for a wide range of students at a variety of achievement levels to participate in the mathematical experience by being simultaneously challenging and attainable is key. Doing so allows learners with varying degrees of understanding to begin and stretch their mathematical understanding. As noted in Principles to Actions: Ensuring Mathematical Success for All, “Effective teaching of mathematics engages students in solving and discussing tasks that promote mathematical reasoning and problem solving and allow multiple entry points and varied solution strategies.” (National Council of Teachers of Mathematics, 2014, p. 17)
I believe that we should use challenging tasks early and often. I now advocate that we begin a unit with a challenging task. As noted, challenging tasks should not be postponed until the end of an instructional unit, rather they can be used to sustain learning throughout the lesson (National Council of Teachers of Mathematics, 2016). But introducing the task is not enough. That brings me to my fourth lesson, all students must participate in their learning during the lesson.
To prepare all students when I introduce a task, I am purposeful in stating that all students are expected to participate during the exploration and discussion of the task. When mathematics teachers use challenging tasks, they set the stage for their students to engage in a productive discussion where mathematical connections are being made (Smith & Stein, 2018). By having students participate in mathematical discourse, teachers position their students as capable contributors during instruction (Bartell et al. 2017). Asking questions like, what is the same, what is different, what is efficient about the mathematical reasoning allows students to make the deep mathematical connections that we strive for in our classrooms.
In sum, the four lessons that I have learned by implementing tasks that promote mathematical reasoning in the middle school classroom are: (1) context matters; (2) tasks should be challenging but attainable; (3) tasks should be used early in a unit; and (4) all students are expected to participate. The tasks we select for our students tell them what we believe they can do in the mathematics classroom and have the potential to empower students to see themselves as mathematically capable.
George J. Roy is a Professor of Mathematics Education at the University of South Carolina (USC). He received his undergraduate degree in Mathematics from Rollins College and attained his Master of Education in Mathematics Education from the nationally recognized Lockheed Martin/University of Central Florida Academy for Mathematics and Science at University of Central Florida (UCF). George went on to receive his Ph.D. in Education with an emphasis in Mathematics Education from UCF. Prior to his position at USC, he taught middle school mathematics for eight years in Orange County Florida Public Schools. During his public school tenure, he achieved a National Board of Professional Teaching Standards certification in Early Adolescence Mathematics. Recently, Dr. Roy was a member of the middle school author team of NCTM’s policy book titled, Catalyzing Change in Middle School Mathematics: Initiating Critical Conversations.
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Works Cited
Bartell, T., Wager, A., Edwards, A., Battey, D., Foote, M., & Spencer, J. (2017). Toward a framework for research linking equitable teaching with the standards for mathematical practice. Journal for Research in Mathematics Education 48, (1): 7–21.
Battista, M. T. (2017). Mathematical reasoning and sense making. In Reasoning and Sense Making in the Mathematics Classroom: Grades 3–5. National Council of Teachers of Mathematics.
National Center on Education and the Economy (NCEE) (2013). What does it really mean to be college and work ready? The mathematics and English literacy required of first-year community college students. NCEE.
National Council of Teachers of Mathematics (2014). Principles to actions: Ensuring mathematical success for all. National Council of Teachers of Mathematics.
National Council of Teachers of Mathematics (2016). High Expectations: A Position of the National Council of Teachers of Mathematics. Retrieved October 25, 2023, from https://www.nctm.org/Standards-and-Positions/Position-Statements/High-Expectations/
National Council of Teachers of Mathematics (2020). Catalyzing change in middle school mathematics: Initiating critical conversations. National Council of Teachers of Mathematics.
Organization for Economic Co-Operation and Development (OCED) (2016). Equations and Inequalities: Making mathematics accessible to all. Paris: OCED Publishing: Program for International Student Assessment (PISA).
Smith, M. S., & Stein, M. K. (2018). 5 Practices for Orchestrating Productive Mathematics Discussions. 2nd ed. National Council of Teachers of Mathematics.
Stein, M. K., & Smith, M. S. (2011). Mathematical tasks as a framework for reflection: From research to practice. Designing and Enacting Rich Instructional Experiences. National Council of Teachers of Mathematics.
