By Glenn Waddell Jr., posted March 27, 2017 —

The mathematics majors in my teaching program are all competent in algebra, calculus, and beyond, but when I start mixing their math knowledge up, they run into walls. Topics at the elementary school level allow preservice teachers the fun of teaching things like T tables and function machines. But challenging them to think about why T tables are so useful to beginning learners is an interesting task. For example, the T-table below looks simple, right?

Take the input shape, add two sides. But what happens when the two inputs—a circle and a vertical line segment—are included at the top of the table? The output for the line may be easy; that is the triangle. But what happens with the circle? Is it even a valid question to insert the circle or line into the input table? Why?

The table on the left (below) gets to the heart of asking good questions that require a strong understanding of definitions.

For the image pairs on the right, a rule gives the output shape by doubling the number of sides in the input shape. But what happens when the circle and line are included?

College students find themselves stumped by a fourth-grade introductory function problem. They realize that they have been doing math but not often thinking about what the math means, and—more important—what the definitions they have been using actually mean. This cognitive dissonance creates questions that are interesting for them and their future learners. Questions such as these arise:

•     Is the circle a shape with no sides? Or infinite sides?

•     Does the line have any sides at all?

•     So, is it zero, or is it none?

•     And what is the difference between zero and no sides for multiplication?

•     What would a two-sided shape look like?

•     If multiplication is just repeated addition, why do the two tables have different holes on the output?

Additional questions of domain and co-domain arise:

•     Can the circle or line be included in the domain?

•     Are they the same type of objects as the rest of the shapes? Why? Why not?

•     In the second table, the co-domain is tied to even numbers; but in the first, it is not?

Around this time, the preservice teachers realize that I never used the word polygon, which has a specific definition. I kept saying shape, which allowed them to create their own definitions. This conversation that began in terms of teaching an elementary school class must turn to address how to use these insights to develop students’ mathematical thinking. What definitions will the learners attach to the shapes if the teacher does not tell them what to think? What outcomes will the learners derive when they are allowed to define the shapes for themselves? How will the class reconcile the different definitions if different definitions are created? How will the teacher ask questions to prompt deeper thinking if the learners are staying at the surface level? And finally, how are the Common Core’s Standards for Mathematical Practice (SMP) used here? Will learners be “reasoning abstractly and quantitatively” (SMP 2)? Will the teacher allow for students to “construct viable arguments and critique the reasoning of others” (SMP 3)? The joy of mathematics can be found when the simple problems we take for granted are tweaked to encourage deeper and creative thinking.

REFERENCE

Common Core State Standards Initiative (CCSSI). 2010. Common Core State Standards for Mathematics (CCSSM). Washington, DC: National Governors Association Center for Best Practices and the Council of Chief State School Officers. http://www.corestandards.org/wp-content/uploads/Math_Standards.pdf

Glenn Waddell Jr. is a Master Teacher for NevadaTeach, a UTeach replication program at the University of Nevada–Reno. He is also currently a doctoral student at UNR, looking forward to comps and dissertation proposals in the next year. He previously taught algebra 1, 2, and 3 for nine years in Washoe County School District and has been an active participant in the MTBoS since 2011. He blogs at http://blog.mrwaddell.net and tweets at @gwaddellnvhs.