As we consider our role in preparing future teachers, we take to heart our responsibility to address the human context of mathematics in our teaching. In 2017, the Association of Mathematics Teacher Educators (AMTE) released the Standards for Preparing Teachers of Mathematics (SPTM) where they describe well-prepared beginning teachers as able to understand that “mathematics is a human endeavor” (p. 9). Teachers are called on to engage in interpersonal interactions that interweave their mathematical expertise and their skills for probing student thinking, finding meaning in students’ perspectives, and responding to students in a way that nurtures their assets and guides their understanding (Ball et al., 2008). The challenge is to create opportunities for preservice teachers to engage in developing these skills and to grow in their understanding of mathematics as a human endeavor in all their mathematics courses.

META Math Project

Through our work on a project called “The Mathematical Education of Teachers as an Application of Undergraduate Mathematics” (META Math), we have developed and researched the use of tasks in four undergraduate mathematics courses taken by secondary preservice teachers: abstract algebra, calculus, discrete mathematics, and introductory statistics. These tasks address two types of connections between undergraduate mathematics and teaching secondary mathematics (see Table 1, Arnold et al., 2020) and have been designed in a way to hold the human context of mathematics on par with the mathematics content (Álvarez et al., 2020). These two types of connections, classified as School Student Thinking and Guiding School Students’ Understanding, align with Indicator C.1.5 – Analyze Mathematical Thinking in the SPTM, which emphasizes the need for preservice teachers to understand the mathematical work of others, analyze different approaches, and respond appropriately (AMTE, 2017). 

Table 1. Two connections to teaching secondary mathematics. 

Figure 1 displays an example of such a task from a lesson on the binomial theorem in an undergraduate discrete mathematics course. In this task, undergraduates first analyze the work of a hypothetical high school student, Hannah, who made some errors while applying the binomial theorem to expand (x - 3y)⁴. Undergraduates are asked to not only identify the errors but to also describe what mathematical understanding Hannah demonstrates through her work. This focus on describing Hannah’s understanding reflects our aim for these tasks to promote teaching practices that nurture students’ assets and to offer alternatives to deficit-perspectives that only focus on lack of understanding. Lastly, undergraduates pose questions to Hannah to help her revise her work and guide her mathematical understanding.

Figure 1. Hannah’s task.

Implementation

Approximately 200 undergraduates across 11 classes engaged in our tasks. To investigate how these tasks support preservice teachers in understanding the work of others and guiding students’ understanding, we interviewed the instructors who implemented these tasks, collected undergraduates’ written responses to them, and invited a subset of undergraduates to participate in interviews. We sought to select as many preservice teachers as possible, but those who completed interviews were a mix of preservice teachers, non-teaching mathematics majors, and non-mathematics majors. Most expressed interest in teaching in some form (e.g., tutoring or plans to teach in graduate school). During an interview, undergraduates re-examined their work on the tasks and explained their thought processes, discussed alternative approaches, and described how examining student work and proposing guiding questions for students was beneficial to their own learning. In the instructor interviews, they described their impression of the materials and how they prepared undergraduates to understand the mathematical work of others.

Analyzing the Mathematical Work of Others

Across all four content areas, instructors and undergraduates alike reported finding value in the tasks that involved analyzing a hypothetical student’s work and commented on the novelty of these tasks in content courses. Many instructors stated that they had not worked with these types of tasks in other contexts, and they found them to be the best part of the lesson due to the high level of engagement and valuable class discussion. Undergraduates often described how these tasks prompted reflection on their own behavior and mathematical understanding. The benefits they described ranged from deepening their own mathematical understanding; making the lesson more engaging; thinking more critically about a problem; having opportunities to see different approaches and multiple perspectives; helping them understand errors other learners make; and giving preservice teachers opportunities to practice skills relevant to their future careers. One preservice teacher noted how these tasks emphasized that “everyone understands things differently,” and that is something she will see when she teaches.

Overall, we found that these tasks encouraged all undergraduates to think about the interplay between human beings and mathematics content. For instance, many viewed Hannah (the character in the task from Figure 1) as a real person, and their written responses revealed respect for her mathematical work. During interviews, preservice teachers emphasized the importance of acknowledging what the student does understand. One said that these tasks “help [her] recognize, oh the students make certain mistakes, but they also do things correctly, too.”

Responding to Student Understanding

In some tasks, undergraduates were also prompted to respond in ways that guide students’ understanding by posing or evaluating a set of questions to ask a hypothetical student (see Figure 2). This was new and challenging for undergraduates because they had either not had practice with posing questions or seen it intentionally modeled by their instructors. Collectively, we saw undergraduates pose a variety of questions that we categorized as advancing, assessing, or other. Sample questions they wrote include: “Why is the expanded form not a mirror of the coefficients?”, “What is (3y)²?”, and “What are you using as your a and b when applying the binomial theorem on (a + b)ⁿ?” Undergraduates valued asking questions that would help guide students’ understanding and described how this was better than simply telling the student a correct solution path. Although difficult, many undergraduates said these kinds of tasks helped them “think more critically about the problem.” Some instructors also spoke about how their undergraduates surprised them with some thoughtful questions and stated these kinds of tasks provided them with a new way to assess their students’ mathematical understanding.

Figure 2. Tasks that ask undergraduates to pose (left) or evaluate (right) questions to a hypothetical student.

We found it beneficial to scaffold undergraduates’ development of this skill by first asking them to evaluate a set of questions. This helped undergraduates better understand the kinds of questions that are helpful to guide students’ understanding, thereby improving their own problem-solving skills and questions for students.

Mathematics is a Human Endeavor

Mathematics content courses play a significant role in secondary teacher preparation, and these courses can simultaneously develop preservice teachers’ understanding of mathematics content and their understanding of mathematics as a human endeavor. One method of developing this dual understanding is by integrating tasks that involve human beings as characters in standard mathematics major courses. We see in these tasks a way for secondary preservice teachers—human beings—to use their mathematical skills to validate students’ thinking, guide their understanding, and recognize that when students make errors, they are often basing their reasoning on justifications that make sense to them.

Acknowledgments

The META Math project is supported by the National Science Foundation Division of Undergraduate Education (DUE-1726624). Any opinions, findings, conclusions, or recommendations are those of the authors and do not necessarily reflect the views of the NSF.

References

Álvarez, J. A. M., Arnold, E. G., Burroughs, E. A., Fulton, E. W., & Kercher, A. (2020). The design of tasks that address applications to teaching secondary mathematics for use in undergraduate mathematics courses. Journal of Mathematical Behavior. 60: 100814.

Arnold, E. G., Burroughs, E. A., Fulton, E. W., & Álvarez, J. A. M. (2020). Applications of Teaching Secondary Mathematics in Undergraduate Mathematics Courses. TSG33 of the 14th International Congress on Mathematical Education. Shanghai, China: International Mathematical Union. Available at arXiv:2102.04537

Association of Mathematics Teacher Educators. (2017). Standards for preparing teachers of mathematics. Raleigh, NC: Author. Available online at amte.net/standards

Ball, D. L., Thames, M. H., & Phelps, G. (2008). Content knowledge for teaching: What makes it special? Journal of Teacher Education, 59(5), 389-407.