By Benjamin Braun, Editor-in-Chief, University of Kentucky; Priscilla Bremser, Contributing Editor, Middlebury College; Art Duval, Contributing Editor, University of Texas at El Paso; Elise Lockwood, Contributing Editor, Oregon State University; and Diana White, Contributing Editor, University of Colorado Denver.
Editor’s note: This is the fifth article in a series devoted to active learning in mathematics courses. The other articles in the series can be found here.
Facts, methods, and insights all are essential to all of us, all enter all our subjects, and our principal job as teachers is to sort out the what, the how, and the why, point the student in the right direction, and then, especially when it comes to the why, stay out of his way so that he may proceed full steam ahead.
— Paul Halmos (Halmos, pp. 848-854)
Because of our passion and love for our subject, mathematicians want to share with students the joy, excitement, and beauty of doing mathematics. Our natural human impulse is to do so by telling students about the ways we have come to understand our discipline, to shed light on the subtleties that surround most mathematical ideas, and to explain the fundamental insights of our field. As we have discussed in our previous articles in this series, there is strong evidence that these goals of inspiring students and helping them deeply learn mathematics are often most effectively reached through the use of active learning techniques. Yet there are some good reasons why we might choose to tell students about mathematics when the time is right. In this article we will explore the act of instructor “telling” and discuss some of the roles that telling can play in active learning environments. We seek to balance our inclination to tell students about math, which is inherently passive for the students, with our desire to foster students’ active engagement with mathematical ideas. By doing so, we can simultaneously acknowledge the value of telling while challenging the idea that traditional telling is the best or only way to communicate mathematics with students.
Telling to Transmit Information
The way in which one thinks about instruction is shaped by how one believes students learn. For example, if one believes that learning occurs as a result of direct transmission of information from instructor to student, and that students learn by a process of directly taking in bits of information that their instructors say or write, then telling is the natural mechanism to help students learn. However, learning is not this simple: almost every teacher has experienced telling a student a certain mathematical fact, during class or during office hours — \((a^2+b^2)\) does not equal \((a+b)^2\) is a classic example — only to have them demonstrate on a test that they have not learned it. This kind of experience, which happens all too often, suggests that it is not enough for students simply to be told information if we want to produce deep and meaningful learning. As a result of such experiences, many mathematics educators and mathematicians now draw upon learning philosophies in which students construct (mathematical) meanings based on their own experiences (e.g., Glasersfeld, 1995; Piaget, 1954; Steffe & Thompson, 2000). These theories of learning align more closely with instructional techniques that reinforce active learning, as discussed in Part II of this series.
However, a part of teaching mathematics involves injecting new ideas and concepts into the classroom. We cannot and do not expect students to come up with brand new formulas, conventions, and concepts without any guidance. Therefore, at times instructors must introduce new bits of information. The question of how best to do this, though, is a tough one. If we want to be able to share new ideas and yet also help students actively engage with material and learn, how might we best introduce new material? One possible answer to this question is through judicious telling, a notion introduced by Smith (1996) in which teachers try to minimize the amount of direct telling they do, encouraging active engagement among students, and yet still occasionally tell when they deem it necessary. This perspective acknowledges the fact that we want to facilitate active engagement when we can, but that there may be aspects of mathematics that students simply need to be told (such as useful terminology, ways of representing mathematical ideas, etc.). This underscores the idea that the goal is not for an instructor never to tell, but rather to avoid telling when students might otherwise be able to actively engage with and understand an idea.
Even in a classroom in which active student engagement is valued and fostered, there are situations in which judicious telling may be particularly productive. One such time is directly following student activities in which students have had the opportunity to engage actively, e.g. through group problems, classroom voting, clickers, or other techniques referred to in Part III of this series. By first engaging students in tasks that stimulate their curiosity and help them to think about the mathematics, we are creating intellectual need in students, where by intellectual need we mean when a student encounters an intrinsic problem that they genuinely understand, appreciate, and are curious about. In his framework of DNR-based instruction, Harel (2007) places the intellectual need of a student in equal importance to the integrity of the content being taught. After establishing this intellectual need through a student activity, it can be effective to use focused telling to solidify resulting ideas and insights.
Another time that may be particularly appropriate for judicious telling is during a wrap up of key ideas. A potential problem with active learning environments is that students can be in situations in which there is rarely closure or a sense of completion for a task or idea. Students may be encouraged to engage, ask questions, think critically, etc., but if there is never any resolution for them, this can be problematic. Telling after students have extended engagement with material could help to solidify, clarify, and confirm ideas they have been developing. Indeed, failure to (judiciously) tell can lead to confusion, and there is something to be said about instruction that fosters resolution of ideas for students. As previously discussed in Part III of this series, the MAA Calculus study (Bressoud et al., 2015) found that a factor in determining whether students persisted in calculus is that “good” teaching can be more important than “ambitious” teaching. Thus, there is value in consistent instruction, and instruction that fosters active engagement is perhaps more effective when other foundational aspects of teaching (such as wrapping up or resolving mathematical discussions or questions) are in place.
Gaining a Sense of Efficacy Through Telling
A well-crafted mathematical lecture goes beyond a simple sequencing of facts. Recent research from the perspective of discourse analysis and instructor pedagogical moves (Artemeva and Fox, 2011; Fukawa-Connelly, 2012) shows that when given by an expert practitioner, a mathematical “chalk talk” is quite complex and relies on practices that are surprisingly stable across social and cultural boundaries. The creation of a complex mathematical lecture, where ideas are carefully introduced and developed so as to build upon each other in meaningful and insightful ways, is one source through which mathematicians provide themselves with evidence of their own mathematical knowledge, understanding, and development. Thus, the crafting of rich lectures contributes to mathematicians’ feelings of efficacy in their discipline.
This positive influence of instructor telling on sense of efficacy has been observed at the K-12 level as well. Smith (1996) points out that the act of telling provides teachers with a sense of efficacy, and he discusses the fact that reforms that move away from lecturing and telling can leave teachers without a clear source of efficacy. He astutely points out that “Telling mathematics allows teachers to build a sense of efficacy by (a) defining a manageable mathematical content that they have studied extensively and (b) providing clear prescriptions for what they must do with that content to affect student learning. The current reform [in the 1990’s at the K-12 level] removes both supports (p. 388).” We can apply this line of thought to our discussion of active learning in university classrooms. The active learning reforms in which many university instructors are engaging can be similarly disorienting, for exactly the reasons Smith lists. In considering when to tell, then, we must consider that instructors’ sense of self-efficacy might be affected by changes in their teaching practices.
The key observation in this context is that while the creation and delivery of a rich lecture contributes positively to instructors’ sense of self-efficacy, this contribution only involves students in a marginal way, if at all. John Mason describes the difference between the student experience of instructor telling and the instructors’ experience of telling as follows:
Yves Chevallard introduced the term didactic transposition to describe the way in which the intuitions and experiences of an expert are trimmed and edited for teaching purposes, so that what learners encounter is often little more than refined formal definitions, proofs of theorems, and examples of applications of techniques. Expert awareness is transposed or transformed into training of behavior. The result is that no appeal is made to learner’s emotions, learners’ powers are not called upon, and mathematical themes remain implicit. The pleasure and insight achieved by the expert in organizing the topic and ‘making sense’ leaks away and is lost to the learner, who experiences merely behavior training. (Mason, p. 259)
When considering active learning techniques and the role telling plays in them, we must be attentive to the need for teaching techniques that are both a positive influence on instructors’ sense of efficacy and a positive influence on student learning. Judicious telling is one example that would allow for instructors to maintain some sense of efficacy while still generally trying to limit direct telling focused on information transfer. There are a number of other ways in which teachers may gain efficacy even while reducing the amount of traditional telling in their classrooms in favor of more active learning techniques. Smith provides alternatives for sources of efficacy, which he calls “new moorings of efficacy” (p. 396). He suggests that, rather than finding their efficacy in traditional telling, teachers could learn to find efficacy in other kinds of classroom activities. These include activities such as choosing problems, predicting student reasoning, generating and directing discourse, pushing for more explanation, asking for questions that extend, and getting immediate feedback from students on what was just told. A high level of expertise and care is needed to construct meaningful sequences of student tasks, see for example the Journal of Inquiry-Based Learning and textbooks such as Active Calculus.
Lobato, Clarke, and Ellis (2005) have built on Smith’s (1996) work by arguing for a reformulation of traditional telling, noting that some key elements of telling can be captured in a way that allows students to be active participants in their mathematical learning. These authors reformulate telling as initiating, which they define as “the set of teaching actions that serve the function of stimulating students’ mathematical constructions via the introduction of new mathematical ideas into a classroom conversation” (p. 110), and as eliciting, “an action intended to ascertain how students interpret the information introduced by the teacher” (p. 111). They contend that some of the pedagogical results that telling accomplishes (like introducing new material into a classroom or communicating mathematical ideas with students) can still be accomplished by alternative means. By considering reformulations of telling such as asking probing questions, implementing carefully designed tasks, and encouraging students to share ideas, teachers can still accomplish goals of traditional telling without sacrificing a commitment to active learning.
Telling to Achieve Coverage
Another reason that instructors might traditionally tell is related to the issue of coverage. Covering all of the necessary course material in an allotted time is a perennial concern for instructors at all levels in most disciplines, and there is no denying that telling can be an efficient way of getting through material. If we fall a day behind in our syllabus, a natural response is to try to transmit the material quickly and efficiently so the students can at least be exposed to what they need to see. However, the example of students believing that \((a+b)^2=a^2+b^2\) should not be far from our minds — telling in order to cover more material is not always effective for students.
By exclusively considering course content coverage, and responding to content coverage with telling, we risk forgetting the many other elements of student learning that active learning addresses. For example, in the 2004 and 2015 CUPM Curriculum Guides, content goals only form one portion of the overall set of goals for mathematics students and programs — equally important are goals such as (from the 2015 guide) “assess the correctness of solutions, create and explore examples, carry out mathematical experiments, devise and test conjectures… [and] approach mathematical problems with curiosity and creativity and persist in the face of difficulties.” By recognizing that “coverage” should refer to both content and mathematical processes and practices, we can alleviate our tendency to tell to achieve coverage. A number of experienced teachers and programs have reached effective balances in this regard. For example, the IBL community has established the Journal of Inquiry-Based Learning to serve as a source of refereed course notes for IBL classes. The calculus program at the University of Michigan has been carefully developed by course coordinators with a strong conceptual focus; this also supports new instructors, including graduate students and postdocs, who are trained to quickly become proficient using active learning techniques.
In addition to the recognition that content topics are not the exclusive subject of coverage, recent research suggests that coverage of material is actually less important for student persistence and achievement in mathematics than the effective use of active learning techniques. In a study that found generally positive long-term impacts of inquiry-based courses on undergraduates in mathematics, Kogan and Laursen (2014) state that “‘covering’ less material in inquiry-based sections had no negative effect on students’ later performance in the major. Evidence for increased persistence is seen among the high-achieving students whom many faculty members would most like to recruit and retain in their department.” Thus, although we acknowledge the difficulties that issues of coverage present, and while judicious telling might be appropriate when coverage is important, we would argue that active engagement can be more effective for student learning than resorting to traditional telling.
It is difficult to measure student learning, especially if we want to assess high-cognitive demand tasks. Hence instructors often depend on proxies: scores on exams that may be testing lower-level tasks, student course evaluations, and the perceived quality of a lecture, among others. Given the evidence supporting active learning, we must move beyond information transmission, instructor’s sense of efficacy, and achieving coverage as proxies for student learning. By expanding our vision of instructor telling, we can develop methods of telling that are better aligned with student learning, and we can train ourselves to look for signs that students have been helped in richer ways. A common theme in the reflections from Part IV of our series, unintended yet clearly present in hindsight, is that the use of active learning techniques has increased our own sense of satisfaction and efficacy as teachers. By re-conceptualizing telling, and moving toward more mature forms of telling, we can find richer ways of recognizing how to help students succeed.
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Bressoud, D., V. Mesa, C. Rasmussen. Insights and Recommendations from the MAA National Study of College Calculus. MAA Press, 2015.
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Fukawa-Connelly, T. P. (2012). A case study of one instructor’s lecture-based teaching of proof in abstract algebra: making sense of her pedagogical moves. Educational Studies in Mathematics, 81(3), 325-345.
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