By Priscilla Bremser, Contributing Editor, Middlebury College
“Can you recommend a good math tutor?” I hear this question from friends with children in local schools, academic support staff at my institution, and my own students. Once or twice I’ve even heard it from a student on the first day of class. Although tutoring has much in common with other educational settings, it presents its own opportunities and challenges. In this post, I explore why one-on-one instruction is so appealing as a supplement to classroom instruction, and how effective tutors make the most of tutoring sessions.
As Lepper and Woolverton point out in setting the stage for “The Wisdom of Practice: Lessons Learned from the Study of Highly Effective Tutors” [2, p. 138], “tutorials provide a venue for learning that is inherently more individualized, more immediate, and more interactive than most common school settings.” Specifically, individualization ensures more focused attention from both tutor and tutee. Immediacy allows for instantaneous feedback. Interactivity means that the tutor can make real-time decisions and adjustments as the student’s comprehension level and emotional state become more clear.
The authors go on to identify specific practices of expert tutors. While the overview is limited to studies of tutors for elementary school students studying mathematics, many of the effective practices it describes are also applicable to secondary and college mathematics settings. For example, “our best tutors seem to prefer a Socratic to a more didactic approach” [2, p. 146]. Naturally this approach involves asking questions and providing hints rather than providing quick answers. It also includes making a distinction between “productive” and “nonproductive” errors [p. 147] and responding accordingly. A productive error is one that the student can self-correct, with the long-term learning benefits that ensue, while a nonproductive error is best corrected immediately by the tutor.
Readers of How People Learn and related works will recognize a metacognition theme in this observation of Lepper and Woolverton: “more effective tutors are more likely to ask students to articulate what they are learning, to explain their reasoning and their answers, and to generalize or relate their work in the tutoring session to other contexts and problems.” An expert tutor, then, guides the interaction not only for strong communication with the tutee, but also to strengthen and reinforce learning.
The question of tutor-student communication is a complex one. In a research review, Graesser et al. [1, p.418] point out five “illusions” that tutors may hold. These are the illusions of grounding, feedback accuracy, discourse alignment, student mastery, and knowledge transfer. They categorize the misunderstandings that tutors often have about their students’ thinking. Have you ever been asked whether you understood something, and said “yes” even though you weren’t sure? You were giving inaccurate feedback, and your questioner may not have caught on. Of course even a sincere “yes, I understand” may be inaccurate, as “it is the knowledgeable students who tend to say ‘No, I don’t understand.’ This result suggests that deeper learners have higher standards of comprehension” (p. 414).
For an example of poor discourse alignment, note that “tutors sometimes give hints, but the students do not realize they are hints” [p. 418]. Now that is a reality check. In our previous post, Jess Ellis Hagman wrote, “Mathematics education research is the systematic study of the teaching and learning of mathematics.” Sometimes a seemingly small detail emerging from such study can have profound implications.
More from Graessner et al.: “A good tutor is sufficiently skeptical of the student’s level of understanding. … A good tutor assumes that the student understands very little of what the tutor says and that knowledge transfer approaches zero … (E)xpert tutors are more likely to verify that the student understands what the tutor expresses by asking follow-up questions or giving follow-up troubleshooting problems” [1, p. 419]. I recall working with an algebra student who insisted that he understood the relationship between the graphs of $y = x^2$ and $y = x^2 +2$, even though his graphs intersected. Rather than pointing at the intersection and explaining my concern, I should have suggested that he add the graph of $y = x^2 + 1$ and tell me what he noticed.
Given recent research on the effects of students’ emotions and mindsets on learning, how do good tutors attend to those factors? For one thing, while they are supportive and kind, they are sparing with praise. When these tutors do offer compliments, they refer to the work, not the person. The compliment might be an indirect one, such as a simple, “That was a hard problem you just did.” Good tutors also find ways to turn control over to their students by, for example, letting the tutee choose between two equally challenging problems .
Many of the above observations about effective tutoring, and potential pitfalls, are relevant to considerations of classroom instruction, especially active learning environments in which instructors have frequent, though short, interactions with individual students and small groups. In addition, faculty office hours are often sequences of tutoring sessions. Occasionally I’ve had the sense that a meeting with a student didn’t go well because I said too much or corrected an interesting mistake too soon. The research seems to confirm my impressions.
Still, tutoring is different from classroom instruction in significant ways. Most obviously, perhaps, tutoring usually happens when someone determines that special intervention is required. A student is struggling, or not doing as well as expected. Perhaps the student’s parents see tutoring as a way to improve grades or test scores for college applications. Under these conditions, it is especially important for the tutor to attend to the student’s affective state.
Additionally, although the appeal of tutoring as a remedy springs from the one-on-one nature of tutoring sessions, there are usually other people on the periphery. There’s the classroom instructor, who may have recommended tutoring, or may not know that it is happening. Perhaps the student’s parents are involved. Many school districts coordinate tutoring programs in cooperation with local organizations. It seems reasonable to conclude that communication challenges come along with those added relationships.
For one thing, the tutor may not know or understand the instructor’s learning objectives for the student. A peer tutor for my Calculus I students may have taken AP Calculus in high school, which can be a very different course from mine. A volunteer tutor in a public school might remember shortcuts for working with fractions, while the teacher wants to Nix the Tricks.
Further, the tutor might not have a deep understanding of the relevant mathematical content. As a sophomore in college, I signed up to be a peer tutor. A junior came to me for help with multivariable calculus. She was baffled by parametric curves, which hadn’t been covered in my multivariable course the previous year. At the time I was mortified, feeling somehow that I’d failed personally. But she and I tried to work through that section of the textbook together, which (I now recognize) was probably good for both of us. According to , “tutors in …same-age and cross-age collaborations tend to learn more than the tutees” (p. 412). It’s probably important that I knew what I didn’t know about parametric curves. In contrast, a colleague once overheard a peer tutor say, “the individual terms of the series go to zero, so it has to converge” in our department common room. Fortunately, our peer tutors now undergo appropriate training before they start.
Can I recommend a good math tutor? Yes, but I would want that tutor to get training first. It wouldn’t hurt to also read  and ! (Other resource suggestions are welcome in the comments.) Good tutors know that showing and telling should be used sparingly, and only after careful listening.
Thanks to Steve Klee for directing me to .
 Graesser, A. C., D’Mello, S., & Cade, W. (2011). Instruction based on tutoring. Handbook of research on learning and instruction, 408-426.
 Lepper, M. R., & Woolverton, M. (2002). The wisdom of practice: Lessons learned from the study of highly effective tutors. Improving academic achievement: Impact of psychological factors on education, 135-158.