Stanford University psychology lecturer Mark Lepper and paediatrician Maria Woolverton surveyed a group of tutors who specialized in primary and secondary school mathematics to understand what makes for an effective tutor. In their study, they discovered seven key characteristics shared by the most competent and successful tutors, leading them to introduce what they term the INSPIRE model. The traits of the best mathematics tutors are as follows:

**(1) Intelligent**

The most successful tutors have an excellent command of the subject matter, in this case, mathematics. Their command of the subject allows them to very rapidly integrate and synthesize skills and techniques from various topics and apply them flawlessly to any question. They also possess a keen intuitive understanding of how students think and are able to adapt their own teaching methods to suit their tutees.

**(2) Nurturant**

The best tutors always establish good rapport with tutees and are able to empathize with their tutees’ struggles in tackling challenging problems or in having to manage a large volume of classroom work. These tutors pay close attention to the level of motivation or level of frustration on the part of their students and are quick to offer support where needed, or adjust the pace and scope of the lesson where appropriate.

**(3) Socratic**

The good tutors do not simply give out the solution to math problems, but also engage their students with continual prompting and probing, trying to elicit from the student possible approaches to the given problem. When tutees encounter roadblocks, these tutors do not start off immediately with the full answer, but give hints that are initially very general, and only get more specific if students are unable progress further by themselves.

**(4) Progressive**

The best math tutors are able to quickly assess their student’s level of understanding at the start and are able to structure the lessons in a way that leads the student from simpler problems to more challenging questions in a progressive manner. They engage tutees in what most people would calling “drilling” or “repetition”, deliberately repeating a particular problem-solving technique several times in the course of a lesson until it becomes familiar to the student.

**(5) Indirect**

The best tutors never criticize tutees directly or overtly, but instead draw attention to mathematical errors by implication and questioning, so that the tutees themselves are led to rethink their approach to the problem or recognize the flaws in their own working. Thus, errors are recognized and corrected mainly by student self-analysis, as opposed to tutor-centred criticism.

**(6) Reflective**

Effective tutors often ask students to describe or articulate what they are doing, what formulas they are trying to use, or what method they are employing. This is found to help the student remember the entire process better, and also assist them in generalizing their understanding to multiple contexts and situations.

**(7) Encouraging**

Effective tutors regularly motivate and encourage their tutees by creating an inquisitive atmosphere, piquing their curiosity with mathematical problems relevant to their daily lives, and providing frequent, positive feedback.