### How to be a good maths tutor

By KM WONG

“First, do no harm,” a pledge for doctors attributed to the ancient Greek physician Hippocrates, is not exactly out of place here as a first reminder for tutors. Under no circumstances should tutees ever be treated or spoken to in ways that would hurt their feelings, confidence and self-esteem. With a cautionary note to begin, it is intended to highlight the potential of a tutor-tutee relationship which has to be handled always with care. On the brighter side, being a tutor can be very rewarding when the tutee makes gradual progress and eventually learns to be independent in acquiring knowledge and skills.

In a sense, a successful tutor is to work themselves out of the job as much as they can. The long term goal of tutoring is to help students become independent learners who no longer need tutors. So goes the proverbial saying here too: “Give a man a fish, and you feed him for a day; show him how to catch fish, and you feed him for a lifetime.” Given the task in this perspective, the usually presumed prerequisite of being a tutor, namely content knowledge of the subject, is obviously insufficient. There has to be something more to it—not just fish, so to speak! Apart from helping connect the dots and explain a difficult point or solve a problem in hand, the tutor has to impart to the student a subtle understanding of how to learn and how to approach an unfamiliar problem in general. Mastering the art of being a good tutor requires therefore content knowledge, experience, social skills and an understanding of the nature of learning.

Let us begin from the beginning. To build up a trustful relationship with the student is essential to effective tutoring, as all social interactions (with contents and emotions) are mediated through this relationship. In order to establish a good rapport with the student, you have to bring along the following items in your toolkit to a tutoring session, in addition to the required subject- specific knowledge—if you think you miss any of these, it is never too late to cultivate and get it ready:

- patience: the ability to wait something out, or endure something tedious or painstakingly slow, and still remain calm, not only during a specific session but sometimes also over a period of time when the student’s progress is not as swift as you wish,
- empathy: the ability to put yourself in students’ shoes, to understand how they perceive and feel about a situation and to communicate your understanding in order to gain their trust and willingness to speak frankly their problems,
- an open mind: the readiness to understand and accept, without passing judgement, other people’s attitude and point of view,
- sincerity and readiness to help: an honest desire to find suitable ways to help students learn,
- a positive, supportive attitude: being optimistic and encouraging at all times,
- a positive outlook: the belief that things can be changed through a course of sensible actions, and
- last but not least, humour: the ability to express things in funny ways which help relax the stressful learning situation at times, or facilitate the communication of constructive criticism.

Coming to a tutoring session, tutors have to recognise that first, successful learning can take many different paths, and your student may not necessarily share the same path as yours. So be open and sensitive to gauge their path of learning based on your subject knowledge, and guide them according to their personal perception and understanding as much as you can empathise. Second, students have different learning styles (aka cognitive styles). Some prefer things expressed in words, some like to picture situations visually more often, and some feel more at home in sounds and rhythms. So keep an open mind and see which forms of interaction can achieve the best result of communicating the desired content for a specific learner.

In the subject of mathematics, what students most frequently ask is how a particular mathematical problem can be solved. On another occasion the tutor may like to choose a specific problem in order to demonstrate the use of some relevant mathematical concepts, relationships, formulae, or theorems. Let us go through in detail what a tutor should be doing in such a situation.

- If you are making the choice, start with problems which you judge to be easy enough for your student to make a start but not necessarily be able to work through to the end. You can classify difficulty levels of problems roughly according to the concepts, relationships, and complexity of steps (computation, classification, comparison, decision, etc.) involved.
- You can also ask the student if they have any specific problems they want to work through with you as a start. Ask the student why a problem is chosen, so you know what and how to address in the solution process.
- Sometimes you may want to start from a problem your student can solve. In that case, ask the student to show you the working and explain the steps. The more the student can talk about it, the better they can master it, and the more you understand their level of knowledge and skills too.
- Don’t ever make comments (before the student has solved a problem) like “this question is simple,” “this one is straightforward,” or “this is a difficult question.” You can never be sure how your student takes in such comments—e.g., if they have no clue at all, but since you say this question is simple, they just feel stupid. You don’t want such unwelcome psychological effects, do you?
- With problems posed by students themselves (from their homework or textbooks) or by you, students of course do not see an immediate way out. Begin by asking the student to identify clearly the given conditions or describe what is given and what exactly is the problem asking If anything in the problem is missed by the student, help them clarify and make sure they have a correct understanding of the problem context, knowns and unknowns first.
- Next, ask the student what they can do to solve the problem by prompting them to ponder the following: what topic this problem concerns; what concepts, relationships, formulae or theorems may be relevant to the situation; can you draw a diagram to represent the problem situation; can you think of any technique or method you know that can help solve this problem; does this problem resemble any textbook examples or problems you have successfully solved before; if the problem is similar to (but not exactly the same as) an example or a problem you have solved, what are the similar parts and what are the differences; can the problem be divided into sub- problems so that some sub-problems are familiar and readily solvable; search for links connecting the known and/or unknown items by relating to the concepts and relationships in the topic; what are the possible options to try here (name them clearly); what can you begin with, as a try; what do you do next; … More useful guiding questions can be devised according to the specific problem in hand.
- In some cases where hints are desperately needed, you may point to a formula, a theorem, a textbook example or a problem they have previously solved to continue the dialogue. Ask them to compare and name the similarities and differences. Learning and insight (the “aha” experience) basically come from seeing (in their own eyes, not yours!) analogies they can connect with.
- Try to avoid solving the problem directly for the student as much as the situation allows, but instead, guide them step by step to the final answer by appropriate questioning, pointing out overlooked connections, and refining their responses and ideas. Help them compare options where necessary, and discuss with them what they know and what they don’t know yet at different stages of the solution process.
- Ask open questions to prompt for students’ idea, explanation, reason, justification or understanding at different stages to move the process forward. Ask what, when, where, how, why, Don’t ask for simple yes/no answers: such tend to be leading questions, and students will guess or answer according to your non-verbal cues. No meaningful learning takes place in the latter.
- When the student makes any mistake or error in the process, point it out and ask if they can correct it themselves. Help them put it right again if necessary. At the same time, it is sensible to check whether it is just a careless mistake, or a (systemic) mistake from wrong understanding of concepts or relationships. Some common mistakes are procedural—for example, when too many steps are involved, some students will lose track of some loose ends. Some other mistakes emerge from so-called misconceptions or sometimes incomplete knowledge of the mathematical method concerned (e.g., restrictions or conditions of application being ignored), which can only manifest themselves when the student applies such (misunderstood or incomplete) knowledge to situations.
- In some situations, you may just tell the student that there is something wrong in a certain part of their working and ask them to identify exactly where it is. To learn to stand aside and find out one’s own mistake is an important skill to cultivate too. In fact, depending on how skilful and confident the student has become in a certain topic, you can sometimes just let the student go on with errors made already until they recognise the issue themselves. So much the better!
- Make the best use of students’ mistakes to enhance learning. While accepting that to err is human, do help the student see the positive side of it when better understanding can be achieved through honest dealing with mistakes. Encourage the student to learn from their mistakes and see that proper learning grows through making mistakes. Help the student cultivate a positive attitude towards making mistakes.
- A problem can very often be solved in several different ways, though perhaps only with minor differences in substance insofar as most school maths problems are concerned. When the student takes an approach different from your presumed solution, keep an open mind and do guide them along as much as their steps are reasonable and make sense to them. Sometimes they may have learned it from their teacher who has taught them this way in class. As long as their solution comes out from their own perception and understanding, they will benefit from solving it in their own way (even if it looks “stupid” to you—but don’t say it!). If it makes good sense to them, it will anchor well in their personal knowledge of the topic.
- When the student has finally solved a problem under your guidance, ask them to look carefully again at the solution and briefly describe the steps involved in the solution, and the mathematical concepts and relationships applied. This recap helps the student grasp in a nutshell what they have done and relate the solution to what they have learned. Depending on the richness of the specific problem, you can ask further questions to enhance learning: what have you learned from solving this problem; any new discovery on your part; does the solution make good sense to you; why can’t you apply the method there in that example directly here—what is missing; review your solution to see if some steps are actually redundant and can be skipped; can you shorten your solution by going directly from here to there; can you compare this problem with the one you solved earlier here—what is the difference; can you think of a different approach to solve this problem now; …
- In this evaluation process, you can also highlight that the problem has been mostly solved by the student on their own, point out their success in choosing the relevant strategies or making the right decision in pursuing a perceived connection to the standard method, as the case may be. These encouraging remarks can help the student build up their confidence in approaching unfamiliar problems in future.
- If the tutor and the student do not just stop at the satisfaction of finding a correct answer and follow the suggestions above, something great will emerge in the long run. Learning is, in essence, perceiving, making and reorganising connections, building a rich network to connect initially unrelated things in various ways and orders, drawing analogies between apparently different things, … So this last step is actually a crucial phase when the student can learn most.
- An additional way you can try to foster learning is to present another similar problem (or an “isomorphic problem,” i.e., with possibly only some numbers or parameters changed) for the student to attempt on their own. This can check whether the student can carry out similar procedures to solve a problem of more or less the same type. You can adjust the amount of changes to parameters in a problem to suit students of different ability levels.
- During and after the solution process, encourage the student to ask questions too, regarding whatever aspects in the solution process, or details about the mathematical concepts or methods involved. Ensure the student that you sincerely welcome questions and there are definitely no stupid questions.
- When you come across anything you are unsure about regarding details in the mathematical concepts or methods, you should never muddle through or say anything that may confuse the student. Point it out clearly and honestly, and promise you will check it out and explain next time. This demonstrates a proper attitude to learning as well.

After a tutoring session with examples explained and problems solved, no one will dispute the importance of doing more exercises afterwards for consolidation of learning. Practice makes perfect, as the proverb says. Tutors can advise the students to work out some more questions in their textbooks or specially prepared worksheets. But before assigning such drill and practice, do make sure the student has mastered the correct procedure with enough understanding for those types of problems. While conceptual understanding usually improves with more practice (by recognising the relevance of more parameters in similar problem situations), let alone the positive contribution of more confidence to cognitive functioning, practising a wrongly perceived procedure too much, on the other hand, can risk the reinforcement of misconceptions or systemic procedural mistakes, which unfortunately may persist in future—just imagine the difficulty of getting a cart with wheels out of a field with many deep soil ruts that run counter to the direction you want!

As a fundamental principle, it is always more fruitful to let the student do the talking rather than the tutor wherever possible. Thinking mathematically can be cultivated by working orderly, logically, systematically, and sensibly with full awareness at every “twist and turn” according to mathematical concepts and rules. To help streamline messy thoughts, the tutor, acting like a tour guide, asks the student to think aloud along the way as much as possible, prompts and cues the student to express what they are thinking, guessing, doubting or making decisions. It is normally difficult for most students to go beyond some very vague expressions at the start, especially because of the lack of confidence and the fear of making mistakes, part of which can be relieved in a trustful tutor-tutee relationship. With supportive examples of verbal expression and enough encouragement from the tutor, students can usually make a breakthrough at some point to talk about what they are pondering with some precision. Of course, this achievement in clear articulation certainly takes some time, which may vary much from person to person, and tutors must remain patient and positive throughout.

Tutors can also ask the student about how they feel in their maths class, in doing maths homework and in tackling unfamiliar maths questions. Do they usually defer their maths homework until the last moment, are they ready to try difficult questions, or do they think they are not smart enough for maths? Issues concerned are like motivation to learn, mathematics anxiety, emotions, confidence, self-esteem, fear of success or of failure, etc. This so-called affective aspect of maths learning is a fuzzy area psychologists have no systematic and thorough knowledge yet, but is certainly playing a significant part in student learning. By establishing a conversation in this aspect and gaining a better understanding of the student, tutors may be able to play a supportive role when the time comes.

In this short article intended for beginning mathematics tutors, issues like students with special educational needs (SEN), mathematics anxiety (aka maths phobia), dyscalculia, and visuospatial deficit cannot be addressed. When you suspect a student may actually have some form of learning disability because their performance goes poorly below some reasonable threshold, you are advised to refer the student to the programme coordinator of LearnerThon so that professional consultation and diagnostics can be arranged with special education teachers and educational psychologists.

##### Summary

- A tutor-tutee relationship has a great potential for good or for bad, so must always be handled with the utmost care.
- The long term goal of tutoring is to help students become independent learners. Therefore, apart from teaching subject knowledge, tutors have to guide them in mastering how to learn and approach difficult problems on their own.
- Aside from subject knowledge, a good tutor also needs patience, empathy, an open mind, sincerity and readiness to help, a positive, supportive attitude, a positive outlook, and humour.
- Students have different learning styles and may proceed along different learning paths, so tutors should be open to various learning modes and opportunities and guide them accordingly.
- Techniques in guiding students to solve a maths problem:
- Establish a supportive learning environment for the student to feel safe and comfortable to tell their thoughts and ask any questions without fear of judgement.
- Beware of any possible negative psychological effects before you make any comment.
- For teaching purposes, choose problems which are not too straightforward but still within reach of students’ capability.
- Guide students to solution by prompting for their thoughts as much as possible, using appropriate open-ended questions, and reminding them to relate to, or compare with, or connect concepts, procedures and previously solved problems.
- Avoid as much as possible any direct demonstration of solution – encourage a minimal contribution from students even in the worst case.
- Handle students’ mistakes wisely: make use of opportunities in different ways to identify misconceptions, enhance proper learning, and cultivate a positive attitude towards making mistakes.
- The more students can talk, ponder and explain on their own, the more they have learned from a task. In other words, the less the tutor talks, the more the students have learned, so is the dictum: “Less is more. “
- Successful solution of a problem is not an end: guide the student to review and summarise the solution procedures, evaluate the whole solution process, and compare with previously solved problems where applicable. This step reflects how much the student can learn from and go beyond the task. Tutors’ encouraging remarks can enhance student’s confidence in future problem-solving.
- Choose similar problems with changed parameters to assess students’ mastery and understanding of the solution procedures where necessary.
- Assign exercises for drilling and practice according to your assessment of the student’s capability.
- If you come across anything you are not too sure, be honest and promise you will come back to it after you have checked it up.

- Practice of solving routine or non-routine maths problem contributes positively to students’ mastery of the related concepts and procedures, and their confidence too, provided that they have acquired the correct procedures with enough understanding.
- Tutors can discuss with students their personal emotions and feelings in maths learning on suitable occasions. Such self-awareness can contribute positively to their learning in the long term.
- Referral is necessary if tutors come across students with special educational needs, maths anxiety, dyscalculia, visuospatial deficit and the like, which are beyond tutors’ competency to address.