Seven principles

Seven principles for good practice in undergraduate education.

A great deal of research on education in universities is synthesised in the "Seven Principles for Good Practice in Undergraduate Education" of Chickering and Gamson (1987). These are all things that we know from many studies to be essential features of best practice across subjects. The points discussed below tell us things about what effective tutoring should look like. 

1. Encouraging contact between students and faculty 

To perform well, students should feel part of a mathematics community. A message we try and convey to students is that we are all mathematicians, working together, getting stuck on problems and so on - it is just that tutors and lecturers are further down that route. Modern universities with large classes can be an anonymous place. Students do better if they feel they know their teachers and believe that they care about their progress.   

• Learn (and use) the names of your students if you possibly can. 
• Be friendly and approachable.    
• Convey the idea that you are enjoying exploring mathematics with your group. 
• Encourage students to approach lecturers, their Personal Tutor, the Student Learning Advisor, etc.  

2. Developing reciprocity and cooperation among students 

All the research shows that a large majority of students work better in groups. While competition can encourage students to do well, it also has a serious downside: fear of failure can cause students (even strong ones) to be reluctant to ask for help and may lead them to be very conservative in their thinking and not to contribute in class for fear of being wrong. 

• Try and find ways of making activities into group ones. 
• Encourage students to help and explain things to each other. 
• Encourage learning for its own sake rather than as a route to being the best in the class. 
• Actively promote discussion and cooperation. 
• Value all contributions to discussion and do not specially focus on students you have decided are better. 

3. Encouraging active learning 

This is one of the most important ideas. Reading something and transcribing it into notes, listening to a lecturer or tutor explaining something and copying it down without thinking are "passive learning". Active learning on the other hand is thinking hard about something, exploring consequences and connections, etc. Cognitive Scientist and Educationalist Daniel Willingham has a memorable phrase which is very relevant: "Memory is the residue of thought". Things we study passively we largely forget on quite a short timescale whereas things we have thought deeply about stick far better. 

The problem is that passive study is easy and comforting: students go to a tutorial and (say) copy down what a tutor writes on a board, intending perhaps to make sense of it later. The student feels they have accomplished something because they have some nice neat notes. The tutor feels satisfied because the students are. It is all too easy for both parties. 

• Take every opportunity to engage students in thinking about ideas and problems. For example, it can be appropriate to ask students what they think the answer to a question will be in advance. Or ask them to look at each other's arguments and comment on them.   

• Resist the temptation to spend much time "telling", i.e. don't normally show students how to do whole problems. (If you do spend time on long explanations, students may nod and look as though they follow what you are saying, but they will often just be taking the line of least resistance: it's easier for them and you than admitting they haven't a clue what you mean!) 

• Turn conversations into questions: if a student says "I don't know how to get started on this question", don't just start it for them. Rather, ask them if they know what all the words in the question mean, or ask (for example) if they can think of a simple example of what they are trying to prove in general.    

And a quick word about what active learning does NOT mean. It does not mean that students have to be left to themselves to work through questions. The idea is that they manage to do some work which is perhaps a little more difficult than they could easily manage without a tutor because you are there asking them questions that guide them in the right direction or causing them to notice mistakes they are making and nudging them towards spotting good approaches. 

4. Giving (prompt) feedback 

This is such a central part of teaching we have a separate section on it.  For now, let's just observe that it is important that in the end students discover whether their lines of thought are right or wrong.   It's better if they establish this for themselves, separately or as a group,  but if they don't it is better to tell them than to leave them in doubt. 

5. Emphasizing time on task 

We know that in many different areas the main factor contributing to expertise is the amount of time spent in (useful) practice. In mathematics more even than in other academic areas there is a "myth of talent", an idea that you have some sort of native ability at the subject which is much more important than how much studying you do. It's not a helpful belief, and there isn't even much truth in it.   

• Try and convey to students that if they study hard and take advantage of all the resources available, they can and will learn. 
• We tell students that for each contact hour we would expect an additional 2 hours of study, whether that is alone or with classmates. 

6. Communicating high expectations 

It is important that students believe they can succeed, and you can help them in this by communicating high expectations to them.   

• Expect students to turn up, work hard and so on. But be supportive and not too judgemental when they don't live up to those ideals. 
• Expect that students can and will learn to be good mathematicians, but remember that this can be a difficult journey for some. 
• When students fall short of your expectations, remember always to criticise the behavior and not the person.  Many students, if they perceive what you are saying as an attack on them will simply resist believing what you are saying.  
• It is good to communicate to students that working hard, attending close to 100% etc are normal.  Students who are not doing these things often imagine that the majority of the class are also not engaging well (and may misattribute their own lack of progress to their being less good at the subject). 

7. Respecting diverse talents and ways of learning 

• Be tolerant of students, and remember that they may not have the same background or interests as you. 

• Listen to students and try to understand what they are saying and their point of view. 

• Remember that learning is a joint activity of learner and teacher, and when it is not happening it may be that both parties need to do better.