In our study group, we read Tommy Dreyfus’ paper Why Johnny Can’t Prove. In the paper, he describes how beginning university mathematics students find it difficult to see the difference between different kinds of mathematical reasoning.
Often, teachers and textbooks use both formal and informal explanations to justify claims. Both are needed, and for an experienced mathematician it is easy to see which one is which. Students, on the other hand, may get confused and cannot form a clear picture on what is meant by a proof.
This is how we came up with the idea of a proof hat. It makes it possible to distinguish between different kinds of justifications. When teacher writes a formal justification for a claim, he/she wears the proof hat. When a topic is discussed more informally, the proof hat is not used.
It seems that the proof hat makes students pay attention to the different levels of justification, since if the hat is missing, they will let the teacher know. And it definitely brings cheerfulness to the classroom.
Instructors working with students. The instructors wear orange vests so that the students can spot them.
At our department we are using the Extreme Apprenticeship method that emphasises students’ active work. Students are assisted in their work by instructors who are usually senior students.
The instructors support students’ own thinking and teach them study skills. They try, at the same time, not to spoil the joy of discovery for the students. This kind of teaching can sometimes be difficult, especially if you are used to thinking that a good teacher is one who can explain everything thoroughly to the students. That is why our instructors take part in training that lasts throughout the semester. The training consists of weekly meetings in which the pedagogy of instruction is discussed.
Together with the instructors, we have put together some guidelines. The list has changed during the years, as we have all learned more about good instruction. This is what the guidelines look like at the moment:
- Listen. Encourage the student to talk, and listen to what he/she says. Let the student’s needs lead your instruction.
- Guide individually. Students are different. Some may need help with the basics and require very concrete advice. Others are just asking for a small hint. Try to find out what the student needs, and address that need.
- Let the student do and discover. The aim is that the student works towards a solution with the support of the instructor. Guide in such a way that the student can have ‘aha’ moments.
- Be encouraging. Students may be very insecure in what comes to mathematics, and feel that they are not doing well enough. Be encouraging and try to find something good in the student’s work.
- Be active. Circulate among students on your own initiative and say hi to them. It is easier for the students to ask questions if the instructor has opened the conversation.
- Divide your attention. Do not let one student take too much of your time. Note also that sometimes it is good to let the student think about the problem on his/her own.
- Help the student in reading the course material. Reading mathematics is difficult for the students, and they may try to use the instructor as a data bank. The instructor should encourage the students to read the course material and advise them how to do this.
- The instructor does not need to know everything. He/She can investigate the topic together with the student. This way, the student sees how a more experienced mathematics student works.
- Teach study skills. The aim of guidance is not only to help the student to solve a task, but to show how mathematicians tackle problems they face.
- Do not take emotional outbursts personally. React to students’ problems and outbursts with compassion and empathy, but do not let them worry you too much. The reason behind outbursts can, for example, be the insecurity of the student.
- Encourage co-operation. Students should learn to discuss mathematics. Encourage students to collaborate, especially if many of them are working alone on the same problem.