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Instructor-led problem solving
- It is important to give students some control over the tutorial – for example, by asking them to nominate the problems to be done during the tutorial, and by getting them to provide their ideas, solutions and methodologies.
- It is important to continually solicit feedback from students during the tutorial in a patient and encouraging manner.
- One idea that can work to get over the long, awkward silences at the start of the lesson when no students respond to anything is for you to start working through a problem. As you go, you can start asking simple questions for them to answer in the chat, and gradually ask for more and more input, until students are settled and participating.
- You can ask students by name for their response, but you need a plan B in case they don’t answer. If you do this, be careful not to distress students by embarrassing them publicly.
- Giving students “a few minutes” to do a question – silently, online – is not effective. “A few minutes” ends up being less than one minute because it feels like a long time when you are waiting silently, but it is not long enough for students to do the problem. It is not a good use of the communal time. When you do this in a classroom, there is some collaboration – students see what other students are doing – but online everyone is on their own and there is no benefit to spending class time on individual work that you cannot see, and cannot provide feedback on
- It is important to model what you want the students to do. It is not enough to talk through the reasoning used to solve a problem. You need to:
- Talk through where in the lecture notes/textbook to refer to / where to find information on the theory you need to do the problem – it’s not your job to go through all the theory, but students do need to know how to look it up for themselves.
- Discuss how to approach the problem.
- Write up the solution in a way that would get full marks.
- It is a bad idea to cut corners when writing up problems in tutorials. It is better to make explanations more succinct and write out a full, beautiful solution than to give lengthy amazing explanations with an incomplete written solution or just a few calculations scribbled down. Students need the whole mathematical process – including mathematical communication – modelled for them.
- Ask the students straightforward questions (“What theorem do we use here?” “How do we simplify this?”), but also ask them higher-order questions (“How does this work on Boolean Algebra relate to our previous section of work on Sets?” “Could you have solved this problem without using Calculus?” “What are the implications of this result?”). Higher order questioning leads to deeper understanding.
Student group-work tutorials
A number of different approaches are in use:
- Students solve new problems in breakout room groups
- Students have pre-solved problems reviewed in breakout room groups
- Students have pre-solved a set of problems. They then solve a similar set of problems in breakout room groups.
Method 2 requires students to be put into pre-allocated rooms. Methods 1 and 3 can use either pre-allocated or randomly allocated rooms.
Observations were that presenting students with new problems to solve was more effective than having them review pre-solved problems.
Hence, method 2 was not as effective as the other two methods. In method 2, students did not make good use of the group time. Mostly they chatted or waited silently until the tutor came to assess their work. They did not discuss their work amongst themselves. After tutor assessment, they either went back to chatting/waiting silently, or left the class. There were also indications that the whole group often did not work through the pre-solved problems.
The main factor in how effective the other group-work tutorials were was dependent on the effectiveness of the tutor’s methodology. Good classroom techniques – clear expectations, explicit routine, effective questioning, and approachable demeanour – appeared to be by far the most important factor.
It seemed unimportant whether students were pre-allocated or randomly allocated to their groups.