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A number of subjects in Economics depend on interconnected levels of increasing complexity, and to miss one step can result in not understanding a whole theory or model. Using appropriate pace and emphasis is very important, particularly as explanations of proofs, diagrams, graphs, tables and charts may otherwise become incomprehensible.

When we are familiar with a topic, we tend to speak faster, and international students in particular may struggle to follow. So it is important that we consider our pace and ensure that we do not go too fast. If we emphasise the most important points we allow international students to take notes and give them the chance to focus. Emphasising can be done by slowing down or pausing; by repeating or rephrasing what we said before; by changing the tone of our voice; and by signalling ‘what I am going to say next is very important/ relevant/ interesting’; by writing on the board (e.g. if we refer to authors and their work during a lecture or tutorial, it is advisable that we write the references down so that international students can take note, otherwise they may simply miss the references altogether); or by using circles or arrows.

Emphasis is also achieved with non-verbal cues. These cues are highly culture-dependent, but international students take cues from lip reading, and from our facial and body expressions to understand the message. So we should face the students when we are talking and avoid covering our mouths. Standing with our back to the light, low lighting; background noise, ‘chalk and talk’ or ‘talk and walk’ can impact on students’ ability to see and hear us, and our voice projection can be compromised. These barriers to following the message are worse for international students. We like to say that ‘chalk and talk’ should not be ‘talk to the chalk’!

If we are lively, use a good pace, face the students and use emphasis it is more likely that international students will feel motivated regarding our subject!

Proofs can be extremely hard to follow for international students. Students may not be familiar or not recognise the pronunciation of expressions and jargon particularly when the rationale seems counter-intuitive; and it is enough to mishear or miss a few words for the rationale to be lost, or totally misinterpreted. Struggling to read, copy, understand and write what is being said can be exhausting and result in international students being unable to follow. Particularly if this happens from the beginning of the explanation it can be very frustrating, and de-motivating.

Most proofs are the result of elaborate processes that involve trial and error and time to mature possibilities until an elegant, useful result is achieved. Yet, they are frequently presented as if they were self-evident. Instead of saying ‘*this is easy; trivial; basic, as we know;*’, it is far more constructive to recognise that the proof is complex and that it requires dedication to become familiar with it. To expect that the students understand immediately the elaborations of proofs and are thus able to ask informed questions is not realistic.

Tell students that we do not simply memorise proofs. In reality we need to understand the basic premises of the proof and revise them. This can help students be realistic about the effort, thinking and time to reflect required to learn proofs.

Most importantly, students have to understand that proofs are not there simply to make their lives difficult. We should clarify this each time we are presenting a proof.

If the students should have covered the basic principles elsewhere we should guide them to where the basic principles were discussed previously ‘*these concepts were addressed in lecture/seminar X and you may want to revisit your notes*’, preferably in good time before the lecture or tutorial.

If we plan to talk for the following 15 minutes, we should inform the students and acknowledge that some might struggle to follow everything. We can tell them that after that period, we will be doing a pause so that they can discuss their notes with their neighbours and we’ll be asking some groups to share in plenary what they remember. This can motivate students to pay attention as they are given a challenge and the promise of interaction!

We can use slideshows very successfully to support explaining proofs, and there are many lectures available on the web and provided by publishing houses that we can adapt. The danger is that showing a series of equations appearing simultaneously with text can result in information overload; and make us go faster inadvertently. International students will struggle to follow.

Another aspect to be aware of is that at times lecturers or teachers write diagrams and maths on the whiteboard, but only vocalise, without ever writing down, related fundamental intuitive arguments. International students may benefit greatly if we left space on the whiteboard to write down, even if only in note form, these arguments (Long and Barnett, 2008).

Pacing explanations and giving time for the students to take notes including a few moments of silence, when moving from one step to another, can signal that something different is to come. It can also give a moment of rest to all.

Clear signposts can be invaluable as the students are not simply presented with a lengthy proof that seems to go on and on with no direction.

So when we teach the proof we can:

- Explain the context of the proof. Why it is important to learn it – what are the practical or value-based implications?
- Show what the whole proof looks like in
*one go*(single slide or OHT, etc.). - Explain how we are presenting the proof in small stages, to ease understanding. We can assign each stage with a number of steps characterised by a particular attribute, for example, an assumption.
- Show the initial equation and the end equation. ‘We are starting here and we will end up here’ and why the final equation is particularly useful. Then present the constituent steps of the proof without going into the detail. ‘From this we will get to this/ from there we will get to there/ and finally from whatever we will get to whatever.’ Only then go on into the detailed explanations.
- Clarify where we are in relation to the whole.
- Repeat/reiterate important points, and signal ‘This is really important; interesting; useful.’

In our experience, this approach is equally beneficial when explaining models, theories and sets of instructions.

Using mixed visual inputs instead of relying on one single input alone to signal different stages of a proof adds variety to the explanation and helps tired students refocus. Slide shows, laminated A4 posters, flipcharts, and the boards can all be used to mark different points of a proof.

For example, we can print in landscape the initial and final equations on coloured A4 paper in a clear large font and use it as a poster (which we can laminate to reuse), and keep it on show until the end of the proof. Alternatively, we can use a visualiser.

Diagrams, graphs, charts and tables are indispensable in Economics. A common assumption is that because they are visual, they are easy to understand. In our experience this is not at all the case as typically there is a lot more information than is needed to make our point. Instead of illustrating the message in an attractive way, students can become distracted by subsidiary information and this leads to confusion rather than clarification.

Even simple instructions can be difficult for international students to follow in the context of a lecture but frequently they feel embarrassed to ask us to repeat them. For example, instructions like ‘if we now look at the top left hand corner of the graph’ or even ‘when x equals 10 and y equals 30 we can see’. International students may still be trying to write what we said before, so they may not even be aware of what we are expecting them to look at. Can we stop for a few seconds to indicate we need their attention? International students will also need more time or non-verbal reinforcement to reason ‘top’ followed by ‘left hand’ or ‘x=10’ and ‘y=30’ and to look in that direction.

Supporting the verbal message with extra signs of what we are referring to in diagrams, graphs, charts and tables can be very helpful. Signs include gestures or movement (pointing in the general direction alone can be too vague); or visual indicators (different colour; a shape like an arrow or circle).

With slideshows, we can point an arrow, or circle the area of interest. Also, when explaining diagrams, graphs, tables and charts we have to ensure that we look at the students, as discussed in section 3.

It is important to be very clear about what we want the students to learn. What visual cues can we add to make absolutely clear what is relevant (e.g. using an arrow or circle)? Can we simplify the diagrams, graphs, charts and tables? Are font size and type clear? Can we refer to the trends in a graph in terms of shape or can we use lines with shapes/dots, etc.? What about the language used? Is it accessible and can they refer to a glossary?

Tools including FEELE, Me:tal, WinEcon, and Embedding Threshold Concepts can be used to support the learning of students from a range of different backgrounds.

The complexity of language (verbs, nouns, adjectives), and nuances used when describing trends, degree and rates of change, movements on a graph (upward, downward or horizontal) can be a deterrent for international students to remain motivated. For example, a decrease can be referred to as a decline, drop, fall or slide.

International students have to learn to present and describe information using various words and expressions. However, initially, we can limit the range of words and expressions used, and we can provide a list for students to refer to in the lecture.

We can plan time in our lecture so that students work in mixed groups on proofs, diagrams, graphs, tables and charts with their neighbours as explained in section 12.