2 Effective feedback

Feedback has been shown to make an enormous difference to learning (Hattie and Timperley, 2007) when it clearly provides answers to these three questions:

  • Where am I going?
  • How am I doing?
  • Where to next?

First the student needs to be very clear about the purpose or objective of the learning (‘Where am I going?’). Feedback on ‘How am I doing?’ has been shown to lead to students displaying enhanced engagement and motivation to close the gap between where they are going and where they currently are in their learning. However, they will only be able to employ that motivation if they can close the gap further by knowing the answer to the third question, ‘Where to next?’

Giving and receiving feedback requires much skill from both teacher and student alike. It requires a classroom ethos that allows students to expose their thoughts without fear of ridicule from anyone and where the focus is entirely on everyone learning and improving together. Within such an ethos it is possible for teachers or other students to give feedback focused on ‘How am I doing?’ and ‘Where next?’, and for students to listen to and act on that feedback.

If students try out their ideas together, they can receive frequent feedback from one another. The feedback will not be of the same quality as the teacher’s, because students will not have the depth of subject knowledge nor the ability to make connections that the teacher has; however, the opportunity to gain feedback quickly often outweighs the issue of quality.

The purpose of the next activity is to learn more about triangles, but it also gives the class the opportunity to try out using the three questions for effective feedback (‘Where am I going?’, ‘How am I doing?’ and ‘Where to next?’). You can also apply the ideas from this activity to other mathematical topics.

Activity 2: What happens if …?

Figure 2 How many different triangles can the students make?

Every student, or group of students, should have in front of them a triangle made from sticks.

Ask your students to discuss the following questions in groups:

  • Using the same three sticks, how many different triangles can you make? Why? What happens if you interchange the position of two sticks?
  • What happens to the triangle if you replace a stick with a shorter or longer one?
  • What happens if you increase or decrease the angle between two of the sticks?
  • Measure the three sides and the three angles. Arrange these measurements in decreasing order. What do you observe? Do other students observe the same thing? State your observation as a result for the triangles.
  • Formulate a result about two triangles, the lengths of whose corresponding sides are the same. Do you think this statement is true for all triangles? Can you extend this statement to other polygons?

Also ask the students to reflect during and after their discussions on the questions:

  • Where am I going?
  • How am I doing?
  • Where to next?

Then ask them to share their findings with the class. Tell the groups to include their thoughts on ‘Where am I going?’, ‘How am I doing?’ and ‘Where to next?’ Tell the students listening to other groups and to prepare and give feedback on these questions.

Then ask different students to present their thinking to the class. Other students should be encouraged to comment. Try to ensure that students from different groups all get an opportunity to speak.

Case Study 2: Mrs Chadha reflects on using Activity 2

I used Activity 2 straight after Activity 1. The students spent quite a lot of time trying to make different triangles and I had to keep reminding them about the sides meeting at the ends of the sticks. Eventually they all felt that they had answers to the first four questions. This time I asked them to present their results to another group and to try and convince them that they were correct. I introduced ideas about feedback to the class, asking them to think before they started to listen to one another about the purpose of the lesson or ‘Where am I going?’, which I had talked about several times by this time. I then told the listening group to provide feedback on ‘How am I doing?’ and ‘Where to next?’ to the group presenting their ideas. After five minutes I reminded the class to change over from listening to presenting.

After that, we had a class discussion. I asked first, ‘What was easy?’ The class said that they found it very easy to talk about their triangles and the sides, but it wasn’t until someone remembered about reflection and rotation that they could say why the triangles were the same – but different! They also came up with the word ‘similar’ to describe the triangles, which I was very pleased to hear. We discussed the mathematical meaning of ‘similar’ and I asked each group to write down their own mathematical definition of what ‘similar’ means in mathematical triangles and shapes.

I then asked the students what they found difficult. They said that giving proper feedback was hard, but that it really made them think. Trying to give a useful reply to ‘How am I doing?’ and ‘Where to next?’ was not easy, but meant that you had to think about what was said. I replied, ‘I know!’ This too turned out to be an interesting session and I think that the debate must have helped a lot of students, because they looked quite happy after the class.

You may also want to have a look at the key resource on ‘Involving all [Tip: hold Ctrl and click a link to open it in a new tab. (Hide tip)] ’.

Pause for thought

  • What responses from students were unexpected? Why?
  • What questions did you use to probe your students’ understanding?
  • Did you modify the task in any way? If so, what was your reasoning for this?

If you have access to the internet you could also use dynamic geometry software (such as the free-to-download programme GeoGebra) to create similar tasks.

1 Talking and learning in mathematics

3 Cooperative learning