3 Cooperative learning
In order to maximise the opportunities to learn in a crowded classroom, students should help one another as they did in Mrs Chadha’s class in Case Study 2. However, cooperative learning is rarely deployed in secondary mathematics classrooms and where it is, it is often by default rather than purposeful. Slavin et al. (2003) reviewed a great deal of evidence and concluded that ‘cooperative learning is one of the greatest success stories in the history of modern research’ (p. 177). The four main reasons (Wiliam, 2011) for this success seem to be:
- Motivation: Students help one another learn because it is in their own interests to do so. This has the effect of increasing all-round effort, leading to more success in learning and therefore more motivation to work on challenging ideas.
- Social cohesion: Students help their peers because they are part of the same group and it matters to them that the group succeeds.
- Personalisation: If a particular student is having difficulties, it is likely that there will be another in the group who can help out. Where groups are well-structured it is not always the same people helping or receiving help.
- Cognitive elaboration: Those who contribute to discussions are forced to think through the ideas and clarify them for themselves and others.
If students are to get the help they need in a large class, they must be available to help one another. Students teaching one another can be surprisingly effective: in one study students learned almost as much when peer-tutored as they did from one-to one instruction from their teacher, possibly because they feel less intimidated asking questions of a peer (Schacter, 2000).
The purpose of the next activity is to ask the students to work together cooperatively in order to make connections with other mathematical ideas to solve a problem.
Activity 3: Making connections
For this activity you need at least three sticks for each student. If this is not possible, use three sticks for each pair of students or even groups of three students. Each student (or pair of students) picks three sticks at random. Make sure they don’t get to choose the lengths of their sticks.
Tell your students:
- You can now slide your sticks in order to ‘cut’ a portion of the longest side of the triangle you have created so that the three lengths form a right-angled triangle.
- Discuss in your groups how you will find the exact length that needs to be cut. What fact(s) do you use to find this length?
- Measure the three sticks you used to form the right-angled triangle as accurately as possible.
- Find the ratios of the lengths of each of the three sticks to each other.
- Find another student for whom any of these ratios coincide.
- Investigate with the other student what is common between yourtriangles that resulted in the ratios coinciding.
- Formulate and present your findings in class.
Case Study 3: Mrs Chadha reflects on using Activity 3
I had taught Pythagoras’ theorem to the class a while ago and wanted to move onto trigonometry. Now that the class was much happier talking about triangles together and using the correct terminology, I thought I would also be able to use this exercise to judge how well they remembered Pythagoras and whether or not I needed to teach it again.
I told them that the purpose of this session was to work with right-angled triangles, to make connections to other mathematical ideas they had learned and to extend what they knew. That meant that they should note for themselves when they made connections so that they could share them with the class and that I would ask them at the end about anything new that they had learned.
They naturally formed into the groups that they had used yesterday when doing Activities 1 and 2 and got hold of their sticks. They started talking about how they could be sure that they had the correct length to ‘cut’ – I didn’t let them actually cut the stick, just mark them with a biro. They first started talking about measuring, saying if we measure 90° exactly then that will be correct, so they got their protractors out and started to measure. I wanted to see what would happen as they talked together, so I just listened in to the various groups. I heard some people saying ‘Check it’, ‘Oh no, it’s moved’ and ‘It’s difficult to do this’. I asked the whole class to think about what connections they could be making at that point with other mathematics that they knew, as they seemed engrossed with measuring. I told them all to be quiet and think for 30 seconds, then to go back to work.
The quiet thinking time did the trick and someone started saying, ‘We must find the hypotenuse, so what if we use, umm p, p …’ Someone else came in with ‘Pythagoras – now what was that!’ I saw several people look Pythagoras’ theorem up in their textbook. It seemed they were just double-checking as within a very few minutes, they were squaring and trying to find the square roots. I was so pleased that all I had had to do was to give them time to think and then the whole class supported one another to use Pythagoras’ theorem easily and naturally to find an exact answer.
I felt that most of the class had a good understanding of Pythagoras but I saw that Pavendeep and a couple of others were confused. I asked them to talk to me once the rest of the class had started Part 2 and it turned out they had missed the lessons on Pythagoras’ theorem. Together we made a plan about how they would make up that knowledge, including using the textbook, using the internet and coming back to tell me what they knew next week, before their next lesson.
I asked for volunteers who were ready to come out and say how they had achieved the task. I also asked them if there were some different ways that they had done the activity. After that they were grouped again in fours and the second part of the activity was done, along with a discussion following that as to what was happening.
They set about Part 2 with a will. They knew there was something to find out and they wanted to find it. They did the arithmetic, double-checking that they had the correct answer. (When do they check normally? Normally they are just trying to finish the exercise!) They could not compare the ratios when they kept them as fractions, so they realised that they would have to do something else. Some said we could make equivalent fractions; some found it easier to represent in decimal forms. Luckily there were several groups with isosceles triangles of different sizes and some 30-, 60- and 90-degree triangles as well. So groups of students formed naturally with the same answers and they could see that if the angles were the same, the ratios for all the sides were the same – some even saw that the sides were doubled in one case. This was the perfect start for learning about the trigonometry ratios but that had to wait for the next lesson. First we had some questions to answer so that they could see that they had successfully fulfilled the purpose of the lesson.
 Pause for thought How often do you use problem solving to evaluate students’ learning? Do you think Mrs Chadha found out what each of her class could do? Many of them would have received some help in remembering how to use Pythagoras’ theorem, does that mean that they did not know about it? |
Observing your class solving problems can help you know who can understand how to use a mathematical concept, who can only follow an algorithm and is therefore lost out of context, and who has no idea at all. Perhaps the most important thing here is that the students find out what they can and cannot do. You as the teacher, can then give really good feedback on the ‘Where to next?’ question, just as Mrs Chadha did.
For more detail, read Resource 2, ‘Assessing progress and performance’.
2 Effective feedback