In this unit you will think about how to introduce fractions to your students.
Some students can see fractions as a very difficult topic to understand. There are many reasons for this, but making sure that your students have rich and varied experiences of working with fractions will help them to develop their understanding.
In this unit you will explore the fact that a fraction only has meaning when looked at in relation to a whole, and consider how to help your students to get to know about different ways to read the symbolic representations of fractions.
Through activities you will also think about the value of asking your students interesting and challenging questions, of getting your students to ask questions themselves and talking about fractions.
This unit links to the teaching requirements of the NCF (2005) and NCFTE (2009) outlined in Resource 1.
One of the reasons fractions can seem so difficult is that there is a lot to understand. For example, half of something can be smaller than a quarter of something else. An example of this is ‘half of six is three’ and ‘a quarter of sixteen is four’. So learning about fractions by folding pieces of paper or by dividing circles may mislead students, especially if the paper is always the same size. Students must be taught to ask ‘A fraction of what?’
Developing an understanding of fractions is not so different from learning to understand other mathematical concepts. For example, very young children are offered many different experiences as they learn to generalise the concept of ‘three’.
Despite being older when they learn about fractions, elementary students similarly will need a great many rich and varied experiences if they are to begin to develop a good understanding of fractions.
Many students will have had experiences that help them to develop some understanding of fractions. In her research, Nunes (2006) found that primary school students already have insights into fractions when solving division problems:
They understand the relative nature of fractions: if one student gets half of a big cake and the other gets half of a small one, they do not receive the same amount. They also realise, for example, that you can share something by cutting it in different ways: this makes it ‘different fractions but not different amounts’. Finally, they understand the inverse relation between the denominator and the quantity: the more people there are sharing something, the less each one will get.
Encouraging the students to talk about fractions and use the vocabulary will help them understand some of the difficult vocabulary associated with fractions. The questions you use should show the students how important the correct vocabulary is, so that everyone knows what is being referred to.
First, model some ways of talking about fractions and drawing attention to how words are used. Then focus on getting your students talking. The more the students use the words themselves, the more they will build their understanding of fractions. Asking the students to make up questions to ask one another is a good way to get them talking. Another way is to ask the students to explain the reasoning they used to arrive at their answers.
The first activity is for you to think about issues of learning fractions in your classroom.
Think about what your students need to know about fractions, and make some notes on the different ideas. Use your textbooks. If you have a multigrade class, you will need to think about what different students need to know about fractions:
For each of the ideas associated with fractions, write down how the vocabulary associated with those ideas and the way it is used to express ideas. For example ‘half of ten’ means ‘divide 10 by 2’, but it can also mean ‘multiply 10 by ’. The students might also see , which has the same outcome and is thus equivalent in meaning but which may also be expressed as ‘10 divided by 2’ or ‘10 shared between 2 people’.
Think about some specific students in your class. What activities might help them to understand the different ways that fractions can be expressed and the different meanings given to those interpretations?
The second activity focuses on students physically representing the concepts of fractions. This is also called embodiment. You will ask them to use their bodies to represent mathematical ideas. If the students move themselves to make fractions of a whole, they will begin to develop their concept of what a fraction is and how they can work with fractions.
Before attempting to use the activities in this unit with your students, it would be a good idea to complete all, or at least part, of the activities yourself. It would be even better if you could try them out with a colleague as that will help you when you reflect on the experience. Trying them for yourself will mean you get insights into a learner’s experiences, which can, in turn, influence your teaching and your experiences as a teacher.
First create a space, and ask eight students to come to the front of the class or somewhere where the rest of the class can see them.
|Video: Using questioning to promote thinking|
This is the account of a teacher who tried Activity 1 with his elementary students.
First, I invited eight students to come to the front of the class and to form themselves into a rectangular shape where the rest of the class could see them. I then asked student Anoushka to come and divide these eight students in half, which was easy to do.
I then asked the class if the group of eight students could be divided in half in another way. This proved to be a little challenging, as the students were used to mathematics questions having just one answer, so they wondered at first if Anoushka was wrong. They needed clarification about what ‘different’ meant here. Of course, whichever way they divided the students in half, there were always four students in each half. Since this was the answer I was looking for, I gave them time to talk about these ideas.
Next, I asked student Nita to come to the front and divide the group into quarters. This time the students were able to suggest different ways to achieve this, and they were happy there would always be two students in each part.
I then asked another group of students to come to the front, this time with six students. This time I asked them to divide themselves into half in two ways. I asked ‘Do you always get the same answer?’ ‘Yes sir!’ they said. Then I asked ‘What other fraction can you divide yourselves into?’ They tried to divide themselves into quarters but they could not, but what they did find was that they could divide themselves into three parts and discussed what this fraction was called.
I then put the class into groups of 12 and asked them what fractions they could embody in their groups. One group came up with twelfths, but most worked happily on halves, quarters, thirds and sixths.
When you do such an exercise with your class, reflect afterwards on what went well and what went less well. Consider the questions that led to the students being interested and being able to progress, and those you needed to clarify. Such reflection always helps with finding a ‘script’ that helps you engage the students to find mathematics interesting and enjoyable. If they do not understand and cannot do something, they are less likely to become involved. Use this reflective exercise every time you undertake the activities, noting as Mrs Rawool did some quite small things that made a difference.
Pause for thought
Good questions to trigger such reflection are:
Teachers ask a lot of questions in their work – some research suggests that teachers ask up to 400 questions every day when they are teaching! The better the questions that teachers ask, the better their teaching will be.
Much research has been undertaken about good questions, for example by Wragg and Brown (2001) and Hattie (2008).The research concludes that effective questions:
Pause for thought
Reflect on the questions you asked in the last lesson you taught.
Activity 3 asks you to first prepare for asking effective questions, and then to try out these questions when teaching your students.
If you can do this part of the activity with another teacher, you may find that it is easier.
Think about the next lesson in which you will teach on fractions. What is it you want the students to know? Write some notes about that now.
What previous knowledge do you think they will need in order to understand the ideas you want them to learn? Write a question which will enable you to know whether or not they have that prior knowledge. For example, you could ask your students: ‘Can you give me an example of …? And another? And another? And another? And another?’ Asking for more examples could help you to find out the extent of their knowledge and some of the students’ misconceptions.
Think about some of the ways that fractions are used in the real world. Write a question that might interest or engage the students because it is based on something they know about and use.
Now write an easy question for the particular topic you have to teach and then write a hard question. Write a sequence of questions that will challenge your students – but not too much!
Think about all the ways that misconceptions can happen in fractions. Write two or three questions that will help you check whether or not your students have these misconceptions. You can find some examples of such questions in Case Study 2. It is also important to think ahead about how you might respond to your students’ answers in the best way to reinforce learning and extend their thinking. You can use Resource 2 to help you think about some ideas for how to receive your students’ responses.
Now write a question that will encourage your students to reason their way to a solution. For example, ‘Your big sister never believes what you say. How will you convince her that your method works?’
Now you have written these questions, use them with a class.
Did you think that the class learned more because they used these questions?
Don’t forget to use real objects to allow your students to work with ideas on fractions and to approach challenging questions through a process of reasoning.
Video: Planning lessons
When thinking about Part 1 of Activity 3, I decided I would use my normal introduction to fractions by demonstrating fractions on the blackboard as usual, but to be very precise and repetitive in the questions and instructions I was going to use. I wrote them down on a piece of paper, and put them on my desk so I would not forget them.
These are the questions and prompts I prepared:
I drew the circle using chalk. I then invited students to come to the blackboard, and asked them the questions. Having the questions written down really helped me to focus and helped to avoid diversions from what I had intended to do. I also noticed that as a result there was less ‘teacher talk’ and more student talk and student work.
Pause for thought
Mary Budd Rowe (1986) researched the ‘wait time’ that teachers allowed after asking a question. ‘Wait time’ is the length of quiet time that teachers allow after asking a question before they expect a student to answer, or before they rephrase the question or even answer the question themselves. Her team analysed 300 tape recordings of teachers asking questions over six years. They found the mean wait time was 0.9 seconds.
If you ask a question that requires the students to think, are you really giving them enough time to think, or are you only giving them time to instantly react?
The teachers in Budd Rowe’s research were trained so that they became able to increase their wait time to between three and five seconds. The increased wait time resulted in:
In other words, the students had more time to think and that increased the level (and quality) of discussion that went on in the classroom, which in turn meant the teachers learnt more about their students thinking and were able to act on any misconceptions. Increasing wait time is not easy to do and can feel odd when you start, but if your students are to think, they must be given sufficient time.
Activity 4 asks you to experiment with increasing the wait time in your classroom in a similar way.
Like the teachers in Budd Rowe’s research, in your next lesson, increase your wait time for students to respond to five seconds. After the lesson, reflect on whether you observed:
The next activity links together many of the ideas that have been discussed so far. It suggests that you:
This activity is an example of the kinds of rich activity that students need to build their understanding of fractions. For this task you will need a quantity of paper plates, or card cut into rectangles of the same size.
Arrange the students to work in groups of three or four, and give them a pile of paper plates or cards. You may want to look at the key resource ‘Using groupwork’ to help you prepare for this.
Make sure everyone is able to do this before carrying on.
The idea is to give everyone time to play a little with fractions and to think about what fractions are.
Now move on to problems that mix the two ideas.
I gave each group 12 paper plates. The plates were to help to support the students’ thinking that finding fractions is about sharing out equally.
First, I set them the task of dividing the plates into quarters. I asked several of the groups to talk about the process of dividing into quarters. Then I asked them to divide their 12 plates into thirds. When they had done this, I once again asked the students to explain how they did it. I made sure that everyone was comfortable sharing out the objects that they were working with, in this case the plates. The students enjoyed working and collaborating together in groups and completing the task.
I then decided that the class was ready for a more challenging question. I gave each group one more plate, so that they then had 13 plates, and again asked them to divide the plates into quarters and then thirds. This time the students discovered that they needed to subdivide the extra plate in order to share out the plates equally into quarters and thirds.
This time I spent more time on the feedback session in order to make sure that everyone understood the reason why one of the plates had to be subdivided. I then asked the class to divide the plates into thirds, and this time I offered them scissors as well. Several students gave some good reasons why they needed to divide up the extra plate, but working in groups helped them all to try out their ideas first before telling the whole class.
Pause for thought
This unit has focused on teaching fractions, but you have also looked at how to ask questions that require students to think and the importance of giving students sufficient time to think.
In studying this unit you have thought about how to enable your students to develop their ideas about fractions and about the necessity of providing rich and varied activities if students are to learn, understand and use ideas about fractions.
You have also seen how reflecting on learning, and how learning happens, is important in becoming better at teaching.
Pause for thought
Identify three techniques or strategies you have learnt in this unit that you might use in your classroom, and two ideas that you want to explore further.
This unit links to the following teaching requirements of the NCF (2005) and NCFTE (2009) and will help you to meet those requirements:
The more positively you receive all answers that are given, the more students will continue to think and try. There are many ways to ensure that wrong answers and misconceptions are corrected, and if one student has the wrong idea, you can be sure that many more have as well. You could try the following:
Value all responses by listening carefully and asking the student to explain further. If you ask for further explanation for all answers, right or wrong, students will often correct any mistakes for themselves, you will develop a thinking classroom and you will really know what learning your students have done and how to proceed. If wrong answers result in humiliation or punishment, then your students will stop trying for fear of further embarrassment or ridicule.
It is important that you try to adopt a sequence of questioning that doesn’t end with the right answer. Right answers should be rewarded with follow-up questions that extend the knowledge and provide students with an opportunity to engage with the teacher. You can do this by asking for:
Helping students to think more deeply about (and therefore improve the quality of) their answer is a crucial part of your role. The following skills will help students achieve more:
As a teacher, you need to ask questions that inspire and challenge if you are to generate interesting and inventive answers from your students. You need to give them time to think and you will be amazed how much your students know and how well you can help them progress their learning.
Remember, questioning is not about what the teacher knows, but about what the students know. It is important to remember that you should never answer your own questions! After all, if the students know you will give them the answers after a few seconds of silence, what is their incentive to answer?
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