Finding factors and multiples is an essential part of mathematics. The use of these concepts starts from an early age, when young students are working on multiplication and sharing. In school mathematics it is built on over the years and these concepts are used in very high-level mathematics as well.
In this unit, you will think about your teaching of the concepts of factors and multiples, and use the ideas of highest common factor (HCF) and lowest common multiple (LCM). During the activities you will also think about how to extend your students’ ability to think through mathematical ideas and make connections between mathematical concepts. The textbook is often a mathematics teacher’s most valuable resource, but it can constrain teaching. In this unit you will think about using the textbook more creatively.
This unit links to the teaching requirements of the NCF (2005) and NCFTE (2009) outlined in Resource 1.
Making connections between mathematical concepts is a very important part of understanding mathematics as a subject. Research suggests that teachers who make connections in their teaching are more successful than those who do not (Askew et al., 1997). Making connections is also often a joyous part of mathematics. Making connections is often lost when students are using textbook questions because the focus is on completing questions as quickly as possible, and these questions are usually only about one aspect of a concept, such as listing all factors of a number.
When students use textbooks, the purpose of the learning is not always made clear to them. They can also get so bogged-down in completing the problems correctly that they lose any perspective of the learning that is supposed to take place.
Pause for thought
Think about a recent mathematics lesson in your classroom. What mathematics were your students learning? To what extent were they thinking mathematically? To what extent were they making connections between mathematical concepts and ideas? Why do you think this is?
The activities in this unit are based on problems and examples as they can be found in any textbook. Additional questions will then be asked to move the students from mechanically finding answers to really thinking about what they are doing, such as:
When students are given the opportunity to think about the process of learning and making connections, they will learn to learn. However, students might initially be uncomfortable with these kinds of questions, as they might not have been required to think in this way before. Therefore, it will be necessary to have support for students, for example through working with their peers in groups or pairs.
Students study and use factors and multiples from an early age. In secondary school, students are required to study the HCF (highest common factor) and LCM (lowest common multiple) of numbers. They have to be able to apply this knowledge when working with expressions. These topics and concepts are thus studied at different times, and in different years. Therefore, students can fail to see connections between the different aspects that they study, and their knowledge can become fragmented as a result. Students often rely on memorisation at each stage rather than understanding the underlying principles leading to a lack of appreciation of the power of factors and multiples.
Activity 1 aims to address this fragmentation by focusing on the mathematical thinking processes involved in finding factors of numbers and expressions. Students are asked to make connections between what a factor is, between factors of numbers and factors in expressions. The activity requires students to work in pairs or small groups and exchange their ideas with other students.
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 the activities yourself will mean that you get insights into a learner’s experiences that can in turn influence your teaching and your experiences as a teacher. When you are ready, use the activities with your students. After the lesson, think about the way that the activity went and the learning that happened. This will help you to develop a more learner-focused teaching environment.
Tell your students:
When your students have attempted these tasks in pairs or small groups, bring the class together and ask different students to share their definitions of a factor. Use these definitions as a starting point for exploring their answers to the other questions. Pay careful attention to who answers your questions – is it always the same students? How can you develop the confidence of other students to share their thinking?
Resources 2 and 3, ‘Using groupwork’ and ‘Talk for learning’, have more information on these subjetcs.
This is the account of a teacher who tried Activity 1 with her secondary students.
I put students in groups of four so that they could offer mutual support and to give them more opportunities to come up with a rich collection of ideas. I organised the groups by putting a notice on the wall with the names of students in each one. I tried to have a mix in each group of students who were confident with learning maths and students who were not so confident. My aim was that the more confident students would support the others.
The students came up with the factors very easily in the case of pure numbers, but there were a lot of arguments about the ones involving expressions. This happened in most of the groups, so I thought it might be helpful to discuss these questions as a whole class. First I asked them to prepare a short presentation for the whole class; this is what they came up with:
I really wanted the students to notice the similarity between factors of numbers and expressions. As a first step I asked, ‘In what way are non-prime factors different from prime factors?’
They were a bit baffled by that, so I added, ‘OK, look at the factors of 60 as an example. Now start your sentence with: “Non-prime factors are different because …”.’ There were several volunteers to have a go at answering this, but it sounded neither clear nor concise – which I did not think would encourage good discussions or learning. I decided to give them a ‘speaking frame’, like they sometimes give writing frames in language learning. So I told them, ‘Try saying it first to a partner and then I will ask you to say it to the whole class.’
They practised verbalising their thinking in that way with each other for a few minutes. I also noticed that some were writing what they had said down in their exercise books, probably so as not to forget what to say when asked. The speaking frame really seemed to have helped because when we came to sharing their sentences, their responses were clear and concise and they used mathematical language in a precise way. They had managed to both notice and express that non-prime factors were numbers that could be decomposed further into products of other numbers until only prime factors were used. I was seriously impressed! But they still had not made the connection between these kinds of factors and the factors of expressions.
My first instinct was simply to tell them – to just pass on my knowledge to them. But they were so full of enthusiasm and engagement from being asked to come up with their ideas that I could not just tell them. I wanted them to discover the answer themselves, and experience the joy of discovering – seeing the beauty of mathematical structures and their connectivity. But what question could I pose so that they could become aware of this connection between factors of numbers and factors of expressions? So many questions and approaches came up in my mind, but they were too complicated, or simply disguised ‘telling them what it is’ questions! What if I just told them what I really wanted them to do? So I said:
I really want you to discover the connections and similarities and differences between finding factors of numbers, and finding factors of expressions. For example, between the factors of 60 and the factors of 3xy. And I don’t just want to tell you, I want you to think about it and discover it for yourself. So think for a moment about all these discussions we have had during this lesson, about the difference between prime and non-prime factors, and look at the factors you have found for 60 and 3xy. What is the same, what is different? Have you got all the factors for all? Have a go at it in your groups.
And they did find the missing factors for 3xy. They found the factors for 3x4 – 27x2: from re-writing it as 3x2(x2 – 9) and then going onto identifying the factors as 3, x, x2, x2 – 9, (x2 – 9), x2(x2 – 9) and 3x2(x2 – 9).
Only then did we move on to think about methods used. We had to do that in the next lesson, because we ran out of time. This was not a problem, because in the next lesson I was able to ask them to think back to the previous lesson. I asked them to look in their exercise books at what they had done and what their thinking had been, asking them to re-enter that thinking. Coming up with descriptions of methods went rather smoothly, probably because we had done so much thinking about it already. We shared the different methods with the whole class, and they corrected each other if the method was lacking or overcomplicated. We actually ended up with a ‘Suggested method of Mrs T’s Class’, and wrote that on a large piece of paper that was then displayed on the wall.
When you do such an activity with your class, reflect afterwards on what went well and what went less well. 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 Kapur did, some quite small things that made a difference.
Pause for thought
After the lesson, try to find time to talk about these questions with a fellow teacher:
The next activity develops the practice of asking students to think about the methods they have used. This activity is, again, very similar to an activity that can be found in textbooks. The difference is that it gives students a mixture of problems in terms of having to find common multiples and factors in one activity, and mixing up numbers and expressions. The other difference is the request to make notes on the methods they have used. The aims of these modifications to the textbook activity are to make students aware of connections between topics, noticing the differences and sameness, and for the mathematical thinking processes involved to become explicit. Again, to help students engage with this new way of working, it might help to let them work in pairs or small groups when you facilitate this activity in your classroom.
Tell your students:
The students did the first question with great confidence. The second one provoked some discussion but the third one was left by most. For this third question I gave a hint of getting factors within the root sign, and then some of them got the answer almost at once. The fourth question resulted in a bit of discussion, but they came up with an answer. However, for the last question, some pairs came up with a2 – b2 as the common factor and a3 – b3 as the multiple.
They described their methods in terms of an algorithm. They kept repeating that they had given the rule and they had learned that this was the rule and that I had told them so! I tell you, this did cause some soul-searching on my part! But I insisted and kept asking how they knew they were allowed to do each step and why they were doing each step. I asked them to imagine their little sister keeping asking that question, ‘Why?’, and that she would not be happy with ‘Because I tell you’ as an answer.
Pause for thought
You have now engaged with a number of activities that help your students to think about the mathematical processes involved in finding common factors and multiples, in order to analyse and describe their own methods.
The next activity shows another way to encourage students to think critically about what they are doing and what they are leaning. This activity asks students to critically evaluate the working methods of some fictitious students and come up with reasons why the students’ way of working is (or is not) mathematically valid. Using work from fictitious students often works well in exposing possible misconceptions, because it avoids emotional reactions and feelings of embarrassment. Since it is not work from the students themselves or any of their classmates, they can treat it without concern for anyone who may be embarrassed or upset about what they are saying. The task is further enriched because one of the methods offered is not incorrect, as such, but could be – that is, it has limitations.
Tell your students that students Meena, Deepak, Aditya and Aneesh were asked to solve the following:
Discuss each of these methods with your partner.
Then bring the class together and ask different pairs of students to present their responses. For each response, ask if other students agree and encourage them to share their answers. In this type of class discussion your role as a teacher is to facilitate, not to give your own opinions. So you could use phrases like ‘Does everyone agree?’ and ‘Who would like to give a different view?’ You should encourage the students to listen to each other and build on each other’s understanding.
The students were grouped in pairs so that there could be input and ideas from both students. I decided not to go for bigger groups this time because I thought the work needed quite detailed reading and discussion. Pair groupings offer a closer working environment where the work can be more thoroughly examined by each student.
I could hear a lot of arguing, with students questioning each other and asking for reasons when they disagreed. After they had had sufficient time to get to grips with each item, we discussed the methods of the fictitious students one by one at the blackboard. For each one I asked a student to come to the blackboard to be the teacher and guide the discussions. The first attempt was slightly chaotic, with some students shouting out, but after I told them there was to be no shouting out at all and hands-up only, the discussion really improved and seemed valuable. I stood at the side, sometimes offering another thought, but with actually very little intervention on my part.
Pause for thought
You can adapt this activity in many different areas of the maths curriculum. Think about the topics you will be teaching over the next month. Which topics do students usually find very difficult? Do you have examples of students’ work from previous years? Plan an activity like Activity 3 to use with your class. Share your ideas with other teachers in your school or cluster. In this way you can develop a folder of interesting activities for your teaching based on the textbook.
The last activity in this unit shows how you can help students to tackle some harder questions and to identify the different places in their mathematics course where they have to use HCF and LCM, and, hence, get equivalent numbers before they can add or subtract. They will also be asked to notice what is the same and what is different in these questions, so they can become aware of which methods and processes to use in which situation. Asking students to engage in this way with textbook questions encourages them to think about the mathematical thinking processes involved.
Tell your students to look at the following problems:
When most of your students have found some more exercises, stop their discussions and ask them to share their ideas with the whole class. Again, your role is as a facilitator, challenging their thinking and encouraging them to ask each other for clarification.
Pause for thought
Again, you can use this exercise for different topics to help your students to build mathematical connections. This will develop their confidence with mathematics and in particular when they are asked to solve an unfamiliar problem.
This unit has explored ideas on teaching multiples and factors in order to illustrate how one idea is threaded through many parts of the mathematical curriculum. The requirements for teaching from the NCF (2005) and NCFTE (2009) are often seen as ambitious goals, but the ideas in this unit show that using textbook questions creatively will allow students to perceive relationships and to see structures. If the students use reasoning, argue the truth or falsity of statements, and look for similarities and differences, they will begin to understand and see the web of ideas in mathematics. Understanding that fluency with one idea met in the early years of schooling is necessary in order to work with much more complex ideas later helps students understand why they are being asked to engage with new ideas that seem to have no immediate application.
This unit links to the following teaching requirements of the NCF (2005) and NCFTE (2009), and will help you to meet those requirements:
Groupwork is a systematic, active, pedagogical strategy that encourages small groups of students to work together for the achievement of a common goal. These small groups promote more active and more effective learning through structured activities.
Groupwork can be a very effective way of motivating your students to learn by encouraging them to think, communicate, exchange ideas and thoughts, and make decisions. Your students can both teach and learn from others: a powerful and active form of learning.
Groupwork is more than students sitting in groups; it involves working on and contributing to a shared learning task with a clear objective. You need to be clear about why you are using groupwork for learning and know why this is preferable to lecturing, pair work or to students working on their own. Thus groupwork has to be well-planned and purposeful.
When and how you use groupwork will depend on what learning you want to achieve by the end of the lesson. You can include groupwork at the start, the end or midway through the lesson, but you will need to allow enough time. You will need to think about the task that you want your students to complete and the best way to organise the groups.
As a teacher, you can ensure that groupwork is successful if you plan in advance:
The task that you ask your students to complete depends on what you what them to learn. By taking part in groupwork, they will learn skills such as listening to each other, explaining their ideas and working cooperatively. However, the main aim is for them to learn something about the subject that you are teaching. Some examples of tasks could include the following:
Groups of four to eight are ideal but this will depend on the size of your class, the physical environment and furniture, and the attainment and age range of your class. Ideally everyone in a group needs to see each other, talk without shouting and contribute to the group’s outcome.
You can set up routines and rules to manage good groupwork. When you use groupwork regularly, students will know what you expect and find it enjoyable. Initially it is a good idea to work with your class to identify the benefits of working together in teams and groups. You should discuss what makes good groupwork behaviour and possibly generate a list of ‘rules’ that might be displayed; for example, ‘Respect for each other’, ‘Listening’, ‘Helping each other’, ‘Trying more than one idea’, etc.
It is important to give clear verbal instructions about the groupwork that can also be written on the blackboard for reference. You need to:
During the lesson, move around to observe and check how the groups are doing. Offer advice where needed if they are deviating from the task or getting stuck.
You might want to change the groups during the task. Here are two techniques to try when you are feeling confident about groupwork – they are particularly helpful when managing a large class:
At the end of the task, summarise what has been learnt and correct any misunderstandings that you have seen. You may want to hear feedback from each group, or ask just one or two groups who you think have some good ideas. Keep students’ reporting brief and encourage them to offer feedback on work from other groups by identifying what has been done well, what was interesting and what might be developed further.
Even if you want to adopt groupwork in your classroom, you may at times find it difficult to organise because some students:
To become effective at managing groupwork it is important to reflect on all the above points, in addition to considering how far the learning outcomes were met and how well your students responded (did they all benefit?). Consider and carefully plan any adjustments you might make to the group task, resources, timings or composition of the groups.
Research suggests that learning in groups need not be used all the time to have positive effects on student achievement, so you should not feel obliged to use it in every lesson. You might want to consider using groupwork as a supplemental technique, for example as a break between a topic change or a jump-start for class discussion. It can also be used as an ice-breaker or to introduce experiential learning activities and problem solving exercises into the classroom, or to review topics.
Talk is a part of human development that helps us to think, learn and make sense of the world. People use language as a tool for developing reasoning, knowledge and understanding. Therefore, encouraging students to talk as part of their learning experiences will mean that their educational progress is enhanced. Talking about the ideas being learnt means that:
In a classroom there are different ways to use student talk, ranging from rote repetition to higher-order discussions.
Traditionally, teacher talk was dominant and was more valued than students’ talk or knowledge. However, using talk for learning involves planning lessons so that students can talk more and learn more in a way that makes connections with their prior experience. It is much more than a question and answer session between the teacher and their students, in that the students’ own language, ideas, reasoning and interests are given more time. Most of us want to talk to someone about a difficult issue or in order to find out something, and teachers can build on this instinct with well-planned activities.
Planning talking activities is not just for literacy and vocabulary lessons; it is also part of planning mathematics and science work and other topics. It can be planned into whole class, pair or groupwork, outdoor activities, role play-based activities, writing, reading, practical investigations, and creative work.
Even young students with limited literacy and numeracy skills can demonstrate higher-order thinking skills if the task is designed to build on their prior experience and is enjoyable. For example, students can make predictions about a story, an animal or a shape from photos, drawings or real objects. Students can list suggestions and possible solutions about problems to a puppet or character in a role play.
Plan the lesson around what you want the students to learn and think about, as well as what type of talk you want students to develop. Some types of talk are exploratory, for example: ‘What could happen next?’, ‘Have we seen this before?’, ‘What could this be?’ or ‘Why do you think that is?’ Other types of talk are more analytical, for example weighing up ideas, evidence or suggestions.
Try to make it interesting, enjoyable and possible for all students to participate in dialogue. Students need to be comfortable and feel safe in expressing views and exploring ideas without fear of ridicule or being made to feel they are getting it wrong.
Talk for learning gives teachers opportunities to:
Not all responses have to be written or formally assessed, because developing ideas through talk is a valuable part of learning. You should use their experiences and ideas as much as possible to make their learning feel relevant. The best student talk is exploratory, which means that the students explore and challenge one another’s ideas so that they can become confident about their responses. Groups talking together should be encouraged not to just accept an answer, whoever gives it. You can model challenging thinking in a whole class setting through your use of probing questions like ‘Why?’, ‘How did you decide that?’ or ‘Can you see any problems with that solution?’ You can walk around the classroom listening to groups of students and extending their thinking by asking such questions.
Your students will be encouraged if their talk, ideas and experiences are valued and appreciated. Praise your students for their behaviour when talking, listening carefully, questioning one another, and learning not to interrupt. Be aware of members of the class who are marginalised and think about how you can ensure that they are included. It may take some time to establish ways of working that allow all students to participate fully.
Develop a climate in your classroom where good challenging questions are asked and where students’ ideas are respected and praised. Students will not ask questions if they are afraid of how they will be received or if they think their ideas are not valued. Inviting students to ask the questions encourages them to show curiosity, asks them to think in a different way about their learning and helps you to understand their point of view.
You could plan some regular group or pair work, or perhaps a ‘student question time’ so that students can raise queries or ask for clarification. You could:
You may be pleasantly surprised at the level of interest and thinking that you see when students are freer to ask and answer questions that come from them. As students learn how to communicate more clearly and accurately, they not only increase their oral and written vocabulary, but they also develop new knowledge and skills.
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