Discuss how work on algebra might be developed in Year 7. You must reflect on what pupils might be reasonably expected to know, and what common misconceptions they might bring with them from previous work on Numbers and the number system, Calculations and Solving problems. You must also identify and analyse the difficulties that Year 7 pupils may have with algebra. In addition you should also provide, with justification, effective teaching strategies that could be implemented to tackle the identified difficulties.
When I started in my school and was given a Year 7, Set 3 maths class, I was wondering how I was going to embark on teaching the topic of algebra. I needed to find out the problems children have when working with algebra. It is very common that some students will have misconceptions about a topic, no matter how well it is taught. However, I feel that an important part of planning is to accustom yourself with the common misconceptions children make and try and teach the children about them in the classroom before they have the chance to get it wrong in the first place. I think that misconceptions should not be left out until the pupil's makes the mistake as they will become harder to explain as time goes on. I believe that they should be discussed repeatedly to reinforce that pupils understand it.
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Some evidence suggests that teaching these misconceptions about algebra at a young age may lead to a deeper understanding of these misconceptions than other teaching methods, such as avoiding the misconceptions (Swan, 1983). So, studying these misconceptions and thinking about the most useful way of dealing with them can prove to be the most helpful way in planning to teach a new topic.
During Key Stage 1 and 2, pupils' progress on their knowledge and understanding of mathematics is done by means of practical activity, investigation and discussion. They learn how to read, count, write and order numbers 1 to 100 and beyond. Children of this age develop a variety of calculating maths mentally and use these throughout their primary school days in different settings. They start to understand mathematical language by using it to talk about their methods of doing a question or when solving a problem.
Children of this age should be able to use methods to solve problems involving number. They also should be able to use different methods to solve a problem and look for ways to overcome difficulties if they encounter them. As well as this, children should be able to communicate verbally, using reasonably good language when dealing with vocabulary related to number and data. They should also be able to display answers in a way, so they are able to justify their methods and give an explanation for doing that method when solving problems.
Children, before leaving primary school, should be able to make and explain number patterns, investigate and record patterns connected to addition and subtraction, and also know patterns and know how to explain patterns of multiples of 2, 5 and 10. Children should be able to understand addition and know that it can be done in any order, and as well as this they should understand subtraction as 'take away' and recognise that subtraction is the inverse of addition. Children should know before entering Year 7 that the symbol '=' represents equality and they also should understand multiplication as repeated addition. Pupils of this age should know that halving is the inverse of doubling and should be able to find one half and one quarter of shapes and small numbers of objects. They should be beginning to understand division as repeated subtraction.
All of the above, children 'should' be able to do before entering Year 7, however some pupils believe they 'can't' do maths and their difficulties are more often than not because of misunderstandings of concepts that we, as teachers, don't give a second glance to. If we were made more aware of these misconceptions, wouldn't we be able to help our pupils a lot more? We all know that recognising and understanding our mistakes can be the most important thing we do and can be a great learning experience. The most common of these misconceptions come from algebra and number. I am going to outline some of these misconceptions that children might bring with them from Year 6 to Year 7 now.
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(This data was taking from the Key Stage 2 National Curriculum Mathematics website).
1. A number with three digits is always bigger than one with two:
Some children believe that 2.26 is bigger than 3.5 because it's got more digits. Why is this? This is the case because for the first few years of them learning about maths, they only came across whole numbers, where the "digits" rule works.
2. When you multiply two numbers together, the answer is always bigger than both the original numbers:
Again, another "rule" that works for whole numbers, but falls apart when one or both of the numbers you multiply is less than one.
3. Which fraction is bigger: Â½ or Â¼?
Most pupils will say that Â¼ is bigger than Â½ because they know that 4 is bigger than 2. However, this shows us that there is a gap in their knowledge about what the bottom number, the denominator, of a fraction does. It divides the top number, the numerator.
So therefore, by me knowing all the knowledge the children should know when starting Year 7 and some of the misconceptions related to that, I need to introduce this new topic of algebra to students very carefully so to reduce the possibility of misconceptions occurring. I will introduce the topic slowly and use a wide range of vocabulary to reinforce the pupils understanding of the subject. I will also ask questions to the pupils to ensure they understand each new word we meet.
When starting algebra for the first time with my Year 7 class, the first misconception I came across was the meaning of letters.
There are many misconceptions that children have relating to the meaning of letters in algebra. Probably the most common is, they believe that letters represent objects and that this letter should be the first letter of that object (Booth, 1984). For example, students were asked to put the following statement into algebraic form:
The number of books in a library.
Pupils who have this misconception would say that 'b' stands for books and that it has to be 'b' because the word books starts with the letter 'b'.
Kuchemann (1981) has recognized six groups of letter usage. He regards the example above as 'letter as object' whereby children consider the letter as shorthand for the object's name.
After consulting the Framework Maths textbook, I feel the activity it suggests for this type of 'letter as object' task, reinforces the misconception rather than trying to eliminate it. The textbook suggests an exercise that consists of a number of statements that children are asked to write down in short form. In each question, key words and numbers are highlighted and pupils are asked to use these to write statements in short form.
The total money raised in a sponsored walk at £5 for each mile.
When the pupils were asked to explain what the letters stand for, many told me that 'm' stands for 'miles' instead of 'the number of miles'. Therefore, in order to try and avoid this from happening in the future, I will ask the pupils to write what each letter stands for underneath each statement. This opinion is supported by who states:
"Having the pupils write out in words what each variable represents not only provides a visual reminder that the variables represent numbers, but is also very much in keeping with the increased emphasis on verbalisation in the standards."
Another common misconception concerning the meaning of letters is what Kuchemann (1981) expresses as 'letter evaluated'. This is when pupils assign a certain number to a letter from the beginning. One way they might do this is by matching a pattern between letters, with a pattern between numbers. For example, a = 1, b= 2 c= 3, etc. This is because of their position in the alphabet. The reason why pupils do this is because they consider maths to be a practical subject and in this way, the aim is to find answers with only numbers in it (Booth, 1984). As a result, even if the children do use the right method and write it down correctly, they still do not consider that a 'method statement' such as 'd + h' can be an answer to a maths question.
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Looking ahead, the next section I saw in the textbook was substitution. The textbook had an activity in it whereby the students were asked to substitute values for letters by means of a dice and to move around a board-game. I knew I would be covering this with my Year 7 class so I had to make sure that the pupils fully understood what to do and stop them from believing that if there is a 'b' in the answer then you must substitute in the value 2.
Some pupils just completely ignore the letter when working with algebra. They may believe that the letters do in fact stand for numbers but all the same, they handle them as objects. For example, if children were asked to simplify the expression 3a + 4b + 2a, they will start inventing rules to try and answer the question. They may start by counting all the numbers and all the letters up separately and end up with false answers such as 9ab or 9aba (Booth, 1984). When I am covering the section on collecting terms with my class, I will deal with this misconception by asking students whether they think their answer is correct and why?
A lot of misconceptions occur when asking students to collect terms and substitute values in place of letters. One common misconception in relation to this is that the pupils believe the answers should always be single expressions and have numbers present in the answers. For example, if an answer is x + y, students would then replace this with a joined term of xy and then substitute numerical values into this. This joined term may be read in a 'place value' meaning as in arithmetic. For example, if z = 3, the term '4z' may be interpreted as 43 (Booth, 1984).
There are many incidents when pupils may become confused when working with algebra. Now that I am aware of some of the problems my Year 7 pupils may face when I am covering the rest of the topic, I can use these to my advantage. I plan to use a lot of discussion in my lessons and this not only gives an opportunity for pupils to voice their opinions but also gives an opportunity for assessment, and more importantly a learning experience for me as a trainee teacher. I feel that pupils talking in groups or pairs can be very effective. It can help pupils to express opinions and talk through concerns or difficulties they might have with a certain topic. I also think that by having a whole class discussions, pupils will gain a much deeper understanding of the concepts, than if we were to just continuously do questions from the textbook.
Although explanation is necessary when dealing with algebra, I feel that it is important that this teaching style should not dominate all lessons. Lots of pupils have different learning styles which need to be catered for. One way to do this is to use a variety of teaching styles and activities, which you can do when dealing with algebra. While working on the rest of this topic in the coming weeks, I anticipate that pupils will expose all the misconceptions I have just discussed above. I am hoping that the teaching styles I am going to use will help to eliminate a lot of the misconceptions quickly. The only way I feel I can investigate the effectiveness of my teaching on these misconceptions is by evaluating all of my lessons thoroughly.