Posts Tagged ‘Algebra’
Key points:
 This article is full of ideas to support my action research with children when the time comes. I intend to use some of the ideas within to help structure my input and form the base and review tasks.
 This article also grabs me as it contains citations of other authors I have read so far – not necessarily the same articles/sources as the Kaput writing I have read is a later date than this article, however the 2008 Kaput source seems to be a development of the 2001 piece used here. These connections confirm to me that my thinking and research is along the right lines and hasn’t been as unfocussed as I first feared.
 “Difficulties occur with adolescent students stem from a lack of early experiences in the elementary school” – relates to functional thinking and how students find it difficult to spot generalisations easily. They lack apporpriate language to describe what’s happening, generally focus on a single data set rather than comparing information and have “an inabilty to visualise spatially or complete patterns.” (Warren, 2000). The researchers found that children had limited experience with visual growth patterns and had rarely used arithmetic for anything other than finding answers.
 It continues to state that recording data in a table inhibited the children’s thinking, encouraging “single variational thinking, finding relationships along the sequence of numbers instead of find the relationship between the pairs.” … “The patterns chosen here were those where links between the pattern and its position were visually explicit…to focus in particular on the relationship between the position number and the pattern.” The article gives examples of the patterns used (shown below) and describes the aims of the questions in detail.
Main Reference:
 Warren, E., Cooper, T. (2007) ‘Generalising the pattern rule for visual growth patterns: Actions that support 8 year olds’ thinking’ Educational Studies in Mathematics, Vol. 67, No. 2, pp 171185
Citations:
 Kaput, J., Blanton, M. (2001) ‘Algebrafying the elementary mathematics experience’ in Chick, H., Stacey, K., Vincent, J. and Vincent, J. (eds.) The future of the teaching and learning of algebra. Proceedings of the 12th ICMI study conference. Melbourne: ICMI, Vol. 1, pp. 5794
 Warren, E. (2000) ‘Visualisation and the development of early understanding in algebra’ in Nakahara, T. Koyama, M. (eds.) Proceedings of the 24th conference of the International Group for the Psychology of Mathematics Education. Hiroshima. Vol. 4, pp.273280
Over the course of the next couple of weeks in school, I intend to begin my work with the children. I haven’t really finished my reading yet, but I have a solid foundation to begin from – also I have the time available to me in school over these two weeks and after that it will be increasingly difficult to be able to do these sorts of things.
Session 1: Show a pattern made from multilink cubes. 1^{st}, 2^{nd} and 3^{rd} stage of the pattern. Following this concrete modelling, introduce the use of an inputoutput table to organise data about the number of blocks used for each stage of the pattern. The table helps quantify the pattern so that children see both the growing pictures and the growing numbers in the table. They can note the change from stage to stage and work to explain how the change in the table matches the change in the picture of the growing pattern. Finally, children can try to write a general rule that will work for any stage of the pattern without having to build it or know how many blocks were used in the stage before it. This is an important abstraction of the pattern and the rule must make sense to children and be in their own words or in their own mathematical notation that reflects the level of their current understanding. These may not be accurate at this stage – but that’s ok (and the whole point of the sessions…)
Fir Tree investigation: uses pattern block triangles to create growing fir trees. Children must extend the pattern, complete the table of values and describe the 10th tree.
Session 2: use ideas from algebraic infants. Modelling how a family of dogs can be constructed. Legs, shoulders, bum, body, head – with only legs and body growing, Children need to show each stage in a table for the dog. They then create their own animal, growing it in the same way. Multilink cubes needed.
Session 3: Tables & Chairs investigation challenges students to find a rule to describe the relationship between the number of small square restaurant tables placed together in a line and the number of diners that can be seated at the larger table if only one person sits on each side. Can model with shapes if needed (squares and circles). As before, children make a table to shapes used and differences to aid thinking.
Session 4: Part 1: Ships Ahoy. Children look at a simple pattern of horn blows and predict future patterns.
Part 2: Rockets. Children look at a pattern for building rockets. They are challenged to see how many parts will be needed for the 50^{th} stage. Whiteboard/paper drawings probably best – stickers might be helpful though?!
Session 5: Task Cards. Children choose a task card then construct it using squares or triangles. Encourage them to look at the patterns and decide how it is growing. Create and complete a table to show the growth pattern. Use the tiles to make the next two shapes in the pattern. Be prepared to explain how the pattern is growing. If there is time, choose another task card. Cards attached at the end.
Session 5 – Challenge Cards (PDF)
Session 6: Same as session 1. Can the children identify the growing pattern and express the rules algebraically?
Show a pattern made from multilink cubes. 1^{st}, 2^{nd} and 3^{rd} stage of the pattern. This pattern must be similar but not the same as session 1. Following this concrete modelling, introduce the use of an inputoutput table to organise data about the number of blocks used for each stage of the pattern. The table helps quantify the pattern so that children see both the growing pictures and the growing numbers in the table. They can note the change from stage to stage and work to explain how the change in the table matches the change in the picture of the growing pattern. Finally, children can try to write a general rule that will work for any stage of the pattern without having to build it or know how many blocks were used in the stage before it. This is an important abstraction of the pattern and the rule must make sense to children and be in their own words or in their own mathematical notation that reflects the level of their current understanding.
Hexagon dragon investigation: dragons made from equal numbers of hexagons and triangles, each new term adding on more. Requires children to extend the pattern, create an input/output table to describe the growing pattern, then draw and/or describe the 10th dragon in words. An extra challenge asks students to generate a rule for this pattern so that Miguel can figure out how many blocks he will need to build a dragon of any size.
Each of these sessions will be completed during the children’s Numeracy time. I am likely to choose 4 or 5 children to work with from across ability groups and a mix of genders. I will take copies of their written work and record conversations to help me analyse their progress and thinking over the course of the sessions – I feel that this will give me the data I need to go further. Session 6 is a repeat of session 1 in order to set a baseline and see the progress, if any, the children have made in between.
Clearly, this is such a small scale research project that results can’t be read into too much. However, it is a beginning for my school to look at how we can use algebra in wider contexts. I have been careful to choose similar tasks throughout the sessions as I will only have at most half an hour to complete the tasks with them.
Key points:
 Algebra is a third of the Malaysian Secondary Mathematics curriculum.
 The article lists a large range of sources relating to misconceptions in the understanding of algebra. It states that, because algebra is an extension of previously acquired mathematical learning, it is linked to the use of symbolic representations and that “Warren (2003) felt that understandings of basic arithmetic operations could assist sucessful transition from arithmetic to algebra.”
 “Students use the equals sign in both contexts: arithmetic and algebra. The concept of the equals sign in the framework of arithmetic is that of a ‘do something’ signal (Bodin & Capponi, 1996). Many students tried to add “= 0″ to algebraic expressions when they were asked to simplify (Kieran, 1997).”
 The article cites Booth, 1981, as a refernce for children not understanding the idea of a letter as a variable, and they rather see each letter as representing a digit.
 The article looks in detail at a range of errors (12 types) made by students. The examples given are from the secondary curriculum and so may not apply to any errors made by children in a primary school – but a worthwhile source to help identify the sorts of errors that could be made.
Main Reference:
 Lim, K. (2009) ‘An Error Analysis of Form 2 (Grade 7) Students in Simplifying Algebraic Expressions: A Descriptive Study’ Electronic Journal of Research in Education Psychology, Vol. 8, No. 1, pp 139162
Key points:
 The article discusses how algebraic equations can be shown as visual sentences. The example of x+y=4, where x>y is used. The Reception aged children are given two rules to colour them in: “they have to colour in four snails, and the number of browncoloured snails must be more than the number of yellowcoloured ones.”
 The author states how remarkable it is that the children of this age can complete this algebraic idea and that staff argue it should make it easier for the children to manipulate equations later in life.
 The approach to teaching also requires children to discuss their ideas in groups, challenging each other’s answers, explaining their reasoning and arguing with the teacher who deliberately makes mistakes to generate such discussion.
 It seems that both children and teachers are capable of exceeding perceived expectations through innovative thinking. Clearly, this is just one example and it’s hard to draw conclusions but it would be interesting to see where those children of 2006 are now in Year 5.
Main Reference:
 Mansell, W. (2006) Algebra at the age of four. TES. [Online] [Accessed on 17th May 2011] http://www.tes.co.uk/article.aspx?storycode=2264060
Original Article:
 TES Online: Algebra at the age of four (Published in The TES on 21 July, 2006. Accessed on Tuesday, 17th May 2011)
Key points:
 That children struggle to make the move from being in a situation where “not knowing answers (to arithmetic calculations) is treated negatively, and then suddenly introduced to algebra in which not knowing is treated positively as an opportunity to use symbols, as a way of working with not knowing.”
 “Whenever a learner solves a problem, there is available the question ‘What is the method that was used?’, which in intimately tied up with the question ‘What can be changed about the problem and still the same technique or method will work?’ or ‘What is the class of problems which can be solved similarly?’ S. Brown and M. Walter (1981) suggest asking ‘What if … something changed?’ or ‘What if not…?’ Watson and Mason advocate explicitly asking learners to consider what dimensions of possible variation and corresponding ranges of permissible change they are aware of (Mason & JohnstonWilder, 2004; Watson & Mason, 2004) as stimulus to becoming aware of, and even expressing features of, the general class of problem of which the ones considered are representative.”
 Awareness of and Expressing Generality: “Picturepattern sequences (Mason, 1988b; Mason, Graham, Pimm & Gowat, 1983; South Notts, n.d.) provide just one context for generalizing.” Learners are shown a sequence of pictures, and then specify a method describing how the pattern is growing through each term in the sequence, extrapolating these ideas to fit further terms in the sequence
 “Getting learners to make use of their powers is not simply an approach to algebra or even approach to mathematics. It is mathematics.”
Main Reference:
 Mason, J. (2008) ‘Making Use Of Children’s Powers To Produce Algebraic Thinking’ in Kaput, J., Carraher, D. and Blanton, M. (eds.) Algebra In The Early Grades. New York: Lawrence Erlbaum Associates, pp. 5794
Citations:
 Brown, S. and Walter, M. (1982) The art of problem posing. Philadelphia: Franklin Institute Press.
 Mason, J. (1988b) Expressing generality [project update]. Milton Keynes: Open University Press.
 Mason, J. and JohnstonWilder, S. (2004). Designing and using mathematical tasks. Milton Keynes: Open University Press.
 Mason, J., Graham, A., Pimm, D. and Gowar, N. (1985). Routes to, roots of algebra. Milton Keynes: Open University Press.
 South Notts Project. (n.d.) Material for secondary mathematics. Nottingham: Shell Centre, University Of Nottingham.
 Watson, A. and Mason, J. (2004) Mathematics as a constructive activity: The role of learnergenerated examples. Mahwah: Lawrence Erlbaum Associates.
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Key points:
 “Recent research on the status of student knowledge based in the traditional arithmeticthenalgebra regime has pointed to specific obstacles to algebra learning that computational arithmetic creates for the learning of algebra. For example, limited approaches to equality and the “=” sign in arithmetic as separator of procedure from result (Kieran, 1992) and now known to interfere with later learning in algebra (Fujii, 2003; MacGregor & Stacey, 1997).”
 The majority of the chapter discusses the uses of algebra and tries to define it – “algebra needs to be described both through a snapshot of its structure and function in mathematics today and in mathematically mature individuals, and through a dynamic picture of its evolution historically and developmentally.” “Most attempts to describe algebra historically…tend to be oriented toward progress in solving equations, where the origin of the equations might be the problem situations or simply assertions about numbers or measurement quantities, often surprisingly similar across millennia (e.g., Katz, 1995).”
 The article continues to describe two core aspects of algebra – generalisation and “syntactically guided action on symbols within organized systems of symbols” (which I take to mean reasoning). When these two core aspects are introduced to children is another area of discussion – with practitioners giving reasonable arguments for each aspect to be given favour.
 Movement from arithmetic to algebra depend on the understanding of the “=” sign. Children must realise that sign shows equivalence: 18 plus 3 is the same as 3 plus 18 just as a plus b is the same as b plus a.
Main Reference:
 Kaput, J. (2008) ‘What Is Algebra? What Is Algebraic Reasoning?’ in Kaput, J., Carraher, D. and Blanton, M. (eds.) Algebra In The Early Grades. New York: Lawrence Erlbaum Associates, pp. 518
Citations:
 Fujii, T., (2003) ‘Probing students’ understanding of variables through cognitive conflict problems: Is he concept of a variable so difficult for students to understand?’ in Pateman, N., Dougherty, B. and Zilliox, J. (eds.) Proceedings of the 27th Conference of the International Group for the Psychology of Mathematics Education, Vol. 1, pp. 4966. Honolulu: University of Hawaii.
 Katz, V. (1995) ‘The development of algebra and algebra education’ in Lacampagne, C., Blair, W. and Kaput, J. (eds) The algebra initiative colloquium, Vol. 1, pp. 1532. Washington, DC: U.S. Department of Education, Office of Educational Research and Improvement.
 Kieran, C. (1992) ‘The Learning and teaching of school algebra’ in Grouws, D. (ed) Handbook of research on mathematics teaching and learning, pp. 390419. New York: Macmillan.
 MacGregor, M., and Stacey, K. (1997) ‘Students’ understanding of algebraic notation: 1115′ Educational Studies in Mathematics, Vol. 33, No. 1, pp. 119.
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“This paper describes a lesson taught to Judy’s Y2 class in a rural Hertfordshire primary school. It shows how meaningful mathematics can be made accessible to young children and, importantly, how fundamental issues of variant and invariant properties might be developed. An important component of this understanding is the position to term relationship – the relationship (or rule) between the position, n, of a number in a sequence and the number itself. In short, it is our view that young children, when given appropriate opportunities, can operate at levels substantially higher than the curriculum would indicate was likely.”
The teacher in this case made an animal, a dog, using linking cubes. The general shape was expanded upon to make two larger animals in a ‘family’ of three dogs.
“We discussed the dogs we had made. My questions and prompts were intended to alert them to an awareness of those elements of the dogs that remained constant (invariant) and those that changed. In respect of those elements that changed the intention was to encourage their understanding as to the systematic, and therefore predictable, nature of that change.”
The key to the children’s understanding seems to be labelling each of the parts of the dog, head, legs, shoulders, body and bottom. Once they could apply these labels to each of the three dogs, they could see how the animal was made up. They then used this knowledge to predict the make up of the fourth dog.
After this, they applied their knowledge to their own animals – working out the 4th and 10th members of the family. Some did this in tables, some did this in sentences, but they all could do it regardless of their mathematical abilities. Some were making generalised statements linking the term in the sequence to the body parts. All good stuff and remarkable when you consider that these children are in Year 2…
Reference:

Andrews, P., Sayers, J. (2003) ‘Algebraic Infants’ Mathematics Teaching, Vol. 182, pp. 1822
The article gives a history of the mathematics landscape in the UK since 1837. Algebra has been part of this throughout.
We see algebra as a key tool to help solve problems now, but historically “problem solving was seen as a specialized skill, only for mathematicians in opening up new fields.”
Key points:
 “A problem solver needs a rich, connected understanding of mathematics and the abilty to see patterns of similarity and association, as well as the skills to carry out the planned attack, and to check that the results make sense in the context of the problem.”
 “A Royal Commision, reporting on the state of mathematics teaching in nine leading Public (i.e., private) Schools in 1837, noted that the typical two weekly hours of mathematics consisted of Arithmetic, a little manipulative Algebra, and “Euclid“, learned by rote.”
Reference:

Burkhardt, H., Bell, A. (2007) ‘Problem solving in the United Kingdom’ ZDM, Vol. 39, no. 5, pp. 395403
Key points:
 “Early algebraic thinking in a primary context is not about introducing formal algebraic concepts into the classroom but involves reconsidering how we think about arithmetic.”
 “arithmetic thinking focuses on product (a focus on arithmetic as a computational tool) and algebraic thinking focuses on process (a focus on the structure of arithmetic) (Malara & Navarra, 2003)”
 “The aim was to assist 5 yearolds to come to an understanding of the structure of equations, and in particular the use of the equal sign.”
 “When asked to find the unknown for 7 + 8 = ? + 9, many students express this as 7 + 8 = 15 + 9 = 24.”
 “Three add 2 is different from 4 add 2. There is a different number on each side. Three add 2 is not equal to 4 add 2. It is different from 4 add 2″. Arithmetic thinking is required when computing the value of the two expressions. Algebraic thinking is required when placing the appropriate language between the expressions. Equals is only applicable if the two expressions are equivalent, that is, have the same value.”
 “It is interesting that both Brianna and Ethan spontaneously introduced symbols as short hand for the object: p for pears and b for bananas or an iconic picture of each. In the secondary context this is commonly referred to ‘fruit salad’ algebra where the letter stands for an object instead of variable, and is thinking that we want to discourage. For example, a common misconception in the secondary context is that 3a + 3b stands for 3 apples and 3 bananas instead of a and b standing for any number. In the early years it is important to make the distinction between how we verbally describe number problems and how we represent these problems with symbols. While we say, “Three cars and five trucks,” the convention is to represent this as 3 and 5. Number sentences are made up of numbers.”
References:
 Malara, N., Navarra, G. (2003) ‘ArAl Project: Arithmetic pathways towards favouring prealgebraic thinking.’ Bologna, Italy: Pitagora Editrice.
 Warren, E., Mollinson, A., Oestrich, K. (2009) ‘Equivalence and Equations in Early Years Classrooms.’ Australian Primary Mathematics Classroom, Vol. 14, no. 1, pp. 1012
The majority of the next few posts are going to be based on my findings from a variety of reading.
Part of the MaST course is to produce a couple of assignments, based on personal research, designed to meet the aims and objectives of the programme, reflecting school based activities, readings, synthesis of theory and practice, analysis of teaching and learning in mathematics and personal and collegiate professional development.
Assignment 1:
Select an area of mathematics or an issue in mathematics teaching that they have identified as an area for development and write a report of a mathematical enquiry (equivalent to 4500 words) that you have undertaken that critically explores content, concepts and relationships in the chosen area.
My chosen area is based around algebra and how children use it to solve problems.