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Emily Cleary - Finding generalisations in mathematical problem-solving activities

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emily Cleary
27 March 2014


I am a maths specialist in an independent school and I work with children aged 5 to 9. I work with learners at both ends of the ability scale, working towards overcoming barriers with children who find maths difficult, and creating more challenging tasks for the stronger mathematicians.


 A letter had gone home to parents of all the children in the Year 4 top stream inviting their children  to take part in an after-school club which was going to introduce early algebra. Four children signed up for the club. In an independent school you are competing against a lot of attractive sporting clubs, so I could make the assumption that I had a group of learners who were particularly keen. On the day of the particular task been discussed here, there were only two children present.


Teaching algebra is the focus of the task and I am approaching it from the point of view of the learners building up skills to enable them to carry out mathematical tasks and to use their findings to create generalisations.  

I am hoping to produce a range of tasks designed to promote an environment where the children carry out investigations, record their results, look for patterns and begin to create their own formulas.  I intend do this over period of two terms with a small group of learners which will allow them to concentrate on developing generalisations rather than identifying a particular pattern.



 The task was taken from Patterns and Algebra, Barbara Berman and Fredda Friederwitzer, (2002).

Learners use Cuisenaire rods to build some square that ‘grow’. They build the first square using one white Cuisenaire rod.  To build the next square they must build around the first square and record how many white square they used to build the next square. They continue building squares and recoding their results. Then they analyse the data to look for patterns to enable them to predict the number of white rods they would need to add to a 14-cm square to produce a 15cm square

Learning objective of the task

  • Identity properties of squares
  • Create concrete models of ‘square numbers’
  • Find, record and extend number patterns
  • Use patterns to make predictions
  • Wanted students to find a formula for their construction


I worked with two high-achieving Year 4 pupils, X and Y . They both really enjoyed the physical aspect of making the squares, one learner used the wooden rods to make the square while the other child used the manipulative rods on the interactive whiteboard to create virtual squares.  The children alternated the activities

They recorded their results onto a ready-made table and by about the 3rd set of results I began questioningthem about patterns which were beginning to emerge in their data. Once their attention was drawn to this they were both very eager to fill in their predictions for the rest of the record sheet.  However, they were still very keen to continue making the physical model for each square.

When I asked them what number of white squares they would use to make the 14th and 15th square they used the data on the record sheet and quickly used well-established arithmetic skills to calculate the correct number of white squares.

When I extended the activity and asked how many squares they would need to make the 145th square, there was a pause and a realisation that they could not calculate this based on their existing arithmetic skills. X  began to look carefully at the squares in front of her and used the physical model to create a general rule.  She noticed they added the same number of white rods to the horizontal side of the square and the same plus one rod extra to the vertical side to create the next square. With this realisation she created a rule which would work for every square.

I became aware of a lack of participation from Y and he became very passive and he seemed deflated when X took the initiative. He is a very intuitive mathematician and this line of investigation seemed to make him uncomfortable.  This disturbance made me realise he may be a child, who is used to succeeding at mathematical tasks and he will need lots of experience to become comfortable in problem-solving situations.



The learners became absorbed in the activity, particularly with the concrete and the virtual manipulatives  and it was difficult for me the teacher to achieve the intended learning outcome.

I found it very difficult to hold back and not to intervene.

I found it difficult to ask the questions without feeling that my questions were too leading.

 How do I know the children were connecting with the intended learning experience and how could I measure whether it was successful?

Does it have a detrimental effect on learners if I take them out of their comfort zone?

I worked with all the most able learners, how would it work with more reluctant learners and would they struggle with looking for and finding patterns?

These types of tasks would need to be regular practice in order to arrive at a situation where the learners were experts at looking for patterns and confident about making generalisations based on their findings.



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Kevin Carroll
7:07pm 28 March 2014


I found your narrative fascinating. I like the way the design included the use of physical models to find the theoretical rule. You ask a very interesting question on the effect on taking learners out of their comfort zone. I expect one could write a book on that question alone.


Dave Martin
10:36pm 30 March 2014

Like Kevin I found this fascinating Emily.

Two things struck me, the difficulty of posing the right questions which could be planned for in the learning design and striking the appropriate level challenge which could not.

At a parents eveing I remember a conversation with a maths teacher demonstrating the value of algebra. I had jokingly suggested the reason why so many more parents were coming to talk to me was because his subject lacked relevance.

He gave me the problem of a frog leaping from lily pad to lily pad across a pond to reach the island in the centre. Like your students I could do it arithmetically when the numbers were quite small. When he made the numbers very big I had to develop a simple equation. For me a light bulb moment that never occured when I was a pupil myself.

alex martin
8:16pm 17 March 2017

I must say that this is one of the most amazing leraning approaching that I have ever heard off. Im glad that the tool worked and it can be verey useful in teaching mathmatics.Im a researcher with an essay typer and I must say that you did a very great job in this study. The experimental approach that you applied is just amazing. If children continue to apply these tactics, they will definetly be better placed to learn even more better. Thanks for the great article

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