Scenario: Maeve discussing the use of formal standard algorithms
Country: UK
Grade (student age): Preservice teacher
Contributed by: Gwen Ineson, Brunel University, UK
Context – professional
The session described below took place in week five of a one year postgraduate teacher training programme. Trainees had not completed either of their sustained school experiences but they had undertaken an observation period of two weeks in primary classrooms as preparation for the course. They had received 14 hours of mathematics input, which included an overview of the national curriculum, planning and assessment as well as subject knowledge relating to calculation, measures and shape. A feature of the course is the strong emphasis on a range of calculation approaches for the number operations.
It would be useful to explain what is in the Irish curriculum for the third class in relation to multiplication.
Scenario
Maeve was part of a group of 24 students on a one year postgraduate teacher training programme. Trainees had been asked to use a range of approaches to solve 3.4 x 4.9 and they were providing examples of different approaches to solving this item. Approaches included the standard formal algorithm, partitioning, the area or grid method and rounding and adjusting. The diagrams below are examples of these approaches:
Algorithm  Partitioning  
3 . 4  3.4 x 4.9 = (3 x 4) + (3 x 0.9) + (0.4 x 4) + (0.4 x 0.9)  
x  4 . 9  
Grid method Rounding and adjusting
3 
0.4 
3.4 x 4.9 = (3.4 x 5)  (3.4 x 0.1) 

4 

0.9 
During discussion Maeve asked why we teach all these different strategies and explained a scenario in Ireland (where she had done some supply teaching) where she had explained the long multiplication algorithm to a group of upper primary aged pupils. She explained that her approach was to count how many places there were after the decimal point (in this case 2), then complete the problem ignoring the decimal point (using a formal algorithm). She then told the pupils to put the decimal point back into the answer, two places from the right. When probed why this was the case she was unable to explain, but queried why that was necessary. She felt very strongly that the alternative strategies that had previously been discussed by other trainees would result in confusing pupils and she didn't understand why we didn't teach the algorithm straight away. Her reasoning was that the other strategies were confusing and there were too many of them, whereas her approach was simple once learnt and always gave the right answer.
Knowledge Quartet Coding Commentary
Contributed by: Gwen Ineson, Brunel University, UK
Knowledge Quartet Dimension: Foundation
Knowledge Quartet Code: Concentrates on procedures (COP)
Scenario: Maeve discussing the use of formal standard algorithms
During discussion Maeve expressed doubts about alternative approaches to calculation. She felt completely comfortable with her strategy and although she couldn't explain why, she knew that 'it worked'. She admitted to feeling rather lost when other trainees were explaining their approaches and hadn't observed teachers using alternative strategies with their pupils.
She felt that the algorithm had worked for her and therefore it would work for other pupils. She was reluctant to engage with alternative strategies and dismissed them as ‘too complicated’. Of course, many alternative approaches to calculation that are taught in the primary years are more complicated if considering them procedurally. What Maeve failed to understand was that there is an attempt to teach pupils to develop relational understanding (Skemp, 1978) and it is therefore not sufficient to be able to follow a procedure.
Maeve's belief that mathematics is about rules which have to be remembered will impact on her approach to teaching mathematics (Goulding, Rowland and Barber, 2002). Despite other trainees discussing the value of alternative approaches, Maeve could not understand why there had been a shift away from teaching a procedure that she had used successfully all her life.
Maeve, may ‘cling’ to this procedural approach because of an insecurity about using more conceptually oriented approaches suggesting some weakness in her Foundation knowledge. This seems to be supported by her admission of being ‘lost when other trainees were explaining their approaches’. This is also indicative of her beliefs about mathematics teaching and learning, i.e. that mathematics is a series of rules to be learned.
References
Goulding, M., Rowland, T. & Barber, P. (2002). Does it matter? Primary teacher trainees‟ subject knowledge in mathematics, British Educational Research Journal. 28 (5), pp. 689704.
Skemp, R. R. (1978). Relational understanding and instrumental understanding. Mathematic Teaching, 77, pp. 20–26.