Background Information

Traditionally children have learnt to carry out pencil and paper vertical algorithms at an early age. For many children all they actually learn when doing this is the procedure for carrying out the operation and not the understanding behind it. In particular, Constance Kamii identified that many Grade 4 students do not really have place value to 100 even though they appear to be able to perform additions such as 36 + 45 correctly. In Young Children Reinvent Arithmetic (p 83) we read:

When we listen to children using the algorithm to do

For example, we can hear them say, "Nine and six is fifteen. Put down the five; carry the one. One and two is three, and one more is four … The algorithm is convenient for adults, who already know that the 2 of 29 stands for 20. However, for Primary children, who have a tendency to think that the "2" means two, and so on, the algorithm serves to reinforce this error.

Kamii specifically blames the early introduction of the algorithm for this lack of understanding, suggesting that most children are only carrying out single digit additions or subtractions when they do pencil and paper algorithms. Our experience in many classrooms is similar. We have found a large number of children even in Grade 4 who can add 3 and 4 but not 30 and 40. Often they say that 30 plus 40 is too hard. This indicates that they are not thinking place value as they compute. Silly errors in computation bare this out as an error to 36 plus 45 can result in the answer 711 when the regrouping (trading) has not been carried out. Children are so used to just adding up the columns that they do not even have an expected answer in mind. Funny isn't it that in the real world we are more often interested in the estimate or ball-park figure that the actual amount. For instance if shopping with a $50 note you see a top for $27 and a belt for $17 rounding and adding is used to quickly establish if there are sufficient funds. Front end addition or subtraction establishes the expected answer. Students who are only taught the traditional algorithm tend not to have in the head strategies for such a situation. In fact many adults use a pencil and paper algorithm in the head because they do not have the range of efficient mental strategies or short cuts available.

Chunking is one such mental strategy and it is now being widely used, particularly in Europe, as a first introduction to addition and subtraction for the following reasons:

• place value is built into the process
• starting with the tens ensures that the answer will be in the ball-park
• students have to actively think and process rather than just recall a procedure
• sophisticated mental strategies can be applied (e.g. for 36 plus 45 a student can chunk 36 and 40 to make 76 and then simply add on the 5 to make 81). This is closer to a real world strategy than vertical addition.

Comments

Chunking is not meant to replace the pencil and paper algorithm. Rather it is meant to be a foundation from which children can develop place value, number sense, estimation and confidence before they use larger numbers where the algorithm becomes an important tool. At the end of the day we want to develop people who can select from a wide range of strategies and tools which they actually understand rather than simply regurgitate procedures without any understanding of what they are doing.

The Early Numeracy Research Project emphasised that addition and subtraction have been and will continue to be a central focus of primary mathematics instruction. However, the report stresses that in recent years there has been a shift from a focus on pen and paper algorithms and error diagnosis to building upon young children's intuitive knowledge and strategies. This shift in thinking has been described by Sugarman as moving from showing children what to do towards showing them how to think.

There is strong justification for this emphasis on mental strategies. Northcote and McIntosh reported on 200 volunteers who completed a survey of all calculations used in a 24-hour period. They found that almost 85% of all calculations performed by adults involved some form of mental mathematics, while only 11% were written. They also found that estimates were sufficient for around 60% of all calculations, with exact answers therefore being required 40% of the time. Addition and subtraction were the most common operations used, 46% and 43% respectively.

But is this imbalance between mental methods, which chunking embodies, and formal methods reflected in classroom practice? Hopefully we are at least heading in the right direction!

References

Clarke, D, et al, (2001) Early Numeracy Research Project, http://www.sofweb.vic.edu.au/eys/num/ENRP/About/index.htm

Kamii, C with Baker Housman, L (2000) Young Children Reinvent Arithmetic Teachers (2nd Edition), College Press, NY

Northcote, M., & McIntosh, M. (1999) in What mathematics do adults really do in everyday life? Australian Primary Mathematics Classroom, 4(1), (pp19-21)

Sugarman, I. 1997 Teaching for strategies. In I. Thompson (Ed.), Teaching and learning early number (pp. 142-153). Open University Press, UK.