Decomposition has been recommended as the dominant method for teaching K-6 subtraction for many years now. But has anything really changed? Have teachers switched comfortably? Has the use of decomposition led to any increase in student understanding and ability? And if not, why not?
The majority of experienced teachers have come from a system where subtraction was taught as “add 10, add 10” or the old “borrow and pay back” method. Up until 1989, the NSW Syllabus did not specify decomposition as a recommended method. Brief explanations were provided in the previous 1972 NSW Maths Syllabus for teaching 2-digit subtraction in Year 3, 3-digits in Year 4 and so on, with equal addends as the preferred method (add 10, add 10). It was usually left to textbook writers to fill in the gaps and they usually reinforced borrow and pay back or equal addends. Neither of these methods can be explained using concrete materials. They are more abstract strategies.
Many younger teachers of today grew up using the decomposition method. The 2002 NSW Maths Syllabus hedged it’s bets and recommended decomposition but suggested that the ‘add 10, add 10’ strategy could be used in Stage 3. The latest version of the ACARA Curriculum doesn’t specify which written strategy to use to solve algorisms but the NSW Board of Studies Syllabus recommends decomposition. The major advantage of decomposition is that it maintains a close link between place value language, structured material, the written digits and the algorism.
But does it achieve these goals in practice?
Any research that looks at the use of decomposition today may be missing the reality. What structured material was used? Was the language linked to this structure? Did the material reflect what was written in the algorism? Did the students have mastery of quick recall facts prior to using a written algorism? Did the school work as a consistent whole or were teachers left to their own devices where students experienced a conflicting range of guidance from class to class and class to home?
After working in close co-operation with teachers and their programs over the last 30 years, six issues emerge:
- the importance of counting and place value
- mental computation and the role of quick recall facts
- the use of structured materials
- the need for a consistent language-based approach
- the actual sequence of skills to be tackled
- the need for flexible thinking
- the need for a whole school approach
The importance of counting and place value
We need to believe that numbers make sense and not just see them as ink markings on paper.
eg. 37 can be described as “thirty-seven”, “three tens and seven ones”, “less than 50″, “more than 30” …
We can model this number using counters but of course it is structured materials which help us to model the number as we write it. Our Hindu-Arabic system uses place value, so that 3 to the left stands for 3 groups of ten and 7 on the right stands for 7 ones. We could just push our counters into 3 groups of 10 with 7 extras but this becomes cumbersome with large numbers.
From Year 1 onwards, students learn to structure our counting system by using a variety of structured materials, such as 10 beans glued to a stick (beansticks), 10 paddle pop sticks with an elastic band (popstick bundles) or to use the Dienes Base 10 blocks (MAB) where 10 is represented as a ”long”. This is then linked to language structures so that 37 is “3 groups of 10 and 7 ones” not just “thirty seven”.
A focus on deepening counting and place value understanding is the key to success. If we don’t have a good grounding then we have a shaky basis for later calculations.
Mental Computation and the role of quick recall facts
To do 2-digit subtraction most effectively it is vital to have addition and subtraction facts to 20 not at your fingertips by counting on or back but as automatic recall. Think of all the Year 3 and 4 (or older) students you know with their hands under their desks secretly manipulating their fingers to work out 13 – 7. In other words they are expected to work with more advanced calculations before they are comfortable and fluent with their basic facts. Just because they have tackled facts in Year 1 and 2 doesn’t mean we can assume the facts have been memorised. Regular, daily practice for up to 10 minutes of all the major maths facts, not just addition and subtraction, will be necessary right up to Year 6 and beyond. Successful teachers are also using homework “drill’ sheets to develop individual speed and accuracy to supplement this maintenance time at school. Most students from late Year 2 onwards need to focus on quick recall of addition and subtraction facts to 20 and then to 99. Most students from late Year 4 onwards need to focus on quick recall of multiplication and division facts to 100 as well.
The use of structured materials
Although the Dienes Base 10 blocks (either wooden or plastic) are in regular use in classrooms, we need to question their effectiveness. If introduced as early as Year 1 when children are only tackling numbers to 100 then the power of the “flats” and “blocks” is not seen. There are so many alternative structures, such as “beansticks” or “popstick bundles”, suitable for numbers up to 99, that it may be better to leave the Base 10 materials until 4-digit numbers are the focus. That way older students don’t feel they are being treated like babies by using material they were familiar with in Year 1.
Similarly, students who want to represent numbers greater than say 2000, struggle with a limited number of the Base 10 blocks and with limited spaces on their desks. In other words, the blocks can be a nuisance. Why not switch in to a smaller material such as counters, unifix or multilink, where colour becomes the Base 10 representation. Yellow can represent ones, blue the tens and so on. There are always oodles of these small counters or cubes in schools and their small size means that space is not a problem. For those children who still need a structure, the wide range of colours can represent any number, even up to a million, something quite awkward with the Base 10 blocks, where a million would be 10 blocks wide by 10 blocks long by 10 blocks high (1 cubic metre).
The use of a consistent language for mathematics
When staff use inconsistent language in a mathematics session it can be difficult for students to fully understand the concepts being taught. Your staff need to discuss, for example, all the different ways they might talk about adding and subtracting. Add 10 add 10 is a less common expression in subtraction nowadays. Borrow and payback is ancient and the least effective. Regrouping or decomposition links to student’s previous place value experiences.
The actual sequence of skills
Then there is the problem of curriculum expectation verses where your particular students are actually at along the learning continuum. In Year 4, for example, what is the point of attempting 4-digit subtraction with trading if many of your students can’t yet mentally add and subtract 2-digit numbers to and from 100? We need to differentiate learning so that groups of students work together along the curriculum continuum that best suits their skills and understandings.
The need for flexible thinking
Even though we have seen the need for consistent approaches, students also need to be flexible in their approach to mathematical thinking. There are many strategies to mentally calculate an answer in addition and subtraction. Just as there are many different algorisms that help a student calculate the correct answer using pencil and paper.
Students need to see numbers as active.
4503 is 4503 ones
4 thousands, 5 hundreds, 0 tens, 3 ones
An alternative way to name this number is to say 4 thousands and 50 tens and 3 ones.
Or even 4 thousands, 503 ones.
Why is this useful?
Once you have estimated your answer to a real-life problem, you need a strategy to calculate your answer.
And for 3-digit subtraction and above, this usually involves a written algorism.
If I want to purchase a 3 piece lounge set for $4503 and I only have $1675 saved up, how much more do I need?
e.g. 4 5 0 3
– 1 6 7 5
If I can’t manage this calculation mentally, and I don’t have a calculator handy, I can use pen and paper. I need to trade a ten for 10 ones, but rather than just saying I have “no tens” I can say I have 50 tens. So when I trade one ten in this problem I’m now left with 49 tens and 13 ones.
Regular oral practice at renaming numbers can be part of the daily maintenance program.
e.g. 4 5 0 3
– 1 6 7 5
This problem now becomes 13 – 5 = 8
And 9 tens – 7 tens leaves 2 tens.
And so it goes on.
And yes this is still complicated thinking but it skips quite a few steps while still maintaining place value meaning.
Need for a whole school approach
School staff these days are so inundated with curricula requirements that thinking about whether there is a whole school approach in Maths is only one of many pressing thoughts. The last decade has seen a heavy emphasis on literacy skills and, with the current political situation, we’re likely to see this trend continue to at least the Year 2020.
Teachers often make daily decisions in isolation from their peers. Mental computation, for example, is sometimes interpreted only as “drill” in the basic facts. There is usually a skimpy hierarchy of skills shown in any K-6 curriculum, so teachers fill in the gaps with their own ideas and interpretations. There are Year 1 and 2 teachers who give students examples like 459 – 225, even though the students are really only tackling this as single digit problems and developing a false concept of what subtraction is all about. Some Year 3 teachers encourage their students to subtract 3 and 4 digits with trading, for example, even though their students can not yet mentally add and subtract 2-digit numbers to 100. This is seen as good self-esteem for both the teacher and the students, even though it is usually taught as pencil and paper calculation. And there are still some Year 6 teachers who discourage students from using decomposition in subtraction at all, who resort to the old “borrow and payback” because it worked for them in their school days.
In other words there are lots of gaps and overlaps. There isn’t even a consistent way to write an algorism and disagreements can break out over where to put the ‘-” sign, for example. One Year 6 student recently reported to his mother that he couldn’t understand certain examples in the Year 7 entrance test he had just sat as the subtraction sign had been placed on the left and he was used to it being always on the right of the algorism. He had no idea what maths concept to apply.
A whole school approach keeps everything in perspective. There are only 40 weeks in a teaching year. In practice, there are probably only a maximum of 36 weeks where Maths can be taught, allowing for sport, choir, education week, excursions … the list goes on. Of these 36 weeks, you still have to teach Maths for about 60 minutes a day. That’s 5 hours per week, or 180 hours per year. And each grade then has to figure out how to fit in Number & Algebra, Measurement & Geometry and Statistics and Probability. Of course Maths can be integrated with other KLA’s but in practice it is seen as a separate entity. Staff need to work co-operatively to ensure their Term Overviews are updated regularly and realistically spread the load.
Once your staff work together on a teaching/learning sequence to suit the needs and abilities of their students, low/medium/high block goals can be clearly specified so each teacher can see where to go next in the sequence. A language-based proforma can be developed to fit the specific use of concrete materials. Mental strategies, estimation skills and real-life examples can be convincingly emphasised.
The whole school can work together with a common purpose, understanding the importance of real-life applications, mental strategies and problem-solving as the basis for sound mathematical understanding.