|Year||Initial stage||Final stage||Time in|
|11||5/6||11 students - stage 4||2 students - stage 5|
9 students - stage 6
Royal Road School was one of a number of schools that were chosen to participate in an exploratory study of how to accelerate the learning for students in maths.
As part of our programme, Royal Road School choose three main points to focus on for effective mathematics teaching (from the Effective Pedagogy in Mathematics by Glenda Anthony and Margaret Walshaw). Our school selected assessment for learning, mathematical language and tools and representations. The approaches that we selected help to engage learners and lead to desirable outcomes.
Looking at achievement data, Royal Road School selected two groups of year 5 and 6 ‘at-risk’ students to work with in three morning sessions a week.
We decided that the ‘cause for concern’ students (1 stage below) can mostly be dealt with, within the effective classroom teaching programme. Students ‘at risk’ (2 or more stages below) needed the “specialist teacher intervention”.
Eleven students at stage 4 formed two groups of students (one group of six and one group of five).
A description of the teaching programme
Our initial programme consisted of lessons that involved a mixture of memory activities, knowledge building through the use of COSDMBRICS, followed by strategy teaching. Lessons started with 10-15 minutes of memory activities/games, approximately 20 minutes of knowledge building and 20 minutes of addition and subtraction strategy teaching at the end.
In total each group was taught for an hour three times a week. The mix of memory, knowledge and strategy teaching engaged each group of students for the hour duration. I decided to focus first on strategy teaching in addition and subtraction as I felt that, before we could progress in other areas of maths, we had to gain a thorough understanding of basic addition and subtraction strategies first.
For memory teaching, I used a variety of approaches. I found that using computers for activities was highly motivating and put a positive spin on the beginning of our lessons. The students also enjoyed seeing something for five seconds then recreating it with either materials or drawing it. Another highlight was holding up tens frames and the students being able to instantly recognise/subitise what number was on the card and the number of spaces to make ten. The students became competitive in a friendly way with each other, which resulted in the students striving to do even better. Being able to subitise the dot patterns on the tens frame card and recognising combinations to ten became very beneficial later on in part-whole strategy teaching.
I used a combination of the COSDMBRICS programme, and a variety of activities from the plans that were offered on the training day (moving students from stage 4-5 planning focusing on basic facts and the place value plans). Students worked on their knowledge gaps that were identified through the NumPA tests. In particular we worked on being able to count forwards and backwards in tens from a given number, for example, 3, ordering and reading numbers, making ten, recognising patterns for example 3 + _ = 10 so 13 + _ = 20 etc, and recognising how many tens and ones are in numbers.
Once we completed knowledge activities around making ten, we started to learn the strategy of splitting numbers and the students grasped the concept rather quickly. By using a range of materials to teach and consolidate patterns to 10 (for example tens frames, fingers, number fans and cards) the students were able to quickly recognise how many they needed to add on to solve the equation. At first, the students struggled with explaining the process they would take to solve problems. By modelling/working through examples together, and then giving the students the opportunity to talk and listen to their peers, the students soon could articulate their mathematical thinking clearly. We then worked on splitting numbers to solve problems to the nearest tidy number, for example 35 + 6. When we moved onto subtraction, the students didn’t grasp the idea of how to solve these problems as quickly. The students didn’t appear as confident and we reverted back to recognising patterns. Within a few weeks all students could confidently split numbers up to solve problems.
Once the students grasped the strategy of splitting numbers, we moved onto jumping the number line and to using the reversibility strategy, for example 56 – 28 = ? to 28 + ? = 56, and finally to the strategy of rounding and compensating. I noticed that when solving problems using these stage six strategies, the students often needed to go back to using a pencil and paper. They could not hold the process they took (especially ones that involved more than three steps) in their heads. This is where I strongly agree that memory activity should be practised daily. Students could however solve problems accurately and talk about how they did this using the paper as a reminder.
As I was working with the students, I noted that some students were working at a faster pace than others. Because of this, I decided to rearrange the groups. My first group consisted of four year 6 students who picked up new concepts quickly. My second group consisted of a mixture of year 5 and 6 students who understood new concepts quickly, but worked at a slower pace. My final group was a group of three (one year 6 and two year 5 students) who needed to manipulate materials for a longer period of time to understand new concepts. I found the smaller groups more manageable and it was easier to spend some one-to-one time with the students who required extra help. Each group now had approximately 40 minutes each and the structure of the lessons became more flexible depending on the student needs.
Once I felt the students were confident solving addition and subtraction problems, I used the remaining two weeks to continue to build knowledge of place value and to introduce the multiplication and division domain. My goal was to move the lower two groups from using repetitive addition to using their basic 2x, 5x and 10x tables to solve multiplication problems. My advanced group were able to more quickly derive multiplication facts from their existing known facts and progressed to using the doubling and halving strategy.
Analysis of the data / student progress
The most significant shift demonstrated in the data is the progress made by all of the 11 students from stage 4 to either stage 5 or stage 6 in addition and subtraction (two students to stage 5 and nine to stage 6). This progress in learning reflects the emphasis of the targeted programme on addition/subtraction strategies and related knowledge teaching over a duration of seven weeks.
With the exception of the same two students who only progressed one stage in add/sub all the other students progressed at least one stage, some two stages in multiplication/division. Proportion and ratios stayed the same as this was not taught during the programme.
The progress students made as a result of this targeted teaching programme can be attributed to the following:
- Consistent knowledge practice using a range of materials (tens frames, card games, number fans) with particular emphasis on combinations to ten and patterns around these combinations, for example 10 – 6 = 4, 20 – 6 = 14, 50 – 6 = 44; 3 + 7 = 10, 13 + 7 = 20, as well as 13 + _ = 20 all visually displayed
- Basic fact practice weekly like doubles, halves, addition/subtraction facts using dice
- Developing language skills and sentence starters to enable students to express their problem solving strategies (The teaching sessions allowed the students ample opportunity to discuss their solutions and strategies used in a think pair share format.)
- Students engaging positively in the maths sessions and student success continually being celebrated within the group (Students were aware of their progress and were eager to practice at home to consolidate their new knowledge - they even requested homework during the holidays! Their attitude to maths changed from negative to positive and they were always keen to come to the sessions and they showed this same enthusiasm in their own classroom maths sessions, as reported by class teachers.)
- The specialist teacher continually communicating with the classroom teachers, letting them know the work covered each week and also finding out how the individual students were coping in their classroom maths sessions
The principles focused on for effective pedagogy in maths
Assessment for learning
I chose this principle as an important focus in that it reflects my style of teaching. I am continually adjusting my teaching programme to accommodate the student’s progress during each lesson. I needed to be flexible in grouping the students as I found they progressed at different rates as the programme went on, necessitating group changes from time to time. My lessons had to be flexible as well. The dynamics of the student discussion took different lengths of time and listening to their responses allowed me to ascertain if they understood the problems and each other’s strategies for solving them.
Their discussion and actions allowed me to follow their thinking processes and reasoning so that I could redirect them where necessary. Assessment was on-going throughout each session and determined where to next in my planning. I think this was a key to the success of the programme as I was able to tailor the learning specifically to each child in the group.
I found that an essential part of the programme initially was developing and building student mathematical vocabulary, including words like rounding up, tidy numbers, ‘times’ five groups of three, add, and subtract (as opposed to ‘take away’) and introducing terms like, ‘adding on ten’ or ‘ten less than’, ‘one more than’ or ‘one before’. I was surprised by the number of basic words and mathematic terms that needed clarifying for these year 5/6 students. I now realise in my own classroom teaching the importance of introducing these terms and phrases early in the schooling and not making assumptions about what I think students might already know!
Another important aspect of mathematical language was the use of context in posing problems and using words that directly relate to the students’ experiences culturally and socially. Throughout the programme, these contexts helped to engage the students in the problems presented.
Also it was essential to allow the students plenty of time to do the ‘talking and thinking out loud’ and much of the newly introduced language was used in these conversations.
Tools and representations
Materials were used daily to help the students to express their thinking and to solve the maths problems. Knowledge was built up with the help of tens frames, card games, dice, iceblock sticks, place value flip book, blocks and number cards. Most of these materials were also used to develop strategies, with the additional inclusion of laminated number lines and representations of number lines drawn by students in their books. The materials were essential for the students to visualise and represent what they were doing when solving the problems. By manipulating the materials, the students were able to mathematically prove their solutions and become totally involved with the strategies being taught. In addition, computers were used once a week to practice memory games, which the students found to be motivating.
Overall, students in the targeted learning group at Royal Road School benefited from the programme as they showed progress in their mathematical ability over the 10 weeks.
Nine students moved two stages in addition and subtraction with the remaining two moving one stage.
In addition to this, four students also moved two stages in the multiplication and division domain while five students moved one stage.
We also had four students who at the beginning of this targeted learning programme had no knowledge of place value and they have currently moved either four or five stages in this area. Another five students moved one stage in their place value understanding.
The achievement of these students is attributed to the teaching of memory skills, knowledge building and strategy teaching with an emphasis on using materials, to on ongoing assessment and to the use of mathematical language (including communicating their own mathematical ideas with others).
The accelerated achievement that this intervention has allowed brings hope that this will continue into 2011 and the years to come.