Operation Maths: A Unique Approach to Problem-Solving
Category : About Operation Maths
In this post, we will look specifically at the Operation Maths approach to problem-solving in the senior end books (3rd to 6th classes). In a subsequent post, we will look at how this approach develops in the junior end books (infants to second classes).
Presenting children with an abundance of mathematical problems does not automatically transform them into competent and confident problem-solvers. Rather, the children must be explicitly taught a range of problem-solving
strategies and they must be facilitated in applying and practising the strategies repeatedly in a range of different contexts.
Operation Maths has an integral multilayered approach to problem-solving throughout the 3rd to 6th class books:
- A variety of key problem-solving strategies is introduced, explored and applied to various real-life contexts in a developmental and spiral way through the classes (i.e. bar model drawing, empty number lines, T-charts , branching etc)
- Regular Work It Out! sections throughout the chapters in the pupils books provide the children with opportunities to apply and hone their problem-solving skills.
- Let’s Investigate! sections at the end of the Pupils’ Books where the focus is on open-ended problems
- Thematic revision spreads with a strong problem-solving focus.
- Extra problem-solving in Early Finisher photocopiables.
All of this happens as part of a larger problem-solving approach based on the acronym RUCSAC. This approach, which can be used as a whole school problem-solving approach, is also reinforced and explained for both children and parents on a convenient French flap/bookmark on the Discovery Book (see images from flaps below), which encourages the children to use RUCSAC as an aid when problem-solving.
Problem-solving skills
The ability to reason mathematically is fundamental to being able to solve mathematical problems. However, reasoning mathematically requires not just one, but a number of mathematical skills e.g. being able to
• Work through a problem in a systematic way
• Predict an answer
• Identify the relevant information and understand what type of answer is being sought
• Visualise the problem mentally or being able to represent the components of the problem in either a pictorial or abstract (using only numbers and symbols) way.
• Plan or decide what approach to take
• Work to get an answer
• Check that the answer is suitable and accurate.
What is fundamentally different about the Operation Maths approach to problem solving is that the children are being taught specific strategies to develop the aforementioned skills, in a spiral and progressive way, in order to equip the children with the necessary skills for them to become capable and confident problem-solvers.
Central to the Operation Math approach to problem solving is RUCSAC. This clear, sequential approach enables the children to work through problems in a systematic way, while simultaneously utilising the mathematical skills that are being developed with and throughout the chapters.
RUCSAC and the Specific Strategies taught in Operation Maths
RUCSAC is an acronym, where each letter represents one of the six distinct phases of this problem-solving approach (see below). However, this more than just a clever mnemonic, as each of these phases is supported by the development of specific strategies throughout the programme, which support this approach. These specific strategies are as follows:
Read – Estimation strategies:
- Reasonable answer: Would you predict a bigger or smaller answer? How many digits would you expect in the answer
- Front-end estimation: Look at the digits at the front of the numbers
- Rounding: Round each number to the place of the highest value digit e.g. tens, hundreds, thousands.
- Rounding to fives: (only in OM6): Usually we round to the nearest tenth, unit, ten, etc. But if the number(s) involved are approximately in the middle, it is more efficient to round them to the nearest five tenths, 5, 50 etc. to get a more accurate estimate. (OM6, Pupils Book p 30)
Underline – Colour coding operational vocabulary:
- Identifying specific phrases, colour coding them, and recording them on in the Discovery Book. This forces the child to engage with the language of problems and to decode them. However, this only suits word problems which contain obvious operational vocabulary or that which can be easily inferred.
Create – Creating visual representation to show the information in the problem, as part of a CPA approach:
- Using concrete materials (e.g. counters, cubes, children etc.)
- Using bar model drawings
- Using empty number lines
- Using T-charts (OM4 to OM6)
- Making/completing a table, grid, list etc.
- Creating number sentences (and/or equations with variables in OM6)
Select – Selecting a suitable and efficient approach:
- Using a mental method, e.g. petitioning, sequencing, compensating etc.
- Using a written method e.g. a formal algorithm, jottings, branching
- Using guess and test
Answer – Answering the question:
- The teaching panels demonstrate how to layout and position work clearly and sequentially
- Children are encouraged to “show your thinking”
Check – Checking answer(s):
- Comparing the answer to the estimate, e.g. does it look reasonable?
- Using the inverse to check.
Furthermore, as part of this approach, specific visual strategies are introduced and repeatedly used where appropriate:
- Empty Number lines
- Bar Models
- T-charts
Empty Number Line (ENL)
Simply, a horizontal line, initially with no numbers or markings that helps develop a child’s number sense, their ability to visualise numbers and to compute mentally.
Also known as a blank or open number line, empty number lines can be used to show elapsed time, operations, skip counting, fractions, decimals, measures, money (making change) and much more (see image below).
While, strictly speaking the number line should initially start empty (i.e. no numbers or markings), in Operation Maths, some of the required numbers and/or markings have been provided, to act as scaffolding for the child. Ultimately, it in envisaged, that as the child grows more confident of this structure, he/she should be able to construct an empty number line from scratch in order to help solve other problems. I is also hope that through using this structure the child would be able to develop this ability to visualise numbers in such a way and, in doing so, enhance their ability to compute mentally.
Bar Models
These are simply drawing(s) that resemble bars, (like that seen in bar graphs), that are used to illustrate number relationships. There are two main types, part-whole bar models and comparison bar models.
Part-whole model: which can represent a whole amount that is subdivided into smaller parts. In Operation Maths these are used to represent:
- Addition/subtraction: where a whole amount has been subdivided into two or three amounts/parts and either the value of one of the parts or the whole/total is required
- Multiplication/division: where a whole amount has been subdivided into equal amounts/parts and either the value of one/some of the parts or the whole amount is required
- Fractions, ratios, decimals and percentages: Where a whole amount has been subdivided into equal amounts/parts and either the value of one/some of the parts or the whole amount is unknown.
Comparison models: which are used when comparing two or more quantities. In Operation Maths these can be used to represent:
- Addition/subtraction and Multiplication/division: where two amounts are being compared and the value of one of the amounts or the difference between the amounts or the total value of the amounts is being sought.
- Fractions, ratios, decimals and percentages: Where two or three amounts are being compared and the value of some of the amounts, the difference between the amounts or the total is unknown. This can also be a very effective way to calculate selling price and cost price when given percentage profit/loss
T-charts
A T-chart is simply a table, usually divided into two columns, giving it a T-shape. They can be used as a means to aid calculations and/or to identify patterns and connections within problems .
Other strategies
Other strategies used in Operation Maths which promote the visualising and decoding of problems include:
• Using number bonds and branching
• Making lists
• Using “guess and test” (also known as Trial & Error)
• Using the process of elimination (e.g. logic problems)
While the ten-frame is the most common arrangement, multiples can be used to demonstrate numbers beyond ten eg 35 could be shown using three full ten-frames and five on a fourth frame. For exploring numbers up to five (eg with junior infants), a five-frame could be used; however, it is perfectly acceptable to use a ten frame and limit your use to just the numbers up to five (ie the top row).
Through repetitive use of the ten frame, the children should already be developing an understanding of the numbers to combine to make these important benchmarks eg 7 + 3 = 10, 4 + 1 = 5 etc. The children can record the benchmarks using number sentences and/or branching number bonds (see opposite). Branching bonds are more visual and less abstract than number sentences alone as it is easier to visualise how 4 and 6 are combined to make 10 and they do not necessitate the use of operational symbols.
Sorting & Shop e-Manipulative
The Sorting & Shop e-Manipulative allows the teacher to easily drag and drop shapes, animals, fruit, classroom objects, shop items, upper and lowercase letters, and numbers onto a workspace. It can be used blank or with various backgrounds, including frames, sets, 2×2, 5×5 grids etc . Of all the backgrounds, the shop background is particularly useful as it allows the teacher to create a shop scene with price tags, coins and sale tags, which can be used to explore a wide range of mathematical scenarios such as using small amounts of money in infants right up to scenarios involving percentage increase and decrease in the senior classes.
The 100 Square e-Manipulative is another extremely versatile tool. It can be used in numbers only, counters only or counters and numbers. You can very easily hide and reveal individual cells, whole sections of the grid or the entire grid. I have gotten a huge amount of use of out it recently, with first and second classes, using it in numbers only mode, hiding all the numbers and just revealing one number. I then ask the children what number comes after/before this, what numbers is missing above/below etc. This is particularly good to assess the children’s ability to identify numbers around the decuples/decades (ie 30, 40, 50 etc) which are widely recognised as hurdles for many children.
The Counting Stick e-Manipulative replicates the physical counting stick that a teacher might use in the classroom. The teacher can set the starting value and the steps value, and reveal or hide numbers along the counting stick. Decimal and negative numbers may also be shown on the Counting Stick e-Manipulative and two counting sticks can be shown at the same time, in order to compare various numbers.
The teacher can use the Fractions e-Manipulative to present fraction bars (linear models), fraction circles and pizzas (both area models). The teacher can change the fraction that is shown on screen, randomise fractions and hide or show the fraction
Analogue and digital clocks are provided with the Clock e-Manipulative. The teacher can choose to show one analogue clock, one digital clock, two analogue clocks, two digital clocks or an analogue and a digital clock at the same time.
Ready to go activities are already set up within each e-Manipulative with pre-programmed questions that appear on screen, meaning that the teacher doesn’t have to waste time looking in a book for the accompanying questions. The questions can also be answered on the children on their MWBs, thereby encouraging whole-class participation.
Create activities are so called because the teacher can open the e-Manipulatives and choose how to use it to best suit them, their class and the concept at hand. There are suggestions for Create activities printed in the TRB which show how the tools can be re-used in infinite ways to achieve a countless number of specific learning outcomes. And the Ready to go activities themselves will also provide the teachers with examples of how each e-Manipulative may be used.
Not only have these activities been written especially for Operation Maths but Operations Maths is only the only maths scheme available currently in Ireland with integrated programming (coding) activities. Each activity is integrated with the Pupils’ Books, comes with step-by-step instructions for teachers and pupils and highlights the connection between maths and coding in an easy-to-follow, visual manner.