Students will experience success if you nurture their confidence so that they can work independently, take risks and persevere. The opposite will occur if students believe that they are no good at maths and will be very likely to give up before they have really tried.

Research shows that there is a strong link between confidence and achievement in mathematics. That’s why nurturing confidence is at the heart of our 11+ materials at 11+ Practice Papers.

When preparing our 11 plus maths resources and online testing we have tapped into the knowledge and experience of leading mathematics education researchers and 11 plus teachers across different regions:

- We closely examined the new National Curriculum for Mathematics (including Key Stage 2)
- We appraised the best performing countries in mathematics and their approaches to learning.
- We reviewed some recent and most commonly cited research papers in mathematics education.
- We audited the most respected and well-used mathematics resources, both print and digital.

As a result we’ve developed a brand-new approach to 11 Plus Mathematics, designed to nurture confidence and raise achievement.

__The 10 key principles of 11+ Practice Papers__

- Fluency
- Mathematical Reasoning
- Problem Solving
- Progression
- Reflection
- Multiplicative Reasoning
- Modelling
- CPA – Concrete, Pictorial and Abstract
- Relevance
- Linking

This blog will explain each of these principles in greater details, including how they raise confidence and achievement and how they are the cornerstones of what we do at 11+ Practice Papers.

We offer a range of learning materials which are specifically designed to foster confidence, visualise concepts, improve understanding and reflect on learning and progress made.

Students who do not master a topic are not encouraged to continue regardless. We expect them to reflect on mistakes, re-learn concepts and redo questions or tests until they have mastered their specific area of difficulty.

Students who do master a specific topic area are not encouraged to simply do more of the same. They are challenged by enhancing the breadth and depth of their knowledge.

__Fluency__

This is when we recall familiar mathematical facts automatically without much thought.

According to Psychologists the amount of brain power we have to apply to mathematics learning and problem solving is limited at any one time. This is why the instant retrieval of information and facts helps use release space in our brain to engage with the more challenging elements of a problem. Therefore, students who possess good mathematical fluency are very likely to achieve well in 11 plus mathematics test.

The new National Curriculum cites developing fluency as an important tool and asset in progressing students to access higher level mathematics.

__Problem Solving__

When students have to undertake multi-step thought processes and working out to tackle 11 plus questions, this is called 11 plus mathematics problem solving.

__How problem-solving fosters confidence and enhances achievement in the 11 plus exam__

Research suggests that when students are trained and taught techniques for 11 plus mathematics problem solving they develop a form of resilience and urge to ‘have a go’ at tasks. Through experiencing success, they acquire confidence in their own ability and become empowered to face unfamiliar problems in the 11 plus using such techniques that they have acquired. Consequently, these students are more likely to succeed in exams and future studies and jobs.

As a result, these students are better equipped to succeed in **11 plus tests **and exams and consequently in future study and employment. It comes as no surprise that problem solving is a vital element of the curriculum in high performing countries.

The new National Curriculum cites problem solving as a central feature in progressing students to access higher level mathematics.

__Mathematical Reasoning__

When students apply logical thinking to solve 11 plus mathematics problems, this is classed as mathematical reasoning.

__How reasoning fosters confidence and enhances achievement in the 11 plus exam__

It is only when students can reason when solving 11 plus questions that they are demonstrating the power of understanding. Many students can often state and display what they can do in mathematics but struggle to reason. When students can explain a particular approach to solving 11 plus mathematics problems, why their approach may be better than another and how they came to their final answer, you can rest assured that are secure at reasoning.

It comes as no surprise that there is strong correlation between the 11 plus achievement of children and their ability to reason.

The new National Curriculum cites reasoning mathematically as a key feature in progressing students to access higher level mathematics.

__Progression__

It is vital that mathematical teaching for the 11 plus is done correctly. Work needs to be carefully planned and topics covered and revisited in a spiral curriculum which sees the same topics revisited throughout with each encounter increasing in complexity and reinforcing previous learning. This will ensure that students’ progression is scaffolded and secure through the entire 11 plus process.

__How progression fosters confidence and enhances achievement in the 11 plus exam__

Success in 11 plus mathematics is dependent on offering students just the ‘right’ amount of challenge, at just the ‘right’ moment. This means ensuring all the necessary prior knowledge, skills and understanding are in place, and gradually building to enable students to progress.

Research suggests that when students are made explicitly aware of this progression, not just topic-by-topic, but lesson-by-lesson, then their confidence and performance improves in the 11 plus exam.

__Reflection (Metacognition)__

This is when students ponder and deliberate about the mental strategies and processes involved when attempting to solve 11 plus exam questions.

__How reflection fosters confidence and enhances achievement in the 11 plus exam__

As they attempt 11 plus mathematics questions, if students reflect on *what *they are doing, *how *they are doing it, and *why *they are taking a particular course of action, then they gain valuable insights into the way that they learn. If, afterwards, they are also encouraged to consider the understanding they gained, what they found easy or difficult, the mistakes that they made and the merits of different approaches, they can confidently adjust how they do things in the future. Research demonstrates that students who regularly reflect in this manner demonstrate greater perseverance and success at solving mathematics problems in the 11 plus exam.

__Multiplicative reasoning__

This is when students use multiplication or division to compare values and work out one value when given the other. These usually arise from proportional situations and often involve the application of ratio, proportion, fractions, decimals and percentages.

__How multiplicative reasoning fosters confidence and enhances achievement in the 11 plus exam__

It is important for students to know if, when and why relationships are multiplicative is essential to the understanding of many 11 plus concepts, among them percentages, functions, proportion, ratio, area, volume and sequences. This is why multiplicative reasoning is an important element of the new National Curriculum for mathematics and plays an important part in progressing to higher level mathematics.

When students can see multiplicative connections in diagrams and bar models, it is believed by researchers to increase their confidence in mathematics. This also strengthens understanding as to what may be happening to one quantity as the other changes. It enables students to choose the correct calculations and solve mathematical and real world context problems in the 11 plus exam.

__Modelling__

Mathematical modelling is the attempt to describe and understand a real world situation in mathematical terms.

__How modelling fosters confidence and enhances achievement in the 11 plus exam__

It is important to provide opportunities for students to model and obtain insights into real world scenarios. This has proven to have a positive motivational effect. There is further evidence to suggest that modelling deepens students’ understanding of key mathematical concepts when devising, testing and evaluating. Not only will this improve students’ performance in the 11 plus exam, it will also improve their performance in other maths related fields and future careers.

The new National Curriculum cites modelling realistic situations as an important feature in progressing students to access higher level mathematics.

__Concrete–Pictorial–Abstract (CPA)__

The learning of mathematics with objects, followed by pictures, then notation is called the CPA (Concrete-Pictorial-Abstract) approach.

__How CPA fosters confidence and enhances achievement in the 11 plus exam__

Students usually start with manipulating objects such as blocks, sticks and dice to make sense of mathematics. They then move on to representing this pictorially with diagrams, graphs, charts or bar models. Finally, they move onto the abstract ideas which are expressed as numbers, symbols and letters.

Students have benefited greatly in learning and extending their knowledge of 11 plus mathematics with the use of bar modelling. For parents, you can find an introduction of bar modelling on this YouTube link: https://www.youtube.com/watch?v=0V5vg3Zxo5k

CPA is freely and readily used to teach mathematics in high performing countries. Research shows that it afford students space and time to develop their conceptual understanding which in turn leads to a much improved performance in the 11 plus test.

__Linking__

When students find learning and concept within mathematics and other subjects, then this is referred to as linking.

__How linking fosters confidence and enhances achievement in the 11 plus exam__

It is important for students to realise that mathematics is not isolated from other subjects. Research proves that when students find links between two concepts and subjects, their knowledge and understanding deepens and their progression advances. This is due to consolidating previous learnt concepts and applying them in new contexts. Consequently, as students confidence increases in mathematics, they are more willing to apply this learning and knowledge in unfamiliar contexts in other subjects such as Verbal and Non-Verbal reasoning. As a result their performance in the real 11 plus exam improves.

__Progress with confidence with 11+ Practice Papers__

Our innovative, well-researched and proven 11+ online tests and resources at www.11practicepapers.co.uk embeds evidence based approaches throughout to create confident and numerate students able to pass the real 11+ exam anywhere in the UK. We welcome you to visit and join our ever increasingly 11+ family.