Explanation

Using fingers

Finger counting is generally accepted as an excellent tool for young children as an introduction to numbers up to 10 and for counting forwards and backwards. It also helps them associate numbers with an actual physical quantity.

There are numerous articles debating at which point finger counting begins to become a hindrance for mathematical progress – year 3 being around the age which most would say that other methods should be employed.

Some issues with continuing use of fingers:

• Accuracy – errors when counting
• Delaying mental maths which is necessary for later years
• Number range is limited
• Hindrance of abstract thinking and visualisation
• Slower calculations
• Inability to see beyond one-by-one counting

Similar methods to finger counting are drawing dots or tally marks – used when numbers are over 10. Practice over time using these methods do not lead to a significant improvement in accuracy and speed.

In contrast, using mental maths from an early age results in a significant improvement in speed and accuracy. Along with this comes an improved number sense and greater cognitive load capacity. Tasks which students initially find hard and mentally draining to work out become increasingly easy with practice.

What is Maths

Many would think of maths as a complex mix of numbers and letters with symbols that would fill them with fear and dread. Surprisingly there is very little “maths” even in the most difficult of questions. Any question can be broken down into foundation and formula.

What is foundation and formula? I shall use cooking as an example. Give a recipe to a professional chef and a regular cook and you will find that the results are very different even though they follow the same “formula”.

The foundation in cooking includes some of the following:

• Knife skills
• Cooking methods
• Control of flavour and seasoning
• Temperature and time control

Trainee chefs start with vegetable preparation , such as chopping and peeling. Weeks or months are spent doing this – on the surface it seems a waste of time, but this task builds fundamental skills and confidence in cooking and ensures a solid foundation for more complex cooking tasks.

With a solid foundation, chefs can make up their own recipes and are able to create a tasty meal from leftovers. Regular cooks who have very little foundation often have to memorise countless recipes and are at a loss when an ingredient is missing.

Foundation and formula in maths

The foundation of maths has commonly been understood to include the following:

• Arithmetic – addition, subtraction, multiplication and division
• Number theory – Intergers, rational numbers and real numbers
• Algebra – variables and equations
• Geometry – Shapes and spatial relationships

I would argue that arithmetic is the most important of the four listed. Number theory, algebra and geometry are closely related to a strong grounding in arithmetic and ability to visualise.

When numbers are associated with a object that can be seen like coins, other mathematical concepts are more easily learned:

• Number theory – integers are whole and complete, rational numbers are part. Seeing the 4 coins fill only 4 out of the 5 spaces – whole, part, fractions, percentages and decimals are seen.
• Algebra – a coin represents an object. It can be easily replaced by an apple, a cat, an x or y. Each object can be valued at any amount. E.g an apple could be 10p, £2 etc.
• Geometry – spatial awareness is improved with visualisation from a young age.

So in conclusion, basic arithmetic and how numbers are related to each other through a physical quantity or visual method is the foundation required for maths.

What is the formula in maths? Formula are the rules that needs to be followed. Examples:

• BIDMAS – order of operations
• Adding, multiplying, dividing fractions
• Area and volume of shapes
• Pythagoras Theorem
• Factorising quadratics
• Integrals and derivatives

If I was to list all the rules it would go on for pages…

Large proportions of the maths syllabus from primary to tertiary level require students memorising “formulas” or processes in order to answer the question.

As we move from primary to tertiary level, the percentage of questions which test the true knowledge of the topic increases. Learning formula and processes rarely help in answering such questions.

Procedural vs Conceptual understanding in maths

Procedural understanding involves following a sequence of steps, rules and formulas to achieve the right answer. A few examples:

• Subtracting 9 from 32, borrowing the 1 from 3 to give to the 2.
• Calculating the area of a triangle, half base times height
• Adding fractions by ensuring denominators are the same.

Conceptual understanding refers to a deep understanding of mathematical ideas instead of following rules. This can be achieved using visual aids, and hands-on learning. Using the same examples above:

• Subtracting 9 from 32, cubes can be used to demonstrate why we need to borrow a block of 10 cubes.
• Drawing a rectangle and dividing it diagonal to demonstrate why half base times height is the area.
• Adding fractions by representing them as portions of a cake. Differing denominators results in different sized slices of cake.

It is generally accepted that having a balance of both conceptual and procedural understanding of maths is required for success in maths. Both types of understanding are mutually beneficial, increases in one increases the other.

Having a strong procedural understanding without conceptual results in being able to answer questions quickly and accurately. However if the problem is altered, the solution may not be seen. This is because procedural understanding in built through “doing” the procedures.

Having a strong conceptual understanding without procedural results in being able to apply their knowledge to various similar problems. Lack of knowledge of correct procedures may result in inaccuracies and slow speed.

Foundation and formula KS1 to KS4

Procedural and conceptual understanding, foundation and formula, coupled with the differing mathematical abilities of students – where do we begin?

How and when conceptual and procedural understanding should be introduced is debatable. It can depend on the age and ability of the student.

In order to solve questions, students must be able to process information/numbers mentally and this effort is known as cognitive loading. This ability to handle the load varies from student to student, but through utilising certain techniques and through practice it can be improved.

An example:

Q1. 5 + 9 =

Q2. 24 x 17 =

The second sum requires more mental effort and places a larger cognitive load on the mind.

The working memory, a temporary storage space for data has a limited capacity, can be overloaded when the cognitive loading is too high. An example:

Q3. 1274 x 3629 =

To make things even more complex, we also have long term memory which helps reduce the cognitive load on the working memory.If a student, answering Q2, had memorised his 17 times table, then drawing that information from his long term memory would place less cognitive loading on his working memory.

When the working memory is overloaded the transfer of knowledge to long term memory is hindered.

By completing many questions, spending hours doing maths homework, we are able to build procedural understanding and also increase the long term memory of what is being practiced. This technique is utilised in Asia where many students spend hours doing homework every day. This approach is very similar to traditional teaching in maths from 50 years ago where primary level students were told to practice adding and subtraction, multiplication and division, with very little of anything else. Emphasis was not placed on conceptual understanding, and the reasons why procedures had to be followed was often discovered at a later age. This is certainly true for me when learning to add fractions – I only realised many years later why the denominator had to be the same.

The method of doing lots of practice may not be the most efficient way of learning as it’s time-intensive. It is also only effective when the number of topics being learned is limited. In a modern day environment where there is very little practice with multiple topics being learned within a short space of time, success is only achieved by the most able of maths students.

This practice of following procedures is often replaced by introducing conceptual understanding from a younger age. When conceptual understanding is rushed and also not backed up with procedural understanding then the benefits are limited. When time is taken to build from young the foundations of conceptual understanding of basic maths functions (with the procedures to reinforce this), then the foundations for learning more complex topics (such as fractions) will be built.

Many educators believe that if complex topics are learned from a young age, students will be at an advantage. In reality the basic foundations of maths will not be properly learned and they will almost certainly not understand the other more complex topics that follow. In contrast when the foundation is properly learned, the more complex topics will be easier to understand as the cognitive loading will be far less due to increased mental maths ability.

Learning basic arithmetic visually through a system of 5 and 10 frames:

• Increases the ability to do basic arithmetic mentally. By practicing the method, over time a student is able to do complex mental arithmetic.
• Cognitive loading is decreased, allowing working memory to work on the “formula”.
• By seeing maths visually, topics such as fractions, ratios, algebra, percentages, decimals become part of the long term memory.

When the foundation of maths, basic arithmetic, is strong and practiced in the correct way, then through regular practice on various questions from KS1 to KS4 the following are strengthened:

• Mental arithmetic
• Procedural understanding allowing the student to spend more mental resources memorising critical steps instead of basic maths
• Conceptual understanding through visual maths. Visual quantities aid understanding in functional maths like lengths, weights, volumes, size.
• Speed and accuracy increases resulting in completion of tests easily within the allocated time.