### Recommended websites to help your child with maths -

www.thenumberrace.com (recommended for ages 4-8) is a free download which can help with maths in a number of ways -

**Strengthen the brain mechanisms of number processing**

What are these brain mechanisms?

Our brain can process numbers in several different ways: visually as digits (“3”), verbally as number words (“three” - written or spoken), and concretely as a quantity (♥♥♥) or a position along a mental number line. Each of these is a different way in which the brain represents numbers, and there are specific brain circuits for handling each representation.

Different arithmetic tasks rely on different representations of number in the brain. For example, the digit representation is used when reading numbers written as digits or when writing them. The verbal representation is used when talking or listening to someone saying numbers, and also for storing multiplication facts in our memory (“three times five is fifteen”). The quantity representation is used to decide which of two numbers is larger, or to quickly approximate quantities.

Our brain can also transform numbers from one representation to another. For example, when we read aloud the number 5, our brain must understand the digit, transform it to its verbal representation, and instruct our speech system to say aloud the word “five”. At the same time, the brain also transforms 5 into a quantity, and we get a sense of how large the number 5 is.

**Why is it important to strengthen these brain mechanisms?**

The ability to handle the different representations of numbers is the cornerstone of numeric literacy. For example, if we could not transform digits into number words quickly and efficiently, reading digits aloud would be difficult for us. Being able to transform numbers into the quantity representation is especially important, because we usually see or hear numbers as digits or words, but it is the quantity representation that makes us understand the “meaning” of a number and have a sense of how large it is.

Like in many other domains, practice makes perfect: if we practice the brain in transforming numbers among representations, it processes numbers faster and faster, with fewer errors, and with less effort.

Whereas many mathematical games focus just on calculation skills, The Number Race is one of only a few games that were specifically designed to teach and practice the various representations of numbers and the transformations between them, with a special focus on the quantity representation.

**How does The Number Race accomplish that?**

The game presents numbers in all representations: they are written as digits; they are narrated as spoken number words; and they are visualized as quantities, by displaying sets of objects. The player has to choose the greater of two numbers, starting with concrete sets, and gradually moving through spoken and written numbers, to written numbers only. Comparing numbers encourages processing quantity and transforming the numbers from their symbolic representation to the quantity representation. Once the player response, all three formats of numbers are reinforced, and the concrete sets are placed in one-to-one correspondence. Finally the tokens the player has won are moved to a racetrack, which demonstrates how numbers are mapped to a number-line-like structure.

Establish the mental number line

The Number Race teaches children to build up a mental number line by using a racetrack (similar to a game board) to map numbers to space.

**Why is it important to establish a mental number line?**

When we think about numbers, we often imagine them on a mental “number line”. When we do this, we are essentially using our brain’s ability to represent space to help us understand numbers. By picturing numbers on the number line, we can understand their relative size. The number line also helps us understand the meaning of addition and subtraction, and plan strategies for adding or subtracting past decade boundaries.

**How does The Number Race help build a mental number line?**

In The Number Race, players win tokens which they place onto a racetrack, or linear number board, with squares numbered up to 40, in order to move their player forward. The game teaches children to “count on” the number of tokens they have won from the square they are at, as they would in a board game. Research has shown that playing linear, numbered, board games this way has huge benefits for establishing the mental number line.

The Number Race teaches counting of numbers 1-40, including “counting on”.

- Why is it important to teach counting?
Counting is a fundamental skill which allows children to work with numbers, particularly to learn to add and subtract. Learning to count progresses through a series of stages. When children first learn to recite the counting sequence, they often do so inflexibly and without really understanding its purpose. Eventually they learn to flexibly count (e.g. stop and start at different numbers, or count by twos or tens), as well as to fully understand the purpose of counting.

Young children first learn arithmetic using counting strategies. These strategies start off slow and inefficient, and progressively become faster and more efficient. For instance when asked to give the answer to 2 + 5, children will first go through a phase of “counting all” with their fingers, in which they count “one, two” fingers, then “one, two, three, four, five” fingers, then “one, two, three, four, five, six, seven” fingers, to get the answer. Eventually they will learn to “count on” from the larger number, “six, seven”. However to be able to do this, they need to quickly identify the larger number, and to use their counting flexibly.

In the long run, children will memorize some frequent addition sums, e.g. 4+3=7, 2 + 8 = 10. But they will continue to use counting for subtraction, for more complicated sums, or as a back-up strategy if memory retrieval fails.

- How does The Number Race teach counting?
In The Number Race, players win tokens which they place onto a racetrack with squares numbered up to 40, in order to move their player forward. The game teaches children to “count on” the number of tokens they have won from the square they are at, as they would in a board game. Thus children have repeated opportunities to practice flexible counting with numbers 1 - 40. The Number Race reinforces the associated addition facts, e.g. if a player is on square 7, and wins 3, the game tells them “7 + 3 = 10” as their player is moved forward.

Teach and practice early addition and subtraction

The Number Race teaches and practices early addition and subtraction facts, focusing on concrete sets and the meaning of the facts.

- Why is it important to teach addition and subtraction facts?
When young children are asked to add two numbers, they initially use counting strategies: e.g. when presented with the exercise 2 + 5, they count on “six, seven” – or even count all the way from 1.

Eventually they learn the more efficient strategy of memory retrieval: adults just remember that 2 + 5 = 7. However this requires a large amount of practice to become fluent. At advanced levels, The Number Race provides this practice, and pushes children to recall facts faster and faster.

- How does The Number Race teach addition and subtraction facts?
The initial levels of The Number Race ask the player to compare sets of objects or numbers 1-10. In advanced levels, however, the player cannot see the number of objects in each set, and each set is annotated with an addition or subtraction exercise. To know which set is larger, the player must solve the addition or subtraction (or on the most difficult level, both!).

If the difference between the compared sets is very large, estimating the result will be enough. However the software will gradually push the player to compare sets closer and closer together, so that he/she will have to calculate the exact answer. The emphasis is on numbers 1-10, which are trained for fluency, but adding a single digit number to numbers 10 – 40 is also taught in the context of moving players along the board (e.g. “27 + 3 = 30”).

Encourage Fluency (Automatic Processing)

- What is fluency?
Our brain can operate in different modes. Some tasks that we do require attention - for example, playing chess. Other tasks are performed automatically, i.e., with no need to allocate attention to them - for example, walking.

Our attention resources are limited and the brain can allocate its full attention only on one task at a time. For example, most of us cannot handle two chess games at the same time. The situation is different when it comes to automatic tasks: we can usually perform several such tasks simultaneously - e.g., we can walk, eat, and tighten a loose button in our shirt, all at the same time.

Many operations may require a lot of attention when we learn them, and then gradually become automatic. For example, think about learning how to ride bicycle.

The Number Race aims to achieve fluency in quantity and simple arithmetic, so that calculation and number sense become effortless, and cease to place a heavy burden on our attention.

- Why is fluency important?
First of all, fluent processing is usually quicker. If your calculation is fluent, you get to the result more quickly.

Another importance of fluency lies in the fact that our attentional resources are limited. If a child can’t calculate automatically, he/she has to spend a lot of attention resources into the calculation process. Once arithmetic is fluent, the child can concentrate his/her full resources on other tasks - such as understanding a math or physics problem.

- How does The Number Race help reach fluency?
The Number Race adjusts its level of difficulty according to the player’s performance of the player, and maintains an average success rate of 75%.

The adaptive algorithm adjusts the numerical distance between the quantities to be compared and the length of the response deadline. The algorithm also adjusts the format in which the answers to choose from are shown (sets of objects, symbolic numbers, or addition or subtraction exercises).

**What is dyscalculia? **

Dyscalculia is a learning disability in mathematics. It can be a selective difficulty in math that is not necessarily accompanied by a general cognitive deficit. Dyscalculia usually results from an impairment in the brain circuits involved in mathematical cognition.

Several brain mechanisms are involved in mathematical cognition, and impairment in different brain mechanisms may result in different kinds of dyscalculia, with different symptoms. For example, if you have impairment in the brain region responsible for processing digits, you may find it difficult to read or write the number 35 but have no difficulty with the words "thirty five". If you have impairment in the brain region responsible for understanding quantities, you may be able to read and write numbers, but may have difficulties understanding the quantities they represent.

You can learn more about dyscalculia on The Number Race author Dr. Anna Wilson’s website

How can The Number Race help children with dyscalculia?

Think about physical injuries: if your leg got injured and you can't walk, you can still be taught how to walk again, and you would probably need to practice a lot. The physiotherapist may give you some exercises to practice specific muscles.

Similarly, when the brain is impaired in a brain region involved in mathematical cognition, it can still be taught to function better. Practice may partially restore the function of the impaired circuit, and alternative regions (nearby or in the opposite hemisphere) can also be trained. This is what The Number Race aims to do.

Often children with dyscalculia have had a lot of negative experiences with mathematics. Because it adapts to the user, The Number Race provides positive experiences with math and numbers. By building up a model of the child’s knowledge and skills, the software is able to present problems that children can do 75% of the time, but which still challenge them.

We are certainly not saying that any brain impairment can be cured, and that any child with dyscalculia can become fluent in math. Still, many children who experience difficulties in math can gain from training tools such as The Number Race and The Number Catcher.

The Number Race has been scientifically tested

www.multiplication.com (all ages) useful for helping learn multiplication tables.

www.woodlands-junior.kent.sch.uk/maths (all ages).This website has a range of topics, such as 'Telling the Time', 'Shape', 'Divisions', 'Fractions' etc.