The Number Catcher has several goals: (click to expand)
Step-by-step teaching of addition and subtraction
Addition exercises can be solved using various strategies. The Number Catcher takes its player, step by step, from basic calculation up to adult strategies.
What calculation strategies are there?
- Young children usually use counting: when presented with the exercise 8+5, they count “nine, ten, eleven, twelve, thirteen” - or even count all the way from 1.
- Older children use the complement to 10 strategy. To solve 8+5, they first figure out that 8+2=10, then add 3. This strategy means that the child is capable of retrieving the answers to several equations: 8+?=10, then 5-2=?, and finally 10+3=?.
- Adults usually use memory retrieval, which is the quickest and most efficient strategy: we just remember that 8+5=13. Many adults with good calculation skills usually memorize the number pairs that sum up to 10, and sometimes up to 20.
How does The Number Catcher teach these calculation strategies?
- In the first levels of The Number Catcher, the game teaches the counting strategy. When the child chooses a box, the objects are loaded on the truck one by one, and as each object is loaded on to the truck, the game says aloud the total number of objects on the truck. This part strengthens the counting routine and gives it a meaning: children learn that the recitation of count words increments quantities by a specific amount.
- Subsequent game levels practice the complement to 10 strategy. The game requires filling the vehicles by multiples of 10. If the carriage has 8 flowers on it and you want to get to 13 flowers, first you have to put 2 flowers to fill an exact decade, and only then can you put the remaining 3 flowers on the carriage. This makes the player practice the complement to 10 strategy.
- The adult strategy, memory retrieval, is trained in several ways. Addition exercises are presented to the player with their results, both visually and verbally (by the narrator), to encourage rote learning. On top of that, as you progress in the game, the boxes are presented in ways that require increasing degree of memorization: in the first game levels, the physical size of each box corresponds to the quantity it represents. In more advanced game levels, all boxes have the same size, so the player must solve the problems without relying on visual cues. In the most advanced levels, each box is labeled with addition and subtraction exercises (e.g., a box with 8 items may be labeled as “5+3” or as “12-4”). To figure out the number of objects in the boxes, the player has to solve several addition and subtraction exercises very quickly, which is possible only by memorizing the addition and subtraction facts.
- What calculation strategies are there?
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 Catcher is one of only few games that were specifically designed to practice not only calculation, but also the more fundamental level – the various representations of numbers and the transformations between them, with special focus on the quantity representation.
How does The Number Catcher 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, arranged in a number-line-like formation.
The score you get in the game is based on the time it took you to complete a level and on the number of errors you made. To achieve high scores in the game, the player must be able to handle numbers in all representations and to move from one representation to another quickly and correctly. This way, the game cements the links among the brain circuits that handle the various representations of numbers.
- What are these brain mechanisms?
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 Catcher aims to achieve fluency in 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 Catcher help reach fluency?
The Number Catcher includes a learning algorithm that automatically adapts to the child’s level of knowledge. As a result, the game’s level of difficulty is always demanding but not frustrating. This feature is highly motivating and encourages continuous practice and improvement.
- Adaptive level of difficulty: The levels of the game are organized in an increasing level of difficulty. Our software continuously monitors the player’s accuracy and response speed, and allows moving on to the next level only when performance is good enough.
- Adaptive speed: when you play faster, boxes fall into the screen faster, which makes the game more challenging. The higher speed allows the player to achieve higher scores, but also leaves him/her less and less time to calculate, and therefore encourages fluency.
The adaptive speed also serves another purpose: a fast-pace game maximizes the number of exercises encountered per minute and minimizes “cognitive idle time”, thereby increasing the learning effect.
- Multiple solutions: The game is designed to have more than one correct solution at most times, and yet some solutions are better than others. To achieve high scores, the player must learn to choose the best solution, and this requires quick evaluation and comparison of several strategies. The player is therefore encouraged to calculate faster and faster, using the optimal strategy.
- Dual-task technique: As the child reaches advanced game levels, he encounters new riddles: the spatial organization of the boxes in the container must be considered to prevent them from piling up; the color should be considered if the player wishes to get the color bonus; and there are special game elements such as clocks and bombs. These elements require some of the player's attention, and encourage the advanced player to practice number processing and calculation with less and less attention, i.e., increasingly automatically.
- What is fluency?
Focus on multi-digit numbers
What’s the difference between single-digit and two-digit numbers?
Processing multi-digit numbers is more complicated than processing single digits: to correctly process a multi-digit number, the brain needs not only to identify all the digits in the number, but also to process the relations between the digits. For example, “25” is not the same as “52”, although they contain the same digits.
Why is it important to exercise two-digit numbers?
The invention of multi-digit numbers is a major achievement in our culture. It rests on a very clever idea, which took centuries to emerge: the very same digit (say 1) can represent different quantities according to its spatial position (in 1, 10, or 100). This is the “base 10 principle” – a difficult idea for children to grasp. It takes some training for them to develop an intuitive understanding of base 10 notation. With training, the brain develops efficient brain circuits for recognizing multi-digit numbers.
What does The Number Catcher do?
The first levels present only single-digit numbers, so even a child with modest knowledge of numbers can start playing. Quite quickly, however, the game moves on to two-digit numbers (up to the number 39), which are the main focus of The Number Catcher.
If your child is just beginning with numbers and is still struggling with single digits, it is possible that he/she would find the game too demanding. If this is the case, you may consider starting with a game like The Number Race, which focuses on smaller numbers.
- What’s the difference between single-digit and two-digit numbers?
Help children with dyscalculia
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 difficulty. Current research suggests that dyscalculia may be the result of mild impairment in the brain areas involved in mathematical cognition (i.e., small differences in structure and/or function of these brain areas). However this impairment may be able to be remediated, especially at a young age.
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.
How can The Number Catcher 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 work 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 Catcher aims to do.
All children have an early intuition of numbers, but this intuititon has to be strengthened and linked with a sense of space and with knowledge of number words and digits. Research in cognitive neuroscience has shown that mathematical games can improve the number sense, especially in children who suffer from a learning disability in mathematics (dyscalculia).
We are certainly not saying that dyscalculia can be fully “cured”, and that all children with dyscalculia can become highly fluent in math. Still, many children who experience difficulties in math can gain from training tools such as The Number Catcher and The Number Race.
- What is dyscalculia?