Division is a repeating issue that sneaks up on us more often that we would like it to. In fact, long division was the final straw that led me to the realisation that there is more than just difficulty in understanding mathematics in my daughter. Last week, the ghost of division came back in the form of reducing fractions to their simplest form. She just turned off when taught to reduce by mental division. (What was I thinking?) From the past two years, I found that wording my explanations to her was important as much as angling the concept to a form that she can understand. Thankfully there are many doors to most mathematics concepts. 1. Highest Multiple of Denominator I have skipped cross multiplication with Sofi as she seemed confused with it at the moment. Instead, we multiply the numerators and denominators to get the final, usually improper fraction. Let's say the answer is 18/5. - "What is the highest multiple of 5 I can put in 18?" - child answers 15 - I write down 15/5 and " What is the balance when I take 15 from 18 ?', child answers 3. - I write down 3/5. So now we have 18/5= 15/5+3/5 - "What is 15 over 5?" Child answers 3 (child is aware that the line separating the numerator and the denominator can also mean divide). I then write 3+3/5= 3 3/5
2. Highest Multiple of Denominator without addition - I ask the child to find the most number of the denominator we can find in the numerator just like before. - Instead of writing out into 2 sets of fractions that are added, I write the number of multiples we get straight away and write the balance beside it. - "The most number of fives we can find in 18 is 3." - I write down 18/5= 3 - "What is the balance after we take away 15?" Child answers 3. - I write down 3/5 beside 3 to get the final answer.
Hopefully next week we can cross divide. : )
Dyscalculia, An introduction
‘Developmental dyscalculia is a condition that affects the ability to acquire arithmetical skills. Dyscalculic learners may have difficulty understanding simple number concepts, lack an intuitive grasp of numbers, and have problems learning number facts and procedures. Even if they produce a correct answer or use a correct method, they may do so mechanically and without confidence.’ (DfES 2001)
Gifford (2005) found that dyscalculia affects about 5 per cent of school-age children and it may be caused by a brain deficit in the parietal lobe (Butterworth 1999). It may also co-exist with other learning difficulties such as dyslexia and ADHD.
At present, the closest available program in Singapore possibly helpful for them is the Math programme run by DAS for dyslexic children ‘with specific areas of difficulty that can affect their mathematical performance’. I have yet to find any program specific to dyscalculics, except one that claims the effectiveness of neurofeedback.
I have also not yet come across any studies on the extent of awareness of and support for dyscalculia in Singapore and how it affects dyscalculics in their learning of Mathematics and in their educational and personal progress as a whole.
Some people doubt that dyscalculia is even a learning difficulty and perceive children weak at math being so for not practicing and doing enough drills. While this may be true, there are many other reasons why someone does poorly in math and this includeds poor teaching and math anxiety. As explained earlier, dyscalculia is also a neurological condition as opposed to other possible reasons for poor math performance.
Patients usually experience a stark difference in their mathematical performance compared to their abilities in other areas. More symptoms of dyscalculia are:
An inability to subitise
An inability to estimate
A tendency not to notice patterns
A problem with all aspects of money
A marked delay in learning to read a clock to tell time . (Bird 2009)
As can be seen in the list, dyscalculia not only affect the academic achievements of a child but his ability to independently survive in life. They may shy away from making purchases on their own outside and depend on adults for it. It is challenging to keep up with schedules and deadlines when you face difficulties trying to make sense of a watch or clock.
Dyscalculics sometimes experience days as sequences and turn to using digital watches to help them work with other people who, unlike them, work the day around the clock for most. A casual survey of a dyscalculic online support group found that such inconveniences extend to adult life.
Low confidence and the feeling of social isolation due to their 'strangeness' is quite typical in their sharings on social media where they also lament the difficulty they face in trying to pursue their college education due to having to pass certain math modules.
To help identify a child at risk of dyscalculia, family and teachers may analyse gaps in their performance compared to average standards. Dewker (2004) suggested ‘that pupils who are at national curriculum level 1 at age 7 and level 3 at age 11 can be considered to have maths difficulties’.
This is comparable to DSM-IV-TR’S definition of Mathematics Disorder as ‘calculation or math reasoning ability falling significantly below the level expected on the basis of age and measured intelligence, assuming that the student has received adequate education.’ (Krasa & Shunkwiler 2009)
Growing research show that those who have dyscalculia alone ‘might have a specific deficit in the nonverbal ‘number sense’ system’. (Hulme and Snowling 2009). Advanced imaging techniques and studies of those with acquired dyscalculia due to brain injury could provide a better picture of how our brain process numbers. Abnormalities in the Intraparietal Sulcus (IPS) have been found in children with isolated math difficulties. These studies were however done on a small scale. (Krasa & Shunkwiler 2009).
Landerl, Bevan and Butterworth (2004) found that dyscalculics were ‘less accurate and slower at arithmetic operations.’ While Emerson & Babtie (2014) believed awareness of counting and pattern and the understanding of the structure of the number system, dyscalculics can be numerically competent.
The option to have more time during exams would thus be helpful to dyscalculics. However, overall, dyscalculics may be challenged by the fast rate at which Mathematics are taught in Singapore.
It is interesting to note that higher intelligence is related to ‘the timing of the cortical sheet’s thickening…Peak thickness, after which shrinkage to adult levels occurred, typically began between age seven and nine in normal or above-average children, but was delayed until age eleven in the highest-IQ children. ‘ (Aamodt & Wang 2009) t is interesting to explore, that given time, seemingly weak students may peak at a much later age, and at higher levels than their average peers. It is of concern, however, that this may be disrupted and go unnoticed with the need to go through a high-pressure critical exam around the same period.