A Framework for Diagnosing Dyscalculia

This paper outlines the development of mathematical items which evolves from making sense of mathematics through perception, operation, and reason. There are three main constructs in this framework, and it involves seven tests namely simple reaction time, short term memory, number sense, matching Items, dot enumeration, number comparison and arithmetic. This study is concerned with how a group of students struggle with basic numerical skills which may be caused by dyscalculia. One of the most important contributions of this study, in addition to its wider theoretical and practical applications, is that it has come out with a set of standardized instruments which is called the Malaysian Dyscalculia Instrument (MDI) for primary school students. This tool could be further used widely in the fields of psychology and education, especially special education and particularly research studies related to education.


INTRODUCTION
This study described the development of a framework for diagnosing dyscalculia. It is based on a fundamental theory of how humans learn to think mathematically from early childhood to adult including mathematicians. In general, this framework is developed based on the concept of numerosity (Butterworth,2002), theories of cognitive development in mathematical thinking (Tall, 1995;Tall, 2007) and symptoms and causes proposed by several researchers such as Murphy (2006), Geary (2006) , Gersten et. al., (2008), andShalev &Von Aster (2007).
Consequently, the researchers concluded four constructs of MDI as shown in Table 1.

II. MATERIALS AND METHODS
There are not many studies about diagnosing Dyscalculia.
A Dyscalculia screener developed by Butterworth (2003) is widely used to diagnose and identify students who suffer from Dyscalculia. The development of the Dyscalculia screener is based on the concept of numerosity proposed by Butterworth (2005 Each of the constructs has a specific role and the order of these constructs is based on the theory of cognitive development in mathematical thinking proposed by Tall (2007), that is to classify the learners into the appropriate level of basic numeracy skills.
Simple Reaction Time is a test to measure psychomotor response time. Recorded response time will take into account to identify the actual cognitive processing time (Butterworth, 2003). Figure 1 displays the chronology of a screenshot for the items in simple reaction time. At first, the screen will blank for a few seconds and then black dots will appear on any area ofthe screen. Once the black dots appeared, the pupils have to press a particular button as soon as possible to obtain the best response time. This process was repeated for ten times for both left and right hands respectively. The response times for the next six tests were adjusted to take this measure into account.

Figure 1. Simple Reaction Time
It is commonly believed that most individuals can hold between five and nine items of information in the short-term memory span at one time (Westwood, 2004).
Working memory is the short-term memory (Banddeley, 2002) and the concept of working memory evolved from earlier concepts of Short-Term Memory (Lervag & Hulme, 2013) Figure 2 shows the chronology of items in short term memory. Students are asked to memorise the pictures which will appear then disappear in a few seconds on the left and then to the right of the screen in a row (shown in Figure 2).After that, they have to give an answer to indicate whether the right screen or the left screen has more black dots.

Figure 2. Short Term Memory
The idea of numerosity (Butterworth, 1999) involves familiar consequences such as two sets of things might have the same numerosity as the other, or a smaller or larger numerosity. This innate ability that we are born with a sense of numbers is called Number Sense (Butterworth, 2002). Students with learning disabilities seem to have problems in many aspects of basic number sense, such as the difficulty in understanding quantity (Gersten et al., 2008). They are slower in completing very simple quantity tasks.   better skills in counting numerals and using symbols (Butterworth, 2002). Learning the basic counting sequence, "one, two, three and four ..." is not difficult and almost all children including dyscalculic students can learn this (Geary, 2006). However, it is not only about the ASM Science Journal, Volume 13, 2020 3 sequence but also involves the ability to assign each counted object and represents the quantity of items in the counted set. Figures 5 shows an item of Dot Enumeration in this study. This is a very straightforward task and researchers expect all the pupils will be able to get most of the correct answers. The critical factor is how long they need to get the correct answer for each problem.

Figure 5. Dot Enumeration
According to Geary (2006), dyscalculic students often do not know basic number names (e.g. "7" = "Seven"), and they have difficulties in discriminating which number is larger or smaller. They have difficulties in completing simple quantity tasks such as comparing two numbers (e.g. Which is bigger, 6 or 8?). Hence, Number Comparison ( Figure 6) becomes a crucial construct to test the brain areas (affected by dyscalculia) which are specialised to deal with the idea of quantity. This is known as sense of ordered numericities (Butterworth, 2002).
This task asks pupils to select the larger of two numbers which requires a fluent understanding of numerals. for 3+2 (Geary, 2006). They have a great difficulty in memorising simple addition, subtraction, and multiplication facts. Thus, the Arithmetic test is required (Butterworth, 2003) to measures the basic arithmetic skills. This task consists of number sentences that involve addition and subtraction only. The number sentence is presented on the screen with an answer (e.g. 3+2=4 in Figure 7). The pupil has to judge as quickly as possible whether 4 is the correct answer or not.

III. RESULTS AND DISCUSSIONS
The numerosity concept (Butterworth, 1999) Generally, this framework was developed based on the sense of numerosity (Butterworth, 2003), coupled with the theory of making sense of mathematics (Chin & Tall, 2012).
The framework basically consists of two parts namely perceptions of objects and actions on objects (operation).
Perception refers to humans learn mathematics through their perception or senses. Meanwhile, operation refers to physical actions, such as counting, which involves the compression of actions to thinkable concepts (see Figure   8).

V.
1st Conference of the European Philosophy of Science