A Beginner’s Guide to Understanding Dyscalculia
Mathematics is a complicated and layered academic area that requires a child to grasp the basics before applying them to concepts and then build upon those concepts as they progress through school. Not only must a child build on the concepts, but there is an assumption that speed and efficiency are a direct result of one’s understanding of those fundamentals. Mathematics also requires strong spatial and reasoning skills. As a result, a child who struggles in math, may also have difficulty in other areas of life that require spatial awareness or reasoning. Examples include: understanding maps or directions; estimating distance, height, or weight; managing time or money; and using logic to solve problems.
In this blog, we will dive into understanding dyscalculia, a challenge in understanding numbers and mathematical constructs. This impairs children’s ability to apply even basic math concepts, impacting not only their academic success, but various general life skills.
What is Dyscalculia?
As mentioned in my previous blog on Understand Learning Disorders, students with difficulties in mathematics may be diagnosed with a Specific Learning Disorder with Impairments in Mathematics. The term dyscalculia is often attributed to children that have deficits in all four areas of mathematics:
Number sense
Memorization of arithmetic facts
Accurate or fluent calculation
Accurate math reasoning
Let’s look at each of these areas separately.
Number Sense
When we read, we understand that a letter is a symbol of a sound and a word carries meaning in that it represents an object or concept. We use symbols as a way to communicate constructs. This is also the case with numbers. The symbol “5,” when written or spoken, carries meaning in that it defines a quantity. Children with dyscalculia frequently struggle to comprehend that a number represents an actual quantity. To them, it is simply two straight lines connected to a curve. If a child has difficulty grasping that there is relationship between a symbol and a quantity, they will also struggle to understand how each number and quantity relate to each other. In other words, there is no concept of where a number belongs on a number line.
Let’s stick with the number 5 for our example. If a child is confused on how 5 represents a quantity, then that same child would certainly not understand that 5 is larger than 3 but smaller than 8. However, that same child could likely look at three piles of apples and point to the smallest pile (3 apples) and distinguish that from the largest pile (8 apples). It’s important to note that the words I used to describe the piles weren’t in any numerical form. By providing a different construct for the child to understand, the child would be able to connect to the material. This means you may see a difference if you ask your child to point to the pile with 3 apples instead of the smallest pile.
This disconnect and inability to “speak the language” of mathematics causes great confusion and significantly hampers a person’s ability to grasp even the most basic mathematical constructs. For children with dyscalculia, understanding simple math concepts (i.e., addition, subtraction, multiplication, or division) is very challenging. Therefore, comprehending higher-level constructs (i.e., performing multi-step problems or inserting letters as placeholders for numbers) feels nearly impossible. As a result, these children quickly get left behind.
Memorization of Arithmetic Facts
We’ve already established that children with dyscalculia have great difficulty learning to speak the mathematical language and that this impacts their ability to grasp the foundational concepts of addition, subtraction, multiplication, and division. As a result, there is a lot of thought and brain power put into basic calculations.
When faced with a math problem, 3 + 3 = ?, the child may take several moments to provide a correct answer. Similar to children with dyslexia who struggle to read sight words and instead have to take the longer path of decoding words every time, children with dyscalculia must take a longer path to the answer. This may include counting on their fingers or silently in their head. They are having to remind themselves what each of those symbols mean, how they are connected, and what to do with them. Due to the lengthy process, many children are also unsure about their final response and may repeat the process several times to ensure they are providing the correct answer.
Therefore, when this child is attempting to memorize a math fact, to simply see 2 + 6 and know that the answer is 8, they typically find this extremely challenging. They are being asked to memorize something that has no meaning to them. It is as if I were to ask you to memorize a word in another language without knowing what it actually means and being able to recall it upon demand. Without the mathematical language, without the meaning, the brain struggles to file the information. Where do you store something when you don’t know what it is or where it belongs? For me, those items usually end up in the trash or become part of the clutter on my desk.
Memorization of math facts leads to automaticity. Automaticity is when things occur mechanically, without conscious thought. Like how you may be reading these words without actually thinking of what each letter sounds like individually within the word; no need to sound it out and blend it together. This allows you to focus on the comprehension of the sentence and the information in this blog as a whole. When our conscious mind doesn’t have to work on these foundational pieces, it is free to put all of its energy towards the higher-level constructs. This is why there is so much focus put on memorizing math facts in elementary school. It prepares students for the more complex math classes to come in middle school and high school.
Accurate or Fluent Calculation
Without automaticity of those basic math facts, the brain of a child with dyscalculia is working extremely hard on the very first steps of a problem and typically exhausts itself before making it through the whole problem. Automaticity also allows you to know without a doubt that the answer to 3 + 4 = 7. This makes adding larger numbers and completing more than one operation within one problem much easier. In contrast, those with dyscalculia are constantly having to solve the basic components of the problem (6 + 3 = 9) before starting the whole process over with the next step in the problem and are never quite sure that they did the calculation correct. Since mathematics builds upon itself, if a mistake is made on the first calculation, the rest of the problem will inherently be wrong even if no further mistakes are made; the wrong turn is costly and results in arriving at a different destination.
As a result, those with dyscalculia frequently struggle to consistently calculate answers correctly and when they do, it takes a lengthy time. You may find that your child is only able to get through 3 of the 10 problems on their test, or that they are spending over an hour on math homework that is designed to be completed in 15 minutes. Other children may accurately calculate 5 + 7 in one instance and then incorrectly two problems later.
Accurate Math Reasoning
Math reasoning is using the knowledge you have acquired of math constructs and applying them to new and unique mathematical problems. In reading, you develop phonics skills and learn phonics rules so that you can apply them to new words you come across. The same thing occurs in math; you learn basic foundational concepts, mathematical rules, and formulas that allow you to interact with numbers and solve the many problems that can be constructed. Children with dyscalculia often struggle in developing these foundational skills, so they lack the framework to know how to approach a new and unique problem. Often, rather than learning a construct, they simply learned how to solve that one particular math problem.
For example, a student may struggle when applying a construct or formula to a new problem. They may understand the example as the teacher walked through the steps, but are completely lost when faced with a problem with new numbers. These students find themselves in the starting block with each problem they face, even though it is the same mathematical concept that should be applied.
Solve.
3y + 2 = 11
The student has to remember that: y is a placeholder for a number, they should solve for y, 3y indicates multiplication, subtraction is the reverse of addition, division is the reverse of multiplication, and to follow the order of operations. Whew! That’s a lot to actively think about. Now the next problem looks like this:
5x – 4 = 6
To someone without dyscalculia, they might begin to approach the problem similarly to the one before without really noticing the differences. For the student with dyscalculia, their mind is reeling…“Why is there an x? Doesn’t x mean multiply? Is that subtraction or a negative? Where do I start? Why is this so hard?”
The educational belief is that repetition of applying the mathematical concept will result in the child learning the concept (similar to writing a spelling word 10 times so that it “sticks”). However, this isn’t the case for students with dyscalculia. In contrast, completing 20 homework problems often causes frustration and takes a lengthy time. The student knows that the problems are all similar, but just can’t see how they are similar. The problems don’t become easier the more that they are practiced.
Now let’s see how things can get even further complicated. After a few different problems like the ones listed above, the problems take a shift:
13 = 41 – 4p
Wait….WHAT?!?! There is nothing in this problem that looks similar to the ones above it because all of the pieces have been moved around to unrecognizable positions. That’s the reaction for a child with dyscalculia. Without the framework to understand how the numbers and operations are connected, the child simply has no understanding of how to approach this problem. They don’t have the mathematical map that guides them through the problem.
Another challenge comes from choosing which mathematical formula should be applied to the problem at hand. Do you remember the formula for calculating the area of a square? What about for a circle? A triangle? Each one has a specific formula to calculate area. Just as it is challenging for a student with dyscalculia to memorize math facts, it’s also challenging to memorize formulas and recall them on homework or tests. Let’s not forget that the student must also remember which formulas go with which problems and what the letters represent.
Additional Challenges (setting up math word problems, copying problems accurately)
In addition to the challenges within these four areas, children with dyscalculia frequently struggle with word problems. In word problems, it is very rare to have something stated straight forward: “Sally wants to know…what is the answer to 3 plus 7?”
Instead, other words are used to imply math constructs. “Sally has 3 toys and her mom bought 7 more. How many toys are there in total?” For you to correctly answer this, you have to understand that starting with 3, then buying 7 more, and asking for a total, are all indicators that you should add the two numbers for the answer. A child with dyscalculia reads that problem and thinks: there is a 3 and a 7….what do I do with them…
But don’t count this kid out just yet! There is a pretty good chance this child would correctly guess to add the two numbers. This is likely to happen for two reasons:
Addition is often the easiest concept to grasp and is focused on heavily in the early grades.
The 3 came before the 7 and often times children with dyscalculia keep the numbers in the same order as they are found in the problem. Manipulating them around makes the problem much more challenging and many children learn that you subtract the smaller number from the larger number (for a minute let’s forget about using negative numbers).
Although guessing on word problems can be a good strategy, it quickly becomes ineffective as word problems become more complicated. In some word problems, you have to complete multiple steps to find an answer. In others, there are more numbers than are needed (Sally has 5 books, John has 8 books, and Carla has 2 books. How many more books does Sally have than Carla?). And in others, you must remember and apply formulas to find the answer. Pretty quickly you can see how things would get derailed and giving up feels much simpler and less stressful.
Due to their weak spatial abilities, students struggling with dyscalculia also have difficulty copying problems correctly. Although they may accurately read 53 off the board, they may write 35. Or they may write something that sounds similar: 217 may become 270 (seventeen and seventy are close in sound). They may also struggle to attend to the mathematical signs, especially with negative numbers. Very young students may also struggle with reversing numbers (i.e., writing a single-digit number backwards).
Characteristics of Dyscalculia
Below are typical features of dyscalculia seen in students. However, some of these characteristics are more common in certain ages and grade levels than others.
Difficulty memorizing basic math facts
Inability to correctly place a number on a number line
Has difficulty counting
Counts on fingers past a developmentally appropriate age
Frequently uses tally marks on scratch paper to complete basic calculations
Struggles to understand place value
Has difficulty aligning numbers based on place value (i.e., putting numbers in the wrong column)
Has difficulty understanding monetary value
Difficulty determining tip or understanding sales percentages
Has difficulty estimating distance
Struggles to tell time
Has difficulty creating a budget or maintaining a schedule
Becomes anxious or frustrated easily on tasks involving math
Transposes numbers (in very young children may also include number reversals)
Struggles greatly with mental math
Inconsistently applies formulas and math concepts (may correctly complete it on one assignment but be unable to complete it the next day)
Difficulty with motor sequencing (i.e., dance steps, athletic plays/routes)
Difficulty with direction and remembering spatial layouts
Managing Dyscalculia
Similar to managing dyslexia, learning math using a multi-sensory approach tends to be the most effective way in helping children with dyscalculia gain a mastery of basic math concepts. By helping the student connect with the material in as many ways as possible (sight, sound, touch, movement), the student has a greater chance of understanding the meaning of the mathematical symbols and concepts. We must build up their mathematical language so that they can better interact with numbers.
Tutoring can also provide assistance. Frequently, children with dyscalculia need additional teaching, repetition, and individualized attention to ensure that they understand the foundations and rules of mathematics. This need will likely continue throughout the child’s schooling as math classes become increasingly challenging and new concepts continue to be introduced. Providing students with alternative memory strategies to ensure that they retain math formulas and rules is very helpful. They will also benefit from learning strategies to limit their mistakes and check their work, particularly on exams.
What students will benefit from the most are accommodations to ease the burdens experienced within the classroom, particularly in middle and high school math classes. The most useful accommodations include: extended time, access to a formula sheet, and allowing the use of a calculator.
Examples of common accommodations:
Extended time
Permission to use a calculator in all math classes
Permission to take basic level math courses to fulfill requirements
Requiring the use of scratch paper on math assignments to minimize careless errors
Permission to use a formula sheet on tests
Minimizing the amount of information required to copy off the board
One-on-one instruction
Testing in a distraction-reduced setting
Providing opportunities for study time and homework completion at school
Opportunities for test corrections to ensure child understands errors
Access to study guides and tutorials