When you think of “numbers” – what comes to mind?
For most adults, what we picture for numbers is really what we call numerals, such as “7” or “28”. These numerals are symbols which we use to represent quantities, or numbers. Numerals themselves are in many ways random, and on their own, can be completely meaningless. It is only when they are connected to visual representations of quantity that they take on meaning.
Numerals are a code, and their value comes in connecting the code to the actual meaning. The process of decoding math, then, is the process of seeing the code and activating the corresponding visual representation of quantity in the brain. The process of encoding math is seeing a visual representation of quantity and translating it into the corresponding code, or symbols, that represent that quantity.
This process is similar to reading, where letters are a code that represent sounds, and in order to read, one must learn the code well enough to decode, or translate the combinations of letters into sounds and words. It is also similar to writing and spelling, where one must be able to encode, or translate the sounds of words into the written code.
In the United States, math has historically been taught with an emphasis on language (“seven”) and symbols (“7”). The problem with this approach is that it is fundamentally devoid of number sense. We now know from neuroscience that language, both oral and written (symbolic), occur primarily in the left hemisphere of the brain. The left hemisphere functions best with linear thinking and routine tasks.
Number sense, by contrast, occurs in the right hemisphere of the brain. The right hemisphere is for visual-spatial thinking, which is multi-dimensional and not linear. It includes the brain’s ability to “see” and activate pictures and hold them in the mind. We are all born with an innate number sense which exists more specifically in the intraparietal sulcus (IPS) of the right hemisphere.
For many years, it was believed that we were born as “blank slates”, and that all mathematical understanding had to be constructed, or learned, through life experiences. It was also believed that math was primarily a function of language, and that mathematical understanding was developed by learning language. We now know that these beliefs are not true. We, and all animals in fact, are born with what neuroscientists refer to as a number sense, or an innate “sense” for quantities, similar to an innate sense for taste or smell.
This innate sense of number falls into two categories: an exact and an approximate sense for numbers. The exact system is our innate ability to subitize, or to see a small quantity and instantly know what it is without counting. The approximate system is our ability to see larger quantities, and not know exactly how much each each one is, but to be able to identify which one is more and which one is less.
These systems form the foundation for mathematical learning. Research is clear that we are generally able to subitize up to about 4 items, and our ability to quickly subitize is increased when the objects are simple (e.g. dots), there is strong contrast between the foreground and background, and most importantly: the dots are arranged into standard or “canonical” arrangements.
What are these “canonical” arrangements? Think of what appears on dice or dominoes. There are multiple possible arrangements for each number, and some are easier to see than others. It is important to know that the brain can really only subitize or instantly “see” up to 3 items in a straight line. Once there are 4, the brain typically needs to do a quick count, or break the 4 into groups of 2 and 2. This challenge, however, is alleviated when the 4 dots are arranged into a square pattern, rather than a straight line. Think again of the dot arrangements on dice – there is never more than 3 dots in a straight line. Most young children can see these arrangements and quickly identify the corresponding quantity, without needing to stop and count.
If our goal is to support students in effectively and efficiently visualizing quantities in ways that they can manipulate, then it is essential to use arrangements that the brain can easily work with. This means using specific arrangements for each numeral, and over time teaching more than one arrangement so that students have options for the visual representation that works best for the particular purpose at hand.
Interestingly, these dot configurations are primarily being used with students who have dyscalculia. Dyscalculia occurs in about 5-7% of the population and is considered to be a core deficit in the innate number sense, or the ability to subitize. Since these students have difficulty subitizing quantities, it is absolutely essential to use the arrangements that are easiest for the brain to visualize. The process of teaching and practicing these arrangements develops the innate number sense and gives students a visual tool to use when making sense of number. Research has shown that effective instruction increases the activation in areas of the brain that were previously underactivated compared to their peers.
Often, however, it is thought that these dot arrangements are just for people with dyscalculia. Again, research is clear that the reason these arrangements are so effective is because it is how the brain best understands and visualizes number. Which means – these arrangements are not just best for people with dyscalculia – they are the best for everyone. All of our brains can more easily picture these dot arrangements, and therefore basing our math instruction on these patterns improves number sense and mathematical learning for everyone. So the type of instruction that is absolutely essential for those with dyscalculia, due to their innate difficulty with subitizing, is actually the best instruction for everyone.
What are the implications for instruction then? Early number concept, and then number concepts on an ongoing, progressive basis, should be taught through these dot arrangements and with specific types of manipulatives. Currently, instruction tends to emphasize counting and 1-to-1 correspondence, which encourages students to always count individual items, rather than subitize. This emphasis on counting has actually taught students to not trust their innate ability to subitize, and instead to use less efficient strategies for identifying number. A research study on kindergarten students showed that at the end of kindergarten, they were actually less effective at subitizing than they were at the beginning. Given that number books for young children generally tell them to “count” the quantity, rather than subitize, this outcome is not surprising.
Counting, fundamentally, is a sequence of words, which means it occurs primarily in the left hemisphere, and by itself is devoid of meaning. The concept of number only occurs when a visual representation of quantity is activated in the intraparietal sulcus of the right hemisphere. It is also important to recognize that counting individual dots or items generally does not create a visual of the whole in students’ minds, especially not for those with weak visual processing abilities. Students need to possess strong innate number sense, which requires strong innate visual processing abilities, to create these visuals on their own.
It wasn’t that long ago that “subitizing” was a foreign word to many teachers. Now, more and more educators are aware of the concept and are using subitizing tasks such as “quick images” in their classes. Often students are shown a group of dots and asked how they see it and how they are decomposing it into smaller “subitizable” groups, and then recomposing these smaller groups into the larger whole.
While this is an improvement from a lack of subitizing instruction, the true power of subitizing emerges when it is used as the foundation for teaching addition and subtraction, and later for multiplication and division. Currently, most addition and subtraction instruction still goes back to the ideas of counting all, then counting on or back as the basis for addition and subtraction.
The problem with this is that students are not using their innate abilities to subitize in order to develop number sense and efficient strategies. Instead, the process of repeatedly practicing counting is developing inefficient neural pathways that become the student’s automatic response and do not, on their own, automatically translate at some point to more efficient strategies. There are many middle and high school students who when adding or subtracting, still revert to counting on or counting back. Whatever the brain repeatedly practices becomes a neural pathway, and the more often it is traversed, the more automatic it becomes. If we want students to use a more efficient and number sense-based approach, they must be taught that way.
Take a moment to look at the dot patterns below as used by Chris Woodin at the Landmark School in Beverly, MA. Notice how the configurations for 1-5, then 6-10, use these canonical arrangements and build on each other.
There are many different versions of dot arrangements used by dyscalculia experts around the world, and each has its advantage depending on which skill and number sense concept a student is learning.
These are in far contrast to the 5- and 10-frames that are ubiquitously used in the United States. While these frames may have some benefits, the problem is that they are not “subitizable” on their own, as the brain cannot effectively picture 5 dots in a straight line. The frames also rely on a never-ending scaffold, which is the frame itself. Many students learn that there are 5 spaces in a row, so if it is full, the quantity is 5. But if they see 5 objects in a row in real life, does this mean they can immediately identify it as 5? Generally not. What they learned is the scaffold of the frame, not the quantity itself.
Many skeptical educators who have tentatively tried these canonical dot arrangements have quickly been surprised by their power. Students who previously struggled can suddenly identify quantities much more quickly. The challenge, then, becomes strengthening these visual representations and mapping them on to numerals as the symbolic representations in a meaningful, connected way.
This brings us back to the concept of decoding math. Now that we have taken number understanding out of the “code” of symbolic form and developed effective visual representations for number, we want to practice encoding these visuals into the symbolic form, so that later, students can effectively decode when they see a number represented in its symbolic form.
One more important thing to understand here is that the brain possesses an innate sense of quantity, and an innate ability for language, but it does not have an innate ability for symbolic form. What that means is that the brain is born with neurons intended to identify visual quantities and to learn language, but it is not born with neurons intended to identify symbols such as letters and numerals. Since there are no innate neurons for identifying symbols, the theory is that the brain repurposes neurons intended to recognize facial characteristics and uses them to identify symbols.
This is a process that must be developed in the brain and takes varying amounts of time for different students. It also means that if math is taught primarily through symbolic form, if students have not yet developed – or repurposed – these neurons, they will not be able to access instruction. Symbolic form is developed over time and becomes more automatic as students become older. In adults, this symbolic understanding is often automatic, and so it can create a misunderstanding that it is the same for children. Adults can benefit from recognizing that children do not yet have the automatic understanding and retrieval of number in symbolic form that they themselves do.
The process of learning number is the process of developing strong visual representations of quantity and mapping it to the number word and the number symbol, so that students can move easily between these three representations.
The process, then, is one of decoding – and encoding – math.
If you are interested in learning more, check out our Decoding Math online course at lordmath.com/online-learning.
The Decoding Math course is available in multiple options, including a 30-hour fully online, self-paced version; a shorter 15-hour Foundations version; and a 45-hour version that offers 15 hours of “live” Zoom instruction and the option for 3 graduate credits.