A “granny summary” of the article by Ella Shalit and Dror Dotan
Exploring the linguistic complexity of third-grade numerical literacy
The world around us is full of numbers. The time, the supermarket bill, our credit card number, the recipe for the kids’ birthday cake, the balance on our bank account, the code that the bank texted us (which somehow never actually arrives) so we can view this balance, and so on. Most of us are probably unaware of this, but we need to read and say numbers every day, several times a day. The ability to handle these numbers – to read, write, say, and understand them – is a central skill in modern life, and it is needed for virtually any activity involving numbers. And that’s not all: as it turns out, reading and writing numbers are also crucial skills to learn how to do arithmetic.
The problem starts because reading and writing numbers is not just important, but also hard. Yes, you may think that it’s trivial (“what’s the big deal about reading 503?”), but it’s really not. Even adults make many errors when they read numbers, and children – well, children make A LOT of errors, and even after they have learned to read numbers, it could take several years to become fully fluent. For example, in one study, we showed that the ability to comprehend 2-digit numbers quickly and automatically is not fully developed even by the 4th grade. Yes, this was not a typo: we’re talking about numbers smaller than 100, and 10-year-old children. In short, number reading is a challenging skill to acquire, and for some children very challenging.
Why is it so hard? This question is still a mystery. But hopefully, after reading this granny summary, you may begin understanding why.
Teachers and education systems know that numerical literacy is important. For example, in several countries, understanding the number system is one of the two main topics in the elementary school math curriculum (the second topic is basic arithmetic). The pedagogical methods for teaching this topic vary from place to place, but the principle is similar. In our country, many children learn about “the house of numbers”. It’s a pedagogical technique for teaching the decimal number system – i.e. that there is a unit digit, a decade digit, a hundred digit, etc.; that the digit string is divided to triplets (and the verbal number accordingly); that 1 decade is worth 10 units; and so on.
Does this kind of teaching work? When children learn about “the house of numbers”, they clearly gain an understanding of the decimal number system, but does this make them better ‘number readers’? And if the answer is no (spoiler: the answer is no), what should we do so that they learn how to read numbers? These are the questions we set out to examine.
Many of our daily activities are things that we do automatically. But for most activities, there used to be a time when we could not do this automatically. Think about driving a car. If we’ve been driving for several years, we would get into the car and start driving, without thinking about it too much. But recall what you had to learn until you became so proficient. So many things to learn, so much practice, so many stages along the way, so many driving tests. In many domains, including in driving, we can roughly classify the knowledge needed into two: conceptual knowledge about the activity in question – e.g., knowing the car, knowing how to fuel it and to put air in the tires, being familiar with the road signs, understanding what the gas, break, and clutch pedals are (clutch?), and why it’s important to blink. Procedural knowledge is the “practical” knowledge of how precisely we act: which muscles we need to move so that the car starts going, for how long you should press the brake pedal to make the car stop, how to fit in traffic, etc. (here’s a nice YouTube video that demonstrates this with bicycle).
In mathematics too, there is conceptual knowledge and procedural knowledge. For example, in the exercise 23+64, the conceptual knowledge tells us why we must add decades with decades (20+60) and units with units (3+4) rather than mixing them. Procedural knowledge tells us the precise series of actions we should take – e.g., add the decade digits, then add the unit digits, and then merge the decade and unit sums.
What’s interesting is that procedural and conceptual knowledge exist even in mathematical operations that to us, as adults, seem to be automatic. Reading numbers – i.e., seeing 503 and saying ‘five hundred and three’ – is something we do almost without thinking, yet it requires some preliminary knowledge. The relevant conceptual knowledge is, for example, understanding the “house of numbers”. We know that the digit 3 means different things at the decade or unit positions, even if it is the same digit. We also know that the words ‘thirteen’ and ‘thirty’ indicate completely different quantities, although both are related to the digit 3. Procedural knowledge is the set of rules we must know to read the number. For example, that 3 is pronounced ‘three’ when in the unit position and ‘thirty’ when in the decade position; that you say the words according to a left-to-right order of digits, even in languages such as Hebrew, in which you read words from left to right (but not in languages such as German); that you don’t say the digit 0 when it is part of a multi-digit number; etc.
When schools teach the decimal number system, they teach conceptual knowledge, but in several countries (including Israel) they don’t teach the procedural knowledge of how to read numbers (more precisely: they do teach this for numbers up to 100 but not thereafter). Is the conceptual knowledge sufficient to learn number reading? If I learned about the house of numbers, and I am an ace in tasks such as saying that the decade digit of 8736 is 3, will I know how to read numbers?
To examine this, we tested 127 children in the 3rd and 4th grades who read aloud, in Hebrew, multi-digit numbers. We also had them perform school-like tasks that reflect conceptual knowledge of the decimal number system. We found that there was not much correlation between a child’s performance in the conceptual knowledge task and in the number reading task. In contrast, the procedural knowledge task did correlate with number reading ability. We concluded that to be able to read numbers, the critical thing is not conceptual knowledge but procedural knowledge. However, a minimal level of conceptual knowledge may be necessary: the children who performed very poorly at the conceptual knowledge task also read numbers poorly.
This conclusion would perhaps not surprise the drivers among us: we know that learning about cars is important, but to actually know how to drive your car, you must have procedural knowledge and sufficient practice. Similarly, to learn how to read numbers, you must learn the specific rules dictating how this is done, and then – practice. But this distinction, between procedural and conceptual knowledge, is a bit under the radar of education systems. Schools typically teach the conceptual knowledge but not the procedural knowledge. Perhaps this is why children make so many errors when they read numbers, and almost 10% of Hebrew-speaking adults have dysnumeria, a learning disorder that disrupts number reading.
Another common assumption made by education systems is that numbers should be taught from small to large. This order makes a lot of sense in mathematics. For example, calculations with smaller numbers are undoubtedly simpler. But is this assumption correct also for number reading? As it turns out, not always. To examine this, we compared the children’s reading of 4-digit numbers versus 5-digit numbers. This is an interesting comparison because in Hebrew, unlike English, the rules for verbalizing 5-digit numbers are not “an extension” of the 4-digit rules; they are different. We observed that some children could read the 4-digit numbers and had difficulties with 5-digit numbers, which is predictable, but critically, about the same amount of children showed an opposite and surprising pattern: they could read the 5-digit numbers and had difficulties with 4-digit numbers. This finding is even more surprising if we take into account that these children, who were in the 3rd grade, have already learned at school the 4-digit numbers but not the 5-digit numbers.
So what happened? We concluded that there are separate procedural rules for reading 4-digit and 5-digit numbers, you must learn each rule separately, and it is possible to learn either of the two first. In other words, it’s not always the size (of the number) that matters. Some children learned one rule, other children learned the other rule, and because the school did not teach the rules explicitly, the specific rule each child learned was determined by other factors.
The bottom line is that not only do we need both conceptual and procedural knowledge, but the procedural knowledge is super specific: in Hebrew, you need to learn how to say 4-digit numbers, and – as a separate rule – how to say 5-digit numbers. Similarly, you must learn each irregular number form separately.
So we see that like many other fundamental skills, number reading is a skill that requires several kinds of learning, as well as practice. We hope that this study will help improve how this important skill is learned in elementary schools and is incorporated in the curriculum.
Interested to know more? The full article is here.