Feb 24 2022

Number Reasoning and Language

Think of the number 23. What just happened in your brain? You have some sense of how large a number 23 is. You are reading, so probably the first representation was the written form, and then you may have said “twenty-three” to yourself. You probably didn’t imagine exactly 23 objects laid out, but could have gotten there if you thought long enough. Perhaps you chunked it in your mind – imagining two rows of 10 followed by a row of three. You may have also thought of three weeks plus an extra weekend, or perhaps, staying on the verbal side, phrases like “23 skidoo”.

Our brains do not always work through complete abstraction or direct representation. As the primate brain evolved more and more complexity and ability, it acquired new functions partly through workarounds. For example, there is the phenomenon of “embodied cognition.” Mentally we start with concrete physical concepts and then use them to get to more abstract concepts. The notion, for example, that your boss is “above” you in the hierarchy derives from the more concrete concept of literally being physically above someone else. A “big” idea is not physically big, but it is metaphorically big. An argument may be weak or strong, and someone may be bright or dim. We hold onto physical concepts in order to anchor more abstract concepts.

Neuroscientists have also been studying how our brains handle numbers, comparing human numerical ability to other animals. One key finding is that our brains handle very small numbers differently than larger numbers. Many animals, including humans, have direct hard-wired representation of small numbers in their brains. For humans that number is about 6. We can easily extrapolate this hard-wired representation of numbers to about a dozen. Beyond that our brains need abstract concepts and language in order to reason about numbers, because larger amounts are not directly represented in our brains. Our brains seem to count as – 1, 2, 3, 4, 5, 6, more than 6. We have a sense of relative amounts, that one number of objects is larger than another, but not exact amounts.

So how, then, do we learn complex math and how to deal with arbitrarily large numbers? That is a focus of research, and the focus of a recent study that sheds some light on the question. The researchers looked at members of the Tsimane’ of Bolivia, a population of about 13,000 people. They have a farming and foraging society and don’t have a pressing need for the mathematics of large numbers. However, they do have modern education and learn math. A prior study showed that they learn math as quickly as children from industrialized societies, although they do tend to start at an older age. Their language contains words for numbers only up to 100, and after that they have borrowed words from Spanish. However, there is also a lot of variability among individuals in terms of how much education they have received. Some individuals can count only to 10, others to 20, and still others to 40 or more.

This population, therefore, provides the opportunity to study the effects of language on mathematical ability. Subjects were given the task of matching numbers of objects, what the researchers call “orthogonal matching”. Objects were placed in a horizontal row. Subjects had to place the exact same number of objects in a vertical row, so that they could not simply match the objects one-for-one. The researchers found that the ability to perform this task was directly related to the ability to verbally count to a specific number. So if the subject could count to 10, they started making mistakes in the task at around 8-9 objects. But if they could count to 15, they started making mistakes around 13-14 objects. Essentially once they got to the limits of their counting they shifting to number estimation and started making mistakes. The authors conclude:

The results show that these behavioral switch points were bounded by participants’ verbal count ranges; their representations of exact cardinalities were limited to the number words they knew. Beyond that range, they resorted to numerical approximation. These results resolve competing accounts of previous findings and provide unambiguous evidence that large exact number concepts are enabled by language.

This may seem obvious in retrospect, but nothing is really obvious when it comes to how our brains function. It is useful to confirm specific details of the strategies that our brains use when completing a task. We may not always be consciously aware of every strategy – some are conscious constructs, but many are subconscious. Confirming the details of how our brains process information and complete tasks is helpful to educational strategies. It may also inform computer science and provide inspiration for computational algorithms. Also, it can help design user interfaces. This is a critical technology, but one that is more easy to notice when it fails than when it succeeds.

Understanding the inherent strategies we use to access and process information can inform how best to optimally interface with information, to minimize cognitive load and maximize efficiency. When done well the experience can be seamless and feel intuitive. When done poorly it can be frustrating and error-prone. The difference often resides in understanding exactly how our brains work.


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