Research is a creative process. You never know where you will find inspiration. The inspiration for my dissertation study came from an article in the New Yorker by John Colapinto called The Interpreter: Has a Remote Amazonian Tribe Upended Our Understanding of Language?. The article is about linguist Dan Everett’s controversial efforts to understand the language structures of the Piraha tribe. I read the article a year before I started my doctoral studies. Everett challenges Chomski’s  language acquisition theory, claiming that the Piraha language does not include a recursive structure. Everett also claims that the language reflects the Piraha culture and provides insight into why they do not use number words and counting principles. Number systems are based on recursion (N + 1), so it follows that a language without recursive structure might also lack a counting system.

One prediction that this makes in Pirahã follows from the suggestions of people who worked on number theory and the nature of number in human speech: that counting systems—numerical systems—are based on recursion, and that this recursion follows from recursion in the language. This predicts in turn that if a language lacked recursion, then that language would also lack a number system and a counting system. I’ve claimed for years that the Pirahã don’t have numbers or accounting, and this has been verified in two recent sets of experiments, one of which was published in Sciencethree years ago by Peter Gordon, arguing that the Pirahã don’t count, and then a new set of experiments which was just carried out in January by people from Brain and Cognitive Sciences at MIT, which establishes pretty clearly that the Pirahã have no numbers, and, again, that they don’t count at all. (Everett, 2007)

The argument is rooted in Whorf’s hypothesis, which is the question of whether language drives culture and cognition, or culture and cognition drive language. If someone has a strong Whorfian orientation, they believe we cannot initiate thought about a construct unless we have the language for it.

Initially, I was interested in Everett’s claims about the relationship between numbers and language but eventually I became interested in the ways in which people conceptualize numbers prior to the development of number words and formal operations. Specifically, I wanted to know whether discrete visual models could change the development of number representation.

So What?

About 5% of the population experiences dyscalculia, or the inability to represent and manipulate numbers. Currently, there are no known research-based interventions for dyscalculia. If it is possible to intervene and initiate appropriate number activities before children learn number words and Arabic numerals and before they experience frustration and failure, this would be a remarkable breakthrough.

About Number Sense

Number sense is an intuitive understanding of quantity and the relationships between quantities. My dissertation research focused on the impact of visual models on our ability to represent integers.

The study was designed to explore the cognitive foundations of number sense. Specifically, I wanted to know what types of visual models enhance the development of number sense in young children. This is important because before children are exposed to numerals, they represent quantity using a logarithmic scale. On a logarithmic scale, the distance between 2 and 4 is represented as the same as the distance between 4 and 8.


Eventually, children have to convert their quantity representations into a standard linear scale (where the distance between 4 and 8 is twice that of 2 and 4) so they can handle numerals and perform computations.  When children fail to make the “logarithmic-to-linear shift”, they have difficulty with math.

Numbers are difficult because of the sheer number of concepts they represent:

  • Numbers can name things, as in player number 52 or bus number 7.
  • Numbers are infinite. We can keep adding one more, and we will always have a new number. We can talk about numbers we can’t visualize or represent, such as a billion, or a googol.
  • Numbers are abstract. I can have 3 balls, 3 balloons, or 3 wishes.
  • Numbers take their meaning from their position. Eight is a big number on a 1 – 10 scale and a small number on a 1 – 100 scale.

Number sense develops over a long period of time. In fact, children are still developing and refining their number sense well into their teens. Some experts argue children begin to fine tune their number sense as they learn to count. However, our quantity and language centers operate independently of one another in the brain. Children can perform simple operations on sets of objects before they learn to speak! It should be possible to stimulate positive development in number sense before children learn the formal counting procedures. I believe the right kinds of experiences at a young age can make a big difference in how children approach math in school.

Dissertation Proposal


Pre-symbolic representations of number are conceptualized on an approximate, logarithmic scale. The precision of this “number sense” is strongly associated with math achievement. The process by which number sense is mapped to a linear, symbolic numberline is unknown.  The goal of this study was to investigate whether structured visual modeling supports log-to-linear mapping and generalization to untrained non-symbolic number tasks in a group of preschool children. Participants were recruited from a college lab school in the northeastern United States. For 8 weeks, children in two treatment groups played games with manipulatives designed to support their visualization of quantity, relative magnitude, and continuous magnitude[1]. Students in the treatment groups were compared to controls on an independent criterion measure of number sense, the linearity of number representations, and the capacity to generalize. Children in the magnitude-only group improved the linearity of their constructions on trained scales. Children in the continuous magnitude group improved number sense as measured by an independent quantity comparison task. Children in the continuous magnitude group also improved the linearity of their numberline estimates and generalized this skill to an untrained scale of reference. Implications for the use of mental models of continuous extent in math instruction and number sense development at the presymbolic level were discussed.

Keywords: Approximate Number System, ANS acuity, ANS precision, mental number line, number sense, numerical cognition


[1] Continuous magnitude is the same thing as continuous extent. These terms refer to the total surface area of a set of objects. These terms also refer to the length of a set when that set is displayed in units along a number line. Continuous variables do not have a discrete beginning or end. Continuous variables are not countable.

The Dr. Seuss Version of Sara’s Dissertation


Small numbers—you don’t count them, but your brain just knows they’re there[1] Fish

Serendipitous when foraging, or when a tree is bare.

Big numbers–now they are hard—an educated guess,

Are guided by the ratio of one-to-one (or less)[2]

When given sets too fast to count, of unknown quantity,

You make a guess that’s pretty close with margins one to three [3]

The closer sets align in size the harder it becomes,

A guy named Weber tested it with weights and sounds and sums[4]

Animals can do it, too—marsupials and rats.

But you can’t teach them how to count (no actuary cats!)[5]

Now that’s no problem in the wild, with fight or flight decisions,

But in math class it isn’t good- there’s no room for revisions.

To learn to count is arduous, a really awful skill

But quantity comparison is quite run of the mill[6]

They measure this with groups of stuff and ask you to compare

The failure rate on such a task –exceptionally rare

But when you do it indicates a lifetime full of woe

To add, subtract, or multiply—it’s painful and it’s slow[7]

Add number words and it’s a chore to map them out and see

That number words are magnitudes—the relativity![8]

Now chimpanzees can learn some nouns (that’s if you give them luncheon)

But parrots trained for twenty years forget successor function[9]

Humans don’t because they talk with generative flair,

They string the words together like a verb and object pair

But when they learn the count words most confusing is the goal,

Of magnitude to linear– a round peg in square hole[10]

If we plotted all your guesses they would clump around a mean

Precision dropping as you go in steps from where you’ve seen[11]

You need a yardstick that will track your mental number line

If you can see it all works out and counting is just fine

But take away your counting list and tell your brain to guess,

You’d hit the target often (sometimes more, and sometimes less)

The yardstick for your number line, compressed and analog

Requires a mental map of space as when Mark Devlin’s dog[12]

Triangulates trajectories to hunt and fetch his ball

The shortest path he takes to go but gets it when you call

Are language maps the thing we need to give exact account

Of all the stuff we want to track to get the right amount?

Or is it space or something like precision in your sense

Of numbers and their counterparts, or am I being dense?[13]

[1] Subitizing
[1] The Approximate Number System is guided by Weber’s Law
[1] Set-based quantification, or the small number (subitizing) system, has an upper bound of 3-4
[1] Weber’s Law is cross-perceptual
[1] Human and animal number capacities are separated by language (ie., a 1, 2, 3… count scheme) only
[1] ANS precision is measured by set comparison tasks
[1] Poor ANS precision is linked to poor math achievement and dyscalculia (Halberda & Feigenson, 2008)
[1] Numbers take their meaning from relative magnitude or position in space (ordinality)
[1] The idea that each number is equal to itself plus all those that came before it; and the concept that the next number in the count sequence is always N + 1
[1] Logarithmic to linear mapping must occur before people can use integers
[1] Numerosity-selective neurons have “noise” in their activations around a target number. Numbers that are close together are more difficult to discriminate
[1] Devlin discusses dogs’ ability to automatically triangulate the shortest route when retrieving a ball
[1] It is unknown whether integers are mapped via language or space models

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