How the Abstract Becomes Concrete

Irrational Numbers Are Anchored on Natural Numbers and Perfect Squares

Abstract

Mathematical cognition research has largely emphasized concepts that can be directly perceived or grounded in visuospatial referents. These include concrete number systems like natural numbers, integers, and rational numbers. Here, we investigate how a more abstract number system, the irrationals denoted by radical expressions like , is understood across three tasks. Performance on a magnitude comparison task suggests that people interpret irrational numbers – specifically, the radicands of radical expressions – as natural numbers. Response times and strategy self-reports during number line estimation reveal that the spatial locations of irrationals are determined by anchoring to neighboring perfect squares. Perfect squares also facilitate the evaluation of complex arithmetic expressions. These converging results align with a constellation of related anchoring phenomena spanning tasks and number systems of varying complexity. Accordingly, we propose that the task-specific recruitment of more concrete representations, formal anchoring, is an important mechanism for understanding and teaching mathematics.

Irrational Numbers Are Anchored on Natural Numbers and Perfect Squares

The historical development of number systems can be characterized as a gradual progression from the concrete to the abstract. Natural numbers, which denote the cardinalities of sets, were understood directly and formalized first. They are concrete in the sense that they map directly to quantities in the material world such as the number of berries in one’s hand. According to the mathematician Leopold Kronecker, “God created” this relatively concrete number system, whereas more abstract number systems like the integers, rationals, irrationals, and reals are “the work of man.” In other words, more abstract number systems were discovered later and constructed hierarchically upon the natural numbers. For example, combining pairs of natural numbers with the subtraction operation generates the integers and combining pairs of integers with the division operation generates rational numbers. These number classes are still relatively concrete in the sense that they can be interpreted as corresponding to real world quantities like basement floors for integers and pie pieces for rationals (Van de Walle, Karp, & Bay-Williams, 2010). It is with the irrational numbers, which include √2 and π, that mathematicians discovered a truly abstract number system in the sense of lacking material referents or models that build intuition. Such abstraction engenders many surprising numerical properties. For instance, there are more irrational numbers than natural numbers, integers, or rational numbers. The set of all irrationals is uncountably infinite in cardinality, whereas the latter sets are each countably finite. Perhaps not surprisingly, mathematics teachers encourage students to think of irrational numbers like π by using rational number approximations like 3.14. This often leads to conceptual confusions.

While cognitive science research has emphasized how more concrete number systems are grounded in analog magnitude representations, it has not tested whether more abstract systems like the irrationals are understood similarly. The natural numbers are thought to be grounded in directly perceptible objects and internally represented as continuous magnitudes on a mental number line (Moyer & Landauer, 1967). For less concrete number systems, the necessity of mastering the associated symbol systems has been acknowledged. However, the focus has been on how these symbol systems recruit and restructure magnitude representations. Examples include incorporating the inverse relationship between positive and negative integers into the mental number line and capturing the ratio relationship between the numerators and denominators of fractions more precisely.

Comparatively little attention has been directed towards abstract mathematical concepts that are both imperceptible and incapable of being fully grounded in visuospatial referents. For such concepts, do magnitude representations continue to be heavily recruited or does notation-specific strategic processing become more essential? This paper uses irrational numbers as a test case to address the research question – How are irrational numbers denoted by symbolic radical expressions like √(2 )mentally represented and processed by typical adults? We begin to examine this question by reviewing the literature on natural numbers, integers, and rational numbers.

Natural Numbers

The natural numbers are the set of countable numbers {0,1,2,…}. Their mental representation has been studied extensively using the magnitude comparison (MC) task. When comparing which number in a pair is greater or lesser, response time decreases as the difference between the numbers increases. For instance, people can judge the greater or lesser number in the pair (2, 9) more quickly than in the pair (5, 6). This well-replicated performance pattern is known as the distance effect. It manifests whether participants compare symbolic numbers or visuospatial numerosities like sets of dots. The distance effect is commonly interpreted as evidence that natural numbers are represented in part as psychophysically scaled magnitudes. It has been observed as early as infancy with numerosities and kindergarten with symbolic numbers. Over development, children’s representation of numerical magnitude changes such that response times decrease overall and the slope of the distance effect decreases. These changes indicate improved precision of magnitude representations.

The magnitude representation of natural numbers and its development have also been probed by the number line estimation (NLE) task. In the bounded version of this task, participants are presented with a number line with the left and right endpoints labeled (e.g., as 0 and 100, respectively) and the middle segment left blank. Numbers are presented one at a time and in random order. The goal is to mark the position of each number on the number line. Performance is typically measured by average absolute error – the absolute difference between the correct position of the number and the position selected by the participant. Higher error values indicate worse performance. Over development, children’s accuracy on the NLE task improves gradually. There is a rich debate in the literature about the representations and processes that underlie performance on this task and its change over time. Some researchers have proposed that children initially represent natural numbers in a compressed, logarithmically spaced fashion. Over development, this incorrect representation improves to incorporate linear spacing. Some researchers have argued that the representation is proportional between critical landmarks such as the midpoint. Others have proposed that the representation is piecewise linear between critical landmarks like place-value boundaries (e.g. 10).

Individual differences in the magnitude representations of natural numbers, as indexed by the slope of the distance effect in the MC task, predict variation in mathematical achievement for elementary and middle school students. Likewise, NLE performance predicts mathematical achievement in elementary school students, although not older students learning more advanced topics. Instructional studies support a causal link between numerical magnitude representations and arithmetic skills. Young children who were tutored on the magnitudes of small natural numbers via a board game performed better on simple arithmetic problems than their peers who received a control lesson.

Integers

The natural numbers combined with their additive inverses, the negative numbers, form the integers {…, -2, -1, 0, 1, 2, …}. Comprehending the meaning of this number system requires engaging in additional symbolic processing. For instance, when judging the greater number in the pair (-3, -9), it is necessary to inhibit the whole number interpretations of “-3” as “3” and “-9” as “9”. Children initially compare such integer pairs by using a symbolic strategy. First, they convert the negative numbers to natural numbers by ignoring the minus signs. Next, they compare the numerical magnitudes and then reverse this judgment. This strategy results in a distance effect when comparing pairs of negatives numbers. But when comparing mixed integer pairs like (-4, 9), judgments can be made by noticing that positives are greater than negatives. The result is the absence of a distance effect in elementary school children. Over development, this strategic processing of integers is substituted by accessing a restructured magnitude representation that incorporates the symmetry of positives and negatives around the zero point. Adults represent the negative number line as a reflection of the positive number line, incorporating the additive inverse property x+-x=0. This shift manifests as an inverse distance effect when comparing mixed integer pairs – as the distance between the integers increases, so does the responses time.

As with natural numbers, NLE tasks have also uncovered a logarithmic-to-linear shift in the mental representation of integers. When estimating the positions of positive numbers in the range 0 – 1000 and negative numbers in the range -1000 – 0, second graders exhibit logarithmic magnitude representations. By contrast, fourth and sixth graders exhibit linear representations for both ranges. This logarithmic-to-linear shift extends to larger number ranges and older people. Middle school students’ estimates of integers in the negative range -10000 – 0 and the combined range -1000 – 1000 are linear, though with lower accuracy for negative numbers. Performance on NLE tasks like these depends partly on recognizing the symmetry of the integers around the zero point.

The inverse distance effect for the MC task and symmetry effects on the NLE task led Tsang, Blair, Bofferding, and Schwartz (2015) to design a manipulative that encourages learners to incorporate the zero point in their magnitude representation of the integers. Instruction with the manipulative improved arithmetic problem-solving for elementary school students on difficult items like 3+x=0.

Rational Numbers

Rational numbers are the set of numbers that can be written as a ratio of two integers such that the denominator is non-zero. Magnitude comparison with symbolic fractions and decimals, as well as with non-symbolic ratios, produces the distance effect. Thus, rational numbers are also thought to be integrated on the mental number line. As children begin to understand fraction magnitudes, the slope of the distance effect is nearly zero. Over time, fraction magnitude information becomes more accessible and the slope increases. By adulthood, the slope stabilizes. These behavioral changes are paralleled by the discovery and use of strategies to facilitate performance. For instance, children and young adults often report using unit fractions like ½ as anchors to perform fraction magnitude comparison.

In contrast to natural numbers and integers, NLE tasks suggest that the representation of rational numbers may not undergo a logarithmic-to-linear developmental shift. When 10-year-olds and adults perform NLE with decimal proportions or fractions, both age groups exhibit highly linear estimation patterns. Improvements have been observed over development. On average, 8th graders make more accurate estimates of fractions than 6th graders. At the individual level, a higher percentage of 8th graders than 6th graders exhibit linear (vs. logarithmic) representations. These performance gains may result in part from using unit fractions like ½ as anchors. For example, Siegler and Thompson (2014) found that children who reported using such anchors to segment the number line produced more accurate estimates.

The precision of rational number magnitude representations is also associated with higher-level mathematical skills. Symbolic fraction magnitude representations predict achievement across a range of domains and measures – fraction arithmetic, algebra, grade school standardized exams, and high school mathematical achievement. Similarly, accuracy on NLE tasks using decimals predicts mathematics achievement in elementary school students. Even nonsymbolic ratio precision predicts college students’ knowledge of fractions and algebra. Causal support for the link between magnitude knowledge and arithmetic skills comes from interventions that devote more instructional time to mastering the magnitudes of fractions. Results show that emphasizing magnitude comprehension boosts arithmetic skills for high and low achievers.

Irrational Numbers

Irrational numbers are incommensurable. Unlike rational numbers, they cannot be expressed as ratios of integers such that the denominator is non-zero. Their decimal expansions are infinitely long, non-repeating, and ostensibly random. When they were first proposed, the notion seemed outlandish. According to a famous anecdote, the Greek mathematician who first proved their existence was drowned at sea for challenging the ratio doctrine of numbers. It was not until the late 1800s that irrationals were formalized and properly integrated onto the real number line (Dedekind, 1963/1888). Two subclasses can be distinguished – algebraic irrationals and transcendental numbers. Algebraic irrationals, like ∛20, are the solutions to polynomial equations and can be denoted by expressions of the form √(y&x). Transcendental numbers, like π and e, are not the solutions to any polynomial equation.

Just one other study has investigated the cognitive bases of algebraic irrationals. It explored whether the same human “number sense” that allows us to compare the magnitudes of natural numbers, integers, and rational numbers extends to irrationals. Participants performed magnitude comparisons with irrationals of the form √(y&x). Both the root y and the radicand x could vary. The researchers tested whether participants compared the magnitudes of irrationals by accessing their holistic magnitudes or by focusing on the root and radicand components. Comparisons were faster when both numbers contained a common root component, as in the pair (√(9&12),√(9&63)). Likewise, comparisons were faster when both numbers contained a common radicand component, as in the pair (√(10&34),√(13&34)). When both the roots and radicands differed, as in the pair (√(10&34),√(15&68)), response times were slower. Unexpectedly, response times for the common component pairs were not predicted by the distances between the root components or radicand components. These mixed results require explanation. One possibility is the use of irrational numbers denoted by complex radical expressions that do not occur in contexts such as solving quadratic equations. Another is the highly expert sample composed of mathematics graduate students and professors. It remains unclear how typical adults understand irrational numbers like √2.

The stimuli we used in our study are also algebraic irrationals, though we focus on square root expressions of the form √x. Such expressions are often encountered when solving quadratic equations. They denote irrational numbers when the radicand x is not a perfect square (e.g., √2) and natural numbers when the radicand is a perfect square (e.g., √(9 )). In the following sections, we will collectively refer to irrational numbers and perfect squares as radical expressions because both contain the radical sign.

Research Questions and Hypotheses

First, we asked whether radical expressions are represented as continuous magnitudes integrated on the mental number line much like natural numbers, integers, and rational numbers. We refer to this proposal as the mental number line hypothesis. An alternative proposal is that people use processes that capitalize on the discrete structure of radical expressions. For instance, when judging the greater or lesser number in the pair (√(3 ), √(8 )), people may ignore the radical signs and only compare the radicand components (3, 8) using their magnitude representations of natural numbers. This is possible because when x and y are non-negative, judgments of (√(x ), √(y )) and (x, y) are equivalent. We refer to this as the equivalence strategy hypothesis.

Second, we investigated whether people process irrational numbers by strategically anchoring them on more concrete concepts like natural numbers and perfect squares. Specifically, we hypothesized the possible use of a multiplication strategy on the MC task whereby perfect square pairs like (√4, √(9 )) are transformed to computationally analogous “tie” multiplication problems like 2 × 2 and 3 × 3. Such a transformation might facilitate comparison because tie problems are processed more quickly than non-tie problems of comparable size. On the NLE task, irrational numbers might be estimated in relation to perfect squares – the landmark strategy. For example, people may estimate the positions of irrational numbers like √(3 ) by referencing the positions of neighboring perfect squares like √(1 ) and √(4 ). Finally, people may leverage their knowledge of perfect squares during arithmetic problem-solving. When simplifying √72, an inefficient strategy would be to decompose the radicand into its prime factors and then shift pairs of common factors outside the radical sign – the prime factorization strategy. For example:

√72 = √(2×36)

= $$\sqrt{2*2*18}$$

= √(2×2×2×9)

= √(2×2×2×3×3)

= 2√(2×3×3)

= 2×3√2

= 6√2

A more efficient approach is to factor the radicand into its largest perfect square factors and directly reduce these – the perfect squares factorization strategy:

√72 = √(2×36)

= 6√2

Third, we investigated whether individual differences in the mental representation and processing of irrationals explain variation in conceptual and procedural knowledge of this number system. Some researchers have proposed that magnitude representations are at the core of numerical and arithmetic performance (Link, Nuerk, & Moeller, 2014; Siegler, 2016). Contra this view, we hypothesize that the influence of magnitude representations on arithmetic problem-solving attenuates as the abstraction of number systems increases. Arithmetic performance with irrationals may depend less on magnitude representations and more on symbolic strategies.