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.

How the Abstract Becomes Concrete:

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.

Method

Participants

Overall, 81 undergraduate students from a large Midwestern university were recruited via flyers posted across campus. We posted flyers in buildings housing diverse departments and those frequented by students from different majors to represent the university population faithfully. Their ages ranged from 18 to 24 years (M = 20.5, SD = 1.7). Our sample consisted of more females than males (57 vs. 24). This is more skewed than the overall percentage of female vs. male undergraduates at the university (52% vs. 48%). However, it is consistent with the percentage of females in the college of education (61% vs. 39%), which is where the laboratory is physically located. On a questionnaire, 30.3% of participants self-reported that they were enrolled in a major emphasizing quantitative and analytic skills such as science, technology, engineering, mathematics, finance, or economics. This percentage is consistent with the distribution of such majors at the university. Experimental sessions spanned about 60 minutes and all participants received $12 in compensation. The protocol was approved by the local Institutional Review Board.

Design and Materials

We employed a within-subjects experimental design, with each participant completing all levels of the four experimental tasks. See Table 1 for an overview of the design and materials. All experimental materials are available via the Open Science Framework (https://osf.io/6s9pc/#).

Magnitude comparison task.

The first task was magnitude comparison and its stimuli were defined by four variables – type, distance, size, and perfect square status. The type variable refers to whether a number pair consisted of natural numbers or radical expressions. The natural number pairs formed from the combinations of 0, 1, 2, …, 9 with same-number pairs like (3,3) excluded (90 pairs). We were particularly interested in pairs where both numbers were perfect squares, like (4, 9) or (√(4 ),√(9 )). Hence, we included an additional copy of each of the 12 perfect square pairs in the stimulus set. In total, 102 natural number pairs were used in each of blocks 1 and 2. Participants judged the greater of two numbers in the first block and the lesser in the second block. Natural number pairs were mirrored by 102 pairs of radical expressions formed in the same way from √(0 ), √(1 ), √2, …, √(9 ). These pairs were used in blocks 3 and 4. Block 3 required greater judgments and block 4 required lesser judgments. Within each block, number pairs were presented in random order. Overall, there were 408 trials.

For the distance and size variables, we used two definitions to evaluate the equivalence strategy and mental number line hypotheses (research question 1). The distance variable was the absolute difference between the numbers. The first definition of distance focused on the radicand components. Thus, the distance of the pair (√(4 ), √(9 )) was |4-9|=5. The second definition focused on the actual magnitude values of the radical expressions. Thus, the distance of the pair (√(4 ), √(9 )) was |√4-√9|=1. The size variable was defined as the average of the two numbers. The first definition again used the radicand components. Thus, the size of the pair (√(4 ), √(9 )) was ((4+9))⁄2=6.5. The second definition again used the actual magnitude values of the radical expressions. Thus, the size of the pair (√(4 ), √(9 )) was ((√4+√9))⁄2=2.5. The perfect square variable was defined by whether both natural numbers or radicands were perfect squares like (4, 9) or (√(4 ), √(9 )) or not. Two dependent variables, accuracy and response time (RT) in milliseconds, were collected.

Number line estimation task.

The second task was number line estimation (NLE) and it included one within-subjects variable (type) with four levels – natural numbers, one-digit radicals, perfect square radicals, and two-digit radicals. For each type, participants saw 11 numbers. The data from the smallest number and the largest number were not analyzed because these were one of the poles of the unmarked number line (i.e. 0 and 10 in all but one case). The first block consisted of the natural numbers 0, 1, 2, …, 10. The values 0 and 10 were not analyzed. The second block consisted of the one-digit radicals √(0 ), √(1 ), √2, …, √(10 ). The values √(0 ) and √(10 ) were not analyzed, meaning all analyzed stimuli had one-digit radicands. The third block included the perfect squares √(0 ), √(1 ), √4, …, √100. The values √(0 ) and √(100 ) were not analyzed. The fourth block included the two-digit radicals √(0 ), √(10 ), √20, …, √100. The values √(0 ) and √(100 ) were not analyzed, meaning all analyzed stimuli had two-digit radicands. These stimulus sets include one confound. The values √(1 ), √(4 ), and √(9 ) are shared across two types – one-digit radicals and perfect square radicals. This was unavoidable; it was forced by the distribution of perfect squares. The 11 stimuli of each type were presented in random order within a block. There were 44 trials overall.

Two dependent variables, average absolute error and linearity of estimates, were formed. Average absolute error was the absolute difference between a participant’s selected position and the target position. Linearity of estimates was the relationship between participants’ selected positions and target positions as estimated by linear regression (R2).

Strategy questionnaire.

The third task was the strategy questionnaire for the NLE task. Participants completed an eight-item paper-and-pencil strategy questionnaire composed of two prompts for each of the four blocks – natural numbers, one-digit radicals, perfect square radicals, and two-digit radicals. All prompts were in the form “How did you decide where to place x?”, where x was a value from one of the four blocks. See Table 1 for examples. The independent variable was number type (natural numbers, one-digit radicals, perfect square radicals, and two-digit radicals) and the dependent variable was self-reported estimation strategies.

Irrationals knowledge test.

The forth task was the irrationals knowledge test. We constructed this novel paper-and-pencil test to measure conceptual and procedural knowledge of irrational numbers. Prior studies have investigated conceptual and procedural knowledge of rational numbers and arithmetic equivalence. We operationally define conceptual knowledge of irrational numbers as the ability to define this class, categorize particular numbers as members of it, and reason about its properties in relation to other number classes. We operationally define procedural knowledge of this class as the ability to simplify radical expressions and perform arithmetic operations on them.

Many items were adapted from prior studies documenting students’ misconceptions about irrational numbers. Conceptual items were distributed across three subsections. They required participants to (1) define the rational, irrational, and real number classes and classify numbers like π as either rational or irrational, (2) answer questions about the density of rational and irrational numbers, and (3)reason conceptually about the results of arithmetic operations on arbitrary rational and irrational numbers. Most of these items were in the selected response format. The procedural items were distributed over two subsections. Subsection (4) required performing simple arithmetic operations to reduce radical expressions and subsection (5) required evaluating complex arithmetic expressions by performing operations on multiple radical expressions. These were all fill-in-the-blank items. Example items from all five subsections are in Table 1; the full test is in the supplementary materials. Correct items earned one point each and partial credit was not awarded. Two dependent variables were computed – accuracy on each section and problem-solving strategies on subsection (5) – complex arithmetic.

Procedure

After obtaining informed written consent, participants were seated alone in a quiet laboratory room in front of a Windows PC with an extended keyboard, a mouse, and a monitor measuring 55.6 cm diagonally. Participants completed the tasks in a fixed order.

Magnitude comparison task.

Participants first completed the MC task, implemented in the experimental design program OpenSesame 2.9.5. In blocks 1 and 2 (natural numbers), they made greater and lesser comparisons, respectively. In blocks 3 and 4 (one-digit radicals), they made greater and lesser comparisons, respectively. Within each block, trials were presented in random order. The background was black.

Each trial began with a blank screen for 250 ms, after which a white fixation dot was shown in the center of the screen for 1000 milliseconds, followed by a blank screen for 500 milliseconds. A number pair was presented next, with each number offset about 5 centimeters on either side of the center. All numbers were displayed in white and in the font Cambria Math (point 80). Participants indicated the greater or lesser number by pressing the key below the target number – “Z” for the number on the left and “M” for the number on the right. The stimuli were visible until participants made a response. Feedback was then provided to discourage participants from trading accuracy for speed. Correct responses were followed by Correct in green and incorrect responses by Incorrect in red. This feedback was displayed for 500 milliseconds. Participants required about six minutes to complete each of the four blocks.

Number line estimation task and strategy questionnaire.

Next, participants completed the NLE task written in JavaScript and hosted online (www.psycholab.org/FreeLab). They estimated the positions of natural numbers in block 1, one-digit radicals in block 2, perfect square radicals in block 3, and two-digit radicals in block 4. Within each block, trials were presented in random order. Participants were instructed to be as accurate as possible and that speed was irrelevant. On each trial, a natural number or radical expression was shown above a number line. These elements were shown in black against a white background. The number line was only labeled at the endpoints, zero on the left and ten on the right. The middle was left blank. Participants estimated the number’s position on the line by left-clicking with a computer mouse. To prevent them from rushing through the task, a blank screen was shown for three seconds between trials and clicks during this interval were ignored. Feedback was not provided to avoid learning effects. Overall, participants needed about six minutes to complete this task. Afterward, they were given the strategy questionnaire to complete at their own pace.

Irrationals knowledge test.

Finally, participants completed the irrationals knowledge test without feedback or the use of electronic devices. They completed the five sections in order: (1) number concepts, (2) density concepts, (3) operation concepts, (4) simple arithmetic, and (4) complex arithmetic. Approximately 25 minutes were allotted to respond.

Afterward, demographic information was collected, participants were debriefed, and compensated with $12 in cash.