## Abstract

A great many students at a major research university make basic conceptual mistakes in responding to simple questions about two successive percentage changes. The mistakes they make follow a pattern already familiar from research on the difficulties that elementary school students have in coming to terms with fractions and decimals. The intuitive core knowledge of arithmetic with the natural numbers makes learning to count and do simple arithmetic relatively easy. Those same principles become obstacles to understanding how to operate with rational numbers.

This article is part of a discussion meeting issue ‘The origins of numerical abilities’.

## 1. Introduction

Typical pre-school children have an approximate number sense (ANS) that enables them to discriminate between arrays with sufficiently different numerosities and to perform basic arithmetic computations on numerosities and other perceived quantities [1–4]. What appears to be the same system is present in a great many non-verbal animals and is manifest already in human infants [5,6].

When children start school, most of them also have some knowledge of verbal counting and the relation of its principles to the operations of addition, subtraction and ordering [4,7–11]. We have proposed that the non-verbal mathematical system for natural number facilitates learning of the verbal and notational natural number system because they share a common structure [12,13]. Recent evidence for this thesis is provided by Hurst *et al*. [14] and Odic *et al*. [15]. This proposal emerges from the broader view that the mind actively assimilates data that fit into available structures [12,16,17] and resists assimilating data that do not fit. ‘Numerons’ is Gelman & Gallistel's [13] term for the quantity-representing symbols in the ANS. On the ANS-facilitation hypothesis, the basic number words and the written symbols for the digits map to the corresponding numerons, and the basic terms for arithmetic operations map to operations on numerons within the ANS. That is why intuitions rooted in the ANS facilitate the acquisition of counting and verbalized simple arithmetic reasoning, reasoning of the form ‘subtracting *x* from *y* reverses the effect of adding *x* to *y*’ [4].

Numerons themselves must be generated as needed [18]. What constitutes the basic verbal digits and the principles for generating the number words beyond those for the basic digits varies dramatically from culture to culture [19,20]. Thus, the diverse culturally evolved systems for producing words for numerons do not map simply to the brain's scheme for generating ever-greater numerons. (We assume that scheme is culture-independent and universal.) It is therefore not surprising that mastery of the mathematical language and the mathematical manipulation of written symbols takes a long time and depends strongly on the extent of a child's experience with large numbers [21,22].

An example of the difficulties that arise in constructing this mapping is that some children pass through a stage where they think that ‘hundred’ is another decade word. They reject that ‘hundred and ten’ follows ‘hundred and nine’ on the grounds that ‘we used it {“hundred”] already’. And, of course, the British ‘billion’ does not refer to the same number as the American ‘billion’. And many languages have startling late-occurring irregularities, such as ‘soixante-dix’ and ‘quatre-vingts’. Thus, children's intuitions about numbers, which derive from the ANS, outrun their skill in talking about them. Children in the early grades often understand—or can easily be led to understand—that the numbers continue indefinitely long before they have mastered the system for generating indefinitely large count words [22]. They do not learn that numbers continue indefinitely—no conceivable experience could teach them that; rather, they intuit it because it is a property of the ANS (ever larger numerons may be generated without limit).

As children move on in the elementary grades, they also start to learn about fractions and decimals in connection with the products of division. The learning of fractions and decimals is a major challenge for many pupils [23]. Many middle-school students and even a goodly number of college students have never come to terms with fractions and decimals [24–29].

In our thesis, a central reason for the strong challenge that the learning of fractions and decimals presents to many pupils is that the mapping from the culturally evolved verbal and written symbols for these numbers to the non-integer numerons in the ANS is complex. Single fractions are generally referred to by word strings composed of two different numbers (e.g. ‘three-fourths’). The written symbols for fractions have three parts: two digits (or, worse yet, two multi-digit strings) separated by the symbol for division (e.g. ‘517/103’). Gallistel [18] assumes that numerons that reference non-integer quantities have the same structure as those that reference integers. We concur and assume that the internal structure of a numeron for a proportion like 517/103 is no different from the internal structure of the numerons for, say, 4. But, as just noted, the structures of the verbal and written symbols for these quantities are very different from the single words or single symbols that refer to the basal integers. Thus, it is difficult to map these word strings and symbol strings to the corresponding pre-linguistic symbols (the numerons). In our thesis, the obstacle the learner confronts in developing a bidirectional mapping between the verbal and written symbols for non-integer quantities and the corresponding numerons in the ANS is closely analogous to the readily apparent obstacles to learning the bidirectional mapping between fractions and decimals. It is far from obvious that 517/103 = 5.01941748. Nor is it obvious why the fractional representation is complete while decimal representation has no fixed termination, despite the fact that both notations reference the same proportion. Our thesis is that the difficulty of developing a bidirectional mapping between the rational numbers and the ANS is a major source of learners' difficulty in coming to terms with the rational numbers.

There is extensive work on the conditions that support learning about fractions and decimals, but there is very little on percentages. This is surprising given that percentage changes are arguably the numerical format most commonly encountered in adult life in numerate cultures. They are used on a daily basis to describe and advertise changes in the prices of goods of all kinds from clothes to houses and from stocks to airfares. It also describes voting preferences, taxation rates, interest rates, fractions of populations and much else.

In this paper, we report our recent research on the extent to which college students do and do not reason appropriately about simple percentage problems. We focus on college students on the assumption that their performance is indicative of the extent to which lengthy education and extensive daily experience instil mastery of simple reasoning involving percentages. In accord with recent large-survey market research on customers’ understanding of percentages [30], we find that many students do not reason appropriately, even when given unlimited time and asked to write out their reasoning. The failures we report fall into a pattern that resembles what is seen when the rational numbers are first introduced in elementary school. We then discuss how our results reinforce our thesis about how the intuitive understanding of quantity provided by the ANS sometimes facilitates and sometimes obstructs the mastery of the culturally evolved systems for representing and reasoning about quantity.

Perhaps, our results may be taken to reveal the use of an inappropriate heuristic, if one understands by ‘an heuristic’ a principle of reasoning that is easy to execute and valid in some, but not in other apparently similar circumstances. Kahneman & Tversky came up with many of their examples of systematic, widespread and resilient cases of erroneous reasoning when they realized that even they were drawn to conclusions they knew to be false [31]. If this way of thinking about our results is taken to imply that subjects really know better but err only from haste and inattention, then we do not think it applies. In experiment 2, no participant remarked that their answer was wrong, even though subjects were asked to take their time and show their work. Nor did providing numerical examples for the hint group lead individuals to say something like ‘Oh, what a silly mistake I have been making’. In our discussion, we offer an account of why the tendency to reason in this way has such a tenacious and widespread hold on reasoning about percentage changes. Our account connects the misuse of the errors to a body of research on pupils' difficulty in mastering fractions and decimals, and our account has pedagogical implications.

## 2. Methods

### (a) Study 1: online survey

Percentage problems are generally presented as adding or subtracting some percentage. When the changes are sequential, the presentation in terms of adding and subtracting makes it easy to overlook the crucial fact that what is added or subtracted is obtained by multiplication. When the cost of a litre increases by 50%, the new price is 1 + 0.5 = 3/2 times the old price. When that price then drops by 50%, the new price is (0.5 × 1.5 = 0.75) of the original price. Although the same percentage was both added and subtracted, the second operation did not restore the original price, because the quantity multiplied to obtain the subtrahend was not the same as the quantity multiplied to obtain the addend.

Our first study of this topic used an online survey to collect data from 1629 undergraduates taking introductory psychology course at a large state university (807 females). We asked each student one of the two versions of each of the following three questions: (i) the $100.00 cost of an item decreases/increases by 50% and later increases/decreases by 50%; what is the final cost? (ii) The $*y* cost of an item decreases/increases by 50%, and then increases/decreases by 50%; what is the final cost of the item in terms of *y*? (iii) The $*y* cost of an item decreases by 30%, then increases by 30%; what is the final cost in terms of *y*?

The students in this survey study were at least 18 years old and were enrolled in the introductory psychology course at a large state university. Our questions were posed in an online screening protocol composed of questions from members of the department who may wish to use students as participants in an experiment. The screen allows experimenters to identify participants who might be suitable for their experiment. Included in the screen were questions about a student's Scholastic Aptitude Test (SAT) score and the level to which a student's mathematical education had progressed.

The 4887 responses to the test items (1629 participants × 3 questions) were separately coded by two coders inter-rater reliability was greater than 99.88% on coding for accuracy (correct or incorrect). A third coder independently coded items for which there were disagreements. This information was used to resolve the differences. All three coders had extensive mathematical backgrounds.

### (b) Study 2: paper-and-pencil format

Approximately 50% of the responses in our study 1 were erroneous. We thought that the students might have answered our questions with little thought, using inappropriate intuitive heuristics. Because the two changes were in opposite directions by the same percentage, the questions invited the casual assumption that the second change bounced the price back to its initial value. We designed our second, more experimental, study to encourage participants to take their time and think carefully and to show their reasoning.

The initial 134 participants were at least 18 years old and were recruited from the undergraduate psychology courses described above. Their participation counted towards their psychology course required research participations. Six participants voluntarily withdrew and four participants were excluded due to incomplete data, leaving *N* = 124 (81 females). Fifty-seven participants (44 females) had no calculus background, 46 (27 females) took calculus I and 21 (10 females) took calculus II or further advanced maths courses. All IRB requirements were met.

Participants were told that our goal was to look at how undergraduates solve these problems. All items were presented in a paper–pencil booklet format with space for participants to show their work, without a calculator, but with no time constraints. Participants were encouraged to answer all questions to the best of their ability and to use the ample space provided on sheets of paper to show their work. They were asked to keep moving forward and not return to a previous section (e.g. the pre-test which was referred to as ‘Section A’) once they moved on to the experimental phase (referred to as ‘Section B’) and then the final Section C. No feedback was given during any phase; we periodically walked by participants to monitor their progression through the problems.

There were two experimental conditions for Section B, number and variable (*y*), that compared whether a numerical value, e.g. 30%, instead of variable (*y*%), filled the slot for answer. In both conditions, individuals answered eight pre-test and eight post-test questions with the same general structure: ‘There are *y* units, *y* is increased by ___% *y* is then decreased by ___%. What is the final value, in terms of *y*?’ Various different percentages appeared in the blanks for different problems (table 1).

In all three phases of this study, participants answered questions of the general form used in our initial survey study, with the important alteration that the percentages by which the base was increased/decreased differed from the percentages by which it subsequently decreased/increased. We made this change in order to discourage the intuitive and fast assumption that the two changes would cancel out. In the first phase (Section A) and last phase (Section C), all questions were posed in their algebraic form, with the unknown starting cost specified by ‘*y*’. In the middle phase (Section B), half the participants saw these questions with the base specified numerically, while the other half continued to view them only in their algebraic form. We thought that working the problems in their numerical format would lead to recall of how to do these kinds of problems. If so, this should promote more appropriate solutions when the questions were once again posed in their algebraic form.

## 3. Results

### (a) Study 1

The frequencies of wrong answers in our large initial online study were surprisingly high. Performance was better given numerical values as opposed to algebraic representations: for the respective correct responses for the 50%—numerical, 50% algebraic and 30% algebraic, the respective mean number correct answers, 75.14, 50 and 37.6% (*χ*^{2}(1, *N* = 1629) = 152.39, *p* < 0.0001). Even under the most favourable circumstances when the base was $100 and both changes were by 50%, 25% of the 1629 students answered incorrectly. When the base was some unspecified quantity denoted by ‘*y*’ and the percentage changes were 50%, 45% of students answered incorrectly. When the base was also ‘*y*’ and the percentage changes 30%, 62% of students answered incorrectly.

The most frequent answer for the online computer study was that the final amount was the same as the initial amount. For example, ‘it remains “*x*” dollars, the original cost at the start’, or ‘back to the original price’. Apparently, the errors were produced by a strong tendency for a participant to assimilate percentage problems to the arithmetic structure that comes most naturally and earliest in development, namely, the addition and subtraction of ‘natural numbers’. With natural numbers, it is of course true that subtracting *x* reverses the effect of adding *x* and vice versa. Even pre-school children appreciate this principle [4,32].

Because the questions in our first study were presented online among a large set of other unrelated questions, it was possible that students gave answers they could retrieve quickly with little or no thought. In other words, they may have relied on an inappropriate heuristic, ‘thinking fast rather than slowly’ [31]. We tested this explanation in the second study, by giving students time and asking them to show their thinking by writing out their answers. To encourage knowledge of everyday relevant contexts, the problems involved everyday expenses (sporting event tickets, supermarket or department purchases, weight changes etc.). As already stressed, the increases and decreases were by different percentages, for example, a 40% increase, followed by a 70% decrease.

### (b) Study 2: paper and pencil

We knew from the study 1 online survey that questions posed in numerical form promoted more correct answers. We thought, therefore, that a brief sequence of questions posed in numerical form, and coming between two sets similar in algebraic format, might serve as a hint as to the general way of solving this kind of question for an algebraically specified quantity. This expectation proved erroneous: participants in the experimental condition did no better in the third phase than they did in the first phase (figure 2). Their experience of working the same problems from numerically specified bases during the middle phase had no effect on their ability to work the problems algebraically.

The overall results in the second study, the paper-and-pencil test, were much the same as in the online survey: pooling across the first and last phases, where a total of 16 questions were posed in algebraic form to each student, about 60% of the questions were answered incorrectly.

Moreover, the performance of the students in the experimental condition did not improve from the first phase to second (post-hint) phase (figure 1). If a student did well in the first phase, they did well in the second; if they did not do well in the first, they did not do well in the second—whether they were in the control condition or the experimental condition (figure 2).

In the first study, the online survey, students were not asked to show their work, but some did so anyway. Those that did show work gave a higher percentage of wrong answers than those that did not. We conclude from this that those that did show work in the first study were struggling. In the second study, despite the request to show work, some students did not show any work. Only 5% of their answers were correct. Of those who showed one to two lines of work, 44% of them were correct. Students who showed more than two lines were correct 68% of the time. Thus, switching to a paper-and-pencil format and telling participants to show their work did promote more thought. Nonetheless, the main result still holds. The overall success rate was remarkably low. Detailed coding of each solution ruled out straightforward computational errors. Conceptual errors dominated. Even when thinking more slowly, students at the college level make these errors with unexpected frequency.

#### (i) Types of error

The most common type of error was conceptual, involving the inappropriate use of natural number arithmetic. Performance pooled across the 16 questions revealed that, on average, a full 33% of participants incorrectly used natural number arithmetic. There was a large gap between errors of that type and the other types: about 3% of participants incorrectly multiplied and/or divided the given percentages, 1.5% gave incomplete responses and about 1% were incorrect due to a missing decimal or bracket error. A closer look at the natural-number-arithmetic errors revealed three sublevels (table 2). Figure 3 shows that 20–30% of participants made a level 2 error, and 13–24% of participants made a level 1 error. Only 3% of the participants made a level 3 error, an inappropriate operation in the final step. Participants making a level 3 error did get the individual parts of the problem correct.

#### (ii) Location of error

The problem posed in all of these questions naturally breaks into two steps, each with a substep: (1a) Compute the first percentage times the base amount. (1b) Add/subtract the result to/from the base amount to obtain the new base. (2a) Compute the second percentage times the new base. (2b) Subtract/add the result from/to the new base. It sometimes happened that a student set up the problem correctly—correctly writing out the two steps just described—but then made an algebraic/arithmetical mistake in obtaining the result. We scored the pencil and paper work shown for all those cases in which the final answer was wrong and work was shown (figures 4 and 5). The most common score for type of error was complete failure: all parts of the problem were incorrect. Forty to 45 of the 124 participants made errors in this category (figure 4, solid curve). The next most common failure was to get the second part of the problem wrong. Twenty to 35 participants made errors in this category (figure 4, dashed curve). Getting only the first step wrong was infrequent, as was making only an error in the final simplification (dash-dot and dotted curves in figure 4).

When the type of error was broken down by participant (figure 5), 26 out of 124 participants were wrong on all parts of the problems on almost every question (the purple bars in figure 5 that span almost the entire *y*-axis). Seven participants got the second step wrong on at least 15 out of 16 questions (the red bars that span most of the *y*-axis). Thus, slightly more than 25% of the participants were consistently unable to deal with these problems. The remaining students occasionally got the first step right, but then failed at the second step. Ten students sometimes got both steps right on some trials, but got one or the other step wrong on other trials (bars with both blue and red parts in figure 5). Errors on final simplification occurred with repeatedly in only two students (bars with green parts in figure 5).

### (c) Correlations with maths level and SAT score

In the first study (computer online survey), we asked respondents to report their SAT scores. Not surprisingly, there was a strong correlation between these self-reported scores and percentage correct answers to the questions. Less than 20% of those with scores below 550 answered correctly the algebraic question involving changes of 30% each way; 70–80% of those with scores above 700 answered these questions correctly. In the second study, we obtained both self-reported SAT scores and level of maths courses taken (no calculus, calculus I, calculus II and higher). The level of maths taken correlated strongly with correct performance. Students with no calculus answered correctly only a little more than 20% of the questions; whereas students who had taken maths through calculus II and, in some cases, beyond answered correctly on 75% of the questions. What is perhaps more notable, however, is the converse: students with a relatively advanced education in mathematics still answered 25% of these questions incorrectly. Completed maths course level correlated extremely well with self-reported SAT score. However, adding the SAT score to the regression model, after entering the maths level, actually weakened the model. Male participants in both studies did a little better than female participants. Gender was, however, confounded to some extent with how far a student had progressed in their high school mathematics curriculum. We did not attempt to tease these factors apart.

From this second study, we conclude that many college students cannot properly work a two-step percentage change problem.

## 4. Discussion

Our results bring out once again the contrast between the relatively rapid progress of learning when the material to be learned maps readily into an existing conceptual structure and the much slower and uncertain progress when it does not. In the latter case, the learner must either create a new conceptual structure in order to assimilate the material or construct a complex mapping of the material to an existing structure or structures. We believe that the difficulty in mastering reasoning about verbal and written rational numbers is of the latter kind. The ANS provides an appropriate structure, but the verbal and written symbols for non-integer quantities do not map easily into that structure. The two different systems—fractions and decimals—do not map readily into one another [25] and neither maps readily into the ANS system. While down very deep the brain's representation of both integer and non-integer quantities may itself have a complex structure (cf. [18]), we believe that these pre-linguistic symbols for quantities (the numerons) present themselves to the map-construction process as unitary entities. The problem then is to map the non-unitary conventional symbol strings to these unitary underlying symbols.

### (a) When the pre-existing structure helps

Learning verbal and symbolic representations of the natural numbers builds on the pre-verbal system for representing and reasoning about quantity. Learning to count and find answers to addition and subtraction problems begins in infancy [21,33–35], and it develops without formal schooling in most numerate cultures. It builds on the conceptual foundation provided by the pre-verbal and pre-linguistic system for estimating number and reasoning about numerosity. This system is pre-linguistic because it is well developed in a very wide range of non-verbal animals ([36–41] see also many other papers in this special issue). This pre-verbal system plays an important but utterly unconscious role in adult numerical reasoning [42–44].

We begin the discussion with a brief review of some of the evidence that the principles implicit in the pre-verbal processes of number estimation and arithmetic reasoning guide the development of counting skill and numerical reasoning in pre-school children and into early schooling.

#### (i) The more numerous the set, the bigger the number word

Most 3-year olds are poor counters. However, when they are shown cards with item arrays of varying numerosity between 2 and 19 and asked how many, their estimates are generally accurate for sets of two and three and declines slowly for larger numerosities [45–47]. More importantly, there is a strong correlation between the numerosity on the card and the ordinal range of the numbers they give as their estimate. For 4, the range is from 3 to 5; for 9, it is from 7 to 10; for 19, it is from 10 to 24 ([46, fig. 5.1, p. 60]). This implies that, despite their poor counting, 3-year olds already understand that the number words are ordered and that position in this order increases with the magnitude of the numerosity being estimated.

#### (ii) Surprise and counting when numerosity is changed surreptitiously

When 2- and 3-year olds are participants in a ‘magic’ experiment, in which the numerosity of a set of two or three items is surreptitiously decreased or increased, they are greatly surprised by the change in numerosity, and they begin to count to confirm this change; if the change was a decrease, they look around for the missing item or conjecture that ‘Jesus took it’ ([46], ch. 10). When the items are simply rearranged or even when obviously different items are substituted one-for-one, there is much less surprise, no counting and no indications that a subtraction or addition is conjectured as an explanation. These results imply that numerosity is the defining attribute of these sets under the conditions of the experiment. They imply that 2- and 3-year olds already understand that numerosity is verbally estimated by counting and that it is altered by subtraction and addition, but not by rearrangement or change of item identity. Again, the 2- and 3-year-old participants in these experiments were generally poor counters, which means that an understanding of the principles that govern counting and the principles that govern the most basic combinatorial operations on numbers—ordering, addition and subtraction—precede skill at counting. These principles include the inference principle that a decrease in numerosity implies a subtraction and an increase implies an addition.

#### (iii) Error detection

Toddlers from an anglophone environment, who still do not count at all, prefer listening to a correctly ordered count of an array of six fish when it is in English but not when it is in French; whereas toddlers from a Francophone environment prefer the correctly ordered count over the incorrectly ordered count only when it is in French [48]. This implies that children who have not yet begun to count already understand that counts come from an ordered list and have already formed a correct enough idea of the proper order to recognize violations.

Three-year olds distinguish between errors that affect the outcome of a count versus deviations from convention. For example, double counting an item or skipping it affects the outcome, whereas counting from right to left is unconventional but has no effect on the outcome [49–51]. Although neither children nor adults can readily articulate the principles that govern a proper count—anymore than they can articulate the principles that govern a well-formed sentence—they nonetheless recognize violations of those principles at an early age.

#### (iv) Cardinality from idiosyncratic count lists

Transiently stable but idiosyncratically ordered count lists can appear in the spontaneous counting of 2.5-year olds [52]. When children with an idiosyncratic list count to confirm numerosity, they treat the final word in their count as the representation of the cardinality of the set. This is so even when its position is idiosyncratic (not in accord with its position in the conventional count list). This implies that before they have mastered the conventional count, they nonetheless understand the principle that the final symbol in a count sequence represents the cardinality of the set.

#### (v) Transfer based on numerical order

Two-and-half- to 3-year olds taught that the plate with two is the winner, as opposed to the plate with one, spontaneously assume that a plate with four is the winner over a plate with three [52]. This implies that children recognize the ordering of numerosities prior to the development of counting skill.

#### (vi) Distinguishing essential principles from conventional practices

To be valid, a verbal count must honour three principles: (i) a one-to-one mapping of the count words to the items in the set; (ii) the use of a stably ordered list of count words; and (iii) the last word represents the cardinality of the set. These are the essential principles. There are also conventions, such as counting from left to right, not skipping around, and so on. Pre-school children readily violate the conventions while adhering to the principles, which again suggests that they grasp the principles [50].

#### (vii) The ease with which the potential infinity of the numbers is induced

Cantor caused a storm in nineteenth-century mathematics by deducing (or positing) ‘realized’ or ‘completed’ infinities, such as the set of the natural numbers and the set of real numbers, and by proving that the latter was more numerous than the former. Before Cantor, most mathematicians recognized only Aristotle's potential infinity—an iterative process in which a further iteration was always possible. A pre-eminent example of a potential infinity was the ever-increasing numbers produced by iterating the addition of 1 to an arbitrarily chosen initial number. Psychologists often call this the successor principle. Among its implications are that there is no largest or last number.

Given the abstract nature of the successor principle and its empirically unverifiable implications, it is startling that some kindergarten and most second-grade children already affirm the potential infinity of the numbers when answering the following questions [53]: ‘Can people always add to make a bigger number, or is there a number so big we couldn't make it any bigger?’ ‘If we count and count will we ever get to the end of the numbers?’ ‘Is there an end to the numbers?’ ‘What if we cheat and start counting from a really high number, then could we get to the end of the numbers?’ ‘Is there a last number?’ ‘So if someone came up to you and said, “I think a googol is the last number there is.” Would you believe him?’

It has been suggested that children induce the successor principle and the linear order of the numbers after learning one by one the referents of the first three to five number words and observing that those referents—those first few numerosities—are ordered [7]. The postulated induction cannot be mathematical induction, because the successor principle is the basis for mathematical induction. Moreover, it has been carefully and extensively explained by other authors that the postulated empirical or bootstrapping induction begs the question how children arrive at this conclusion. It begs the question by implicitly assuming that a linear ordering is the only possible or, at least, the only plausible conclusion from these few empirical facts [54,55]. Adding one to a two's complement fixed point number may unendingly increase the number or it may generate a circular sequence of signed numbers [18]. Which it does depends on whether or not the leftmost bit in the result (i.e. the most significant than) is discarded when it lies to the left of the leftmost bit in the addends. In angular calculations, it is discarded, with the result that adding generates a circular system of numbers. When it is not discarded, adding generates an ever-increasing linear system of numbers. How many additions are required before one returns to 01 (the symbol for one) in the circular system depends on how many bits the system allocates to the representation of a number—in essence on the size of the circle. Elsewhere, the same point—that there are other just as plausible conclusions that may be drawn—is made by reference to the times on a clock [54,55]. Thus, the fact that kindergarten children draw the same conclusions as pre-Cantorian mathematicians and philosophers of mathematics such as Aristotle suggests that the mind is steered to this conclusion by the fact that the principle is already manifest in the operation of the pre-linguistic system for reasoning about discrete quantities.

### (b) Nevertheless skill acquisition matters

It is equally clear, however, that the principles alone will not take one far in the absence of learning how to put the principles into practice in a variety of situations. Children must, for example, master the count list of their language. This learning has both a rote component, which varies from culture to culture depending on the base or bases (many languages have mixed bases), and a generative component, which, when mastered, enables the generation of ever-greater number names. Thus, children must learn to count accurately and reliably within a language-specific framework, a process that can takes years [56]. In the process of mastering both the rote and generative aspects of language-based counting, they must also learn the mapping from the count words to the numerons, the pre-verbal symbols for numerosity, which the latest results suggest are generated by a pre-verbal counting process [57,58].

Children must also learn to put the results of counting into effect. This, too, proves surprisingly difficult, even in some seemingly straightforward contexts. One such context is the ‘Give *n*’ task, in which children are asked to give the experimenter *n* items. A child's ability to do this reliably develops slowly and step by step, rather than all at once. Thus, a child that counts easily to four or five or even higher may only reliably give *n* when *n* ≤ 2 or, at a somewhat later age, when *n* ≤ 3, and at a still later age when *n* ≤ 4 [59]. These robust results have led some investigators to the conclusion that children's mastery of counting and of mathematical reasoning principles is not guided by an early grasp of the principles that must be honoured by the practice [7]. On this view, children first learn the referential meanings of the first several count words in just the same way that they learn the meanings of non-count words like ‘cow’ and ‘tree’. Eventually, when they have mastered the referential meanings of a sufficient number of the words that refer to small numerosities, they—somehow—induce the ordering of the numbers, the successor principle, and all the other principles for counting and numerical reasoning. The problem with this view has always been with the ‘somehow’—that is, proponders' conclusions from those purely referential data in the absence of any principles that constrain the induction [54,55].

There is also abundant evidence that counting is built into many idiosyncratic self-developed algorithms that young children have for solving basic arithmetic problems [46,60,61]. Being able to count both up and down and silently and quickly is central to the success of these algorithms, which is why successful arithmetic reasoning depends on the counting skills that come only with practice.

### (c) When the pre-existing structure becomes an obstacle

There is extensive evidence that the very principles that make possible the early progress in expressing quantitative reasoning in language become a serious obstacle to further progress. Every elementary school teacher knows that learning to deal with fractions and decimals—the conventional symbols for representing the non-integer quantities—is a huge hurdle, a hurdle that many students have not cleared by the time they reach college and may, therefore, never clear. The conceptual problem with the rational number is there is no next number. Given that the successor principle is the foundation for verbal counting and for writing the natural numbers, how can there be numbers for which there is no next and numbers that one can never finish writing out? And, how can it be that one and the same operation, namely multiplication, sometimes increases the quantity and sometimes decreases it? And if 10 is bigger than 5, how can it be that 1/10 is less than 1/5? The ‘1’ appears in both, so it cannot be that which reverses the order. And why the ‘/’ is there at all is a bit of a mystery; it is a symbol for an operation, not for a quantity, so what is it doing in the midst of a symbol for a quantity? And what is one to make of a signed number and of the crazy rules that apply to these mysterious so-called numbers? Auden [62] captured this perplexity in a couplet: ‘Minus times minus is plus; the reason for this, we need not discuss’.

What is now clear is that the intuitive principles that guide and facilitate the learning of counting, addition and subtraction in pre-school children continue to be an obstacle to the proper understanding of more advanced mathematical reasoning, particularly the reasoning required to incorporate the rational numbers into arithmetic reasoning.

### (d) Pedagogical implications

Errors can inform us about what to teach students. Understanding rational number, for example, requires formal instruction and dedicated work on the part of the learner. In learning, students will undoubtedly make mistakes and quite possibly fail to recognize those mistakes. As a result, these problems will be treated as novel cases to which to apply their knowledge of natural numbers.

In order to successfully introduce two-step percentage change problems, one-step percentage change problems must first be mastered. Only then do we suggest the following inclusions in a learning framework: (i) compare and contrast models promote deeper understanding of the links between abstract concepts and solution models [63,64], increase overall conceptual understanding [65–67] and encourage flexibility in problem-solving [68]. (ii) Predictions of the final answer allow students to set an expected estimate and later see if their answer violates their expectation [4]. (iii) Dispel false cognates in mathematical language (e.g. ‘increase’ does not always imply addition) and define operational processes that correspond to the language used. (iv) Practice standard textbook problems *and* derive real world examples, using multiple representational formats.

In sum, with or without schooling, almost everyone has a sense of numbers. They are part of a knowledge structure in our core domain. A conceptual change must occur, for children and adults alike, to make the shift from natural number concepts to those dealing with rational numbers (e.g. percentages). Two things need to be communicated and deeply understood: (i) what do these new numbers represent, that is, why do we need them? (ii) How can we make them behave, that is, fit in with the natural numbers? Where do they fit in the ordering? How do we add and multiply them? Ultimately, developing a more successful framework for teaching rational numbers is critical in helping people understand that rational numbers, especially percentages, are *not* natural numbers.

## Data accessibility

This article has no additional data.

## Competing interests

We declare we have no competing interests.

## Funding

Funding was provided by Rutgers University Internal Grant to R.G.

## Footnotes

One contribution of 19 to a discussion meeting issue ‘The origins of numerical abilities’.

- Accepted September 25, 2017.

- © 2017 The Author(s)

Published by the Royal Society. All rights reserved.